ریسک و بازده

Risk and Re turn Prop er ties of Port fo lios Based on

Directional Forecasts

by Kath ryn A. Wilkens, Worces ter, MA 01609; Jean L. Heck, and Ste ven J. Co chran, De -

part ment of Fi nance, Col lege of Com merce and Fi nance, Vil la nova Uni ver sity, Vil la nova,

PA 19085

Abstract

In this study, a for mula is de rived for the pe riod spe cific beta (mar ket risk) for a port fo lio

of fi nan cial as sets that has been formed on the ba sis of di rec tional fore casts. This is an im -

por tant con tri bu tion to the lit era ture since meas ur ing the risk of an ac tively man aged port -

fo lio is prob lem atic due to the fact that man ag ers may change fund risk con di tional on

mar ket ex pec ta tions. The period- specific na ture of the meas ure is a sig nifi cant ad van tage

since his tori cal fund re turns are not re quired and the beta is not in flu enced by prior fund

re turns’ de via tions from the bench mark. The meth od ol ogy em ployed al lows for the de vel -

op ment of a time se ries of fund be tas that per mits in ves ti ga tion into a number of im por tant

em piri cal is sues. This study is also of prac ti cal in ter est from the per spec tive of risk man -

age ment and for both port fo lio per form ance and at tri bu tion. Fi nally, there are many ac tive

strate gies based on di rec tional fore casts and the ap proach used here en com passes a sig nifi -

cant pro por tion of these.

I. In tro duc tion

Dur ing pe ri ods of sub stan tial mar ket de cline, there is of ten in creased at ten tion paid to

stock pick ing abili ties and to al ter na tive in vest ments that have a low cor re la tion to tra di -

tional as sets. There is, how ever, sub stan tial evi dence that un sta ble fund be tas may be as so -

ci ated with such ac tive in vest ment strate gies and that these time- varying be tas in tro duce a

bias in per form ance meas ures. The im pli ca tion, of course, is that it may be dif fi cult to

meas ure the risk level and per form ance of an ac tively man aged port fo lio. The pri mary

con tri bu tion of this ar ti cle to the lit era ture is that a for mula is de rived for the period-

specific beta (mar ket risk) for a port fo lio of fi nan cial as sets that has been formed on the ba -

sis of di rec tional fore casts for in di vid ual se cu ri ties rela tive to a bench mark. The period-

specific na ture of the meas ure is a sig nifi cant ad van tage since his tori cal fund re turns are

not re quired. Con se quently, fund re turns’ past de via tions from the bench mark are not in -

cluded in the cal cu la tion of the cur rent pe ri od’s fund beta and, in this sense, the beta es ti -

mate is not di luted.

The frame work used in this ar ti cle should be of in ter est to both aca dem ics and prac -

ti tio ners. The meth od ol ogy em ployed al lows for the de vel op ment of a time se ries of fund

be tas that per mits in ves ti ga tion into a broad range of im por tant theo reti cal and em piri cal

is sues, such as the ma nipu la tion of fund risk in re sponse to chang ing eco nomic con di tions

and the possi ble existence of market inefficiencies.1 This study is also of prac ti cal in ter est

from the per spec tive of risk man age ment and for both port fo lio per form ance and at tri bu -

tion and, thus, has im pli ca tions for the ef fi cient al lo ca tion of in vest ment funds among

man ag ers. Fi nally, there are many ac tive strate gies based on di rec tional fore casts and the

ap proach de vel oped here can be ap plied to a sig nifi cant pro por tion of these. With mi nor

modi fi ca tions, the frame work used in this ar ti cle for ana lyz ing fund risk and re turn can ac -

commodate many investment styles, analytic approaches, factor expo sures, bench marks,

Managerial Finance 58

as set classes and other key risk com po nents. It is ro bust enough to ap pro pri ately meas ure

risk for tra di tional mu tual funds as well, al though the pri mary con tri bu tion here is the ap -

pli ca tion to funds where the man ager is mak ing ex plicit di rec tional fore casts.

The re main der of this ar ti cle is or gan ized as fol lows. A re view of the lit era ture on di -

rec tional fore cast ing, where the fo cus is on its im por tance and al ter na tive meth od olo gies,

and on per form ance analy sis and time- varying risk for ac tively man aged funds is pre -

sented in Sec tion II, fol lowed by the de vel op ment of the meth od ol ogy in Sec tion III. In

Sec tion IV, the for mu las for the beta of an ac tively man aged port fo lio at any point in time

are pro vided and a sum mary of the ba sic steps re quired in any em piri cal analy sis us ing this

meth od ol ogy are given. Sec tion V con cludes and of fers di rec tion for fu ture em piri cal re -

search in this area.

