1. University at Albany School of Business Fall 2006 Professor Ross Miller Copyright 2006 by Ross M. Miller. All rights reserved Fin 525Week 10 Portfolio Optimization

2. Professor Ross Miller Fall 2006 1 The Idea Behind Portfolio Optimization  Get the maximum expected return with the minimum risk (volatility)  It should be clear that we usually need to take more risk, especially market risk, in order to get more return, so there must be some trade-off between risk and return  We use a utility function to quantify this trade-off and optimize the portfolio by maximizing utility relative to the any constraints we face

3. Professor Ross Miller Fall 2006 2 Individual Choice and Risk Aversion  Most people are risk-averse  Financial theory only applies to a world dominated by risk-averse people  Risk aversion means that for investments having the same expected return, lower volatility is preferred to higher volatility

4. Professor Ross Miller Fall 2006 3 Volatility and Risk Aversion  Chapter 6 of BKM is about quantifying risk aversion  Various utility functions can be used to express this trade-off  The utility function that BKM favor is:

5. Professor Ross Miller Fall 2006 4 Properties of U = E(r) -.005Aσ 2  More expected return is always better by the same amount  More risk (σ) is always worse and increasingly so  Not everyone has the same value for A  Most people do not have preferences that are this simple, but this is generally an adequate representation of the tradeoff between risk and return

6. Professor Ross Miller Fall 2006 5 Some Comments about BKM Chapter 6  BKM use “scenarios” to illustrate “risky” situations Scenarios can be useful and are frequently used by financial planners, rating agencies, etc. Scenarios do not naturally present themselves in the real world  In recent years, a reasonable value of A is much greater than BKM suggest

7. Professor Ross Miller Fall 2006 6 BKM Chapter 7 in One Slide: Maximizing Utility with Two Assets: One Safe and One Risky

8. Professor Ross Miller Fall 2006 7 What Did We Need to Know to Create the Diagram on the Previous Slide  The expected return for each asset  The volatility of each asset (zero for the safe asset)  The correlation between the safe asset and any other asset is zero, so that can be ignored here

9. Professor Ross Miller Fall 2006 8 Portfolio Optimization by Mean-Variance Analysis  This approach is known as mean-variance analysis because all we need to know about portfolios is their mean (expected) return and the variance (square of the volatility) of that return  This analysis extends in a straightforward (if messy) manner to any number of assets

10. Professor Ross Miller Fall 2006 9 Computing the Expected Return for a Portfolio  This is the “easy” part  The expected return for a portfolio is the dollar-weighted average of the expected returns for each holding in the portfolio:  The weights add up to 1 (all money is used)  If selling short is prohibited, no weights are negative

11. Professor Ross Miller Fall 2006 10 Portfolio Expected Return: BKM Example from Table 8.1  Debt: E(r) = 8% and Volatility = 12%  Equity: E(r) = 13% and Volatility = 20%  If we have w D = 0.82 and w E = 0.18, then E(r) = 0.82(8%) + 0.18(13%) = 8.9%

12. Professor Ross Miller Fall 2006 11 Where Do the Expected Returns for the Pieces of the Portfolio (Assets) Come From  Using the average return over recent history is a bad idea  Using analysts projections is a better idea, but you really have to trust those projections  Getting the asset’s beta—either yourself with a regression or with published numbers—and applying CAPM is a safe approach

13. Professor Ross Miller Fall 2006 12 Expected Returns from CAPM  Currently 5.1% is a good number for the risk-free rate  Currently 10.6% is a good number for the return on the S&P 500 (the “market” for many funds, including all of the GE stock funds)  Bond funds are not measured relative to the stock market, but have a beta computed relative to a bond index  The Lehman Brother Aggregate Bond index can be assumed to have an expected return of 5.5%

14. Professor Ross Miller Fall 2006 13 Variance, Covariance, and Correlation  Variance is the square of the standard deviation, which is the same as “volatility”  The algebra works neatly on variances, so we operate on them and convert to a volatility (by taking a square root) at the least step when we need it to plug into the utility function  The “correlation terms” that are included are technically covariances:

15. Professor Ross Miller Fall 2006 14 That Portfolio Variance Formula Again

16. Professor Ross Miller Fall 2006 15 Portfolio Volatility: BKM Example from Table 8.1 and p. 231-2  Debt: E(r) = 8% and Volatility = 12%  Equity: E(r) = 13% and Volatility = 20%  If correlation = 0.30 and w D = 0.82 and w E = 0.18

17. Professor Ross Miller Fall 2006 16 Finding Covariances  For this, you cannot grab other people’s numbers off of the Internet  You need to get historical prices and create historical holding period returns from them Fortunately, total (not excess) returns are used You do, however, have to determine the frequency and amount of data to use More on this later

18. Professor Ross Miller Fall 2006 17 Computing Covariances in Excel  Excel computes covariance directly with the COVAR function  The covariance of a variable with itself is simply the variance—the VAR function in Excel—so one can create formulas using only COVAR to keep things simple  The problem with covariance is that it is less intuitive than correlation because it is not limited to a bounded range

