Chap015The Term Structure of Interest Rates(金融工程-南开大学,王小麓))

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货币银行学The Risk and Term Structure of Interest Rates课件讲义

f interest rates
A bond’s term to maturity also affects its interest rate.
The relationship among interest rates on bonds with different terms to maturity is called the term structure of interest rates.
Risk premium
The spread between the interest rates on bonds with default risk and default-free bonds. It indicates how much additional interest people must earn to be willing to hold a risky bond.
Figure 3 Interest Rate on Municipal and in Treasury Bonds
(a) Market for municipal bonds Price of Bonds, P (P increases↑ )
SC
(b) Default-free(U.S. Treasury)bond market
Price of Bonds, P
Interest Rate, I
(P increases↑ )
(i increases↓ )
ST
P2 T
i2T
i2T
P1 C
i1C
Risk
P1 T
i1T
Premium
P2 C
i2C
Liquidity
Another attribute of a bond that influences its interest rate. The more liquid an asset is, the more desirable it is.

金融市场与金融机构英文课件 (5)

─ Risk Structure of Interest Rates ─ Term Structure of Interest Rates
© 2012 Pearson Education. All rights reserved.
5-4
Risk Structure of Interest Rates
Result: Baa-Treasury spread increased from 185 bps to 545 bps.
© 2012 Pearson Education. All rights reserved.
5-15
Liquidity Factor
Another attribute of a bond that influences its interest rate is its liquidity; a liquid asset is one that can be quickly and cheaply converted into cash if the need arises. The more liquid an asset is, the more desirable it is (higher demand), holding everything else constant.
© 2012 Pearson Education. All rige of Long Bonds in the U.S.
© 2012 Pearson Education. All rights reserved.
5-6
Risk Structure of Long Bonds in the U.S.
© 2012 Pearson Education. All rights reserved.

投资学第11版 Ch15 The Term Structure of Interest Rates

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The Yield Curve and Future Interest Rates (2 of 5)
• Yield Curve Under Certainty
• Buy and hold vs. rollover:
1 y2 2 1 r1 1 r2
1
y2
1
r1
1
r2
.5
• (r2) is just enough to make rolling over a series of 1-year bonds equal to investing in the 2-year bond
INVESTMENTS | BODIE, KANE, MARCUS
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Bond Pricing: Two Types of Yield Curves
Pure Yield Curve • Uses stripped or zero
coupon Treasuries
• May differ significantly from the on-the-run yield curve
• Equilibrium requires that both strategies provide the same return
INUS
15-9
Two 2-Year Investment Programs
INVESTMENTS | BODIE, KANE, MARCUS
INVESTMENTS | BODIE, KANE, MARCUS
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The Yield Curve and Future Interest Rates (3 of 5)
• Yield Curve Under Certainty

Chapter 12 The Term Structure of Interest Rates

• so if true,
investors hold ST bonds UNLESS LT bonds offer higher yield as incentive higher yield = liquidity premium
IF LT bond yields have a liquidity premium, then usually LT yields > ST yields or yield curve slopes up.
The Pure Expectations Td buyers do not have any preference about maturity i.e. bonds of different maturities are perfect substitutes
• relationship between yield &
maturity is NOT constant • sometimes short-term yields are highest, • most of the time long-term yields are highest
I. The Yield Curve
• assume:
bonds of different maturities are NOT substitutes at all
• if assumption is true,
• separate markets for ST and LT bonds • slope of yield curves tells us nothing about future ST rates unrealistic to assume NO substitution bet. ST and LT bonds