II. Lit era ture Re view

Di rec tional Fore cast ing

While there is no gen eral agree ment as to the best meth od olo gies to use in ana lyz ing the

good ness of eco nomic fore casts, most stud ies that have as sessed their ac cu racy have fo -

cused on quan ti ta tive er rors (see, for ex am ple, Clements and Hen dry (1993, 1995), Arm -

strong and Fil des (1995), Makri dakis and Hi bon (1995), Ol ler and Ba rot (2000), and Pons

(2000)). How ever, sev eral re cent stud ies have re sulted in a re newed in ter est in di rec tional

analy sis, in clud ing those by Lai (1990), Schnader and Stek ler (1990), Pe sa ran and Tim -

mer mann (1992a, 1994), Ash et al. (1998), and Pons (2001).2 In Lai (1990), the Hen riks -

son and Mer ton (1981) non para met ric test of di rec tion is used to evalu ate sur vey fore casts

for several curren cies.3 Lai (1990) ar gues that while un bi ased fore casts are pref er able to

bi ased fore casts, the lat ter may be use ful if they are on the cor rect side of the price change

more of ten than not. This im plies that for mal tests of di rec tion are de sir able. Birche nall et

al. (1996), in con sid er ing eco nomic pol icy, takes a po si tion simi lar to that of Lai (1990).

While not ing that quali ta tive state ments con cern ing eco nomic pol icy are dif fi cult to make,

they are im por tant com po nents in the for mu la tion and im ple men ta tion of pol icy. Birche -

nall et al. (1996) ar gues that the in abil ity of tra di tional fore cast ing meth ods to pre dict the

down turn ex pe ri enced in many coun tries in the early 1990s and the sub se quent dif fi culty

in dat ing the on set of re cov ery has in creased the in ter est in al ter na tive meth od olo gies that

con sider the di rec tion of move ment in the busi ness cy cle.

Schnader and Stek ler (1990) em ploy two meth ods to de ter mine whether fore casts of

real GNP are use ful to us ers. The first is the Hen riks son and Mer ton (1981) non para met ric

method for de ter min ing the con di tions un der which a mar ket tim ing fore cast has value.

The sec ond is the c2 test of in de pend ence in a 2x2 con tin gency ta ble re lat ing the di rec tion

of ac tual and pre dicted changes. Pe sa ran and Tim mer mann (1992a) de velop a

distribution- free pre dic tive fail ure test for de ter min ing the ac cu racy of di rec tional fore -

casts which dif fers from the test de rived by Hen riks son and Mer ton (1981).4 In Pe sa ran

and Tim mer mann (1994), a gen er ali za tion of the Hen riks son and Mer ton (1981) test of

mar ket tim ing from the 2x2 case to the case with n cate go ries is de rived.5 The gener alized

test is used to evalu ate the mar ket tim ing per form ance of a two- fund in vest ment strat egy in

the pres ence of trans ac tion costs, where the strat egy fo cuses on the switch ing of funds be -

tween two as sets, a stock fund and a bond fund. Ash et al. (1998) and Pons (2001) em ploy

tests based on Mer ton (1981) and Hen riks son and Mer ton (1981) to evalu ate the ra tion al ity

and use ful ness of fore casts for the im plicit GDP price de fla tor made by the IMF for the G7

Volume 30 Number 8 2005 59

coun tries and for fore casts of vari ous eco nomic vari ables made by the OECD for the G7

coun tries, respectively.

Acar (1998) and Acar and Satchell (1998) de rive re sults for the ex pected re turns of,

as well as dis tri bu tional char ac ter is tics for port fo lios of re turns from, di rec tional fore casts.

A theo reti cal ap proach is taken in or der to avoid the limi ta tions of period- specific em piri -

cal find ings that may not gen er al ize across as sets classes or time pe ri ods. With re spect to

port fo lio risk, it is shown that con ven tional port fo lio the ory does not ap ply to ac tive strate -

gies. Re sults are gen er al ized for a sub class of the el lip ti cal fam ily of dis tri bu tions and it is

ar gued that knowl edge of theo reti cal ac tive port fo lio re turns has nu mer ous po ten tial ap pli -

ca tions, such as con struct ing new tests of the ran dom walk from the joint prof it abil ity of

ac tive strate gies. Fi nally, while Acar (1998) and Acar and Satchell (1998) are suc cess ful at

de riv ing the vari ance of a port fo lio of ac tive strate gies, the use of their re sults de mands

knowl edge of the cor re la tion be tween the vari ous ac tive strat egy sig nals.6

Per form ance Analy sis and Time- Varying Risk

Sev eral is sues re lated to port fo lio per form ance analy sis and time- varying risk have re -

ceived con sid er able in the lit era ture. Mer ton (1981), Hen rikk son and Mer ton (1981), and

Fer son and Schadt (1996) dis cuss the dif fi culty of meas ur ing port fo lio risk when tra di -

tional fund man ag ers ma nipu late be tas con di tional on mar ket ex pec ta tions. Sepa rat ing the

ef fects of se cu rity se lec tion, where price move ments of in di vid ual stocks are fore cast rela -

tive to stocks in gen eral, from mar ket tim ing, which in volves the fore cast ing of the gen eral

stock mar ket rela tive to fixed in come se cu ri ties, has been a re lated con cern in meas ur ing

in vest ment per form ance.7 Vari ous mod els have been de vel oped that ad dress the con tri bu -

tions of se cu rity se lec tion and mar ket tim ing to port fo lio re turn, in clud ing those of Jensen

