《金融理论与公司政策(第四版)》课后答案

t
RT =
1 [b(t,T)r(t) − A(t,T)] T−t
Equation (8.6) suggests that the entire yield curve can be obtained as a function of r(t) once the three process parameters: the long-term mean of the short-term rate, µ, the speed of adjustment, ν, and the instantaneous standard deviation, σ are specified. The term structure can be upward-sloping, downward-sloping or humped depending on the values of the various parameters. In this question, the current short-term rate, r(t) is 2%, the long-run mean, µ, is 6%, the instantaneous standard deviation, σ, is 10% and the rate of adjustment, ν, is 0.4. These set of parameters result in the following term structure of interest rates: Time to Maturity in Years (T – t) 1 2 3 4 5 Value of a ZeroCoupon Bond [B(t, T)] 0.9745 0.9443 0.9139 0.8848 0.8572 Interest Rate [tRT] 2.58% 2.87% 3.00% 3.06% 3.08%
2 σ2 ( b(t,T )−T + t) ν µ− 2 2 2 b (t,T) σ − 4ν ν2
A (t, T) = e
Since B(t, T) = e− (t RT )(T− t) , where tRT is the interest rate at time t for a term of (T – t), we get:
t
RT =
1 [b(t,T)r(t) − A(t,T)] T−t
where b(t, T) and A(t, T) are as defined above. In this question, the current short-term rate is 2%, the long-run mean is 6%, the instantaneous standard deviation is 10% and the rate of adjustment is 0.4. These set of parameters result in the following term structure of interest rates: Time to Maturity in Years (T – t) 1 2 3 4 5 Value of a ZeroCoupon Bond [B(t, T)] 0.9734 0.9373 0.8963 0.8531 0.8095 Interest Rate [tRT] 2.70% 3.24% 3.65% 3.97% 4.23%
2 3
f = 17.03%
(b) The rate of interest on a bond held from the beginning of the third year and held to the beginning of the fifth year is a two-year rate of interest equal to the product of the expected forward rates during the third and fourth years. The formula is given below:
The two-year rate of interest is 36.93%. The average one-year rate is (1.3693).5 –1 = 17.02%. 2. Vasicek (1977) assumes the following mean-reverting process for the short-term rate (Eq. 8.4): dr = ν(µ – r)dt + σdz where µ is the long-term mean of the short-term rate, ν is the speed of adjustment of the short-term rate to the long-term mean and σ is the instantaneous standard deviation. There is mean reversion in the process specified in equation (8.4) since a higher (lower) current short-term rate, r, as compared to the long-run mean, µ, impliea a negative (positive) drift, which, in turn, implies that, an average, the short-term rate will decrease (increase) towards the long-run mean. The price at time t of a zero-coupon bond that pays $1 at time T is then given by (Eq. 8.5): B(t, T) = A(t, T)e
b(t, T) 0.8242 1.3767 1.7470 1.9953 2.1617
A(t, T) 0.9907 0.9707 0.9464 0.9208 0.8951
3. Cox, Ingersoll and Ross (1985) (CIR) propose a mean reverting process for the short rate where the standard deviation of the changes in interest rates are proportional to the square root of the level of the rate. In the model, the short-term interest rate process is Eq. (8.7): dr = ν(µ − r)dt + σ( r )dz The price of a zero-coupon bond in this framework is given by the equation (8.8): B(t,T) = A(t,T)e − b(t,T)r(t)
h(0) = h* (0) = 1 The above equations when divided imply that (Eq. 8.11):
B(t,i + 1,T) h(T − t) = B(t,i, T) h*(T − t)
The no-arbitrage condition implies that (Eq. 8.12) B(t,i,T) = [πB(t + 1,i + 1,T) + (1 − π)B(t + 1,i,T)]B(t,i, t + 1) where π is the probability associated with the perturbation h(τ). Combining the no arbitrage condition with the perturbation equations yields the following constraint on the perturbation function (Eq. 8.13): πh(τ) + (1 − π)h*(τ) = 1 Ho and Lee also show that for path independence to hold such that the order of the perturbations is not important, h(τ) and h*(τ) have to satisfy the following conditions(Eq. 8.14): h( τ) =
Chapter 8
The Term Structure of Interest Rates, Forward Contracts, and Futures
1. (a) Assuming that the unbiased expectations hypothesis (as given by Eq. 8.2) is valid, we can solve the problem by computing the following ratio: (1 + 0 R 3 )3 (1 + 0 R1 )(1 + 1f 2 )(1 + 2 f 3 ) = = 1 + 2f3 (1 + 0 R 2 ) 2 (1 + 0 R1 )(1 + 2 f 2 ) (1.15)3 1.5209 = = 1.1703 = 1 + 2 f 3 (1.14) 2 1.2996 Therefore, the implied forward rate for the third year is
b(t, T) 0.8231 1.3705 1.7326 1.9712 2.1284
A(t, T) 0.9895 0.9634 0.9279 0.8874 0.8447
4. Ho and Lee propose a binomial model for bond prices of all following form:
2γe( ν+γ )(T−t)/2 A(t, T) = 2 γ + ( ν + γ ) (e γ (T− t) − 1) γ = ν 2 + 2σ 2
The interest rate from t to T, tRT, would be given wk.baidu.comy Equation (8.9):
Chapter 8
The Term Structure of Interest Rates, Forward Contracts, and Futures
93
where B(t, i, T) is the price of a bond at time t and state i that pays $1 at time T. Note that B(0, 0, T) is the initially observed term structure of bond prices. The evolution of bond prices is based on perturbation functions h(τ) and h*(τ) such that (Eqs. 8.10a, 8.10b, 8.10c) B(t − 1,i,T) B(t,i + 1,T) = h(T − t) B(t − 1,i,t) B(t,i,T) = B(t − 1,i,T) h * (T − t) B(t − 1,i,t)
92
Copeland/Shastri/Weston • Financial Theory and Corporate Policy, Fourth Edition
where b(t, T) =
2 γ + (ν + γ )(e γ (T− t) − 1)
2 νµ/σ 2
2 (er(T−t) − 1)
(1 + 2f3 ) (1 + 3f 4 ) =
=
(1 + 0 R 4 ) 4 (1 + 0 R 2 ) 2
(1 + 0 R 1 )(1 + 1f 2 )(1 + 2 f 3 )(1 + 3f 4 ) (1 + 0 R 1 )(1 + 1f 2 )
=
(1.155)4 1.7796 = = 1.3693 (1.140) 2 1.2996
–b(t, T)r(t)
Chapter 8
The Term Structure of Interest Rates, Forward Contracts, and Futures
91
where r(t) = the short rate at t b(t, T) = 1 − e− v(T − t) v