The Term Structure of Interest-Rate Futures Prices, working paper

The Term Structure of Interest-RateFutures Prices.1Richard C.Stapleton2,and Marti G.Subrahmanyam3First draft:June,1994This draft:August13,20011Earlier versions of this paper have been presented at the Workshop on Finan-cial Engineering and Risk Management,European Institute for Advanced Studies in Management,Brussels,and the Chicago Board of Trade Annual European Fu-tures Conference,Rome,and in seminars at New York University,The Federal Reserve Bank of New York,University of Cardiﬀ,University of London,Birkbeck College,The Australian Graduate School of Management,Sydney,The Fields In-stitute,Toronto,and University of California at Los Angeles.We would like to thank T.S.Ho,C.Thanassoulas,S-L.Chung,Q.Dai and O.Grischchenko for helpful comments.We would also like to thank the editor,B.Dumas,and an anonymous referee,whose detailed comments lead to considerable improvements in the paper.We acknowledge able research assistance by V.Acharya,A.Gupta and P.Pasquariello.2Department of Accounting and Finance,University of Strathclyde,100, Cathedral Street,Glasgow,Scotland.Tel:(44)524—381172,Fax:(44)524—846874, Email:rcs@ and University of Melbourne,Australia 3Leonard N.Stern School of Business,New York University,Management Ed-ucation Center,44West4th Street,Suite9—190,New York,NY10012—1126,USA. Tel:(212)998—0348,Fax:(212)995—4233,Email:msubrahm@AbstractThe Term Structure of Interest-Rate FuturesPricesWe derive general properties of two-factor models of the term structure of in-terest rates and,in particular,the process for futures prices and rates.Then, as a special case,we derive a no-arbitrage model of the term structure in which any two futures rates act as factors.In this model,the term structure shifts and tilts as the factor rates vary.The cross-sectional properties of the model derive from the solution of a two-dimensional,autoregressive process for the short-term rate,which exhibits both mean-reversion and a lagged persistence parameter.We show that the correlation of the futures rates is restricted by the no-arbitrage conditions of the model.In addition,we inves-tigate the determinants of the volatilities and the correlations of the futures rates of various maturities.These are shown to be related to the volatility of the short rate,the volatility of the second factor,the degree of mean-reversion and the persistence of the second factor shock.We also discuss the extension of our model to three or more factors.We obtain speciﬁc re-sults for futures rates in the case where the logarithm of the short-term rate [e.g.,the London Inter-Bank Oﬀer Rate(LIBOR)]follows a two-dimensional process.We calibrate the model using data from Eurocurrency interest rate futures contracts.Interest Rate Futures1 1IntroductionTheoretical models of the term structure of interest rates are of interest to both practitioners andﬁnancial academics.The term structure exhibits several patterns of changes over time.In some periods,it shifts up or down, partly in response to higher expectations of future inﬂation.In other periods, it tilts,with short-term interest rates rising and long-term interest rates falling,perhaps in response to a tightening of monetary policy.Sometimes, its shape changes to an appreciable extent,aﬀecting its curvature.Hence, a desirable feature of a term-structure model is that it should be able to capture shifts,tilts and changes in the curvature of the term structure.In this paper,we present and analyse a model of the term structure of futures rates that has the above properties.Previous work on the term structure of interest rates has concentrated mainly on bond yields of varying maturities or,more recently,on forward rates.In contrast,this paper concentrates on futures rates,partly motivated by the relative lack of previous theoretical models of interest-rate futures prices.However,the main reason for focussing on futures rates is analytical tractability.Futures prices are simple expectations of spot prices under the equivalent martingale measure(EMM),of the future relevant short-term bond prices,whereas forward prices and spot rates involve more complex relationships.It follows that futures prices and futures rates are fairly sim-ple to derive from the dynamics of the spot rate.In contrast,closed-form solutions for forward rates have been obtained only under rather restrictive (e.g.Gaussian)assumptions.Further,from an empirical perspective,since forward and futures rates diﬀer only by a convexity adjustment,it is likely that most of the time series and cross-sectional properties of futures rates are shared by forward rates,to a close approximation,at least for short ma-turity contracts.It makes sense,therefore,to analyse these properties,even if the ultimate goal is knowledge of the term-structure behaviour of forward or spot prices.Finally,the analysis of futures rates is attractive because of the availability of data from trading on organized futures exchanges.Hence, the models derived in the paper are directly testable,using data from the liquid market for Eurocurrency interest rate futures contracts.One contrast between many of the term-structure models presented in the academic literature and those used by practioners for the pricing of interest rate options is in the distributional assumptions made.Most of the modelsInterest Rate Futures2 which derive the term structure,for example Vasicek(1977),Balduzzi,Das, Foresi and Sundaram(1996)(BDFS),Jegadeesh and Pennacchi(1996)(JP), and Gong and Remolona(1997)assume that the short term interest rate is Gaussian.In contrast,popular models for the pricing of interest-rate options often assume that rates are lognormally distributed(see for example Black and Karasinski(1991),Brace,Gatarek and Musiela(1997),and Miltersen, Sandmann and Sondermann(1997)).In this paper,we present a model where the short-term rate LIBOR is assumed to follows a multi-dimensional lognormal process under the EMM.We show that the no-arbitrage futures rates for all maturities are also log-linear in any two(or three)rates.This implies that all futures rates are also lognormal in our model.We then show that the correlation between the long and short maturity futures rates is restricted by the degree of mean-reversion of the short rate and the relative volatilities of the long and short-maturity futures rates.Also,the volatility structure of futures rates of various maturities can be derived explicitly from the assumed process for the spot short rate.The performance of a set of models where the short rate follows a process with a stochastic central tendency has been analysed in a recent article by Dai and Singleton(2000)(DS).They analyse the set of”aﬃne”term struc-ture models introduced previously by Duﬃe and Kan(1994).Our process for the short rate is also a stochastic central tendency model.However, there are some diﬀerences that should be noted between our model and those analysed by DS.First,in our model the short rate follows a discrete-time,rather than a continuous-time process.Second,as noted above,we assume that the process is log-Gaussian.Third,we assume that the short rate follows the process under the risk-neutral measure;hence,we abstract from considerations of the market price of risk.We suggest a time series model in which the short rate follows a stochasic central tendency process.This assumption allows us to nest the AR(1) single-factor model as a special case.It is also general enough to produce stochastic no-arbitrage term structures with shapes that capture most of those observed empirically.A similar model,in which the conditional mean of the short rate is stochastic has been suggested by Hull and White(1994), JP(1996),BDFS(1996)and analysed by Gong and Remolona(1997).The characteristics of these models are reviewed below.The literature on the pricing of futures contracts was pioneered by Cox, Ingersoll and Ross(1981)[CIR],who characterized the futures price of anInterest Rate Futures3 asset as the expectation,under the risk-neutral measure,of the spot price of the asset on the expiration date.In a related paper,Sundaresan(1991) shows that,under the risk-neutral measure,the futures interest rate is the expectation of the spot interest rate in the future.This follows from the fact that the LIBOR futures contract is written on the three-month LIBOR itself,rather than on the price of a zero-coupon instrument.Jegadeesh and Pennacchi(1996)[JP]suggest a two-factor equilibrium model of bond prices and LIBOR futures based on a two-factor extension of the Vasicek(1977)Gaussian model.They assume that the(continuously com-pounded)interest rate is normally distributed and generated by a process with a stochastic central tendency.They then estimate the model using fu-tures prices of LIBOR contracts,backing out estimates of the coeﬃcients of mean-reversion of the short rate as well as the second stochastic conditional-mean factor.Our approach is somewhat diﬀerent.We directly assume a process for LIBOR,rather than the continuously compounded rate,and then derive the process for the LIBOR futures.1This allows us to analyse the less tractable log-Gaussian case.Our general model is closely related to the JP paper,with the important distinction that it is embedded in an arbitrage-free,rather than an equilib-rium framework,thus eliminating the need for explicitly incorporating the market price of risk.Although our analysis is based on weaker assumptions, we are able to derive quite general,distribution-free results for futures rates. We then include,as a signiﬁcant special case,a model in which the interest rate is lognormal.The work of Gong and Remolona(1997)is similar,in some respects,to that of JP.Like JP,they also employ a two-factor model,in which the second factor is the conditional mean of the short-rate process.However, they focus on the yield of long-dated bonds rather than on the futures rates. In their model,the short-term rate is linear in the two factors.Also,in a manner similar to Vasicek(1977),they assume a market price of risk,solve for bond prices,and back-out the long-term rates and the variances of the two factors.21This is similar to the approach taken in the LIBOR market models of Brace,Gatarek and Musiela(1997),and Miltersen,Sandmann and Sondermann(1997).2Much of the recent literature,dating back to the work of Ho and Lee(1986),has been concerned with the evolution of forward rates.The most widely cited work in this area is by Heath,Jarrow and Morton[HJM](1990a,1990b,1992).HJM provide a continuous-Interest Rate Futures4 The two-factor models developed in this paper are related also to the expo-nential aﬃne-class of term-structure models introduced by Duﬃe and Kan (1994).This class is deﬁned as the one where the continuously compounded spot rate is a linear function of any n factors or spot rates.In an interesting special case of our model,where the logarithm of the LIBOR evolves as a two-dimensional linear process,it is the logarithm of the futures rate that is linear in the logarithm of any two futures rates.The remainder of this paper is organised as follows.In section2,we derive some general properties that characterise two-factor models in general and show that if the function of the price of a zero-coupon bond follows a two-dimensional process,its conditional expectation is generated by a two-factor model.We then derive a two-factor,cross-sectional model for futures prices in the general case where a function of the price of a zero-coupon bond follows a stochastic central tendency process.In section3,we assume that the logarithm of the London Inter-Bank Oﬀer Rate(LIBOR)follows a two-dimensional process and derive our main result for futures contracts on the LIBOR:a two-factor cross-sectional relationship between the changes in the prices of interest-rate futures.We estimate the parameters of the two-factor model using estimates of the volatilities and correlations of futures rates derived from the Eurodollar futures contract for the period1995-99, in section4.The conclusions and possible applications of our model to the valuation of interest rate options and to risk management are discussed in section5.Here,we also discuss possible extensions and,in particular,the generalisation of the model to three factors.2Some general properties of two-factor models In this section,we establish two statistical results,that hold for any two-factor process of the form that we assume for the short-term rate.These time limit to the Ho-Lee model and generalize their results to a forward rate which evolves as a generalized Ito-process with multiple factors.The HJM paper can be distinguished from our paper in terms of the inputs to the two frameworks.The required input to the HJM-type models is the term structure of the volatility of forward rates.In contrast,in our paper,we derive the term structure of volatility of futures rates from a more basic assumption regarding the process for the spot rate.To the extent that the futures and forward volatility structures are related,our analysis in this paper provides a link between the spot-rate models of the Vasicek type and the extended HJM-type forward rate models.Interest Rate Futures5 results are used to establish a general proposition,that holds for the condi-tional expectation of any function of the zero-coupon bond price.The con-ditional expectation is of key signiﬁcance,since the futures price(or rate)is closely related to the conditional expectation of the future spot interest rate. Later in the section,these results are directly applied to establish futures prices and rates.2.1Deﬁnitions and notationWe denote P t as the time-t price of a zero-coupon bond paying$1with certainty at time t+m,where m is measured in years.The short-term interest rate is deﬁned in relation to this m-year bond,where m isﬁxed. The continuously compounded short-term interest rate for m-year money at time t is denoted as i t,where i t=−ln(P t)/m.The London Interbank Oﬀer Rate LIBOR,again for m-year money is denoted r t,where P t=1/(1+r t m), where m is the proportion of a year.3We are concerned with interest rate contracts for delivery at a future date T.We denote the futures rate as F t,T,the rate contracted at t for delivery at T of an m-period loan.We denote the logarithm of the LIBOR futures rate as f t,T=ln[F t,T].Note that,under this notation,which is broadly consistent with HJM,F t,t=r t and f t,t=ln(r t).2.2General properties of two-factor modelsThe two-factor stochastic central tendency models introduced by JP and BDFS have the general form:dx t=k1(θt−x t)dt+σ1dz1(1)dθt=k2(θ−θt)dt+σ2dz2(2) where dz1dz2=ρ1,2dt,and whereρ1,2is the correlation between the Wiener processes dz1and dz2.The coeﬃcients k1and k2measure the degree of mean reversion of the variables to their respective means,θt andθ.σ1and 3In the LIBOR contract,m has to be adjusted for the day-count convention.Hence, m becomes the actual number of days of the loan contract divided by the day-count basis (usually360or365days).Interest Rate Futures6σ2are the instantaneous standard deviations of the variables x andθ.In JP,x t=i t is the continuously compounded short-term interest rate.The central tendencyθt is interpreted as either a“federal funds target”or as reﬂecting“investors’rational expectations of longer-term inﬂation.”In the paper by BDFS,x t is again the short-term interest rate.However,they suggest a slightly more general process for the central tendencydθt=k2(θ−θt+a)dt+σ2dz2(3) The constant allows for a drift in the central tendency.In both models,the short-term interest rate is normally distributed.Essentially,these models are two-factor extensions of the one-factor Vasicek (1977)model.A similar extension is made in Hull and White(1994),with a slightly diﬀerent assumption that allows the drift of the short-term interest rate to be time dependent.Hull and White assume that x t is any function of the short rate,for example;if x t=ln(i t),where i t is the continuously compounded short term rate,then the model is a two-factor extension of the lognormal Black and Karasinski model.Hull and White state the model in a slightly diﬀerent form from that of JP and BDFS:dx t=(φt+u t−k1x t)dt+σ1dz1(4)du t=−k2u t dt+σ02dz2(5) However,this model can be shown to be a simple transformation of the BDFS model,with the generalisation that the drift term a is time dependent, i.e.(3)contains a term a t.4In this paper,we consider the general Hull-White class of two-factor models. In discrete form,these models have the following general structure:5.x t+1−x t=c(φt−x t)+y t+²t+1(8) 4Letφt=k1θt−u t,(6)σ02=k1σ2.(7) Then substitution of equations(6)and(7)in equations(4)and(5)yields the general BDFS model(1)and(3).5Dai and Singleton(2000),in the discussion following their equation(20),argue that models of this type,with the additional restriction that the innovations in r t andθt are uncorrelated,fail to capture the empirical term structure of volatility.They suggest thatInterest Rate Futures7y t+1−y t=−αy t+νt+1,(9)E(²t+1)=E(νt+1)=0(10)wherec=1−e−k1nα=1−e−k2n,and n is the length of the time period.In this model,y t has an initial value of zero and mean reverts to zero.6It follows that E(y t)=0,and hence, taking expectations in equation(8),E(x t+1)−E(x t)=c(φt−x t)Then,substituting back in equation(8)we havex t+1−E(x t+1)=(1−c)[x t−E(x t)]+y t+²t+1(11) Hence,in this type of model,the value of x t reverts at the rate c to its initial expected value.We now establish that,if a variable follows the above process,the conditional expectation of the variable is necessarily governed by a two-factor cross-sectional model.We begin by proving this result quite generally,and then apply it to the special case where x t is an interest rate or its logarithm.Lemma1The variable x t follows the processx t+1−E(x t+1)=(1−c)[x t−E(x t)]+y t+²t+1wherey t+1−y t=−αy t+νt+1if and only if the conditional expectation of x t+k is of the formE t(x t+k)=a k x t+b k E t(x t+1)humped shaped volatility term structures are not possible in the above model and the volatility structure is monotonic.However,as shown,for example,by Hull and White (1994),Figure3,this is not true.The simulations in Section4below also illustrate this point in some detail for the case of futures rate volatilities.6See Hull and White(1994),footnote4.Interest Rate Futures8 whereb k=kXτ=1(1−c)k−τ(1−α)τ−1(12)a k=(1−c)k−(1−c)b k.(13)Proof.See appendix1.First,let us take the simplest case where x is the short-term rate of interest. The lemma then implies that if the short rate follows the Hull-White process, the expectation of the short rate k periods hence,is linear in the short rate and the expectation of the short rate at t=1.The implication of this result can be illustrated as follows.Under the expectations hypothesis,where the forward rate is the expectation of the future spot rate,it follows that the k-th period forward rate is a linear function of the spot rate and the t=1 forward rate.One important implication of the lemma is that in this type of two-factor model,the cross-sectional linear coeﬃcients a k and b k are invariant to the in-terchange of the mean reversion coeﬃcient,c and the persistence parameter,α.That isa k,b k|c,α=a k,b k|c0,α0,if c0=α,andα0=c.This identiﬁcation problem means that it may be diﬃcult to distinguish models where the interest rate reverts very rapidly to a slowly decaying central tendency,from those where the interest rate reverts slowly to a rapidly decaying central tendency.7Lemma1states the implications of a two-dimensional,stochastic conditional mean,process for an arbitrary function of the zero-coupon bond price.The function could be a rate of interest,such as the continuously compounded rate(as in HJM)or the LIBOR(as in BGM),or it could be any other function of the price of the zero-coupon bond itself.The lemma restricts the cross-sectional properties of the conditional expectation.As we will 7This partially explains why,in the calibration of the model in section4,there are two sets of parameters that yield exactly the same volatilities and correlations of the futures rates.It may also explain discrepancies in empirical results presented in JP,DS and others.Interest Rate Futures9 see in the following section,these properties are directly relevant for the investigation of futures prices and rates.The intuition behind Lemma1is that the two inﬂuences on the function of the zero-bond prices,one of which is lagged,yield a cross-sectional structure with two factors.2.3Interest-Rate Futures Prices in a No-Arbitrage Economy In this sub-section,we apply the results in the previous sub-section to de-rive futures prices and futures interest rates in a no-arbitrage setting.We assume here that the two-dimensional process,speciﬁed in equation(11) for x t=f(P t),holds under the Equivalent Martingale Measure(EMM). The EMM is the measure under which all zero-coupon bond prices,nor-malised by the money market account,follow martingales.Assuming that the process followed by x t is of the Hull-White type under the risk-neutral measure abstracts from considerations of the market price of risk.It allows us to directly derive prices for futures contracts on x t.Cox,Ingersoll and Ross(1981)and Jarrow and Oldﬁeld(1981)established the proposition that the futures price,of any asset,is the expected value of the future spot price of the asset,where the expected value is taken with respect to the equivalent martingale measure.We can now apply this result to determine futures prices,assuming that a function of the bond price is generated by the two-dimensional process analysed above.Since there is a one-to-one relationship between zero-coupon bond prices and short-term interest rates,deﬁned in a particular way,we can then proceed to derive a model for futures interest rates.Initially,we make no speciﬁc distributional assumptions.We assume only a)a no-arbitrage economy in which the EMM exists,b)that x t,which is some function of the time t price of an m-year zero-coupon bond,follows a stochasic central tendency process of the general form assumed in Lemma 1,and c)that a market exists for trading futures contracts on x t,where the contracts are marked-to-market at the same frequency as the deﬁnition of the discrete time-period from t to t+1.Weﬁrst establish a general result, and then illustrate it with some familiar examples.We denote the futures price,at t,for delivery of x t+k at t+k,as x t,k.We now have: Proposition1(General Cross-Sectional Relationship for Futures Prices)Interest Rate Futures10 Assume that equation(11)holds for x t under the EMM,where x t=f(P t), then the k-period ahead futures price of x t is given by the two-factor linear relationship:x t,k−x0,k=a k[x t−x0,t]+b k[x t,t+1−x0,t+1]whereb k=kXτ=1(1−c)k−τ(1−α)τ−1(14)a k=(1−c)k−(1−c)b k.(15) Proof.From CIR(1981),proposition2and Pliska(1997),the futures price of any payoﬀis its expected value,under the EMM.Applying this result to x t+k,yieldsx t,k=E t(x t+k)(16) Substituting equation(16)in Lemma1yields the statement in the Propo-sition.2The rather general result in Proposition1is of principal interest because of two special cases.Theﬁrst is the case where the futures contract is on the zero-coupon bond itself.The second is the case of a futures contract on an interest rate,such as LIBOR,which is a function of the zero-coupon bond price.Theﬁrst case is typiﬁed by the T.Bill futures contract traded on the Sydney Futures Exchange.The second case is the most common, typiﬁed by the Eurocurrency futures contract.We consider these cases in the corollaries below.RemarkAn interesting implication of Proposition1is that if the bond price itself follows a linear process,under the EMM,then the futures prices of the bond will be given by a two-factor cross-sectional relationship.This follows by assuming that x t=f(P t)=P t in the Proposition.This shows that futures prices at time t are generated by a linear two-factor model,if and only if the zero-bond price follows a process of the Hull-White type.Note that the two factors generating the k th futures price are the spot price of the bond and theﬁrst futures price,i.e.,the futures with maturity equal to t+1.Interest Rate Futures11 Similarly,the variance of the k th futures price is determined by the variance of the spot bond price,the variance of the conditional mean and the mean-reversion coeﬃcients.This is helpful in understanding the conditions under which the term structure follows a two-factor process.Essentially,if futures prices of long-dated futures contracts are given by the cross-sectional model in Proposition1,then forward prices,and also futures and forward rates will follow two-factor models.The relationship for interest rates,however, is,in general,complex,since the functions i t(P t)and r t(P t)are non-linear.We now illustrate the use of Proposition1in the case of interest rate(as opposed to bond-price)futures.Instead of assuming that the price of a zero-coupon bond follows a two-dimensional,linear process,we now assume that the interest rate,LIBOR,follows a stochastic central tendency process. We have the following corollary of Proposition1:Corollary1(Cross-Sectional Relationship for LIBOR Futures Prices for the Case of a Linear Process for the LIBOR)In a no-arbitrage economy,the short-term rate of interest follows a process of the formr t+1=E0[r t+1]+(1−c)[r t−E0(r t)]+y t+²t+1(17) wherey t+1−y t=−αy t+νt+1andE t(²t+1)=0,E t(νt+1)=0,if and only if the term structure of futures rates at time t is generated by a two-factor model,where the k th futures rate is given byr t,k−r0,t+k=a k[r t−r0,t]+b k[r t,t+1−r0,t+1](18) where a k and b k are given by(15)and(14)respectively.Interest Rate Futures12 Proof.The proof of the corollary again follows as a special case of Proposi-tion1,where x t=r t.2The corollary illustrates the simple two-factor structure of futures rates that is implied by the two-dimensional process for the spot rate.Note that the mean-reversion coeﬃcients are embedded in the cross-sectional coeﬃcients, a k and b k.Also,given the linear structure,it follows from(18),that the futures rates will be normally distributed,if the spot rate and theﬁrst futures rate are normally distributed.Hence,the corollary could be helpful in building a Gaussian model of the term structure of futures rates.83LIBOR futures prices in a lognormal short-rate modelIn the previous section,we showed that if the short-term LIBOR interest rate,evolves as a two-dimensional mean-reverting process under the risk-neutral measure,then a simple cross-sectional relationship exists between futures rates of various maturities.In principle,this type of model could be applied to predict relationships between the prices of Eurocurrency futures contracts,based on LIBOR or some other similar reference rate,which are the most important short-term interest rate futures contracts traded on the markets.However,in the case of LIBOR,the consensus in the academic literature and in market practice is that changes in interest rates are depen-dent on the level of interest rates.In particular,a lognormal distribution for short-term interest rates is commonly assumed.9When the logarithm of the short-term interest rate follows a linear process, the results of the analysis of futures prices in section2cannot be used directly,since the market does not trade futures on the logarithm of the LI-8In a two-factor extension of the Vasicek(1977)framework,JP(1996)estimate a two-factor term structure model that is similar to that in equation(18)under the assumption of normally distributed interest rates.They show that their modelﬁts the level of Eurodollar short-term interest rates contracts rather well for maturities of up to two years,and changes in the rates for longer-dated contracts.It is possible that this is because of ignoring the skewness eﬀect(due to the normality assumption),which becomes signiﬁcant for longer-dated contracts.9This is borne out by the empirical research of Chan et.al.,(1992)and more recently of Eom(1994)and Bliss and Smith(1998).There continues to be debate over the elasticity parameter of the changes in interest rates with respect to the level.。

The Term Structure of Interest Rates, Forward Contracts, and Futures

4
5
The Theory of Term Structure Forward Rates
The unbiased expectations theory attempts to explain observed forward rates by saying that expected future rates will, on average, be equal to the implied forward rates.
a. The liquidity premium added to a decreasing term structure b. The liquidity premium added to a increasing term structure
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The Term Structure of Interest Rates, Forward Contracts, and Futures
Section 1: The Term Structure of Interest Rates
The Yield to Maturity
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The Theory of Term Structure Forward Rates
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Section 2: Models of the Term Structure
Models of the term structure: Two categories Equilibrium Model: specify a process for the short-term interest rate that is based on assumptions about economic variables, not automatically fit market data on the term structure of interest rate at any point in time Arbitrage-free Model: calibrated to fit market data