(1972) and Hen riks son and Mer ton (1981). Within the con text of the con ven tional mean-

variance Capi tal As set Pric ing Model (CAPM), Jensen (1972) shows that the sepa rate con -

tri bu tions of se cu rity analy sis and mar ket tim ing can not be iden ti fied un less, for each pe -

riod, the mar ket tim ing fore cast, the port fo lio ad just ment cor re spond ing to the fore cast,

and the ex pected re turn on the mar ket are known. On the other hand, the para met ric test of

di rec tional move ment in Hen riks son and Mer ton (1981), which as sumes a CAPM frame -

work, per mits the iden ti fi ca tion of the sepa rate con tri bu tions of se cu rity analy sis and mar -

ket tim ing to a port fo lio’s re turn us ing only ex post port fo lio and mar ket re turns. Pe sa ran

and Tim mer mann (2002) note that al though tac ti cal as set al lo ca tion at tempts by fund man -

ag ers de pend on di rec tional fore casts, the is sue of sign pre dict abil ity when the dis tri bu -

tional char ac ter is tics of the re turn gen er at ing pro cess are not time in vari ant has not been

widely in ves ti gated in the lit era ture. These prob lems are mag ni fied when ac tively man -

aged non tra di tional funds are ana lyzed. For ex am ple, as noted in Schnee weis (1996), there

ex ists a well- documented non line ar ity in the risk of hedge funds, that is, they tend to have

posi tive be tas in up mar kets and nega tive be tas in down mar kets. This is the case even

though hedge funds largely fol low “ab so lute re turn” strate gies where re turns are of ten de -

rived from rela tive value plays and strate gies that are not re lated to broad mar ket moves

(Un dis cov ered Man ag ers, 2001).

Al though varia tions ex ist, most of the mod els that have been de signed to cap ture or

ad dress ef fects of chang ing fund be tas rely on his tori cal fund re turns. These in clude the

quad ratic re gres sion ap proach of Trey nor and Ma zuy (1966), the switch ing re gres sion

meth od ol ogy of Kon and Jen (1978), the up- and down- market be tas of Mer ton (1981) and

Hen riks son and Mer ton (1981), the sto chas tic dis count fac tor meth od ol ogy of Han sen and

Managerial Finance 60

Ja ga na than (1991) and Chen and Knez (1996), and the con di tional per form ance evalua tion

meth od ol ogy of Fer son and Schadt (1996).8 In con trast to these ap proaches, which fo cus

on vola til ity (stan dard de via tion) over time, the meth od ol ogy pre sented here is based on

dis per sion sta tis tics for the bench mark uni verse for a par ticu lar in ter val of time. While

vola til ity is a time se ries sta tis tic, that is, the stan dard de via tion of re turns on a se cu rity or

port fo lio over suc ces sive pe ri ods of time, dis per sion is the single- period cross- sectional

stan dard de via tion within a group of as sets or as set class. Al though vola til ity and dis per -

sion are re lated, it is pos si ble that high (low) lev els of vola til ity can be ac com pa nied by

low (high) lev els of dis per sion.9

Re cent stud ies that use dis per sion sta tis tics in clude Bes simbinder et al. (1996), who

em ploy se cu rity re turn dis per sion as a proxy for company- specific in for ma tion flows, and

Sol nik and Rou let (2000), who de velop a new meth od ol ogy for us ing the cross- sectional

dis per sion in re turns to im prove es ti mates of global cor re la tion or the cor re la tions be tween

coun try mar kets. The cross- sectional method in Sol nik and Rou let (2000) is dy namic and

pro vides in stan ta ne ous in for ma tion on chang ing lev els of global mar ket cor re la tion.10

Since global as set al lo ca tion should be struc tured along the lines that maxi mize global dis -

per sion and mini mize global cor re la tion, Sol nik and Rou let (2000) point out that us ing dis -

per sion in es ti mat ing cross- sectional cor re la tion has im pli ca tion in the cur rent de bate

con cern ing the rela tive im por tance of coun try and in dus try al lo ca tion in global as set man -

age ment. de Silva et al. (2001) em ploy port fo lio the ory to de velop a link be tween se cu rity

dis per sion and fund dis per sion. Spe cifi cally, de Silva et al. (2001) find that, al though the

cor re la tion is not con stant, most of the in tertem po ral varia tion in fund dis per sion re sults

from changes in se cu rity dis per sion.11 It is ar gued that in for ma tion em bed ded in se cu rity

re turn dis per sion, as well as in for ma tion on re turn means, is use ful in per form ance evalua -

tion. Fund re turns in ex cess of the stock in dex bench mark should be cor rected by a

period- specific dis per sion sta tis tic that ad justs for changes in re turn dis per sion over time.12

The cor rec tion pro posed is to the al pha es ti mate for the het ero sce das tic ity pres ent in non -

con stant lev els of re turn dis per sion and, to the ex tent that the mar ket is in ef fi cient, the cor -

rected al pha will be a bet ter in di ca tor of mana ge rial tal ent that the un cor rected al pha.

