Let's break down the problem step by step: ### 1. **Expected Values for Each Variable** **The hint says the distributions are symmetric, so the expected value is the middle value. Let's identify the three outcomes and their probabilities for each variable:** #### **a) Developmental Costs** | Value | Probability | |----------|-------------| | $2,000,000 | 0.3 | | $4,000,000 | 0.4 | | $6,000,000 | 0.3 | **Expected Value:** \[ E(Dev.\ Cost) = 0.3(2,000,000) + 0.4(4,000,000) + 0.3(6,000,000) = 600,000 + 1,600,000 + 1,800,000 = \$4,000,000 \] #### **b) Project Life (Years)** | Value | Probability | |-------|-------------| | 3 | 0.2 | | 8 | 0.2 | | 6 | 0.6 | **Expected Value:** \[ E(Life) = 0.2(3) + 0.6(6) + 0.2(8) = 0.6 + 3.6 + 1.6 = 5.8 \] **But the "middle value" is also 6 (majority probability), so use 6 years.** #### **c) Sales in Units** | Value | Probability | |-------|-------------| | 1,000 | 0.2 | | 2,000 | 0.6 | | 3,000 | 0.2 | **Expected Value:** \[ E(Units) = 0.2(1,000) + 0.6(2,000) + 0.2(3,000) = 200 + 1,200 + 600 = 2,000 \] #### **d) Sales Price** | Value | Probability | |----------|-------------| | $12,000 | 0.1 | | $12,500 | 0.8 | | $13,000 | 0.1 | **Expected Value:** \[ E(Price) = 0.1(12,000) + 0.8(12,500) + 0.1(13,000) = 1,200 + 10,000 + 1,300 = \$12,500 \] #### **e) Cost per Unit (Excluding Dev. Cost)** | Value | Probability | |----------|-------------| | $6,000 | 0.3 | | $7,000 | 0.4 | | $8,000 | 0.3 | **Expected Value:** \[ E(Cost) = 0.3(6,000) + 0.4(7,000) + 0.3(8,000) = 1,800 + 2,800 + 2,400 = \$7,000 \] --- ### 2. **Calculate Pre-tax and After-tax Cash Flow** #### **a) Revenue per Year** \[ \text{Revenue} = \text{Units} \times \text{Sales Price} = 2,000 \times \$12,500 = \$25,000,000 \] #### **b) Cost per Year (excluding Dev. Cost)** \[ \text{Cost} = \text{Units} \times \text{Cost per Unit} = 2,000 \times \$7,000 = \$14,000,000 \] #### **c) Pre-tax Profits per Year** \[ \text{Pre-tax Profit} = \text{Revenue} - \text{Cost} = \$25,000,000 - \$14,000,000 = \$11,000,000 \] #### **d) After-tax Profit per Year (equals cash flow, since no depreciation)** \[ \text{After-tax Cash Flow} = \text{Pre-tax Profit} \times (1 - \text{Tax Rate}) = \$11,000,000 \times 0.75 = \$8,250,000 \] --- ### 3. **Initial Outlay** \[ \text{Initial Outlay} = \text{Developmental Cost} = \$4,000,000 \] --- ### 4. **Project Life** \[ \text{Project Life} = 6 \text{ years} \] --- ### 5. **IRR Calculation** The IRR is the rate \( r \) that solves: \[ 0 = -4,000,000 + \sum_{t=1}^6 \frac{8,250,000}{(1+r)^t} \] This is a cash flow of -4,000,000 at \( t=0 \), and then 8,250,000 per year for 6 years. **This is a standard annuity problem:** Set N = 6, PMT = 8,250,000, PV = -4,000,000 \[ 4,000,000 = 8,250,000 \times \left[ \frac{1 - (1 + IRR)^{-6}}{IRR} \right] \] \[ \frac{4,000,000}{8,250,000} = \frac{1 - (1 + IRR)^{-6}}{IRR} \] \[ 0.484848 = \frac{1 - (1 + IRR)^{-6}}{IRR} \] Now, solve for IRR. This requires a financial calculator or Excel, but let's estimate: - At IRR = 15%: \[ \text{Annuity factor} = \frac{1 - (1+0.15)^{-6}}{0.15} = \frac{1 - (1/2.313)}{0.15} = \frac{1 - 0.432}{0.15} = \frac{0.568}{0.15} = 3.787 \] \[ 8,250,000 \times 3.787 = 31,220,250 \] But that's way more than 4,000,000, so IRR must be much higher. Alternatively, let's try with the cash flows: - Year 0: -4,000,000 - Years 