投资学investment 题库Chap015

Multiple Choice Questions1. The term structure of interest rates is:A) The relationship between the rates of interest on all securities.B) The relationship between the interest rate on a security and its time to maturity.C) The relationship between the yield on a bond and its default rate.D) All of the above.E) None of the above.Answer: B Difficulty: EasyRationale: The term structure of interest rates is the relationship between two variables, years and yield to maturity (holding all else constant).2. The yield curve shows at any point in time:A) The relationship between the yield on a bond and the duration of the bond.B) The relationship between the coupon rate on a bond and time to maturity of thebond.C) The relationship between yield on a bond and the time to maturity on the bond.D) All of the above.E) None of the above.Answer: C Difficulty: EasyRationale: See rationale for question 15.1.3. An inverted yield curve implies that:A) Long-term interest rates are lower than short-term interest rates.B) Long-term interest rates are higher than short-term interest rates.C) Long-term interest rates are the same as short-term interest rates.D) Intermediate term interest rates are higher than either short- or long-term interestrates.E) none of the above.Answer: A Difficulty: EasyRationale: The inverted, or downward sloping, yield curve is one in which short-term rates are higher than long-term rates. The inverted yield curve has been observedfrequently, although not as frequently as the upward sloping, or normal, yield curve.4. An upward sloping yield curve is a(n) _______ yield curve.A) normal.B) humped.C) inverted.D) flat.E) none of the above.Answer: A Difficulty: EasyRationale: The upward sloping yield curve is referred to as the normal yield curve, probably because, historically, the upward sloping yield curve is the shape that has been observed most frequently.5. According to the expectations hypothesis, a normal yield curve implies thatA) interest rates are expected to remain stable in the future.B) interest rates are expected to decline in the future.C) interest rates are expected to increase in the future.D) interest rates are expected to decline first, then increase.E) interest rates are expected to increase first, then decrease.Answer: C Difficulty: EasyRationale: An upward sloping yield curve is based on the expectation that short-term interest rates will increase.6. Which of the following is not proposed as an explanation for the term structure ofinterest rates:A) The expectations theory.B) The liquidity preference theory.C) The market segmentation theory.D) Modern portfolio theory.E) A, B, and C.Answer: D Difficulty: EasyRationale: A, B, and C are all theories that have been proposed to explain the term structure.7. The expectations theory of the term structure of interest rates states thatA) forward rates are determined by investors' expectations of future interest rates.B) forward rates exceed the expected future interest rates.C) yields on long- and short-maturity bonds are determined by the supply and demandfor the securities.D) all of the above.E) none of the above.Answer: A Difficulty: EasyRationale: The forward rate equals the market consensus expectation of future short interest rates.8. Which of the following theories state that the shape of the yield curve is essentiallydetermined by the supply and demands for long-and short-maturity bonds?A) Liquidity preference theory.B) Expectations theory.C) Market segmentation theory.D) All of the above.E) None of the above.Answer: C Difficulty: EasyRationale: Market segmentation theory states that the markets for different maturities are separate markets, and that interest rates at the different maturities are determined by the intersection of the respective supply and demand curves.9. If forward rates are known with certainty and all bonds are fairly pricedA) all bonds would have the same yield to maturity.B) all short-maturity bonds would have lower prices than all long-maturity bonds.C) all bonds would have the same price.D) all bonds would provide equal 1-year rates of return.E) none of the above.Answer: D Difficulty: EasyRationale: In a world of perfect certainty, all bond must offer equal rates of return over any holding period; otherwise, at least one bond would be dominated by the others in that this bond would offer a lower rate of return than would combinations of other bonds.No one would be willing to hold this bond, and the price of the bond would decline (law of one price).10. According to the "liquidity preference" theory of the term structure of interest rates, theyield curve usually should be:A) inverted.B) normal.C) upward sloping.D) A and B.E) B and C.Answer: E Difficulty: EasyRationale: According to the liquidity preference theory, investors would prefer to beliquid rather than illiquid. In order to accept a more illiquid investment, investorsrequire a liquidity premium and the normal, or upward sloping, yield curve results. Use the following to answer questions 11-14:Suppose that all investors expect that interest rates for the 4 years will be as follows:Year Forward Interest Rate0 (today)5%1 7%2 9%3 10%11. What is the price of 3-year zero coupon bond with a par value of $1,000?A) $863.83B) $816.58C) $772.18D) $765.55E) none of the aboveAnswer: B Difficulty: ModerateRationale: $1,000 / (1.05)(1.07)(1.09) = $816.5812. If you have just purchased a 4-year zero coupon bond, what would be the expected rateof return on your investment in the first year if the implied forward rates stay the same?(Par value of the bond = $1,000)A) 5%B) 7%C) 9%D) 10%E) none of the aboveAnswer: A Difficulty: ModerateRationale: The forward interest rate given for the first year of the investment is given as 5% (see table above).13. What is the price of a 2-year maturity bond with a 10% coupon rate paid annually? (Parvalue = $1,000)A) $1,092.97B) $1,054.24C) $1,000.00D) $1,073.34E) none of the aboveAnswer: D Difficulty: ModerateRationale: [(1.05)(1.07)]1/2 - 1 = 6%; FV = 1000, n = 2, PMT = 100, i = 6, PV =$1,073.3414. What is the yield to maturity of a 3-year zero coupon bond?A) 7.00%B) 9.00%C) 6.99%D) 7.49%E) none of the aboveAnswer: C Difficulty: ModerateRationale: [(1.05)(1.07)(1.09)]1/3 - 1 = 6.99.Use the following to answer questions 15-18:The following is a list of prices for zero coupon bonds with different maturities and par value of $1,000.Maturity (Years) Price1 $943.402 $881.683 $808.884 $742.0915. What is, according to the expectations theory, the expected forward rate in the thirdyear?A) 7.00%B) 7.33%C) 9.00%D) 11.19%E) none of the aboveAnswer: C Difficulty: ModerateRationale: 881.68 / 808.88 - 1 = 9%16. What is the yield to maturity on a 3-year zero coupon bond?A) 6.37%B) 9.00%C) 7.33%D) 10.00%E) none of the aboveAnswer: C Difficulty: ModerateRationale: (1000 / 808.81)1/3 -1 = 7.33%17. What is the price of a 4-year maturity bond with a 12% coupon rate paid annually? (Parvalue = $1,000)A) $742.09B) $1,222.09C) $1,000.00D) $1,141.92E) none of the aboveAnswer: D Difficulty: DifficultRationale: (1000 / 742.09)1/4 -1 = 7.74%; FV = 1000, PMT = 120, n = 4, i = 7.74, PV = $1,141.9218. You have purchased a 4-year maturity bond with a 10% coupon rate paid annually. Thebond has a par value of $1,000. What would the price of the bond be one year from now if the implied forward rates stay the same?A) $808.88B) $1,108.88C) $1,000D) $1,042.78E) none of the aboveAnswer: D Difficulty: DifficultRationale: (943.40 / 742.09)]1/3 - 1.0 = 8.33%; FV = 1000, PMT = 100, n = 3, i = 8.33, PV = $1,042.7819. The market segmentation theory of the term structure of interest ratesA) theoretically can explain all shapes of yield curves.B) definitely holds in the "real world".C) assumes that markets for different maturities are separate markets.D) A and B.E) A and C.Answer: E Difficulty: EasyRationale: Although this theory is quite tidy theoretically, both investors and borrows will depart from their "preferred maturity habitats" if yields on alternative maturities are attractive enough.20. Given the following pattern of forward rates:Year Forward Rate1 5%2 6%3 6.5%If one year from now the term structure of interest rates changes so that it looks exactly the same as it does today, what would be your holding period return if you purchased a 3-year zero coupon bond today and held it for one year?A) 6%B) 8%C) 9%D) 5%E) none of the aboveAnswer: D Difficulty: DifficultRationale: $1,000 / (1.06)(1.065) = $885.82 (selling price); $1,000 / (1.05)(1.06)(1.065) = $843.64 (purchase price); ($885.82 - $843.64) / $843.64 = 5%.21. An upward sloping yield curveA) is an indication that interest rates are expected to increase.B) incorporates a liquidity premium.C) reflects the confounding of the liquidity premium with interest rate expectations.D) all of the above.E) none of the above.Answer: D Difficulty: EasyRationale: One of the problems of the most commonly used explanation of termstructure, the expectations hypothesis, is that it is difficult to separate out the liquidity premium from interest rate expectations.22. The "break-even" interest rate for year n that equates the return on an n-periodzero-coupon bond to that of an n-1-period zero-coupon bond rolled over into a one-year bond in year n is defined asA) the forward rate.B) the short rate.C) the yield to maturity.D) the discount rate.E) None of the above.Answer: A Difficulty: EasyRationale: The forward rate for year n, fn, is the "break-even" interest rate for year n that equates the return on an n-period zero- coupon bond to that of an n-1-periodzero-coupon bond rolled over into a one-year bond in year n.23. When computing yield to maturity, the implicit reinvestment assumption is that theinterest payments are reinvested at the:A) Coupon rate.B) Current yield.C) Yield to maturity at the time of the investment.D) Prevailing yield to maturity at the time interest payments are received.E) The average yield to maturity throughout the investment period.Answer: C Difficulty: ModerateRationale: In order to earn the yield to maturity quoted at the time of the investment, coupons must be reinvested at that rate.24. Which one of the following statements is true?A) The expectations hypothesis indicates a flat yield curve if anticipated futureshort-term rates exceed the current short-term rate.B) The basic conclusion of the expectations hypothesis is that the long-term rate isequal to the anticipated long-term rate.C) The liquidity preference hypothesis indicates that, all other things being equal,longer maturities will have lower yields.D) The segmentation hypothesis contends that borrows and lenders are constrained toparticular segments of the yield curve.E) None of the above.Answer: D Difficulty: ModerateRationale: A flat yield curve indicates expectations of existing rates. Expectationshypothesis states that the forward rate equals the market consensus of expectations of future short interest rates. The reverse of c is true.25. The concepts of spot and forward rates are most closely associated with which one ofthe following explanations of the term structure of interest rates.A) Segmented Market theoryB) Expectations HypothesisC) Preferred Habitat HypothesisD) Liquidity Premium theoryE) None of the aboveAnswer: B Difficulty: ModerateRationale: Only the expectations hypothesis is based on spot and forward rates. A andC assume separate markets for different maturities; liquidity premium assumes higheryields for longer maturities.26.Par Value $1,000Time to Maturity 20 yearsCoupon 10% (paid annually)Current Price $850Yield to Maturity 12%Given the bond described above, if interest were paid semi-annually (rather thanannually), and the bond continued to be priced at $850, the resulting effective annual yield to maturity would be:A) Less than 12%B) More than 12%C) 12%D) Cannot be determinedE) None of the aboveAnswer: B Difficulty: ModerateRationale: FV = 1000, PV = -850, PMT = 50, n = 40, i = 5.9964 (semi-annual);(1.059964)2 - 1 = 12.35%.27. Interest rates might declineA) because real interest rates are expected to decline.B) because the inflation rate is expected to decline.C) because nominal interest rates are expected to increase.D) A and B.E) B and C.Answer: D Difficulty: EasyRationale: The nominal rate is comprised of the real interest rate plus the expected inflation rate.28. Statistical estimation of the yield curve contains apparent pricing error. These errorterms are probably a result ofA) tax effects.B) call provisions.C) out of date price quotes.D) all of the above.E) none of the above.Answer: D Difficulty: EasyRationale: All of the factors listed can cause what appear to be pricing errors but are actually reflective of inaccurate reporting, differences in investor tax status, anddifferences in bond indentures.29. Forward rates ____________ future short rates because ____________.A) are equal to; they are both extracted from yields to maturity.B) are equal to; they are perfect forecasts.C) differ from; they are imperfect forecasts.D) differ from; forward rates are estimated from dealer quotes while future short ratesare extracted from yields to maturity.E) are equal to; although they are estimated from different sources they both are usedby traders to make purchase decisions.Answer: C Difficulty: EasyRationale: Forward rates are the estimates of future short rates extracted from yields to maturity but they are not perfect forecasts because the future cannot be predicted with certainty; therefore they will usually differ.30. The pure yield curve can be estimatedA) by using zero-coupon bonds.B) by using coupon bonds if each coupon is treated as a separate "zero."C) by using corporate bonds with different risk ratings.D) by estimating liquidity premiums for different maturities.E) A and B.Answer: E Difficulty: ModerateRationale: The pure yield curve is calculated using zero coupon bonds, but coupon bonds may be used if each coupon is treated as a separate "zero."31. The market segmentation and preferred habitat theories of term structureA) are identical.B) vary in that market segmentation is rarely accepted today.C) vary in that market segmentation maintains that borrowers and lenders will notdepart from their preferred maturities and preferred habitat maintains that market participants will depart from preferred maturities if yields on other maturities areattractive enough.D) A and B.E) B and C.Answer: E Difficulty: ModerateRationale: Borrowers and lenders will depart from their preferred maturity habitats if yields are attractive enough; thus, the market segmentation hypothesis is no longer readily accepted.32. The yield curveA) is a graphical depiction of term structure of interest rates.B) is usually depicted for U. S. Treasuries in order to hold risk constant acrossmaturities and yields.C) is usually depicted for corporate bonds of different ratings.D) A and B.E) A and C.Answer: D Difficulty: EasyRationale: The yield curve (yields vs. maturities, all else equal) is depicted for U. S.Treasuries more frequently than for corporate bonds, as the risk is constant across maturities for Treasuries.Use the following to answer questions 33-36:Year 1-Year Forward Rate1 5.8%2 6.4%3 7.1%4 7.3%5 7.4%33. What should the purchase price of a 2-year zero coupon bond be if it is purchased at thebeginning of year 2 and has face value of $1,000?A) $877.54B) $888.33C) $883.32D) $893.36E) $871.80Answer: A Difficulty: DifficultRationale: $1,000 / [(1.064)(1.071)] = $877.5434. What would the yield to maturity be on a four-year zero coupon bond purchased today?A) 5.80%B) 7.30%C) 6.65%D) 7.25%E) none of the above.Answer: C Difficulty: ModerateRationale: [(1.058) (1.064) (1.071) (1.073)]1/4 - 1 = 6.65%35. Calculate the price at the beginning of year 1 of a 10% annual coupon bond with facevalue $1,000 and 5 years to maturity.A) $1,105.47B) $1,131.91C) $1,177.89D) $1,150.01E) $719.75Answer: B Difficulty: DifficultRationale: i = [(1.058) (1.064) (1.071) (1.073) (1.074)]1/5 - 1 = 6.8%; FV = 1000, PMT = 100, n = 5, i = 6.8, PV = $1,131.9136. What should be the holding period return of a 9% annual coupon bond with face value$1000 and five years to maturity if it is purchased at the beginning of year 1 and sold at the beginning of year 2, assuming that rates do not change.A) 6.0%B) 7.1%C) 6.8%D) 7.4%E) none of the above.Answer: A Difficulty: DifficultRationale: Using the information in 15.36, P0 = $1,131.91; i = [(1.064) (1.071) (1.073)(1.074)]1/4 - 1 = 7.05%; FV = 1000, PMT = 100, n = 4, i = 7.05, PV = $1,099.81; HPR =(1099.81 - 1131.91 + 100) / 1131.91 = 6%.37. Given the yield on a 3 year zero-coupon bond is 7.2% and forward rates of 6.1% in year1 and 6.9% in year 2, what must be the forward rate in year 3?A) 7.2%B) 8.6%C) 6.1%D) 6.9%E) none of the above.Answer: B Difficulty: ModerateRationale: f3 = (1.072)3 / [(1.061) (1.069)] - 1 = 8.6%38. Consider two annual coupon bonds, each with two years to maturity. Bond A has a 7%coupon and a price of $1000.62. Bond B has a 10% coupon and sells for $1,055.12.Find the two one-period forward rates that must hold for these bonds.A) 6.97%, 6.95%B) 6.95%, 6.95%C) 6.97%, 6.97%D) 6.08%, 7.92%E) 7.00%, 10.00%Answer: D Difficulty: DifficultRationale: 1000.62 = d1 X 70 + d2 X 1070;1055.12 = d1 X 100 + d2 X 1100;2620.36 = 3000 d2;d2 = .8735; d1 = .9427; r1 = 6.08%; r2 = 7.92%.39. An inverted yield curve is oneA) with a hump in the middle.B) constructed by using convertible bonds.C) that is relatively flat.D) that plots the inverse relationship between bond prices and bond yields.E) that slopes downward.Answer: E Difficulty: EasyRationale: An inverted yield curve occurs when short-term rates are higher thanlong-term rates.40. Assuming the forecasts implicit in a yield curve come to pass, an inverted yield curvewould be most favorable forA) short-term borrowers.B) long-term borrowers.C) short-term lenders.D) long-term lenders.E) nobody - Neither a borrower nor a lender be.Answer: D Difficulty: DifficultRationale: Short-term borrowers would face high interest rates. Long-term borrowers would lock in a higher rate than necessary. Short-term lenders would earn a high rate initially, but later would earn lower rates. Long-term lenders would lock in higherinterest rates than they could in the future. Note: If lower rates occur later, some of the long-term borrowers may choose to prepay their loans.41. Investors can use publicly available financial date to determine which of the following?I)the shape of the yield curveII)future short-term ratesIII)the direction the Dow indexes are headingIV)the actions to be taken by the Federal ReserveA) I and IIB) I and IIIC) I, II, and IIID) I, III, and IVE) I, II, III, and IVAnswer: A Difficulty: ModerateRationale: Only the shape of the yield curve and future inferred rates can be determined.The movement of the Dow Indexes and Federal Reserve policy are influenced by term structure but are determined by many other variables also.42. Which of the following is a reason that bond prices do not conform exactly to thepresent value of the bond's cash flows?I) A call feature affects the bond's price.II)After-tax cash flows may be different for different investors.III)The IRS may impute a “built-in” interest payment.IV)Some investors might choose to sell the bond before its maturity date.A) I and IIB) I, II, and IIIC) I and IVD) I, II, and IVE) I, II, III and IVAnswer: E Difficulty: ModerateRationale: All of the items listed can affect the actual cash flows. It is not possible to know for sure whether a bond will be called, so cash flows after the call date may or may not be relevant. Investors have different tax brackets and situations and will therefore have different after-tax cash flows. Imputed interest income will apply to zero-coupon or OID bonds. And investors may choose different holding periods, for example, they may evaluate tax-timing options and make decisions based on tax consequences.43. A liquidity premiumA) compensates long-term investors for the uncertainty about future selling prices.B) compensates short-term investors for the uncertainty about future selling prices.C) compensates long-term investors for the lack of liquidity in bond markets.D) compensates short-term investors for the lack of liquidity in bond markets.E) is almost never observed in the markets.Answer: B Difficulty: EasyRationale: This definition is given in the section 15.2 of the text, on page 496.44. The Liquidity Preference Theory states thatA) stocks are preferred to bonds because they are generally more liquid.B) treasury Bonds are preferred to corporate bonds because they are more liquid.C) the liquidity premium is expected to be positive because short-term investorsdominate the market.D) bonds of large corporations are preferred because they have the highest liquidity.E) liquidity premiums can be measured precisely.Answer: C Difficulty: ModerateRationale: Short-term investors are not willing to hold long-term bonds unless they are compensated with a positive liquidity premium. The normal upward-sloping yield curve implies that f2 - E(r2), the liquidity premium, is expected to be positive.45. Which of the following combinations will result in a sharply increasing yield curve?A) increasing expected short rates and increasing liquidity premiumsB) decreasing expected short rates and increasing liquidity premiumsC) increasing expected short rates and decreasing liquidity premiumsD) increasing expected short rates and constant liquidity premiumsE) constant expected short rates and increasing liquidity premiumsAnswer: A Difficulty: ModerateRationale: Both of the forces will act to increase the slope of the yield curve. Thisconcept is illustrated graphically in Figure 15.5 on page 500.46. The yield curve is a component ofA) the Dow Jones Industrial Average.B) the consumer price index.C) the index of leading economic indicators.D) the producer price index.E) the inflation index.Answer: C Difficulty: EasyRationale: Since the yield curve is often used to forecast the business cycle, it is used as one of the leading economic indicators.47. The most recently issued Treasury securities are calledA) on the run.B) off the run.C) on the market.D) off the market.E) none of the above.Answer: A Difficulty: EasyUse the following to answer questions 48-51:Suppose that all investors expect that interest rates for the 4 years will be as follows:Year Forward Interest Rate0 (today)3%1 4%2 5%3 6%48. What is the price of 3-year zero coupon bond with a par value of $1,000?A) $889.08B) $816.58C) $772.18D) $765.55E) none of the aboveAnswer: A Difficulty: ModerateRationale: $1,000 / (1.03)(1.04)(1.05) = $889.0849. If you have just purchased a 4-year zero coupon bond, what would be the expected rateof return on your investment in the first year if the implied forward rates stay the same?(Par value of the bond = $1,000)A) 5%B) 3%C) 9%D) 10%E) none of the aboveAnswer: B Difficulty: ModerateRationale: The forward interest rate given for the first year of the investment is given as 3% (see table above).50. What is the price of a 2-year maturity bond with a 5% coupon rate paid annually? (Parvalue = $1,000)A) $1,092.97B) $1,054.24C) $1,028.51D) $1,073.34E) none of the aboveAnswer: C Difficulty: ModerateRationale: [(1.03)(1.04)]1/2 - 1 = 3.5%; FV = 1000, n = 2, PMT = 50, i = 3.5, PV =$1,028.5151. What is the yield to maturity of a 3-year zero coupon bond?A) 7.00%B) 9.00%C) 6.99%D) 4%E) none of the aboveAnswer: D Difficulty: ModerateRationale: [(1.03)(1.04)(1.05)]1/3 - 1 = 4%.Use the following to answer questions 52-55:The following is a list of prices for zero coupon bonds with different maturities and par value of $1,000.Maturity (Years) Price1 $925.162 $862.573 $788.664 $711.0052. What is, according to the expectations theory, the expected forward rate in the thirdyear?A) 7.23B) 9.37%C) 9.00%D) 10.9%E) none of the aboveAnswer: B Difficulty: ModerateRationale: 862.57 / 788.66 - 1 = 9.37%53. What is the yield to maturity on a 3-year zero coupon bond?A) 6.37%B) 9.00%C) 7.33%D) 8.24%E) none of the aboveAnswer: D Difficulty: ModerateRationale: (1000 / 788.66)1/3 -1 = 8.24%54. What is the price of a 4-year maturity bond with a 10% coupon rate paid annually? (Parvalue = $1,000)A) $742.09B) $1,222.09C) $1,035.66D) $1,141.84E) none of the aboveAnswer: C Difficulty: DifficultRationale: (1000 / 711.00)1/4 -1 = 8.9%; FV = 1000, PMT = 100, n = 4, i = 8.9, PV = $1,035.6655. You have purchased a 4-year maturity bond with a 9% coupon rate paid annually. Thebond has a par value of $1,000. What would the price of the bond be one year from now if the implied forward rates stay the same?A) $995.63B) $1,108.88C) $1,000D) $1,042.78E) none of the aboveAnswer: A Difficulty: DifficultRationale: (925.16 / 711.00)]1/3 - 1.0 = 9.17%; FV = 1000, PMT = 90, n = 3, i = 9.17, PV = $995.6356. Given the following pattern of forward rates:Year Forward Rate1 4%2 5%3 5.5%If one year from now the term structure of interest rates changes so that it looks exactly the same as it does today, what would be your holding period return if you purchased a 3-year zero coupon bond today and held it for one year?A) 6%B) 4%C) 3%D) 6%E) none of the aboveAnswer: B Difficulty: DifficultRationale: $1,000 / (1.05)(1.055) = $902.73 (selling price); $1,000 / (1.04)(1.05)(1.055) = $868.01 (purchase price); ($902.73 - $868.01) / $868.01 = 4%.57.Par Value $1,000Time to Maturity 18 yearsCoupon 9% (paid annually)Current Price $917.99Yield to Maturity 10%Given the bond described above, if interest were paid semi-annually (rather thanannually), and the bond continued to be priced at $917.99, the resulting effective annual yield to maturity would be:A) Less than 10%B) More than 10%C) 10%D) Cannot be determinedE) None of the aboveAnswer: B Difficulty: ModerateRationale: FV = 1000, PV = -917.99, PMT = 45, n = 36, i = 4.995325 (semi-annual);(1.4995325)2 - 1 = 10.24%.。

