For an annual bond with coupons $c =40$ and 10 years maturity $n=10$ ,face value $P_p=1000$ and interest rate $y = 0.08$ calculate the Macaulay Duration defined as :
$$ D= \frac{1}{P} \left( \frac{ 1\times c}{(1+y)} +\frac{ 2\times c}{(1+y)^2} +\frac{ 3\times c}{(1+y)^3}+ \dots + \frac{ n\times (c+P_p)}{(1+y)^n} \right) $$
The writer states that the resulted $D =8.1184 $ and the price of the bond $P = 731.5967$.
My question is: How he found these resulted values? First he finds $P$? And how ? What formula he uses ?
Any help would be appreciated.
The price $P$ matches the (unscaled) present values of coupons and the face value, given by:
$$\begin{align*} P &= \frac{ c}{(1+y)} +\frac{ c}{(1+y)^2} +\frac{ c}{(1+y)^3}+ \dots + \frac{ c+P_p}{(1+y)^n}\\ &= \frac cy \left[1-(1+y)^{-n}\right] + P_p(1+y)^{-n}\\ &= \frac{40}{0.08}\left(1-1.08^{-10}\right) + 1000\cdot 1.08^{-10}\\ &\approx 268.4033 + 463.1935\\ &\approx 731.5967 \end{align*}$$