I am supposed to calculate the following problem:

The Company issued 5,000 dollars voucher bonds with a half-year return of 100$ and a maturity of 8 years. Determine the duration of the bond if the normal annual yield of the probable bond is 6%.

I have tried to calculate it with the formula:

$$\frac{\sum_{i=1}^{15}\frac{150}{1,03^{i}}+\frac{150+5000}{1,03^{16}}.16}{\sum_{i=1}^{15}\frac{150}{1,03^{i}}+\frac{150+5000}{1,03^{16}}}$$ but it is not correct.

Can anyone help me?


2 Answers 2


With semiannual coupon payments the relationship between bond price and yield is given by

$$P = \sum_{j=1}^n C (1 + y/2)^{-j} + F(1+y/2)^{-n},$$

where $F$ is the face value redeemed at maturity, $C$ is the coupon amount paid semiannualy, $y$ is the yield, and $n$ is the number of coupon payments.

Modified duration is defined as

$$D = - \frac{1}{P} \frac{\partial P}{\partial y} = \frac{\sum_{j=1}^n j\frac{C}{2}(1 + y/2)^{-(j+1)} + n\frac{F}{2}(1+y/2)^{-(n+1)}}{\sum_{j=1}^n C (1 + y/2)^{-j} + F(1+y/2)^{-n}}$$

In this case compute duration using $C = 100$, $F = 5000$, $y = 0.06$, and $n = 16$.

You can also avoid working with the sums using a more compact expression for the bond price. Using the closed form expression for a geometric sum $\sum_{j=1}^n \alpha^j = \frac{\alpha - \alpha^{n+1}}{1-\alpha}$, where $\alpha = (1+y/2)^{-1}$, we get

$$P = \frac{2C}{y}\left[1 - (1+y/2)^{-n} \right]+ F (1+y/2)^{-n} $$


The correct formula for the duration is,

$$\frac{\sum_{i=1}^{16}\frac i2\cdot\frac{100}{1.03^{i+1}}+\frac{16}2\cdot\frac{5000}{1.03^{17}}}{\sum_{i=1}^{16}\frac{100}{1.03^{i}}+\frac{5000}{1.03^{16}}}$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.