19. Professor Ross Miller Fall 2006 18 Doing the Double Summation in Excel  The BKM Approach Create a covariance matrix with weights on the border Multiply the mess out  The Miller Approach Avoid the mess entirely by using MMULT and SUMPRODUCT functions in Excel If you understand matrix algebra, this is the natural way to do it using Excel matrix functions If not, it is still much easier to extend to larger examples

20. Professor Ross Miller Fall 2006 19 Inputs and Outputs to Portfolio Optimization  Inputs Utility function (or just the parameter A) The expected return for each asset The variance for each asset The covariance between each pair of assets  Outputs The share (weight) to be invested in each asset The expected return of the portfolio The expected risk (volatility) of the portfolio The utility of the portfolio (not that anyone looks at it)

21. Professor Ross Miller Fall 2006 20 Why Bother with Portfolio Optimization?  Portfolio optimization is usually effective at eliminating bad choices  Portfolio optimization find synergies that are difficult to find by intuition alone  Portfolio optimization makes sure that risks are appropriately rewarded  Portfolio optimization enforces discipline and makes difficult choices easier  Portfolio optimization is an easy way to gain a competitive edge

22. Professor Ross Miller Fall 2006 21 An Excel Example on WebCT (Vanguard3Funds.xls)Vanguard3Funds.xls  Choose an allocation among three Vanguard funds S&P 500 stock index fund (VINIX)VINIX Intermediate-term bond index fund (VBTIX)VBTIX Short-term bond index fund (VBISX)VBISX  This is a version of the standard allocation among stocks, bonds, and cash Historical money market fund returns are harder to get, so the short-term bond fund was used instead

23. Professor Ross Miller Fall 2006 22 Going From This Example to Your Project  Expand both the expected return and the risk calculations to deal with more than three assets  Get current historical data for the assets from Yahoo Finance! Try both weekly and monthly data going back various periods, for example, 1 or 2 years of weekly data vs. 3 or 5 years of monthly data (Morningstar uses 3 years of monthly data)  See what happens if you change A in the utility function

24. Professor Ross Miller Fall 2006 23 An Easy Way to Compute Total Returns that Takes Splits and Dividends into Account  Returns can be computed directly from the “Adj Close” column for historical prices in Yahoo! Finance  You can use the holding period return (HPR) formula and ignore the dividends because the adjusted closing price already reflects them  This is exactly what was done to compute the weekly returns in the Google statistics spreadsheet and on Vanguard3Funds.xls

25. Professor Ross Miller Fall 2006 24 CAPM, Alpha, Mutual Fund Expenses, and Portfolio Optimization  A conservative estimate for the expected return of a mutual fund involves taking the return generated by CAPM from its beta and making two adjustments Deduct the expense ratio Possibly adjust for alpha, usually by taking a further deduction  What is going on? Expenses are like money down the drain Commissions are not included in the published expense ratio, but are reflected in a negative alpha

26. Professor Ross Miller Fall 2006 25 Comments on the BKM International Portfolio Example  Using historical returns over a limited number of years to project future expected returns is a bad idea  The basic math is the same as the Vanguard example, only that show all the intermediate calculations rather than use SUMPRODUCT and MMULT to collapse the variance calculation down to a single cell  It is only designed to generate the efficient frontier, but can be adapted to give an optimal portfolio

27. Professor Ross Miller Fall 2006 26 Using Solver Add-In for Portfolio Optimization  Make sure that under Tools – Add-Ins menu, that “Solver Add-in” is checked  Choose “Solver…” from Tools menu (if you have not done the previous step, this choice will not appear)  Maximize the cell with utility in it by changing the cells with the weights in them  Under “Options” check “Assume non-negative” so that only zero and positive weights are considered  Add constraints so that all weights are <=1 and the sum of the weights =1

28. Professor Ross Miller Fall 2006 27 A Warning About Excel’s Solver  It finds the optimum by successive trial and error  It is possible if it starts in the “wrong place” for it to “get lost” and generate a bizarrely bad portfolio  Two ways to avoid this: 1.Do each optimization twice from two very different starting point 2.Once the Solver generates an optimum, run the Solver again to make sure nothing changes

29. Professor Ross Miller Fall 2006 28 Observations About the Results  Optimal portfolios rarely use all available assets  Choice of portfolio depends greatly on choice of A Efficient frontier can be traced out by varying A and plotting the values for Return and Risk that result An alternative is to maximize the Sharpe ratio, which gives the best ratio of excess return to risk  Choice of portfolio also depends greatly on the expected returns of the assets

30. Professor Ross Miller Fall 2006 29 What Do You Maximize if the Portfolio is a Small Proportion of Your Assets ?  Then utility maximization is problematic because you are not looking at the big picture  DO NOT TRY USING A=0 in the utility function, THIS WILL NOT WORK!!!  The Sharpe ratio (known in the textbook as the reward-to-variability ratio) is an improvement, but still ignores the fact that any specific risk you take on can be at least somewhat diversified away in the rest of the portfolio  Alpha (the risk-adjusted excess return from the CAPM regression) is usually the best bet

31. Professor Ross Miller Fall 2006 30 For Week 11  Work on your 401(k) project and bring any and all questions to class  Read RJW Chapters 6 and 17 in depth and look over Chapters 15 and 16 in preparation for the capital budgeting material