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A SIMPLIFIED MODEL FOR PORTFOLIO ANALYSIS*WILLIAM F. SHARPEtUniversity of Washington

This paper d^cribes the advantages of using a particular model of the rela-tionships among securities for practical applications of th e Markowitz portfolioanalysis technique. A computer program has been developed to take full ad-vantage of the model: 2,000 securities can be analyzed at an extremelylow costaa little as 2%of that associated with standard quadratic pro-gramming cod^. Moreover, preliminary evidence suggests that the relativelyfew parameters used by the model can lead to very nearly th e same results ob-tained with much larger sets of relationships among securities. The possi-bility of low-cost analysis, coupled with a likelihood that a relatively smallamount of information need be sacrificed make the model an attractive candi-date for initial practical applications of the Markowitz technique.1. Introduction

Markowitz has si^ested that the process of portfolio selection be approachedby (1) making probabilistic estimates of the future performances of securities,(2) analyzii^ those estimates to determine an efficient set of portfolios and(3) selecting from that set the portfolios best suited to the investor's preferences[1, 2, 3]. This paper extends Markowitz' work on the second of these three stagesportfolio analysis. The preliminary sections state the problem in ite general formand describe Markowitz' solution technique. Theremainder of the paper presentsa simplified model of the relationships among securities, indicates the manner inwhich it allows the portfolio analysis problem to be simplified, and provides evi-dence on the costs as well as the desirability of using the model for practicalapplications of the Markowitz technique.

2. The Portfolio Analysis ProblemA security analyst has provided the following predictions concerning the future

returns from each of N securities:Ei = the expected value of Ri (the return from security i)Ca throi^h C ; Cy represents the covariance between Ri and Ej (as

usual, when i = j the figure is the variance of Ri)* Received December 1961.t The author wishes to express his appreciation for the cooperation of the staffs of boththe Western Data Processing Center at UCLA and the Pacific Northwest Research Com-puter Laboratory at the University of Washington where theprogram was tested. Hisgreatest debt, however, is to Dr. Harry M. Markowitz of the RAND Corporation, with

whom he was privileged to have a number of stimulating conversations during the peatyear. It is no longer possible to segregate the ideas in this paper into those which were his,those which were the a utho r's , and those which were developed jointly. SuflSce it to say th at

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2 7 8 WILLIAM F. SHABPEThe portfolio analyas problem is as follows. Given such a set of predictionsdetermine the set of efficient portfolios; a portfolio is efficient if none other giveither (a) a higher expected retum and the same variance of retum or (b) alower variance of retu m and the same expected retum .Let Xi represent the proportion of a portfolio invested in security i. Then theexpected re tum (E ) and variance of ret um (V ) of any portfolio can be expressedin terms of (a) th e basic da ta (i-values and C . Figure 1 illustrates this relationship for a particula r value of X. The linfe shows the com binations of E and V which give 4> = 4n, where ^ = \kE Vthe othe r lines refer to la i ^ r values of 0( 0j > ^ ) . Of all possible portfolios,one will maximize the value of ^;' in figure 1 it is portfolio C. The relationshipbetween this solution and the portfolio analysis problem is obvious. The E, Vcombination obtained will be on the boundary of the set of attainable combina-tions; moreover, the objective function will be tangent to the set at that point.Since this function is of the form

0 = X ?- F* Since cash can be included as one of the securities (explicitly or imp licitly) this assump-

tion n eed cause no lack of realism.* Th is is th e standard formulation. Casra in which short sales are allowed require a differ-ent approach.

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A SIMPLIFIED MODEL FOB PORTFOLIO ANALYSIS 279

PiGTTBB 1

the slope of the boundary at the point must be X; thu s, by varying Xfrom -|- to 0, every solution of the portfolio analysis problem can be obtained .For any given value of X the problem described in this section requires themaximization of a quadratic function, (which is a function of X,-, X / , andXiXj terms) subject to a linear cons traint (23< ^i 1) > with the variables re-stricted to non-negative values. A num ber of techniques have been developed tosolve such quadratic programming problems. The critical line method, developedby Markowitz in conjunction with his work on portfolio analysis, is particu larlysuited to this problem and was used in the program described in this paper.3 . The Critical Line Method

Two imp orta nt characteristics of the set of efficient portfolios make system atic

