Math and Finance help: Present value and comparing annual yield

Question

One business has a large amount of debt on which it pays an annual yield of 10%.

 

A competitor has no debt, and any excess money goes into the bank and earns an annual yield of
2%.

Each of these businesses bids on a contract which pays $20,000 in two years and three months. These bids have different present values for the different companies. Calculate the difference between these present values, rounding to the nearest dollar. [Hint: for both businesses the yield is treated as positive.]

 

The parts I've bolded are tripping me up. I know that the formula is f = p (1+y)ⁿ but since one company is paying and the other is earning annual yields, do I input the percents any differently than I regularly would? I plugged in the percents as 1.1% and 1.02% respectively when figuring out each present value, is that correct?

Answer 1

The key is that both interest rates are treated as positive, regardless of whether the company is paying interest on debt or earning interest on savings.

We use the present value formula:

PV = FV / (1 + r)^t

Where:

1. FV = $20,000
2. r = annual interest rate expressed as a decimal
3. t = 2 years and 3 months = 2.25 years

Company 1 (10% annual yield):

PV = 20,000 / (1.10)^2.25

PV = $16,139.74

Company 2 (2% annual yield):

PV = 20,000 / (1.02)^2.25

PV = $19,128.44

Difference:

$19,128.44 - $16,139.74 = $2,988.71

Final Answer: $2,989

The important distinction is that 10% is written as 0.10, giving us 1.10 in the formula, while 2% is written as 0.02, giving us 1.02.

We don't use negative rates for the company paying interest because the problem specifically tells us to treat both yields as positive.

The company with the higher interest rate has the lower present value because money received in the future is worth less today when the discount rate is higher.