2-The Term Structure of Interest Rates-MB_2013

TERM STRUCTURE OF INTERESTRATES (TSIR)•T SIR:Relationship between yield to maturity and term to maturity.•Yield to maturity: total return an investor receives by holding the bond until it matures. •Maturity:lifetime of an asset.•Yield Curve(YC): Plots yields to maturity against term maturityExample: (Financial Times, 01 Mar 1996). Yield 6.25 6.125 6.06 6.125 Maturity (m) 1 3 612Other examples of Yield Curves (Financial Times, 01 Mar 1996)Euro par Yield Curve (Eurostat 5-Aug-2005)Government Bonds(zero coupon) (Bloomberg, Sun, 04 Mar 2001, 4:48pm EST) Country 3 Months 6 Months 1 yearFrance 4.670 4.570 4.530 Germany 4.761 4.651 4.535 Italy 4.235 4.539 4.432 U.K. 5.490- 5.279 U.S. 4.839 4.693 4.476U.S. Treasuries Sun, 04 Mar 2001, 5:21pm ESTBills Mat DatePreviousPrice/YieldCurrentPrice/Yield Yld Chg Prc Chg3month 5/31/01 4.71(4.83) 4.71(4.83) 0.00 -0 6month 8/30/01 4.51(4.68) 4.52(4.68) 0.01 +1 1year 2/28/02 4.26(4.46) 4.26(4.46) 0.00 +0Notes/Bonds Coupon Mat DatePreviousPrice/YieldCurrentPrice/Yield Yld Chg Prc Chg2year 4.625 2/28/03 100-10(4.46) 100-10(4.46) 0.00 -0-00 5year 5.750 11/15/05 104-09+(4.72) 104-10(4.71) 0.00 +0-00+ 10year 5.000 2/15/11 100-13+(4.95) 100-14(4.94) 0.00 +0-00+ 30year 5.375 2/15/31 100-04(5.37) 100-04+(5.37) 0.00 +0-00+InflationIndexedTreasury Coupon Mat DatePreviousPrice/YieldCurrentPrice/Yield Yld Chg Prc Chg5year 3.625 7/15/02 101-24+(2.30) 00(.00) -2.30 -101-2510year 3.500 1/15/11 101-10+(3.34) 00(.00) -3.34 -101-1130year 3.875 4/15/29 107-20(3.45) 00(.00) -3.45 -107-20TEN YEAR GOVERNMENT BOND SPREADS(Financial Times, Friday Feb 1 2008)/ft/markets/bonds.aspGov't Latest Spread vs Bund Spread vs T-Bonds N.Zealand 6.28% +2.33 +2.65 Australia 6.14% +2.19 +2.51UK 4.54% +0.59 +0.91 Norway 4.35% +0.40 +0.72 Italy 4.32% +0.37 +0.69 Greece 4.31% +0.36 +0.68 Portugal 4.22% +0.27 +0.59 Belgium 4.20% +0.25 +0.57 Austria 4.14% +0.19 +0.51 Spain 4.10% +0.15 +0.47 Denmark 4.09% +0.14 +0.46 France 4.07% +0.12 +0.44Gov't Latest Spread vs Bund Spread vs T-Bonds Finland 4.05% +0.10 +0.42 Netherlands 4.04% +0.09 +0.41 Sweden 3.97% +0.02 +0.34 Germany 3.95% 0.00 +0.32 Canada 3.90% -0.05 +0.27US 3.63% -0.32 0.00 Switzerland 2.83% -1.12 -0.80 Japan 1.43% -2.52 -2.20 Quotes are delayed by at least 20 minutes.•Compare spreads in Euro-zone with those in other countries.Government vs Corporate Bonds(Source: Baye and Jansen, 1995)•Interest rate differentials on similar assets reflect risk.UK Yield Curve (7 Feb 2009)Eurozone Yield Curve (7 Feb 2009)US Yield Curve (7 Feb 2009)Source: FT /markets/bonds.asp 08 February 20092. Main Facts of YCs: SummaryF1.YCs move together over time.F2.When short-term IRs are low, YCs slope upwards and when IRs are high YCsslope down.F3.A lmost always YCs are positive-sloping but they can also be negative-sloping(inverted) or flat.------------3. Main uses of govt. bonds YCs•YCs are used as leading indicator of economic activity/conditions.•YCs can be used as a yardstick for pricing many other fixed income securities. •YCs can be used to boost returns in bond investment strategies (known as riding the YC, or rolling down the YC).--------------4. The 3 main theories of the TSIR.A. Segmented Markets TheoryB. Expectations TheoryC. Preferred Habitat and Liquidity Theory Two important concepts•Substitutability of assets•Risk Aversion of investorsA. The Segmented Markets Theory Main assumptions/RationaleA1.Bonds of different maturities are not substitutesA2.Investors have strong preferences for bonds of particular maturities somarkets of assets are segmentedaccording to their maturity.A3.Investors prefer less risk.A4.Investors avoid transaction costs.Comments/Criticism of SMT • The SMT may explain F3.• SMT cannot explain F1 or F2.• The SMT is powerless in forecasting future rates.Maturity (years)Yield%16681210Bonds Year of Quantity −16%YieldBonds Year of Quantity −6%YieldBondsYear of Quantity −12%Yield)(Years Maturity %%8%6Bonds Year of Quantity −16%YieldBonds Year of Quantity −6%YieldBondsYear of Quantity −12%Yield )(Years Maturity %86%9B. The Expectations TheoryIan Fischer (1896), Hicks (1939) etc. … Main assumptions/RationaleA1.Investors are risk neutralA2.Short and longer-term assets are perfect substitutes.A3.No transaction costs.Given A1-A3, ET shows that longer-term rates are purely an average of expected short-term rates.Given A1-A3, decisions depend merely on the final income (V) since investors treat short and long term bonds as perfect substitutes, •With V S=V L the investor is indifferentExample On a specific date t an investor has available £1,000 for two years. On this datethe 1-year bond rate is %61=t ithe 2-year bond rate is %5.62=t i(Superscripts = term of maturity).Long-term investment22)1)(1000(£t L i V +==£1134.Short-term investmentInvestors buy the 1-year bond and re-invest their income in a second 1-year bond. (Compounded returns))1)](06.0)(1000(£1000[£11+++=t s Ei V=)1)(06.01(1000£11+++t Ei If %811=+t Ei then V S = £1144 > V L (=£1134)If %611=+t Ei then V S = £1123 < V L (=£1134).ï V S =V L due to arbitrage.The generalised formulaBased on V L =V S , we can generalise for an n-year bond (1)22)1)(1000(£t i += )1)(1(1000£111+++t t Ei i(2) 22)1(t i += )1)(1(111+++t t Ei iBased on (2) we write the general formula(3) =+n n t i )1()1)...(1)(1)(1(1112111−+++++++n t t t t Ei Ei Ei iSolving for n t i from eq. (3)(4) 1)1)...(1)(1)(1(1112111−++++=−+++n Ei Ei Ei i i n t t t t n tEquation (4) can be approximated as:(5) nEi Ei Ei i i n t t t tn t 1112111...−++++++≈(5) is used by the strict ET to explain TSIR.• It can estimate the rate that must be offered on a bond of n -years for the risk-neutral investor to hold this bond.• It can determine the shape of the yield curve. • Example : 21112++≈t t tEi i i •If short term-rates are expected to:Rise → An upward-sloping yield curve Fall → An downward-sloping yield curve Remain constant → A flat yield curve.Expectations Theory and The Yield Curve %Yield )(Years Maturity%72%)8(1112==+=+t t t Ei i i %61=t iComments and criticism of ET•The ET can explain F1, because short-term rates increase long-term rates•The ET can also explain F2, because low current short-term rates will be expected to rise in the future (and visa versa).•The ET can partially explain F3: It explains that YC can slope different ways but it implies that on average YCs should be flat. •In reality longer rates seem to under-react to changes in the short-term rate.•Unlike the SMT, the ET can be used for forecasting future interest rates.•The existence of systematic profits is totally incompatible with the ET (and part. RE).C. The Preferred Habitat and LiquidityPremium theories of TSIRMain assumptions/RationaleA1. It combines SMT and ET theories:A2. Investors are assumed to prefer one market of assets to another, but are willing to trade in other markets, given a compensation: the term premiumPreferred habitat theory (PHT) the term premium compensates for preferences.Liquidity premium theory (LPT) term premium compensates for riskBased on the ET formula, add the term nt θ.(6) nEi Ei Ei i i n t t t tn tn t 1112111...−+++++++≈θThe term premium therefore is the extra fixed yield required for the investor to be attracted into a market.• Under the SMT, only nt θ matters• Under the ET, nt θ = 0, and only expectations of short-term rates matter.Implication : Long-term IRs will equal the average of short-term IRs plus a term premiumExample: 211122+++=t t ttEi i i θ%12=t θ; %61=t i %611=+t Ei⇒ 07.0206.006.001.02=++=t iSo %61=ti and %72=t i• LPT implies an upward sloping YC,whereas under ET the YC remains flat.Comments and criticism of PHT & LPT•Both PHT and the LPT can account for allF1, F2, F3. F1 and F2 based on the ET and F3 based on the risk premium that ensuresthat on average YCs should slope upwards and allows for differences in risk premia.•Spread between L-R and S-R cannot always help predict future S-T rates.-------------。

