Risk and Rates of Return Chapter 8 Risk and Rates of Return Stand-Alone Risk Portfolio Risk Risk and Return: CAPM/SML

Stock Market Level, 2000-2016, 2000=100

Apple, Inc. and S&P 500 Monthly Adjusted Price 2000-2016, 2000=100

Apple & S&P 500 Monthly Returns, 2000-2016

Variance of Apple vs Variance of S&P500 Standard deviation of Apple capital gain in decade shown is 12.8% a month (not annualized) (arithmetic mean 3.47% a month, geometric mean 2.65% a month) 1.0347^123=65, 1.0265^123=25 Standard deviation of S&P 500 return in decade shown is 4.7% (arithmetic mean capital gain mean 0.01%, geometric mean -0.16% a month, meaning we’ve lost money)

What is risk? Risk refer to the chance that some unfavorable event will occur Risk is the potential that a chosen activity will lead to a loss. It results from the variability and the uncertainty of the outcome. Many of our activities in life entail risk (e.g. going to school, investing in the stock market)

Investment Returns The rate of return on an investment can be calculated as follows: Investors like returns & dislike risk Prefer investment with high expected return & low risk For example if $1,000 is invested and $1,100 is returned after one year, the rate of return for this investment is: ($1,100 – $1,000)/$1,000 = 10%.

What is investment risk? Two types of investment risk: Stand-alone risk Portfolio risk Diversifiable risk: can be eliminated by proper diversification Market risk component: cannot be eliminated Investment risk is related to the probability of earning a low or negative actual return. The greater the chance of lower than expected, or negative returns, the riskier the investment. The expected rate of return need to be high enough to compensate the investor for its perceived risk No investment will be undertake unless the expected rate of return need to be high enough to compensate the investor for its perceived risk Systematic Vs. unsystematic (company specific)

Probability Distributions A listing of all possible outcomes, and the probability of each occurrence. Can be shown graphically. Expected Rate of Return Rate of Return (%) 100 15 -70 Firm X Firm Y

Selected Realized Returns, 1926-2010 Source: Based on Ibbotson Stocks, Bonds, Bills, and Inflation: 2011 Classic Yearbook (Chicago: Morningstar, Inc., 2011), p. 32.

Hypothetical Investment Alternatives Economy Prob. T-Bills HT Coll USR MP Recession 0.1 5.5% -27.0% 27.0% 6.0% -17.0% Below avg 0.2 -7.0% 13.0% -14.0% -3.0% Average 0.4 15.0% 0.0% 3.0% 10.0% Above avg 30.0% -11.0% 41.0% 25.0% Boom 45.0% -21.0% 26.0% 38.0%

How do the returns of High Tech and Collections behave in relation to the market? High Tech: Moves with the economy, and has a positive correlation. This is typical. Collections: Is countercyclical with the economy, and has a negative correlation. This is unusual.

Do T-bills promise a completely risk-free return? Why is the T-bill return independent of the economy? T-bills will return the promised 5.5%, regardless of the economy No, T-bills do not provide a completely risk-free return, as they are still exposed to inflation. Although, very little unexpected inflation is likely to occur over such a short period of time. T-bills are also risky in terms of reinvestment risk. T-bills are risk-free in the default sense of the word.

Calculating the Expected Return

Summary of Expected Returns High Tech 12.4% Market 10.5% US Rubber 9.8% T-bills 5.5% Collections 1.0% High Tech has the highest expected return, and appears to be the best investment alternative, but is it really? Have we failed to account for risk?

Calculating Standard Deviation In other words, the lesser the variability, the lesser is the risk SD is a measure of variability Using historical returns to measure SD Problems with historical SD: a) how far we should go back b) is what happened in the past going to repeat itself exactly? A measure of how far the actual is likely to deviate from the expected return The smaller the STD DEV, The tighter the probability distribution the smaller the risk is. The lower the risk

Standard Deviation for Each Investment σHT = 20% σColl = 13.2% σM = 15.2% σUSR = 18.8%

Comparing Standard Deviations USR Prob. T-bills HT 0 5.5 9.8 12.4 Rate of Return (%)

Comments on Std Dev as a Measure of Risk Standard deviation (σi) measures total, Stand-alone, risk. The larger σi is, the lower the probability that actual returns will be close to expected returns. Larger σi is associated with a wider probability distribution of returns. If probability distribution is normal: The actual return between 1 STD DEV of expected return 68.26% of time

Comparing Risk & Return Security Expected Return, Risk, s T - bills 5 .5% 0.0% High Tech 12.4 20.0 Collections* 1.0 13.2 US Rubber* 9.8 18.8 Market 10.5 1 5.2

Coefficient of Variation (CV) A standardized measure of dispersion about the expected value, that shows the risk per unit of return Capture the effect of both risk and return Better measure for evaluating risk in situation which investments have substantially different expected return If we had two investments with similar returns but different risk, how we choose? Same risk but different return? What if it was different return and different risk?