III. A Base Model: Static Fund Risk and Re turn

The ob jec tive of this ar ti cle is to de velop an ap pro pri ate risk meas ure for an ac tively man -

aged port fo lio based on di rec tional fore casts while re main ing within an as set pric ing para -

digm with which in sti tu tional in ves tors are fa mil iar. Al though this is not an em piri cal

study, it should be of in ter est to both prac ti tio ners and aca dem ics whose main in ter est is in

em piri cal re search. The base model de scribes risk and re turn for a given time pe riod and,

as stated above, his tori cal fund re turns are not re quired in the deri va tion. By re peat ing the

pro ce dure out lined be low for mul ti ple time pe ri ods, a time se ries of fund al phas and fund

be tas may be ob tained. In de riv ing the base model, the fol low ing sim pli fy ing as sump tions

are made: (1) no trans ac tions costs, (2) a single- period frame work where all trans ac tions

are con ducted on the first day of the in vest ment pe riod, (3) no ad di tional funds flow into or

out of the port fo lio intra- period, (4) the port fo lio is equal- weighted, (5) for the rele vant in -

vest ment pe riod, 50% of the se cu ri ties in the bench mark uni verse will un der per form the

bench mark and 50% of the se cu ri ties will out per form the bench mark, (6) the fund man ager

con sid ers only the di rec tion of se cu rity per form ance rela tive to the bench mark and not the

mag ni tude of the rela tive per form ance, and (7) in con trast to fund be tas, in di vid ual se cu -

rity be tas are rela tively sta ble over time.

Volume 30 Number 8 2005 61

Managerial Finance 62

Bench mark Re turn and Port fo lio Return

RO and RU are de fined as the av er age re turns of the se cu ri ties that out per form and un der -

per form, re spec tively, the over all bench mark mean re turn, B. The dis tri bu tional as sump -

tions per tain ing to B, RO, and RU are il lus trated in Ex hibit 1. The bench mark re turn is the

average of RO and RU and is de fined as

B = 50%(RO)+50%(RU) (1)

where13

RO = B + sBE(Z) (2)

RU = B - sBE(Z) (3)

and

E(Z) = 2/ 2p. (4)

The port fo lio re turn, RP, de pends on the weights in se cu ri ties rela tive to the bench -

mark re turn and can be ex pressed as

RP = wRORO + wRURU. (5)

The port fo lio weights, wRO and wRU, are de ter mined by the fund mana ger’s skill and

the level of turn over. With full turn over and per fect skill, the ex pec ta tion is that wRO = 1

and wRU = 0. An un skilled man ager would, on av er age, choose wRO = . 5 and wRU =. 5 and

the port fo lio would earn the bench mark re turn. Given both the dis per sion sta tis tics on the

uni verse of se cu ri ties the man ager is se lect ing from and the ob served port fo lio re turn, the

weights wRO and wRU can be in ferred from the as sump tion that their sum equals one.

Port fo lio Risk

Hav ing de fined the weights in se cu ri ties with av er age re turns RO and RU above, we note

that the beta of the un con strained port fo lio is de fined as

RU B RO

Prob(x <= RU)=.25, Prob(x <= B) = .5, Prob(x <= RO) = .75

Ex hibit 1

Prob (x <= RU) =.25, Prob (x <= B) = .5, Prob (x <= RO) = .75

bP = wRO bRO + wRUbRU (6)

where14

bRO = wRO-b- + wRO+b+ (7)

bRU = wRU-b- + wRU+ b+. (8)

The quan ti ties b- and b+ rep re sent the av er age be tas of all as set be tas that are less than and

greater than the bench mark, where the bench mark beta is de fined as bB = b=1, and are de -

fined as

b b sb += + E Z ( ) (9)

b b sb -= - E Z ( ) (10)

and where, as given above, E(Z) = 2/ 2p. The dis tri bu tional as sump tions as so ci ated with

bB, b+, and b- are pre sented in Ex hibit 2.

