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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 Stock Return Breakdown With Scenarios

Problem:

A stock is selling today for $40 per share. At the end of the year, it pays a dividend of $2 per share and sells for $44. a. What is the total rate of return on the stock? b. What are the dividend yield and percentage capital gain? c. Now suppose the year-end stock price after the dividend is paid is $36. What are the dividend yield and percentage capital gain in this case?

To solve this, we need to calculate the total rate of return, dividend yield, and percentage capital gain. Let's work through each part.

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Part a: Total rate of return

The total rate of return ($R$) is given by:

$$R = \frac{\text{Dividend} + (\text{End Price} - \text{Initial Price})}{\text{Initial Price}}$$

Here:

• Initial Price = $40

• Dividend = $2

• End Price = $44

$$R = \frac{2 + (44 - 40)}{40} = \frac{2 + 4}{40} = \frac{6}{40} = 0.15 \text{ or } 15\%.$$

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Part b: Dividend yield and percentage capital gain

1. Dividend Yield ($DY$):
   The dividend yield is calculated as:

   $$DY = \frac{\text{Dividend}}{\text{Initial Price}}$$ $$DY = \frac{2}{40} = 0.05 \text{ or } 5\%.$$
   

2. Percentage Capital Gain ($CG$):
   The percentage capital gain is:

   $$CG = \frac{\text{End Price} - \text{Initial Price}}{\text{Initial Price}}$$ $$CG = \frac{44 - 40}{40} = \frac{4}{40} = 0.10 \text{ or } 10\%.$$
   

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Part c: Dividend yield and percentage capital gain (End Price = $36)

1. Dividend Yield ($DY$):
   The formula remains the same:

   $$DY = \frac{\text{Dividend}}{\text{Initial Price}}$$ $$DY = \frac{2}{40} = 0.05 \text{ or } 5\%.$$
   

2. Percentage Capital Gain ($CG$):
   In this case, the End Price is $36. The percentage capital gain is:

   $$CG = \frac{\text{End Price} - \text{Initial Price}}{\text{Initial Price}}$$ $$CG = \frac{36 - 40}{40} = \frac{-4}{40} = -0.10 \text{ or } -10\%.$$
   

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Summary of Results:

Scenario	Total Rate of Return	Dividend Yield	Percentage Capital Gain
End Price =$44	15%	5%	10%
End Price = $36 (after dividend)	—	5%	-10%

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