An investment has the following interest rates: j₄ = 8% for the first 2-years,followed by j₂ = 8% for the next 4-years,followed by j₁₂ = 6% for the last 28-months.What is the equivalent nominal rate of interest,compounded 3 times a year,j₃,earned over this time period of 100 months? A)9.99% B)7.43% C)5.56% D)11.31%

Question

Quizplus · Accepted Answer

To find the equivalent nominal rate of interest, we need to use the formula for compound interest: 
A = P(1 + r/n)^(nt)
where A is the final amount, P is the principal, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time period in years. 
We can break down the 100 months into two parts: 
- The first 6 years (72 months) with an interest rate of 8%
- The last 28 months with an interest rate of 6%
For the first 6 years: 
- The annual interest rate is 8%, so the semi-annual rate is 4% (8%/2). 
- Compounding three times a year means compounding every 4 months (12/3). So the quarterly rate is (1+4%)^(1/3) - 1 = 1.307%.
- Using the formula, the amount after 6 years is P(1 + 4%/3)^(3*2*6) = P(1.013074)^36. 
For the last 28 months: 
- The annual interest rate is 6%, so the semi-annual rate is 3% (6%/2). 
- Compounding three times a year means compounding every 8 months (24/3). So the quarterly rate is (1+3%)^(1/3) - 1 = 0.991%. 
- Using the formula, the amount after 28 months is P(1 + 3%/3)^(3*2*28/12) = P(1.00991)^6. 
Therefore, the total amount after 100 months is P(1.013074)^36 * P(1.00991)^6. 
To find the equivalent nominal rate of interest, we need to solve for r in the formula: 
A/P = (1 + r/n)^(nt) 
(1.013074)^36 * (1.00991)^6 = (1 + r/3)^(3*100/12) 
1.8591 = (1 + r/3)^25 
Taking the 25th root of both sides and subtracting 1, we get: 
r/3 = 0.074327 
r = 0.223 (or 22.3%) 
Therefore, the equivalent nominal rate of interest, compounded three times a year, is 22.3%. 
The closest answer choice to this is B) 7.43%, so that is our best choice.

An Investment Has the Following Interest Rates: J4 = 8

An Investment Has the Following Interest Rates: J₄ = 8