The port fo lio weights in as sets with betas av er ag ing b- and b+, that is, wRU- , wRU+,

wRO- and wRO+, are de fined as a per cent age of the area of the ‘re gres sion el lipse’ (Galton,

1886) that en closes most points about the em pir i cal mar ket line. With an il lus tra tive

bench mark re turn of 10%, a risk pre mium of 6%, and a risk-free rate of 4%, the rel e vant

ar eas of the el lipse are shown in Ex hib its 3a and 3b. The av er age beta of the as sets rep re -

sented by the shaded area in Ex hibit 3a is b-. The lighter shaded area in Ex hibit 3b rep re -

sents wRO-, the pro por tion of all as sets with a beta less than one that out per form the

bench mark. The darker shaded area in Ex hibit 3b rep re sents wRO+, the pro por tion of all as -

sets with a beta greater than one that out per form the bench mark. These pro por tion are

given as

wRO- = Area(A+B)/Area(A+B+C+D) (11)

wRO+ = Area(C+D)/Area(A+B+C+D). (12)

Volume 30 Number 8 2005 63

b - b b+

Prob(x <= b-)=.25, Prob(x <= b) = .5, Prob(x <= b+) = .75

Ex hibit 2

Prob (x ) .25, Prob(x ) .5, Prob(x ) .75 <= = <= = <= = - + b b b

b b b - +

IV. Re sults

Equa tions (6)-(12) de fine the single- period port fo lio beta. The re la tion ships pre sented be -

low in equa tions (13)-(23) per mit the cal cu la tion of the beta, given only the ob served re -

Managerial Finance 64

Ex hibit 3a

Bench mark Beta Weights

bB = 1 = (Area(B+A+E+F)/To tal Area)b-+ (Area(C+D+H+G)/To tal Area) b+ = .5b + .5b+

Ex hibit 3b

Out per form ing Beta Weights

bRO = wRO -b- + wRO+b+

wRO = Area(A+B)/Area(A+B+C+D)

wRO+ = Area(C+D)/Area(A+B+C+D)

turn on the port fo lio and ba sic sta tis tics on the un der ly ing uni verse of se cu ri ties. Based on

sym me try con di tions, it must be that wRO- = wRU+ and wRO+= wRU-. The deri va tion for these

quan ti ties is based on stan dard cal cu lus and is pre sented in the Ap pen dix. The quan ti ties

wRO-= wRU+) and wRO+ (= wRU-) are de fined as

wRO- = wRU+ = .25 – c + d (13)

wRO+ = wRU- = .25 + c – d (14)

where

d b

b b

= + - +

ÁÁ

˜˜

ÿ -

= 1

2 2 2

2 2

1 * *

R a

Sin

(15)

a RP

1 1 2

(16)

R b

1 1 2 (17)

b b* *

= - +

ÁÁ

˜˜

ÿ - -

= b

Sin

a 2 2

2 2

**R** (18)

** =

a b

2 2 2 (19)

R b

a b

** = -

2 2 2 q

(20)

and

= -

ÎÍ

°ÿ

- Tan Tan RP

1 ( ) . (21)

The only em piri cal in puts re quired for com put ing bRO and bRU are a, b, and the slope

of the em piri cal mar ket line, ob tained from the re gres sion of ex- post re turns against be tas.

This slope, de noted above by RP, is the mar ket risk pre mium and is de fined as the re turn

on the mar ket mi nus the risk- free rate. The stan dard er ror of the es ti mate for the em piri cal

mar ket line de ter mines the length of a, whereas the stan dard de via tion of the in di vid ual be -

tas de ter mines the length of b. The quan ti ties a and b are de fined as

Volume 30 Number 8 2005 65

a = 3(stan dard er ror) = 3 2 2 s B f B r - - ( ) (22)

b = 3(sb). (23)

Given the port fo lio’s beta, the port fo lio al pha can be cal cu lated as

a = Rp - (rf + bp(RP)) (24)

where a is the port fo lio al pha, Rp is the port fo lio re turn, rf is the risk- free rate, and (rf +

bp(RP)) is the re quired re turn on the port fo lio.

For any em piri cal study em ploy ing the frame work pre sented here, the fol low ing

steps should be un der taken in or der to de ter mine period- specific be tas and al phas:

1. Using historical returns, calculate betas for all securities in the bench mark uni -

verse. This is con sis tent with sim pli fy ing as sump tion #7 given above, that is, in

con trast to fund betas in di vid ual se cu rity betas are rel a tively sta ble over time.

Calculate sb the stan dard de vi a tion of the in di vid ual se cu rity betas.

2. For the time pe riod for which the fund is be ing ana lyzed, cal cu late the bench -

mark re turn and dis per sion sta tis tics (B and sB) from the re turn data for the

bench mark uni verse.

3. Us ing the bench mark re turn, B, and the risk-free rate, rf, for the same time pe -

riod, along with sb from step 1, cal cu late a and b (equa tions (22)-(23)).

4. Calculate wRU- , wRU+ , wRO- , and wRO+ us ing equa tions (13)-(21).

5. Calculate bRO and bRU us ing equa tions (7)-(10).

6. Es ti mate RO and RU us ing B, sB, and equa tions (2)-(4).

7. In fer wRO and wRU from the ac tual port fo lio re turn, RP, and equa tion (5).

8. Cal cu late the port fo lio al pha (equa tion (24)) and beta (equa tion (6)) for the time

pe riod that the fund is be ing ana lyzed.

