# calculate the number of coupon payments

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by (280 points)

Thank you for the question. I am not quite sure which part of the solution you don't understand so I'll play it safe and explain the question first and then talk you through each step of the solution thereafter.

So the question basically requires you to find n such that the present value of the n semi-annual coupon payments is 39.04 when they are valued at an interest rate of 14% p.a. effective.

Since n is the actual number of payments, we have to adjust our effective interest rate such that it matches the frequency at which the payments are paid (semi-annually) then we can apply our usual annuity formulae. So the first line of the solution is basically converting i = 14% p.a. effective to $$\frac{i^{(2)}}{2}$$ using the familiar, $$1+i = (1+\frac{i^{(p)}}{p})^p$$ Setting p=2, it is easy to verify that $$\frac{i^{(2)}}{2} = (1+14\%)^\frac{1}{2}-1$$ The solution just calls this j.

The next line of the solution says that the present value of the coupons is equal to 5.5 multiplied by an annuity of n payments payable at an effective rate of j.

This makes perfect sense because the value of each coupon is R5.5 (per R100 nominal) since the coupon rate is R11 per R100 nominal for the whole year, there are only 2 payments during any year so each coupon is R5.5. We multiply this by the annuity in order to find the present value of the n semi-annual payments of R5.5 which is precisely what we want.

The next line equates 39.04 to the expression above (we are told that the present value of the payments is 39.04).

The hard work is done. The rest of the solution uses ordinary algebra to solve for n. (Albeit they skip a few steps, it shouldn't be too difficult to follow what they did). If you can't follow what they did, please leave a comment and I will fill in the details.