Portfolio Optimization

Transcription

1 13 Portfolio Optimizatio 13.1 Itroductio Portfolio models are cocered with ivestmet where there are typically two criteria: expected retur ad risk. The ivestor wats the former to be high ad the latter to be low. There is a variety of measures of risk. The most popular measure of risk has bee variace i retur. Eve though there are some problems with it, we will first look at it very closely The Markowitz Mea/Variace Portfolio Model The portfolio model itroduced by Markowitz (1959), see also Roy (1952), assumes a ivestor has two cosideratios whe costructig a ivestmet portfolio: expected retur ad variace i retur (i.e., risk). Variace measures the variability i realized retur aroud the expected retur, givig equal weight to realizatios below the expected ad above the expected retur. The Markowitz model might be mildly criticized i this regard because the typical ivestor is probably cocered oly with variability below the expected retur, so-called dowside risk. The Markowitz model requires two major kids of iformatio: (1) the estimated expected retur for each cadidate ivestmet ad (2) the covariace matrix of returs. The covariace matrix characterizes ot oly the idividual variability of the retur o each ivestmet, but also how each ivestmet s retur teds to move with other ivestmets. We assume the reader is familiar with the cocepts of variace ad covariace as described i most itermediate statistics texts. Part of the appeal of the Markowitz model is it ca be solved by efficiet quadratic programmig methods. Quadratic programmig is the ame applied to the class of models i which the objective fuctio is a quadratic fuctio ad the costraits are liear. Thus, the objective fuctio is allowed to have terms that are products of two variables such as x 2 ad x y. Quadratic programmig is computatioally appealig because the algorithms for liear programs ca be applied to quadratic programmig with oly modest modificatios. Loosely speakig, the reaso oly modest modificatio is required is the first derivative of a quadratic fuctio is a liear fuctio. Because LINGO has a geeral oliear solver, the limitatio to quadratic fuctios is helpful, but ot crucial. 357

2 358 Chapter 13 Portfolio Optimizatio Example We will use some publicly available data from Markowitz (1959). Eppe, Gould ad Schmidt (1991) use the same data. The followig table shows the icrease i price, icludig divideds, for three stocks over a twelve-year period: Growth i Year S&P500 ATT GMC USX For referece later, we have also icluded the chage each year i the Stadard ad Poor s/s&p 500 stock idex. To illustrate, i the first year, ATT appreciated i value by 30%. I the secod year, GMC appreciated i value by 29%. Based o the twelve years of data, we ca use ay stadard statistical package to calculate a covariace matrix for three stocks: ATT, GMC, ad USX. The matrix is: ATT GMC USX ATT GMC USX From the same data, we estimate the expected retur per year, icludig divideds, for ATT, GMC, ad USX as , , ad , respectively. The correlatio matrix makes it more obvious how two radom variables move together. The correlatio betwee two radom variables equals the covariace betwee the two variables, divided by the product of the stadard deviatios of the two radom variables. For our three ivestmets, the correlatio matrix is as follows: ATT GMC USX ATT 1.0 GMC USX The correlatio ca be betwee 1 ad +1 with +1 beig a high correlatio betwee the two. Notice GMC ad USX are highly correlated. ATT teds to move with GMC ad USC, but ot early so much as GMC moves with USX.

3 Portfolio Optimizatio Chapter Let the symbols ATT, GMC, ad USX represet the fractio of the portfolio devoted to each of the three stocks. Suppose, we desire a 15% yearly retur. The etire model ca be writte as: MODEL:!Miimize ed-of-period variace i portfolio value; [VAR] MIN = * ATT * ATT * ATT * GMC * ATT * USX * GMC * ATT * GMC * GMC * GMC * USX * USX * ATT * USX * GMC * USX * USX;! Use exactly 100% of the startig budget; [BUD] ATT + GMC + USX = 1;! Required wealth at ed of period; [RET] * ATT * GMC * USX >= 1.15; Note the two costraits are effectively i the same uits. The first costrait is effectively a begiig ivetory costrait, while the secod costrait is a edig ivetory costrait. We could have stated the expected retur costrait just as easily as: * ATT * GMC * USX >=.15 Although perfectly correct, this latter style does ot measure ed-of-period state i quite the same way as start-of-period state. Fas of cosistecy may prefer the former style. The equivalet sets-based formulatio of the model follows: MODEL: SETS: ASSET: AMT, RET; COVMAT(ASSET, ASSET): VARIANCE; SETS DATA: ASSET = ATT GMC USX;!Covariace matrix ad expected returs; VARIANCE = ; RET = ; TARGET = 1.15; DATA! Miimize the ed-of-period variace i portfolio value; [VAR] MIN COVMAT(I, J): AMT(I) * AMT(J) * VARIANCE(I, J));! Use exactly 100% of the startig budget; ASSET: AMT) = 1;! Required wealth at ed of period; ASSET: AMT * RET) >= TARGET;

4 360 Chapter 13 Portfolio Optimizatio Whe we solve the model, we get: Optimal solutio foud at step: 4 Objective value: E-01 Variable Value Reduced Cost TARGET AMT( ATT) AMT( GMC) AMT( USX) RET( ATT) RET( GMC) RET( USX) Row Slack or Surplus Dual Price VAR E BUDGET RETURN The solutio recommeds about 53% of the portfolio be put i ATT, about 36% i GMC ad just over 11% i USX. The expected retur is 15%, with a variace of or, equivaletly, a stadard deviatio of about We based the model simply o straightforward statistical data based o yearly returs. I practice, it may be more typical to use mothly rather tha yearly data as a basis for calculatig a covariace. Also, rather tha use historical data for estimatig the expected retur of a asset, a decisio maker might base the expected retur estimate o more curret, proprietary iformatio about expected future performace of the asset. Oe may also wish to use cosiderable care i estimatig the covariaces ad the expected returs. For example, oe could use quite recet data to estimate the stadard deviatios. A large set of data extedig further back i time could be used to estimate the correlatio matrix. The, usig the relatioship betwee the correlatio matrix ad the covariace matrix, oe could derive a covariace matrix.

