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A Stock Has A Beta Of 2.57 And An Expected Return Of 28.0%. The T-bill Rate Is 7.6%. Calculate The Portfolio

A Stock Has A Beta Of 2.57 And An Expected Return Of 28.0%. The T-bill Rate Is 7.6%. Calculate The Portfolio

Understanding how to calculate the expected return of a portfolio is a fundamental aspect of investment analysis, especially for investors aiming to optimize their asset allocation. When given specific data points such as a stock’s beta, its expected return, and the risk-free rate (represented here by T-bills), investors can utilize various financial models to determine the appropriate portfolio composition. This article provides a comprehensive guide to calculating the expected return of a portfolio, emphasizing the importance of beta and risk premiums in such calculations.

Understanding Key Financial Concepts

Before diving into calculations, it’s crucial to understand the core concepts involved in portfolio expected return estimation.

Beta: A Measure of Systematic Risk

Beta (β) quantifies a stock's sensitivity to market movements. A beta of 1 indicates that the stock’s price moves in line with the overall market. A beta greater than 1 suggests higher volatility than the market, whereas less than 1 indicates lower volatility.
  • Beta of 2.57 signifies that the stock is significantly more volatile than the average market, implying that it tends to amplify market movements.

Expected Return and the Capital Asset Pricing Model (CAPM)

The CAPM provides a theoretical framework to determine the expected return of an asset based on its systematic risk:

\[
E(Ri) = Rf + \betai \times (E(Rm) - R_f)
\]

Where:


  • \( E(R_i) \) = Expected return of the asset

  • \( R_f \) = Risk-free rate (T-bill rate)

  • \( \beta_i \) = Beta of the asset

  • \( E(R_m) \) = Expected return of the market


This model emphasizes the relationship between risk and return, implying that higher risk (beta) should be compensated with higher expected returns.

Calculating the Expected Return of the Stock

Given data:


  • Beta (\( \beta \)) = 2.57

  • Expected return of the stock (\( E(R_i) \)) = 28.0%

  • T-bill rate (\( R_f \)) = 7.6%


The goal is to find the expected market return (\( E(R_m) \)) based on the provided data or to understand how the stock’s return relates to market expectations.

Using the CAPM formula rearranged:

\[
E(Ri) = Rf + \betai \times (E(Rm) - R_f)
\]

Plugging in the known values:

\[
28.0\% = 7.6\% + 2.57 \times (E(R_m) - 7.6\%)
\]

To find \( E(R_m) \):

\[
28.0\% - 7.6\% = 2.57 \times (E(R_m) - 7.6\%)
\]

\[
20.4\% = 2.57 \times (E(R_m) - 7.6\%)
\]

\[
E(R_m) - 7.6\% = \frac{20.4\%}{2.57}
\]

\[
E(R_m) - 7.6\% \approx 7.93\%
\]

\[
E(R_m) \approx 7.6\% + 7.93\% = 15.53\%
\]

Interpretation:


  • The expected market return based on this stock’s parameters is approximately 15.53%.

  • The high beta indicates high systematic risk, which is compensated with a high expected return of 28%.


Constructing the Portfolio: Determining Asset Weights

To calculate the overall expected return of a portfolio, we need to determine the weights of individual assets within it. Typically, investors aim to balance risk and return by adjusting these weights.

Suppose the investor is considering a portfolio composed of:


  • The high-beta stock (with beta = 2.57)

  • A risk-free asset (T-bills at 7.6%)


The key question: What proportion of the portfolio should be invested in the stock versus the risk-free asset to achieve a desired return or risk profile?

Using the Capital Market Line (CML)

The CML demonstrates the risk-return trade-off of efficient portfolios combining the risk-free asset and the market portfolio. The formula for the expected return of a portfolio combining these two assets is:

\[
E(Rp) = Rf + \frac{\sigmap}{\sigmam} \times (E(Rm) - Rf)
\]

Where:


  • \( E(R_p) \) = Expected return of the portfolio

  • \( R_f \) = Risk-free rate

  • \( \sigma_p \) = Portfolio standard deviation

  • \( \sigma_m \) = Market standard deviation


Note: Since the specific standard deviations are not provided here, a more straightforward approach involves the concept of leveraging or deleveraging the portfolio based on beta.

Calculating Portfolio Beta and Expected Return

The beta of the portfolio (\( \beta_p \)) is the weighted sum of individual asset betas:

\[
\betap = ws \times \betas + wf \times \beta_f
\]

Where:


  • \( w_s \) = weight of the stock in the portfolio

  • \( \beta_s = 2.57 \)

  • \( w_f \) = weight of the risk-free asset (T-bills)

  • \( \beta_f = 0 \) (since risk-free assets have zero beta)


The sum of weights:

\[
ws + wf = 1
\]

The expected return of the portfolio:

\[
E(Rp) = ws \times E(Rs) + wf \times R_f
\]

Given that:


  • \( E(R_s) = 28.0\% \)

  • \( R_f = 7.6\% \)


Suppose the investor aims to achieve a specific portfolio expected return—for example, matching the stock’s expected return (28%) or a lower/higher target.

