Risk and Return An Overview Dr. C. Bulent Aybar Professor of International Finance

Mary Smith, a financial analyst for Chargers Products, a manufacturer of stadium benches must evaluate the risk and return of two assets, X and Y. The firm is considering adding these assets to its diversified asset portfolio. To assess the return and risk of each asset, Mary gathered data on the annual cash flow and beginning- and end-of-year values of each asset over the immediately preceding 10 years, 2000-2009.

Data Asset X Asset Y Value Year Cash Flow Beginning Ending 2000 1,000 20,000 22,000 1,500 2001 21,000 1,600 2002 1,400 24,000 1,700 2003 1,800 2004 1,900 23,000 2005 26,000 2,000 2006 25,000 2,100 2007 2,200 2008 27,000 2,300 2009 30,000 2,400

Returns on Asset X and Y We can calculate the returns earned in each period in the past by using the above holding period return model

Expected Return E(RX)=11.74% E(RY)=11.14 Year Return (X) Return (Y) 2000 15.00% 7.50% 2001 2.27% 8.00% 2002 20.95% 13.50% 2003 -1.25% 8.57% 2004 13.18% 13.81% 2005 20.00% 13.64% 2006 2.69% 9.13% 2007 4.00% 13.91% 2008 21.25% 13.75% 2009 19.26% 9.60% Mary's investigation suggests that both assets, on average, will tend to perform in the future just as they have during the past 10 years. She believes that the expected annual return can he estimated by finding the average annual return for each asset over the past 10 years. E(RX)=11.74% E(RY)=11.14

Volatility sX =8.9% sY=2.78% CVX=0.76 CVY=0.25 Mary believes that each asset's risk can be assessed in two ways: in isolation and as part of the firm's diversified portfolio of assets. The risk of the assets in isolation can be found by using the standard deviation and coefficient of variation of returns over the past 10 years. Year Return (X) Return (Y) 2000 15.00% 7.50% 2001 2.27% 8.00% 2002 20.95% 13.50% 2003 -1.25% 8.57% 2004 13.18% 13.81% 2005 20.00% 13.64% 2006 2.69% 9.13% 2007 4.00% 13.91% 2008 21.25% 13.75% 2009 19.26% 9.60% sX =8.9% sY=2.78% CVX=0.76 CVY=0.25

Portfolio Risk and Return An investment portfolio is any collection or combination of financial assets. If we assume all investors are rational and therefore risk averse, that investor will ALWAYS choose to invest in portfolios rather than in single assets. Investors will hold portfolios because he or she will diversify away a portion of the risk that is inherent in “putting all your eggs in one basket.” If an investor holds a single asset, he or she will fully suffer the consequences of poor performance. This is not the case for an investor who owns a diversified portfolio of assets.

Portfolio Return The return of a portfolio is a weighted average of the returns on the individual assets from which it is formed and can be calculated as follows:

Risk of a Two Asset Portfolio Assume that we construct a portfolio of asset A and B. Each asset is allocated 50% weight in the portfolio: Expected Return STD Asset-A 11.80% 17.80% Asset-B 15.40% 36.50% Covariance(A, B) 0.01975 Correlation (A, B) 0.304

Covariance and Correlation In calculation of portfolio risk or standard deviation, we used a new variable that we did not consider before. This variable is “Covariance”. Covariance is a measure of co-movement between two assets. It can be calculated as: The lower the covariance of asset returns, the more effective is the diversification of a portfolio in terms of risk reduction.

Correlation The correlation coefficient is calculated by dividing the Covariance by the products of the standard deviations of respective assets This normalization by the standard deviations has the effect of making correlation a dimensionless number that varies from −1, which signifies a perfectly linear negative relation, to +1, which indicates a perfectly linear positive relation.

