固定收益证券试题及部分答案

国际经济贸易学院研究生课程班《固定收益证券》试题班级序号：学号：姓名：成绩：1）Explain why you agree or disagree with the following statement: “The price of a floater will always trade at its par value.”Answer:I disagree with the statement: “The price of a floater will always trade at its par value.” First, the coupon rate of a floating-rate security (or floater) is equal to a reference rate plus some spread or margin. For example, the coupon rate of a floater can reset at the rate on a three-month Treasury bill (the reference rate) plus 50 basis points (the spread). Next, the price of a floater depends on two factors: (1) the spread over the reference rate and (2) any restrictions that may be imposed on the resetting of the coupon rate. For example, a floater may have a maximum coupon rate called a cap or a minimum coupon rate called a floor. The price of a floater will trade close to its par value as long as (1) the spread above the reference rate that the market requires is unchanged and (2) neither the cap nor the floor is reached. However, if the market requires a larger (smaller) spread, the price of a floater will trade below (above) par. If the coupon rate is restricted from changing to the reference rate plus the spread because of the cap, then the price of a floater will trade below par.2）A portfolio manager is considering buying two bonds. Bond A matures in three years and has a coupon rate of 10% payable semiannually. Bond B, of the same credit quality, matures in 10 years and has a coupon rate of 12% payable semiannually. Both bonds are priced at par.(a) Suppose that the portfolio manager plans to hold the bond that is purchased for three years. Which would be the best bond for the portfolio manager to purchase?Answer:The shorter term bond will pay a lower coupon rate but it will likely cost less for a given market rate.Since the bonds are of equal risk in terms of creit quality (The maturity premium for the longer term bond should be greater),the question when comparing the two bond investments is:What investment will be expecte to give the highest cash flow per dollar invested?In other words,which investment will be expected to give the highest effective annual rate of return.In general,holding the longer term bond should compensate the investor in the form of a maturity premium and a higher expected return.However,as seen in the discussion below,the actual realized return for either investment is not known with certainty.To begin with,an investor who purchases a bond can expect to receive a dollar return from(i)the periodic coupon interest payments made be the issuer,(ii)an capital gainwhen the bond matures,is called,or is sold;and (iii)interest income generated from reinvestment of the periodic cash flows.The last component of the potential dollar return is referred to as reinvestment income.For a standard bond(our situation)that makes only coupon payments and no periodic principal payments prior to the maturity date,the interim cash flows are simply the coupon payments.Consequently,for such bonds the reinvestment income is simply interest earned from reinvesting the coupon interest payments.For these bonds,the third component of the potential source of dollar return is referred to as the interest-on-interest components.If we are going to coupute a potential yield to make a decision,we should be aware of the fact that any measure of a bond’s potential yield should take into consideration each of the three components described above.The current yield considers only the coupon interest payments.No consideration is given to any capital gain or interest on interest.The yield to maturity takes into account coupon interest and any capital gain.It also considers the interest-on-interest component.Additionally,implicit in the yield-to-maturity computation is the assumption that the coupon payments can be reinvested at the computed yield to maturity.The yield to maturity is a promised yield and will be realized only if the bond is held to maturity and the coupon interest payments are reinvested at the yield to maturity.If the bond is not held to maturity and the coupon payments are reinvested at the yield to maturity,then the actual yield realized by an investor can be greater than or less than the yield to maturity.Given the facts that(i)one bond,if bought,will not be held to maturity,and(ii)the coupon interest payments will be reinvested at an unknown rate,we cannot determine which bond might give the highest actual realized rate.Thus,we cannot compare them based upon this criterion.However,if the portfolio manager is risk inverse in the sense that she or he doesn’t want to buy a longer term bond,which will likel have more variability in its return,then the manager might prefer the shorter term bond(bondA) of thres years.This bond also matures when the manager wants to cash in the bond.Thus,the manager would not have to worry about any potential capital loss in selling the longer term bond(bondB).The manager would know with certainty what the cash flows are.IfThese cash flows are