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?单项选择题
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Question

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?单项选择题

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BlackTom AI · Accepted Answer

Question restatement: Gilles Lebouder is offered a job with end-of-year payment of $50,000, with salaries rising 5% per year for 40 years, discount rate 8%. We are asked for the present value of his lifetime salary.
Option set: The provided option is a single choice: a. $1,126,571. There are no other options listed to compare against.
Approach to evaluation: When payments grow at a constant rate (g) and are discounted at a rate (r), the present value of a growing annuity with the first payment P at t = 1 for n periods is P times the growing annuity factor: PV = P * [1 - ((1+g)/(1+r))^n] / (r - g).
Plugging in the numbers: P = 50,000, g = 0.05, r = 0.08, n = 40.
Compute the growth discount ratio: (1+g)/(1+r) = 1.05/1.08 ≈ 0.972222...
Raise to the 40th power: (0.972222...)^40 ≈ exp(40 * ln(0.972222...)) ≈ exp(40 * (-0.0281)) ≈ exp(-1.124) ≈ 0.324.
Now compute the numerator: 1 - 0.324 ≈ 0.676.
Denominator: r - g = 0.08 - 0.05 = 0.03.
Thus the factor is 0.676 / 0.03 ≈ 22.533...
Multiply by P: 22.533... * 50,000 ≈ 1,126,650.
With a more precise calculation the value aligns closely with the provided figure of $1,126,571, so the option listed is consistent with the mathematical result.
Why this value is correct: It accounts for the annual 5% salary growth, discounts each future payment back at 8%, and aggregates over 40 years. The growing annuity formula captures both the time value of money and the increasing payments.
Why there are no other options to evaluate here: Since only one option is given, there is no alternative choice to refute or compare. In a broader sense, if there were other choices, you would compare them to the computed PV and identify which lies closest to the calculated value, considering rounding.
Practical note: Small differences may arise due to rounding in intermediate steps or in the display of the final PV, but \$1,126,571 is the consistent ballpark figure given the standard formula and inputs.