Chapter 9 Summary

Key Concepts Summary

The relationship between risk and return is an essential element of financial analysis. Because investors tend to behave in a risk-averse manner, we anticipate that higher risk investments should have higher expected returns. In this chapter, we formalize that analysis by introducing measures of risk (standard deviation and beta) and expected return. Standard deviation captures the dispersion of possible returns (total risk) and is best used when evaluating individual investments in isolation or poorly diversified portfolios. Beta captures the risk of securities relative to the overall market (market risk) and is best used when evaluating individual investments within a portfolio. A portfolio represents a collection of securities held together and is an essential tool for diversifying away firm-specific risk. The lower the correlation between a pair of securities, the more potential diversification benefits there are. This is reminiscent of the “don’t put all your eggs in one basket” cliché. While diversification can virtually eliminate the impact of firm-specific risk, it cannot eliminate market risk. Therefore, investors should be compensated for the degree of market risk that they are exposed to in their investments. The Security Market Line (SML) formalizes the risk-return tradeoff with respect to beta, the risk-free rate, and the market risk premium. It hypothesizes that higher beta stocks should generate higher returns and beta should be the only factor that systematically differentiates returns. Unfortunately, the SML has not held up well to empirical testing which has led to the introduction of newer attempts to formalize the relationship between risk and return. While the specific nature of risk and return is not finalized, it is safe to say that, in general, higher risk should be rewarded with higher expected returns. However, it is important to remember that expected returns are not the same as realized returns due to the nature of risk.

Glossary of Terms

Beta. A measure of a stock’s volatility in relation to the overall market. It indicates the stock’s sensitivity to market movements.

Capital Asset Pricing Model (CAPM). A model used to determine the expected return on an asset based on its beta, the risk-free rate, and the expected market return.

Correlation. A statistical measure that describes the extent to which two securities move in relation to each other.

Diversifiable Risk (Firm-Specific Risk). The portion of an investment’s risk that can be eliminated through diversification by holding a portfolio of various assets. It is specific to a company or industry.

Diversification. An investment strategy that involves holding a variety of assets to reduce the overall risk of the portfolio. It aims to mitigate the impact of poor performance in any single investment.

Equilibrium. A state in which the required return equals the expected return, leading to stable prices as the market is in balance.

Expected Return. The average return an investor anticipates earning from an investment, based on the probabilities of different outcomes.

Firm-Specific Risk. Risk factors that affect a particular company or industry, such as business operations, management decisions, and specific events.

Market Index. A statistical measure that tracks the performance of a group of stocks, representing a portion of the market. Examples include the S&P 500 and the Dow Jones Industrial Average.

Market Risk. The overall risk that affects the entire market or a large segment of it, influenced by economic, political, and social factors.

Market Risk Premium. The additional return expected from holding a risky market portfolio instead of risk-free assets. It is the difference between the expected return on the market and the risk-free rate.

Market-to-Book Ratio. A financial ratio comparing the market value of a company’s stock to its book value. A low ratio can indicate undervaluation, while a high ratio can indicate overvaluation.

Non-Diversifiable Risk (Market/Systematic Risk). The risk inherent to the entire market or market segment that cannot be eliminated through diversification. It is associated with factors that affect all securities, such as economic events.

Required Return. The minimum return an investor expects to achieve by investing in a particular asset, given its risk level.

Risk refers to the possibility of an unfavourable event occurring.

Risk Aversion. The preference for lower risk and the desire to avoid uncertainty in investment decisions. Risk-averse investors prefer investments with lower risk, even if they offer lower returns.

Risk-Free Rate. The theoretical rate of return of an investment with zero risk, often approximated by the yield on a 10-year Treasury bond.

Risk Premium. The return in excess of the risk-free rate that investors require as compensation for the additional risk of an investment.

Security Market Line (SML). A graphical representation of the Capital Asset Pricing Model (CAPM) that shows the relationship between the expected return of a security and its beta with the market.

Standard Deviation. A measure of the dispersion of returns for a given security or market index. It quantifies the amount of variation or volatility from the average return.

