This is a problem from Ross's Elementary Mathematical Finance book:
A five-year $\$10,000$ bond with a $10\%$ coupon rate costs $\$10,000$ and pays its holder $\$500$ every six months for five years, with a final additional payment of $\$10,000$ made at the end of those ten payments. Find its present value if the interest rate is $6\%$. Assume the compounding is monthly.
I have a solution, but I don't fully understand it. It might be incorrect, as I have found a mistake in these solutions before. Here it is:
Let $S$ denote the present value. Then,
$$S = -10000 + \sum_{i=1}^{10} \frac{500}{(1 + r/2)^{i}} + \frac{10000}{(1 + r/2)^{10}}.$$
Plugging in $r = 0.06$ yields $1706.04$, which is our desired result.
So, I want to understand how they got this answer. I get that the $-10000$ term comes from the initial down payment that one makes to purchase the bond. Then, the holder is paid $\$500$ for each of the $10$ six-month intervals, so I guess that's where the sum comes from? We're dividing the rate $r$ by $2$ in the equation since we're looking at six-month intervals? Where does the "compounded monthly" part come into play?
Present value is discounting by the rate of interest. If I follow it correctly, you would do it as:$$\frac{10000\cdot10\%\cdot5+10000}{(1+0.06)^{60}}= \frac{15000}{1.06^{60}}\approx 454.72$$
but apply it to each payment instead. the division by 2 is probably a conversion rate to semiannual non compound interest. and i is the number of semiannums actually. There is a possible error in the book, it should care about the non compounding monthly rate.