By re peat ing these steps for suc ces sive pe ri ods, a time se ries of al phas and be tas

may be ob tained. As noted above, there are nu mer ous ac tive strate gies based on di rec tional

fore casts and the meth od ol ogy pre sented here can be ap plied to many of these. The ba sic

frame work de vel oped in this study for ana lyz ing fund risk and re turn can, with mi nor

modifications, accommodate many investment styles, analytic approaches, and asset

classes. Fur ther more, it is ro bust enough to ap pro pri ately meas ure risk for tra di tional mu -

tual funds, al though the pri mary con tri bu tion here is the ap pli ca tion to funds where the

manager is making explicit directional forecasts.

V. Con clu sions and Ave nues for Fur ther Re search

In this ar ti cle, a new ap proach to the study of time vary ing fund be tas is ad vanced. Ro bust

risk meas ures are de vel oped that ad dress a well- known de fi ciency in port fo lio the ory, that

Managerial Finance 66

is, pe ri odic modi fi ca tions in port fo lio weights change the un der ly ing re turn dis tri bu tion of

the port fo lio which, in turn, makes it dif fi cult to ap pro pri ately meas ure port fo lio re turn and

risk. Fo cus ing on ac tively man aged funds that rely on di rec tional fore casts, the re sults ob -

tained here not only have prac ti cal im pli ca tions for per form ance meas ure ment and risk

man age ment, but also make an im por tant con tri bu tion to the em piri cal meth od ol ogy lit era -

ture. Be cause the al pha and beta meas ures de vel oped here do not rely on his tori cal fund re -

turns and ap ply to a sin gle time pe riod, a time se ries of fund risk lev els can be ob tained.

One of the more im por tant ar eas for fu ture re search us ing this meth od ol ogy is the

study of the sta tis ti cal prop er ties of and eco nomic re la tion ships as so ci ated with the time

se ries of fund be tas. Sev eral im por tant is sues can be ad dressed. The re la tion ship be tween

changes in fund be tas and changes in eco nomic fac tors, in ves tor sen ti ment, and in other

risk meas ures, such as im plied and his tori cal vola til ity, over time can be in ves ti gated. Evi -

dence con cern ing any pat terns that ex ist in the fund be tas (and al phas) may also be ob -

tained, as well as in for ma tion as to whether these pat terns vary, for ex am ple, across fund

types. It can be noted that such pat terns, if they were to ex ist, may be used to de velop a

fund clas si fi ca tion scheme that is more mean ing ful than some of the cur rent cate go ri za -

tions, such as self- reporting by eq uity hedge funds. Fu ture re search can ex am ine how

trans ac tions costs and data fre quency im pact the find ings. Simi lari ties in the time se ries

be hav ior of the fund be tas and al phas ob tained us ing the meth od ol ogy de vel oped here and

those from other ap proaches, such as the con di tional per form ance evalua tion meas ures of

Fer son and Schadt (1996), can be de ter mined. Ad di tion ally, com pari son of fore cast ing

abil ity, that is, the abil ity to pre dict fu ture re turns, of these vari ous ap proaches can be

made.

Ex ten sions to the base model pre sented here will per mit its ap pli ca tion to funds util -

iz ing lev er age and short sales, which are com monly em ployed in many types of funds

based on di rec tional fore casts such as hedge funds and other al ter na tive in vest ment ve hi -

cles. The base model can be ex tended to al low for mul ti ple as set classes and, no ta bly, the

rec og ni tion of cash im pacts (i.e., fund flows, bor row ing, and cash drag ef fects). The end

re sult will be meas ures of port fo lio risk and re turn that are con sis tent with the frame work

of tra di tional fund man age ment but can be ap plied to a wide va ri ety of less tra di tional

funds.

The frame work pre sented in this study is not in con sis tent with oth ers that have been

pre sented in the lit era ture. It is well known that hedge funds and vari ous al ter na tive in vest -

ments are char ac ter ized by high turn over, the pur pose of which is to earn high re turns (un -

ad justed for risk). Gla zier and Wilkens (1999) de velop a model of man ager per form ance

that de scribes how well turn over is used in beat ing a bench mark. Their re sults in di cate a

posi tive re la tion ship be tween re turn and turn over when the man ager is skilled, even un der

gen er ous trans ac tions costs as sump tions. Al though the Gla zier and Wilkens (1999) per -

form ance meas ure ac counts for risk only in the sense that it can dis tin guish be tween sta tis -

ti cally sig nifi cant and in sig nifi cant re turns and their meth od ol ogy does not al low for

lev er age or for short sales, their frame work is con sis tent with that pre sented here. Com bin -

ing as pects of the Gla zier and Wilkens (1999) re turn de com po si tion with the re sults of this

ar ti cle may be use ful in de scrib ing skill, turn over, and the risk and re turn prop er ties of

funds that em ploy di rec tional fore casts.

