- A legal settlement offers a choice between receiving $500,000 today or receiving a single payment of $800,000 in five years. At what approximate discount rate would an individual be financially indifferent between the two options?
Answer: 9.86%
Indifference means the present value of the future payment is equal to the payment offered today. We need to solve for the interest rate (r) where PV = FV / (1+r)ⁿ. So, $500,000 = $800,000 / (1+r)⁵. Rearranging gives (1+r)⁵ = $800,000 / $500,000 = 1.6. Then, (1+r) = (1.6)1/5 ≈ 1.09856. Therefore, r ≈ 0.09856 or 9.86%. - An investor deposits $50,000 today. The investment is expected to earn a nominal annual return of 10%. If the average annual inflation rate over the investment period is 3%, what is the expected future value of the investment in 7 years, expressed in today's (real) dollars?
Answer: $79,160
This is a two-step problem. First, calculate the nominal future value: FV_nominal = $50,000 × (1 + 0.10)⁷ ≈ $97,435.86. Second, discount this nominal future value back to today's dollars using the inflation rate to find the real future value: FV_real = FV_nominal / (1 + inflation)⁷ = $97,435.86 / (1.03)⁷ ≈ $79,160.03.
- An investment of $15,000 is made into an account that earns an annual interest rate of 5%, compounded continuously. What will be the value of the investment after exactly 8 years?
Answer: $22,377
This problem requires the formula for continuous compounding: FV = PV × ert. Here, PV = $15,000, r = 0.05, and t = 8. FV = $15,000 × e0.05×8 = $15,000 × e0.4 ≈ $15,000 × 1.49182 ≈ $22,377.37. - An investment doubles in value in exactly 10 years. If this same annually compounded rate of return continues, what will be the value of a $100,000 investment after 30 years?
Answer: $800,000
If the investment doubles in 10 years, it means the value is multiplied by 2 every 10 years. A 30-year period contains three 10-year doubling periods. Therefore, the initial investment will double three times. Value = $100,000 × 2 × 2 × 2 = $100,000 × 2³ = $100,000 × 8 = $800,000.
- An entrepreneur invests $25,000 in a startup, aiming to triple the investment. If the investment grows at a rate of 12% per year, compounded annually, approximately how many years will it take for the investment to reach the $75,000 target?
Answer: 9.69 years
This problem requires solving for the number of periods (N). The formula is FV = PV × (1 + r)ⁿ. $75,000 = $25,000 × (1 + 0.12)ⁿ. This simplifies to 3 = (1.12)ⁿ. To solve for N, use logarithms: N = ln(3) / ln(1.12) ≈ 9.69 years.
- A rare coin was purchased for $5,000. Exactly ten years later, it was sold for $13,000. Assuming the value of the coin grew at a constant annual rate, what was the annually compounded rate of return earned on this investment?
Answer: 10.03%
This problem requires solving for the interest rate (r) in the future value formula. FV = PV × (1 + r)ⁿ. So, $13,000 = $5,000 × (1 + r)¹⁰. Rearranging gives (1 + r) = ($13,000 / $5,000)^(1/10) = (2.6)0.1 ≈ 1.1003. Therefore, r ≈ 0.1003 or 10.03%.
