You want to start a business that you believe can produce cash flows of $44,000, $61,000, and $80,000 at the end of each of the next three years, respectively. At the end of three years you think you can sell the business for $200,000. At a discount rate of 9.7 percent, what is this business worth today?

Question

BlackTom AI · Accepted Answer

Question restatement:
You’re evaluating a business idea with expected cash flows of $44,000 in year 1, $61,000 in year 2, and $80,000 in year 3, plus a potential sale price of $200,000 at the end of year 3. With a discount rate of 9.7%, you want to determine the present value (worth today) of this business. The answer choices provided include a single option: $302,897.

Step-by-step analysis of the cash flows and the present value calculation:
- The present value (PV) of each amount is found by dividing the future amount by (1 + r)^t, where r is the discount rate (9.7% or 0.097) and t is the year in which the cash flow occurs.
- Year 1 cash flow: 44,000 received at t = 1. The discount factor is (1.097)^1 = 1.097. PV1 = 44,000 / 1.097 ≈ 40,100. The exact figure is around 40.1k, acknowledging small rounding differences.
- Year 2 cash flow: 61,000 received at t = 2. The discount factor is (1.097)^2 ≈ 1.203. PV2 = 61,000 / 1.203 ≈ 50,700. This value reflects the time value of money two years out at roughly a 9.7% rate.
- Year 3 cash flow: 80,000 plus the 200,000 sale price both occur at t = 3. The discount factor is (1.097)^3 ≈ 1.319. PV3 from the operating cash flow is 80,000 / 1.319 ≈ 60,700. The PV of the sale proceeds is 200,000 / 1.319 ≈ 151,700. The sum of these two PVs represents the year-3 inflows discounted back to today.
- Summing the present values:
  PV1 ≈ 40,100
  PV2 ≈ 50,700
  PV3 operating ≈ 60,700
  PV3 sale ≈ 151,700
  Total ≈ 40,100 + 50,700 + 60,700 + 151,700 = 302,200 (approximate, subject to rounding differences).

Common sources of small variation in this kind of calculation come from rounding the discount factors (1.097)^t to different decimal places, which slightly shifts each PV component. If you tighten the rounding or use a financial calculator or software with exact power calculations, you typically land very close to the given option around the low to mid 302k range.

Reasoning about the single option present:
- The problem provides only one possible value for the present worth: $302,897. In a typical finance exercise, this would be the result of summing the discounted cash flows as shown above with precise arithmetic.
- If you compare the approximate total calculated above (roughly $302,200) to the provided option ($302,897), the difference is small and attributable to rounding of the discount factors and intermediate PVs. With exact calculations (no rounding at each step until the final sum), you would converge on the option near $302,897.

What could be misunderstood or miscalculated in alternative approaches:
- Ignoring the sale value at year 3 would understate the present value by about $151,700, since the sale contributes a large portion of the total present value.
- Discounting each cash flow incorrectly (for example, using 9.7% as a simple annual rate without compounding, or using the wrong power t for each cash flow) would yield a noticeably different result and could lead to a value that does not match the given option.
- Excessive rounding early in the calculations can push the final number away from the precise present value, so keeping sufficient precision until the final addition is preferable.

Summary note:
- The calculation combines the PVs of three annual cash inflows plus the PV of the sale at year 3, all discounted at 9.7%. The resulting present value aligns with the provided option when precise arithmetic is used, yielding a value in the vicinity of $302,897.

FA25-BL-BUS-F307-1134

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