In sum mary, the pri mary con tri bu tion of this ar ti cle to the lit era ture is that a for mula

is de rived for the period- specific beta for a port fo lio of fi nan cial as sets that has been

Volume 30 Number 8 2005 67

formed on the ba sis of di rec tional fore casts for in di vid ual se cu ri ties rela tive to a bench -

mark. The period- specific na ture of the meas ure is a sig nifi cant ad van tage since his tori cal

fund re turns are not re quired, fund re turns’ past de via tions from the bench mark are not in -

cluded in the cal cu la tion of the cur rent pe ri od’s fund beta, and the beta es ti mate is not di -

luted. The frame work for risk and re turn de vel oped here pro vides a ba sis for re search into

a number of ar eas as so ci ated with ac tively man aged funds and dy namic strate gies. Ad di -

tion ally, al though no sin gle meas ure of risk for ac tively man aged port fo lios is cur rently

agreed upon in the lit era ture, it is hoped that this ar ti cle makes prog ress in this im por tant

area.

Managerial Finance 68

Endnotes

1. As noted in Mer ton (1981), vio la tions of mar ket ef fi ciency (e. g., evi dence of su pe rior

fore cast ing skills on the part of port fo lio man ag ers) would have sig nifi cant im pli ca tions

for the the ory of fi nance, in clud ing those per tain ing to op ti mal port fo lio choice and the

equi lib rium valua tion of se cu ri ties.

2. The foun da tion for tests on the di rec tion of mar ket fore casts was de vel oped by Mer ton

(1981) and Hen riks son and Mer ton (1981). Spe cifi cally, the ba sic mar ket tim ing model in

Mer ton (1981) is used in Hen riks son and Mer ton (1981) to de velop both para met ric and

non para met ric tests of mar ket tim ing. These tests per mit a de ter mi na tion of whether mar -

ket tim ing fore casts, that is, fore casts of when stocks will out per form bonds or vice versa,

will earn su pe rior re turns.

3. The sur vey ex change rate fore casts were made by Money Mar ket Serv ices, Inc. The cur -

ren cies are de nomi nated in U.S. dol lars and are the Brit ish pound, Ger man mark, Swiss

franc, and Japa nese yen.

4. For a dis cus sion of the dif fer ences be tween the Pe sa ran and Tim mer mann (1992a) pre -

dic tive fail ure test and the Hen riks son and Mer ton (1981) non para met ric test, see Pe sa ran

and Tim mer mann (1992b). Ad di tion ally, Pe sa ran and Tim mer mann (1992a) show that

their pre dic tive fail ure test is as ymp toti cally equiva lent to the c2 goodness- of- fit test based

on con tin gency ta bles in the 2x2 case. In gen eral, the pre dic tive fail ure and c2 test are not

equiva lent, with the lat ter be ing more con ser va tive as a test of pre dic tive per form ance even

in large sam ples. Lai (1990) and Pe sa ran and Tim mer mann (1992a) show that non para met -

ric tests of di rec tional fore cast ing, such as the Hen riks son and Mer ton (1981) and Pes ran

and Tmmer mann (1992a) tests, pos ses sev eral de sir able prop er ties. In gen eral, these tests

im pose no dis tri bu tional as sump tions and per mit non sta tion ary con di tional prob abili ties

through time but pos sess well- defined fi nite sam ple dis tri bu tions. This means that the tests

will be ro bust with re spect to pos si ble non sta tion ari ties in the vari ables of in ter est.

5. There are nu mer ous situa tions in which the n- category case is rele vant, in clud ing that

where a risk neu tral in vestor is con sid er ing which of n as sets to pur chase on the ba sis of

their ex pected re turns.

6. Un der the ran dom walk as sump tion, both buy and hold and ac tive strate gies earn zero

profit. How ever, when more than one di rec tional strat egy is used, the re turn dis tri bu tion is

non- normal. For ex am ple, if two fore casts are used to time the mar ket, the dis tri bu tion is a

mix ture of two nor mal laws and the mix ture co ef fi cient de pends on the cor re la tion be -

tween the two fore casts.

7. One of the ma jor ap pli ca tions of mod ern port fo lio the ory has been to pro vide a frame -

work for the meas ure ment of in vest ment per form ance. Within this frame work, it is com -

mon prac tice to par ti tion fore cast ing skills into two com po nents, se cu rity se lec tion and

mar ket tim ing. See Fama (1972) for a dis cus sion.

8. Kon and Jen (1978) use the Quandt (1972) switch ing re gres sion meth od ol ogy to ex am -

ine chang ing lev els of market- related risk over time for mu tual fund port fo lios. They find

evi dence that mu tual funds do have dis crete changes in the risk lev els they choose, which

is con sis tent with fund man ag ers in cor po rat ing mar ket tim ing into their in vest ment strate -

gies. Trey nor and Mauzy (1966) add a quad ratic term to the stan dard CAPM to test for

Volume 30 Number 8 2005 69

mar ket tim ing abil ity. They find that the hy pothe sis of no mar ket tim ing abil ity can be re -

jected at the 95% level for only one of 57 open- end funds ex am ined.

