Let's break down the problem and answer step-by-step, as requested. --- ### **Given Data** - **Depreciation of old computer:** \$281,000 per year - **Old computer is fully written off in:** 3 years - **Can sell old computer now for:** \$450,000 - **Can sell old computer in 2 years for:** \$130,000 - **Operating cost savings from new computer:** \$315,000 per year - **Tax rate:** 22% - **Discount rate:** 12% --- ## **Step 1: Calculating EAC (Equivalent Annual Cost) for Old and New Computer** ### **A. EAC of Old Computer** #### **1. Cash flows if we keep the old computer for 2 years:** - **Year 0:** No purchase (already owned), but opportunity cost is the foregone sale value: **\(-\$450,000\)** - **Year 1 & 2:** No explicit data for operating costs, so ignore or assume negligible incremental cost unless specified. The main opportunity cost is the lost sale value. - **Year 2:** Sell for \$130,000 (cash inflow) #### **2. Net Cash Flows:** | Year | Cash Flow | |------|--------------| | 0 | \(-\$450,000\) | | 2 | \(+\$130,000\) | #### **3. NPV of Keeping Old Computer (Opportunity Cost):** \[ \text{NPV}_{\text{old}} = -450,000 + \frac{130,000}{(1.12)^2} \] \[ \text{NPV}_{\text{old}} = -450,000 + \frac{130,000}{1.2544} \] \[ \text{NPV}_{\text{old}} = -450,000 + 103,666 \] \[ \text{NPV}_{\text{old}} = -346,334 \] #### **4. EAC Formula:** For a 2-year life: \[ \text{EAC} = \frac{\text{NPV}}{\text{PVIFA}(r, n)} \] Where PVIFA (Present Value Interest Factor for Annuity) = \(\frac{1 - (1 + r)^{-n}}{r}\) For 12% and 2 years: \[ \text{PVIFA}(12\%, 2) = \frac{1 - (1.12)^{-2}}{0.12} = \frac{1 - 0.7972}{0.12} = \frac{0.2028}{0.12} = 1.690 \] \[ \text{EAC}_{\text{old}} = \frac{-346,334}{1.690} = -\$204,997 \text{ per year} \] --- ### **B. EAC of New Computer** #### **1. Savings from new computer:** - Operating cost savings: \$315,000 per year (pre-tax) - Tax shield: 22% of savings \[ \text{After-tax savings} = 315,000 \times (1 - 0.22) = 315,000 \times 0.78 = \$245,700 \text{ per year} \] #### **2. Opportunity cost (foregone sale of old computer):** - Cost to "buy" new is also the lost opportunity to sell old: **Initial outlay = \$450,000** #### **3. Net Cash Flows:** Assume the new computer lasts at least as long as the analysis period (2 years, for comparison). | Year | Cash Flow | |------|--------------------------| | 0 | \(-\$450,000\) | | 1 | \(+\$245,700\) | | 2 | \(+\$245,700\) | Assume no residual value (as not specified). #### **4. NPV of New Computer:** \[ \text{NPV}_{\text{new}} = -450,000 + \frac{245,700}{1.12} + \frac{245,700}{(1.12)^2} \] \[ \text{NPV}_{\text{new}} = -450,000 + 219,375 + 195,871 \] \[ \text{NPV}_{\text{new}} = -450,000 + 415,246 = -\$34,754 \] #### **5. EAC of New Computer:** \[ \text{EAC}_{\text{new}} = \frac{\text{NPV}}{\text{PVIFA}(12\%, 2)} = \frac{-34,754}{1.690} = -\$20,570 \text{ per year} \] --- ### **a-1. Final Answers:** **EAC (Equivalent Annual Cost):** - **Old computer:** \(-\$204,997\) per year - **New computer:** \(-\$20,570\) per year --- ## **Step 2: NPV of Replacing the Computer Now** \[ \text{NPV}_{\text{replace now}} = \text{NPV}_{\text{new}} - \text{NPV}_{\text{old}} \] Since both NPV values are negative (they represent costs), but the new computer is less costly: \[ \text{NPV}_{\text{replace now}} = (-34,754) - (-346,334) = 311,580 \] **Interpretation:** A positive NPV means replacing now is **better** by \$311,580 (in present value terms) over the next two years compared to waiting. --- ### **a-2. Final Answer:** - **NPV of replacing the computer now:** **\$311,580** --- ## **Summary Table** | Option | NPV | EAC (per year) | |---------------|-------------|---------------| | Old Computer | \(-\$346,334\) | \(-\$204,997\) | | New Computer | \(-\$34,754\) | \(-\$20,570\) | | Replace Now | **\$311,580** | — | --- ### **Explanation of Each Step** 1. **Calculate NPV for both options** using the relevant cash flows (opportunity cost for old; cost & after-tax savings for new). 2. **Calculate EAC** by spreading the NPV over the analysis period (2 years) using the present value annuity factor. 3. **Compare EACs** to see which is lower (new computer). 4. **Find the NPV of the replacement decision**: The difference in NPVs shows the net benefit of switching now. --- ### **Final Recommendation** **Replace the old computer now**. The new computer has a much lower equivalent annual cost, and the NPV of replacing now is positive (\$311,580), indicating it's the economically better choice.

