Jacqueline deposited $20,000 at 9% effective annual interest rate. She will make equal withdrawals at the end of each year over a 6-year term. What would be the amount of equal withdrawal?

Question

BlackTom AI · Accepted Answer

We start by identifying the financial setup: a present value of 20,000 is to be amortized by equal end-of-year withdrawals over 6 years at an annual effective interest rate of 9%. 
Option analysis often begins with computing the present value factor for an ordinary annuity, a-angle-6 at i = 9%. The formula is a-angle-n = (1 - (1+i)^(-n)) / i. 
First, evaluate (1+i)^n: (1.09)^6. A stepwise approximation: 1.09^2 = 1.1881, 1.09^3 ≈ 1.295; squaring 1.295 gives roughly 1.677. Thus (1.09)^6 ≈ 1.677, so (1.09)^(-6) ≈ 1 / 1.677 ≈ 0.596. 
Next, compute a-angle-6 ≈ (1 - 0.596) / 0.09 ≈ 0.404 / 0.09 ≈ 4.489. This is the present value factor for a level annuity of 6 payments. 
To find the annual withdrawal A that amortizes 20,000, use 20,000 = A * a-angle-6, so A ≈ 20,000 / 4.489 ≈ 4,457.7. 
Now compare with the provided options: 
- Option "$2,658.40": This value is far smaller than the computed A and would leave a large remaining balance, so it cannot be correct. It might correspond to an over-discounted or misapplied factor.
- Option "$4,090.27": This is somewhat close but under the precise calculation; if you test it by multiplying by a-angle-6 ≈ 4.489, you’d get about 18,370, leaving a balance, indicating it’s too small. It’s not aligned with the PV of 20,000 at 9% for 6 years.
- Option "$4,458.40": This value aligns with the calculation A ≈ 4,457–4,458 range. Multiplying 4,458.40 by a-angle-6 ≈ 4.489 yields a product very near 20,000, which matches the required present value. This makes it a strong candidate.
- Option "$2,438.90": Like the first small option, this is clearly too small to amortize 20,000 over 6 years at 9% and would not extinguish the balance by year 6. It’s inconsistent with the required annuity amount.
In summary, the reasoning hinges on using the present value of an ordinary annuity with i = 9% and n = 6 to solve for A. The approximate a-angle-6 value around 4.489 leads to A ≈ 20,000 / 4.489 ≈ 4,457.7, which is best matched by the option "$4,458.40". The other options either underestimate the payment, or are far enough off that they would not extinguish the loan under the given interest rate and term.

Jacqueline deposited $20,000 at 9% effective annual interest rate. She will make equal withdrawals at the end of each year over a 6-year term. What would be the amount of equal withdrawal?单项选择题

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