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2 8 0 WILLIAM F. SHABPE

described in te rm s of a smaller set of comer portfdioa. Any point on the E, V curv(other than the po ints associated with com er portfolios) can be obtained w itha portfolio constructed by dividing the total investment between tiie two ad-jacent comer portfolios. For exam ple, the portfolio which gives E, V combinationC in Figure 1 might be some linear com bination of th e two com er portfolios withE, V combinations shown by points 2 and 3 . This characteristic allows iJie ana lystto restrict his atten tion to comer portfolios rath er t ha n the complete set ofefficient portfolios; the latter can be readily derived from the former.The second characteristic of the solution concerns the relationships amongcomer portfolios. Two comer portfolios which are adjacent on the E, V curveare related in the following m ann er: one portfolio will contain either (1) all thesecurities which appear in the other, plus one additional security or (2) all bu tone of the securities which appear in the other. Thus in moving down the E, Vcurve from one comer portfolio to the next, the quantities of the securities inefficient portfolios will vary until either one drops out of the portfolio or anotherenters. The point a t which a change takes place marks a new comer portfolio.The major steps in the critical line method for solving the portfolio analysisproblem are:1. The comer portfolio with X =

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282 VILLIAH 7.be the same at each point along the line). Finally, A^+t indicates the expectedvalue of / and Qn+i the variance around t ha t expected va lue.The diagonal m odel requires the followii^ predictions from a security analyst:1) values of Ai, Bi and Q,- for each of N securitira2) values of A^+i and Qn -i for the index / .The number of estimates required from the analy st is thu s greatiy reduced : from5,150 to 302 for an analysis of 100 securities and from 2,003,000 to 6,002 for ananalysis of 2,000 securities.Once the parameters of the diagonal model have been specified all the inputsrequired for the standard portfolio analysis problem can be derived. The rela-tionships are:

Ei = 4< -f .B

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A SmPUFIED MODEL FOR POBTrOLIO ANALYSIS 283

Probabi l i ty FIGURE 2Defining Xn+i as the weighted average responsiveness of Rp to the level of / :

n+1 2-1im.1and substituting this variable and the formula for the determinants of /, weobtain:

^.(^< + ^.O + Xn+Mn+l -I- Cn+l)H+l

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284 WILLIAM F. SHABPE

The expected retum of a portfolio is thus:rr+lE = Y

while the variance is:'if+iV - Yx'o

This formulation indicates the reason for the use of the parameters An+i andQn+i to describe the expected value and variance of the future value of 7. Italso indicates the reason for calling this the "diagonal model". The variance-covariance matrix, which is full when N securities are considered, can be ex-pressed as a matrix with non-zero elements only along the diagonal by includingan (n + l)st security defined as indicated. This vastly reduces the number ofcomputations required to solve the portfolio analysis problem (primarily in8tep 2 of the critical line method, when the variance-covariance m atrix mu st beinverted) and allows the problem to be stated directly in terms of the basicparameters of the diagonal m odel:

Maximize: X^ FWhere: f? = E

N+l

Subject to: Xi ^ 0 for all i from I to N

i1

6. The Diagonal Model Portfolio Analysis CodeAs indicated in the previous section, if the portfolio analysis problem is ex-pressed in terms of the basic parameters of the dif^onal model, computing timeand memory space required for solution can be greatiy reduced. This sectiondescribes a machine code, writte n in the FORTRAN language, which takes fulladvantage of the characteristics of the diagonal model. It uses the critical linemethod to solve the problem stated ia the previous section.The computing time required by the diagonal code is considerably smallerthan that required by standard quadratic programmii^ codes. The RAND QP

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A SIMPLIFIED MODEL FOR PORTFOLIO ANALYBIS 2 8 5

code' required 33 minutes to solve a 100-security example on an IBM 7090computer; the same problem was solved in 30 seconds with the diagonal code.Moreover, the reduced storage requirements allow many more securities to beanalyzed: with the IBM 709 or 7090 the RAND QP code can be used for nomore than 249 securities, while the diagonal code can analyze up to 2,000securities.