5 Portfolio Optimizatio Chapter Efficiet Frotier ad Parametric Aalysis There is o precise way for a ivestor to determie the correct tradeoff betwee risk ad retur. Thus, oe is frequetly iterested i lookig at the tradeoff betwee the two. If a ivestor wats a higher expected retur, she geerally has to pay for it with higher risk. I fiace termiology, we would like to trace out the efficiet frotier of retur ad risk. If we solve for the miimum variace portfolio over a rage of values for the expected retur, ragig from to , we get the followig plot or tradeoff curve for our little three-asset example: Figure 13.1 Efficiet Frotier Notice the kee i the curve as the required expected retur icreases past This is the poit where ATT drops out of the portfolio Portfolios with a Risk-Free Asset Whe oe of the ivestmets available is risk free, the the optimal portfolio compositio has a particularly simple form. Suppose the opportuity to ivest moey risk free (e.g., i govermet treasury bills) at 5% per year has just become available. Workig with our previous example, we ow have a fourth ivestmet istrumet that has zero variace ad zero covariace. There is o limit o how much ca be ivested at 5%. We ask the questio: How does the portfolio compositio chage as the desired rate of retur chages from 15% to 5%?

6 362 Chapter 13 Portfolio Optimizatio We will use the followig slight geeralizatio of the origial Markowitz example model. Notice a fourth istrumet, treasury bills (TBILL), has bee added: MODEL:! Add a riskless asset, TBILL;! Miimize ed-of-period variace i portfolio value; [VAR] MIN = * ATT * ATT * ATT * GMC * ATT * USX * GMC * ATT * GMC * GMC * GMC * USX * USX * ATT * USX * GMC * USX * USX;! Use exactly 100% of the startig budget; [BUD] ATT + GMC + USX + TBILL = 1;! Required wealth at ed of period; [RET] * ATT * GMC * USX * TBILL >= 1.15; Alteratively, this ca be also modeled usig the sets formulatio: MODEL: SETS: ASSET: AMT, RET; COVMAT(ASSET, ASSET): VARIANCE; SETS DATA: ASSET= ATT, GMC, USX, TBILL;!Covariace matrix; VARIANCE = ; RET = , 1.05; TARGET = 1.15; DATA! Miimize the ed-of-period variace i portfolio value; [VAR] COVMAT( I, J): AMT( I)* AMT( J) * VARIANCE( I, J));! Use exactly 100% of the startig budget; AMT) = 1;! Required wealth at ed of period; ASSET: AMT * RET) >= TARGET;

7 Portfolio Optimizatio Chapter Whe solved, we fid: Optimal solutio foud at step: 8 Objective value: E-01 Variable Value Reduced Cost ATT E E-07 GMC USX E-07 TBILL Row Slack or Surplus Dual Price VAR E BUD RET Notice more tha 34% of the portfolio was ivested i the risk-free ivestmet, eve though its retur rate, 5%, is less tha the target of 15%. Further, the variace has dropped to about from about What happes as we decrease the target retur towards 5%? Clearly, at 5%, we would put zero i ATT, GMC, ad USX. A simple form of solutio would be to keep the same proportios i ATT, GMC, ad USX, but just chage the allocatio betwee the risk-free asset ad the risky oes. Let us check a itermediate poit. Whe we decrease the required retur to 10%, we get the followig solutio: Optimal solutio foud at step: 8 Objective value: E-02 Variable Value Reduced Cost ATT E GMC E-06 USX E E-06 TBILL Row Slack or Surplus Dual Price VAR E BUD RET E This solutio supports our cojecture: as we chage our required retur, the relative proportios devoted to risky ivestmets do ot chage. Oly the allocatio betwee the risk-free asset ad the risky assets chage. From the above solutio, we observe that, except for roud-off error, the amout ivested i ATT, GMC, ad USX is allocated i the same way for both solutios. Thus, two ivestors with differet risk prefereces would evertheless both carry the same mix of risky stocks i their portfolio. Their portfolios would differ oly i the proportio devoted to the risk-free asset. Our observatio from the above example i fact holds i geeral. Thus, the decisio of how to allocate fuds amog stocks, give the amout to be ivested, ca be separated from the questios of risk preferece. Tobi received the Nobel Prize i 1981, largely for oticig the above feature, the so-called Separatio Theorem. So, if you oticed it, you must be Nobel Prize caliber.

8 364 Chapter 13 Portfolio Optimizatio The Sharpe Ratio For some portfolio p, of risky assets, excludig the risk-free asset, let: R p = its expected retur, s p = its stadard deviatio i retur, ad r 0 = the retur of the risk-free asset. A plausible sigle measure (as opposed to the two measures, risk ad retur) of attractiveess of portfolio p is the Sharpe ratio: (R p - r 0 ) / s p I words, it measures how much additioal retur we achieved for the additioal risk we took o, relative to puttig all our moey i the risk-free asset. It happes the portfolio that maximizes this ratio has a certai well-defied appeal. Suppose: t = our desired target retur, w p = fractio of our wealth we place i portfolio p (the rest placed i the risk-free asset). To meet our retur target, we must have: ( 1 - w p ) * r 0 + w p * R p = t. The stadard deviatio of our total ivestmet is: w p * s p. Solvig for w p i the retur costrait, we get: w p = ( t r 0 ) /( R p r 0 ). Thus, the stadard deviatio of the portfolio is: w p * s p = [( t r 0 ) /( R p r 0 )] * s p. Miimizig the portfolio stadard deviatio meas: Mi [( t r 0 ) /( R p r 0 )] * s p or Mi [( t r 0 ) * s p /( R p r 0 )]. This is equivalet to: Max ( R p r 0 ) /s p. So, regardless of our risk/retur preferece, the moey we ivest i risky assets should be ivested i the risky portfolio that maximizes the Sharpe ratio.