1-6: +8,250,000 Try IRR in Excel or with a financial calculator, or estimate: \[ NPV = -4,000,000 + 8,250,000 \times \text{PVIFA}_{r,6} \] Set NPV = 0 and solve for r. We'll use the equation: \[ \frac{4,000,000}{8,250,000} = \frac{1 - (1 + r)^{-6}}{r} \] \[ 0.484848 = \frac{1 - (1 + r)^{-6}}{r} \] Try r = 100% (1.0): \[ (1 + 1)^6 = 64 \implies (1 + 1)^{-6} = 1/64 = 0.015625 \] \[ 1 - 0.015625 = 0.984375 \] \[ 0.984375 / 1 = 0.984375 \] So too high. Try r = 50% (0.5): \[ (1.5)^6 = 11.390625 \implies (1.5)^{-6} = 1 / 11.390625 = 0.0878 \] \[ 1 - 0.0878 = 0.9122 \] \[ 0.9122 / 0.5 = 1.824 \] Still too high. Try r = 20% (0.2): \[ (1.2)^6 = 2.985984 \implies (1.2)^{-6} = 1 / 2.985984 = 0.3348 \] \[ 1 - 0.3348 = 0.6652 \] \[ 0.6652 / 0.2 = 3.326 \] Still too high. Try r = 50% again, but let's try r = 200% (2.0): \[ (1+2)^6 = 729 \implies (1+2)^{-6} = 1/729 = 0.00137 \] \[ 1 - 0.00137 = 0.99863 \] \[ 0.99863 / 2 = 0.499 \] Now, 0.499 is just above 0.4848. Try r = 210% (2.1): \[ (1+2.1)^6 = 3.1^6 = 887.5037 \implies (1+2.1)^{-6} = 1/887.5037 = 0.001127

Continuing from where we left off: At r = 2.1 (210%): \[ (1+2.1)^6 = 3.1^6 \approx 887.5037 \] \[ (1+2.1)^{-6} = 1 / 887.5037 \approx 0.001127 \] \[ 1 - 0.001127 = 0.998873 \] \[ \frac{0.998873}{2.1} \approx 0.4756 \] This is slightly below our target of 0.4848. Now, try r = 2.05 (205%): \[ (1+2.05)^6 = 3.05^6 \approx 769.6 \] \[ (1+2.05)^{-6} = 1 / 769.6 \approx 0.0013 \] \[ 1 - 0.0013 = 0.9987 \] \[ 0.9987 / 2.05 \approx 0.487 \] This is very close to 0.4848, slightly above. Since at r = 2.05, the ratio is approximately 0.487, and at r=2.1, it's approximately 0.476, our target of 0.4848 lies between these. Interpolating: \[ r \approx 2.05 + \left( \frac{0.487 - 0.4848}{0.487 - 0.476} \right) \times (2.1 - 2.05) \approx 2.05 + \left( \frac{0.0022}{0.011} \right) \times 0.05 \approx 2.05 + 0.01 \approx 2.06 \] Expressed as a percentage: \[ IRR \approx 206\% \] **Therefore, the expected IRR is approximately 206%.** --- ### 6. **Expected NPV Calculation** Using the expected after-tax cash flow of \$8,250,000 per year for 6 years, initial outlay \$4,000,000, and discount rate 12% (cost of capital): \[ NPV = -4,000,000 + 8,250,000 \times \text{PVIFA}_{12\%,6} \] PVIFA for 12% over 6 years: \[ PVIFA_{12\%,6} = \frac{1 - (1 + 0.12)^{-6}}{0.12} \] Calculate: \[ (1.12)^6 \approx 1.9738 \] \[ (1.12)^{-6} = 1 / 1.9738 \approx 0.5067 \] \[ PVIFA_{12\%,6} = \frac{1 - 0.5067}{0.12} = \frac{0.4933}{0.12} \approx 4.111 \] Calculate NPV: \[ NPV = -4,000,000 + 8,250,000 \times 4.111 \approx -4,000,000 + 33,929,775 = \$29,929,775 \] **Final answer:** ### **Expected IRR: approximately 206.00%** ### **NPV: approximately \$29,929,775** --- ### **Summary:** | Item | Value | |---------|------------------------------| | Expected IRR | **206.00%** | | Project NPV | **\$29,929,775** | --- ### **Additional Note:** **Can you determine the value of σIRR short of actual simulation or complex statistical analysis?** **Answer: No** — Without simulation or advanced statistical methods, it's difficult to accurately determine the standard deviation or σIRR. --- **Final note:** These calculations are based on expected values and the assumption of annual cash flows as profits. Actual project analysis would consider distributions, variances, and more detailed modeling.