Introduction & Learning Objectives
This guide is your interactive companion to understanding the core concepts of **Return, Risk, and the Security Market Line (SML)**, based on **Chapter 13 of Ross's textbook**. This chapter explores the fundamental relationship between risk and required return, which is a cornerstone of modern finance.
Learning Objectives (Ross, page 427)
- Show how to calculate expected returns, variance, and standard deviation.
- Discuss the impact of diversification.
- Summarize the systematic risk principle.
- Describe the security market line and the risk-return trade-off.
1. Expected Returns, Variance, and Diversification
Calculating Expected Return & Variance (Ross, page 429)
The **expected return** is the weighted average of possible returns, where the weights are the probabilities of each return occurring. The **variance** measures the dispersion of these returns around the expected return.
Exam Hint: Be able to perform these calculations for both single assets and portfolios, as demonstrated in **Problem 13.1** and **13.2** of the self-test problems.
Systematic vs. Unsystematic Risk (Ross, page 437)
**Systematic risk** (or market risk) affects a large number of assets. Examples include changes in GDP, interest rates, or inflation. **Unsystematic risk** (or asset-specific risk) affects a single asset or a small group of assets. Examples include a company's lawsuit or a new product launch.
The Principle of Diversification:
Diversification can **eliminate unsystematic risk**, but it **cannot eliminate systematic risk**. This is a core concept that explains why investors are only rewarded for bearing systematic risk.
2. Beta, the SML, and the CAPM
Beta Coefficient ($\beta$) (Ross, page 442)
The **beta coefficient** measures an asset's systematic risk relative to an average asset. By definition, an average asset has a beta of 1.0. An asset with a beta of 0.5 has half the systematic risk of the market, while a beta of 2.0 has twice the systematic risk.
Exam Hint:
You must be able to calculate a **portfolio beta** by taking the weighted average of the individual asset betas. This was tested in **Problem 11** and **12** from the basic problems.
The Security Market Line (SML) (Ross, page 450)
The **Security Market Line (SML)** is a graphical representation of the relationship between systematic risk ($\beta$) and expected return. In a well-functioning market, all assets must plot on this line, meaning they offer the same reward-to-risk ratio. The slope of the SML is the **market risk premium**.
Exam Hint:
Be able to use the CAPM to find the required return of an asset given its beta, or to solve for a missing beta or market premium. This was tested in **Question 4(b) from the May 2025 exam** and **Question 1(ix) from the Jan 2025 exam**.
3. Problems & Solutions
Problem Statement: There are two assets and three states of the economy:
| State of Economy | Probability | Stock A Return | Stock B Return |
|---|---|---|---|
| Recession | 0.20 | -0.15 | 0.20 |
| Normal | 0.50 | 0.20 | 0.30 |
| Boom | 0.30 | 0.60 | 0.40 |
What are the expected returns and standard deviations for these two stocks?
Solution:
Expected Return:
$E(R_A) = (0.20 \times -0.15) + (0.50 \times 0.20) + (0.30 \times 0.60) = 25\%$
$E(R_B) = (0.20 \times 0.20) + (0.50 \times 0.30) + (0.30 \times 0.40) = 31\%$
Standard Deviation:
$\sigma_A^2 = 0.20(-0.15 - 0.25)^2 + 0.50(0.20 - 0.25)^2 + 0.30(0.60 - 0.25)^2 = 0.07$
$\sigma_A = \sqrt{0.07} = 26.46\%$
$\sigma_B^2 = 0.20(0.20 - 0.31)^2 + 0.50(0.30 - 0.31)^2 + 0.30(0.40 - 0.31)^2 = 0.0049$
$\sigma_B = \sqrt{0.0049} = 7\%$
Problem Statement: You have $10,000 to invest. Your choices are Stock X with an expected return of 12.4% and Stock Y with an expected return of 10.1%. If your goal is to create a portfolio with an expected return of 10.85%, how much money will you invest in Stock X? In Stock Y?