9 Portfolio Optimizatio Chapter The followig illustrates for whe the risk free rate is 5%: MODEL:! Maximize the Sharpe ratio; MAX = ( *ATT *GMC *USX )/ (( * ATT * ATT * ATT * GMC * ATT * USX * GMC * ATT * GMC * GMC * GMC * USX * USX * ATT * USX * GMC * USX * USX)^.5);! Use exactly 100% of the startig budget; [BUD] ATT + GMC + USX = 1; The solutio is: Optimal solutio foud at step: 7 Objective value: Variable Value Reduced Cost ATT E-04 GMC USX E-04 Notice the relative proportios of ATT, GMC, ad USX are the same as i the previous model where we explicitly icluded a risk free asset with a retur of 5%. For example, otice that, except for roud-off error: / = / Importat Variatios of the Portfolio Model There are several issues that may cocer you whe you thik about applyig the Markowitz model i its simple form: a) As we icrease the umber of assets to cosider, the size of the covariace matrix becomes overwhelmig. For example, 1000 assets implies 1,000,000 covariace terms, or at least 500,000 if symmetry is exploited. b) If the model were applied every time ew data become available (e.g., weekly), we would rebalace the portfolio frequetly, makig small, possibly uimportat adjustmets i the portfolio. c) There are o upper bouds o how much ca be held of each asset. I practice, there might be legal or regulatory reasos for restrictig the amout of ay oe asset to o more tha, say, 5% of the total portfolio. Some portfolio maagers may set the upper limit o a stock to oe day s tradig volume for the stock. The reasoig beig, if the maager wats to uload the stock quickly, the market price would be affected sigificatly by sellig so much. Two approaches for simplifyig the covariace structure have bee proposed: the sceario approach ad the factor approach. For the issue of portfolio ervousess, the icorporatio of trasactio costs is useful.

10 366 Chapter 13 Portfolio Optimizatio Portfolios with Trasactio Costs The models above do ot tell us much about how frequetly to adjust our portfolio as ew iformatio becomes available (i.e., ew estimates of expected retur ad ew estimates of variace). If we applied the above models every time ew iformatio became available, we would be costatly adjustig our portfolio. This might make our broker happy because of all the commissio fees, but that should be a secodary objective at best. The importat observatio is that there are costs associated with buyig ad sellig. There are the obvious commissio costs, ad the ot so obvious bid-ask spread. The bid-ask spread is effectively a trasactio cost for buyig ad sellig. The method we will describe assumes trasactio costs are paid at the begiig of the period. It is a straightforward exercise to modify the model to hadle the case of trasactio costs paid at the ed of the period. The major modificatios to the basic portfolio model are: a) We must itroduce two additioal variables for each asset, a amout bought variable ad a amout sold variable. b) The budget costrait must be modified to iclude moey spet o commissios. c) A additioal costrait must be icluded for each asset to eforce the requiremet: amout ivested i asset i = (iitial holdig of i) + (amout bought of i) (amout sold of i) Example Suppose we have to pay a 1% trasactio fee o the amout bought or sold of ay stock ad our curret portfolio is 50% ATT, 35% GMC, ad 15% USX. This is pretty close to the optimal mix. Should we icur the cost of adjustig? The followig is the relevat model: MODEL: [VAR] MIN = * ATT * ATT * ATT * GMC * ATT * USX * GMC * ATT * GMC * GMC * GMC * USX * USX * ATT * USX * GMC * USX * USX; [BUD] ATT + GMC + USX +.01 * ( BA + BG + BU + SA + SG + SU) = 1; [RET] * ATT * GMC * USX >= 1.15; [NETA] ATT =.50 + BA - SA; [NETG] GMC =.35 + BG - SG; [NETU] USX =.15 + BU - SU; The BUD costrait says the total uses of fuds must equal 1. Aother way of iterpretig the BUD costrait is to subtract each of the NET costraits from it. We the get: [BUD].01 * (BA + BG + BU + SA + SG + SU) + BA + BG + BU=SA + SG + SU; It says ay purchases plus trasactio fees must be fuded by sellig.

11 Portfolio Optimizatio Chapter For referece, the followig is the sets formulatio of the above model: MODEL: SETS: ASSET: AMT, RETURN, BUY, SELL, START; COVMAT( ASSET, ASSET):VARIANCE; SETS DATA: ASSET = ATT, GMC, USX; VARIANCE = ; RETURN = ; START = ; TARGET = 1.15; DATA [VAR] MIN COVMAT(I, J): AMT(I) * AMT(J) * VARIANCE(I, J)); ASSET(I): AMT(I) +.01 * ( BUY(I) + SELL(I))) = 1; ASSET: AMT * RETURN) >= ASSET(I): [NET] AMT(I) = START(I) + BUY(I) - SELL(I);); The solutio follows: Optimal solutio foud at step: 4 Objective value: E-01 Variable Value Reduced Cost ATT GMC USX BA E BG E-02 BU E-02 SA E-02 SG E-02 SU E Row Slack or Surplus Dual Price VAR E BUD RET NETA E-02 NETG E-02 NETU E-02 The solutio recommeds buyig a little bit more ATT, either buy or sell ay GMC, ad sell a little USX.

