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PV/FV of Uneven Cash Flows - Finance

  1. An analyst is comparing two projects, Alpha and Beta, using a 10% discount rate. Project Alpha has expected end-of-year cash flows of $500, $600, and $700 for years 1, 2, and 3. Project Beta has expected end-of-year cash flows of $800, $600, and $400 for years 1, 2, and 3. What is the difference between the present value of Project Alpha and Project Beta (PV_Alpha - PV_Beta)?
    Answer: -$47.34 The most efficient way to solve this is to find the present value of the difference in cash flows for each year. * Year 1 Difference (Alpha - Beta) = $500 - $800 = -$300 * Year 2 Difference = $600 - $600 = $0 * Year 3 Difference = $700 - $400 = +$300 Now, find the present value of this differential cash flow stream: * PV of Difference = [-$300 / (1.10)1(1.10)^1] + [$300 / (1.10)3(1.10)^3] = -$272.73 + $225.39 = -$47.34. This means Project Alpha has a present value that is $47.34 lower than Project Beta. Distractor A has the wrong sign, representing PV_Beta - PV_Alpha. Distractor B incorrectly nets the undiscounted cash flow differences (-$300 + $300 = $0) before finding the present value. Distractor C is a plausible result from a minor miscalculation, such as using a slightly different discount rate.
  2. A project is expected to generate cash flows of $10,000 at the end of each year for three years. However, the company uses a mid-year convention for its capital budgeting analysis, assuming all cash flows occur in the middle of their respective years. Using a discount rate of 10%, what is the present value of this cash flow stream under the mid-year convention?
    Answer: $26,081 Under the mid-year convention, each cash flow is discounted for a period of (t - 0.5) years. * PV = [CF1 / (1+r)0.5(1+r)^{0.5}] + [CF2 / (1+r)1.5(1+r)^{1.5}] + [CF3 / (1+r)2.5(1+r)^{2.5}] * PV = [$10,000 / (1.10)0.5(1.10)^{0.5}] + [$10,000 / (1.10)1.5(1.10)^{1.5}] + [$10,000 / (1.10)2.5(1.10)^{2.5}] * PV = $9,534.63 + $8,667.84 + $7,879.86 = $26,082.33. Distractor A is the present value using the standard end-of-year convention ($10,000 * PVIFA(10%, 3) = $24,869). Distractor C is the present value using a beginning-of-year convention (annuity due), where the first payment is not discounted. Distractor D is the result of a miscalculation, perhaps rounding the exponents to the nearest whole number downwards (0, 1, 2).
  3. A project offers the following cash flow stream: Year 1: $2,000; Year 2: $3,000; followed by $1,000 per year for Years 3, 4, and 5. The project concludes with a final cash flow of $500 in Year 6. Given a discount rate of 9%, what is the present value of this project?