Risk of a Two Asset Portfolio by using Correlation Coeff. Assume that we construct a portfolio of asset A and B. Each asset is allocated 50% weight in the portfolio: Expected Return STD Asset-A 11.80% 17.80% Asset-B 15.40% 36.50% Covariance(A, B) 0.01975 Correlation (A, B) 0.304 Covariance (A,B)

Asset Correlations and Effectiveness of Diversification Note that portfolio standard deviation of 22.6% is not the simple weighted average of the individual asset standard deviations. Weighted Average STD=(0.5)x(0.1780)+(0.5)x(0.3650)=27.15% This is larger than the portfolio standard deviation of 22.6. This means that by combining two assets together, we were able to reduce risk by about 17%. We owe this to the low correlation between the asset returns. The lower the correlation, the higher the diversification benefits.

Portfolio Theory and Asset Return Correlations The extent to which risk is reduced by portfolio diversification depends on the correlation of assets in the portfolio As the number of assets increases, portfolio variance becomes more dependent on the covariances (or correlations) and less dependent on individual asset variances. The risk of an asset when held in a large portfolio depends on its return correlation with other assets in the portfolio.

Risk of a Portfolio When asset returns are perfectly correlated, portfolio variance falls between the highest and lowest risk asset variances. When the correlation between asset returns are zero, portfolio variance can be reduced to a level lower than the variance of the least risky asset, but to a level above zero. When the asset returns are perfectly negatively correlated, portfolio variance can be reduced up to zero.

Let’s go back to our previous example where we compared two assets!!

Systematic and Unsystematic Risk A general but intuitive model for asset returns is “Market Model”. In the market model alpha is a constant, beta measures the sensitivity of the company returns to an index that reflects the returns on all risky assets in the economy. While it is not possible to capture this in precision, a broad index such as S&P 500 is an acceptable proxy. The epsilon is an error term that captures everything else that is captured by the variation in the index. According to market model, a stock’s returns are driven by “Market Risk” captured in changes in market index. All firm specific variation is captured in an error term:

AT&T vs SP500

Types of Risk Total risk is measured by standard deviation of asset returns. Systematic or Non-diversifiable risk is the portion of an asset’s risk attributable to market factors that affect all firms; cannot be eliminated through diversification. Diversifiable risk is the portion of an asset’s risk that is attributable to firm-specific, random causes; can be eliminated through diversification. Also called unsystematic risk. Total Risk is simply the combination of a security’s non-diversifiable risk and diversifiable risk. Because any investor can create a portfolio of assets that will eliminate virtually all diversifiable risk, the only relevant risk from an investor perspective should be non-diversifiable risk.

Beta: Systematic Risk Indicator Beta of a firm i, is product of its relative volatility with respect to market index and its correlation with the market. Higher the relative volatility of the firm and its correlation with the market, higher is its beta!

Systematic Risk Ratio Variance of Ri can be written as follows: Accordingly “Systematic Risk Ratio” is: Systematic risk ratio tells us what percentage of the firm risk is systematic. Unsystematic portion of the risk is captured in the following:

Un-systematic Risk Ratio Risk Decomoposition Un-systematic Risk Ratio ? Systematic Risk Ratio Total Risk (Volatility)

Total, Systematic (Non-diversifiable) and Non-Systematic Risk Applying some sophisticated quantitative techniques, Mary estimated betas for assets X and Y of 1.60 and 1.10, respectively. She also estimated S&P index volatility as 2%. What are the respective total risks of each investment option? What percentage of the total risk can be attributed to systematic risk? What percentage can be attributed to unsystematic risk?

Total and Systematic Risk Ratio of X Total risk or volatility of asset X is 8.9%! 35% of its volatility or total risk is attributable to market risk or systematic risk factors. Total risk or volatility of asset Y is 2.78%! 63% of its volatility or total risk is attributable to market risk or systematic risk factors.

CAPM and Expected Returns Beta (X) 1.60 Beta (Y) 1.00 Risk Free Rate 0.07 EMPR 0.05 Required Return Expected Return Invest R(X) 15.00% 11.74% No R(Y) 12.00% 11.14% Required returns from assets X and Y are higher than expected returns based on historical data. This suggests that expected returns remain below required returns. A rational investor should avoid both assets.