spent when received,the manager would know exactly how much money could be spent at certain points in time.Finally,a manager can try to project the total return performance of a bond on the basis of the panned investment horizon and expectations concerning reinvestment rates and future market yields.This ermits the portfolio manager to evaluate thich of several potential bonds considered for acquisition will perform best over the planned investment horizon.As we just rgued,this cannot be done using the yield to maturity as a measure of relative ing total return to assess performance over some investment horizon is called horizon analysis.When a total return is calculated oven an investment horizon,it is referred to as a horizon return.The horizon analysis framwor enabled the portfolio manager to analyze the performance of a bond under different interest-rate scenarios for reinvestment rates and future market yields.Only by investigating multiple scenarios can the portfolio manager see how sensitive the bond’s performance will be to each scenario.This can help the manager choosebetween the two bond choices.(b) Suppose that the portfolio manager plans to hold the bond that is purchased for six years instead of three years. In this case, which would be the best bond for the portfolio manager to purchase?Answer:Similear to our discussion in part(a),we do not know which investment would give the highest actual relized return in six years when we consider reinvesting all cash flows.If the manager buys a three-year bond,then there would be the additional uncertainty of now knowing what three-year bond rates would be in three years.The purchase of the ten-year bond would be held longer than previously(six years compared to three years)and render coupon payments for a six-year period that are known.If these cash flows are spent when received,the manager will know exactly how much money could be spent at certain points in timeNot knowing which bond investment would give the highest realized return,the portfolio manager would choose the bond that fits the firm’s goals in terms of maturity.3） Answer the below questions for bonds A and B.Bond A Bond BCoupon 8% 9%Yield to maturity 8% 8%Maturity (years) 2 5Par $100.00 $100.00Price $100.00 $104.055(a) Calculate the actual price of the bonds for a 100-basis-point increase in interest rates.Answer:For Bond A, we get a bond quote of $100 for our initial price if we have an 8% coupon rate and an 8% yield. If we change the yield 100 basis point so the yield is 9%, then the value of the bond (P) is the present value of the coupon payments plus the present value of the par value. We have C = $40, y = 4.5%, n = 4, and M = $1,000. Inserting these numbers into our present value of coupon bond formula, we get:41111(1)(10.045)$40$143.5010.045nr P C r ⎡⎤⎡⎤--⎢⎥⎢⎥++===⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦The present value of the par or maturity value of $1,000 is:4$1,000$838.561(1)(1.045)n M r ==+ Thus, the value of bond A with a yield of 9%, a coupon rate of 8%, and a maturity of 2 years is: P = $143.501 + $838.561 = $982.062. Thus, we get a bond quote of $98.2062. We already know that bond B will give a bond value of $1,000 and a bond quote of $100 since a change of 100 basis points will make the yield and couponrate the same, For example, inserting Thus, the value of bond A with a yield of 9%, a coupon rate of 8%, and a maturity of 2 years is: P = $143.501 + $838.561 = $982.062. Thus, we get a bond quote of $98.2062. We already know that bond B will give a bond value of $1,000 and a bond quote of $100 since a change of 100 basis points will make the yield and coupon rate the same, For example, inserting(b) Using duration, estimate the price of the bonds for a 100-basis-point increase in interest rates.Answer:To estimate the price of bond A, we begin by first computing the modified duration. We can use an alternative formula that does not require the extensive calculations required by the Macaulay procedure. The formula is:211(100/)1(1)(1)n n Cn C y y y y Modified Duration P +⎡⎤--+⎢⎥++⎣⎦=Putting all applicable variables in terms of $100, we have C = $4, n = 4, y = 0.045, and P = $98.2062. Inserting these values, in the modified duration formula gives: 212451(100/)$414($100$4/0.045)11(1)(1)0.045(1.045)(1.045)98.2062n n C n C y y y y Modified Duration P +⎡⎤⎡⎤---+-+⎢⎥⎢⎥++⎣⎦⎣⎦=== ($1,975.308642[0.161439] + $35.664491) / $98.2062 = ($318.89117 + $35.664491) / $98.2062 = $354.555664 / $98.2062 = 3.6103185 or about 3.61. Converting to annual number by dividing by two gives a modified duration of 1.805159 (before the increase in 100 basis points it was 1.814948). We next solve for the change in price using the modified duration of 1.805159 and dy = 100 basis points = 0.01. We have: ()() 1.805159(0.01)0.0180515dP Modified Duration dy P=-=-=- We can now solve for the new price of bond A as shown below:(1)(10.0180515)$1,000$981.948dP P P+=-= This is slightly less than the actual price of $982.062. The difference is $982.062 – $981.948 = $0.114. To estimate the price of bond B, we follow the same procedure just shown for bond A. Using the alternative formula for modified duration that does not require the extensive calculations required by the Macaulay procedure and noting that C = $45, n = 10, y = 0.045, and P = $100, we get:21210111(100/)$4.5110($100$4.5/0.045)11(1)(1)0.045(1.045)(1.045)$100n n C n C y y y y Modified Duration P +⎡⎤⎡⎤---+-+⎢⎥⎢⎥++⎣⎦⎣⎦== ($791.27182 + $0) / $100 = 7.912718 or about 7.91 (before the increase in 100 basis points it was 7.988834 or about 7.99). Converting to an annual number by dividing by two gives a modified duration of 3.956359 (before the increase in 100 basis points it was 3.994417). We will now estimate the price of bond B using the modified durationmeasure. With 100 basis points giving dy = 0.01 and an approximate duration of 3.956359, we have:()() 3.956359(0.01)0.0395635dP Modified Duration dy P=-=-=-Thus, the new price is(1 – 0.0395635)$1,040.55 = (0.9604364)$1,040.55 = $999.382.This is slightly less than the actual price of $1,000. The difference is $1,000 – $999.382 = $0.618.(c) Using both duration and convexity measures, estimate the price of the bonds for a 100-basis-point increase in interest rates. Answer:For bond A, we use the duration and convexity measures as given below. First, we use the duration measure. We add 100 basis points and get a yield of 9%. We now have C = $40, y = 4.5%, n = 4, and M = $1,000. NOTE. In part (a) we computed the actual bond price and got P = $982.062. Prior to that, the price sold at par (P = $1,000) since the coupon rate and yield were then equal. The actual change in price is: ($982.062 – $1,000) = $17.938 and the actual percentage change in price is: $17.938 / $1,000 = 0.017938%. We will now estimate the price by first approximating the dollar price change. With 100 basis points giving dy = 0.01 and a modified duration computed in part (b) of 1.805159, we have:()() 1.805159(0.01)0.01805159dP Modified Duration dy P=-=-=- This is slightly more negative than the actual percentage decrease in price of1.7938%. The difference is 1.7938% – ( 1.805159%) = 1.7938% + 1.805159% = 0.011359%. Using the 1.805159% just given by the duration measure, the new price for bond A is:(1)(10.01805159)$1,000$981.948dP P P+=-= This is slightly less than the actual price of $982.062. The difference is $982.062 – $981.948 = $0.114. Next, we use the convexity measure to see if we can account for the difference of 0.011359%. We have: convexity measure (half years) =2232121212(1)(100/)11(1)(1)(1)n n n d P C Cn n n C y dy P y y y y y P ++⎡⎤⎡⎤+-⎡⎤=--+⎢⎥⎢⎥⎢⎥+++⎣⎦⎣⎦⎣⎦ For bond A, we add 100 basis points and get a yield of 9%. We now have C = $40, y = 4.5%, n = 4, and M = $1,000. NOTE. In part (a) we computed the actual bond price and got P = $982.062. Prior to that, the price sold at par (P = $1,000) since the coupon rate and yield were then equal. Expressing numbers in terms of a $100 bond quote, we have: C = $4, y = 0.045, n = 4, and P = $98.2062. Inserting these numbers into our convexity measure formula gives:convexity measure (half years) = 342562$412($4)44(5)(100$4/0.045)1116.93250.045(1.045)0.045(1.045)(1.045)$98.2062y ⎡⎤⎡⎤-=⎡⎤--+=⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦ 2216.9325() 4.2331252Convexity Measure inm period per year TheConvexity Measure in years m === Adding the duration measure and the convexity measure, we get 1.805159% + 0.021166% = 1.783994%. Recall the actual change in price is: ($982.062 – $1,000) = $17.938 and the actual percentage change in price is: $17.938 / $1,000 = −0.017938 or approximately 1.7938%. Using the 1.783994% resulting from both the duration and convexity measures, we can estimate the new price for bond A. We have:Pr (1)(10.01783994)$1,000(0.9819484)$1,000$982.160dP New ice P P=+=+-== Adding the duration measure and the convexity measure, we get 1.805159% + 0.021166% = 1.783994%. Recall the actual change in price is: ($982.062 – $1,000) = $17.938 and the actual percentage change in price is: $17.938 / $1,000 = −0.017938 or approximately 1.7938%. Using the 1.783994% resulting from both the duration and convexity measures, we can estimate the new price for bond A. We have:()() 3.056359(0.01)0.0395635dP Modified Duration dy P=-=-=- This is slightly more negative than the actual percentage decrease in price of -3.896978%. The difference is (-3.896978%)-(-3.95635%)=0.059382%Using the -3.95635%just given by the duration measure, the new price for Bond B is: (1)(10.0395635)$1,040.55$999.382dP P P+=-=This is slightly less than the actual price of $1,000. This difference is $1,000-$999.382=$0.618We use the convexity measure to see if we can account for the difference of 00594%. We have:2232121212(1)(100/)1()1(1)(1)(1)n n n d P C Cn n n C y Convexity Measure half years dy P y y y y y P ++⎡⎤⎡⎤+-⎡⎤==--+⎢⎥⎢⎥⎢⎥+++⎣⎦⎣⎦⎣⎦ For Bond B, 100 basis points are added and get a yield of 9%. We now have C=$45, y=4.5%, n=10, and M=$1,000. Note in part (a), we computed the actual bond price and got P=$1,000 since the coupon rate