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Chapter 5

Chapter 5. Risk and Return. After studying Chapter 5, you should be able to:. Understand the relationship (or “trade-off”) between risk and return. Define risk and return and show how to measure them by calculating expected return, standard deviation, and coefficient of variation.

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Chapter 5

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  1. Chapter 5 Risk and Return

  2. After studying Chapter 5, you should be able to: • Understand the relationship (or “trade-off”) between risk and return. • Define risk and return and show how to measure them by calculating expected return, standard deviation, and coefficient of variation. • Discuss the different types of investor attitudes toward risk. • Explain risk and return in a portfolio context, and distinguish between individual security and portfolio risk. • Distinguish between avoidable (unsystematic) risk and unavoidable (systematic) risk and explain how proper diversification can eliminate one of these risks. • Define and explain the capital-asset pricing model (CAPM), beta, and the characteristic line. • Calculate a required rate of return using the capital-asset pricing model (CAPM). • Demonstrate how the Security Market Line (SML) can be used to describe this relationship between expected rate of return and systematic risk. • Explain what is meant by an “efficient financial market” and describe the three levels (or forms) of market efficiency.

  3. Risk and Return • Defining Risk and Return • Using Probability Distributions to Measure Risk • Attitudes Toward Risk • Risk and Return in a Portfolio Context • Diversification • The Capital Asset Pricing Model (CAPM) • Efficient Financial Markets

  4. Defining Return • Return is defined as “the total gain or loss experienced on an investment over a given period of time”. • It is measured as follows:

  5. Defining Return Income received on an investment plus any change in market price, usually expressed as a percent of the beginning market price of the investment.(see eqn 5.1 p.98) Dt+ (Pt – Pt - 1) R = Pt - 1

  6. Defining Return • Where: • R = Actual, expected or required rate of return during the period t • Dt = Cash flow received from the investment in the time period [t – 1 to t] • Pt = Price of the asset at time t • Pt-1 = Price of the asset at time t – 1

  7. Return Example The stock price for Stock A was $10 per share 1 year ago. The stock is currently trading at $9.50 per share and shareholders just received a $1 dividend. What return was earned over the past year? $1.00 + ($9.50 – $10.00) = 5% R = $10.00

  8. Another Example

  9. Defining Risk Risk is defined as “the chance of financial loss”. It is the variability (difference) of returns from those that are expected. What rate of return do you expect on your investment (savings) this year? What rate will you actually earn? Does it matter if it is a bank CD or a share of stock?

  10. The Risk-and-Return Trade-off • Investments must be analysed in terms of both their return potential as well as their riskiness or variability. • Historically, its been proven that higher returns are accompanied by higher risks.

  11. Risk Assessment Of A Single Asset – Probability Distribution • Provides a more quantitative, yet behavioural, insight into an asset’s risk. • Probability is the chance of a particular outcome occurring. • Can be graphed as a model relating probabilities and their associated outcomes. • The expected value of a return R [the most likely return on an asset] can be calculated by:

  12. Probability Distribution • A probability distribution is a model that relates probabilities to the associated out comes. • The simplest type of probability distribution is the bar chart, which shows only a limited number of outcome-probability coordinates.

  13. Probability Distribution • The bar charts for Shuia Na Ltd’s assets A and B are shown in Slide 13. • Although both assets have the same most likely return, the range of return is much more dispersed for asset B than for asset A—16% versus 4%.

  14. Probability Distribution

  15. Probability Distribution • If we knew all the possible outcomes and associated probabilities, a continuous probability distribution could be developed. This type of distribution can be thought of as a bar chart for a very large number of outcomes. • Slide 15 shows continuous probability distributions for assets A and B. Note that although assets A and B have the same most likely return (15%), the distribution of returns for asset B has much greater dispersion than the distribution for asset A. • Clearly, asset B is more risky than asset A.