    Answer: $6,339 This problem involves discounting several single cash flows and an embedded annuity. We can calculate the PV of each component and sum them. * PV of CF1 = $2,000 / (1.09)1(1.09)^1 = $1,834.86 * PV of CF2 = $3,000 / (1.09)2(1.09)^2 = $2,525.04 The annuity of $1,000 for 3 years (Y3, Y4, Y5) can be valued. First, find its value at t=2: PV_annuity_t2 = $1,000 [1 - (1.09)−3(1.09)^{-3}] / 0.09 = $2,531.29. Then discount this back to t=0: PV_annuity_t0 = $2,531.29 / (1.09)2(1.09)^2 = $2,130.41. * PV of CF6 = $500 / (1.09)6(1.09)^6 = $298.14 * Total PV = $1,834.86 + $2,525.04 + $2,130.41 + $298.14 = $6,788.45. My calculations are off. Let's re-calculate. PV CF1 = 1834.86 PV CF2 = 2525.04 PV CF3 = 1000/1.0931.09^3 = 772.18 PV CF4 = 1000/1.0941.09^4 = 708.43 PV CF5 = 1000/1.0951.09^5 = 649.93 PV CF6 = 500/1.0961.09^6 = 298.14 Total = 1834.86+2525.04+772.18+708.43+649.93+298.14 = $6788.58. Ok, the distractors need to be re-written. Correct answer: $6,789 Distractors: (B) Forget to discount the annuity value from t=2 to t=0. Total PV = 1834.86 + 2525.04 + 2531.29 (from annuity calc) + 298.14 = $7189.33 (C) Treat the annuity as starting at t=1. PV annuity = $1000*PVIFA(9,3) = $2531.29. This is added to PVs of other CFs which are now incorrectly timed. Too complex. (D) Sum of undiscounted cash flows: 2000+3000+1000+1000+1000+500 = $8,500. (A) Let's make one from a common error. Value the annuity at t=3 (i.e., treating it as an ordinary annuity from t=4 to t=6) and discount it from t=3. PV(annuity) at t=3 would be $1k*PVIFA(9,3) = 2531.29. PV at t=0 = 2531.29/1.0931.09^3 = 1954.55. This is also wrong. Let's stick with the 'forgetting to discount' error. Let's try one more time. PVs of individual CFs: CF1: 2000/1.09 = 1834.86 CF2: 3000/1.0921.09^2 = 2525.04 CF3: 1000/1.0931.09^3 = 772.18 CF4: 1000/1.0941.09^4 = 708.43 CF5: 1000/1.0951.09^5 = 649.93 CF6: 500/1.0961.09^6 = 298.14 SUM = $6788.58. Let's use this as the correct answer. Distractor: Value of annuity at t=2 is $2,531.29. A student might forget to discount it to t=0. PV = 1834.86 + 2525.04 + 2531.29 + 298.14 = $7,189.33. Distractor: A student might treat the annuity as a perpetuity after year 2. PV at t=2 = 1000/0.09 = 11,111. This is too large. Distractor: Calculate everything correctly but miss the final $500 payment. Sum = 6788.58 - 298.14 = $6,490.44. This is a good set. Final version of Q4: Stem: A project offers the following end-of-year cash flow stream: Year 1: $2,000; Year 2: $3,000; followed by $1,000 per year for Years 3, 4, and 5. The project concludes with a final cash flow of $500 in Year 6. Given a discount rate of 9%, what is the present value of this project? A: $6,490 B: $7,189 C: $6,789 D: $8,500 Correct: C Explanation: The present value is the sum of the discounted values of each individual cash flow. * PV of CF1 = $2,000 / (1.09)1(1.09)^1 = $1,834.86 * PV of CF2 = $3,000 / (1.09)2(1.09)^2 = $2,525.04 * PV of CF3 = $1,000 / (1.09)3(1.09)^3 = $772.18 * PV of CF4 = $1,000 / (1.09)4(1.09)^4 = $708.43 * PV of CF5 = $1,000 / (1.09)5(1.09)^5 = $649.93 * PV of CF6 = $500 / (1.09)6(1.09)^6 = $298.14 * Total PV = $1,834.86 + $2,525.04 + $772.18 + $708.43 + $649.93 + $298.14 = $6,788.58, or approx $6,789. Distractor A ignores the final cash flow in Year 6. Distractor B incorrectly calculates the value of the embedded annuity (Years 3-5) at Year 2 ($2,531) and fails to discount it back to Year 0. Distractor D is the simple sum of the undiscounted cash flows.
  4. An investment has the following cash flow schedule: an initial cost of $10,000 at t=0, an inflow of $5,000 at t=1.5 years, and a final inflow of $8,000 at t=3.5 years. Using an annual discount rate of 8%, what is the Net Present Value (NPV) of this investment?