and yield were then equal. Prior to that, the price sold at P=$1,040.55. Expressing numbers in terms of a $100 bond quote, we have C=$4.5, y-0.045, n=10 and P=$100. Inserting these numbers into our convexity measure formula gives:310211122($4.5)12($4.5)410(11)(100$4.5/0.045)1()1(0.045)(1.045)(0.045)(1.045)(1.045)$100Convexity Measure half years ⎡⎤⎡⎤-⎡⎤=--+⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦ 7,781.03[0.01000]77.8103==The convexity measure (in years)=2277.810319.4525642convexitymeasureinm period per year m == Note. Dollar Convexity Measure=Convexity Measure (years) times P=19.452564($100)=$1,945.2564.The percentage price change due to convexity is 21()2dP convexity measure dy P = Inserting in the values, we get 21(77.8103)(0.01)0.000974632dP P == Thus, we have 0.097463% increase in price when we adjust for convexity measure.Adding the duration measure and convexity measure, we get -3.9563659%+0.097263% equals -3.859096%. Recall the actual change in price is ($1,000-$1,040.55)=-$40.55 and the actual new price is(1)(10.03859096)$1,040.55(0.9614091)$1,040.55$1,000.394dP P P-=-== For Bond A. This is about the same as the actual price of $1,000. The difference is $1,000.394-$1,000=$0.394. Thus, using the convexity measure along with the duration measure has narrowed the estimated price from a difference of -$0.618 to $0.394.(d) Comment on the accuracy of your results in parts b and c, and state why one approximation is closer to the actual price than the other.Answer:For bond A, the actual price is $982.062. When we use the duration measure, we get a bond price of $981.948 that is $0.114 less than the actual price. When we use duration and convex measures together, we get a bond price of $982.160. This is slightly more than the actual price of $982.062. The difference is $982.160 – $982.062 = $0.098. Thus, using the convexity measure along with the duration measure has narrowed the estimated price from a difference of $0.114 to $0.0981. For bond B, the actual price is $1,000. When we use the duration measure, we get a bond price of $999.382 that is $0.618 less than the actual price. When we use duration and convex measures together, we get a bond price of $1,000.394. This is slightly more than the actual price of $1,000. The difference is $1,000.394 – $1,000 = $0.394. Thus, using the convexity measure along with the duration measure has narrowed the estimated price from a difference of −$0.618 to $0.394As we see, using the duration and convexity measures together is more accurate. The reason is that adding the convexity measure to our estimate enables us to include the second derivative that corrects for the convexity of the price-yield relationship. Moredetails are offered below. Duration (modified or dollar) attempts to estimate a convex relationship with a straight line (the tangent line). We can specify a mathematical relationship that provides a better approximation to the price change of the bond if the required yield changes. We do this by using the first two terms of a Taylor series to approximate the price change as follows:2221()(1)2dP d P dP dy dy error dy dy =+++ Dividing both sides of this equation by P to get the percentage price change gives us: 22211()(2)2dP dP d P error dy dy P dy P dy P =+++The first term on the right-hand side of equation (1) is equation for the dollar price change based on dollar duration and is our approximation of the price change based on duration. In equation (2), the first term on the right-hand side is the approximate percentage change in price based on modified duration. The second term in equations(1) and (2) includes the second derivative of the price function for computing the value of a bond. It is the second derivative that is used as a proxy measure to correct for the convexity of the price-yield relationship. Market participants refer to the second derivative of bond price function as the dollar convexity measure of the bond. The second derivative divided by price is a measure of the percentage change in the price of the bond due to convexity and is referred to simply as the convexity measure.(e) Without working through calculations, indicate whether the duration of the two bonds would be higher or lower if the yield to maturity is 10% rather than 8%.Answer: Like term to maturity and coupon rate, the yield to maturity is a factor that influences price volatility. Ceteris paribus, the higher the yield level, the lower the price volatility. The same property holds for modified duration. Thus, a 10% yield to maturity will have both less volatility than an 8% yield to maturity and also a smaller duration. There is consistency between the properties of bond price volatility and the properties of modified duration. When all other factors are constant, a bond with a longer maturity will have greater price volatility. A property of modified duration is that when all other factors are constant, a bond with a longer maturity will have a greater modified duration. Also, all other factors being constant, a bond