13.1 Expected Returns and Variances
This section explains how to analyze returns and variances using information about future possible returns and their probabilities, moving beyond historical data.
Expected Return
The expected return, \(E(R)\), on a security is the return that investors, on average, expect to earn over the coming period. It's calculated as the sum of the possible returns multiplied by their respective probabilities.
\(E(R) = \sum_{i=1}^{n} P_i \times R_i\)
where \(P_i\) is the probability of a return \(R_i\).
The risk premium is the difference between the expected return on a risky asset and the return on a risk-free asset (\(R_f\)).
\( \text{Risk Premium} = E(R) - R_f \)
Calculating the Variance
The variance, \(\sigma^2\), measures the volatility or risk of an investment. It is the weighted average of the squared deviations from the expected return. The standard deviation, \(\sigma\), is the square root of the variance.
\( \sigma^2 = \sum_{i=1}^{n} P_i \times [R_i - E(R)]^2 \)
A higher variance and standard deviation indicate a greater risk.
Textbook Example: Unequal Probabilities
Suppose a stock has a 30% return in a recession and a 10% return in a boom. The probability of a recession is 80%, and a boom is 20%. The risk-free rate is 10%.
Expected Return:
\( E(R) = 0.80 \times 30\% + 0.20 \times 10\% = 24\% + 2\% = \textbf{26\%} \)
Risk Premium:
\( \text{Risk Premium} = 26\% - 10\% = \textbf{16\%} \)
Variance and Standard Deviation:
\( \sigma^2 = 0.80 \times (0.30 - 0.26)^2 + 0.20 \times (0.10 - 0.26)^2 \)
\( \sigma^2 = 0.80 \times (0.04)^2 + 0.20 \times (-0.16)^2 = 0.80 \times 0.0016 + 0.20 \times 0.0256 \)
\( \sigma^2 = 0.00128 + 0.00512 = \textbf{0.0064} \)
\( \sigma = \sqrt{0.0064} = \textbf{0.08 \text{ or } 8\%} \)
13.2 Portfolios
Most investors hold a portfolio of assets. This section explains how to calculate the expected return and variance of a portfolio.
Portfolio Weights and Expected Returns
The portfolio weights are the percentages of the total portfolio's value invested in each asset. They must sum to 1.00.
The expected return on a portfolio, \(E(R_P)\), is a weighted average of the expected returns of the individual assets in the portfolio.
\( E(R_P) = x_1 E(R_1) + x_2 E(R_2) + \dots + x_n E(R_n) \)
where \(x_i\) is the portfolio weight of asset \(i\).
Portfolio Variance
The variance of a portfolio is not a simple weighted average of the individual asset variances. The crucial observation is that combining assets into a portfolio can substantially alter the risk faced by the investor, a concept known as diversification. The portfolio's variance depends not only on the variances of the individual assets but also on how the assets' returns move together (covariance).
13.3 Announcements and Expected Returns
The return on any stock is composed of a predictable part and a surprise part. The surprise is the true source of an investment's risk.
\( \text{Total Return} = \text{Expected Return} + \text{Unexpected Return} \)
\( R = E(R) + U \)
The impact of a news announcement depends on the surprise or innovation part of the announcement, not the part that was already expected or "discounted" by the market.
13.4 Risk: Systematic and Unsystematic
Investment risk can be broken down into two main types. This distinction is fundamental to understanding diversification and risk premiums.
Systematic and Unsystematic Risk
- Systematic Risk: Also called market risk or nondiversifiable risk. It is a risk that affects a large number of assets. Examples include changes in GDP, inflation, or interest rates.
- Unsystematic Risk: Also called unique risk or asset-specific risk or diversifiable risk. It is a risk that affects a single asset or a small group of assets. Examples include a company-specific product launch, a lawsuit, or a strike.
The total risk of an investment is the sum of these two components.
