Math Problem Statement

A stock with a beta of 1.8 has an expected rate of return of 16%. If the market return this year turns out to be 6 percentage points below expectations, what is your best guess as to the rate of return on the stock?

Solution

We can solve this problem using the Capital Asset Pricing Model (CAPM) formula, which relates the expected return of a stock to its beta and the market's expected return:

Ri=Rf+βi×(RmRf)R_i = R_f + \beta_i \times (R_m - R_f)

Where:

  • RiR_i is the expected return on the stock.
  • RfR_f is the risk-free rate.
  • βi\beta_i is the beta of the stock (which measures the stock's sensitivity to market movements).
  • RmR_m is the market return.

Now, in this case, we are told that the market return will be 6 percentage points below expectations, meaning the actual market return (RmR_m) will be:

Rm6%R_m - 6\%

Given the stock's beta of 1.8, this means that for every percentage point the market return changes, the stock return should change by 1.8×market change1.8 \times \text{market change}.

Step-by-step solution:

  1. The change in market return is 6%-6\%.
  2. The stock's return will adjust by 1.8×(6%)=10.8%1.8 \times (-6\%) = -10.8\%.
  3. The expected rate of return for the stock was initially 16%. Since the market return is worse than expected, we subtract the 10.8% adjustment: New stock return=16%10.8%=5.2%\text{New stock return} = 16\% - 10.8\% = 5.2\%

Thus, your best guess for the rate of return on the stock is 5.2%.


Would you like any further clarification or details on this?

Here are 5 questions related to this topic:

  1. How does beta affect a stock’s risk level compared to the market?
  2. What would happen to the stock's return if the market exceeded expectations?
  3. How does the risk-free rate influence the CAPM formula?
  4. Can a stock have a negative beta, and what would that imply?
  5. How would changing the market’s expected return impact the stock's performance?

Tip: A beta above 1 means the stock is more volatile than the market, while a beta below 1 indicates less volatility.

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Math Problem Analysis

Mathematical Concepts

Finance
Capital Asset Pricing Model (CAPM)
Stock Market Volatility

Formulas

R_i = R_f + β_i × (R_m - R_f)
Stock return adjustment = β × Market return change

Theorems

Capital Asset Pricing Model (CAPM)

Suitable Grade Level

Undergraduate Finance/Economics