Question

Your friend would like a loan of $50,000 from you and if you accept to borrow him you expect to earn an expected return similar to your current investments. Hence, you consider the following possibilities: There is a 5% chance that things will go very wrong and that in one year you will get repaid nothing at all; there is a 20% chance that in one year you will only get back $25,000 (with no interest); finally there is a 75% chance the venture will succeed and your friend will repay the entire $50,000 plus the interest. You perceive the likelihood of a default occurring as being random (not related to the state of the economy) and you could be earning a 2% annual rate of return on your money for the next year if you left it safely in AAA rated bonds. Determine the annual interest rate you need to charge your friend on this 1-year loan given the assumed probabilities of default in order to earn, on average, the same return you would if you left this money in AAA bonds. Determine what this rate (in annual terms) for the loan would have to be.

Answer 1

Total amount expected if invested in AAA bonds for 1 year (interest rate of 2%, loan amount = $50,000) = $50,000 * (1+2%) = $51,000

Now, if the amount of $ 50,000 is loaned to the friend, the expected amount at the end of 1 year = $51,000

The scenarios are as follows:

(a) 5% chance that no amount will be received => $0 received at end of 1 year

(b) 20% chance that $25,000 will be received => $25,000 received at end of 1 year.

(c) 75% chance that full amount will be received (to be calculated) (assume 'x' to be the interest rate)

Weighted average amount at the end of 1 year = $51,000 = weighted average of the above scenarios based on the probability:

($0*5%) + ($25,000*25%) + ($50,000*(1+x%) * 75%) = $51,000

0+6,250 + (50,000*(1+x%) * 75% = $51,000

37,500*(1+x%) = 44,750

1+x% = 44,750 / 37,500

1+x% = 1.193333..

=> x = 19.33% approximately

Therefore, the interest rate to be charged from the friend to earn the same return as AAA bonds will be approximately 19.33 %