Top hedge fund manager Sally Buffit believes that a stock with the same market risk as the S&P 500 will sell at year-end at a price of $44. The stock will pay a dividend at year-end of $2.00. Assume that risk-free Treasury securities currently offer an interest rate of 1.9%.

Average rates of return on Treasury bills, government bonds, and common stocks, 1900–2020 (figures in percent per year) are as follows.

Portfolio	Average Annual Rate of Return (%)	Average Premium (Extra return versus Treasury bills) (%)
Treasury bills	3.7	 
Treasury bonds	5.4	1.7
Common stocks	11.5	7.7
What is the discount rate on the stock?

Note: Enter your answer as a percent rounded to 2 decimal places.

What price should she be willing to pay for the stock today?

Note: Do not round intermediate calculations. Round your answer to 2 decimal places.

We will calculate the discount rate and the price Sally Buffit should be willing to pay for the stock today using the Capital Asset Pricing Model (CAPM) and the dividend discount model.

### Step 1: Calculate the Discount Rate

The Capital Asset Pricing Model (CAPM) formula is as follows:

\[
r = r_f + \beta \times (r_m - r_f)
\]

Where:
- $ r $ is the discount rate or required return.
- $ r_f $ is the risk-free rate.
- $ \beta $ is the stock's beta (since it's the same as the market, $ \beta = 1 $).
- $ r_m $ is the expected market return.
- $ (r_m - r_f) $ is the market risk premium.

From the problem:
- $ r_f = 1.9\% $ (risk-free Treasury rate).
- The market risk premium $ (r_m - r_f) = 7.7\% $.

Now, applying the CAPM formula:

\[
r = 1.9\% + 1 \times 7.7\% = 1.9\% + 7.7\% = 9.6\%
\]

So, the discount rate is:

\[
r = 9.60\%
\]

### Step 2: Calculate the Price of the Stock Today

Using the dividend discount model (DDM) for a stock that pays dividends:

\[
P_0 = \frac{D_1 + P_1}{1 + r}
\]

Where:
- $ P_0 $ is the price today.
- $ D_1 = 2.00 $ (dividend at year-end).
- $ P_1 = 44.00 $ (expected price at year-end).
- $ r = 9.6\% $ (discount rate, as calculated above).

Now, let's substitute the values:

\[
P_0 = \frac{2.00 + 44.00}{1 + 0.096} = \frac{46.00}{1.096}
\]

\[
P_0 = 41.97
\]

### Final Answers:
- The discount rate is **9.60%**.
- The price Sally Buffit should be willing to pay for the stock today is **$41.97**.

Would you like to explore further details or ask any questions?

Here are 5 related questions:
1. How does changing the market risk premium affect the stock price?
2. What would happen to the price if the dividend were expected to grow annually?
3. Can you calculate the stock price with a different beta value?
4. How does the CAPM model compare with other models for stock pricing?
5. What is the relationship between the risk-free rate and stock valuations?

**Tip:** The CAPM assumes that all risk is systematic; diversifiable risks are ignored. Always check if this assumption holds when using CAPM.

Top hedge fund manager Sally Buffit believes that a stock with the same market risk as the S&P 500 will sell at year-end at a price of $44. The stock will pay a dividend at year-end of $2.00. Assume that risk-free Treasury securities currently offer an interest rate of 1.9%. Average rates of return on Treasury bills, government bonds, and common stocks, 1900–2020 (figures in percent per year) are as follows. Portfolio Average Annual Rate of Return (%) Average Premium (Extra return versus Treasury bills) (%) Treasury bills 3.7 Treasury bonds 5.4 1.7 Common stocks 11.5 7.7 What is the discount rate on the stock? Note: Enter your answer as a percent rounded to 2 decimal places. What price should she be willing to pay for the stock today? Note: Do not round intermediate calculations. Round your answer to 2 decimal places.

Learn how to calculate the discount rate and the current stock price using the Capital Asset Pricing Model (CAPM) and the Dividend Discount Model (DDM). This problem uses key finance concepts like market risk premium and the dividend discount method. The solution is suitable for undergraduate finance or business students.

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