Illustrating the CV as a Measure of Relative Risk A B Rate of Return (%) Prob. σA = σB , but A is riskier because of a larger probability of losses In other words, the same amount of risk (as measured by σ) for smaller returns

Risk Rankings by Coefficient of Variation CV T-bills 0.0 High Tech 1.6 Collections 13.2 US Rubber 1.9 Market 1.4 Collections has the highest degree of risk per unit of return. High Tech, despite having the highest standard deviation of returns, has a relatively average CV

Example

Investor Attitude Towards Risk Risk aversion: assumes investors dislike risk & require higher rates of return to encourage them to hold riskier securities Risk premium: the difference between the return on a risky asset & a riskless asset, which serves as compensation for investors to hold riskier securities The willingness to take risk varies over time No investment should under take unless the expected rate of return is high enough to compensate for the perceived risk Remember the coin flip? I either pay you 500 or a 1000? Risk premium is the incentive I give you to take the risk (higher salary after coming to school) Higher risk investments will always demand higher expected returns. What will happen if this was not the case?

Portfolio Construction: Risk & Return A portfolio’s expected return is a weighted average of the returns of the portfolio’s component assets Risk decline as the number of assets increases Diversification can reduce risk but cannot eliminated it Impossible for completely riskless stock portfolio Standard deviation is a little more tricky Requires that a new probability distribution for the portfolio returns be constructed. Not weighted average of the STD DEV of the individual Smaller & depend on both STD DEV & correlation Diversification does nothing to reduce risk if the portfolio consists of perfectly positively correlated stocks Most stocks are held in portfolios

Calculating Portfolio Expected Return

An Alternative Method for Determining Portfolio Expected Return Economy Prob HT Coll Port Recession 0.1 -27.0% 27.0% 0.0% Below avg 0.2 -7.0% 13.0% 3.0% Average 0.4 15.0% 7.5% Above avg 30.0% -11.0% 9.5% Boom 45.0% -21.0% 12.0%

Calculating Portfolio Standard Deviation and CV

Comments on Portfolio Risk Measures σp = 3.4% is much lower than the σi of either stock (σHT = 20.0%; σColl = 13.2%). σp = 3.4% is lower than the weighted average of High Tech & Collections’ σ (16.6%). Not the weighted average of individual stocks’ STD Dev The portfolio provides the average return of component stocks, but lower than the average risk On average, Portfolio Risk decline as the number of assets increase Why? Negative correlation between stocks

General Comments about Risk σ  35% for an average stock. Most stocks are positively (though not perfectly) correlated with the market (i.e., ρ between 0 and 1). Combining stocks in a portfolio generally lowers risk. Rational investor choose to hold portfolio not a stock Diversification is only good when assets or stocks are not perfectly correlated

Remmber : Correlation Correlation : The tendency of two variable to move together Correlation coefficient (ρ ): a measure of the degree of relationship between two variable

Returns Distribution for Two Perfectly Negatively Correlated Stocks (ρ = -1.0) If you decide to invest in each stock separately you would get 15% but with risk If you invest in a 50/50 portfolio, you would get the 15% with no risk Buying the portfolio is the correct way to go for rational investors What will happen to this portfolio? Their movements offset each other

Returns Distribution for Two Perfectly Positively Correlated Stocks (ρ = 1.0) Stock M 15 25 -10 Stock M’ 15 25 -10 Portfolio MM’ 15 25 -10 If correlation is perfect, the portfolio has the same risk. Diversification is only good when assets or stocks are not perfectly correlated

Partial Correlation, ρ = +0.35 Most stocks are positively correlated Past data indicates that stocks are 0.35 on average correlated Creating portfolio is a good way then to reduce risk You should buy the portfolio not the individual stocks

Creating a Portfolio: Beginning with One Stock & Adding Randomly Selected Stocks to Portfolio σp decreases as stocks are added, because they would not be perfectly correlated with the existing portfolio. Expected return of the portfolio would remain relatively constant. Eventually the diversification benefits of adding more stocks dissipates (after about 40 stocks), and for large stock portfolios, σp tends to converge to  20%.

