Question

1. Suppose that an investment of $3750 accumulates to $8503.08 at the end of 13 years, then the effective annual interest rate is i=

2. At an effective annual rate of interest of 5.3%, the present value of $10371.57 due in t years is $5300. Determine t=

3.

a) Under simple interest rate of 6%. The effective rate of interest in the 14 th year is i14= %
b) Under simple interest rate of 6%. The effective rate of interest in the 18 th year is i18=    %
c) Under compound interest rate of 6%. The effective rate of interest in the 14 th year is i14= %
d) Under compound interest rate of 6%. The effective rate of interest in the 18 th year is i18=    %

Answer 1

1. The rate i at which the initial investment of $3,750 accumulates to $8,503.08 at the end of 13 years can be found using the below formula:

Maturity amount = Principal * (1 + effective annual rate)^period invested

Rearranging the above formula and substituting the values:

i = (8,503.08 / 3,750)^(1 / 13) - 1
= (2.2675)^(1 / 13) - 1
= 1.065 - 1
= 0.065 or 6.5%

2. The time period t in which the initial investment of $5,300 accumulates to $10,371.57 when invested at 5.3% can be found using the above formula, as shown below:

0.053 = (10,371.57 / 5,300)^(1 / t) - 1
1.053 = (1.957)^(1 / t)
(1.053)^t = 1.957
Taking log on both sides,
log 1.053 * t = log 1.957
t = (log 1.957) / (log 1.053)
= 0.2916 / 0.0224
= 13 years

3. Before answering this question, let's see how simple interest and compound interest work over multiple periods of investment.

When invested at simple interest rate of i% per annum, the interests earned during each period remain the same. So, the effective rate of interest remains the same for the period invested.

When invested at compound interest of i% per annum, however, the interest earned during each period gets invested and forms the base for calculation of interest for the next period. The effective rate is calculated using the formula,

i = (1 + periodic interest rate)^m - 1; where m denotes the number of periods

It is to be noted that if the number of periods in a year is one, then the stated compound interest rate will be the same as the effective rate of interest.

With the above understanding, let's now answer the questions:

a) The simple interest rate is 6%. Since the amount is invested at simple interest rate, the effective rate in the 14th year i14 is 6%. The investment at this rate would have grown by 1 + 0.06 * 14 = 1.84 times in 14 years.

b) By the same logic as above, the effective rate in the 18th year i18 is 6% and the investment at this rate would have grown by 1 + 0.06 * 18 = 2.08 times in 18 years.

c) The compound interest rate is 6%. Assuming this is the annual rate, the effective rate in the 14th year will also be 6%.
    The investment at this rate, however, would have grown by (1 + 0.06)^14 = 2.26 times in 14 years.

d) The compound interest rate is 6%. Here again, assuming this is the annual rate, the effective rate in the 18th year will be 6% and the investment at this rate would have grown by (1 + 0.06)^18 = 2.85 times in 18 years.