12 368 Chapter 13 Portfolio Optimizatio Portfolios with Taxes Taxes are a upleasat complicatio of ivestmet aalysis that should be cosidered. The effect of taxes o a portfolio is illustrated by the followig results durig oe year for two similar growth-ad-icome portfolios from the Vaguard compay. Portfolio S was maaged without (Sas) regard to taxes. Portfolio T was maaged with after-tax performace i mid: Distributios Iitial Portfolio Icome Gai-from-sales Share-price Retur S $0.41 $2.31 $ % T $0.28 $0.00 $ % The tax maaged portfolio, probably just by chace, i fact had a higher before tax retur. It looks eve more attractive after taxes. If the tax rate for both divided icome ad capital gais is 30%, the the tax paid at year ed per dollar ivested i portfolio S is.3 ( ) /19.85 = 4.1 cets; whereas, the tax per dollar ivested i portfolio S is.3.28/13.44 = 0.6 of a cet. Below is a geeralizatio of the Markowitz model to take ito accout taxes. As iput, it requires i particular: a) umber of shares held of each kid of asset, b) price per share paid for each asset held, ad c) estimated divideds per share for each kid of asset. The results from this model will differ from a model that does ot cosider taxes i that this model, whe cosiderig equally attractive assets, will ted to: i. purchase the asset that does ot pay divideds, so as to avoid the immediate tax o divideds, ii. sell the asset that pays divideds, ad iii. sell the asset whose purchase cost was higher, so as to avoid more tax o capital gais. This is all give that two assets are otherwise idetical (presumig rates of retur are computed icludig divideds). For completeess, this model also icludes trasactio costs ad illustrates how a correlatio matrix ca be used istead of a covariace matrix to describe how assets move together: MODEL:! Geeric Markowitz portfolio model that takes ito accout bid/ask spread ad taxes. (PORTAX) Keywords: Markowitz, portfolio, taxes, trasactio costs; SETS: ASSET: RET, START, BUY, SEL, APRICE, BUYAT, SELAT, DVPS, STD, X; SETS DATA:! Data based o origial Markowitz example; ASSET = TBILL ATT GMC USX;! The expected returs as growth factors; RET = ;! S. D. i retur for each asset; STD = ;! Startig compositio of the portfolio i shares; START = ;! Price per share at acquisitio; APRICE = ;

13 Portfolio Optimizatio Chapter ! Curret bid/ask price per share; BUYAT = ; SELAT = ;! Divideds per share(estimated); DVPS = ;! Tax rate; TAXR =.32;! The desired growth factor; TARGET = 1.15; DATA SETS: TMAT( ASSET, ASSET) &1 #GE# &2: CORR; SETS DATA:! Correlatio matrix; CORR= ; DATA! ;! Mi the var i portfolio retur; [OBJ] MIN ASSET( I): ( X( I)*SELAT( I)* STD( I))^2) + 2 TMAT( I, J) I #NE# J: CORR( I, J) * X( I)* SELAT( I) * STD( I) * X( J)* SELAT( J) * STD( J)) ;! Budget costrait, sales must cover purchases + taxes; ASSET( I): SELAT( I) * SEL( I) - BUYAT( I) * BUY( I)) >= TAXES; [TAXC] TAXES >= TAXR ASSET( I): DVPS( I)* X( I) + SEL( I) * ( SELAT( I) - APRICE( I)));! After tax retur requiremet. This assumes we do ot pay tax o appreciatio util we sell; ASSET( I): RET( I)* X(I)* SELAT( I)) - TAXES >= TARGET ASSET(I): START( I) * SELAT( I));! Ivetory balace for each ASSET( I): [BAL] X( I) = START( I) + BUY( I) - SEL( I); );

14 370 Chapter 13 Portfolio Optimizatio Factors Model for Simplifyig the Covariace Structure Sharpe (1963) itroduced a substatial simplificatio to the modelig of the radom behavior of stock market prices. He proposed that there is a market factor that has a sigificat effect o the movemet of a stock. The market factor might be the Dow-Joes Idustrial average, the S&P 500 average, or the Nikkei idex. If we defie: M = the market factor, m 0 = E(M), 2 s 0 = var(m), e i = radom movemet specific to stock i, 2 s i = var(e i ). Sharpe s approximatio assumes (where E( ) deotes expected value): E(e i ) = 0 E(e i e j ) = 0 for i j, E(e i M) = 0. The, accordig to the Sharpe sigle factor model, the retur of oe dollar ivested i stock or asset i is: u i + b i M + e i. The parameters u i ad b i are obtaied by regressio (e.g., least squares, of the retur of asset i o the market factor). The parameter b i is kow as the beta of the asset. Let: X i = amout ivested i asset i ad defie the variace i retur of the portfolio as: var[ X i (u i + b i M + e i )] = var( X i b i M) + var( X i e i ) = ( X i b i ) 2 s o 2 + X i 2 s i 2. Thus, our problem ca be writte: Miimize Z 2 s o 2 + X i 2 s i 2 subject to Z X i b i = 0 X i = 1 X i ( u i + b i m o ) r. So, at the expese of addig oe costrait ad oe variable, we have reduced a dese covariace matrix to a diagoal covariace matrix. I practice, perhaps a half doze factors might be used to represet the systematic risk. That is, the retur of a asset is assumed to be correlated with a umber of idices or factors. Typical factors might be a market idex such as the S&P 500, iterest rates, iflatio, defese spedig, eergy prices, gross atioal product, correlatio with the busiess cycle, various idustry idices, etc. For example, bod prices are very affected by iterest rate movemets.

