If the current rate of interest is 10%, then the present value of an investment that pays $800 per year and lasts 10 years is closest to:

Question

BlackTom AI · Accepted Answer

To evaluate the present value of an annual payment stream, we treat it as an ordinary annuity and apply the formula PV = PMT × [1 − (1 + r)^−n] / r.

First, identify the inputs: PMT = 800, r = 0.10 (10%), n = 10 years. 
Next, compute (1 + r)^−n = 1.10^−10. This is approximately 0.3855. 
Then, subtract from 1: 1 − 0.3855 ≈ 0.6145. 
Now divide by the interest rate r: 0.6145 / 0.10 ≈ 6.145. 
Finally, multiply by the annual payment: 6.145 × 800 ≈ 4,916.

Option analysis:
- "$6,264": This value would imply a larger annuity factor than 6.145, which would occur if either the payment were higher, the rate were lower, or the period were longer. Given the 10% rate and 10 years, the factor is around 6.145, not 6.264. 
- "$12,784": This is roughly double the expected PV for this setup; such a result would require a much larger factor (near 16) or a payment about twice 800, which is inconsistent with the given parameters. 
- "$4,916": This matches the calculated PV using the standard annuity formula with 10% interest over 10 years, yielding a factor near 6.145 times 800. 
- "$3,568": This would correspond to a significantly smaller annuity factor, suggesting a lower rate or shorter horizon than specified. The 10% rate over 10 years does not produce this low PV.

Summarizing the reasoning, the present value calculation aligns with the ordinary annuity formula, producing a PV close to 4,916, while the other options deviate due to mismatches in the annuity factor implied by the rate and term.

If the current rate of interest is 10%, then the present value of an investment that pays $800 per year and lasts 10 years is closest to:单项选择题

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