An appraiser is verifying the sale of a land contract. The investor paid $400,000 for the right to receive a single payment of $532,400 at the end of three years, with no other payments. What is the annual yield rate on this purchase?
Correct Answer
B) 10.00%
The yield is the rate that makes the payment's present value equal the price: ($532,400 ÷ $400,000)^(1/3) − 1 = 1.331^(1/3) − 1 = 10.00%. Averaging the 33.1% total gain over three years gives the simple-interest figure of 11.03%. Measuring the gain against the future payment and averaging gives 8.29%. Compounding over two periods instead of three gives 15.37%.
Why This Is the Correct Answer
Option B is correct because $400,000 × 1.10 × 1.10 × 1.10 = $532,400, so 10.00% equates the price with the discounted payment. In formula terms, 1.331^(1/3) − 1 = 10.00%.
Why the Other Options Are Wrong
Option A: 8.29%
Option A, 8.29%, measures the $132,400 gain against the future payment and averages it: ($132,400 ÷ $532,400) ÷ 3 = 8.29%. The correct rate compounds the price: 1.331^(1/3) − 1 = 10.00%.
Option C: 11.03%
Option C, 11.03%, spreads the gain evenly: ($132,400 ÷ $400,000) ÷ 3 = 11.03%. That is a simple-interest average, and it ignores interest on interest; the compounded rate is 10.00%.
Option D: 15.37%
Option D, 15.37%, takes a square root: 1.331^(1/2) − 1 = 15.37%. The payment arrives after three periods, so the cube root applies and the rate is 10.00%.
Check by compounding forward
Grow the price by the candidate rate once for each year. The right rate lands on the future payment.
How to use: Use this forward check on any yield problem where the answer choices are rates.
Exam Tip
You can test choices without a financial function: $400,000 × 1.10 × 1.10 × 1.10 reaches $532,400 exactly.
Common Mistakes to Avoid
- -Dividing the total gain by the number of years
- -Using the future amount as the base for the return
- -Miscounting the number of compounding periods
Concept Deep Dive
Analysis
Price and future payment are known, and the rate is the unknown. The rate must compound the price into the payment over three periods, so the growth multiple is converted with a root, not divided by three. Testing each option by compounding the price is a reliable check.
Background Knowledge
An internal rate of return is the discount rate at which future benefits equal present value. For one lump sum it can be solved directly, and for several cash flows it is found by calculator or trial.
Real-World Application
An appraiser analyzing the sale of a deferred land payment knows the price paid and the amount due later. Solving for the implied yield gives a market indication of the rate buyers accepted for that kind of deferred benefit.
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