15 Portfolio Optimizatio Chapter Example of the Factor Model The Factor Model represets the variace i retur of a asset as the sum of the variaces due to the asset s movemet with oe or more factors, plus a factor-idepedet variace. To illustrate the factor model, we used multiple regressio to regress the returs of ATT, GMC, ad USX o the S&P 500 idex for the same period. The model with solutio is: MODEL:! Multi factor portfolio model; SETS: ASSET: ALPHA, SIGMA, X; FACTOR: RETF, SIGFAC, Z; AXF( ASSET, FACTOR): BETA; SETS DATA:! The factor(s); FACTOR = SP500;! Mea ad s.d. of factor(s); RETF = ; SIGFAC = ;! The stocks were multi-regressed o the factors;! i.e.: Retur(i) = Alpha(i) + Beta(i) * SP500 + error(i); ASSET = ATT GMC USX; ALPHA = ; BETA = ; SIGMA = ;! The desired retur; TARGET = 1.15; DATA! ;! Mi the var i portfolio retur; [OBJ] MIN FACTOR( J):( SIGFAC( J) * Z( J))^2) ASSET( I): ( SIGMA( I) * X( I))^2) ;! Compute portfolio FACTOR( J): Z( J) ASSET( I): BETA( I, J) * X( I)); );! Budget ASSET: X) = 1;! Retur ASSET( I): X( I )* ALPHA( I)) FACTOR( J): Z( J) * RETF( J)) >= TARGET;

16 372 Chapter 13 Portfolio Optimizatio Part of the solutio is: Variable Value Reduced Cost TARGET X( ATT) X( GMC) X( USX) E Z( SP500) Row Slack or Surplus Dual Price OBJ Notice the portfolio makeup is slightly differet. However, the estimated variace of the portfolio is very close to our origial portfolio Sceario Model for Represetig Ucertaity The sceario approach to modelig ucertaity assumes the possible future situatios ca be represeted by a small umber of scearios. The smallest umber used is typically three (e.g., optimistic, most likely, ad pessimistic ). Some of the origial ideas uderlyig the sceario approach come from the approach kow as stochastic programmig; see Madasky (1962), for example. For a discussio of the sceario approach for large portfolios, see Markowitz ad Perold (1981) ad Perold (1984). For a thorough discussio of the geeral approach of stochastic programmig, see Ifager (1992). Eppe, Marti, ad Schrage (1988) use the sceario approach for capacity plaig i the automobile idustry. Let: P s = Probability sceario s occurs, u is = retur of asset i if the sceario is s, X i = ivestmet i asset i, Y s = deviatio of actual retur from the mea if the sceario is s; = i X i ( u is q P q u iq ). Our problem i algebraic form is: Miimize s P s Y s 2 subject to Y s i X i (u i s q P q u iq ) = 0 (deviatio from mea of each sceario, s) i X i = 1 (budget costrait) i X i s P s u is r (desired retur). If asset i has a iheret variability v i 2, the objective geeralizes to: Mi i X i 2 v i 2 + s P s Y s 2 The key feature is that, eve though this formulatio has a few more costraits, the covariace matrix is diagoal ad, thus, very sparse.

17 Portfolio Optimizatio Chapter You will geerally also wat to put upper limits o what fractio of the portfolio is ivested i each asset. Otherwise, if there are o upper bouds or iheret variabilities specified, the optimizatio will ted to ivest i oly as may assets as there are scearios Example: Sceario Model for Represetig Ucertaity We will use the origial data from Markowitz oce agai. We simply treat each of the 12 years as beig a separate sceario, idepedet of the other 11 years. Because of the amout of data ivolved, it is coveiet to use the sets form of LINGO i the followig model: MODEL:! Sceario portfolio model; SETS: SCENE/1..12/: PRB, R, DVU, DVL; ASSET/ ATT, GMT, USX/: X; SXI( SCENE, ASSET): VE; SETS DATA: TARGET = 1.15;! Data based o origial Markowitz example; VE = ;! All scearios cosidered to be equally likely; PRB= ; DATA! Target edig value; [RET] AVG >= TARGET;! Compute expected value of edig positio; AVG SCENE: PRB * SCENE( S):! Measure deviatios from average; DVU( S) - DVL( S) = R(S) - AVG;! Compute value uder each sceario; R( S) ASSET( J): VE( S, J) * X( J)));

18 374 Chapter 13 Portfolio Optimizatio! Budget; ASSET: X) = 1; [VARI] VAR SCENE: PRB * ( DVU + DVL)^2); [SEMIVARI] SEMIVAR SCENE: PRB * (DVL) ^2); [DOWNRISK] DNRISK SCENE: PRB * DVL);! Set objective to VAR, SEMIVAR, or DNRISK; [OBJ] MIN = VAR; Whe solved, (part of) the solutio is: Optimal solutio foud at step: 4 Objective value: E-01 Variable Value Reduced Cost X( ATT) X( GMT) X( USX) Row Slack or Surplus Dual Price RET BUD OBJ E The solutio should be familiar. The alert reader may have oticed the solutio suggests the same portfolio (except for roud-off error) as our origial model based o the covariace matrix (based o the same 12 years of data as i the above sceario model). This, i fact, is a geeral result. I other words, if the covariace matrix ad expected returs are calculated directly from the origial data by the traditioal statistical formulae, the the covariace model ad the sceario model, based o the same data, will recommed exactly the same portfolio. The careful reader will have oticed the objective fuctio from the sceario model ( ) is slightly less tha that of the covariace model (.02241). The exceptioally perceptive reader may have oticed /11 is, except for roud-off error, equal to The differece i objective value is a result simply of the fact that stadard statistics packages ted to divide by N 1 rather tha N whe computig variaces ad covariaces, where N is the umber of observatios. Thus, a slightly more geeral statemet is, if the covariace matrix is computed usig a divisor of N rather tha N 1, the the covariace model ad the sceario model will give the same solutio, icludig objective value.

