Mr.Hill wishes to borrow 5000. He will repay the loan with a single payment at the end of one year. The lending agency has a "risk free" rate of interest of 13%, but estimates there is a 8% chance that Mr.Hill will not repay the loan. How much should they ask Mr.Hill to repay?
I know the answer should be 6141.30 but have no clue how to. I know if I use compound interest $(1+i)^n$ and I use i=0.13 but how do I include the 8% that he doesn't repay the loan.
On
Just a HINT.
Let us for the moment forget the interest rate.
If a trustworthy person borrowed $5000$ it would be sufficient to ask them to return you $5000$ after one year.
But if the were a chance of $8 $% not to see your money back, would it no be fair to ask them to repay you, in one year, a sum $x$ such that your expected return is still $5000$ ? Then $$ x \times 0.92 + 0 \times 0.08 = 5000 $$ so $x = 5434.79$ Considering the interest should be easy now, as you do not want only $5000$ back, you want $5000 \times 1.13$....
This situation is a good example of contingency payment. To solve the problem, the finance company first determines their expected value to be $(0.92)X +(0.08)(0)$ where $X$ is the amount to be repaid. This is because there is a 92% chance that $X$ will be repaid and a 8% chance the company gets nothing.
The present value of this expected value at the time of loan is : $(0.92X)(1.13)^{-1}$, so we have, $$\frac{0.92X}{1.13}=5000 \implies X = 6141.30$$ as is expected.