Illustrating Diversification Effects of a Stock Portfolio

Breaking Down Sources of Risk Stand-alone risk = Market risk + Diversifiable risk Market risk: portion of a security’s stand-alone risk that cannot be eliminated through diversification. Measured by beta. Diversifiable risk: portion of a security’s stand-alone risk that can be eliminated through proper diversification What is diversifiable risk? Risk caused by random events affecting firms (e.g. product failure, bankruptcy, lawsuits). They are unique to the firm. But when we diversify we offset this risk because when one firm is doing bad the other is doing well. Market risk is the risk that affects all firms in the market to some extent (e.g. war, inflation, recession). Because it happens to all firms it is not diversifiable.

Failure to Diversify If an investor chooses to hold a one-stock portfolio (doesn’t diversify), would the investor be compensated for the extra risk they bear? NO! Stand-alone risk is not important to a well diversified investor Rational, risk-averse investors are concerned with σp, which is based upon market risk. There can be only one price (the market return) for a given security. No compensation should be earned for holding unnecessary, diversifiable risk.

Capital Asset Pricing Model (CAPM) Model linking risk & required returns. CAPM suggests that there is a Security Market Line (SML) that states that a stock’s required return equals the risk-free return plus a risk premium that reflects the stock’s risk after diversification. ri = rRF + (rM – rRF)bi Primary conclusion: The relevant riskiness of a stock is its contribution to the riskiness of a well-diversified portfolio Market portfolio is a portfolio of all stock High administration fees and commissions would be more than offset the benefit for individual investor Index fund Since stand-alone, investors are not compensated for and should not care for if they are holding portfolios. How can we measure the market risk of a firm? Market risk is also called relevant risk, the only risk that matters. Therefore, this is the risk that measures how much a stock and the market are connected

Beta Measures a stock’s market risk, & shows a stock’s volatility relative to the market. Indicates how risky a stock is if the stock is held in a well-diversified portfolio. Theoretically the correct measure of any stock’s risk The most relevant measure of stock’s risk Market risk premium: a measure of additional return over the risk-free rate needed to compensate investor for assuming an average amount of risk It is the slope of SML & reflect the degree of risk in the economy

Comments on Beta If beta = 1.0, the security is just as risky as the average stock. If beta > 1.0, the security is riskier than average. If beta < 1.0, the security is less risky than average. Beta=0, riskless security Most stocks have betas in the range of 0.5 to 1.5. Changing the stocks in a portfolio can change its riskiness What does this beta mean if the market went up or down by 10%?

Calculating Betas Well-diversified investors are primarily concerned with how a stock is expected to move relative to the market in the future. Without a crystal ball to predict the future, analysts are forced to rely on historical data. A typical approach to estimate beta is to run a regression of the security’s past returns against the past returns of the market. The slope of the regression line is defined as the beta coefficient for the security.

Can the beta of a security be negative? Yes, if the correlation between Stock i & the market is negative (i.e., ρi,m < 0). If the correlation is negative, the regression line would slope downward, and the beta would be negative. However, a negative beta is highly unlikely

Illustrating the Calculation of Beta . ri _ rM -5 0 5 10 15 20 20 15 10 5 -5 -10 Regression line: ri = -2.59 + 1.44 rM ^ Year rM ri 1 15% 18% 2 -5 -10 3 12 16

Beta Coefficients for High Tech, Collections & T-Bills ri -20 0 20 40 40 20 -20 HT: b = 1.32 T-bills: b = 0 Coll: b = -0.87 Beta is the slope rM

Comparing Expected Returns and Beta Coefficients Security Expected Return Beta High Tech 12.4% 1.32 Market 10.5 1.00 US Rubber 9.8 0.88 T-Bills 5.5 0.00 Collections 1.0 -0.87 Riskier securities have higher expected returns, so the rank order is OK.

Calculating Portfolio Beta Explain the formula What is w?