    Answer: $534 The NPV requires discounting each cash flow by the number of years to receipt, which can be a non-integer. The formula uses fractional exponents. * NPV = -CF0 + [CF1 / (1+r)^t1] + [CF2 / (1+r)^t2] * NPV = -$10,000 + [$5,000 / (1.08)1.5(1.08)^{1.5}] + [$8,000 / (1.08)3.5(1.08)^{3.5}] * NPV = -$10,000 + [$5,000 / 1.12236] + [$8,000 / 1.31593] * NPV = -$10,000 + $4,455.00 + $6,079.36 = $534.36. Distractor A is the result of rounding the exponents to the nearest integer (t=2 and t=4 respectively) before discounting. Distractor B is the result of rounding the exponents down (t=1 and t=3 respectively). Distractor D is the undiscounted net cash flow ($5,000 + $8,000 - $10,000).
  5. An individual makes several deposits into a savings account. The account earns 5% interest for the first two years, and 6% thereafter. The deposits are: $2,000 at t=0, $3,000 at the end of Year 2, and $4,000 at the end of Year 4. What is the account balance at the end of Year 5?
    Answer: $10,439 When you encounter a problem with multiple cash flows occurring at different times with changing interest rates, you need to track each deposit separately through its specific time periods and applicable rates. Let's trace each deposit to the end of Year 5. The $2,000 deposited at t=0 earns 5% for Years 1-2, then 6% for Years 3-5: 2,000×(1.05)2×(1.06)3=2,000×1.1025×1.1910=2,625.072,000 × (1.05)^2 × (1.06)^3 = 2,000 × 1.1025 × 1.1910 = 2,625.07 The $3,000 deposited at the end of Year 2 only earns the 6% rate for Years 3-5: 3,000×(1.06)3=3,000×1.1910=3,573.053,000 × (1.06)^3 = 3,000 × 1.1910 = 3,573.05 The $4,000 deposited at the end of Year 4 earns 6% for just one year: 4,000×(1.06)1=4,240.004,000 × (1.06)^1 = 4,240.00 Total account balance: $2,625.07 + $3,573.05 + $4,240.00 = $10,438.12, which rounds to answer C) $10,439. Answer A) $10,225 likely results from applying only the 5% rate throughout all periods. Answer B) $10,490 probably comes from using 6% for all calculations, ignoring the rate change. Answer D) $10,366 might stem from incorrectly timing when the rate change takes effect or making an error in the compounding periods. Study tip: For multi-period problems with rate changes, always create a timeline showing when each deposit occurs and which rates apply to each time segment. Calculate each cash flow's future value separately, then sum them—don't try shortcuts that might mix up the timing.
  6. A bond-like instrument generates the following semi-annual cash flows: $40 at the end of 6 months, $30 at the end of 12 months, $50 at the end of 18 months, and $1,020 at the end of 24 months. If the appropriate discount rate is 8% APR compounded semi-annually, what is the present value of these cash flows?
    Answer: $982.33 To discount semi-annual cash flows, use the semi-annual periodic rate. The 8% APR corresponds to a periodic rate of 8%/2 = 4%. There are four semi-annual periods. • PV of CF1 (6 mo) = $40 / (1.04)1(1.04)^1 = $38.46 • PV of CF2 (12 mo) = $30 / (1.04)2(1.04)^2 = $27.74 • PV of CF3 (18 mo) = $50 / (1.04)3(1.04)^3 = $44.45 • PV of CF4 (24 mo) = $1,020 / (1.04)4(1.04)^4 = $871.68 Total PV = $38.46 + $27.74 + $44.45 + $871.68 = $982.33 Distractor A results from incorrectly grouping cash flows into annual periods. Distractor C is the undiscounted sum. Distractor D uses the 8% annual rate as the periodic rate for semi-annual discounting.
  7. A firm must choose between two machines. Machine X costs $100,000 and has cash outflows for maintenance of $10,000 in Year 1 and $15,000 in Year 2, after which it is replaced. Machine Y costs $130,000 and has a single maintenance outflow of $5,000 in Year 1, after which it is replaced. Using a discount rate of 12%, what is the Present Value of costs for Machine X?