19 Portfolio Optimizatio Chapter Measures of Risk other tha Variace The most commo measure of risk is variace (or its square root, the stadard deviatio). This is a reasoable measure of risk for assets that have a symmetric distributio ad are traded i a so-called efficiet market. If these two features do ot hold, however, variace has some drawbacks. Cosider the four possible growth distributios i Figure Ivestmets A, B, ad C are equivalet accordig to the variace measure because each has a expected growth of 1.10 (a expected retur of 10%) ad a variace of 0.04 (stadard deviatio aroud the mea of 0.20). Risk-averse ivestors would, however, probably ot be idifferet amog the three. Uder distributio (A), you would ever lose ay of your origial ivestmet, ad there is a 0.2 probability of the ivestmet growig by a factor of 1.5 (i.e., a 50% retur). Distributio (C), o the other had, has a 0.2 probability of a ivestmet decreasig to 0.7 of its origial value (i.e., a egative 30% retur). Risk-averse ivestors would ted to prefer (A) most ad to prefer (C) least. This illustrates variace eed ot be a good measure of risk if the distributio of returs is ot symmetric: Figure 13.2 Possible Growth Factor Distributios P r o b a b i l i t y (A) (B) (C) (D) Growth Factor Ivestmet (D) is a iefficiet ivestmet. It is domiated by (A). Suppose the oly ivestmets available are (A) ad (D) ad our goal is to have a expected retur of at least 5% (i.e., a growth factor of 1.05) ad the lowest possible variace. The solutio is to put 50% of our ivestmet i each of (A) ad (D). The resultig variace is 0.01 (stadard deviatio = 0.1). If we ivested 100% i (A), the stadard deviatio would be Nevertheless, we would prefer to ivest 100% i (A). It is true the retur is more radom. However, our profits are always at least as high uder every outcome. (If the radomess i profits is a issue, we ca always give profits to a worthy educatioal istitutio whe our profits are high to reduce the variace). Thus, the variace objective may cause us to choose iefficiet ivestmets. I active ad efficiet markets such as major stock markets, you will ted ot to fid ivestmets such as (D) because ivestors will realize (A) domiates (D). Thus, the market price of (D) will drop util its retur approaches competig ivestmets. I ivestmet decisios regardig ew physical facilities, however, there are o strog market forces makig all ivestmet cadidates efficiet, so the variace risk measure may be less appropriate i such situatios.

20 376 Chapter 13 Portfolio Optimizatio Maximizig the Miimum Retur A very coservative ivestor might react to risk by maximizig the miimum retur over scearios. There are some curious implicatios from this. Suppose the oly ivestmets available are A ad C above ad the two scearios are: Sceario Probability Payoff from A Payoff from C If we wish to maximize the miimum possible wealth, the probability of a sceario does ot matter, as log as the probability is positive. Thus, the followig LP is appropriate: The solutio is: MODEL: MAX = WMIN;! Iitial budget costrait; A + C = 1;! Wealth uder sceario 1; - WMIN + A * C > 0;! Wealth uder sceario 2; - WMIN * A * C > 0; Optimal solutio foud at step: 1 Objective value: Variable Value Reduced Cost WMIN A C Row Slack or Surplus Dual Price Give that both ivestmets have a expected retur of 10%, it is ot surprisig the expected growth factor is That is, a retur of 10%. The possibly surprisig thig is there is o risk. Regardless of which sceario occurs, the $1 iitial ivestmet will grow to $1.10 if 50 cets is placed i each of A ad C.

21 Portfolio Optimizatio Chapter Now, suppose a extremely reliable fried provides us with the iterestig ews that, if sceario 1 occurs, the ivestmet C will payoff 1.3 rather tha 1.2. This is certaily good ews. The expected retur for C has just goe up, ad its dowside risk has certaily ot gotte worse. How should we react to it? We make the obvious modificatio i our model: MODEL: MAX = WMIN;! Iitial budget costrait; A + C = 1;! Wealth uder sceario 1; - WMIN + A * C > 0;! Wealth uder sceario 2; - WMIN * A * C > 0; ad re-solve it to fid: Optimal solutio foud at step: 1 Objective value: Variable Value Reduced Cost WMIN A C Row Slack or Surplus Dual Price This is a bit curious. We have decreased our ivestmet i C. This is as if our fried had cotiued o: I have this very favorable ews regardig stock C. Let s sell it before the market has a chace to react. Why the aomaly? The problem is we are basig our measure of goodess o a sigle poit amog the possible payoffs. I this case, it is the worst possible. For a further discussio of these issues, see Clyma (1995).

