Stock Investment Analysis At Equilibrium

Problem:

A stock with a beta of .75 currently sells for $50. Investors expect the stock to pay a year-end dividend of $2. The T-bill rate is 4%, and the market risk premium is 7%. If the stock is perceived to be fairly priced today, what must be investors’ expectation of the price of the stock at the end of the year?

Solution:

We’re given a stock priced at $50 today, with the following details:

Given:

• Beta, $\beta = 0.75$
  

• Current price, $P_0 = 50$
  

• Expected dividend at year-end, $D_1 = 2$
  

• Risk-free rate, $R_f = 4\% = 0.04$
  

• Market risk premium, $R_m - R_f = 7\% = 0.07$
  

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Step 1: Calculate Required Rate of Return using CAPM

$$r = R_f + \beta (R_m - R_f)$$
$$r = 0.04 + 0.75(0.07) = 0.04 + 0.0525 = \boxed{0.0925 \text{ or } 9.25\%}$$

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Step 2: Use the Required Return Formula

For a fairly priced stock:

$$P_0 = \frac{D_1 + P_1}{1 + r}$$

Where:

• $P_1$ is the expected price at the end of the year

Rearranging to solve for $P_1$:

$$P_1 = (1 + r) \cdot P_0 - D_1$$ $$P_1 = (1 + 0.0925) \cdot 50 - 2 = 1.0925 \cdot 50 - 2 = 54.625 - 2 = \boxed{\$52.625}$$

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✅ Answer:

Investors must expect the stock price at year-end to be $52.63 (rounded) if the stock is fairly priced today.

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Firm Valuation With Uncertain Cash Flows

Problem:

You are considering acquiring a firm that you believe can generate expected cash flows of $10,000 a year forever. However, you recognize that those cash flows are uncertain. a. Suppose you believe that the beta of the firm is .4. How much is the firm worth if the risk free rate is 4% and the expected rate of return on the market portfolio is 11%? b. By how much will you overvalue the firm if its beta is actually .6?

Solution:

To value the firm using the Capital Asset Pricing Model (CAPM) and a perpetuity formula, we follow these steps:

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Given:

• Expected cash flows = $10,000 per year forever

• $R_f = 4\% = 0.04$
  

• $R_m = 11\% = 0.11$
  

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Step 1: Use CAPM to Find the Required Return $r$

$$r = R_f + \beta (R_m - R_f)$$

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a. Valuation if β = 0.4

$$r = 0.04 + 0.4(0.11 - 0.04) = 0.04 + 0.4(0.07) = 0.04 + 0.028 = 0.068 = 6.8\%$$

Now value the firm using the perpetuity formula:

$$\text{Value} = \frac{\text{Cash Flow}}{r} = \frac{10,000}{0.068} \approx \boxed{\$147,059}$$

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b. If β = 0.6, what is the actual value?

$$r = 0.04 + 0.6(0.07) = 0.04 + 0.042 = 0.082 = 8.2\%$$
$$\text{True Value} = \frac{10,000}{0.082} \approx \boxed{\$121,951}$$

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Overvaluation:

$$\text{Overvaluation} = 147,059 - 121,951 = \boxed{\$25,108}$$

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✅ Final Answers:

• a. Firm value with β = 0.4: $147,059

• b. Overvaluation if true β = 0.6: $25,108

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Investment Portfolio And Stock Return Analysis

Problem;

Suppose you invest $400,000 in Treasury bills and $600,000 in the market portfolio. What is the return on your portfolio if bills yield 3% and the expected return on the market is 10%? What does the return on this portfolio imply for the expected return on individual stocks with betas of .6?

Let's break this down into two parts:

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Part 1: Portfolio Return Calculation

You invested:

• $400,000 in Treasury bills (risk-free asset) yielding 3%

• $600,000 in the market portfolio with an expected return of 10%

The return on the total portfolio is the weighted average of these two:

$$\text{Portfolio Return} = (w_{\text{bills}} \times R_f) + (w_{\text{market}} \times R_m)$$

Where:

• $R_f = 3\%$
  

• $R_m = 10\%$
  

• $w_{\text{bills}} = \frac{400,000}{1,000,000} = 0.4$
  

• $w_{\text{market}} = \frac{600,000}{1,000,000} = 0.6$
  

$$\text{Portfolio Return} = (0.4 \times 0.03) + (0.6 \times 0.10) = 0.012 + 0.06 = 0.072 = \boxed{7.2\%}$$

