Question

Bonds pay annual coupons at a rate of \(6 \%\) per annum, in arrear, and are redeemable at par. The bonds, redeemable in exactly one, two, three, four and five years respectively, are all priced at R 96 per R100 nominal.
i. Determine the one-year, two-year and three-year spot rates.
[3]
ii. Calculate \(f_{0,1}, f_{1,2}\) and \(f_{2,2}\) where \(f_{t, r}\) is the annual forward interest rate agreed at time 0 for an investment made at time \(t\) (where \(t>0\) ) for a period of \(r\) years.
iii. Comment on the term structure of the spot rates in i. with reference to expectations theory.

Answer 1

(i)
\[
106\left(1+y_{1}\right)^{-1}=96
\]
\[
y_{1}=0.1041666
\]
\[
106\left(1+y_{2}\right)^{-2}+6\left(1+y_{1}\right)^{-1}=96 \Rightarrow 106\left(1+y_{2}\right)^{-2}+6(1+0.104167)^{-1}=96
\]
\[
106\left(1+y_{2}\right)^{-2}=90.56604
\]
\[
y_{2}=0.081858
\]
\[
106\left(1+y_{3}\right)^{-3}+6\left(1+y_{2}\right)^{-2}+6\left(1+y_{1}\right)^{-1}=96
\]
\[
106\left(1+y_{3}\right)^{-3}=85.43965824
\]
\[
y_{3}=0.074522
\]

(ii)
\[
\begin{aligned}
&f_{0,1}=y_{1}=10.41666 \% \\
&\left(1+f_{1,2}\right)^{2}=\frac{\left(1+y_{3}\right)^{3}}{\left(1+y_{1}\right)}=\frac{(1+0.074522)^{3}}{(1+0.1041666)}=1.1236
\end{aligned}
\]
\[
f_{1,2}=0.06
\]
\[
106\left(1+y_{4}\right)^{-4}+6\left(1+y_{3}\right)^{-3}+6\left(1+y_{2}\right)^{-2}+6\left(1+y_{1}\right)^{-1}=96
\]
\[
\begin{aligned}
&\left(1+y_{4}\right)^{-4}=\frac{80.60345}{106}=0.760409917 \\
&\left(1+f_{2,2}\right)^{2}=\frac{\left(1+y_{4}\right)^{4}}{\left(1+y_{2}\right)^{2}}=\frac{(0.760409917)^{-1}}{(1+0.08185797)^{2}}=1.1236 \\
&f_{2,2}=0.06
\end{aligned}
\]
(iii)
- Spot rates are a decreasing function of the term.
- The expectations theory posits that investors are expecting a decrease in short term spot rates.
- This makes short-term investments less attractive and longer-term investments more attractive.
- In these circumstances, yields on short-term investments will rise and yields on longterm investments will fall.