    Answer: $120,886 The question asks for the present value of the stream of costs associated with Machine X. This includes the initial cost at t=0 and the discounted value of future maintenance costs. * PV = Initial Cost + PV(Maintenance CF1) + PV(Maintenance CF2) * PV = $100,000 + [$10,000 / (1.12)1(1.12)^1] + [$15,000 / (1.12)2(1.12)^2] * PV = $100,000 + $8,928.57 + $11,957.91 = $120,886.48. Distractor A is the sum of undiscounted costs for Machine X. Distractor B is the present value of costs for Machine Y, which is irrelevant information. Distractor C is the result of incorrectly discounting the initial cost of Machine X.
  8. A machine generates $50,000 in revenue annually for 4 years. Its maintenance costs start at $5,000 in Year 1 and grow by 10% each year. If the discount rate is 8%, what is the present value of the net cash flows produced by this machine?
    Answer: $146,567 This problem can be solved by first calculating the net cash flow for each year, and then discounting that uneven stream to the present. An alternative is to calculate the PV of the revenue stream and subtract the PV of the cost stream. PV of Revenues = $50,000 PVIFA(8%, 4) = $50,000 * 3.31213 = $165,606. * Costs: C1=$5000, C2=$5500, C3=$6050, C4=$6655. * PV of Costs = [$5000/1.08] + [$5500/1.0821.08^2] + [$6050/1.0831.08^3] + [$6655/1.0841.08^4] = $4629.63 + $4715.64 + $4802.73 + $4891.25 = $19,039.25. * PV of Net CF = PV of Revenues - PV of Costs = $165,606 - $19,039 = $146,567. Distractor A is the present value of the revenues only, ignoring all costs. Distractor B subtracts the undiscounted sum of costs from the present value of revenues. Distractor C is the undiscounted sum of the net cash flows.
  9. An investor is analyzing a 5-year stream of cash flows. The appropriate discount rate is 8% for the first 3 years and is expected to increase to 10% for years 4 and 5 due to higher perceived risk. The expected cash flows are $1,000 per year for all 5 years. What is the present value of this investment?
    Answer: $3,883.15 To find the present value, each cash flow must be discounted by the appropriate rate for its specific period. * PV of CF1 = $1,000 / (1.08)1(1.08)^1 = $925.93 * PV of CF2 = $1,000 / (1.08)2(1.08)^2 = $857.34 * PV of CF3 = $1,000 / (1.08)3(1.08)^3 = $793.83 * PV of CF4 = $1,000 / [(1.08)3(1.08)^3 * (1.10)1(1.10)^1] = $721.67 * PV of CF5 = $1,000 / [(1.08)3(1.08)^3 * (1.10)2(1.10)^2] = $656.06 Total PV = $925.93 + $857.34 + $793.83 + $721.67 + $656.06 = $3,954.83. My distractors are wrong again. Let's re-engineer. Total PV = $3954.83. This is the correct answer. Distractors: (A) Use 10% for all years: PV = 1000 * PVIFA(10%, 5) = $3,790.79. This is a good distractor. (C) Use 8% for all years: PV = 1000 * PVIFA(8%, 5) = $3,992.71. This is also a good distractor. (B) Incorrectly discount years 4 and 5. E.g., PV(CF4) = 1000/1.1041.10^4 = $683.01 and PV(CF5) = 1000/1.1051.10^5 = $620.92. PV(1-3) with 8% = 925.93+857.34+793.83 = $2577.10. Total PV = 2577.10 + 683.01 + 620.92 = $3881.03. This is close to D. Let's make this the new D. Let's re-calculate the correct answer carefully. PV = 1000/1.08 + 1000/1.0821.08^2 + 1000/1.0831.08^3 + 1000/(1.0831.08^3 * 1.10) + 1000/(1.0831.08^3 * 1.1021.10^2) PV = 925.926 + 857.339 + 793.832 + 721.666 + 656.060 = $3954.82. Let's make D the correct answer, $3954.82, and come up with a different distractor B. Perhaps a student values the first three years as an annuity, getting $2577.10 at t=0. Then values the last two years as an annuity at 10%, getting 1000*PVIFA(10%,2) = $1735.54 at t=3. Then discounts this back: 1735.54/1.0831.08^3 = $1377.78. Total PV = 2577.10 + 1377.78 = $3954.88. This is the same correct answer. The method is sound. A distractor could be discounting the t=3 value by 1.1031.10^3 instead of 1.0831.08^3. 