Question

Let 0 < r < 1 be an interest rate. Consider the following compound investment schemes:

Scheme I: Interest is compounded at a rate of r at the end of every month.

Scheme II: Interest is compounded at a rate of 3r at the end of every third month.

Scheme III: Interest is compounded at a rate of 6r at the end of every sixth month.

Question 1 Suppose that you invest R dollars on January 1st. Rank these three schemes in terms of which provides the greatest return after one year. Justify, using calculations and arguments, why you chose this ranking.

Question 2 Let rI, rII, and rIII denote the annual effective rates of the three schemes. Rank these three schemes in terms of their annual effective rates. Justify, using calculations and arguments, why you chose this ranking.

Answer 1

Compound interest rate formula:

• A = Future value of the investment, including interest
• P = Principal investment amount
• r = Annual interest rate
• n = Number of times that interest is compounded per unit t
• t = Time the money is invested

=> Question 1: t = 1 (given)

  

Assuming P = 100, r = 10%;

A ( Scheme I) = 110.47
A (Scheme II) = 133.54
A (Scheme III) = 169.00

from the above results, Returns provided by the Scheme III > Returns provided by the Scheme II > Returns provided by the Scheme I

=> Question 2:

• r = interest rate
• n = number of calculations in a year

Assuming r = 10%;

Annual Effective Rate ( Scheme I) = 0.104 = 10.47%
Annual Effective Rate ( Scheme II) = 0.335 = 33.54%
Annual Effective Rate ( Scheme III) = 0.69 = 69.00%

from the above results, Annual Effective Rate provided by the Scheme III > Annual Effective Rate provided by the Scheme II > Annual Effective Rate provided by the Scheme I