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The question I was given:

"An investor purchases an annuity payable annually in arrear for 20 years. The first annuity payment is R5 000 and subsequent payments increase by R250 each year. The investor, who calculates his purchase price on the basis of an interest rate of 5% per annum effective, draws up a schedule showing the division of each repayment into capital and interest.

Derive expressions for the capital and interest content of the $t^{th}$ annuity payment."

I tried to find the capital portion of the payment by finding the difference between the balance outstanding at time $t-1$ and the balance outstanding at time $t$. Then to find the interest portion of the payment would be the difference between the payment at time $t$ and the capital portion at time $t.$

$i =$ effective interest rate $=$ 5% p.a.; $v =$ discount factor $= 1.05^{-1}$; $d =$ discount rate $= \frac{i}{1+i}$

$$Payment_{Capital} =4750 (\frac{1-v^{20-(t-1)}}{i}) + 250(\frac{\frac{1-v^{20-(t-1)}}{d}-(20-(t-1))v^{20-(t-1)}}{i}) - ( 4750 (\frac{1-v^{20-t}}{i}) + 250(\frac{\frac{1-v^{20-t}}{d}-(20-t)v^{20-t}}{i}) ) $$ $$Payment_{Interest} = 4750 +250t - Payment_{Capital}$$

Can you please tell me if this is correct, any help will be appreciated, thanks.

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For each period's payment the interest part will be $i$ (interest rate) times the amount of outstanding debt at the previous period. To find the capital part just subtract interest payment from the total payment. In your case, if I understood right, you calculated the present value of future payments at time $t$ and $t-1$ and subtracted from each other to find the capital payment, which is correct and then you found interest payment by subtracting capital payment from the total payment, which is also correct. Both ways are identical and will lead you to the correct answer.

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