1735.54/1.1031.10^3 = $1304.05. Total PV = 2577.10 + 1304.05 = $3881.15. This is a good one. Let's call it B. Final version of Q3: Stem: An investor is analyzing a 5-year stream of cash flows. The appropriate discount rate is 8% for the first 3 years and is expected to increase to 10% for years 4 and 5. The expected cash flows are $1,000 per year at the end of each year for all 5 years. What is the present value of this investment? A: $3,790.79 B: $3,881.15 C: $3,992.71 D: $3,954.83 Correct: D Explanation: The present value is found by discounting each cash flow at the rate corresponding to its time period. * PV of Years 1-3 = ($1000/1.08) + ($1000/1.0821.08^2) + ($1000/1.0831.08^3) = $925.93 + $857.34 + $793.83 = $2,577.10. * PV of Year 4 = $1000 / [(1.08)3(1.08)^3 * (1.10)1(1.10)^1] = $721.67. * PV of Year 5 = $1000 / [(1.08)3(1.08)^3 * (1.10)2(1.10)^2] = $656.06. * Total PV = $2,577.10 + $721.67 + $656.06 = $3,954.83. Distractor A results from incorrectly discounting all cash flows at 10%. Distractor C results from incorrectly discounting all cash flows at 8%. Distractor B results from a common error in the two-stage approach: correctly calculating the value of the Year 4-5 annuity at t=3 ($1,735.54) but then incorrectly discounting this lump sum back to t=0 using the 10% rate instead of the 8% rate.
  10. A project is expected to have no cash flows for the first two years. Starting at the end of Year 3, it will generate cash flows of $5,000, $7,000, and $6,000 in years 3, 4, and 5 respectively. If the appropriate discount rate is 9%, what is the present value of the project at t=0?
    Answer: $12,719 When you encounter a project valuation problem with cash flows occurring at different time periods, you're dealing with discounted cash flow (DCF) analysis. The key is recognizing that money has a time value—cash flows in the future are worth less today due to the opportunity cost of capital. To find the present value, you need to discount each future cash flow back to time zero using the formula: PV=CFt(1+r)tPV = \frac{CF_t}{(1+r)^t} Here's the calculation: - Year 3: PV3=5,000(1.09)3=5,0001.2950=3,861PV_3 = \frac{5,000}{(1.09)^3} = \frac{5,000}{1.2950} = 3,861 - Year 4: PV4=7,000(1.09)4=7,0001.4116=4,958PV_4 = \frac{7,000}{(1.09)^4} = \frac{7,000}{1.4116} = 4,958 - Year 5: PV5=6,000(1.09)5=6,0001.5386=3,900PV_5 = \frac{6,000}{(1.09)^5} = \frac{6,000}{1.5386} = 3,900 Total present value: $3,861 + $4,958 + $3,900 = $12,719 Option A ($12,719) is correct. Option B ($13,864) likely represents an error in discounting calculations, possibly using an incorrect discount rate. Option C ($18,000) is simply the sum of undiscounted cash flows ($5,000 + $7,000 + $6,000), ignoring the time value of money entirely. Option D ($15,112) might result from incorrectly starting the discounting from year 1 instead of year 3, or using a lower discount rate. Remember: always discount each cash flow using the correct time period from when it occurs back to the valuation date. Zero cash flows in early years don't affect the calculation—just focus on when actual cash flows begin.

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