Financial Mathematics: Annuity relating to loan

Question

You took a loan of 500,000 which required to pay 25 equal annual payments at 10% interest. The payments are due at the end of each year. The bank sold your loan to an investor immediately after receiving your 6th payment. With yield to the investor of 7% , the price the investor pay was 569,326. Determine the bank's overall return on its investment. (Ans : 0.127)

What i did so far, I calculated the payment by using the annuity formula 500,000 = P(1-1.1^-25 / 0.1) which yield P = 55084.0361.

Then i though the overall return is to calculate the AV for n=25 with 10% interest and use 569326 / AV. But the ans i get is 0.1051.

In this question i totally ignored the 7% because i have no idea what is the use for it. Is it after the bank sold the loan, then the payment after 6th will be charge with 7% interest?

Thank in advance for anyone can solve it.

Answer 1

You are correct in finding the payment

Then it is simple IRR formula:

$CF_0 = -500000$

$CF_1 - CF_5 = 55084.04$

$CF_6 = 55084.04+569326 = 624410.04$

Using a calculator or EXCEL and put int these values and you will find the internal rate of return is $12.7$%

Good Luck

Answer 2

Let $L=500,000$, $n=25$ and $i=10\%$ then $$ P=\frac{L}{a_{\overline{n}|i}}=\frac{500,000}{a_{\overline{25}|10\%}}=55,084 $$ The bank sold the remaing loan after the 6th payment at interest $i'=7\%$, that is at price $$ L'=P\,a_{\overline{n-6}|i'}=569,326 $$ So, for the bank we have the return $r$ is the solution of $$ L=Pa_{\overline{6}|r}+L'v^6 $$ were $v=\frac{1}{1+r}$. Solving numerically you will find $v=0.8873$ and $r=12.7\%$