Engr. Reyes invested an equal amount to two different investments today.

Question

Engr. Reyes invested an equal amount to two different investments today. At the end of 8 years, he will receive an amount in his first investment twice than the second. If the first investment earns 8.55% compounded semi-annually, determine the following:

1. Nominal interest rate compounded quarterly of the second investment, %
2. At what year can he receive an amount on the first investment the same amount he can receive at the end of 5 years on the second investment? (in three decimal places)

Answer 1

Let's take the initial investment amount as "P" for both investments.

First, let's determine the future value (FV) of the first investment after 8 years. We can use the compound interest formula:

FV = P * (1 + r/n)^(n*t)

Where:

FV is the future value

P is the principal amount (initial investment)

r is the annual interest rate (8.55%)

n is the number of compounding periods per year

t is the number of years

For the first investment:

FV1 = P * (1 + 0.0855/2)^(2*8)

FV1 = P * (1.04275)^16

Now, according to the problem, the future value of the first investment is twice the future value of the second investment. Let's denote the future value of the second investment after 8 years as "FV2."

FV1 = 2 * FV2

P * (1.04275)^16 = 2 * P * (1 + r2/4)^(4*8)

Simplifying the equation:

(1.04275)^16 = 2 * (1 + r2/4)^32

Now, we need to solve for the nominal interest rate compounded quarterly of the second investment, denoted as "r2".

Taking the 32nd root of both sides:

(1.04275)^(16/32) = 1 + r2/4

1.02114478 = 1 + r2/4

r2/4 = 1.02114478 - 1

r2/4 = 0.02114478

r2 = 0.02114478 * 4

r2 = 0.08457912

Therefore, the nominal interest rate compounded quarterly of the second investment is approximately 8.458%.

Now let's determine at what year the amount received from the first investment will be the same as the amount received at the end of 5 years from the second investment.

For the first investment:

FV1 = P * (1 + 0.0855/2)^(2*t)

For the second investment:

FV2 = P * (1 + r2/4)^(4*5)

Since we want the two amounts to be equal, we can set them equal to each other and solve for "t":

P * (1.04275)^(2t) = P * (1 + 0.08457912/4)^(45)

Simplifying the equation:

(1.04275)^(2*t) = (1.02114478)^(20)

Take the logarithm of both sides to solve for "t":

2*t * log(1.04275) = 20 * log(1.02114478)

t = (20 * log(1.02114478)) / (2 * log(1.04275))

Calculating the value:

t ≈ 10.656

Therefore, it will take approximately 10.656 years for Engr. Reyes to receive an amount on the first investment equal to the amount he can receive at the end of 5 years from the second investment.