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The current prices on one-year, two-year, and three-year $10,000$ zero-coupon bonds are $9765$, $9428$, and $8986.82$, respectively. Find forward rates $f[0,2]$ and $f[2,3]$ implied by these prices.

This question is from Mathematical Interest Theory Second Edition section $8.3$ #$1$, and the answer provided is $f[0,2]=2.98885\%$ and $f[2,3]=4.90919\%$

I understand the relationship between spot rates and forward rates, however I am unsure how to find the forward rate just given these current prices. Is there a way to find the spot rate from these prices given in order to convert them into the forward rates? Any help would be appreciated, thank you!

Arnaud Mortier
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uytt
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1 Answers1

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The spot rates $r_k$, i. e. forward rates $f_{[0,k]}$, for $k=1,2,3$, are given by $$ \begin{align} 9765 &=\frac{10000}{1+r_1}&\Longrightarrow& & r_1 &=f_{[0,1]}=\frac{10000}{9765}-1\approx 2.40655\%\\ 9428 &=\frac{10000}{(1+r_2)^2}&\Longrightarrow& & r_2&=f_{[0,2]}=\left(\frac{10000}{9428 }\right)^{1/2}\approx 2.98885\%\\ 8986.82 &=\frac{10000}{(1+r_3)^3}&\Longrightarrow& &r_3&=f_{[0,3]}=\left(\frac{10000}{8986.82}\right)^{1/2}\approx 3.62503\%\\ \end{align} $$ The remaining foward rates are given by $$ \begin{align} f_{[1,2]}&=\frac{(1+r_2)^2}{1+r_1}-1\approx 3.57446\%\\ f_{[1,3]}&=\left[\frac{(1+r_3)^3}{1+r_1}\right]^{\frac{1}{3-1}}-1\approx 4.23969\%\\ f_{[2,3]}&=\frac{(1+r_3)^3}{(1+r_2)^2}-1\approx 4.90919\%\\ \end{align} $$

alexjo
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