Question

Suppose you want $10,000 at the end of 4 years. You are evaluating multiple options for investment today.

1. a) Compounding every month at a rate of 9% per year

2. b) Compounding every week at a rate of 8.7% per year

3. c) Compounding every day at a rate of 8.5% per year.

Which of the above options requires least up-front investment? What is that amount?

Answer 1

First we have to calculate Effective annual rate (EAR) in each case

(a) Compounding every month at a rate of 9% per year

Effective annual rate (EAR) = (1 + r/m) ^m – 1

Where,

Effective annual rate (EAR) =?

Where, nominal annual interest rate annual percentage rate (APR); r=9%

Number of compounding per year, m = 12 (monthly compounding, where number of months in a year is 12)

Therefore

EAR= (1 + 9%/12) ^12 - 1

Or EAR= (1 + 0.09/12) ^12 -1 =0.0938 or 9.38%

(b) Compounding every week at a rate of 8.7% per year

Effective annual rate (EAR) = (1 + r/m) ^m – 1

Where,

Effective annual rate (EAR) =?

Where, nominal annual interest rate annual percentage rate (APR); r= 8.7%

Number of compounding per year, m = 52 (weekly compounding, where number of weeks in a year is 52)

Therefore

EAR= (1 + 8.7%/52) ^52 - 1

Or EAR= (1 + 0.087/52) ^52 -1 =0.0908 or 9.08%

(c) Compounding every day at a rate of 8.5% per year

Effective annual rate (EAR) = (1 + r/m) ^m – 1

Where,

Effective annual rate (EAR) =?

Where, nominal annual interest rate annual percentage rate (APR); r= 8.5%

Number of compounding per year, m = 365 (daily compounding, where number of days in a year is 365)

Therefore

EAR= (1 + 8.5%/365) ^365 - 1

Or EAR= (1 + 0.085/365) ^365 -1 =0.0887 or 8.87%

As the Effective annual rate (EAR) is highest for option (a) compounding every month at a rate of 9% per year; therefore this option will require least up-front investment.

The amount calculation:

We need to calculate the present value of the amount required in future (after 4 years) to know the amount needs to be invested today

Formula for present value calculation

PV = FV / (1+i) ^N

Where, FV is the future value of investment =$10,000

Present Value (PV) of the investment =?

i = I/Y = interest rate per year or discount rate = 9%; compounding every month therefore effective annual rate (EAR) = 9.38%

And N is time period = 4 years

Therefore,

PV = $10,000 / (1+9.38%) ^4

= $10,000 / (1+ 0.0938) ^4

= $6,986.14

An amount $6,986.14 need to be invested.