Chapter 7 Portfolio Theory and Asset Pricing

1. Chapter 7Portfolio Theory and Asset Pricing

2. Learning Objectives • Understand how ‘risk’ and ‘return’ are defined and measured. • Understand the concept of risk-aversion by investors. • Explain how diversification reduces risk. • Understand the importance of covariance between returns on assets in determining the risk of a portfolio.

3. Learning Objectives (cont.) • Explain the concept of efficient portfolios. • Explain the distinction between systematic and unsystematic risk. • Explain why systematic risk is important to investors. • Explain the relationship between returns and risk proposed by the capital asset pricing model.

4. Learning Objectives (cont.) • Understand the relationship between the capital asset pricing model and the arbitrage pricing model. • Explain the development of the Fama–French three-factor model.

5. Return • There is uncertainty associated with returns from shares. • Assume we can assign probabilities to the possible returns — given an assumed set of circumstances, the expected return is given by:

6. Expected Return Calculation • Distribution of returns for security

7. Risk • Risk is present whenever investors are not certain about the outcome an investment will produce. • Risk is measured in terms of how much a particular return deviates from an expected return, measured by variance: • We often use standard deviation to measure risk. This is simply the square root of the variance.

8. Continuing with the previous example, risk is given by: Risk Calculation

9. Risk Attitudes • Risk-neutral investor: • One whose utility is unaffected by risk and hence, when choosing an investment, focuses only on expected return. • Risk-averse investor: • One who demands compensation in the form of higher expected returns in order to be induced into taking on more risk. • Risk-seeking investor: • One who derives utility from being exposed to risk, and hence may be willing to give up some expected return in order to be exposed to additional risk.

10. Risk Attitudes (cont.) • The standard assumption in finance theory is risk-aversion. • This does not mean an investor will refuse to bear any risk at all. • Rather, an investor regards risk as something undesirable, but which may be worth tolerating if compensated with sufficient return. • That is, there is a trade-off between risk and return.

11. risk-seeking risk-neutral risk-averse Utility to Wealth Functions Utility U(W ) Wealth (W ) Figure 7.3

12. Investors’ Risk Preferences • Indifference curve • Curve which represents those combinations of expected return and risk that result in a fixed level of expected utility for an investor.

13. Indifference Curves(for a risk-averse investor)

14. Risk of Assets and Portfolios • We now know that the risk of an individual asset is summarised by standard deviation (or variance) of returns. • Investors usually invest in a number of assets (a portfolio) and will be concerned about the risk of their overall portfolio. • Now concerned about how these individual risks will interact to provide us with overall portfolio risk.

15. Portfolio Theory • Assumptions • Investors perceive investment opportunities in terms of a probability distribution defined by expected return and risk. • Investors’ expected utility is an increasing function of return and a decreasing function of risk (risk-aversion). • Investors are rational.

16. Measuring Return for a Portfolio • Portfolio return (Rp) is a weighted average of all the expected returns of the assets held in the portfolio:

17. Portfolio Return Calculation • Assume 60% of the portfolio is invested in security 1 and 40% in security 2. • The expected returns of the securities are 0.08 and 0.12 respectively. • The Rp can be calculated as follows:

18. Portfolio Risk • Portfolio risk depends on: • The proportion of funds invested in each asset held in the portfolio (w). • The riskiness of the individual assets comprising the portfolio (2). • The relationship between each asset in the portfolio with respect to risk, correlation (.

19. Measurement of Portfolio Risk • For a two-asset portfolio, the variance is:

20. Given the variances of security 1 and security 2 are 0.0016 and 0.0036, respectively, and the correlation (1,2) is –0.5: Portfolio Risk Calculation

21. Relationship Measures • Covariance • Statistic describing the relationship between two variables. • If positive, when one of the variables takes on a value above its expected value, the other has a propensity to do the same. • If the covariance is negative, the deviations tend to be of an opposite sign.

22. Relationship Measures (cont.) • Correlation coefficient is another measure of the strength of a relationship between two variables. • The correlation is equal to the covariance divided by the product of the asset’s standard deviations. • It is simply a standardisation of the covariance and for this reason is bounded by the range +1 to –1.

23. Gains from Diversification • The gain from diversifying is closely related to the value of the correlation coefficient. • The degree of risk reduction increases as the correlation between the rates of return on two securities decreases. • Combining two securities whose returns are perfectly positively correlated results only in risk averaging, and does not provide any risk reduction.

24. Gains from Diversification (cont.) • Risk reduction occurs by combining securities whose returns are less than perfectly positively correlated. • When the correlation coefficient is less than one, the third term in the portfolio variance equation is reduced, reducing portfolio risk. • If the correlation coefficient is negative, risk is reduced even more, but this is not a necessary prerequisite for diversification gains.

25. These diversification benefits are greater, the more assets we incorporate into the portfolio. The key is the correlation between each pair of assets in the portfolio. With n assets, there will be a n × n covariance matrix. The properties of the variance–covariance matrix are: It will contain n2 terms. The two covariance terms for each pair of assets are identical. It is symmetrical about the main diagonal which contains n variance terms. Diversification with Multiple Assets

26. Diversification with Multiple Assets (cont.) • For a diversified portfolio, the variance of the individual assets contributes little to the risk of the portfolio. • For example, in a 50-asset portfolio there are 50 (n) variance terms and 2450 (n2 − n) covariance terms. • The risk depends largely on the covariances between the returns on the assets.

27. Systematic and Unsystematic Risk • Intuitively, we should think of risk as comprising: • Systematic risk (market-related risk or non-diversifiable risk): • That component of total risk that is due to economy-wide factors. • Unsystematic risk (diversifiable risk): • That component of total risk that is unique to the firm and may be eliminated by diversification.

