Consider two bonds purchased at the redemption value of 1000, and due in 5 years. One bond has 5% annual coupon rate payable semi-annually and the other has 10% annual coupon rate payable semi-annually. Find the duration of each bond if both bonds were purchased to yield 7% compounded semi-annually.
Find the duration of the 5% bond.
Based on the formula,
$$ duration = \frac {\sum_{t=1}^{n} tv^{t}R_{t}}{ {\sum_{t=1}^{n} v^{t}R_{t}} } $$
Since we know that the time is 5 years where the coupon price is compounded semi-annually, it gives total payment period n = 10.
We also know the coupon rate is 5% compounded semi-annually therefore, the coupon price is 1000 * .05/2 = $25
- Lastly, we know the interest is 7% compounded semi-annually thus, i = 3.5%
Given the formula for duration d, a rough computation would look something like
$$ \frac { (1 * (1.035)^{-1})(25) + (2 * (1.035)^{-2})(25) + ... +(10 * (1.035)^{-10}) (25) + (10 * (1.035)^{-10}) (1000)} { 25[(1.035)^{-1} + (1.035)^{-2} + .... + (1.035)^{-10}{.035}] + 1000(1.035)^{-10}} $$
Which is equivalent to:
$$ \frac { (1 * (1.035)^{-1})(25) + (2 * (1.035)^{-2})(25) + ... +(10 * (1.035)^{-10}) (25) + (10 * (1.035)^{-10}) (1000)} { 25\require{enclose} a_{\enclose {actuarial}{10}{.035} } + 1000(1.035)^{-10}} $$
Therefore, my solution was
d = 8.91523057
However, according to my textbook, the answer is exactly half of what I got
d = 4.45761529
The steps that my textbook showed was not sufficient to confirm on my own.
It basically did everything I did except for the fact that on the numerator, it was multiplied by 0.5 and I have no idea where it decided to multiply by one-half. Can someone please explain?
EDIT:
So after re-reading the question, it specifies that the purchase of both bonds will yield 7%... So I figured if both bonds had the same yield of 7%, then their average yield will also be 7%. However, if I am mistaken about this, then it would mean the purchase of both bonds in conjunction will yield 7%... meaning that both bonds will yield somewhere around 3.5% therefore, I would need to take the half of both bond yields at 7% ... whereby the sum of their yields will provide 7%. This would explain why the answers are exactly half of what I got...
But if any expert happens to read this I'd greatly appreciate any input. Thank you in advance.
The factor of $t$ in the numerator of the duration formula is the time of the coupon payments. Since coupons are paid semiannually, they should really take the values $0.5$, $1$, $1.5$, ..., $5$ rather than $1$ to $10$. That is why your answer is double the correct answer.
What you calculated was the duration of a $10$-year bond with redemption value $1000$, annual coupons at a rate of $2.5\%$, and yield to maturity $3.5\%$ (annual effective).
(With regards to the $7\%$ yield, you're over-thinking it - the question means that both bonds individually yield $7\%$.)