Formula & Symbol Hub

Symbols Used

  • [latex]\beta_{A}[/latex] = the Beta of Stock A
  • [latex]corr_{x,y}[/latex] = the correlation between the returns of stocks [latex]x[/latex] and [latex]y[/latex]
  • [latex]corr_{A,MKT}[/latex] = the correlation between stock A and the overall market
  • [latex]\bar{k}[/latex] = expected return of a stock
  • [latex]\bar{k_{n}}[/latex] = expected return for stock [latex]n[/latex]
  • [latex]\bar{k_{m}}[/latex] = the expected return on the market (often approximated by the S&P [latex]500[/latex])
  • [latex]k_{A}[/latex] = the required return for stock A
  • [latex]k_\textrm{RF}[/latex] = the risk-free rate of interest (often approximated by the yield on [latex]10-[/latex]year Treasury bond)
  • [latex]k_i[/latex] = the return under the [latex]i[/latex]th outcome (state of nature)
  • [latex]k_n[/latex] = the return under the [latex]n[/latex]th outcome (state of nature)
  • [latex]P_i[/latex] = the probability of the [latex]i[/latex]th possible outcome (state of nature)
  • [latex]P_n[/latex] = the probability of the [latex]n[/latex]th possible outcome (state of nature)
  • [latex]\sum_{i=1}^{n}[/latex] = repeatedly sum the expression from [latex]i[/latex] to [latex]n[/latex]
  • [latex]\sigma[/latex] = standard deviation
  • [latex]\sigma_{A}[/latex] = the standard deviation of stock A
  • [latex]\sigma_{MKT}[/latex] = the standard deviation of the overall market
  • [latex]W_{n}[/latex] = the weight (proportion of portfolio) of stock [latex]n[/latex]

Formulas Used

  • Formula 9.1 – Expected Return (Single Security)

[latex]\bar{k}=\sum_{i=1}^{n}P_{i}k_{i}[/latex]

OR

[latex]\bar{k}=P_{1}k_{1}+P_{2}k_{2}+...+P_{n}k_{n}[/latex]

  • Formula 9.2 – Standard Deviation (Single Security)

[latex]\sigma=\sqrt{\sum_{i=1}^{n}P_{i}\left(k_{i}-\overline{k}\right)^2}[/latex]

OR

[latex]\sigma=\sqrt{P_{1}\left(k_{1}-\overline{k}\right)^2+P_{2}\left(k_{2}-\overline{k}\right)^2+...+P_{n}\left(k_{n}-\overline{k}\right)^2}[/latex]

  • Formula 9.3 – Expected Return (Portfolio)

[latex]\bar{k_{p}}=\sum_{i=1}^{n}W_{i}\bar{k_{i}}[/latex]

OR

[latex]\bar{k_{p}}=W_{1}\bar{k_{1}}+W_{2}\bar{k_{2}}+...+W_{n}\bar{k_{n}}[/latex]

  • Formula 9.4 – Standard Deviation (Two-Stock Portfolio)

[latex]\sigma_{p}=\sqrt{W_{1}^2\sigma_{1}^2+W_{2}^2\sigma_{2}^2+2W_{1}W_{2}\sigma_{1}\sigma_{2}corr_{1,2}}[/latex]

Chapter References

Black, F. (1972). Capital Market Equilibrium with Restricted Borrowing. Journal of Business, 45(3), pp. 444-54.

Fama, E. F. and K. R. French (1992). The Cross-Section of Expected Stock Returns.

Journal of Finance, 472, pp. 427-65.

Lintner, J. (1965). The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. Review of Economics and Statistics, 47(1), pp. 13-37.

Sharpe,W. F. (1964). Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. Journal of Finance, 19(3), pp. 423-42.

Statman, M. (1987). How Many Stocks Make a Diversified Portfolio? Journal of Financial and Quantitative Analysis, 22(3), 353-363.

Attribution

“Chapter 7 – Risk Analysis” from Business Finance Essentials by Dr. Kevin Bracker; Dr. Fang Lin; and Jennifer Pursley is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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