3货币时间价值与利率期限结构清华大学绝版金融工程课件

u2
-(1+RC)
u1
e0
Time 0
11
• More generally, consider consumption grow at random rate. Investors maximize their expected utility over many periods.
T
max E[ tu(ct )] t 0
Nonlinear technology
Time 1
u2 u1
(c0 , c1 )
b(1+r)
e1
b
u2
-(1+RC) u 1
e0
Time 0 Investment
opportunity set
10
Linear technology
Time 1
(c0 , c1 )
(1+r)b
e1
b
u2 u1
F$10 (10 1% 5$ )11?5
19
Suppose forward price F = $106 per share
Replicating Stock Using risk-free bond and forward contract
Position Immediate Cash Flow Cash Flow in the Future
r u'(c0) 1(1/1)1 c0u"(c0) dc
u'(c1)
u'(c0) c0
Thus, the real interest rate is given by
r(1/1)1[c0 uu'("c(0 c)0)]d c0 c
Relative risk aversion coefficient 9
16
• Conversion example: Treasury maturity par coupon rate current price
A
1 year 1,000
0
B
2 years 1,000
10%
910.50 982.10
r1109 019 .5 0010 .5009.8% 3 9.8 1 2 0 1 1 r1 0 1 0 (1 0 r2 1 )0 2 0 010 1 9 .80 % 3 0(1 1 r2 1 )20r0 2 11.08%
Yes!
Risk-free interest varies with terms . It’s called the term structure of interests.
13
• Nominal and real interest rates
— nominal interest rate = real interest rate+ inflation
— real interest rate = pure time value+ risk premium
• Compound interest — interest earned on interest
already earned
— Continuously compounding
r — simple rate of return annually
n Ct
t1 (1r)t
n

t1
Ct (1rt)t
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Term Structure of Interest Rates
• Our objective is to value riskless cash flows. • Given the rich set of fixed-income securities
— Risk-free security:
• Substitute in reality: Treasury
15
— Treasury Yield Curve
• Treasury yield curve usually has three forms: upward, flat and downward.
m — times of interest payments annually r 1 — compounding rate of m interest payments m annually
r

1
r1
m
m
m
1
Let m
rlim1rm1er 1 m m
• The form in which this information is expressed depends on the particular market.
4
Determination of Interest Rate
• Four basic factors
1. Capital production ability —— the more the capital’s expected return, the higher the interest rates and vice versa.
• Real interest rates are determined by supply and demand of funds in the economy.
• 3 factors in determining real interest rates:
– Aggregate endowments – Aggregate investment opportunities – Aggregate preferences for different
3. Time preference of consumption —— the stronger preference to current consumption, the higher the risk premium required and the higher the interest rates and vice versa.
traded in the market, their prices provide the information needed to value riskless cash flows at hand.
3
Forms of Interest Rates
• In this market, this information on the time value of money is given in several different forms:
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— Shapes of Yield Curve
upward
flat
downward
• Some theories for the shapes of yield curve
— Unbiased expectations theory
— Liquidity preference theory
— Market segment theory
4. Risk aversion —— the more the risk aversion, the higher the risk premium required and the lower the riskfree interest rates.
5
Theory of Real Interest Rates
• Zero-coupon rates set
— Bills are zero coupon while notes and bonds have coupons. — Zero-coupon rates set can be obtained by conversion.
rt,t1,,n
— Preferred habitat theory
18
Forward Interest
A mini case:
— There is a no-dividend stock and its expected return is 15%. The current price is S0 $100. One year’s risk-free rate rf 5% . What is one year’s forward price of this stock?
s.t. c0 e0 b1 b2 ...bT
ct et (1 r)t bt t 1....T
• Where (b1,...... bT ) is his holdings of discount bonds, (e1,...... eT ) is future endowments, (c1,...... cT ) is future consumption, both can be uncertain.
12
The Benchmark of Interest
— Yield to Maturity (YTM) ? No!
YTM varies with different financial instruments, because the exposure of financial instruments are quite different and the required risk premiums differ from each other. — Risk-free interests ?
CHAPTER TWO: Time Value of Money and Term Structure of Interest
1
Discounted Cash Flow Formula
?
Yes ! ris the
PV
n t1
Ct (1r)t
expected rate of
No ! r is the
return, i.e., the
discount rate that
mean of the
discount rates for different terms
P
V
n t1
Ct (1rt )t
cannot be used for P0 so long period
Let NPVP0t n1(1C tr)t 0
2. Uncertainty of capital production ability —— the more the uncertainty, the higher the risk premium required and the higher the interest rates and vice versa.
Continuously compounding
r* ln1(r)
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Financial Risks and Risk-free Security
• Basic financial risks:
— Default risk — Liquidity risk — Purchase power risk — Interest risk — Foreign exchange risk — Other market risks
consumption path
6
• Consider a representative investor:
– Has endowment of ( e0, e1) – Faces a bond market with interest rate r.
7
• He maximizes his utility over his consumption now and later:
– Spot interest rates – Price of discount bonds (e.g., zero-coupon bonds and
STRIPS) – Prices of coupon bonds – Yield-to-maturity (an average of spot interest rates) – Forward interest rates
max u(c0 ) u(c1)
s.t. c0 e0 b c1 e1 (1 r )b
Where b is the bond holding, u’>0 and u”<0
8
• The optimality condition is
u'(c0)(1r)u'(c1)
or (for c1 c0 dc)