Gilles Lebouder has just been offered a job paying $50,000 at the end of his first year. He expects his annual salary will increase by 5% a year until he retires in 40 years. Given an interest rate of 8%, what is the present value of his lifetime salary?

Question

BlackTom AI · Accepted Answer

We begin by restating the problem in our own words: We need the present value (PV) of Gilles Lebouder’s lifetime salary, given that he starts with a salary of $50,000 at the end of year 1, the salary grows at 5% per year, there are 40 years in total, and the discount rate is 8%.

Option a. $1,126,571

Step 1: Identify the cash flow structure. The salaries form a growing annuity: payments occur at the end of each year for 40 years, with year t payment equal to 50,000 × (1.05)^(t−1). The present value is the sum of each payment discounted back 1 year, 2 years, ..., 40 years at 8% per year.

Step 2: Use the present value of a growing annuity formula. For a growing annuity with initial payment P at the end of year 1, growth rate g, discount rate r, and n periods, the PV is:
PV = P × [1/(1+r)] × [1 − ((1+g)/(1+r))^n] / [1 − (1+g)/(1+r)]
Here, P = 50,000, g = 0.05, r = 0.08, n = 40.

Step 3: Compute the ratio in the exponent. (1+g)/(1+r) = 1.05/1.08 = 35/36 ≈ 0.972222…

Step 4: Evaluate the denominator 1 − (1+g)/(1+r). This is 1 − 35/36 = 1/36 ≈ 0.027777…

Step 5: Put numbers into the formula. PV = 50,000 × [1/1.08] × [1 − (35/36)^40] ÷ (1/36).

Step 6: Simplify the fraction. Dividing by (1/36) is the same as multiplying by 36, and 50,000 × (1/1.08) ≈ 46,296.296. Multiplying 46,296.296 by 36 gives about 1,666,666.67. So PV ≈ 1,666,666.67 × [1 − (35/36)^40].

Step 7: Evaluate (35/36)^40. Since (35/36) ≈ 0.972222…, raising to the 40th power yields roughly 0.324–0.326 (precise value depends on rounding). Therefore, 1 − (35/36)^40 ≈ 0.676–0.676.

Step 8: Final multiplication. 1,666,666.67 × 0.676 ≈ 1,126,000 to 1,127,000. The option given, $1,126,571, falls within this range and matches the more exact computation closely.

Step 9: Concept check. The result makes intuitive sense: discounting over 40 years with a modest growth rate reduces the weight of later salaries, and the growing annuity formula captures this trade-off between growth and discounting. A figure around $1.126 million aligns with the stated inputs.

Notes on why other hypothetical options (not shown) would be incorrect: If an option claimed a present value significantly higher, it would ignore the growing discounting effect or misapply the formula. If it claimed a value much lower, it would imply an incorrect growth/discount handling or an erroneous n. In any case, the correct approach is to use the growing annuity PV formula with P = 50,000, g = 5%, r = 8%, n = 40, which yields approximately $1.126 million.

Gilles Lebouder has just been offered a job paying $50,000 at the end of his first year. He expects his annual salary will increase by 5% a year until he retires in 40 years. Given an interest rate of 8%, what is the present value of his lifetime salary?

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Gilles Lebouder has just been offered a job paying $50,000 at the end of his first year. He expects his annual salary will increase by 5% a year until he retires in 40 years. Given an interest rate of 8%, what is the present value of his lifetime salary?

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