You are considering buy an asset that generates an initial cash flow of $1000 in 5 year's time. The cash flow then grows at a rate of 5% p.a. compounded annually for 10 years. If similar investments are offering a return of 10% p.a. compounded quarterly, how much should this asset cost today?

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We need to determine the present value today of a future cash flow that starts at year 5 and then grows each year at 5% for a total of 11 payments (from year 5 through year 15, inclusive). The discount rate is given as a nominal 10% per year compounded quarterly. To compare the cash flows on an annual basis, convert the quarterly rate to an effective annual rate.

Step 1: Convert the discount rate to an annual effective rate.
- Quarterly rate = 10% / 4 = 2.5% per quarter.
- Effective annual rate = (1 + 0.025)^4 − 1 ≈ 1.1038 − 1 = 0.1038 (about 10.38%).

Step 2: Identify the cash flows.
- First cash flow occurs at year 5 and equals 1000.
- Each subsequent year, the cash flow grows by 5%: year 6 = 1000×1.05, year 7 = 1000×1.05^2, …, up to year 15, giving a total of 11 payments (years 5–15).

Step 3: Express the present value as a discounted growing annuity.
- PV = Sum from t = 5 to t = 15 of [1000 × (1.05)^{t−5}] / (1.1038)^{t}.
- Factor out the t=5 term: PV = 1000 / (1.1038)^5 × Sum_{k=0}^{10} [(1.05)/(1.1038)]^k, where k = t−5.
- Compute the ratio: q = 1.05 / 1.1038 ≈ 0.9512.
- The sum of a finite geometric series with 11 terms is: Sum = (1 − q^{11}) / (1 − q).

Step 4: Calculate the geometric sum.
- q^{11} ≈ 0.9512^{11} ≈ 0.576 (roughly, from successive multiplications or using ln: ln(q) ≈ ln(0.9512) ≈ −0.0501, so ln(q^{11}) ≈ −0.551 ⇒ q^{11} ≈ 0.577).
- Sum ≈ (1 − 0.576) / (1 − 0.9512) ≈ 0.424 / 0.0488 ≈ 8.68.

Step 5: Compute the present value.
- (1.1038)^5 ≈ 1.648 (using the annual effective rate accumulation).
- PV ≈ 1000 / 1.648 × 8.68 ≈ 606.7 × 8.68 ≈ 5260.

Step 6: Compare with the provided options.
- The approximate computed PV is in the vicinity of 5,260 to 5,300, which matches the option around 5,294.14.
- Now quickly check alternatives: the other numbers (7,859.18; 5,471.44; 4,299.48; 4,436.21; 8,010.73) lie farther from the calculated range or would require a noticeably different interpretation (fewer/more payments or a different discounting approach), making them less plausible given the 11-payment assumption and the growth rate provided.

Conclusion: The option that aligns closest to the computed present value under these assumptions is $5,294.14.

FINC5001 (ND) Quiz 3

You are considering buy an asset that generates an initial cash flow of $1000 in 5 year's time. The cash flow then grows at a rate of 5% p.a. compounded annually for 10 years. If similar investments are offering a return of 10% p.a. compounded quarterly, how much should this asset cost today?

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