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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