22 378 Chapter 13 Portfolio Optimizatio Value at Risk I 1994, J.P. Morga popularized the "Value at Risk" (VAR) cocept with the itroductio of their RiskMetrics system. To use VAR, you must specify two umbers: 1) a iterval of time (e.g., oe day) over which you are cocered about losig moey, ad 2) a probability threshold (e.g., 5%) beyod which you care about harmful outcomes. VAR is the defied as that amout of loss i oe day that has at most a 5% probability of beig exceeded. A comprehesive survey of VAR is Jorio (2001). Example Suppose that oe day from ow we thik that our portfolio will have appreciated i value by $12,000. The actual value, however, has a Normal distributio with a stadard deviatio of $10,000. From a Normal table, we ca determie that a left tail probability of 5% correspods to a outcome that is stadard deviatios below the mea. Now: * = So, we would say that the value at risk is $ Example of VAR Let us apply the VAR approach to our stadard example, the ATT/GMC/USC model. Suppose that our risk tolerace is 5% ad we wat to miimize the value at risk of our portfolio. This is equivalet to maximizig that threshold, so the probability our wealth is below this threshold is at most.05. Aalysis: A left tail probability of 5% correspods to probability threshold. We wat to cosider the poit that is stadard deviatios below the mea. Miimizig the value at risk correspods to choosig the mea ad stadard deviatio of the portfolio, so the ( mea * (stadard deviatio)) is maximized. The followig model will do this: MODEL:! Markowitz Value at Risk Portfolio Model(PORTVAR); SETS: STOCKS: AMT, RET; COVMAT(STOCKS, STOCKS): VARIANCE; SETS DATA: STOCKS = ATT GMC USX;!Covariace matrix ad expected returs; VARIANCE = ; RET = ; STARTW = 1.0;! How much we start with; PROB =.05;! Risk threshold, must be <.5; DATA! ;! Get the s.d. correspodig to this risk threshold; PROB Z);

23 Portfolio Optimizatio Chapter With solutio:! Maximize value ot at risk; [VAR] Max = ARET + Z * SD; ARET STOCKS: AMT * RET) ; SD = (@SUM( COVMAT(I, J): AMT(I) * AMT(J) * VARIANCE(I, J)))^.5;! Use exactly 100% of the startig budget; STOCKS: AMT) = STARTW; [VAR] Max = ARET + Z * SD; ARET STOCKS: AMT * RET) ; SD = (@SUM( COVMAT(I, J): AMT(I) * AMT(J) * VARIANCE(I, J)))^.5;! Use exactly 100% of the startig budget; STOCKS: AMT) = STARTW; Variable Value Reduced Cost PROB E Z ARET SD AMT( ATT) AMT( GMC) AMT( USX) E RET( ATT) RET( GMC) RET( USX) Row Slack or Surplus Dual Price E VAR E BUDGET Note that, if we ivested solely i ATT, the portfolio variace would be So, the stadard deviatio would be , ad the VAR would be 1 - ( * ) = The portfolio is efficiet because it is maximizig a weighted combiatio of the expected retur ad (a egatively weighted) stadard deviatio. Thus, if there is a portfolio that has both higher expected retur ad lower stadard deviatio, the the above solutio would ot maximize the objective fuctio above. Note, if you use: PROB =.1988, you get essetially the origial portfolio cosidered for the ATT/GMC/USX problem.

24 380 Chapter 13 Portfolio Optimizatio 13.6 Sceario Model ad Miimizig Dowside Risk Miimizig the variace i retur is appropriate if either: 1) the actual retur is Normal-distributed or 2) the portfolio ower has a quadratic utility fuctio. I practice, it is difficult to show either coditio holds. Thus, it may be of iterest to use a more ituitive measure of risk. Oe such measure is the dowside risk, which ituitively is the expected amout by which the retur is less tha a specified target retur. The approach ca be described if we defie: T = user specified target threshold. Whe risk is disregarded, this is typically less tha the maximum expected retur ad greater tha the retur uder the worst sceario. Y s = amout by which the retur uder sceario s falls short of target. = max{0, T X i u is } The model i algebraic form is the: Mi P s Y s! Miimize expected dowside risk subject to (compute deviatio below target of each sceario, s): Y s T + X i u is 0 X i = 1 (budget costrait) X i P s u is r Notice this is just a liear program. (desired retur).

25 Portfolio Optimizatio Chapter Semi-variace ad Dowside Risk The most commo alterative suggested to variace as a measure of risk is some form of dowside risk. Oe such measure is semi-variace. It is essetially variace, except oly deviatios below the mea are couted as risk. The sceario model is well suited to such measures. The previous sceario model eeds oly a slight modificatio to covert it to a semi-variace model. The Y variables are redefied to measure the deviatio below the mea oly, zero otherwise. The resultig model is: MODEL:! Sceario portfolio model;! Miimize the semi-variace; SETS: SCENE/1..12/: PRB, R, DVU, DVL; ASSET/ ATT, GMT, USX/: X; SXI( SCENE, ASSET): VE; SETS DATA: TARGET = 1.15;! Data based o origial Markowitz example; VE = ;! All scearios happe to be equally likely; PRB= ; DATA! Compute value uder each * X(J));! Measure deviatios from average; DVU( S) - DVL( S) = R(S) - AVG;);! Budget; ASSET: X) = 1;! Compute expected value of edig positio; [DEFAVG] AVG SCENE: PRB * R);! Target edig value; [RET] AVG > TARGET;! Miimize the semi-variace; [OBJ] MIN SCENE: PRB * DVL^2);

26 382 Chapter 13 Portfolio Optimizatio The resultig solutio is: Optimal solutio foud at step: 4 Objective value: E-02 Variable Value Reduced Cost R( 1) R( 2) R( 3) R( 4) R( 5) R( 6) R( 7) R( 8) R( 9) R( 10) R( 11) R( 12) DVU( 1) E DVU( 2) E DVU( 3) DVU( 4) E-01 DVU( 5) E-01 DVU( 6) E-01 DVU( 7) E-01 DVU( 8) DVU( 9) E-01 DVU( 10) E-02 DVU( 11) E-01 DVU( 12) DVL( 1) E-09 DVL( 2) E-09 DVL( 3) E-09 DVL( 4) DVL( 5) X( ATT) X( GMT) E X( USX) Row Slack or Surplus Dual Price BUD DEFAVG E-02 RET OBJ E Notice the objective value is less tha half that of the variace model. We would expect it to be at most half, because it cosiders oly the dow (ot the up) deviatios. The most oticeable chage i the portfolio is substatial fuds have bee moved to USX from GMC. This is ot surprisig if you look at the origial data. I the years i which ATT performs poorly, USX teds to perform better tha GMC.

