A NUMERICAL EXAMPLE FOR PORTFOLIO OPTIMIZATION. In 2003, I collected data on 20 stocks, which are listed below: Berkshire-Hathaway B. Citigroup, Inc.

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1 A NUMERICAL EXAMPLE FOR PORTFOLIO OPTIMIZATION In 3, I collected data on stocks, which are listed below: Sybol ADBE AMZN BA BRKB C CAT CSCO DD FDX GE GLW GM INTC JNJ KO MO MSFT RTN SBC Nae Adobe Systes Aazon.co Boeing Berkshire-Hathaway B Citigroup, Inc. Caterpillar Cisco Dupont Federal Express General Electric Corning General Motors Intel Corporation Johnson and Johnson Coca Cola Altria Microsoft Raytheon Southern Bell Copany FRE 633, Deceber 5, 7 R. Van Slyke p.

2 VOD Vodaphone For each stock, I obtained fro Yahoo Financial the adjusted close values, onthly for the previous year; specifically the closes were for: 3-Nov-3 -Oct-3 -Sep-3 -Aug-3 -Jul-3 -Jun-3 -May-3 -Apr-3 3-Mar-3 3-Feb-3 -Jan-3 -Dec- -Nov- Fro this data we need to estiate two things, the return and the risk. The estiated return: We need to estiates for the total return (the principal plus net return). The first is the estiated yearly return for the asset. For that we siply use the actual annual return (the asset value at the end of the year divided by the asset value at the beginning). for Noveber, to Noveber 3, 3. In order to calculate our risk easure, we will also need an estiate for the onthly return. We assue that the expected returns for each onth are the sae. The siplest ethod is to siply divide the annual return by. So if R is the estiated expected onthly return and R is the annual return, then R = *R. But this is not the way returns usually y operate. If R is the expected onthly return, we would expect the expected return for two onths to be R, and for onths to be R. ( The power of copounding. ) The first ethod can be called arithetic accuulation, while the second can be called geoetric accuulation. There is soe controversy about which of these ethods actually works better in practice. The GAMS files you are given has the geoetric ethod as the default, but coented in are the instructions needed for arithetic ethod as an option. Here we assue the geoetric ethod. To calculate y FRE 633, Deceber 5, 7 R. Van Slyke p.

3 the onthly return per share, R, we used the geoetric ean of the total return for the year. That is siply the th root of the ratio of the closing value on Noveber 3, 3 to the closing value on Noveber,. The risk odel: We will estiate the risk of a portfolio of stocks by soe easure of the volatility of the value. Markowitz used the variance of the portfolio s value for his easure, which is essentially the su of squares of the onthly differences between the actual values and their ean. But we will consider other easures of volatility risk as well. Variance easures the fluctuation of the value about the ean, both above and below the ean. However, for our investents we are usually uch ore concerned with the fluctuations below the ean (the downside risk ); we are only too pleased if the fluctuations are above the ean. If the rando fluctuations are syetric about the ean; for exaple, if we assue they are noral then this is no proble because the fluctuations above the ean are statistically the sae as the fluctuations below so the two easures are related by a constant factor. In any case, we can solve both probles by using different easures of the variation of a portfolio. In addition to the variance, we will consider two additional ethods. In the first we will easure the variation by the su of the absolute values of the differences between the ean and the value. In the second, we will easure it by the largest difference between the ean and its value. In the language of approxiation theory (See the appendix or, for exaple) the variance is an L easure, our first alternative is an L easure, and the last is an L Aong other reasons the L and L can. be solved as linear progras. To illustrate our risk odels let us consider a portfolio of one stock, say, Citigroup, Inc, (Sybol C). prices returns deviations dev. sqd. -Nov Dec Jan Feb Mar Apr FRE 633, Deceber 5, 7 R. Van Slyke p. 3

4 -May Jun Jul Aug Sep Oct Nov One onth predicted return:.6 One year total return.5 L easure (su of the absolute derviations).594 L easure (square root of the su of the squared deviations}. L easure (largest onthly deviation).3 You are provided two GAMS files, L.gs, and L.gs. The first ipleents the Markowitz variance risk odel, the second ipleents the L easure. We have already looked at the Markowitz odel. Now let s look in a little ore detail at the L odel. The L risk odel: In period i, we copare the current adjusted price with the Noveber, th price ties the onthly rate to the i power. We require an expected return of % (R =.) on our $, investent. We look for the portfolio with the given expected return that iniizes the risk as easured by the su of the absolute deviations fro the ean. This leads to the following optiization progra for deterining optial portfolios using the L easure of risk. FRE 633, Deceber 5, 7 R. Van Slyke p. 4

5 which has a non-linear, non differentiable objective, but is equivalent to: which is a linear progra! There is a nuber of odeling issues associated with all this; for exaple, we should probably weight the recent values ore than the older ones. There are straightforward techniques for doing this (see Exercise 5). Appendix: L p nors For any p we can define easure of the size of a vector, called an L p FRE 633, Deceber 5, 7 R. Van Slyke p. 5

