Answer to Question #108539 in Financial Math for Suver

Question

A model of interest rates gives the n-year spot rate of interest as:
y_n=1/(8(1+e^(-0.25n) ) )
Assuming this is correct and that the term structure of interest rates does not change in the future, calculate:

a) The price per £100 nominal of an 8-year zero coupon bond purchased now. (2 marks)
b) The price per £100 nominal of an 8-year zero coupon bond purchased in 4 years’ time. (3 marks)
c) The one-year forward rate at time 4 years. (3 marks)
d) The 3-year par yield. (4 marks)
(Total 12 marks)

Answer 1

when calculating such a present value, the number of periods used for discounting is not the number of years to maturity of the bond, but the number of years multiplied by 2. The discount rate in this case is equal to half the required annual yield.

a)$P=\frac{M}{(1+r)^n}$

n=0

P=100, any number in the zero degree is equal to one.

b)n=8

y_n=1/(8(1+e^(-0.25n) ) )

$y=\frac{1}{8(1+e^{-2})}=\frac{1}{9.0827}=0.1101$

$P=\frac{100}{(1+0.1101)^8}=43.36$

c)$F=\frac{S}{(1+r\times t)^n}=\frac{100}{1+0.01101)^8}=43.36$

calculation of the rate from point b, since the forward rate is one year

d)$y=\frac{1}{8(1+e^{-0.25n})}=\frac{1}{8(1+e^{-1.5})}=0.10219=10.22$