with a lower coupon rate will have greater bond price volatility. Also, generally, a bond with a lower coupon rate will have a greater modified duration. Thus, bonds with greater durations will greater price volatilities.4）Suppose a client observes the following two benchmark spreads for two bonds:Bond issue U rated A: 150 basis pointsBond issue V rated BBB: 135 basis pointsYour client is confused because he thought the lower-rated bond (bond V) should offer a higher benchmark spread than the higher-rated bond (bond U). Explain why the benchmark spread may be lower for bond U.5）The bid and ask yields for a Treasury bill were quoted by a dealer as 5.91% and 5.89%, respectively. Shouldn’t the bid yield be less than the ask yield, because the bid yield indicates how much the dealer is willing to pay and the ask yield is what the dealer is willing to sell the Treasury bill for?Answer:The higher bid means a lower price. So the dealer is willing to pay less than would be paid for the lower ask price. We illustrate this below. Given the yield on a bank discount basis (Yd), the price of a Treasury bill is found by first solving the formula for the dollar discount (D), as follows:()()360d t D Y F = The price is then Price = F-DFor the 100-day Treasury bill with a face value (F) of $100,000, if the yield on a bank discount basis (Yd) is quoted as 5.91%, D is equal to:100()()0.0591($100,000)()$1,641.67360360d t D Y F ===Therefore, price = $100,000 – $1,641.67 = $98,358.33. For the 100-day Treasury bill with a face value (F) of $100,000, if the yield on a bank discount basis (Yd) is quoted as 5.89%, D is equal to:100()()0.0589($100,000)()$1,636.11360360d t D Y F === Therefore, price is: P = F – D = $100,000 – $1,636.11 = $98,363.89.Thus, the higher bid quote of 5.91% (compared to lower ask quote 5.89%) gives a lower selling price of $98,358.33 (compared to $98,363.89). The 0.02% higher yield translates into a selling price that is $5.56 lower. In general, the quoted yield on a bank discount basis is not a meaningful measure of the return from holding a Treasury bill, for two reasons. First, the measure is based on a face-value investment rather than on the actual dollar amount invested. Second, the yield is annualized according to a 360-day rather than a 365-day year, making it difficult to compare Treasury bill yields with Treasury notes and bonds, which pay interest on a 365-day basis. The use of 360 days for a year is a money market convention for some money market instruments, however. Despite its shortcomings as a measure of return, this is the method that dealers have adopted to quote Treasury bills. Many dealer quote sheets, and some reporting services, provide two other yield measures that attempt tomake the quoted yield comparable to that for a coupon bond and other money market instruments.6）What is the difference between a cash-out refinancing and a rate-and-term refinancing?Answer:When a lender is evaluating an application from a borrower who is refinancing, the loan-to-value ratio (LTV) is dependent upon the requested amount of the new loan and the market value of the property as determined by an appraisal. When the loan amount requested exceeds the original loan amount, the transaction is referred to as a cash-out-refinancing. If instead, there is financing where the loan balance remains unchanged, the transaction is said to be a rate-and-term refinancing or no-cash refinancing. That is, the purpose of refinancing the loan is to either obtain a better note rate or change the term of the loan.7）Describe the cash flow of a mortgage pass-through security.Answer:The cash flow of a mortgage pass-through security depends on the cash flow of the underlying mortgage.The cash flow consists of monthly mortgage payments representing interest,the scheduled repayment of principal,and any prepayments. Payments are made to security holders each month.Neither theamount nor the timing,however,of the cash flow from the pool of mortgages is identical to that of the cash flow passed through to investors.The monthly cash flow for a pass-through is less than the monthly cash flow of the underlying mortgages by an amount equal to servicing and other fees.The other fees are those charged by the issuer or guarantor of the pass-through for guaranteeing the issue.The coupon rage on a pass-through,called the pass-through coupon rate,is less than the mortgage rage on the underlying pool of mortgage loans by an amount equal to the servicing and guaranteeing feesThe timing of the cash flow,like the amount of the cash flow,is also different.The monthly mortgage payment is due from each mortgagor on the first day of each month,but there is a delay in passing through the corresponding monthly cash flow to the securityholders.The length of the delay varies by the type of pass-through security. Because of prepayments,the cash flow of a pass-through is also not known with certainty.8）Explain the effect on the average lives of sequential-pay structures of including an accrual tranche in a CMO structure.。