Portfolio construction with modified Sharpe’s method

A n n a C zapkiew icz , M a łgorzata M ach ow ska

P O R T F O L I O C O N S T R U C T I O N W I T H M O D I F I E D S H A R P E ’S M E T H O D

ABSTRACT. In the paper we consider a modification of Sharpe’s method used in classical portfolio analysis for optimal portfolio building.

The key idea of the paper is the modification of the classical approach by application of the errors-in-variable model. We assume that both independent (market portfolio return) as well as dependent (given asset’s return) variables are randomly distributed values related with each other by linear relationship and we build the model used for parameters’ estimation.

For model evaluation we made a comparison of portfolios comprising nine stocks from Warsaw Stock Exchange, which are built using classical Sharpe’s and proposed method.

Key words: Sharpe’s model, errors-in-variable model, estimation, comparison.

I. CLASSIC SH A R PE’S M O D EL

In classic Sharpe’s model return o f k-th asset’s R k is explained by market portfolio return R"' trough characteristic line equation according to the relationship:

R k = Pk (R m - rf ) + rf + s k, where rf is riskless rate o f interest and £k has normal distribution. Knowing the market sensitivity ß k o f k-th asset we receive the formula for the expected value and the variance o f portfolio return ratio Rp :

E (R p) = ß ( E ( R " ') - r / ) + rf , D \ R p) = ß 2D \ R ”) + kj r xt D \ £k), ( i) *-i

’ Ph. D, AGH University o f Science and Technology Faculty o f Management. " MSc, AGH University o f Science and Technology Faculty o f Management.

k - K

where ß = ^ xkß k; xk - Л-th stock’s share in investment portfolio. The problem *-i

of optimal portfolio choice may be reducted to finding тах|^£'(Л /)) - Д 0 г(Лр)]

k - K

with condition: x k > 0, ^ xk = 1, where A - risk/profit exchange ratio * -i

coefficient. In this model the ß k coefficient is estimated by least square method.

II. M O D IFIED SH A R PE’S M O D EL

Let us consider the situation, in which both: dependent variable, being the certain asset’s surplus return ratio and denoted R k as well as independent variable - the market portfolio surplus return ratio R"‘ are disturbed observations related with each other by linear equation, i.e.:

R m - r f = t + e ; R k - r f = ß k t + a k + ö k , (2)

If we assume that random variables

£

Sk

In that case, replication o f measurement o f each pair o f observation: dependent and independent; m times overcomes the nonidentifiability (see Bunke & Bunke(1989)).

The following approach was therefore proposed: returns o f given asset and market portfolio are analysed within periods o f t. To be more specific the period o f one month has been assumed as t.

Let n denotes number o f historical months, m number o f monthly returns during given month and К number o f assets in portfolio. Let R'j' and Ä* are j -th mon-thly return ratios in -the i--th mon-th of portfolio market and k --th asset, respectively and

X

We consider the model:

x u = t y + £y> Y j = А к4 „ + а к + 0 * , i k = \ , . . . , K , ( 3 )

It has been assumed that Sv ~ N (0, cr^ ), <^y ~ , o ) ), etJ ~ N (0, ). All the parameters o f aforementioned distributions are unknown.

Now the issue of optimal portfolio choice is finding: m a x [£ (/? p) - A D 2(/?p)J

k - K

under conditions: x k > 0, ^ xk = 1 with: *-l

£ ( * , ) - Z Xt ( ß t s + rf ) , D \ R p) = ( X Xkß k f a ] + £ x ] a l . (4)

*-1 *-i *-•

where s is the expected value for distribution of monthly returns of market portfolio during the period o f the prognosis.

III. ESTIMATION OF UNKNOWN PARAMETERS First step - Estimation of parameters for given asset

For given к in model (3) the number o f unknown parameters to be estimated increases with n. Unknown parameters are: ß k , a k , , a ] , a ] , 5, . It can be shown, that in this model the maximum likelihood estimators have normal distribution asymptotically with respect to m and n (see Dolby(1976)).

1 '» 1 i 1 "í, 1 " , Let: X ľ =

_m

m

n j

n JmX

=-^zz<*„"t-^zzoŕ-iŕ)*.

»-— Z Z f r - O

* i- - Z Z a ‘- 0 -

ß k =

w k s — w s kУ у XX XX у у

- V Ä

2 (w: _О a k = Y k - ß kX ij,

where A = (w s k —s w k Ý —4 ( s k w k —s k w k )(.s w k — s k w )_{\ x x ^ y y ^XX r yy / rryy * yy vv xy /V°xv rvxy a xy rvxx / •} The estimators o f all the other values are expressed in terms o f ß k (Cox(1976)).These estimators have forms of:

К - л « , * ( А > ' А = ? , ( » ' ( A ) + a ( A ) )

where, Ж А ) = 4 ~ 2 Л < + А Ч , . » ' ( Á ) = < - 2 Ä < + Á !

and p, = (Á "■„ - < ) / Ж ( А ) ,

ł( - ( » £ - А < ) / И А ) .