28. Systematic and Unsystematic Risk (cont.) • Unsystematic risk is removed by holding a well-diversified portfolio. • The returns on a well-diversified portfolio will vary due to the effects of market-wide or economy-wide factors. • Systematic risk of a security or portfolio will depend on its sensitivity to the effects of these market-wide factors.

29. Risk of an Individual Asset • The risk contribution of an asset to a portfolio is largely determined by the covariance between the return on that asset and the return on the holder’s existing portfolio: • Well-diversified portfolios will be representative of the market as a whole, thus the relevant measure of risk is the covariance between the return on the asset and the return on the market:

30. Beta is a measure of a security’s systematic risk, describing the amount of risk contributed by the security to the market portfolio. Cov(Ri , RM) can be scaled by dividing it by the variance of the return on the market. This is the asset’s beta (i): Beta

31. The opportunity set: The set of all feasible portfolios that can be constructed from a given set of risky assets. Construction of a Portfolio

32. Construction of a Portfolio (cont.) • The efficient frontier • Given risk-aversion, each investor will try to secure a portfolio on the efficient frontier. • The efficient frontier is determined on the basis of dominance. • A portfolio is efficient if: • No other portfolio has a higher return for the same risk, or • No other portfolio has a lower risk for the same return.

33. Construction of a Portfolio (cont.) • Investors are a diverse group and, therefore, each investor may prefer a different point along the efficient frontier. • Investor risk preferences will determine the preferred portfolio on the efficient frontier.

34. Value at Risk • A relatively new measure of the riskiness of an asset or portfolio. • Defined as ‘the worst loss that is possible under normal market conditions during a given time period’. • Requires the standard deviation of the return on the asset or portfolio. • Typically assumes returns are normally distributed. • Using the normal distribution and the standard deviation, can calculate a worst-case scenario.

35. Value at Risk (cont.) • Investment of $10m in Curzon has an estimated return of zero and a standard deviation of 20% ($2m). • Assume returns are normally distributed and bad market conditions expected 5% of the time. • Worst outcome under normal conditions is a loss of 1.645 (from normal tables) multiplied by standard deviation of $2m. • Worst outcome is loss of $3.29m or an investment value of $6.71m. • VaR was not used effectively used by NAB in the foreign exchange scandal — poor implementation and execution.

36. The Pricing of Risky Assets • What determines the expected rate of return on an individual asset? • Risky assets will be priced such that there is a relationship between returns and systematic risk. • Investors need to be sufficiently compensated for taking on the risks associated with the investment.

37. The Capital Market Line • Combining the efficient frontier with preferences, investors choose an optimal portfolio. • This can be enhanced by introducing a risk-free asset: • The opportunity set for investors is expanded and results in a new efficient frontier — Capital Market Line (CML). • The CML represents the efficient set of all portfolios that provides the investor with the best possible investment opportunities when a risk-free asset is available.

38. The Capital Market Line (cont.) • The CML links the risk-free asset with the optimal risky portfolio (M). • Investors can then vary the riskiness of their portfolio investment by changing weights in the risk-free asset and portfolio M. • This changes their return according to the CML:

39. The Capital Market Line (cont.)

40. In equilibrium, the expected return on a risky asset i (or an inefficient portfolio), is given by the security market line: The CAPM and the Security Market Line

41. The covariance term is the only explanatory factor in the equation that is specific to asset i. As Cov(Ri,RM) is the risk of an asset held as part of the market portfolio, and M is the risk of the market portfolio, beta (  measures the risk of i relative to the risk of the market as a whole. We can thus write the SML as the CAPM equation: The CAPM and the Security Market Line (cont.)

42. Graphical depiction of CAPM, the security market line. The CAPM and the Security Market Line (cont.)

43. The systematic risk (Beta) of a portfolio is calculated as the weighted average of the betas of the individual assets in the portfolio: Portfolio Beta

44. Risk and the CAPM • The capital market will only reward investors for bearing risk that cannot be eliminated by diversification. • Unsystematic risk can be diversified away, so capital market will not reward investors for taking this type of risk. • However, CAPM states the reward for bearing systematic risk is a higher expected return, consistent with the idea of higher risk requires higher return.

45. Tests of the CAPM • Early empirical evidence was supportive of CAPM in explaining asset pricing. • Roll’s critique (1977) criticised methodology of testing CAPM empirically. • Most tests of the CAPM can only determine if the market portfolio used is efficient. • In response, researchers implemented methodological refinements — CAPM seems untestable, given Roll’s critique. • However, CAPM is a useful tool when thinking about asset returns.

46. Arbitrage Pricing Theory (APT) • Developed primarily as a response to the shortcomings of the CAPM. • Less restrictive assumptions. • Empirically testable. • A model of asset pricing that describes the risk premium for a risky asset as a linear combination of various risk factors.

47. The APT model is based on the following equation describing the returns to asset i: Arbitrage Pricing Theory (APT) (cont.)

48. From the previous equation, we can conclude that in the absence of arbitrage opportunities, the expected return on asset i: Interpret l1 as the risk-free rate and l2 as the risk premium on the risk factor F. This seems like the CAPM, main point is that with APT we have not specified the source of risk — the risk factor F. Arbitrage Pricing Theory (APT) (cont.)

49. Another important point is that the APT can be generalised to explain asset returns as a function of multiple factors: Arbitrage Pricing Theory (APT) (cont.)

50. APT: The Factors • What factors should be included in the model? • How many factors should be included in the model? • It is only with retrospective factor analysis that these questions can be addressed. • These questions are best answered empirically as the theory does not explicitly specify what the risk factor(s) are.