Question

If a stock has a \(\beta\) of \(1.5,\) the return on the market is \(10 \%,\) and the riskfree rate of return is \(5 \%,\) what expected rate of return should this stock offer according to the Capital Asset Pricing Model? If the expected value of the stock is \(\$ 100,\) what price should the stock be selling for today?

Short answer

The expected rate of return is 12.5% and the stock price should be $88.89 today.

Step-by-step solution

Understanding the Problem

We need to calculate the expected rate of return on a stock using the Capital Asset Pricing Model (CAPM). The CAPM formula is: \( E(R_i) = R_f + \beta_i (E(R_m) - R_f) \). We are given: \( \beta = 1.5 \), \( R_f = 5\% \), and \( E(R_m) = 10\% \).

Calculate Expected Rate of Return

Substitute the given values into the CAPM formula: \( E(R_i) = 5\% + 1.5(10\% - 5\%) \). This simplifies to \( E(R_i) = 5\% + 1.5 \times 5\% \). Calculate \( 1.5 \times 5\% = 7.5\% \). Add this to the risk-free rate: \( 5\% + 7.5\% = 12.5\% \). So, the expected rate of return \( E(R_i) = 12.5\% \).

Determine Present Value of the Stock

We know the expected future value of the stock is \\(100, and the expected rate of return is \(12.5\%\). To find the present value \( PV \), use the formula \( PV = \frac{FV}{1 + E(R_i)} \) where \( FV = \\)100 \). Substitute the values: \( PV = \frac{100}{1 + 0.125} = \frac{100}{1.125} \).

Calculate Present Value

Divide \( 100 \) by \( 1.125 \) to calculate the present value: \( PV = 88.89 \). This means the stock should be selling for approximately \$88.89 today.

Key concepts

Beta Coefficient

The beta coefficient, often denoted as \( \beta \), is an important metric in finance that measures a stock's volatility compared to the market as a whole. It helps investors understand how much risk they are taking on when investing in a particular stock.

• Market Benchmark: A beta of 1 indicates a stock's price will move with the market. If the market goes up by 10%, the stock also tends to go up by 10%.
• Higher Beta: A beta greater than 1, like 1.5 in our example, means the stock is more volatile than the market. It should offer a higher potential return to compensate for the extra risk.
• Lower Beta: Conversely, a beta less than 1 implies the stock is less volatile than the market and generally considered less risky.

This concept is crucial in the Capital Asset Pricing Model (CAPM), which uses beta to calculate expected returns, balancing both potential risk and reward for investors.

Risk-Free Rate of Return

The risk-free rate of return, symbolized as \( R_f \), represents the theoretical return on an investment with no risk of financial loss. It is usually based on government bonds from stable countries, like U.S. Treasury Bills, because they are seen as almost risk-free.

• Benchmark for Safe Investments: This rate serves as a baseline for evaluating the attractiveness of riskier investments. For instance, if an investment's expected return is less than the risk-free rate, it may not be worthwhile.
• Part of CAPM: In the CAPM formula, the risk-free rate is added to the risk premium, which is the product of beta and the difference between the market return and the risk-free rate. This helps establish the required return on equity.

In the context of this problem, the risk-free rate is 5%, which serves as the foundation for calculating the stock's expected return, ensuring investors are compensated for any additional risk they undertake.

Expected Rate of Return

The expected rate of return, denoted as \( E(R_i) \), is what investors anticipate earning from an investment over a period, taking into account all risk factors. In the framework of the CAPM, it is calculated using the formula:\[ E(R_i) = R_f + \beta_i (E(R_m) - R_f) \]

• Components: The formula considers the risk-free rate, the stock's beta, and the market return to derive the expected return.
• Risk Premium: The part \( \beta_i \times (E(R_m) - R_f) \) represents the risk premium, which compensates investors for assuming extra risk.
• Example Application: With a beta of 1.5, a risk-free rate of 5%, and a market return of 10%, the expected return is calculated to be 12.5% in the given scenario.

This expected rate of return guides investors in decision-making, offering a benchmark for comparing potentially risky investments against safer alternatives.