Let's solve the problem step by step. --- ## **1. Compute the current price of both securities.** ### **A. Corporate Bond** - **Coupon rate:** 6% - **Face value (assumed):** $100 - **Years to maturity:** 5 - **Yield to Maturity (YTM):** 4.73% **Price of bond** (\( P \)) is given by the present value of coupons plus the present value of the face value: \[ P = \sum_{t=1}^{5} \frac{C}{(1+y)^t} + \frac{F}{(1+y)^5} \] Where: - \( C = 6 \) (annual coupon, 6% of $100) - \( y = .0473 \) - \( F = 100 \) \[ P = 6 \left(\frac{1 - (1+.0473)^{-5}}{.0473}\right) + \frac{100}{(1.0473)^5} \] Calculate each part: - \((1.0473)^5 = 1.2621\) - \( \frac{1}{1.2621} = .7926 \) - \( 1 - .7926 = .2074 \) - \( \frac{.2074}{.0473} = 4.385 \) - \( 6 \times 4.385 = 26.31 \) - \( \frac{100}{1.2621} = 79.26 \) Total Price: \[ P = 26.31 + 79.26 = \$105.57 \] --- ### **B. Delta Corporation Stock** - **Dividend (\( D \)):** $.95 per share - **Required return (\( r \)):** 4.73% - **Current Price (\( P_ \)):** $41.30 (given, let's verify) Using the **Dividend Discount Model** (constant dividend): \[ P_ = \frac{D}{r} \] \[ P_ = \frac{.95}{.0473} = \$20.09 \] But the current market price is $41.30, so the market is using a lower required return or expecting dividend growth. But since the problem says dividends are constant, and the required return is 4.73%, the theoretical price using these numbers is $20.09. We'll use **$41.30** as the market price as specified in the problem. --- ## **2. Table: Expected Price under Recession and Normal State** | Security | Current Price | Price in Recession | Price in Normal | |---------------|--------------|--------------------|----------------| | Bond | \$105.57 | ? | ? | | Delta Stock | \$41.30 | ? | ? | #### Bond Prices: - **Recession:** YTM drops to 4%. Price rises. - **Normal:** YTM stays at 4.73%. Price unchanged. **Bond price in recession (\( y = .04 \)):** \[ P = 6 \left(\frac{1 - (1+.04)^{-5}}{.04}\right) + \frac{100}{(1.04)^5} \] - \((1.04)^5 = 1.2167\) - \( \frac{1}{1.2167} = .8217 \) - \( 1 - .8217 = .1783 \) - \( \frac{.1783}{.04} = 4.457 \) - \( 6 \times 4.457 = 26.74 \) - \( \frac{100}{1.2167} = 82.17 \) \[ P = 26.74 + 82.17 = \$108.91 \] **Bond price in normal:** Stays at \$105.57. --- #### Stock Prices: - **Recession:** Required return increases by 20%. New required return = 4.73% × 1.2 = **5.676%** \[ P = \frac{.95}{.05676} = \$16.74 \] - **Normal:** Required return drops by 10%. New required return = 4.73% × .9 = **4.257%** \[ P = \frac{.95}{.04257} = \$22.32 \] --- **Updated Table:** | Security | Current Price | Price in Recession | Price in Normal | |---------------|--------------|--------------------|----------------| | Bond | \$105.57 | \$108.91 | \$105.57 | | Delta Stock | \$41.30 | \$16.74 | \$22.32 | --- ## **3. Table: Expected Returns under Recession and Normal** \[ \text{Return} = \frac{\text{End Price} - \text{Start Price}}{\text{Start Price}} \] | Security | Recession Return | Normal Return | |---------------|--------------------------------|-------------------------------| | Bond | \((108.91-105.57)/105.57 = 3.17\%\) | \((105.57-105.57)/105.57 = \%\) | | Delta Stock | \((16.74-41.30)/41.30 = -59.48\%\) | \((22.32-41.30)/41.30 = -45.95\%\) | --- ## **4. Expected Return on Portfolio** - **Weights:** 50% Bond, 50% Stock - **Probabilities:** Recession = 40%, Normal = 60% Calculate portfolio returns in each state: - **Recession:** \(.5 \times 3.17\% + .5 \times (-59.48\%) = -28.16\%\) - **Normal:** \(.5 \times \% + .5 \times (-45.95\%) = -22.98\%\) Expected portfolio return: \[ E(R) = .4 \times (-28.16\%) + .6 \times (-22.98\%) = -11.26\% + (-13.79\%) = -25.05\% \] --- ## **5. Will you earn the expected return?** **No**, you will **not necessarily** earn the expected return. The actual return will depend on the realized state of the economy: - If the economy is in recession: you earn **-28.16%** - If the economy is normal: you earn **-22.98%** - The expected return is just the probability-weighted average, **-25.05%**, not a guaranteed outcome. --- ## **Summary Table** | State | Bond Return | Stock Return | Portfolio Return | |------------|-------------|--------------|------------------| | Recession | 3.17% | -59.48% | -28.16% | | Normal | % | -45.95% | -22.98% | | **Expected** | | | **-25.05%** | --- ### **Final Answers (Summary):** 1. **Current Price:** Bond = \$105.57; Stock = \$41.30 2. **Expected Prices:** - Recession: Bond = \$108.91, Stock = \$16.74 - Normal: Bond = \$105.57, Stock = \$22.32 3. **Expected Returns:** - Recession: Bond = 3.17%, Stock = -59.48% - Normal: Bond = %, Stock = -45.95% 4. **Expected Portfolio Return:** **-25.05%** 5. **Will you earn this return?** No, actual return will be -28.16% (recession) or -22.98% (normal), depending on the state of the economy. The expected return is a weighted average, not a guaranteed outcome.