Solution:
Let $w_X$ be the weight of Stock X and $w_Y$ be the weight of Stock Y. We know $w_X + w_Y = 1$.
$E(R_P) = w_X \times E(R_X) + w_Y \times E(R_Y)$
$0.1085 = w_X \times 0.124 + (1-w_X) \times 0.101$
$0.1085 = 0.124w_X + 0.101 - 0.101w_X$
$0.0075 = 0.023w_X \implies w_X = 0.326$ or 32.6%
$w_Y = 1 - w_X = 1 - 0.326 = 0.674$ or 67.4%
Investment in X = $10,000 \times 0.326 = \$3,260
Investment in Y = $10,000 \times 0.674 = \$6,740
Problem Statement: A stock has a beta of 1.19 and an expected return of 12.4 percent. A risk-free asset currently earns 2.7 percent.
a. What is the expected return on a portfolio that is equally invested in the two assets?
b. If a portfolio of the two assets has a beta of .92, what are the portfolio weights?
c. If a portfolio of the two assets has an expected return of 10 percent, what is its beta?
d. If a portfolio of the two assets has a beta of 2.38, what are the portfolio weights? How do you interpret the weights for the two assets in this case? Explain.
Solution:
a. Equally Weighted Portfolio Return:
$E(R_P) = 0.50 \times E(R_{Stock}) + 0.50 \times E(R_{RF}) = 0.50 \times 12.4\% + 0.50 \times 2.7\% = 7.55\%
b. Portfolio Weights for $\beta_P=0.92$:
$\beta_P = w_{stock} \times \beta_{stock} + w_{RF} \times \beta_{RF}$
$0.92 = w_{stock} \times 1.19 + (1-w_{stock}) \times 0$
$w_{stock} = 0.92 / 1.19 = 0.7731 \implies 77.31\% \text{ in stock}$
$w_{RF} = 1 - 0.7731 = 0.2269 \implies 22.69\% \text{ in risk-free asset}$
c. Portfolio Beta for $E(R_P)=10\%$:
$E(R_P) = w_{stock} \times E(R_{stock}) + w_{RF} \times E(R_{RF})$
$0.10 = w_{stock} \times 0.124 + (1-w_{stock}) \times 0.027$
$0.10 = 0.124w_{stock} + 0.027 - 0.027w_{stock} \implies w_{stock} = 0.7526$
$\beta_P = w_{stock} \times \beta_{stock} = 0.7526 \times 1.19 = 0.8956$
d. Portfolio Weights for $\beta_P=2.38$:
$2.38 = w_{stock} \times 1.19 + (1-w_{stock}) \times 0 \implies w_{stock} = 2.38 / 1.19 = 2.0$
$w_{RF} = 1 - 2.0 = -1.0$
This means the portfolio is **200% in stock and -100% in the risk-free asset**. This implies the investor borrowed an amount equal to their initial investment at the risk-free rate to invest in the stock.
4. Minicase: The Beta for Colgate-Palmolive
Joey Moss, a recent finance graduate, has been tasked with calculating the beta for Colgate-Palmolive. His firm currently uses a commercial data vendor, and he's been asked to check if the numbers can be calculated in-house.
Textbook Link: Ross, Chapter 13, Minicase (Page 464)
Solution:
The **market model** is a regression model that attempts to explain a stock's excess return (return minus risk-free rate) based on the market's excess return. The motivation is to find the linear relationship between the two, with the slope being the stock's beta.
**Jensen's alpha** ($\alpha_i$) is the y-intercept of this regression. It measures the asset's performance relative to the SML. A positive alpha means the asset plotted **above** the SML, earning an excess return for its level of risk. The residuals ($\epsilon_t$) represent the unsystematic risk, or the portion of the stock's return not explained by the market's movements.
Solution:
Betas from different sources (like Yahoo! Finance vs. Value Line) can differ because they use different calculation methodologies, different time periods for the data, or different market indices. The choice of daily, weekly, or monthly data can also impact the result. It is a good practice to use multiple sources and a reasonable time frame to get a reliable estimate.