Calculating the present value of bond when face value is not equal to value at maturity I need to calculate PV of the following bond:

*

*Face value is $1000$.

*It pays $50$ every $6$ months.

*Annual rate is $8\%$.

*Value at maturity is $1050$.

*The bond lasts $20$ years.

I know how to calculate the PV when face value is equal to value at maturity and when coupons are payed every year. In that case it will be
$$PV=1000\cdot(50\cdot a_{20}+v^{20})$$
Where $a_{20}=\frac{1-v^n}{i}$ and $v=\frac{1}{1+i}$. Could anyone give me a hint how to do this in more general case?
 A: I don't really understand your formula. In any case, the best way to calculate something like this is to remember and use basic principles, rather than formulas. The advantage of this is that the formulas you learn in textbooks are rarely the ones needed in real life. But if you know how to calculate things from scratch, then you can figure out the PV of any set of future cash flows.

*

*The discount factor at time $t$ years from today, $D(t)$ is equal to the PV of the future cash flow \$1 at time $t$.


*Therefore, The PV of a future cash flow $C$ at time $t$ years from today is equal to $C D(t)$.


*The remaining piece you need is how to calculate $D(t)$ for each of the cash flows. Here, the information you provide is incomplete. The discount factor is calculated using the 8% interest rate, but you don't say what the compounding frequency is. Since the standard convention is semiannual compounding, I'll assume that here. This formula is worth memorizing: If the interest rate is $R\%$, then
$$
D(t) = \left(1 + \frac{R}{100N}\right)^{-Nt}.
$$
Now just use these principles to do your calculation. It's best to use a spreadsheet, so you can check each step of your calculation.