Although the diagonal code allows the total computing time to be greatlyreduced, the cost of a large analysis is still far from insignificant. Thus there isevery incentive to lim it the com putation s to those essential for the final selectionof a portfolio. By taking into account the possibilities of borroAving and lendingmoney, the diagonal code restricts the computations to those absolutely neces-sary for determination of the final set of efficient portfolios. The importance ofthese alternatives, their effect on the portfolio analysis problem and the mannerin which they are taken into account in the diagonal code are described in theremainder of this section.A . The "lending portfolio"

There is some interest ra te (ri) at w hich money can be lent with virtual as-surance that both principal and interest will be returned; at the least, moneycan be buried in the ground {r i = 0). Such an alternative could be included asone possible security {A i = 1 + ri, Bi = 0, Qi = 0) but this would necessitatesome needless computation.' In order to minimize computii^ time, lending atsome pure intere st ra te is taken in to accoun t explicitly in the diagonal code.The relationship between lending and efficient portfolios can best be seen interms of an E, a curve showing the combinations of expected return and standarddeviation of retu rn (= \ / F ) associated with efficient portfolios. Such a curveis shown in Figure 3 (F B C G ); point A indicates the E, a combination attained ifall funds are len t. The relationship between lending money and purchasing po rt-folios can be illustrated with th e portfolio which has the E, a combination shownby point Z. Consider a portfolio w ith Xi invested in portfolio Z and the remainder(1 X J ) lent at the rate ri . The expected return from such a portfolio would be;E = X .B. + (1 - X .) (1 -f r,)

and the variance of retu rn would be:V = X .'F , + (1 - X .)V , -f- 2X .(1 - X .) (cov.,)

The program is described in [4]. Several alte rnative quadratic programming codes areavailable. A recent code, developed by IBM , which uses the critical line method is likely toprove considerably more efficient for the portfolio analysis problem. The RAND code isused for comparison since it is the only standard program with which the author has hadexperience.^ Actually, the d iagonal code canno t accept non-positive values of Qi; thiis if the lendingalternative is to be included as simply another security, it must be assigned a very small

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286 WILLIAM F. SHABPE

E

FlGUBB 3

But, since Vi and cov,i are both zero:V = X.'V,

and the standard deviation of return is:

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A SIUPL IVIED MODEL FOB POBTFOLIO ANAL'TOIS 2 8 7

E below th at aeeodated with portfolio B, th e m (t efficient portfolio will be somecombination of portfolio B and lending. Portfolio B can be termed the "lendingportfolio" since it is the appropriate portfolio whenever some of the investor'sfunds are to be lent at the rate rt . This portfolio can be found readily once theE, a curve is known. I t lies at the point on the curve a t which a ray from(JB = 1 -f f I,

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2 8 8 WILLIAM F . SHABFBto perform an analysis on an IBM 709 computer was obtained by analyzing aseries of runs during which the time required to complete each major s^ m en tof the program was recorded. The approximate time required for the analysiswill be.-

Number of seconds = .6+ .114 X num ber of securities analyzed+ .54 X number of comer portfolios+ .0024 X num ber of securities analyzed X num ber ofcomer portfolios.Unfortunately only the number of securities analyzed is known before theanalysis is begun. In order to estimate the cost of portfolio analysis before it isperformed, some relationship between the number of comer portfolios and thenum ber of securities analyzed m ust be assumed. Since no theoretical relationshipcan be derived and since the total number of comer portfolios could be severaltimes the num ber of securities analysed, it seemed desirable to ob tain some crudenotion of the typical relationship when "reasonable" inputs are used. To ac-complish this, a series of portfolio analyses was performed using inputs generatedby a Monte Carlo model.Data were gathered on the annual returns during the period 1940-1951 for96 industrial common stocks chosen randomly from the New York Stock Ex-change. The returns of each security were then related to the level of a stockmarket index and estimates of the parameters of the diagonal model obtained.These parameters were assumed to be samples from a population oi Ai, Bi andQi triplets related as follows:

Ai = A+ nBi = B + i^Ai -} - rjQi = Q + 0Ai + yBi + rz

where n , ri and r% are random variables with zero means. Estimates for thparameters of these three equations were obtained by regression analysis andestimates of the variances of the random variables determined.' With this in-formation the characteristics of any desired number of securities could begenerated. A random number generator was used to select a value for .4j; thisvalue, together with an additional random number determined the value ofBi ; the value of Q, was then determined with a third random number and thepreviously obtained values of Ai and Bi.Figure 4 shows the relationship between the number of securities analyzed' The computations in this section are based on the assumption that no corner port-folios prior to th e lending portfolio are prin ted . If the analyst chooses to prin t all preceding

portfolios, the estimates given in this section should be miiltiplied by 2.9; intermediatecases can be estimated b y interpolation .' The random variables were considered normally distribu ted; in one case, to b etter ap-

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A SIMPLIFIED MODEL FOB POBTFOLIO ANALYSIS 289Kamber of cornerpo rtfo l ios with" aninterest rate^^