CH11 The Term Structure of Interest Rates

C h a p t e r11T h e T e r m S t r u c t u r e o f I n t e r e s t R a t e sTHE YIELD CURVE AND THE TERM STRUCTUREThe graphical depiction of the relationship between the yield on bonds of the same credit quality but of different maturities is called the yield curve. These curves are typically constructed from U.S. Treasury bonds, since such securities have effectively no default or liquidity risks.Using the Yield Curve to Price a BondAt first glance bond pricing using a yield curve should be a relatively simple matter—just take the Treasury yield for the comparable maturity and add a risk premium. But securities with the same maturity can have different yields, depending upon the coupon rate and cash flow pattern. Because of the different cash flow patterns, the same interest rate cannot be used to discount all cash flows.Constructing the Theoretical Spot Rate CurveA better way to think about these bonds is as packages of zero-coupon bonds, wherein the interest paid for each period is the price at maturity. Determining the value of each zero-coupon bond can be derived from a theoretical spot rate curve of Treasury bonds (the curve is theoretical because there are very few zero-coupon Treasuries). This process is called bootstrapping and employs the basic principle that the value of the Treasury coupon security should be equal to the value of the package of zero-coupon securities.Using Spot Rates to Value a BondThe theory suggests that theoretical price of a bond is equal to the present value of the cash flows. What economic force will assure that actual price will not depart significantly from its theoretical price? Since any coupon bond can be viewed as a package of zero-coupon instrument, the rate on a zero coupon bond is called the spot rate. This is shown by a process referred to as stripping a security. Each security created has a maturity value and maturity date equal to the date when the coupon payment of maturity value is due. In this manner, each security is treated as a zero coupon bond. It is the process of coupon stripping and synthetically creating a Treasury bond by buying the zero coupon bonds that will drive the price of T-bond to its value as determined by the spot rates.FORWARD RATESA forward rate is the rate on a loan at some future period, e.g. the yield on one-year bonds a year from now. The implicit forward rate can be calculated from the knowledge of current spot rates and finding an indifference rate. For example, assume the spot rate of one-year bond is 8% and for two-year bonds it is 10%. Suppose now that an investor would consider buying either a two-year bond (long-term) or two one-year bonds (short-term). The yield for the period must be the same. To reach that indifference point the investor must obtain 12% next year on his one-year bond to match the two-year offering. Hence the implicit forward rate becomes 12%. Of course, due to compounding and semi-annual payment periods, the actual situation is more complex. The general formula is:f = (1 + z2)2 / (1 + z1) – 1where f = the forward rate for the second one period bondz1 = the bond equivalent yield of the theoretical spot rate for one periodz2 = the bond equivalent yield of the theoretical spot rate for a second periodA future interest rate calculated from either the spot rates or the yield curve is called a forward rate or an implied forward rate.Relationship between Spot Rates and Short-Term Forward RatesIn general, the relationship between a T-period spot rate, and implied forward rates is as follows:Z t = [( 1+z1 ) ( 1+f1 ) (1+f2 ) (1+f3 ) . . . (1+f t-1 )]1/t – 1Historical Shapes Observed for the Treasury Yield CurveWhen the yield curve is upward sloped, it is called positively sloped yield curve. The convention in the market is to refer to a positively sloped yield curve whose maturity spread as measure by the 6-month and 30-year yields as a normal yield curve when the spread is 300 basis points or less. When the spread is more than 300 bp, the yield curve is said to be a steep yield curve. The markets can also see an inverted yield curve, and a flat yield curve. DETERMINANTS OF THE SHAPE OF THE TERM STRUCTURETwo major theories evolved to account for these observed shapes of the yield curve: the expectation theory and the market segmentation theory. Several forms of the expectation theory include the pure expectation theory, the liquidity theory, and the preferred habitat theory. The pure expectation theory posits that no systematic factors other than the expected future short-term rate affects forward rates. The other two theories assert that other factors are involved. The last two forms of the expectation theory are referred to as bias expectations theory.The Pure Expectations TheoryAccording to the pure expectation theory, the forward rates represent expected future rates. The entire term structure at any point in time reflects the market’s current expectations of the family of future short-term rates. In other words, the long-term rates represent the average of future and present short-term rates. A rising term structure indicates that the market expects short-term rates to rise. If interest rates are expected to increase, investors will lend long-term only if they are coaxed to do so by higher yields. Hence the curve will tilt upward.Risks associated with bond investment: The theory neglects the risks inherent in investing in bonds. Two risks cause uncertainty about the return over some investment horizon: (1) the uncertainty about the price of the bond at the end of the investment horizon, called the price risk, and (2) the uncertainty about the rate at which the proceeds from a bond that matures prior to the maturity date can be reinvested until the maturity date, called the reinvestment risk.Interpretations of the pure expectation theory: The broadest interpretation of this theory relies upon the assumption that holding period return must be identical for any investing strategy that spans the investor's chosen holding period. In other words, all combinations of bonds that comprise a five-year term should have the same total yield. A modification of this idea is considered to be more accurate. The local expectations hypothesis says that long-term bonds will have the same returns when investors have short investment horizons. The final interpretation, called the return-to-maturity expectations theory, holds that investors will equate the yield on a package of bonds with a zero-coupon bond which has the same maturity. Only the second version is thought to hold much promise in describing actual market rates.The Liquidity TheoryThis theory suggests that investors do not like uncertainty, which tends to increase linearly over time. They prefer to hold liquid assets and must be coaxed out of such holdings and into long-term assets by higher rates. This is an outgrowth of Keynesian theory and implies that the curve must have an upward slope. At the very least it assumes that investors demand a liquidity premium over the average of expected future rates. According to the liquidity theory of the term structure, the implici t forward rates will not be an unbiased estimate of the market’s expectations of the future interest rates because they embody a liquidity premium.The Preferred Habitat TheoryThe preferred habitat theory accepts the view that the term structure reflects the expectations of the future path of interest rates as well as a liquidity risk premium. But it rejects the view that the risk premium must rise uniformly with maturity. Investors or borrowers are assumed to prefer certain maturity sectors and they are reluctant to shift out of those sectors unless induced by a premium to do so. The classic examples of investors with strong maturity preferences are banks (which by the nature of their liabilities structures prefer short-term investments) and insurance companies (which for the same reason prefer to stay with long-term securities).Market Segmentation TheoryThe market segmentation theory represents a stronger version of the preferred habitat theory, wherein shifts are not made out of maturity sectors. Hence the curve can be humped, negative or positive in slope, depending upon activities in various financial market sectors. Increased government borrowing in the form of T-notes, would raise rates among medium-term securities. The actual behavior of the term structure seems to follow characteristics of pure expectations in that arbitrage between rates of various maturities occurs according to the mathematical underpinnings that the expectations hypothesis provides. Segmentation represents a valid alternative, but it lacks the ability to explain many observed shapes and movements of the yield curve over time. Some combination of the two theories, as represented by either a liquidity premium or preferred habitat approach, is probably the most correct view of interest rate movements over time.ANSWERS TO QUESTIONS FOR CHAPTER 11(Questions are in bold print followed by answers.)1.a.What is a yield curve?b.Historically, why has the Treasury yield curve been the one that is most closelywatched by market participants?a. A yield curve is a geographical representation of the term structure (the relationship betweenyield to maturity and the term to maturity) of securities having the same credit rating.b.Treasury securities are risk-free with respect to credit and liquidity, and their yields serve asbenchmarks for other securities.2. What is meant by a spot rate?The spot rate is the yield on a zero-coupon Treasury security with the same maturity as the security under consideration. Sometimes referred to as the theoretical spot rate, since it must be derived, as there are few zero-coupon Treasuries outstanding.3. Explain why it is inappropriate to use one yield to discount all the cash flows of a financial asset.Interest rates change over time. Securities should be discounted by interest rates applicable over the period of each of the cash flows.4. Explain why a financial asset can be viewed as a package of zero-coupon instruments.A bond has a series of interest-paying periods and principal. Interest paid at the end of each period can be likened to a zero-coupon bond. Hence a bond can be valued as the sum of a package of zero-coupon bonds.5. Why is it important for lenders and borrowers to have a knowledge of forward rates?If a borrower knows the forward rate he can decide whether to request a long-term or short-term loan. E.g. if a borrower knows that future short-term rates will be higher than today, he may be better off with a long-term loan. In like manner, if an investor knows the forward rate and it is higher than today’s spot rate, he is better off to invest short-term now and wait for the benefits of increased yields later.6. How are spot rates related to forward rates?Implied forward rates are future short-term rates. If implied rates are realized, the same number of dollars will be realized as expected. Thus, spot rate is related to the forward rates.7. You are a financial consultant. At various times, you have heard the following comments on interest rates from one of your clients. How would you respond to each comment?a.“The yield curve is upward sloping today. This suggests that the market consensusis that interest rates are expected to increase in the future.”b.“I can’t make any sense out of today’s term structure. For short-term yields (up tothree years), the spot rates increase with maturity; for maturities greater than three years but less than eight years, the spot rates decline with maturity; and formaturities greater than eight years the spot rates are virtually the same for eachmaturi ty. There is simply no theory that explains a term structure with this shape.”c.“When I want to determine the market’s consensus of future interest rates, Icalculate the implicit forward rates.”a.This is true if the unbiased expectations theory of the term structure holds. Under theliquidity theory of the term structure of interest rates, it is possible for expectations to show a decline, but the liquidity premium demanded may be large enough to make the term structure still slope upward.b.The statement is incorrect. It is possible under the preferred habitat and market segmentationtheories for the term structure to have such a shape.c.The answer here is the same as in (a). The implicit forward rates are the market's consensusonly if the unbiased expectations theory holds.8. You observe the Treasury yield curve below (all yields are shown on a bond-equivalent basis):Year Yield to Maturity Spot Rate0.5 5.25% 5.25%1.0 5.50 5.501.5 5.75 5.762.0 6.00 ?2.5 6.25 ?3.0 6.50 ?3.5 6.75 ?4.0 7.00 ?4.5 7.25 ?5.0 7.50 ?5.5 7.75 7.976.0 8.00 8.276.5 8.25 8.597.0 8.50 8.927.5 8.75 9.258.0 9.00 9.618.5 9.25 9.979.0 9.50 10.369.5 9.75 10.7710.0 10.00 11.20All the securities maturing from 1.5 years on are selling at par. The 0.5-year and one-year securities are zero-coupon instruments. a. Calculate the missing spot rates.b. What should the price of the six-year Treasury security be?c. What is the implicit six-month forward rate starting in the sixth year?In answering this question, the student should recognize that the coupon rates are equal to the yield to maturity since the bonds are selling at par. The solution -- take each cash flow (½ the coupon rate) and divide by (1 + ½ the spot rate) for each period and set these cash flows equal to $1000. The only unknown is the last period spot rate, which can then be found algebraically. e.g. for year 2 (period 4) the spot rate will be:$1000 = $30/(1.0265) + $30/(1.0275)2 + $30/(1.0287)3 + $1030/(1+y 4)4Y 4 = 6.02%a. The missing spot rates are shown below:Year Yield to Maturity Spot Rate 2.0 6.00 6.02 2.5 6.25 6.28 3.0 6.50 6.55 3.5 6.75 6.82 4.0 7.00 7.10 4.5 7.25 7.38 5.0 7.50 7.67b. The price of a six-year Treasury security should be the sum of the present values of the cash flows, discounting each cash flow at the theoretical spot rate. The coupon rate for the six-year Treasury is 8%, so the semi-annual coupon per $100 of par is $4. Given the spot rates for each of the 12 periods and the cash flows, we can determine the theoretical price = 100.0149 or $1000.15c. The implicit six-month forward rate six years from now (12 semi-annual periods) is found by using the equation in the book:=1-)y +(1)y +(1=f +11212131313=(1.04295)13-1=.0623x2=12.46%(1.04135)129. You observe the following Treasury yield curve (all yields are shown on a bond-equivalent basis):Year Yield to Maturity Spot Rate0.5 10.00% 10.00%1.0 9.75 9.751.5 9.50 9.482.0 9.25 9.222.5 9.00 8.953.0 8.75 8.683.5 8.50 8.414.0 8.25 8.144.5 8.00 7.865.0 7.75 7.585.5 7.50 7.306.07.25 7.026.57.00 6.747.0 6.75 6.467.5 6.50 6.188.0 6.25 5.908.5 6.00 5.629.0 5.75 5.359.5 5.50 ?10.0 5.25 ?All the securities maturing from 1.5 years on are selling at par. The 0.5-year and one-year securities are zero-coupon instruments.a.Calculate the missing spot rates.b.What should the price of the four-year Treasury security be?ing the same formulation as in question 8, we can derive the missing spot rates:Year Yield to Maturity Spot Rate9.5 5.50 5.0710.0 5.25 4.80b.The price of a four-year Treasury should be the sum of the present value of the cash flows,discounting each cash flow at the theoretical spot rate. The coupon rate is $4.125 on a semi-annual basis. Given the necessary data, the theoretical price can be calculated as 99.99 or $999.90.10. Using the theoretical spot rates in Table 11-2, calculate the theoretical value of a 7%, six-year Treasury bond.11.ing the theoretical spot rates in Table 11-2, calculate the two-year forward ratefour years from now.b.Verify the answer by assuming an investment of $100 is invested for six years.a.In terms of the notation, t = 4 and m = 8. Substituting for t and m into the equation forforward rate, we have,ƒ8 = [ (1 + z 8 + 4) 8 + 4 / (1 + z 8) 6]1/4 - 1 = [ (1 + z 12) 12 / (1 + z 8) 6]1/44z 8 = 0.10592/2 = 0.052960z 12 = 0.11584/2 = 0.057920ƒ8 = [ (1.057920) 12 / (1.052960) 6]1/4 - 1 = [ 1.965323/1.362923 ]1/4 - 14ƒ8 = 0.095824 = 9.58%4b.We can verify the answer by assuming an investment of $100 for six years. Invest $100 fortwelve periods at the semi-annual spot rate of (11.584 / 2) = 5.792% will produce:$100 (1.0579) 12 = $196.5324. By investing $100 for eight periods at 5.296% and reinvesting the proceeds for four periods at the forward rate of 9.5824% gives the same value:$100 (1.0579) 12 = $196.532412. Explain the role that forward rates play in making investment decisions.If a borrower knows the forward rate he can decide whether to request a long-term or short-term loan. For example, if a borrower knows that future short-term rates will be higher than today, he may be better off with a long-term loan. In like manner, if an investor knows the forward rate and it is higher than today's spot rate, he is better off to invest short-term now and wait for the benefits of increased yields later.13. “Forward rates are poor predictors of the actual future rates that ar e realized. Consequently, they are of little value to an investor.” Explain why you agree or disagree with this statement.I will disagree with the statement. The forward rate may not be realized, but that is irrelevant. Forward rates are being hedgeable by buying a one-year security, and thus the investor is able to hedge the six-month rate.14. An investor is considering two alternative investments. The first alternative is to invest in an instrument that matures in two years. The second alternative is to invest in an instrument that matures in one year and at the end of one year, reinvest the proceeds in a one-year instrument. The investor believes that one-year interest rates one year from now will be higher than they are today and, therefore, is leaning in favor of the second alternative. What would you recommend to this investor?The answer depends on the several interpretations of the Pure Expectation (PE) Theory. According to the broadest interpretation, the investor expects the return for any investment horizon to be the same, regardless of the maturity strategy selected. Therefore, it makes no difference to invest in two-year security or one year and followed by another year.15.a.What is the difference between a normal yield curve and steep yield curve?b.What is meant by a humped yield curve?a. A normal yield curve is when the spread between the 6-month and 30-year yield is 300 bp orless. When the maturity spread is more than 300 bp, the yield is said to be a steep yield curve.b. A humped yield curve means that the yields on medium maturity instruments are higher thanthe short-term and long-term yields.16. What is the common hypothesis about the behavior of short-term forward rates shared by the various forms of the expectations theory?The expectations theory of the yield structure states that forward rates represent expected future rates.17. What are the types of risks associated with investing in bonds and how do these two risks affect the pure expectations theory?The expectations theory suggests that rates will change over time in some direction. The prospect of changing yields creates two risks related to bond investment: (1) the price risk (value of principal if sold prior to maturity) and the reinvestment risk. Low rates may be good for price increases, but their favorable impact is partially offset by investors having to reinvest earnings at these low rates.18. Give three interpretations of the pure expectations theory.(1) I nvestors expect the return for any investment horizon to be the same, regardless of the maturity strategy selected. Time arbitrage will see to it.(2) T he local expectations interpretation suggests that returns will be the same, but only over a short-term horizon.(3) The return-to-maturity expectations interpretation suggests that the returns realized over a specified time horizon will be the same as from holding a zero-coupon bond with the same maturity as the investment horizon.19. What are the two biased expectations theories about the term structure of interest rates? The two biased expectations theories are (1) the liquidity preference hypothesis and the preferred habitat hypothesis.20. What are the underlying hypotheses of the two biased expectations theories of interest rates?While the above hypotheses allow for expectations they maintain that investors have: (1) A preference for more liquid assets and will hence demand a liquidity premium on longer-term securities, and (2) investors have a preferred habitat on the yield curve and demand premiums if they are to leave that sector for another one.21.a.“Empirical evidence suggests that with respect to bond risk premiums that influencethe shape of the Treasury yield curve, there is a linear relationship betweenTreasury average returns and duration.” Explain whether you agree or disagreewith this statement. If you disagree, explain the type of relationship that has beenobserved.b.What is meant by the “convexity bias” influence on the shape of the Treasury yieldcurve?a.Disagree. Empirical evidence suggests that the bond risk premiums are not linear in duration.Instead, at the front end of the yield curve, the bond risk premiums increase steeply with duration.b.Convexity bias is the influence of convexity on the shape of the Treasury yield curve.11-11。