  16. Probability Distribution

  17. Discrete versus. Continuous Distributions DiscreteContinuous

  18. Risk Measurement – Standard Deviation • A normal probability distribution will always resemble a bell shaped curve.

  19. Determining Expected Return (Discrete Dist.) n R = S ( Ri )( Pi ) R is the expected return for the asset, Ri is the return for the ith possibility, Pi is the probability of that return occurring, n is the total number of possibilities. I = 1

  20. Risk Measurement – Standard Deviation • Measures the dispersion around the expected value. • The higher the standard deviation the higher the risk.

  21. How to Determine the Expected Return and Standard Deviation Stock BW RiPi (Ri)(Pi) -0.15 0.10 –0.015 -0.03 0.20 –0.006 0.09 0.40 0.036 0.21 0.20 0.042 0.33 0.10 0.033 Sum1.000.090 The expected return, R, for Stock BW is .09 or 9%

  22. Determining Standard Deviation (Risk Measure) n s = S ( Ri – R )2( Pi ) Standard Deviation, s, is a statistical measure of the variability of a distribution around its mean. It is the square root of variance. Note, this is for a discrete distribution. i = 1

  23. How to Determine the Expected Return and Standard Deviation Stock BW RiPi (Ri)(Pi) (Ri- R )2(Pi) –0.15 0.10 –0.015 0.00576 –0.03 0.20 –0.006 0.00288 0.09 0.40 0.036 0.00000 0.21 0.20 0.042 0.00288 0.33 0.10 0.033 0.00576 Sum1.000.090 0.01728

  24. Determining Standard Deviation (Risk Measure) n s = S ( Ri – R )2( Pi ) s = .01728 s = 0.1315 or 13.15% i=1

  25. Another example • Shuia Na Ltd, a tennis-equipment manufacturer, is attempting to choose the better of two alternative investments, A and B. Each requires an initial outlay of $10,000 and each has a most likely annual rate of return of 15%. To evaluate the riskiness of these assets, management has made pessimistic and optimistic estimates of the returns associated with each. • The expected values for these assets are presented in the Table on slide 18. Column 1 gives the Pri’s and column 2 gives the ri’s, n equals 3 in each case. The expected value for each asset’s return is 15%. • Slide 19 shows the standard deviations for these assets

  26. Another example – cont. • Before looking at the standard deviations, can you identify which asset is most risky?

  27. Another example – cont.

  28. Which Asset Is Riskier?

  29. Risk Measurement – Coefficient Of Variation • A measure of relative dispersion, useful in comparing the risk of assets with differing expected returns. • The higher the coefficient of variation, the greater the risk. • Allows comparison of assets that have different expected returns.

  30. Coefficient of Variation The ratio of the standard deviation of a distribution to the mean of that distribution. It is a measure of RELATIVE risk. CV = s/R CV of BW = 0.1315 / 0.09 = 1.46

  31. Risk Preferences • Three Preferences: • Risk Averse: Require a higher rate of return to compensate for taking higher risk. • Risk Seeking: Will accept a lower return for a greater risk. • Risk Indifferent: Required return does not change in response to a change in risk.

  32. Risk Attitudes Certainty Equivalent(CE) is the amount of cash someone would require with certainty at a point in time to make the individual indifferent between that certain amount and an amount expected to be received with risk at the same point in time.

  33. Risk Attitudes Certainty equivalent > Expected value Risk Preference Certainty equivalent = Expected value Risk Indifference Certainty equivalent < Expected value Risk Aversion Most individuals are Risk Averse.

  34. Risk Attitude Example You have the choice between (1) a guaranteed dollar reward or (2) a coin-flip gamble of $100,000 (50% chance) or $0 (50% chance). The expected value of the gamble is $50,000. • Mary requires a guaranteed $25,000, or more, to call off the gamble. • Raleigh is just as happy to take $50,000 or take the risky gamble. • Shannon requires at least $52,000 to call off the gamble.