Sccond step - Estimation of unknown parameters for the whole model

For given j we construct vectors: x j , as:

x ? = ( Xxj, . . . X J , y kT = ( Y kv,...Yknj), k = \...K.

The random variables //; = ( х * ,у ^ т,у*т ...у * т) are independent with the same normal distribution and the vector o f expected values // = {sT ,( ß {s + a k)T,

( ß Ks + a k) T) where: s T = (st,...,s n) and with covariance matrix V:

2 2 + c r . ß \ ° ) ■•• ß K ° ] V = V .. . ß l ° ) + *•• ß \ ß x a \ S ® / . ß \ Р к ° 1 ■•• ß W s + ° ] K _

The logarithm o f likehood function for variable f.ij with unknown parameters of: ß t,...,ß k , a t,...,a k , a ] , a ] , crj:,... ,crj , , has a form of:

nt 1 _ W

log L(jUj, 4*) = С ~ “ log(det(F))- ~ ^ Jd JV~'dJ,

/ I jm i

where d / = (//, - / / ) and lF is the set of unknown parameters.

Let Vv denote a matrix composed of matrix V elements after derivation with respect to у / parameters ( у/ being the arbitrary element o f 'F set). The V matrix is symmetric thus:

— \ ogL( f . i , , VY ) = m { - t r ( P V ¥ ) - d Tv V ^ d \ , with P = V \ D - V ) V ^ di y ' [2 J 1 m and D - — S ] d , d j . m j . i It comes that: d Ta V ]d = 0 , — T r(P V fi ) — d p V d - 0 , * 2 d TV - ' d = Q, T r ( P V 2 ) = 0 , T r (P V 2 ) = 0 , T r ( P V 2 ) = 0. s i a j ° S k

From equations d T, V ~ ' d = 0 , d Ts V~xd = 0 we received relationships: a k = Y k - ß k X and

X, +

A

Ук +

1 +

Ax

ßx

+

...AKßK

(5)

where: A t = ß k<r) / <т)к .

From: T r (P V ^ ) = 0 , T r (P V 2 ) = 0 , exploiting certain features o f matrix ah

algebra we obtain:

.+ .,

4 +Äv;- £ £ <у« ~Ał| -gijl.

(6)

From equations (5), (6) and relationship Tr(PV^_) = 0 we calculated cr£2:

' í v v I U ä )

with:

T k ( Ä ) = ( 4 + < ) - 2 ß k (skxy + < ) + Ä 2 ( i e + w „ ),

rtí(Ä,Ä) = Ä2(4 + w^)-2ÄÄ(4+w") + Ä2(5^+w^)>

From relationships (6) eliminating cr2 parameters , we received к o f square equations with cr2 as variable. Applying the Viete’s formulas for each o f them

Substituting c r calculated according to (8) into equation (7), we received

we substitute into one o f к square equations, which gives a basis for calculating

Ultimately the maximum o f likehood function depends only on unknown ß \ , . . . , ß K . The last may be determined numerically starting from initial values equal to their estimators calculated in the first step.

An example portfolio consists o f nine companies listed on the Warsaw Stock Exchange (debica, KGHM, krosno, orbis, PKNorlen, sokolow, TPSA, wolczanka, zywiec - to be more specific). As the market portfolio the portfolio being the base o f WIG index was assumed. The analysis was performed on archival data starting from January 2000 until March 2006.

In practice in portfolio construction it is sufficient to take estimators calculated in the first step. Slight differences in a 2 estimated for various assets have been we received the а г6к in terms o f cr^ for k=2,...,K:

relationship between cr2 and . The parameter a 2 obtained in such a way

corrected by averaging. For estimation o f unknown value o f s, was /*1 It

assumed. It can be shown that: = x /=1

To ilustratc the features o f portfolio constructed with use o f modified method we compared it with that constructed using classical method. It appears that both methods construct quite different contents. Moreover for lambda values, for which there is no significant further portfolio diversification classical method “prefers” (in decreasing order): żywiec, krosno, debica, PKNorlen, sokolow; while the modified one: PKNorlen, żywiec, TPSA, sokolow.

For illustration the expected value and the variance of the portfolio being rebuilt during 14 months have been presented.