\( \text{Total Risk} = \text{Systematic Risk} + \text{Unsystematic Risk} \)
13.5 Diversification and Portfolio Risk
Diversification is the process of spreading an investment across many assets. This reduces risk, but only up to a certain point.
The Principle of Diversification
The principle of diversification states that by combining assets into a portfolio, the unsystematic, or unique, risks tend to cancel each other out. This is because a positive surprise for one asset may be offset by a negative surprise for another.
The systematic risk, however, cannot be eliminated through diversification because it affects all assets in the portfolio to some degree. Therefore, for a well-diversified portfolio, nearly all of the remaining risk is systematic risk.
13.6 Systematic Risk and Beta
The systematic risk principle is the cornerstone of modern finance. It states that the reward for bearing risk depends only on the systematic risk of an investment, because unsystematic risk can be eliminated for free through diversification.
Measuring Systematic Risk: Beta ($\beta$)
The beta coefficient, or \(\beta\), 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 an average asset, while an asset with a beta of 2.0 has twice as much.
The expected return on an asset is directly related to its beta. Higher beta means higher systematic risk, which in turn demands a higher expected return.
Portfolio Betas
The beta of a portfolio is simply the weighted average of the betas of the individual assets in the portfolio.
\( \beta_P = x_1 \beta_1 + x_2 \beta_2 + \dots + x_n \beta_n \)
where \(x_i\) is the portfolio weight of asset \(i\).
13.7 The Security Market Line
In a well-functioning, competitive market, the reward-to-risk ratio must be the same for all assets. This is the fundamental relationship between risk and return. The reward-to-risk ratio is the risk premium divided by the asset's beta.
The Capital Asset Pricing Model (CAPM)
The relationship between expected return and beta is graphically represented by the Security Market Line (SML). The equation of the SML is the Capital Asset Pricing Model (CAPM).
\( E(R_i) = R_f + [E(R_M) - R_f] \times \beta_i \)
where \(E(R_M) - R_f\) is the market risk premium.
The CAPM shows that an asset's expected return is determined by the risk-free rate, the market risk premium, and the asset's amount of systematic risk (its beta).
Chapter Summary and Review
The Security Market Line is a crucial concept because it provides a benchmark against which to compare the expected returns of new investments. The appropriate discount rate for a new project should be the expected return offered in financial markets on investments with the same systematic risk. This minimum required return is known as the cost of capital.
The core logic of this chapter can be summarized as:
- There is a reward for bearing risk (the risk premium).
- Only systematic risk is rewarded, as unsystematic risk can be diversified away.
- Systematic risk is measured by beta (\(\beta\)).
- The relationship between expected return and beta is described by the SML and the CAPM.
Chapter Review and Critical Thinking Questions
Use these questions to test your conceptual understanding of the chapter's topics. Click on each question to reveal the solution.
Solution: Unsystematic risks are diversifiable because they are unique to a particular company or asset. By holding a large number of different assets, the positive and negative unsystematic events tend to cancel each other out. Systematic risks are nondiversifiable because they affect the entire economy, so they cannot be eliminated by simply adding more assets to a portfolio. An investor can control the level of unsystematic risk by diversifying their portfolio, but they cannot control the level of systematic risk, which is inherent in the market.
Solution: Security prices will stay the same if the announcement was already anticipated by the market. The price change only occurs if there is an element of surprise, which is the difference between the actual result and what the market expected. For example, if the market was already forecasting a 2% growth rate, the official announcement will not cause a price change because the information has already been "discounted" into the current price.
Solution:
- a. Short-term interest rates increase unexpectedly: Mostly systematic. This affects the cost of borrowing for almost all companies.
- b. The interest rate a company pays on its short-term debt borrowing is increased by its bank: Mostly unsystematic. This is specific to the company and its relationship with its lender.
- c. Oil prices unexpectedly decline: Mostly systematic. This affects all industries that use oil, from manufacturing to transportation.
