Question

1.You are considering an investment that will pay you $1,200 in one year, $1,400 in two years, and $1,600 in three years, $1,800 in four years, and $11,000 in five years. All payments will be received at the end of the year. • Your opportunity cost of capital (r ) is 10.5%

• Using the present value formula calculate the present value of each of the cash flows by

1. Discounting cash flows using annual compounding

2. Discounting cash flows using monthly compounding

3. Discounting cash flows using continuous compounding • How much would you be willing to pay for the investment using each of the three different compounding scenarios? That is, what is the present value of the cash flows from the investment using each of the three different compounding scenarios? • Which of the three present values is the largest (annual, monthly or continuously compounded returns)? Please explain why this is the case.

2. Tiny’s Quick Loans offers customers a “four for five or I knock on your door” loan. That is, Tiny will lend you $4 today and you repay Tiny $5 in one week when you get paid. What is the effective annual return Tiny is earning in this lending business? What is the APR that you are paying Tiny? (Assume that there are 52 weeks in the year.)

Answer 1

Answer 1(a):

Working:

(1) Annual compounding: Effective annual rate = 10.5%

PV = 1200 / (1 + 10.5%) + 1400 /(1 + 10.5%) ² + 1600 /(1 + 10.5%) ³ + 1800 /(1 + 10.5%) ⁴ + 11000 / (1 + 10.5%) ⁵

= $11302.73

(2) Monthly compounding: Effective annual rate = (1 + 10.5%/12) ¹² - 1 = 11.0203450%

PV = 1200 / (1 + 11.020345%) + 1400 /(1 + 11.020345%) ² + 1600 /(1 + 11.020345%) ³ + 1800 /(1 + 11.020345%) ⁴ + 11000 / (1 + 11.020345%) ⁵

= $11092.83

(3) Continuously compounding: Effective annual rate = e ^0.105 - 1 = 11.071061%

PV = 1200 / (1 + 11.071061%) + 1400 /(1 + 11.071061%) ² + 1600 /(1 + 11.071061%) ³ + 1800 /(1 + 11.071061%) ⁴ + 11000 / (1 + 11.071061%) ⁵

= $11072.66

Answer 1 (b):

The largest present value is when we have returns with annual compounding.

When we increase compounding periods, the effective annual rate increases and the discounting factors (we apply to discount the future cash flows) decrease resulting in lower present value.

Hence we find the present values with monthly compounding lesser than present values with annual compounding and present values with continuous compounding lesser than present values with monthly compounding

Answer 2:

Lends 4 dollar and gets repayment of $5 in one week.

Hence dollar return in one week = $1

% return in one week = 1 / 4 = 25%

Effective annual return = (1 + 25%) ⁵² - 1 = 10947544.25 %

APR = 25% * 52 = 1300.00%

Effective annual return = 10947544.25 %

APR = 1300.00%