Example

Example

The Security Market Line (SML): Calculating Required Rates of Return SML: ri = rRF + (rM – rRF)bi ri = rRF + (RPM)bi Assume the yield curve is flat and that rRF = 5.5% and RPM = rM  rRF = 10.5%  5.5% = 5.0%.

What is the market risk premium? Additional return over the risk-free rate needed to compensate investors for assuming an average amount of risk Its size depends on: The perceived risk of the stock market Investors’ degree of risk aversion Varies from year to year, but most estimates suggest that it ranges between 4% & 8% per year

Calculating Required Rates of Return

Expected vs. Required Returns High Tech 12.4% 12.1% Undervalued Market 10.5 Fairly valued US Rubber 9.8 9.9 Overvalued T-bills 5.5 Collections 1.0 1.15

Illustrating the Security Market Line . Coll HT T-bills USR SML rM = 10.5 rRF = 5.5 -1 0 1 2 SML: ri = 5.5% + (5.0%)bi ri (%) Risk, bi . Vertical & horizontal axis The intercept is the risk-free rate. It is what you would achieve at a beta = 0 The slope is the risk premium The slope therefore reflects the degree of risk aversion. Higher slope higher aversion Both the SML and a stock’s position on the line changes over time because of changes in the interest rate, risk aversion, and a company’s beta Why would beta for a firm change? Things a firm controls (e.g. capital structure, business risk), industry effects (competition)

An Example: Equally-Weighted Two-Stock Portfolio Create a portfolio with 50% invested in High Tech and 50% invested in Collections. The beta of a portfolio is the weighted average of each of the stock’s betas. bP = wHTbHT + wCollbColl bP = 0.5(1.32) + 0.5(-0.87) bP = 0.225

Calculating Portfolio Required Returns The required return of a portfolio is the weighted average of each of the stock’s required returns. rP = wHTrHT + wCollrColl rP = 0.5(12.10%) + 0.5(1.15%) rP = 6.625% Or, using the portfolio’s beta, CAPM can be used to solve for expected return. rP = rRF + (RPM)bP rP = 5.5% + (5.0%)(0.225)

Example

Example

Is it possible to construct a portfolio of real- world stocks that has a required rate if return equal to the risk-free rate?

Factors That Change the SML What if investors raise inflation expectations by 3%, what would happen to the SML? SML1 ri (%) SML2 0 0.5 1.0 1.5 13.5 10.5 8.5 5.5 ΔI = 3% Risk, bi Risk-free rate = real rate + inflation premium Inflation premium = expected inflation rate Inflation premium is a return to compensate investors for loss of purchasing power

Factors That Change the SML What if investors’ risk aversion increased, causing the market risk premium to increase by 3%, what would happen to the SML? SML1 ri (%) SML2 0 0.5 1.0 1.5 ΔRPM = 3% Risk, bi 13.5 10.5 5.5 What if people were not risk averse? Risk did not matter to them? What would the slope of the line be?

Factors That Change the SML As risk aversion increase, so do the this premium & slope of SML A firm influence its market risk by changing composition of assets or/and use of debt The greater the average investor aversion of risk: The steeper the slop of the line The greater the risk premium for all stocks The higher the required rate of returns What if people were not risk averse? Risk did not matter to them? What would the slope of the line be?

Factors That Change the SML

Factors That Change the SML

Verifying the CAPM Empirically The CAPM has not been verified completely. Statistical tests have problems that make verification almost impossible. Some argue that there are additional risk factors, other than the market risk premium, that must be considered. Fama and French’s finding (no relationship between beta and stock returns) CAPM remains to be very widely used in practice (e.g. analysts’ reports)

More Thoughts on the CAPM Investors seem to be concerned with both market risk & total risk. Therefore, the SML may not produce a correct estimate of ri. ri = rRF + (rM – rRF)bi + ??? CAPM/SML concepts are based upon expectations, but betas are calculated using historical data. A company’s historical data may not reflect investors’ expectations about future riskiness.

Some Concluding Thoughts There is a trade-off between risk and return Needs to take higher risk To achieve a higher expected return Diversification is crucial Real returns are what matters, not nominal returns. The risk of an investment depends investment horizon Stocks are very risky on the short-term but their risk is reduced if you hold them over the long-run While historical data can be helpful, there is no guarantee that the past will repeat itself Studies have raised concerns about the validity of CAPM Talk here about stock return forecast. Structural changes.

The (in a Sense Fallacious) Risk/Return Pyramid