6 nor. It is defined by for a n-vector e. The three ost coon L p nors are for p =, p =, and p =. The forula given is not directly defined for p =, but it is fairly easy to show that so we use that as our definition for p =. So e is the noral euclidean distance; e is the su of the absolute values of the vector coponents (soeties called the Manhattan distance), and e is the largest absolute valued coponent. The figure shows the unit ball of each of the three nors, {e: e p }. The L nor is slightly inconvenient for optiization, first because it cannot be represented in a linear progra; secondly because of the square root in its definition. The second proble is not serious, we norally use the square of the L nor. Usually this does not cause any probles because we can use the onotone transforation theore we discussed in the first lecture. Even with this we have non-linearity so we often use the L nor or L nor. Thus for our optial portfolio ethod we are tepted to use the L nor or the L nor instead of the L nor which corresponds to the variance easure of risk. FRE 633, Deceber 5, 7 R. Van Slyke p. 6

7 Exercises: For all the probles in addition to your solutions subit copies of your input files using MyPoly s Digital Dropbox. Be sure to identify each input as to what who you are, what proble it solves, etc. It is a good idea to keep unodified copies of the files L.gs and L.gs, and just work fro copies. th Wednesday evening, Noveber 8 our class will eet in Brooklyn to hear the lecture by Nassi Taleb. It starts at 6:PM in the auditoriu in Dibner. I will arrange for a conference roo before and after his talk, to discuss the probles with you. I expect your draft solutions that night. It will count towards your hoework grade. I also will try to give you soe useful feedback.. Run the GAMS file, L.gs for R =., and M = $,. What is the value for risk (the objective)? What stocks were chosen.,. Change fro the geoetric to the arithetic ean for estiating the expected onthly total return. (Be careful, you have to change the code in two places.). How do the rates copare? What does this say about the power of copounding? 3. The forulation in the file you were given requires that you invest all your oney. Modify the budget constraint in L.gs so that you are only liited to spend no ore than your budget. Do you get the sae answer as in Proble? Explain why you did or didn t. Do this again with R = Using GAMS trace the efficient frontier for R =. to.4 by increents of. for both L and L. What stocks were never chosen; which were always chosen? What difference did the risk odel ake? 5. Now run the L profile considering only the downside risk. What changes in the optial portfolios, if any, did you note for given total target returns? 6. Vanderbei proposed weighting the historic returns, so that the ost recent have a higher influence than the earlier ones. For t our proble, list weights w that su to one, so that each one is 9/th the previous. Hint, review A Useful Trick in the FRE 633, Deceber 5, 7 R. Van Slyke p. 7

8 Single Variable Optiization Lecture. [Vanderbei,, p. 398] 7. Using your weights fro Proble 6, calculate for Citigroup: (a) the weighted arithetic average of Actual Monthly Return. That is, the expected (b) return per onth for Citigroup. The weighted geoetric average, which is given by. 8. Copare the average onthly return for Citigroup as given in the text above, and in Proble 7; I will post an Excel file with the (correct) Citigroup data. 9. Construct an L risk odel that can be solved using linear prograing. Ipleent it as a GAMS file by odifying L.gs and/or L.gs.. Modify the L.gs file to restrict the nuber of assets, k, you can invest in. To be specific, write your GAMS file, to restrict investent to 5 or fewer assets (but you get to choose which 5). Build your odel using ixed, integer and continuous variables. Solve using a ixed integer prograing solver. That is, use a ip algorith. Using your progra trace the efficient frontier for R =. to.4 by increents of.. Copare with your results for Proble 4 (for L risk odel only).. Modify both the L and the L files using integer variables, so that portfolios are constrained to have exact integer nubers of round lots of shares ( shares at tie). List the portfolios, and their iniu risk easures for R =.. Solve using integer prograing. Use a inlp algorith.. If we allow short selling with the L odel (variance); that is we allow the coponents of the x vector to be either positive or negative (short selling), then if we have two optial portfolios (with different target total returns) then an optial portfolio for any third target return can constructed by taking a linear cobination of the first two optial FRE 633, Deceber 5, 7 R. Van Slyke p. 8

9 portfolios. (See the suppleental note: Portfolio Optiization and Quadratic Prograing.) Verify this epirically by odifying L.gs so that x can take on both positive and negative values. Then ake several runs to verify the result. Make the total return equation and the budget equation equalities. You ay need arithetic averages to ake this work. Try this first considering only 7 assets (ADBE, AMZN, BA, BRKB, C, CAT, and CSCO) instead of all assets. Discuss, what happens in both cases. 3. How do the Relaxation Theore, and Monotone Transforation Theore apply to the issues arising in the notes, and probles? References: Chvatal, Vasek, Linear Prograing, Freean, 983. Konno, Hiroshi, and Hiroaki Yaazaki, Mean-Absolute Deviation Portfolio Optiization Model and Its Applications to Tokyo Stock Market, Manageent Science, v. 37, no. 5, May 99. Vanderbei, R., Linear Prograing, Kluwer,. Young, Martin R., A Miniax Portfolio Selection Rule with Linear Prograing Solution, Manageent Science, v. 44, no. 5, May 998. Richard Van Slyke (Revised /5/7) FRE 633, Deceber 5, 7 R. Van Slyke p. 9