Lecture Presentation Software to accompany Investment Analysis and Portfolio Management Eighth Edition by Frank K. Reilly & Keith C. Brown Chapter 7

Chapter 7 - An Introduction to Portfolio Management Questions to be answered: What do we mean by risk aversion and what evidence indicates that investors are generally risk averse? What are the basic assumptions behind the Markowitz portfolio theory? What is meant by risk and what are some of the alternative measures of risk used in investments? How do you compute the expected rate of return for an individual risky asset or a portfolio of assets? How do you compute the standard deviation of rates of return for an individual risky asset? What is meant by the covariance between rates of return and how do you compute covariance?

Chapter 7 - An Introduction to Portfolio Management What is the relationship between covariance and correlation? What is the formula for the standard deviation for a portfolio of risky assets and how does it differ from the standard deviation of an individual risky asset? Given the formula for the standard deviation of a portfolio, why and how do you diversify a portfolio? What happens to the standard deviation of a portfolio when you change the correlation between the assets in the portfolio? What is the risk-return efficient frontier? Is it reasonable for alternative investors to select different portfolios from the portfolios on the efficient frontier? What determines which portfolio on the efficient frontier is selected by an individual investor?

Background Assumptions As an investor, you want to maximize return for a given level of risk. Your portfolio includes all of your assets and liabilities, not just your traded securities. The relationship between the returns of the assets in the portfolio is important. A good portfolio is not simply a collection of individually good investments.

Risk Aversion Given a choice between two assets with equal rates of return, most investors will select the asset with the lower level of risk.

Evidence That Investors are Risk Averse Many investors purchase insurance: –Life –Automobile –Health –Disability The insured trades a known cost (the premium) for an unknown risk of loss The required yield on bonds increases with risk classifications from AAA to AA to A…. Insurance is one of the few things we buy which we know has a negative NPV

But Not Totally Risk Averse... Risk preferences may have to do with the amount of money involved – we are willing to risk small amounts, but we insure against large losses –People buy lottery tickets (negative expected value but the potential loss is small) –But also buy insurance (negative expected value but the potential loss is large)

Which Definition of Risk? Uncertainty of future outcomes –Risk involves both positive & negative outcomes –What we measure with standard deviation Probability of an adverse outcome –Ignore outcomes that are better than expected –Investors only really care about negative surprises. They like positive surprises.

Rates of Return Source: Ibbotson Associates Year Percentage Return Stock Market Index Returns Actual market returns exhibit significant fluctuation around the mean return. Measure the size of the fluctuations with variance & standard deviation

Measuring Risk Return % # of Years Histogram of Annual Stock Market Returns Each bar shows the number of years that the annual return was within that range out a total of 102 years of data.

Measuring Risk: Portfolio Standard DeviationVariance Treasury bills 2.8%7.9 Government bonds 8.2%68.0 Common stocks 20.1%402.6 Period Std. Dev. Of US Stock Market 1931 – % 1941 – % 1951 – % 1961 – % 1971 – % 1981 – % %

StockStandard Deviation StockStandard Deviation Amazon72.9%GE28.2% Dell53.0%Coca-Cola27.3% Reebok52.3%Pfizer24.3% Microsoft47.5%Heinz23.7% Ford43.8%ExxonMobil18.2% Alcan30.2%Nokia54.0% Risk of Individual Common Stocks Standard deviation over the period January 1999 – December, 2003

Markowitz Portfolio Theory Derives the expected rate of return and expected risk for a portfolio of assets Shows that the variance & standard deviation of the rate of return is a meaningful measure of portfolio risk Derives the formula for computing the variance & standard deviation of a portfolio, showing how to effectively diversify a portfolio

Harry Markowitz Nobel Laureate (1990) In 1952, while still a graduate student at Chicago, Markowitz took just one afternoon to convert the notions of risk & return into a set of written rules involving the use of diversification & optimization. These became the building blocks for all future advances in investment theory.

Assumptions of Markowitz Portfolio Theory 1.Investors consider each investment alternative as defined by a probability distribution of expected returns over a holding period. 2.Investors maximize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth. 3.Investors estimate the risk of the portfolio on the basis of the variability of expected returns (assumes that returns are normally distributed). 4.Investors base decisions solely on expected return and risk, so their utility curves are a function of expected return and the expected variance (or standard deviation) of returns only. 5. For a given level of risk, investors prefer higher returns to lower returns. Similarly, for a given level of expected returns, investors prefer less risk to more risk.

Markowitz Portfolio Theory Using these five assumptions, a single asset or portfolio of assets is considered to be efficient if no other asset or portfolio of assets offers a higher expected return with the same (or lower) risk, or lower risk with the same (or higher) expected return.