Let's approach this problem step-by-step, building from the given data, applying the Black-Scholes model, and considering the dividend payment. We will clarify each part and finally compute the value of the European call option. --- ### **Step 1: Understanding the Given Data** - **Current stock price, \( S_0 \):** \$64 - **Dividend:** \$2 paid in 2 months - **Time to expiration, \( T \):** 3 months - **Exercise price, \( K \):** \$58 - **Risk-free interest rate per month, \( r \):** 0.7% or 0.007 - **Volatility per month, \( \sigma \):** 8% or 0.08 --- ### **Step 2: Adjusting for the dividend** Dividends affect the stock price by reducing it at the dividend date. - Since the dividend is paid in 2 months, the **stock price just before dividend payment** is expected to drop by \$2 at that time. - **Net-of-dividend stock price at time 0:** To price an option considering dividends, we can think of the **expected stock price after dividend payment** as: \[ S_{d} = S_0 - D = 64 - 2 = 62 \] But in Black-Scholes, we typically adjust the stock price for dividends by discounting the dividend or by adjusting the stock price. Alternatively, for a single dividend, the **approach is to model the stock price as**: \[ S_{0}^{\text{adj}} = S_0 \times e^{-q T} \] where \( q \) is the dividend yield. Since we have a discrete dividend, not a continuous yield, a common approximation is to reduce the current stock price by the present value of the dividend: \[ S_{0}^{\text{adj}} = S_0 - D \times e^{-r \times t_D} \] where \( t_D = 2 \) months, \( r = 0.007 \) per month. Calculate the present value of the dividend: \[ PV(D) = 2 \times e^{-0.007 \times 2} = 2 \times e^{-0.014} \approx 2 \times 0.9861 = 1.972 \] Adjusted stock price: \[ S_{0}^{\text{adj}} = 64 - 1.972 \approx 62.03 \] **Note:** Since the dividend is paid in 2 months, and the option expires in 3 months, the stock price at expiration will be affected after 2 months, so the pricing model must account for this. --- ### **Step 3: Model setup** **We will treat the problem as**: - The stock price today is approximately \$62.03 (adjusted for dividend). - The dividend reduces the stock price by \$2 at 2 months. - The option expires in 3 months. For simplicity, assume **no early exercise** (European call). The problem asks for the Black-Scholes value, which applies to European options. --- ### **Step 4: Adjusting the parameters for Black-Scholes** The standard Black-Scholes formula requires: \[ d_1 = \frac{\ln(S/K) + (r + \frac{\sigma^2}{2}) T}{\sigma \sqrt{T}} \] \[ d_2 = d_1 - \sigma \sqrt{T} \] where: - \( S \) is the current stock price - \( K \) is the strike price - \( r \) is the risk-free rate (per period) - \( T \) is time to expiration (in years) - \( \sigma \) is volatility (per year) Given the data are monthly, convert to annual: \[ \text{Monthly to annual:} \quad T_{\text{annual}} = 3 \text{ months} \times 12 = 0.25 \text{ years} \] Similarly, volatility per year: \[ \sigma_{\text{annual}} = 0.08 \times \sqrt{12} \approx 0.08 \times 3.4641 \approx 0.2771 \] But the problem suggests defining one period as a month, so we can proceed with months. --- ### **Step 5: Calculate \( d_1 \) and \( d_2 \)** Using monthly parameters: \[ d_1 = \frac{\ln(S/K) + (r + \frac{\sigma^2}{2}) T}{\sigma \sqrt{T}} \] \[ d_2 = d_1 - \sigma \sqrt{T} \] Where: - \( S = 62.03 \) - \( K = 58 \) - \( r = 0.007 \) per month - \( T=3 \) months - \( \sigma = 0.08 \) Calculate: \[ \ln(S/K) = \ln(62.03/58) \approx \ln(1.069) \approx 0.0668 \] Compute numerator: \[ r T + \frac{\sigma^2 T}{2} = 0.007 \times 3 + \frac{(0.08)^2 \times 3}{2} = 0.021 + \frac{0.0064 \times 3}{2} = 0.021 + \frac{0.0192}{2} = 0.021 + 0.0096 = 0.0306 \] Calculate denominator: \[ \sigma \sqrt{T} = 0.08 \times \sqrt{3} \approx 0.08 \times 1.732 = 0.1386 \] Now, \( d_1 \): \[ d_1 = \frac{0.0668 + 0.0306}{0.1386} = \frac{0.0974}{0.1386} \approx 0.703 \] And \( d_2 \): \[ d_2 = 0.703 - 0.1386 = 0.564 \] --- ### **Step 6: Calculate the option price** Using the Black-Scholes formula: \[ C = S \times N(d_1) - K \times e^{-r T} \times N(d_2) \] Where: - \( N(d) \) is the cumulative distribution function (CDF) of the standard normal distribution. Find \( N(d_1) \) and \( N(d_2) \): - \( N(0.703) \approx 0.760 \) - \( N(0.564) \approx 0.714 \) Calculate: \[ e^{-r T} = e^{-0.007 \times 3} = e^{-0.021} \approx 0.979 \] Now the price: \[ C = 62.03 \times 0.760 - 58 \times 0.979 \times 0.714 \] Compute: \[ 62.03 \times 0.760 \approx 47.13 \] \[ 58 \times 0.979 \approx 56.76 \] \[ 56.76 \times 0.714 \approx 40.56 \] Finally: \[ C = 47.13 - 40.56 = \boxed{\$6.57} \] --- ### **Final Answer:** **The Black-Scholes value of the European call option is approximately \(\boxed{\$6.57}\).** --- ### **Summary:** - Adjusted stock price for dividends: approximately \$62.03 - Calculated \( d_1 \) and \( d_2 \) using monthly parameters - Used the Black-Scholes formula to compute the call value - **Result:** **\$6.57** This approach considers the dividend payment's impact and the time frame, leading to an accurate valuation based on the given data.