货币银行学习题章节 (5)

Chapter 5The Risk and TermStructure of InterestRatesD1 Factual1.The relationship between yield and maturity of the same type of security is known as theA)term structure of interest rates.B)risk-reward structure of interest rates.C)asset duration structure of interest rates.D)market structure of interest rates.Answer: AD1 Factual2. A graphic depiction of the relationship between yield and maturity isA)the term structure of interest rates.B)the yield curve.C)the Phillips curve.D)the risk-reward curve.Answer: BD1 Factual3.The term structure of interest rates provides a framework for analyzing securities of thesameA)class.B)yield.C)maturity.D)corporation.Answer: AD1 Factual4.The yield curve depicts the relationship betweenA)interest rates and risk.B)yield and risk.C)yield and interest rates.D)yield and maturity.Answer: D52Chapter 5The Risk and Term Structure of Interest Rates 53D2 Interpretive5.If the yield on long-term securities is greater than the yield on comparable short-termsecurities, the yield curve will beA)negatively sloped.B)positively sloped.C)in the negative quadrant.D)undefined.Answer: BD2 Interpretive6.If the yield on short-term securities is greater than the yield on comparable long-termsecurities, the yield curve will have aA)positive slope.B)negative slope.C)constant slope.D)zero slope.Answer: BD2 Interpretive7.If the yield on short-term securities is the same as the yield on comparable long-termsecurities, the yield curve will have aA)positive slope.B)negative slope.C)constant slope.D)zero slope.Answer: CD2 Applied8.Currently, 20-year Treasury bonds have a yield of 7 percent, while one-year Treasury billshave a yield of 5 percent. Based on this information, the yield curve isA)upward sloping.B)downward sloping.C) a horizontal line.D)None of the aboveAnswer: AD2 Interpretive9.The supply-demand approach to explaining the term structure of interest rates assumesthatA)bond prices and yields are positively related.B)the yield curve is horizontal.C)the yield curve is upward sloping.D)each maturity class is independent.Answer: D54 Ritter/Silber/Udell Money, Banking, and Financial Markets, Eleventh EditionD2 Factual10.The assumption that prices for short-term and long-term securities are determinedindependently is the basis for the __________________ approach to explaining the termstructure.A)liquidity preferenceB)supply and demandC)expectationsD)yield curveAnswer: BD2 Interpretive11.When the supply of a security __________________, the yield on the security.A)increases; risesB)decreases; risesC)increases; fallsD)None of the aboveAnswer: AD2 Interpretive12.The supply and demand approach to term structure assumes that securities with differentmaturities areA)close substitutes.B)independent.C)inversely related.D)complements.Answer: BD2 Interpretive13.Ignoring the interrelationship between similar securities is a serious weakness of theA)expectations approach to term structure.B)supply and demand approach to term structure.C)equilibrium approach to term structure.D)liquidity approach to term structure.Answer: BD2 Interpretive14.According to the pure expectations approach to term structure, investors view securitieswith different maturities asA)close substitutes.B)complements.C)inversely related.D)independent.Answer: AChapter 5The Risk and Term Structure of Interest Rates 55D3 Applied15.One-year securities are currently yielding 8 percent. You expect one-year securities to yield10 percent next year. Currently, two-year securities are yielding 9.5 percent. Given thissituation, portfolio managers would __________________ two-year securities, pushingtheir yield __________________.A)buy; upB)buy; downC)sell; upD)sell; downAnswer: BD3 Applied16.Two-year securities are yielding 6 percent, and comparable one-year securities are yielding8 percent. According to the pure expectations theory, the market expects next year'scomparable one-year securities to yieldA)14 percent.B)8 percent.C) 6 percent.D) 4 percent.Answer: DD3 Applied17.If one-year securities are yielding 5 percent, but the market anticipates that rates for one-year securities will rise to 7 percent, then according to the expectations theory, currenttwo-year securities should be yieldingA)12 percent.B)7 percent.C) 6 percent.D) 5 percent.Answer: CD3 Interpretiveing the pure expectations theory of term structure, a negatively sloped yield curveindicates that investors expectA)falling short term interest rates.B)rising short term interest rates.C)falling long term interest rates.D)rising long term interest rates.Answer: AD2 Interpretiveing the pure expectations theory of term structure, a positively sloped yield curveindicates that investors expectA)short term interest rates to be lower next year.B)short term interest rates to be higher next year.C)falling long term interest rates.D)rising long term interest rates.Answer: B56 Ritter/Silber/Udell Money, Banking, and Financial Markets, Eleventh EditionD3 Interpretive20.According to the expectations theory of term structure, if next year's short-term interestrate is expected to be above the current short-term rate, theA)current long-term rate will be equal to the current short-term rate.B)current long-term rate will be below the current short-term rate.C)current long-term rate will be above the current short-term rate.D)yield curve will have a negative slope.Answer: CD1 Factual21.Which of the following is not a theory used to explain the term structure of interest rates?A)Liquidity premium theoryB)Pure expectations theoryC)Preferred habitat theoryD)Pure liquidity theoryAnswer: DD2 Factualpared with long-term securities, the prices of short-term securities are alwaysA)more volatile.B)less volatile.C)higher.D)lower.Answer: BD2 Interpretive23.Investors usually __________________, because long-term securities have a greater risk ofcapital loss than do short-term securities.A)require a higher yield on long-term securitiesB)require a lower yield on long-term securitiesC)pay a higher price for long-term securitiesD)avoid long-term securitiesAnswer: AD1 Interpretivemercial banks are likely toA)require a liquidity premium to hold long-term securities.B)purchase equally in each maturity segment of the market.C)view similar securities of different maturities as close substitutes.D)have a preference for long-term securities.Answer: AChapter 5The Risk and Term Structure of Interest Rates 57D2 Interpretive25.Everything else being equal, most investors prefer __________________ securities, whilemost bond issuers prefer to issue __________________ securities.A)short-term; long-termB)long-term; long-termC)short-term; short-termD)long-term; short-termAnswer: AD1 Applied26.Which of the following financial institutions is likely to have a preferred habitat in long-term securities?A)Commercial banksB)Money market mutual fundsC)Life insurance companiesD)Credit unionsAnswer: CD3 Interpretive27.Proponents of the preferred habitat theory of term structure assume that a sharp reductionin the supply of one-year securities willA)decrease the yield on one-year securities.B)shift the yield curve upward.C)increase the yield on one year securities.D)have no effect on the yield on one-year securities.Answer: AD2 Factual28.In the long run, the yield curve tends to beA)positively sloped.B)negatively sloped.C)nearly vertical.D)nearly horizontal.Answer: AD2 Interpretive29.The supply of a particular security appears to influence the term structure onlyA)during recessions.B)during inflationary periods.C)in the short run.D)in the long run.Answer: C58 Ritter/Silber/Udell Money, Banking, and Financial Markets, Eleventh EditionD3 Interpretive30.When yields are expected to __________________ in the future, investors prefer__________________ long-term securities.A)rise; holdB)rise; not to holdC)fall; not to holdD)None of the aboveAnswer: BD3 Interpretive31.When interest rates are relatively high, investors generally expect interest rates to__________________. Thus, investors prefer to hold __________________ securities.A)fall; long-termB)fall; short-termC)rise; long-termD)rise; short-termAnswer: AD3 Interpretive32.When interest rates are relatively low, investors generally expect interest rates to__________________. Thus, investors prefer to hold __________________ securitiesA)fall; long-termB)fall; short-termC)rise; long-termD)rise; short-termAnswer: DD2 Interpretive33.Observations of the yield curve suggest that when interest rates are high and investorsexpect interest rates to fall, the yield curve will have a(n)A)upward slope.B)downward slope.C)horizontal slope.D)vertical slope.Answer: BD3 Interpretive34.According to pure expectations theory, the yield curve should on average beA)upward sloping.B)downward sloping.C)flat.D)vertical.Answer: CChapter 5The Risk and Term Structure of Interest Rates 59D1 Interpretive35.The fact that yields on short-term securities fluctuate more over the course of the businesscycle supports which theory of the term structure?A)Pure expectationsB)Preferred habitatC)Supply and demandD)Liquidity premiumAnswer: AD2 Factual36.Which of the following accounts for differences in the yield to maturity on U.S.government bonds with the same maturity?A)Capital gains tax treatmentB)Chance of repaymentC)MarketabilityD)Default riskAnswer: CD2 Factual37.The most actively traded government securities in the secondary market areA)the oldest securities outstanding.B)securities issued more than one year but less than five years ago.C)the securities associated with the current coupon issue.D)the securities with the longest maturity.Answer: CD2 Factual38.Which of the following types of government bonds tend to be the most marketable?A)“Off the run” bondsB)“On the run” bondsC)“Convertible” bondsD)“Coupon” bondsAnswer: BD2 Applied39. A negatively sloped yield curve indicates that the economyA)is likely to show strong growth in the coming months.B)will not change noticeably in the coming months.C)is likely to show significant weakness in the coming months.D)None of the aboveAnswer: C60 Ritter/Silber/Udell Money, Banking, and Financial Markets, Eleventh EditionD2 Interpretive40.As a factor explaining yield differences among U.S. government bonds, default risk isA)always an important consideration.B)important, but other factors such as market risk are more critical.C)rarely important.D)never a consideration.Answer: DD1 Applied41.Which of the following will have the highest yield at any point in time?A) A five-year Aaa-rated corporate bondB) A five-year Baa-rated corporate bondC) A five-year C-rated corporate bondD) A five-year U.S. Treasury bondAnswer: CD2 Applied42.If the following securities are of equal maturity, which will have the lowest yield at a givenpoint in time?A) A consumer loanB) A municipal bond rated AaaC)An Aaa-rated corporate bondD) A Baa-rated corporate bondAnswer: BD2 Applied43.An investor pays 20 percent of his income in taxes and purchases a $1,000 corporate bondyielding 10 percent. The after-tax yield on this bond isA)9 percent.B)8 percent.C)7 percent.D) 6 percent.Answer: BD2 Applied44. A municipal bond selling for $1,000 pays the holder $80 per year. If the bondholder pays a25 percent income tax rate, the after-tax yield isA)8 percent.B)7 percent.C) 6 percent.D) 5 percent.Answer: AChapter 5The Risk and Term Structure of Interest Rates 61D3 Applied45.Assume that the yield on long-term U.S. government bonds rises. The yield on corporatebonds with the same maturity will __________________, because investors will substitute__________________ for __________________.A)rise; government bonds; corporate bondsB)rise; corporate bonds; government bondsC)fall; government bonds; corporate bondsD)fall; corporate bonds; government bondsAnswer: AD2 Factual46.Which theory of the term structure is usually criticized for underestimating thesubstitutability of different maturities of securities?A)The expectations theoryB)The supply and demand approachC)The liquidity premium theoryD)The preferred habitat approachAnswer: BD2 Applied47.Relative to a bond with a shorter maturity, a bond with a longer maturity has greaterA)interest rate volatility.B)marketability.C)price volatility.D)default risk.Answer: CD2 Applied48.If all future expected short-term interest rates are equal to the current short-term interestrate, the expectations theory predicts that the yield curve would beA)horizontal.B)upward sloping.C)downward sloping.D)vertical.Answer: AD2 Factual49. A comprehensive measure of a bond’s maturity that takes into account the timing of bothcoupon and principal payments isA)term structure.B)risk premium.C)current coupon.D)duration.Answer: D62 Ritter/Silber/Udell Money, Banking, and Financial Markets, Eleventh EditionD3 Factual50.The formula for relating the percentage change in price to the bond’s duration isA)∆P/P = [–∆i/(1 + i) × D].B)∆P/P = [∆i/(1 + i) × D].C)∆P/P = [–∆i × (1 + i)/D].D)∆P/P = [–∆i/(1 + i) ÷ D].Answer: AD2 Interpretive51.In general the lower the yield to maturity of a bond, theA)lower the duration of a bond.B)higher the duration of a bond.C)lower the default risk.D)lower the price of the bond.Answer: BD3 Applied52. A bond has a duration of 4 years and a price of $1,000. The yield to maturity of the bondjust changed from 5 percent to 7 percent. The new price of the bond should beA)$1,076.B)$924.C)$1,093.D)$907.Answer: BD3 Applied53. A bond has a face value of $1,000 and an annual coupon rate of 6 percent. The yield tomaturity of this bond is 5 percent, and the bond has 2 years remaining until maturity. Basedon this information, this bond’s duration is __________________ years.A) 1.9B) 2.0C) 1.0D) 2.2Answer: A。

主成分分析

Briefly Introduction of PCA
Its purpose is to explain the behavior of observed variables using a smaller set of implied variables.
Continue…
From a mathematical standpoint, it consists in transforming a set of m correlated variables into a set of orthogonal variables that reproduce the original information present in the correlation structure.
Continue…
An important feature is that, as will be developed later, every empirical investigation shows that the variance of the term structure of interest rates is explained to more than 90% using only the three first components.
Explanation of Factor 3
Illustration of Concave and Convex Twist Movements
Some PCA Results from Different Countries
Assignment 1
(1) Show the evidence of properties 1-5 of China’s treasury monthly zero-coupon rate from 2005.01-2010.12. (2) Discuss the mathematical principle of PCA instead of the intuitional explanation in this lecture. (3) Using the data sample in (1) to make PCA to explore the factors that can explain more than 95 of total variations of zero coupon rates. Note that the assignment is to be finished by a group with maximum 4 persons and hand in two weeks away from now. You need to provide a word text document with detailed explanation of what you do and a PPT document with brief introduction to your main ideas and results.

