Question

1. Calculating EAR. Friendly’s Quick Loans, Inc., offers you “three for four or I knock on your door.” This means you get $3 today and repay $4 when you get your paycheck in one week (or else). What’s the effective annual return Friendly’s earns on this lending business? If you were brave enough to ask, what APR would Friendly’s say you were paying?
2. Valuing Perpetuities. Live Forever Life Insurance Co. is selling a perpetuity contract that pays $1,250 monthly. The contract currently sells for $245,000. What is the monthly return on this investment vehicle? What is the APR? The effective annual return?

Answer 1

Question 1

Calculating EAR

Sol:

Present value (PV) = $3

Future value (FV) =$4

Interest rate (r)

To determine interest rate:

FV = PV * (1 + r)

4 = 3 * (1 + r)

r = 4 /3 - 1

r = 1.3333 - 1 = 0.3333 or 33.3333%

Now 33.3333 is a one week interest rate, for APR we have to multiply it by total number of weeks in a year.

Number of weeks in a year = 52

APR = 33.3333% * 52 = 1,733.3333%

Effective annual return = (1 + APR / 52)^52 -1

Effective annual return = (1 + 1,733.3333 / 52)^52 - 1

Effective annual return = (1 + 0.3333)^52 - 1

Effective annual return = (1.3333)^52 -1

Effective annual return = 3,139,165.1569 or 313,916,515.69%

Question 2

Valuing Perpetuities

Sol:

Monthly payments (P) = $1,250

Present value (PV) = $245,000

Interest rate (r)

PV of a perpetuity = P / r

245,000 = 1,250 / r

r = 1,250 / 245,000 = 0.0051 or 0.5012% per month

Monthly return on this investment vehicle = 0.5012%

Now 0.5012% is a monthly interest rate, for APR we have to multiply it by total number of months in a year.

APR = 0.5012% * 12 = 6.1224%

Effective annual return = (1 + APR / 12)^12 -1

Effective annual return = (1 + 6.1224% / 12)^12 - 1

Effective annual return = (1 + 0.0051)^12 - 1

Effective annual return = (1.0051)^12 -1

Effective annual return = 0.062972 or 6.30%