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:

1. Face value is $$1000$$.
2. It pays $$50$$ every $$6$$ months.
3. Annual rate is $$8\%$$.
4. Value at maturity is $$1050$$.
5. 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?

• The face value doesn't matter at all except to calculate the interest payment if the interest is given as a percentage of face value. It is the redemption value that represents the cash you will receive at maturity. Jun 1 '21 at 19:49

1. The discount factor at time $$t$$ years from today, $$D(t)$$ is equal to the PV of the future cash flow \\$1 at time $$t$$.
2. Therefore, The PV of a future cash flow $$C$$ at time $$t$$ years from today is equal to $$C D(t)$$.
3. 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 $$D(t) = \left(1 + \frac{R}{100N}\right)^{-Nt}.$$