# Macaulay Duration

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,

$$duration = \frac {\sum_{t=1}^{n} tv^{t}R_{t}}{ {\sum_{t=1}^{n} v^{t}R_{t}} }$$

1. Since we know that the time is 5 years where the coupon price is compounded semi-annually, it gives total payment period n = 10.

2. We also know the coupon rate is 5% compounded semi-annually therefore, the coupon price is 1000 * .05/2 = $25 1. 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. ## 1 Answer 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 $$7\%$$.) • So let me get this straight. In the numerator, the time-period t is with respect to the interest conversion per year thus, t=0.5 over a period of n = 10 Oct 10 '20 at 21:44 • More precisely, the values of$t$are the times at which the cash flows occur. That is not necessarily related to the interest conversion period (although in this particular problem both the coupons and interest conversion are semiannual). You could also imagine a problem in which cash flows occur irregularly, say at$t=1, 3, $and$10$. In that case there is no "time period" for$t\$, but rather just the values when the payments occur. I hope that makes sense.
– kccu
Oct 11 '20 at 2:24
• ah yes.. I get it now, you treat t as each time-period of payment that is made. thank you Oct 11 '20 at 3:30