---

Example: Constructing a Portfolio with a Desired Expected Return

Scenario 1: Portfolio expected return is 20%

Using:

\[
E(Rp) = ws \times 28.0\% + (1 - w_s) \times 7.6\%
\]

Set:

\[
20\% = ws \times 28\% + (1 - ws) \times 7.6\%
\]

Solve for \( w_s \):

\[
20\% = 28\% \times ws + 7.6\% - 7.6\% \times ws
\]

\[
20\% - 7.6\% = w_s \times (28\% - 7.6\%)
\]

\[
12.4\% = w_s \times 20.4\%
\]

\[
w_s = \frac{12.4\%}{20.4\%} \approx 0.608
\]

Interpretation:


  • Invest approximately 60.8% of the portfolio in the stock.

  • Remaining 39.2% in T-bills.


Portfolio beta:

\[
\beta_p = 0.608 \times 2.57 + 0.392 \times 0 = 1.565
\]

This portfolio has a beta of approximately 1.565, indicating it is more volatile than the market.

---

Example: Targeting the Market Portfolio

If an investor wishes to construct a portfolio with a beta equal to the market (i.e., beta = 1), they can solve:

\[
\betap = ws \times 2.57 + (1 - w_s) \times 0 = 1
\]

\[
w_s \times 2.57 = 1
\]

\[
w_s = \frac{1}{2.57} \approx 0.389
\]

Portfolio composition:


  • About 38.9% in the stock.

  • About 61.1% in T-bills.


Expected return:

\[
E(R_p) = 0.389 \times 28\% + 0.611 \times 7.6\% \approx 10.89\% + 4.65\% = 15.54\%
\]

This aligns with the previously calculated expected market return, confirming the consistency of the model.

---

Implications for Investors

Understanding the relationship between beta, expected return, and asset allocation empowers investors to make informed decisions. Here are some key takeaways:


  • High-beta stocks like the one discussed (beta = 2.57) offer the potential for higher returns but come with increased risk.

  • Risk-free assets such as T-bills provide stability but lower returns.

  • Portfolio construction involves balancing these assets to match the investor’s risk appetite and return objectives.


Risk-Return Tradeoff


Investors seeking higher returns must accept higher systematic risk, often achieved through leveraged positions in high-beta stocks. Conversely,

Frequently Asked Questions

What does a beta of 2.57 indicate about the stock's risk relative to the market?
A beta of 2.57 indicates that the stock is significantly more volatile than the overall market; it tends to move 2.57 times the market's movements, implying higher systematic risk.
How do you interpret the expected return of 28.0% for the stock?
The expected return of 28.0% suggests that investors anticipate a high return for holding the stock, which compensates for its increased risk as indicated by its beta.
What is the significance of the T-bill rate being 7.6% in this context?
The T-bill rate of 7.6% serves as the risk-free rate in the Capital Asset Pricing Model (CAPM), used as a benchmark to evaluate the stock's additional risk premium.
How can we use the CAPM to determine the stock's expected return?
The CAPM formula is Expected Return = Risk-Free Rate + Beta × (Market Return - Risk-Free Rate). Given the expected return, beta, and risk-free rate, we can estimate the market return or analyze the stock's risk premium.
Given the stock's expected return of 28.0% and beta of 2.57, what is the market risk premium?
Using CAPM: 28.0% = 7.6% + 2.57 × (Market Return - 7.6%). Solving for Market Return: (Market Return - 7.6%) = (28.0% - 7.6%) / 2.57 ≈ 8.4%. Therefore, the market risk premium is approximately 8.4%.
If an investor wants to build a portfolio including this stock, how does the stock's beta impact their portfolio risk?
Since the stock has a high beta of 2.57, including it in a portfolio increases overall portfolio risk and volatility, especially if the portfolio is heavily weighted in such high-beta stocks.
How would you calculate the weight of this stock in a portfolio to achieve a desired expected return?
Using the weighted average return formula: Portfolio Return = w × Stock Return + (1 - w) × Other Assets Return. Rearranged, the weight w = (Desired Portfolio Return - Other Assets Return) / (Stock Return - Other Assets Return). You can adjust w based on the target return.
What other factors should investors consider when evaluating a stock with a high beta like 2.57?
Investors should consider the company's fundamentals, industry stability, economic conditions, and whether the high beta aligns with their risk tolerance, as high-beta stocks can lead to larger gains but also bigger losses.
Can the expected return of 28.0% be justified solely based on the stock's beta and risk-free rate?
Not entirely; while beta and the risk-free rate help estimate expected return via CAPM, other factors like company performance, market conditions, and investor sentiment also influence actual returns and should be considered.

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