If the appropriate discount rate for the following cash flows is 9 percent compounded quarterly, what is the present value of the cash flows?

Year

Cash Flow

1

$790

2

$860

3

$0

4

$1,340

(Please fill out the blank with an integer value without the dollar sign. e.g. 1234 or -1234)

Question

If the appropriate discount rate for the following cash flows is 9 percent compounded quarterly, what is the present value of the cash flows?

Year

Cash Flow

1

$790

2

$860

3

$0

4

$1,340

(Please fill out the blank with an integer value without the dollar sign. e.g. 1234 or -1234)

BlackTom AI · Accepted Answer

Use the quarterly periodic rate r = 9%/4 = 0.0225 and discount each yearly cash flow by the number of quarters until receipt. Present value PV = Σ CF_t / (1 + r)^{quarters} where quarters = years × 4.

1) Relevant concepts
- Periodic rate for compounding: r = nominal annual rate / m (m = 4 quarters).
- Discount factor for a cash flow at year t: (1 + r)^{4t}.
- PV is sum of discounted cash flows.

2) Step-by-step calculations
- Quarterly rate: r = 0.09/4 = 0.0225.
- (1 + r)^4 = (1.0225)^4. Using binomial expansion:
  (1.0225)^4 = 1 + 4(0.0225) + 6(0.0225)^2 + 4(0.0225)^3 + (0.0225)^4
             ≈ 1.09308331879.
- (1 + r)^8 = ((1.0225)^4)^2 ≈ 1.194831045557.
- (1 + r)^16 = ((1.0225)^8)^2 ≈ 1.427621237.

3) Discount each cash flow (showing intermediate steps)
- Year 1 (4 quarters): PV1 = 790 / 1.09308331879 ≈ 722.72.
- Year 2 (8 quarters): PV2 = 860 / 1.194831045557 ≈ 719.77.
- Year 3: CF = 0 → PV3 = 0.
- Year 4 (16 quarters): PV4 = 1,340 / 1.427621237 ≈ 938.63.

4) Sum the present values
PV_total = PV1 + PV2 + PV3 + PV4
         ≈ 722.72 + 719.77 + 0 + 938.63
         ≈ 2,381.12.

5) Why this is correct and why other approaches are wrong
- Correct method: use the nominal 9% with quarterly compounding (period rate 2.25%) and discount by quarters. 
- Common errors: (a) treating 9% as an effective annual rate and discounting by (1.09)^t (this ignores intra-year compounding); (b) using continuous discounting or wrong period counts (e.g., using years as periods instead of quarters). Both produce different PVs and are incorrect here.

Conclusion: The present value of the cash flows, discounted at 9% compounded quarterly, is $2,381.12, which matches the provided answer.

If the appropriate discount rate for the following cash flows is 9 percent compounded quarterly, what is the present value of the cash flows? Year Cash Flow 1 $790 2 $860 3 $0 4 $1,340 (Please fill out the blank with an integer value without the dollar sign. e.g. 1234 or -1234)数值题

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