H H

1,000 2,000ntmber of securities analyzedFlQUBB 4

and the number of comer portfolios with interest rates greater than 3% (anapproximation to the "lending rate"). Rather than perform a sophisticatedanalysis of the^ data, several lines have been used to bracket the results invarious ways. These will be used subsequen tly a s extreme cases, on the p resump-tion that most practical cases will lie within these extremes (but with no pre-sumption th at these limits will never be exceeded). Curve A indicates the average

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290 WILLIAM F. SHABPBaverage (N,/N,) = .37. Curve Hi in d ic at e the highest such relationship: maximum (Np/N.) .63; th e line Z>i indicates the lowest: minimum (Np/N.) = .24The other two curves, Ht and Lt, indicato r^pectively the maTfimum deviationabove (155) and below (173) the number of comer portfolios indicated by theaverage relationship Np = .37 N,.In Figure 5 the total time required for a portfolio analysis is related to thenumber of securities analyzed under various assumptions about the relationship$ Minutes

500 1,000 1,500 2,000

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A BIMFLIFISD MODEL FOE POBTFOLIO ANALYSIS 291between ihe number of comer portfolios and the number of securities analyzed.Each of the curves shown in Figure 5 is based on the corresponding curve inFigure 4; for example, curve A in Figure 5 indicates the relationship betweentotal time and num ber of securities analyzed on the assumption th at the relation-ship between the num ber of comer portfolios and th e number of securities is th atshown by curve A in Figure 4 . For convenience a second scale has been providedin Figure 5, showing the total cost of the analysis on the assumption that anIBM 709 computer can be obtained a t a cost of $300 per hour.

8. The Value of Portfolio Analysis Based on the Diagonal ModelThe assumptions of the dif^onal model lie near one end of the spectrum ofpossible assumptions about the relationships among securities. The model'sextreme simplicity enables the investigator to perform a portfolio analysis at a

very small cost, as we have shown. However, it is entirely possible that this sim-plicity so restricts the security analy st in making his predictions tha t th e value ofthe resulting portfolio analysis is also very small.In order to estimate the ability of the diagonal model to summarize informa-tion concerning the perfonnance of securities a dmple test was performed.Twenty securities were chosen randomly from the New York Stock Exchangeand their perfonnance during the period 1940-1951 used to obtain two sets ofPercent of portfolioinvested

100 r

75

50

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292 W U J J I A M F . SHABPE1 0 0

50

\\

Fio. 6b. Composition of efficient portfolios derived from the analysis of historical data

d at a : (1) the ac tual mean returns, variances of returns and covariances of returnsduring the period and (2) th e param eters of the diagonal model, estimated byregress ion techniques from the performance of the securities during the periodA portfolio analysis was then performed on each set of data. The results aresummarized in Figures 6a and 6b. Each security which entered any of th e efficientportfolios in significant amounts is represented by a particular type of line; theheight of each line above any given value of E indicates the percentage of theefficient portfolio with that particular E composed of the security in questionThe tw o figures th us indica te the com positions of all the efficient portfolioschosen from the analysis of the historical dat a (Figure 6b) and the compositionsof all the portfolios chosen from the analysis of the parameters of the diagonalmodel (Figure 6 a ). Th e similarity of the two figures ndicates tha t the 62 param-eters of the diagonal model were able to capture a great deal of the informationcontained in the complete set of 230 historical relationships. An additional test,using a second set of 20 securities, gave similar results.These results are, of course, far too fragmentary to be considered conclusivebu t they do s i ^ e s t th at the diagonal model may be able to represent the relationships among securities rather well and thus that the value of portfolio analysesbased on the model will exceed their rather nominal cost. For these reasons it

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A SIMPLIFIED MODEL FOB POBTFOLIO ANALYSIS 2 9 3Refermcee

1. MABKOWITZ, H ABBT M. , Portfolio Selection, Efficient Diversification of Investments, NewYork, John W i ley and Sons , In c . , 1959.2. MABKOW ITZ, H ASB T M ., "Port fol io Select ion" , The Journal of Finance, Vol. 12, (March

1952), 77-^1 .3. MABKOW ITZ, H ABBT M . , "The Optimization of a Quadratic Function Subject to LinearCons t ra in t s , " Naval Research Logistics Quarterly, Vol. 3, (March and June, 1956) ,111-133.4. W oL m, P H I I I P , "The Simplex Method for Quadrat ic Programming," Econometrica,Vol. 27, ( Ju ly, 1959), 382-3M.

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