Simple and Compound Interest Problems

Exercise 1: Calculating Monthly Interest Rate

A person invested $12,000 in an institution and received $13,008.00 after seven months. What is the equivalent monthly interest rate that the investor earned?

Solution:

• Initial Capital (C) = $12,000
• Time (n) = 7 months
• Future Value (FV) = $13,008.00

Using the formula: FV / C = (1 + i * n)

13,008.00 / 12,000.00 = (1 + i * 7)

1. 0840 = 1 + 7i

2. 0840 = 7i

i = 0.0840 / 7

i = 0.012 or 1.2% per month

Exercise 2: Equivalent Financial Return

An investment of $15,000 is made for a period of three months at a simple interest rate of 26% per annum. What amount must be invested for two months at a linear rate of 18% per year to achieve the same financial return?

Solution:

Investment 1:

• C = $15,000
• n = 3 months
• i (annual) = 26% (monthly = 26%/12 = 2.17%)

Investment 2:

• n = 2 months
• i (annual) = 18% (monthly = 18%/12 = 1.5%)

Equating the returns (J₁ = J₂):

C₁ * i₁ * n₁ = C₂ * i₂ * n₂

15,000 * 0.0217 * 3 = C₂ * 0.015 * 2

976.5 = C₂ * 0.03

C₂ = 976.5 / 0.03 = $32,550.00

Exercise 3: Effective Annual Cost

A financing is being negotiated at a nominal rate of 72% per year. Determine the effective annual cost of this operation, assuming that the interest is capitalized:

1. Monthly
2. Quarterly
3. Every six months

Solution:

a) Monthly capitalization:

• Monthly rate = 72% / 12 = 6% per month
• Effective Annual Rate = (1 + 0.06)¹² - 1 = 1.0122 -1 = 101.22% per year

b) Quarterly capitalization:

• Quarterly rate = 72% / 4 = 18% per quarter
• Effective Annual Rate = (1 + 0.18)⁴ - 1 = 0.9388 = 93.88% per year

c) Semi-annual capitalization:

• Semi-annual rate = 72% / 2 = 36% per six months
• Effective Annual Rate = (1 + 0.36)² - 1 = 0.8496 = 84.96% per year

Exercise 4: Present Value Calculation

At a current interest rate of 10% per quarter, how much should be invested today to receive $38,500.00 in 28 months?

Solution:

• i = 10% per quarter
• FV = $38,500.00
• n = 28 months = 7 quarters (since each quarter is 3 months, and we use the interest rate period)
• PV = ?

PV = FV / (1 + i)ⁿ

PV = 38,500 / (1 + 0.10)⁷

PV = 38,500 / 1.9487171

PV = $19,756.59

Exercise 5: Monthly Interest Rate

Calculate the monthly interest rate on an investment that grows from $68,700.00 to $82,084.90 in eight months.

Solution:

• PV = $68,700
• FV = $82,084.90
• n = 8 months
• i = ?

FV / PV = (1 + i)ⁿ

82,084.90 / 68,700.00 = (1 + i)⁸

1. 1948195 = (1 + i)⁸

⁸√1.1948195 = 1 + i

2. 0225 - 1 = i

i = 0.0225 or 2.25% per month