27 Portfolio Optimizatio Chapter Dowside Risk ad MAD If the threshold for determiig dowside risk is the mea retur, the miimizig the dowside risk is equivalet to miimizig the mea absolute deviatio (MAD) about the mea. This follows easily because the sum of deviatios (ot absolute) about the mea must be zero. Thus, the sum of deviatios above the mea equals the sum of deviatios below the mea. Therefore, the sum of absolute deviatios is always twice the sum of the deviatios below the mea. Thus, miimizig the dowside risk below the mea gives exactly the same recommedatio as miimizig the sum of absolute deviatios below the mea. Koo ad Yamazaki (1991) use the MAD measure to costruct portfolios from stocks o the Tokyo stock exchage Scearios Based Directly Upo a Covariace Matrix If oly a covariace matrix is available, rather tha origial data, the, ot surprisigly, it is evertheless possible to costruct scearios that match the covariace matrix. The followig example uses just four scearios to represet the possible returs from the three assets: ATT, GMC, ad USX. These scearios have bee costructed, usig the methods of sectio , so they mimic behavior cosistet with the origial covariace matrix: MODEL: SETS:! Each asset has a variable value ad a average retur; ASSET/ATT, GMC, USX/: AMT, RET;! the variace of retur at each sceario (which ca be egative), ad the probability of it happeig; SCEN/1..4/: Y, P;! Retur for each asset uder each sceario; COVMAT( SCEN, ASSET):ENTRY; SETS DATA: ENTRY = ; RET = ; P = ; DATA! Miimize the variace; MIN SCEN: Y * Y * P);! Compute the retur uder each of 4 ENTRY(I,J)*AMT(J)) + MEAN = 0 );! The Budget AMT) = 1;! Defie or compute the AMT * RET) = MEAN; MEAN > 1.15;! Target retur;! The variace of each retur ca

28 384 Chapter 13 Portfolio Optimizatio Whe solved, we get the familiar solutio: Optimal solutio foud at step: 4 Objective value: E-01 Variable Value Reduced Cost MEAN AMT( ATT) AMT( GMC) AMT( USX) E-08 RET( ATT) RET( GMC) RET( USX) Y( 1) E Y( 2) Y( 3) Y( 4) E P( 1) P( 2) P( 3) P( 4) ENTRY( 1, ATT) ENTRY( 1, GMC) ENTRY( 1, USX) ENTRY( 2, ATT) ENTRY( 2, GMC) ENTRY( 2, USX) ENTRY( 3, ATT) ENTRY( 3, GMC) ENTRY( 3, USX) ENTRY( 4, ATT) ENTRY( 4, GMC) ENTRY( 4, USX) Row Slack or Surplus Dual Price E E E E Notice the objective fuctio value ad the allocatio of fuds over ATT, GMC, ad USX are essetially idetical to our origial portfolio example.

29 Portfolio Optimizatio Chapter Hedgig, Matchig ad Program Tradig Portfolio Hedgig Give a bechmark portfolio B, we say we hedge B if we costruct aother portfolio C such that, take together, B ad C have essetially the same retur as B, but lower risk tha B. Typically, our portfolio B cotais certai compoets that caot be removed. Thus, we wat to buy some compoets egatively correlated with the existig oes. Examples are: a) A airlie kows it will have to purchase a lot of fuel i the ext three moths. It would like to be isulated from uexpected fuel price icreases. b) A farmer is cofidet his fields will yield $200,000 worth of cor i the ext two moths. He is happy with the curret price for cor. Thus, would like to lock i the curret price Portfolio Matchig, Trackig, ad Program Tradig Give a bechmark portfolio B, we say we costruct a matchig or trackig portfolio if we costruct a ew portfolio C that has stochastic behavior very similar to B, but excludes certai istrumets i B. Example situatios are: a) A portfolio maager does ot wish to look bad relative to some well-kow idex of performace such as the S&P 500, but for various reasos caot purchase certai istrumets i the idex. b) A arbitrageur with the ability to make fast, low-cost trades wats to exploit market iefficiecies (i.e., istrumets mispriced by the market). If he ca costruct a portfolio that perfectly matches the future behavior of the well-defied portfolio, but costs less today, the he has a arbitrage profit opportuity (if he ca act before this mispricig disappears). c) A retired perso is cocered maily about iflatio risk. I this case, a portfolio that tracks iflatio is desired. As a example of (a), a certai so-called gree mutual fud will ot iclude i its portfolio compaies that derive more tha 2% of their gross reveues from the sale of military weapos, ow directly or operate uclear power plats, or participate i busiess related to the uclear fuel cycle. The followig table, for example, compares the performace of six Vaguard portfolios with the idices the portfolios were desiged to track; see Vaguard (1995): Total Retur Six Moths Eded Jue 30, 1995 Vaguard Portfolio Comparative Idex Portfolio Name Growth Growth Idex Name 500 Portfolio +20.1% +20.2% S&P500 Growth Portfolio S&P500/BARRA Growth Value Portfolio S&P500/BARRA Value Exteded Market Portfolio +17.1% +16.8% Wilshire 4500 Idex SmallCap Portfolio Russell 2000 Idex Total Stock Market Portfolio +19.2% +19.2% Wilshire 5000 Idex Notice, eve though there is substatial differece i the performace of the portfolios, each matches its bechmark idex quite well.