Question

In fund A, $100,000 accumulates at an annual nominal rate of interest j compounded semiannually to $130,666.52 in 4 years. In fund B, $100,000 accumulates at an annual nominal rate of discount k compounded quarterly to $154,531.82 in 5 years. In fund C, $100,000 accumulates at an annual effective rate of interest j in year one and an annual effective rate of interest k in year 2. What is the balance in fund C at the end of year 2?

Answer 1

Answer:

at the end of year 2, the balance of fund C = $116,639.23

Explanation:

to determine the nominal semiannual interest rate j we can use the future value formula:

$130,666.52 = $100,000 x (1 + j)⁸

(1 + j)⁸ = $130,666.52 / $100,000 = 1.3066652

⁸√(1 + j)⁸ = ⁸√1.3066652

1 + j = 1.034000004

j = 0.034000004

effective annual interest j = (1 + 0.034000004)² - 1 = 0.069156 = 6.9156%

to determine the nominal quarterly interest rate k we can use the future value formula:

$154,531.82 = $100,000 x (1 + k)²⁰

(1 + k)²⁰ = $154,531.82 / $100,000 = 1.5453182

²⁰√(1 + k)²⁰ = ²⁰√1.5453182

1 + k = 1.022

k = 0.022

effective annual interest k = (1 + 0.022)⁴ - 1 = 0.090946828 = 9.094682805%

at the end of year 1, the balance of fund C = $100,000 x (1 + 6.9156%) = $106,915.60

at the end of year 2, the balance of fund C = $106,915.60 x (1 + 9.094682805%) = $116,639.23