Figures 1 and 2 present expected return ratio and its variance for both methods, respectively. It turned out that the expected value is greater than resulting from analysis o f the classic model and the variance is lower. To depict the difference in constructing models we present the real monthly return ratio for optimal contents for both portfolios for chosen X coefficient. The verification was done using past market price (exchange ratio) o f the asset in the month, for which the prognosis was made. Picture 3 contains the comparison o f real profits for both discussed methods. The WIG return Rm is also included as the background reference. It may be noticed that low-risk portfolio built using the proposed method gives per average profits greater than that of classic Sharpe’s portfolio.

Ф E n (R p ) — x — E c (R p ) E (R m ) 0,0200 0 ,0180 0 ,0 1 6 0 0 ,0140

_

£

Fig. 2. Expected return ratio for both methods

portfolio nu m b er

— ♦ —— ne w - c l a s s i c

- - ■ A - ■ W IG

Fig. 3. Real profit comparison for both methods.

V. CO NCLUSIO NS

A new method o f portfolio construction has been proposed. The main assumptions is that, the variable explaining the assets’ return ratios is a random variable biased by a random error with normal distribution. The method o f estimators’ construction in errors-in-variable model extended for many variables

has been shown. Comparing parameters o f portfolios constructed with classic Sharpe’s method and the proposed one, we found that expected returns for our method are bigger, while its variance is lower. In terms o f real profits it means that for higher X (portfolios’ diversification is practically “saturated”) are greater for the presented model.

R E FE R E N C E S

B unke O. Bunke H. (1 9 8 9 ). N o n L in e a r R eg ressio n , F u n c tio n a l R e la tio n sh ip s, a n d R o b u s t M e th o d s. N e w York: W iley.

C ox N .R . (1 9 7 6 ). T he lin e a r str u c tu r a l re la tio n f o r se v e r a l g r o u p s o f d a ta . B iom etrika 6 3 ,2 3 1 - 2 3 7 .

D olb y G .R . (1 9 7 6 ). T he u ltra stru c tu ra l rela tio n a sy n th e s is o f th e f u n c t io n a l a n d s tr u c tu r a l re la tio n s. B iom etrika, 63, 3 9 -5 0 .

Elton E.J. Gruber M.J. (1 9 9 8 ). N o w o c z e sn a te o ria p o r tfe lo w a i a n a liz a p a p ie r ó w w a rto ś c io w y c h . W IG Press, W arszawa.

R eiersol O. (1 9 5 0 ) Identifiability o f a linear relation betw een variables w h ich are subject to error, E conom etrica 18, 5 7 5 -5 8 9 .

Sharpe W . F. (1 9 7 0 ). P o rtfo lio th e o ry a n d c a p ita l m a rkets, M cG R A W -H IL L , N e w York.

A n n a C za p k ie w ic z, M a łg o rza ta M a c h o w sk a

BUDOW A PO R T F E L A A K C JI ZA PO M O C Ą Z M O D Y F IK O W A N E J M ETO D Y SH A R PE ’A

W klasycznej jed n oczyn n ik ow ej analizie portfelow ej konstruując op tym aln y portfel papierów w a rtościow ych , w ykorzystuje się m odel je g o b u d ow y zap rop on ow an y przez Sharpe’a. P od staw ą tej teorii jest założenie, że stopa zw rotu d an ego w aloru jest objaśniana stop ą zw rotu portfela rynkow ego poprzez zależn ość liniow ą. W iadom o, że na zm ien n ość c en w a lo ró w w p ły w m ają rów nież inne (często trudne d o zm ierzenia) czynniki rynku. W klasyczn ym podejściu parametry z a leż n o ści p o m ięd zy stop ą zw rotu danego waloru a stop ą zw rotu portfela rynkow ego w y zn aczan e są z m odelu prostej regresji, g d zie zaburzenie lo so w e je s t dopuszczane tylko na w artości zm iennej objaśnianej. W p roponow anym w pracy m odelu o b ecn o ść tych c z y n n ik ó w u w zględ n ion a jest jak o zaburzenie na obu zm ien n ych lo so w y ch w ch od zących do k la sy czn eg o m odelu

Sharpe’a.

Przyjęto, ż e zarów no stopa zw rotu danego w a lo m jak i stopa zw rotu portfela rynkow ego są p ew n ym i zaburzonym i ju ż w artościam i, m ięd zy którym i istnieje zależn ość liniow a.

Dla ilustracji zagadnienia porównano portfele składające się z dziewięciu spółek zbudowane w oparciu o klasyczną metodę Sharpe’a i proponowaną jej modyfikację. Jako portfel rynkowy przyjęto portfel leżący u podstaw indeksu giełdowego WIG. Analizę przeprowadzono na podstawie notowań archiwalnych od stycznia 2000 do marca 2006. Budując wyżej wymienione portfele miesięczne dokonano porównań przebudowując je co miesiąc uzyskując w ten sposób wektor składów do analizy porównawczej.