- d. An oil tanker ruptures, creating a large oil spill: Mostly unsystematic. The primary impact is on the specific company and the local environment.
- e. A manufacturer loses a multimillion-dollar product liability suit: Mostly unsystematic. This is a unique event specific to the manufacturer.
- f. A Supreme Court decision substantially broadens producer liability: Mostly systematic. This affects a large number of companies and their legal and financial risks.
Solution:
- a. Government announces that inflation unexpectedly jumped by 2 percent last month: This would cause stocks in general to change price (systematic).
- b. Big Widget's quarterly earnings report falls in line with expectations: This would cause no change in Big Widget's stock price, as the news was already discounted (no surprise).
- c. Government reports that economic growth agreed with forecasts: This would cause no change in stock prices in general, as there was no surprise.
- d. The directors of Big Widget die in a plane crash: This would cause Big Widget's stock to change price (unsystematic).
- e. Congress approves changes to the tax code that will increase the top marginal corporate tax rate: This would cause stocks in general to change price, as the legislation affects all corporations. However, since it had been debated, the surprise may be minimal.
Solution: The expected return on a portfolio is a weighted average of the expected returns of the assets in the portfolio. Therefore, the portfolio's expected return cannot be greater than the expected return of the best-performing asset, nor can it be less than the expected return of the worst-performing asset. It must lie somewhere in between.
Solution: False. For a well-diversified portfolio, the most important characteristic is the systematic risk of the individual assets in the portfolio (measured by beta). Unsystematic risk (measured by variance) is largely eliminated through diversification, so it does not contribute to the expected return of the portfolio.
Solution: Yes, the standard deviation on a portfolio can be less than that on every asset in the portfolio due to diversification. When asset returns are not perfectly positively correlated, the unsystematic risk cancels out, reducing the overall portfolio risk. The portfolio beta, however, must always be a weighted average of the individual asset betas. Therefore, the portfolio beta cannot be less than the lowest beta in the portfolio or greater than the highest beta.
Solution: Yes, a risky asset could have a beta of zero if its return is uncorrelated with the market. Based on the CAPM, the expected return on such an asset would be equal to the risk-free rate (\(E(R) = R_f\)). It is also possible for a risky asset to have a negative beta if its returns tend to move in the opposite direction of the market. The CAPM predicts that such an asset would have an expected return less than the risk-free rate. This is because holding a negative beta asset provides a hedge against market downturns, and investors are willing to accept a lower return for this risk-reduction benefit.
Solution: Alpha measures the difference between an asset's actual return and the return predicted by the SML for its given level of systematic risk. A positive alpha means the asset has outperformed its expected return, while a negative alpha means it has underperformed. As an investor, you would want to see a high positive alpha on your investments, as it indicates a superior risk-adjusted return.
Solution: "Keep your alpha high and your beta low" is common advice because alpha measures the excess return relative to risk, and a high alpha means you are getting a better return for the risk you are taking. A low beta means you are taking on less systematic risk, which is the only risk that is rewarded in a well-diversified portfolio. Therefore, this advice suggests that investors should seek to maximize their risk-adjusted returns while minimizing their exposure to market volatility.
Solution: The stock market reacts to unexpected news. A massive layoff announcement is often a surprise that indicates the company is taking drastic measures to cut costs and improve profitability. The stock price rises because investors believe the firm's future cash flows and profitability will increase as a result of the cost-cutting. This does not necessarily mean that Wall Street is "cheering on" the layoffs themselves, but rather reacting to the unexpected positive impact on the company's value. From an ethical standpoint, it is a separate issue from the financial impact.
Solution: Earnings announcements are relevant because they provide information that helps investors forecast a company's future performance. If a company's past earnings are higher than expected, it suggests that its future cash flows may also be higher than previously forecast, leading to a revision in the stock price. Accounting earnings, while not the same as cash flow, are a primary source of information for investors. They are used to create the pro forma statements that are the basis for cash flow projections, so they are a relevant input in the valuation process.