Concept of Dominance A B C D Standard Deviation Return A dominates B & C B dominates C D does not dominate

Expected Rates of Return: Single Asset For an individual asset - sum of the possible returns multiplied by the corresponding probability of the return occurring

Expected Rate of Return: Single Risky Asset ProbabilityPossible Rate of Return Expected Return 35%8%2.8% 30%10%3.0% 20%12%2.4% 15%14%2.1% E(R)10.30%

Variance (Standard Deviation) of Returns for an Individual Asset Variance is a measure of the dispersion of returns around the mean If returns are tightly clustered around the mean, variance is low If returns are widely dispersed around the mean, variance is high Standard deviation is the square root of the variance

Variance (Standard Deviation) of Returns for an Individual Investment Where: P i is the probability of R i occurring R i is the i th rate of return

Variance & Standard Deviation: Example Calculate the variance & standard deviation for an asset with the following returns & associated probabilities. ProbabilityReturn 35%8% 30%10% 20%12% 15%14%

Variance & Standard Deviation: Example

Variance & Standard Deviation of Historical Rates of Return The variance & standard deviation we just calculated assumes that we have a distribution of expected returns When we are calculating the variance and standard deviation of historical returns, the probability for each return occurring is the same.

Checking for Normality Many of our standard models in finance assume normality To check a distribution for normality, check for both skewness and kurtosis A normal distribution is completely described by two parameters: –Mean –Variance A normal distribution is perfectly symetrical –68.26% of the observations lie within one standard deviation of the mean –95.44% of the observations lie within two standard deviations of the mean –99.74% of the observations lie within three standard deviations of the mean

Skewness Skewness is present when the distribution is not symmetrical A distribution with positive skewness has a large number of small negative values with a few large positive values – the distribution has a long right tail (why we buy lottery tickets) A distribution with negative skewness has a large number of small positive values with a few large negative values – the distribution has a long left tail (why we buy insurance)

Skewness Positive skewness –Mode < Median < Mean –Investors like positive skewness because the mean is greater than the median Negative skewness –Mean < Median < Mode –to help remember the correct order, this is the same order as they would be listed in the dictionary

Skewness A symmetrical distribution has a skewness of zero. The sign indicates whether the skewness is positive or negative (when we cube the deviation, the sign of the deviation is retained). For a sample size of 100 taken from a normal distribution, a skewness of +/- 0.5 would be considered unusually large. See the exact formula on the next page.

Exact Formula for Skewness For small sample sizes (n < 100), should use the exact formula. For large samples (over 100 observations) the approximation is appropriate.

Example: Calculating Skewness T. Rowe Price Equity Fund, YearReturn % % % % % % % % % % Points to note: 1. The mean is 11.64% 2.5 observations are greater than the mean; 5 observations are smaller than the mean 3.It would appear that the negative deviations approximately offset the positive deviations

Solution Interpretation: A skewness of zero indicates a perfectly symmetrical distribution. Given a value of we can assume the distribution is almost symmetrical.

Skewness: S&P 500 Annual & Monthly Returns, Series# of Periods Arithmetic Mean Std DevSkewnessKurtosis Annual %20.49% Monthly %5.65% Note the difference in skewness between the monthly & annual returns. This is typical of many return series – the nature of the statistic depends heavily on the time period used.

Kurtosis Kurtosis is a statistical measure that indicates whether a distribution is more or less peaked than a normal distribution. For example, observations may be either more tightly clustered around the mean (a more peaked distribution) or more of them may lie in the tails of the distribution –Leptokurtic – distribution is more peaked (lepto is from the word Greek word for slender) & has fatter tails –Platykurtic – the distribution is less peaked than a normal distribution (platy is the Greek word for broad) –Mesokurtic – a “normal” distribution (meso is from the Greek word for middle) A normal distribution has a kurtosis of 3

Calculating Kurtosis Where n is the sample size

Kurtosis The kurtosis of a normal distribution is 3 Excel and many other software packages calculate kurtosis and then subtract 3, giving a measure called excess kurtosis (as calculated on the previous page). An excess kurtosis greater than zero indicates a leptokurtic distribution; an excess kurtosis less than zero indicates a platykurtic distribution.

Moving From One Risky Asset to Several Risky Assets The return on the risky asset portfolio is calculated as a weighted average of the return of the assets in the portfolio –Weights are the market values of each asset divided by the total market value of the portfolio

Expected Rate of Return: Portfolio of Risky Assets Weight (% of Portfolio) Expected Return (Asset i) Expected Portfolio Return 20%10%2.0% 30%11%3.3% 30%12%3.6% 20%13%2.6% E(R)11.50%

Calculating Risk: Two Risky Assets The risk of a single risky asset is calculated as its standard deviation When there are two or more risky assets in a portfolio, we must also incorporate how the individual assets move in relation to each other Thus we need to understand covariance & correlation

A measure of the degree to which two variables “move together” relative to their individual mean values over time –If both returns are typically above their respective means at the same time, the covariance will be positive –If one return is typically above its mean when the other return is below its mean, covariance will be negative –For two assets, i and j, the covariance of their returns is defined as: Covariance of Returns

Covariance of Returns: Example DateWilshire 5000Lehman T Bond Index January %1.77% February %2.00% March %1.50% April %-5.59% May %-0.54% June %0.95% July %1.73% August %3.74% September %0.84% October %1.51% November %-2.19% December %2.31% Mean Monthly Return1.0217%0.6692%

Covariance of Returns: Example Note that we divided by N -1 rather than N, since we are dealing with a sample of the data rather than a population.

