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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.
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(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.
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