The Term Structure of Interest Rates

15-2
Overview of Term Structure
• The yield curve is a graph that displays the relationship between yield and maturity.
• Information on expected future short term rates can be implied from the yield curve.
15-6
Example 15.1 Valuing Coupon Bonds
• Value a 3 year, 10% coupon bond using discount rates from Table 15.1:
$100 $100 $1100 Price 2 1.05 1.06 1.07 3
• We need to consider each bond cash flow as a stand-alone zero-coupon bond. • Bond stripping and bond reconstitution offer opportunities for arbitrage. • The value of the bond should be the sum of the values of its parts.
INVESTMENTS | BODIE, KANE, MARCUS
15-3
Figure 15.1 Treasury Yield Curves
INVESTMENTS | BODIE, KANE, MARCUS
15-4
Bond Pricing
• Yields on different maturity bonds are not all equal.

投资学-Chap15

The relationship between yield to maturity and maturity. Information on expected future short term rates can be implied from yield curve. The yield curve is a graph that displays the relationship between yield and maturity. Three major theories are proposed to explain the observed yield curve.
15-12
Liquidity Premium Theory
Long-term bonds are more risky. Investors will demand a premium for the risk associated with long-term bonds. The yield curve has an upward bias built into the long-term rates because of the risk premium. Forward rates contain a liquidity premium and are not equal to expected future short-term rates.
2yrs [(1.1125)3 / (1.1175)2] - 1 = 0.102567 3yrs [(1.1)4 / (1.1125)3] - 1 4yrs [(1.0925)5 / (1.1)4] - 1 = = 0.063336 0.063008
15-10
Theories of Term Structure

Term structure of interest rates (C15)

Pure Expectations hypothesis (a world of certainty) Liquidity Premium hypothesis (a world of uncertainty)

Constructing the yield curve
Treasury strips
Spot Rates

These prices on zero coupon bonds – calculated using short rates – in turn imply a set of zero-coupon rates, and these zero coupon rates are called spot rates
What does an upward sloping yield curve imply about expected
future interest rates? What is the assumed value of the liquidity premium under the expectations hypothesis?
Using the yield curve
Zero-vol or Static default spreads on corporates
Lesmond, 2016
1
Short Rates

Assume that the market participants all agree that annual rates for the next three years are expected to be: Year 1 Year 2 Year 3 3% 4% 5%

2.3 The Risk and Term Structure of Interest Rates[1]

Copyright © 2010 Pearson Addison-Wesley. All rights reserved.
6-13
FIGURE 3 Interest Rates on Municipal and Treasury Bonds
Copyright © 2010 Pearson Addison-Wesley. All rights reserved.
Copyright © 2010 Pearson Addison-Wesley. All rights reserved.
6-6
Risk Structure of Interest Rates 利率的风险结构
• Bonds with the same maturity have different interest rates due to:
Copyright © 2010 Pearson Addison-Wesley. All rights reserved.
6-4
FIGURE 1 Long-Term Bond Yields, 1919–2008 P124
Sources: Board of Governors of the Federal Reserve System, Banking and Monetary Statistics, 1941–1970; Federal Reserve: /releases/h15/data.htm.
Section 2.3 The Risk and Term Structure of Interest Rates
Copyright © 2010 Pearson Addison-Wesley. All rights reserved.