Covariance and Correlation The correlation coefficient is obtained by dividing the covariance by the product of the individual standard deviations The correlation between the Wilshire 5000 and the Lehman Treasury Bond Index is 0.109

Correlation Coefficient Can vary only in the range +1 to -1. A value of +1 would indicate perfect positive correlation. –This means that returns for the two assets move together in a completely linear manner. A value of –1 would indicate perfect negative correlation. –This means that the returns for two assets have the same percentage movement, but in opposite directions

Measuring Portfolio Return & Risk: 2 Risky Assets

Variance-Covariance Matrix The variance of a two stock portfolio is the sum of these four boxes 2 B 2 B 2 A 2 A σxStock B σx Stock A Stock B Stock A

Example You are holding the following portfolio of two risky assets: Asset AAsset B Return14%8% Standard Deviation 22%14% Proportion of portfolio 40%60% Correlation0.20 Calculate: 1.Return on the portfolio 2.Risk of the portfolio

Example: Solution

Stock 2 Stock 1 2Stock1

Many Risky Assets Portfolio Return on the portfolio is simply a weighted average of the returns of the assets within the portfolio X i = Proportion in asset i R i = Return on asset i

Risk: Many Risky Assets To calculate the variance of the portfolio, use a variance- covariance matrix Asset 1Asset 2Asset 3Asset 4 Asset 1Variance of Asset 1 Covariance of Asset 1 & 2 Covariance of Asset 1 & 3 Covariance of Asset 1 & 4 Asset 2Covariance of Asset 1 & 2 Variance of Asset 2 Covariance of Asset 2 & 3 Covariance of Asset 2 & 4 Asset 3Covariance of Asset 1 & 3 Covariance of Asset 2 & 3 Variance of Asset 3 Covariance of Asset 3 & 4 Asset 4Covariance of Asset 1 & 4 Covariance of Asset 2 & 4 Covariance of Asset 3 & 4 Variance of Asset 4

Variance-Covariance Matrix The variance-covariance matrix shows that the influence of individual asset risk quickly diminishes as the size of the portfolio grows, whereas the influence of covariance grows quickly. For a portfolio of N assets, there are N variance terms and N 2 – N covariance terms

Contribution to Portfolio Risk As N, the number of securities in the portfolio, increases, portfolio variance approaches the average covariance Thus the risk of a well-diversified portfolio depends on the market risk of the securities in the portfolio. Market risk is measured by Beta.

Measuring Risk Portfolio risk falls rapidly as the number of securities in the portfolio rises. A 1970 study by Fama & Lorie found that 80% of the unique risk is diversified away with 8 stocks; 95% with 32 stocks & 99% with 128 stocks Canadian studies have found that substantially more stocks are required in Canada to achieve good diversification, due to the heavy concentration of resource stocks on the TSX.

Estimation Issues Results of portfolio allocation depend on accurate statistical inputs Estimates of –Expected returns –Standard deviation –Correlation coefficient Among entire set of assets With 100 assets, 4,950 correlation estimates Estimation risk refers to potential errors

Estimation Issues With assumption that stock returns can be described by a single market model, the number of correlations required reduces to the number of assets Single index market model: b i = the slope coefficient that relates the returns for security i to the returns for the aggregate stock market R m = the returns for the aggregate stock market έ i = error term (lower case Greek letter epsilon)

Estimation Issues If all the securities are similarly related to the market and a b i derived for each one, it can be shown that the correlation coefficient between two securities i and j is given as:

The Efficient Frontier The efficient frontier represents that set of portfolios with the maximum rate of return for every given level of risk, or the minimum risk for every level of return Frontier will be portfolios of investments rather than individual securities –An exception is the asset with the highest return

Efficient Frontier for Alternative Portfolios Efficient Frontier A B C E(R) Standard Deviation of Return

The Efficient Frontier and Investor Utility An individual investor’s utility curve specifies the trade- offs he is willing to make between expected return and risk –the more risk averse the individual, the steeper the slope of his/her utility curve The slope of the efficient frontier decreases steadily as you move upward These two interactions will determine the particular portfolio selected by an individual investor The optimal portfolio has the highest utility for a given investor It lies at the point of tangency between the efficient frontier and the utility curve with the highest possible utility

Selecting an Optimal Risky Portfolio X Y U3U3 U2U2 U1U1 U 3’ U 2’ U 1’ More Risk Averse Investor Less Risk Averse Investor

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