  35. Risk Attitude Example What are the Risk Attitude tendencies of each? Mary shows “risk aversion” because her “certainty equivalent” < the expected value of the gamble. Raleigh exhibits “risk indifference”because her “certainty equivalent” equals the expected value of the gamble. Shannon reveals a “risk preference” because her “certainty equivalent” > the expected value of the gamble.

  36. Portfolios • A portfolio is a collection of assets. • An efficient portfolio is: • One that maximises the return for a given level of risk. • OR • One that minimises risk for a given level of return.

  37. Portfolio Return • Is calculated as a weighted average of returns on the individual assets from which it is formed. • Is calculated by (Eqn 5.6): • Where: • rp = Return on a portfolio • wj = Proportion of the portfolio’s total dollar value • represented by asset j • rj = Return on asset j

  38. Determining Portfolio Expected Return m RP = S ( Wj )( Rj ) RPis the expected return for the portfolio, Wj is the weight (investment proportion) for the jth asset in the portfolio, Rj is the expected return of the jth asset, m is the total number of assets in the portfolio. J = 1

  39. Determining Portfolio Standard Deviation m m sP = S SWjWk s jk Wj is the weight (investment proportion) for the jth asset in the portfolio, Wkis the weight (investment proportion) for the kth asset in the portfolio, sjkis the covariance between returns for the jth and kth assets in the portfolio. J=1 K=1

  40. What is Covariance? • Covariance is a statistical measure of the degree to which two variables (eg, securities‘ returns) move together. • Positive covariance shows that the two variables move together. • Negative covariance suggests that the two variables move in opposite diiections. • Zero covariance means that the two variables show no tendency to vary together in either a positive or negative linear fashion. • Covariance between security returns complicates the calculation of portfolio standard deviation. • Covariance between securities provides for the possibility of eliminating some risk without reducing potential return.

  41. Covariance sjk = s j s k rjk sj is the standard deviation of the jth asset in the portfolio, skis the standard deviation of the kth asset in the portfolio, rjkis the correlation coefficient between the jth and kth assets in the portfolio.

  42. Correlation • A statistical measure of the relationship, if any, between a series of numbers representing data of any kind. • Three types: • Positive Correlation: Two series move in the same direction. • Uncorrelated: No relationship between the two series. • Negative Correlation: Two series move in opposite directions.

  43. Correlation Coefficient A standardized statistical measure of the linear relationship between two variables. Its range is from –1.0 (perfect negative correlation), through 0 (no correlation), to +1.0 (perfect positive correlation).

  44. Correlation • The degree of correlation is measured by the correlation coefficient.

  45. Diversification • Combining assets with low or negative correlation can reduce the overall risk of the portfolio. • Combining uncorrelated risks can reduce overall portfolio risk. • Combining two perfectly positively correlated assets cannot reduce the risk below the risk of the least risky asset. • Combining two assets with less than perfectly positive correlations can reduce the total risk to a level below that of either asset.

  46. Diversification

  47. Correlation, Diversification, Risk & Return • The lower the correlation between asset returns, the greater the potential diversification of risk.

  48. Portfolio Risk and Expected Return Example You are creating a portfolio of Stock D and Stock BW (from earlier). You are investing $2,000 in Stock BW and $3,000 in Stock D. Remember that the expected return and standard deviation ofStock BWis 9% and 13.15% respectively. The expected return and standard deviation ofStock D is 8% and 10.65% respectively. The correlation coefficient between BW and D is 0.75. What is the expected return and standard deviation of the portfolio?

  49. Determining Portfolio Expected Return WBW = $2,000/$5,000 = 0.4 WD= $3,000/$5,000 = 0.6 RP= (WBW)(RBW) + (WD)(RD) RP = (0.4)(9%) + (0.6)(8%) RP = (3.6%) + (4.8%) = 8.4%

  50. Determining Portfolio Standard Deviation sP = 0.0028 + (2)(0.0025) + 0.0041 sP = SQRT(0.0119) sP = 0.1091 or 10.91% You will not be asked to do this calculation.

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