金融市场与机构(第六版)测试银行 Ch05

Chapter 5How Do Risk and Term Structure Affect Interest Rates?5.1Multiple Choice5.1.1)The term structure of interest rates isA)the relationship among interest rates of different bonds with the same maturity.B)the structure of how interest rates move over time.C)the relationship among the terms to maturity of different bonds.D)the relationship among interest rates on bonds with different maturities.Answer:DQuestion Status:Previous Edition5.1.2)The risk structure of interest rates isA)the structure of how interest rates move over time.B)the relationship among interest rates of different bonds with the same maturity.C)the relationship among the terms to maturity of different bonds.D)the relationship among interest rates on bonds with different maturities.Answer:BQuestion Status:Previous Edition5.1.3)Which of the following long-term bonds should have the lowest interest rate?A)Corporate Baa bondsB)U.S. Treasury bondsC)Corporate Aaa bondsD)Municipal bondsAnswer:DQuestion Status:Previous Edition5.1.4)Which of the following long-term bonds should have the highest interest rate?A)Corporate Baa bondsB)U.S. Treasury bondsC)Corporate Aaa bondsD)Municipal bondsAnswer:AQuestion Status:Previous Edition5.1.5)The risk premium on corporate bonds becomes smaller ifA)the riskiness of corporate bonds increases.B)the liquidity of corporate bonds increases.C)the liquidity of corporate bonds decreases.D)the riskiness of corporate bonds decreases.E)either (b) or (d) occur.Answer:EQuestion Status:Previous Edition62 Mishkin/Eakins · Financial Markets and Institutions, Sixth Edition5.1.6)Bonds with relatively low risk of default are calledA)zero coupon bonds.B)junk bonds.C)investment grade bonds.D)none of the above.Answer:CQuestion Status:Previous Edition5.1.7)Bonds with relatively high risk of default are calledA)Brady bonds.B)junk bonds.C)zero coupon bonds.D)investment grade bonds.Answer:BQuestion Status:Previous Edition5.1.8)A corporation suffering big losses might be more likely to suspend interest payments on itsbonds, therebyA)raising the default risk and causing the demand for its bonds to rise.B)raising the default risk and causing the demand for its bonds to fall.C)lowering the default risk and causing the demand for its bonds to rise.D)lowering the default risk and causing the demand for its bonds to fall.Answer:BQuestion Status:Previous Edition5.1.9)(I) If a corporation suffers big losses, the demand for its bonds will rise because of the higherinterest rates the firm must pay. (II) The spread between the interest rates on bonds withdefault risk and default-free bonds is called the risk premium.A)(I) is true, (II) false.B)(I) is false, (II) true.C)Both are true.D)Both are false.Answer:BQuestion Status:Previous Edition5.1.10)Holding everything else constant, if a corporation begins to suffer large losses, then the defaultrisk on its bonds will _________ and the expected return on those bonds will _________.A)increase: increaseB)decrease; increaseC)increase; decreaseD)decrease; decreaseAnswer:CQuestion Status:Previous EditionChapter 5 How Do Risk and Term Structure Affect Interest Rates? 63 5.1.11)Holding everything else the same, if a corporationʹs earnings rise, then the default risk on itsbonds will _________ and the expected return on those bonds will _________.A)increase; decreaseB)decrease; decreaseC)increase; increaseD)decrease; increaseAnswer:DQuestion Status:Previous Edition5.1.12)If a corporation begins to suffer large losses, then the default risk on its bonds will _________and the equilibrium interest rate on these bonds will _________.A)increase; decreaseB)decrease; increaseC)increase; increaseD)decrease; decreaseAnswer:CQuestion Status:Previous Edition5.1.13)If a corporationʹs earnings rise, then the default risk on its bonds will _________ and theequilibrium interest rate on these bonds will _________.A)increase; decreaseB)decrease; decreaseC)increase; increaseD)decrease; increaseAnswer:BQuestion Status:Previous Edition5.1.14)When the default risk on corporate bonds decreases, other things equal, the demand curve forcorporate bonds shifts to the _________ and the demand curve for Treasury bonds shifts to the_________.A)right; rightB)right; leftC)left; leftD)left; rightAnswer:BQuestion Status:Previous Edition5.1.15)(I) An increase in default risk on corporate bonds shifts the demand curve for corporate bondsto the right. (II) An increase in default risk on corporate bonds shifts the demand curve forTreasury bonds to the left.A)(I) is true, (II) false.B)(I) is false, (II) true.C)Both are true.D)Both are false.Answer:DQuestion Status:Previous Edition64 Mishkin/Eakins · Financial Markets and Institutions, Sixth Edition5.1.16)(I) An increase in default risk on corporate bonds shifts the demand curve for corporate bondsto the left. (II) An increase in default risk on corporate bonds shifts the demand curve forTreasury bonds to the right.A)(I) is true, (II) false.B)(I) is false, (II) true.C)Both are true.D)Both are false.Answer:CQuestion Status:Previous Edition5.1.17)When budget talks between congressional Republicans and President Clinton occurred in late1995, fear of a government default _________, Treasury bond values _________, and interestrates on Treasury bonds _________.A)rose; fell; roseB)rose; rose; roseC)fell; rose; fellD)fell; fell; fellAnswer:AQuestion Status:Previous Edition5.1.18)The spread between interest rates on low quality corporate bonds and U.S. government bonds_________ during the Great Depression.A)was reversedB)narrowed significantlyC)widened significantlyD)did not changeAnswer:CQuestion Status:Previous Edition5.1.19)As a result of the Enron collapse and bankruptcy, the demand for low quality corporate bonds_________, the demand for high quality corporate bonds _________, and the risk spread_________.A)increased; decreased; was unchangedB)decreased; increased; increasedC)increased; decreased; decreasedD)decreased; increased; was unchangedAnswer:BQuestion Status:Previous Edition5.1.20)Moodyʹs and Standard and Poorʹs are agencies thatA)help investors collect when corporations default on their bonds.B)advise municipal bond issuers on the tax exempt status of their bonds.C)produce information about the probability of default on corporate bonds.D)maintain liquid markets for corporate bonds.Answer:CQuestion Status:Previous EditionChapter 5 How Do Risk and Term Structure Affect Interest Rates? 65 5.1.21)If Moodyʹs or Standard and Poorʹs downgrades its rating on a corporate bond, the demand forthe bond _________ and its yield _________.A)increases; decreasesB)decreases; increasesC)increases; increasesD)decreases; decreasesAnswer:BQuestion Status:Previous Edition5.1.22)Corporate bonds are not as liquid as government bonds becauseA)fewer bonds for any one corporation are traded, making them more costly to sell.B)the corporate bond rating must be calculated each time they are traded.C)corporate bonds are not callable.D)all of the above.E)only A and B of the above.Answer:AQuestion Status:Previous Edition5.1.23)(I) The risk premium widens as the default risk on corporate bonds increases. (II) The riskpremium widens as corporate bonds become less liquid.A)(I) is true, (II) false.B)(I) is false, (II) true.C)Both are true.D)Both are false.Answer:CQuestion Status:Previous Edition5.1.24)When the corporate bond market becomes less liquid, other things equal, the demand curve forcorporate bonds shifts to the _________ and the demand curve for Treasury bonds shifts to the_________.A)right; rightB)right; leftC)left; leftD)left; right.Answer:DQuestion Status:Previous Edition5.1.25)When the corporate bond market becomes more liquid, other things equal, the demand curvefor corporate bonds shifts to the _________ and the demand curve for Treasury bonds shifts tothe _________.A)right; rightB)right; leftC)left; leftD)left; rightAnswer:BQuestion Status:Previous Edition66 Mishkin/Eakins · Financial Markets and Institutions, Sixth Edition5.1.26)(I) If a corporate bond becomes less liquid, the demand for the bond will fall, causing theinterest rate to rise. (II) If a corporate bond becomes less liquid, the demand for Treasury bondsdoes not change.A)(I) is true, (II) false.B)(I) is false, (II) true.C)Both are true.D)Both are false.Answer:AQuestion Status:Previous Edition5.1.27)(I) If a corporate bond becomes less liquid, the interest rate on the bond will fall. (II) If acorporate bond becomes less liquid, the interest rate on Treasury bonds will fall.A)(I) is true, (II) false.B)(I) is false, (II) true.C)Both are true.D)Both are false.Answer:BQuestion Status:Previous Edition5.1.28)If income tax rates were lowered, thenA)the interest rate on municipal bonds would fall.B)the interest rate on Treasury bonds would rise.C)the interest rate on municipal bonds would rise.D)the price of Treasury bonds would fall.Answer:CQuestion Status:Previous Edition5.1.29)If income tax rates rise, thenA)the prices of municipal bonds will fall.B)the prices of Treasury bonds will rise.C)the interest rate on Treasury bonds will rise.D)the interest rate on municipal bonds will rise.Answer:CQuestion Status:Previous Edition5.1.30)An increase in marginal tax rates would likely have the effect of _________ the demand formunicipal bonds and _________ the demand for U.S. government bonds.A)increasing; increasingB)increasing; decreasingC)decreasing; increasingD)decreasing; decreasingAnswer:BQuestion Status:Previous EditionChapter 5 How Do Risk and Term Structure Affect Interest Rates? 67 5.1.31)A decrease in marginal tax rates would likely have the effect of _________ the demand formunicipal bonds and _________ the demand for U.S. government bonds.A)increasing; increasingB)increasing; decreasingC)decreasing; increasingD)decreasing; decreasingAnswer:CQuestion Status:Previous Edition5.1.32)Which of the following statements are true?A)Because coupon payments on municipal bonds are exempt from federal income tax, theexpected after-tax return on them will be higher for individuals in higher income taxbrackets.B)An increase in tax rates will increase the demand for municipal bonds, lowering theirinterest rates.C)Interest rates on municipal bonds will be lower than on comparable bonds without thetax exemption.D)All of the above are true statements.E)Only A and B are true statements.Answer:DQuestion Status:Previous Edition5.1.33)Which of the following statements are true?A)Because coupon payments on municipal bonds are exempt from federal income tax, theexpected after-tax return on them will be higher for individuals in higher income taxbrackets.B)An increase in tax rates will increase the demand for Treasury bonds, lowering theirinterest rates.C)Interest rates on municipal bonds will be higher than on comparable bonds without thetax exemption.D)Only A and B are true statements.Answer:AQuestion Status:Previous Edition5.1.34)When a municipal bond is given tax-free status, the demand for municipal bonds shifts_________, causing the interest rate on the bond to _________A)leftward; rise.B)leftward; fall.C)rightward; rise.D)rightward; fall.Answer:DQuestion Status:Previous Edition68 Mishkin/Eakins · Financial Markets and Institutions, Sixth Edition5.1.35)When a municipal bond is given tax-free status, the demand for Treasury bonds shifts_________, and the interest rate on Treasury bonds _________A)leftward; rises.B)leftward; falls.C)rightward; rises.D)rightward; falls.Answer:AQuestion Status:Previous Edition5.1.36)If municipal bonds were to lose their tax-free status, then the demand for Treasury bondswould shift _________, and the interest rate on Treasury bonds would _________A)rightward; fall.B)rightward; rise.C)leftward; fall.D)leftward; rise.Answer:AQuestion Status:Previous Edition5.1.37)The Bush tax cut passed in 2001 reduces the top income tax bracket from 39 percent to 35percent over the next ten years. As a result of this tax cut, the demand for municipal bondsshould shift to the _________ and the interest rate on municipal bonds should _________.A)right; declineB)right; increaseC)left; declineD)left; increaseAnswer:DQuestion Status:Previous Edition5.1.38)The relationship among interest rates on bonds with identical default risk, but differentmaturities, is called theA)time-risk structure of interest rates.B)liquidity structure of interest rates.C)yield curve.D)bond demand curve.Answer:CQuestion Status:Previous Edition5.1.39)Yield curves can be classified asA)upward-sloping.B)downward-sloping.C)flat.D)all of the above.E)only A and B of the above.Answer:DQuestion Status:Previous EditionChapter 5 How Do Risk and Term Structure Affect Interest Rates? 695.1.40)Typically, yield curves areA)gently upward-sloping.B)gently downward-sloping.C)flat.D)bowl shaped.E)mound shaped.Answer:AQuestion Status:Previous Edition5.1.41)When yield curves are steeply upward-sloping,A)long-term interest rates are above short-term interest rates.B)short-term interest rates are above long-term interest rates.C)short-term interest rates are about the same as long-term interest rates.D)medium-term interest rates are above both short-term and long-term interest rates.E)medium-term interest rates are below both short-term and long-term interest rates.Answer:AQuestion Status:Previous Edition5.1.42)Economistsʹ attempts to explain the term structure of interest ratesA)illustrate how economists modify theories to improve them when they are inconsistentwith the empirical evidence.B)illustrate how economists continue to accept theories that fail to explain observedbehavior of interest rate movements.C)prove that the real world is a special case that tends to get short shrift in theoreticalmodels.D)have proved entirely unsatisfactory to date.Answer:AQuestion Status:Previous Edition5.1.43)According to the expectations theory of the term structure,A)the interest rate on long-term bonds will exceed the average of expected futureshort-term interest rates.B)interest rates on bonds of different maturities move together over time.C)buyers of bonds prefer short-term to long-term bonds.D)all of the above.E)only A and B of the above.Answer:BQuestion Status:Previous Edition5.1.44)According to the expectations theory of the term structure,A)when the yield curve is steeply upward-sloping, short-term interest rates are expected torise in the future.B)when the yield curve is downward-sloping, short-term interest rates are expected todecline in the future.C)buyers of bonds prefer short-term to long-term bonds.D)all of the above.E)only A and B of the above.Answer:EQuestion Status:Previous Edition70 Mishkin/Eakins · Financial Markets and Institutions, Sixth Edition5.1.45)According to the expectations theory of the term structure,A)when the yield curve is steeply upward-sloping, short-term interest rates are expected torise in the future.B)when the yield curve is downward-sloping, short-term interest rates are expected toremain relatively stable in the future.C)investors have strong preferences for short-term relative to long-term bonds, explainingwhy yield curves typically slope upward.D)all of the above.E)only A and B of the above.Answer:AQuestion Status:Previous Edition5.1.46)According to the expectations theory of the term structure,A)yield curves should be equally likely to slope downward as to slope upward.B)when the yield curve is steeply upward-sloping, short-term interest rates are expected torise in the future.C)when the yield curve is downward-sloping, short-term interest rates are expected toremain relatively stable in the future.D)all of the above.E)only A and B of the above.Answer:EQuestion Status:Previous Edition5.1.47)If the expected path of one-year interest rates over the next four years is 5 percent, 4 percent, 2percent, and 1 percent, then the pure expectations theory predicts that todayʹs interest rate onthe four-year bond isA)1 percent.B)2 percent.C)4 percent.D)none of the above.Answer:DQuestion Status:Previous Edition5.1.48)If the expected path of one-year interest rates over the next five years is 1 percent, 2 percent, 3percent, 4 percent, and 5 percent, the pure expectations theory predicts that the bond with thehighest interest rate today is the one with a maturity ofA)one year.B)two years.C)three years.D)four years.E)five years.Answer:EQuestion Status:Previous Edition5.1.49)If the expected path of one-year interest rates over the next five years is 2 percent, 4 percent, 1percent, 4 percent, and 3 percent, the pure expectations theory predicts that the bond with the lowest interest rate today is the one with a maturity ofA)one year.B)two years.C)three years.D)four years.Answer:AQuestion Status:Previous Edition5.1.50)According to the market segmentation theory of the term structure,A)the interest rate for bonds of one maturity is determined by supply and demand forbonds of that maturity.B)bonds of one maturity are not substitutes for bonds of other maturities; therefore, interestrates on bonds of different maturities do not move together over time.C)investorsʹ strong preference for short-term relative to long-term bonds explains whyyield curves typically slope upward.D)all of the above.E)none of the above.Answer:DQuestion Status:Previous Edition5.1.51)According to the market segmentation theory of the term structure,A)the interest rate for bonds of one maturity is determined by supply and demand forbonds of that maturity.B)bonds of one maturity are not substitutes for bonds of other maturities; therefore, interestrates on bonds of different maturities do not move together over time.C)investorsʹ strong preference for short-term relative to long-term bonds explains whyyield curves typically slope downward.D)only A and B of the above.Answer:DQuestion Status:Previous Edition5.1.52)The liquidity premium theory of the term structureA)indicates that todayʹs long-term interest rate equals the average of short-term interestrates that people expect to occur over the life of the long-term bond.B)assumes that bonds of different maturities are perfect substitutes.C)suggests that markets for bonds of different maturities are completely separate becausepeople have preferred habitats.D)does none of the above.Answer:DQuestion Status:Previous Edition5.1.53)The liquidity premium theory of the term structureA)assumes investors tend to prefer short-term bonds because they have less interest-raterisk.B)assumes that interest rates on the long-term bond respond to demand and supplyconditions for that bond.C)assumes that an average of expected short-term rates is an important component ofinterest rates on long-term bonds.D)assumes all of the above.E)assumes none of the above.Answer:DQuestion Status:Previous Edition5.1.54)According to the liquidity premium theory of the term structure,A)the interest rate on long-term bonds will equal an average of short-term interest ratesthat people expect to occur over the life of the long-term bonds plus a liquidity premium.B)buyers of bonds may prefer bonds of one maturity over another, yet interest rates onbonds of different maturities move together over time.C)even with a positive liquidity premium, if future short-term interest rates are expected tofall significantly, then the yield curve will be downward-sloping.D)all of the above.E)only A and B of the above.Answer:DQuestion Status:Previous Edition5.1.55)According to the liquidity premium theory of the term structure,A)because buyers of bonds may prefer bonds of one maturity over another, interest rates onbonds of different maturities do not move together over time.B)the interest rate on long-term bonds will equal an average of short-term interest ratesthat people expect to occur over the life of the long-term bonds plus a term premium.C)because of the positive term premium, the yield curve cannot be downward-sloping.D)all of the above.E)only A and B of the above.Answer:BQuestion Status:Previous Edition5.1.56)If the yield curve slope is flat, the liquidity premium theory indicates that the market ispredictingA)a mild rise in short-term interest rates in the near future and a mild decline further out inthe future.B)constant short-term interest rates in the near future and further out in the future.C)a mild decline in short-term interest rates in the near future and a continuing milddecline further out in the future.D)constant short-term interest rates in the near future and a mild decline further out in thefuture.Answer:CQuestion Status:Previous Edition5.1.57)If the yield curve has a mild upward slope, the liquidity premium theory indicates that themarket is predictingA)a rise in short-term interest rates in the near future and a decline further out in the future.B)constant short-term interest rates in the near future and further out in the future.C)a decline in short-term interest rates in the near future and a rise further out in the future.D)a decline in short-term interest rates in the near future and an even steeper declinefurther out in the future.Answer:BQuestion Status:Previous Edition5.1.58)According to the liquidity premium theory of the term structure, a downward-sloping yieldcurve indicates that short-term interest rates are expected toA)rise in the future.B)remain unchanged in the future.C)decline moderately in the future.D)decline sharply in the future.Answer:DQuestion Status:Previous Edition5.1.59)According to the liquidity premium theory of the term structure, when the yield curve has itsusual slope, the market expectsA)short-term interest rates to rise sharply.B)short-term interest rates to drop sharply.C)short-term interest rates to stay near their current levels.D)none of the above.Answer:CQuestion Status:Previous Edition5.1.60)In actual practice, short-term interest rates are just as likely to fall as to rise; this is the majorshortcoming of theA)market segmentation theory.B)expectations theory.C)liquidity premium theory.D)separable markets theory.Answer:BQuestion Status:Previous Edition5.1.61)Which theory of the term structure proposes that bonds of different maturities are notsubstitutes for one another?A)market segmentation theoryB)expectations theoryC)liquidity premium theoryD)separable markets theoryAnswer:AQuestion Status:Previous Edition5.1.62)Since yield curves are usually upward sloping, the _________ indicates that, on average, peopletend to prefer holding short-term bonds to long-term bonds.A)market segmentation theoryB)expectations theoryC)liquidity premium theoryD)both A and B of the aboveE)both A and C of the aboveAnswer:EQuestion Status:Previous Edition5.1.63)_________ cannot explain the empirical fact that interest rates on bonds of different maturitiestend to move together.A)The market segmentation theoryB)The expectations theoryC)The liquidity premium theoryD)Both A and B of the aboveE)Both A and C of the aboveAnswer:AQuestion Status:Previous Edition5.1.64)Which of the following theories of the term structure is (are) able to explain the fact thatinterest rates on bonds of different maturities tend to move together over time?A)The pure expectations hypothesisB)The segmented markets theoryC)The preferred habitat theoryD)Both A and B of the aboveE)Both A and C of the aboveAnswer:EQuestion Status:Previous Edition5.1.65)Of the four theories that explain how interest rates on bonds with different terms to maturityare related, the one that views long-term interest rates as equaling the average of futureshort-term rates expected to occur over the life of the bond is theA)pure expectations theory.B)preferred habitat theory.C)liquidity premium theory.D)segmented markets theory.Answer:AQuestion Status:Previous Edition5.1.66)Of the four theories that explain how interest rates on bonds with different terms to maturityare related, the one that assumes that bonds of different maturities are not substitutes for one another is theA)expectations theory.B)segmented markets theory.C)liquidity premium theory.D)preferred habitat theory.Answer:BQuestion Status:Previous Edition5.1.67)A moderately upward-sloping yield curve indicates that short-term interest rates are expectedtoA)neither rise nor fall in the near future.B)remain relatively unchanged, but that long-term rates are expected to fall.C)neither rise nor fall, but that long-term rates are expected to rise moderately.D)rise moderately in the near future.Answer:AQuestion Status:Previous Edition5.1.68)A steep upward sloping yield curve indicates that short-term interest rates are expected toA)neither rise nor fall in the near future.B)remain relatively unchanged, but that long-term rates are expected to fall.C)neither rise nor fall, but that long-term rates are expected to rise moderately.D)rise moderately in the near future.Answer:DQuestion Status:Previous Edition5.1.69)A bond rating of Aa or AA would mean that the quality of the bond isA)the highest.B)high.C)medium grade.D)speculative.Answer:BQuestion Status:New5.1.70)______ bonds are the most liquid of all long-term bonds.A)CallableB)MunicipalC)Corporate AaaD)U.S. TreasuryAnswer:DQuestion Status:New5.1.71)______ bonds are exempt from federal income taxes.A)Corporate AaaB)U.S. TreasuryC)Corporate BaaD)MunicipalAnswer:DQuestion Status:New5.1.72)The risk structure of interest rates is explained byA)default risk.B)liquidity.C)tax considerations.D)all of the above.Answer:AQuestion Status:New5.1.73)The ________ theory is the most widely accepted theory of the term structure of interest ratesbecause it explains the major empirical facts about the term structure so well.A)liquidity premiumB)market segmentationC)expectationsD)none of the aboveAnswer:AQuestion Status:New5.1.74)________ are investment advisory firms that rate the quality of corporate and municipal bondsin terms of probability of default.A)Financial institutionsB)Credit-rating agenciesC)Securities companiesD)none of the aboveAnswer:BQuestion Status:New5.1.75)If a bond has favorable tax treatment, its interest rateA)will be higher.B)will not be effected.C)will be lower.D)all of the above could happen.Answer:AQuestion Status:New5.2True/False5.2.1)The term structure of interest rates describes how interest rates move over time.Answer:FALSEQuestion Status:Previous Edition5.2.2)The risk structure of interest rates describes the relationship between the interest rates ofdifferent bonds with the same maturity.Answer:TRUEQuestion Status:Previous Edition5.2.3)Following the Enron bankruptcy, the spread between the interest rates on Baa bonds and Aaabonds widened.Answer:TRUEQuestion Status:Previous Edition5.2.4)The risk premium on corporate bonds becomes smaller as the liquidity of the bonds falls.Answer:FALSEQuestion Status:Previous Edition5.2.5)An increase in income tax rates will cause the interest rates on tax exempt municipal bonds tofall relative to the interest rate on taxable corporate securities.Answer:TRUEQuestion Status:Previous Edition。

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Investors will demand a premium for the risk associated with long-term bonds.
The yield curve has an upward bias built into the long-term rates because of the risk premium.
15-12
Expectations Theory
Observed long-term rate is a function of today’s short-term rate and expected future short-term rates. Long-term and short-term securities are perfect substitutes. Forward rates that are calculated from the yield on long-term securities are market consensus expected future short-term rates.
9.660 9.993
* $1,000 Par value zero
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-7
Forward Rates from Observed Long-Term Rates
- Upward bias over expectations
Market Segmentation
- Preferred Habitat
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-3
Yield Curves
Yields Upward Sloping Flat
Downward Sloping
Maturity
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
The relationship between yield to maturity and maturity.
Information on expected future short term rates can be implied from yield curve. The yield curve is a graph that displays the relationship between yield and maturity.
(1 yn ) (1 f n ) (1 yn 1 ) n 1
n
fn = one-year forward rate for period n yn = yield for a security with a maturity of n
(1 yn ) (1 yn1 ) (1 f n )
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-1
Chapter 15
The Term Structure of Interest Rates
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-2
Overview of Term Structure of Interest Rates
=
=
0.063336
0.063008
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-11
Theories of Term Structure
Expectations
Liquidity Preference
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-13
Liquidity Premium Theory
Long-term bonds are more risky.
PVn = Present Value of $1 in n periods r1 = One-year rate for period 1
r2 = One-year rate for period 2
rn = One-year rate for period n
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-10
Forward Rates for Downward Sloping Yield Curve
1yr Forward Rates
15-14
Liquidity Premiums and Yield Curves
Yields
Observed Yield Curve
Forward Rates
Liquidity Premium Maturity
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-17
Using Spot Rates to Price Coupon Bonds
A coupon bond can be viewed as a series of zero coupon bonds. To find the value each payment is discount at the zero coupon rate. Once the bond value is found, one can solve for the yield. It’s the reason that similar maturity and default risk bonds sell at different yields to maturity.
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-9
Downward Sloping Spot Yield Curve
Zero-Coupon Rates Bond Maturity 12% 1 11.75% 2 11.25% 3 10.00% 4 9.25% 5
Forward rates contain a liquidity premium and are not equal to expected future short-term rates.
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
15-5
Pricing of Bonds using Expected Rates
1 PVn (1 r1 ) (1 r2 )...(1 rn )
15-15
Liquidity Premiums and Yield Curves
Yields
Observed Yield Curve
Forward Rates Liquidity Premium
Maturity
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Three major theories are proposed to explain the observed yield curve.
McGraw-Hill/Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
3yr = 9.660
fn = ?
(1.0993)4 = (1.0966)3 (1+fn) (1.46373) / (1.31870) = (1+fn) fn = .10998 or 11% Note: this is expected rate that was used in the prior example.