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Part 2: Expected Return on Individual Stocks with Beta = 0.6

We use the Capital Asset Pricing Model (CAPM):

$$E(R_i) = R_f + \beta_i (R_m - R_f)$$

Where:

• $R_f = 3\%$
  

• $R_m = 10\%$
  

• $\mathrm{<span\; style="font-size:\; medium;"><b>\beta </b></span>}<span\; style="font-size:\; medium;"><b>=</b></span>\mathrm{<span\; style="font-size:\; medium;"><b>0.6</b></span>}$

$$E(R_i) = 0.03 + 0.6 \times (0.10 - 0.03) = 0.03 + 0.6 \times 0.07 = 0.03 + 0.042 = \boxed{7.2\%}$$

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Conclusion

• Your portfolio return is 7.2%.

• A stock with a beta of 0.6 also has an expected return of 7.2%.

This means your portfolio has the same systematic risk (beta = 0.6) as a stock with beta 0.6, and under CAPM assumptions, you'd expect the same return—showing consistency between the portfolio's composition and the expected return for assets with similar risk.

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Stock Valuation Infographic With Expected Return

Problem:

A stock with a beta of .75 currently sells for $50. Investors expect the stock to pay a year-end dividend of $2. The T-bill rate is 4%, and the market risk premium is 7%. If the stock is perceived to be fairly priced today, what must be investors’ expectation of the price of the stock at the end of the year?

We’re given a stock priced at $50 today, with the following details:

Given:

• Beta, $\beta = 0.75$
  

• Current price, $P_0 = 50$
  

• Expected dividend at year-end, $D_1 = 2$
  

• Risk-free rate, $R_f = 4\% = 0.04$
  

• Market risk premium, $R_m - R_f = 7\% = 0.07$
  

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Step 1: Calculate Required Rate of Return using CAPM

$$r = R_f + \beta (R_m - R_f)$$
$$r = 0.04 + 0.75(0.07) = 0.04 + 0.0525 = \boxed{0.0925 \text{ or } 9.25\%}$$

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Step 2: Use the Required Return Formula

For a fairly priced stock:

$$P_0 = \frac{D_1 + P_1}{1 + r}$$

Where:

• $P_1$ is the expected price at the end of the year

Rearranging to solve for $P_1$:

P1​ = (1+0.0925)⋅50−2 = 1.0925⋅50−2 = 54.625−2 = $52.625​

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✅ Answer:

Investors must expect the stock price at year-end to be $52.63 (rounded) if the stock is fairly priced today.

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 Financial Analysis Of Nuclear Plant Project

Problem:

A proposed nuclear power plant will cost $2.2 billion to build and then will produce cash flows of $300 million a year for 15 years. After that period (in year 15), it must be decommissioned at a cost of $900 million. a. What is project NPV if the discount rate is 5%?  b. What if the discount rate is 18%?

Solution:

To calculate the Net Present Value (NPV) of the nuclear power plant project, we follow these steps:

1. Cash inflows:

   • The plant produces cash flows of $300 million per year for 15 years.

2. Cash outflows:

   • The initial cost of building the plant is $2.2 billion (incurred at time $t = 0$).

   • The decommissioning cost of $900 million is incurred in year 15.

3. Discount rate:

   • We calculate the NPV for two discount rates: 5% and 18%.

Formula:

The NPV formula is:

$$NPV = \sum \left( \frac{C_t}{(1 + r)^t} \right) - C_0$$

Where:

• $C_t$ = Cash flow at time $t$(inflows and outflows)

• $r$ = Discount rate

• $C_0$ = Initial cost (at time $t = 0$)

• $t$ = Time period

Step-by-Step Calculation:

• Initial Cost $C_0 = -2.2$ billion (outflow)

• Annual Cash Flow $C_1$ to $C_{15} = 300$ million

• Decommissioning Cost in Year 15 = $-900$ million

Let's compute the NPV using both discount rates.

The Net Present Value (NPV) of the nuclear power plant project is:

• At a 5% discount rate: $480.98 million

• At an 18% discount rate: -$747.69 million

This means that the project is financially viable at a 5% discount rate but results in a negative NPV at an 18% discount rate, indicating it would not be profitable at the higher rate.

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