Suppose that the price of interest rate risk is constant and that (on an annual basis) the spot rate y_t follows the following first order autoregressive process under the risk neutral measure:

y_t+1 =(1- )^N+ y_t +_t+1 (1)

where _t is an i.i.d. normal error term with variance of ².

Recall that the price of an m-period discount bond (D_m,t) is the discounted value of its risk neutral expectation in the next period:

D_m,t = exp(-y_t)E_t^N [D_m-1,t+1]. (2)

(a) Use this to show that the logarithm (d_m,t) of this price is an affine function of the spot rate:

-d_m,t = a_m +b_my_t; m=1,..,M. (3)

where: a₁ =0,b₁ =1.

and

b_m = 1+b_m-1; (4)

a_m = a_m-1+b_m-1(1- )^N - ²b²_m-1; m=2,.,M; (5)

b) Use (1) to derive a linear relationship between y_t and E_t^N [y_t+m], which holds for all m > 0 and is the risk neutral expectation at time t of the interest rate at time t + m:

(c) Define the forward rate f_m,t as the continuously compounded rate that allows an m period discount bond to be rolled over to give the same return as an m + 1 period bond:

f_m,t = d_m,t - d_m+1,t m = 1,.,M (6)

Solve for the m-period forward rate f_m,t in terms of the spot rate y_t and the parameters ,^N , and ².

(d) Compare this with the answer to (b) and comment on your result. (e) Does this model offer a realistic description of the UK Treasury market at the moment?

Suppose that the price of interest rate risk is constant and that (on an annual basis) the spot rate yt follows the following first order autoregressive process under the risk neutral measure: yt+1=(1)N+yt+t+1 where t is an i.i.d. normal error term with variance of 2. Recall that the price of an m-period discount bond (Dm,t) is the discounted value of its risk neutral expectation in the next period: Dm,t=exp(yt)EtN[Dm1,t+1]. (a) Use this to show that the logarithm (dm,t) of this price is an affine function of the spot rate: dm,t=am+bmyt;m=1,,M. where: a1=0,b1=1 and bmam=1+bm1;=am1+bm1(1)N212bm12;m=2,,M; (b) Use (1) to derive a linear relationship between yt and EtN[yt+m], which holds for all m>0 and is the risk neutral expectation at time t of the interest rate at time t+m. (c) Define the forward rate fm,t as the continuously compounded rate that allows an m period discount bond to be rolled over to give the same return as an m+1 period bond: fm,t=dm,tdm+1,tm=1,,M Solve for the m-period forward rate fm,t in terms of the spot rate yt and the parameters ,N and 2. (d) Compare this with the answer to (b) and comment on your result. (e) Does this model offer a realistic description of the UK Treasury market at the moment