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Sam buys an eight-year, 5000 par bond with an annual coupon rate of 5%, paid annually. The bond sells for 5000. Let $d_1$ be the Macaulay duration just before the first coupon is paid. Let $d_2$ be the Macaulay duration just after the first coupon is paid. Calculate $\dfrac{d_1}{d_2}$.

Solution: According to SOA solutions, "This solution employs the fact that when a coupon bond sells at par the duration equals the present value of an annuity-due.

To be honest I don't know why is that so. But suppose that a bond is not selling for par value, how can I solve for the duration of each coupon payment? Should I stick with the definition $$D_{mac} = \dfrac{\sum_{t \in N} tv^tR_t}{\sum_{t \in N}v^tR_t}$$ where $N$ is the set of positive integers and $R_t$ is the payment at time $t?$

I try doing it for $d_0$ in the above problem, but I'm getting $0$ as $t=0$ using the definition.

Any alternatives?

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Selling at par means that $i=r$, where $i$ is the yield to maturity and $r$ is the coupon rate. The Macaulay duration can be written as (see here) $$D=\frac{F(r(Ia)_{\overline{n}\rceil i}+nv^n)}{Fra_{\overline{n}\rceil i}+Fv^n}=\frac{r(Ia)_{\overline{n}\rceil i}+nv^n}{ra_{\overline{n}\rceil i}+v^n}=\frac{r(1+i){a}_{\overline{n}\rceil i}+(i-r)\,nv^n}{r+(i-r)v^n}$$

If $i=r$, we have $$ D=(1+i)a_{\overline{n}|i}=\ddot a_{\overline{n}|i} $$

So, in your question, we have $$ d_0=\ddot a_{\overline{\,8}|5\%},\quad d_1=d_0-1,\qquad d_2=\ddot a_{\overline{\,7}|5\%} $$ and $$ \frac{d_1}{d_2}=\frac{\ddot a_{\overline{\,8}|5\%}-1}{\ddot a_{\overline{\,7}|5\%}}=\frac{1}{1+5\%}\approx 0.9524 $$ where we used the following $$ \frac{\ddot a_{\overline{n}|i}-1}{\ddot a_{\overline{n-1}|i}}=\frac{\frac{1-v^n}{d}-1}{\frac{1-v^{n-1}}{d}}=\frac{1-v^n-d}{1-v^{n-1}}=\frac{v-v^n}{1-v^{n-1}}=\frac{v(1-v^{n-1})}{1-v^{n-1}}=v=\frac{1}{1+i} $$ that is $$ \ddot a_{\overline{n}|i}=1+v\,\ddot a_{\overline{n-1}|i} $$

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