9. For ex am ple, as noted in de Silva et al. (2001), very wide re turn dis per sion be tween

stocks oc curred in De cem ber 1999 but in dex vola til ity was only mod er ate. See Camp bell

et al. (2001) for a dis cus sion of the re la tion ship be tween dis per sion and both in dex and se -

cu rity vola til ity. Camp bell (2001) et al. show that dis per sion is re lated to the in dus try spe -

cific and idio syn cratic por tions of in di vid ual se cu rity vola til ity.

10. Sol nik and Rou let (2000) ar gue that the tra di tional time se ries method of cal cu lat ing

cor re la tions pos sesses sev eral dis ad van tages rela tive to their new meth od ol ogy, in clud ing

re quir ing a long pe riod of ob ser va tions and the use of over lap ping data. Ad di tion ally, the

use of over lap ping data im plies that the cor re la tion es ti mates may change too slowly to be

of practi cal use.

11. The re la tion ship be tween changes in fund dis per sion and se cu rity dis per sion over time

is not con stant for sev eral rea sons. Changes in in vest ment prac tices over time may lead to

in creases or de creases in fund re turn dis per sion even if se cu rity re turn dis per sion is con -

stant. If the number of se cu ri ties that in ves tors hold in their port fo lios in creases over time,

fund dis per sion will de cline. On the other hand, fund dis per sion will in crease if se cu ri ties

held by in ves tors pos sess greater lev els of cor re la tion than in the past.

12. Loun gani et al. (1990) and Brai nard and Cut ler (1993), among oth ers, ar gue that struc -

tural shifts in the econ omy aris ing from sig nifi cant po liti cal or eco nomic dis rup tions lead

to large cor po rate re valua tions, which, in turn, re sult in pe ri ods of large stock re turn dis -

per sion.

13. In de fin ing the mean of RO and RU, we fol low Barr and Sher rill (1999) who pres ent ap -

proxi ma tion for mu las for the mean and vari ance of a trun cated nor mal dis tri bu tion. The

ad van tage of us ing the ap proxi ma tions is that they are eas ily com puted in a stan dard elec -

tronic spread sheet such as Excel. The ex pected val ues of RO and RU may also be de fined as

in Gla zier and Wilkens (1999), where RO = B + sBE(Z) ª RO = N-1(.75,B,sB) and RU = B -

sBE(Z) ª RU = N-1(.25, B,sB) and where N-1(prob, x, sx) is the in verse of the nor mal cu mu -

la tive dis tri bu tion for the speci fied prob abil ity, mean x, and stan dard de via tion s. In this

study, the Barr and Sher rill (1999) for mu la tion has an ad di tional ad van tage in that it can

fa cili tate ex ten sions to the model pre sented, with a par ticu lar em pha sis on port fo lio vari -

ance.

14. It should be noted that wRO- + wRO+ = wRO and wRU- + wRU+ = wRU only if the manager

has perfect forecasting ability. In general, wRO and wRU are inferred from the actual

portfolio return.

Managerial Finance 70

References

Acar, E. Ex pected Re turns of Di rec tional Fore cast ers. In E. Acar and S. Satchell (eds.),

Ad vanced Trad ing Rules. Woburn, MA: Butterworth- Heinemann, 1998.

_______ and Satchell, S. The Port fo lio Dis tri bu tion of Di rec tional Strate gies. In E. Acar

and S. Satchell (eds. ), Ad vanced Trad ing Rules. Woburn, MA: Butterworth- Heinemann,

1998.

Arm strong, J. S. and Fil des, R. Cor re spon dence on the Se lec tion of Er ror Meas ures for

Com pari son among Fore cast ing Meth ods. Jour nal of Fore cast ing 14, 1995, pp. 67- 71.

Ash, J. C. K., Smyth, D. J., and Heravi, S. M. Are OECD Fore casts Ra tional and Use ful?:

A Di rec tional Analy sis. In ter na tional Jour nal of Fore cast ing 14, 1998, pp. 381- 91.

Barr, D. R. and Sher rill, E. T. Mean and Vari ance of Trun cated Nor mal Dis tri bu tions.

Ameri can Stat is ti cian 53, 1999, pp. 357- 61.

Bes simbinder, H., Chan, K. and Se guin, P. J. An Em piri cal Ex ami na tion of In for ma tion,

Dif fer ences of Opin ion, and Trad ing Ac tiv ity. Jour nal of Fi nan cial Eco nom ics 40, 1996,

pp.105- 34.

Birche nall, C., Jensen, H., and Os born, D. R. Pre dict ing US Busi ness Cy cle Re gimes.

Uni ver sity of Man ches ter, School of Eco nomic Stud ies, Dis cus sion Pa per No. 9625.

Brai nard, S. L. and Cut ler, D. M. Sec to ral Shifts and Cy cli cal Un em ploy ment Re con sid -

ered. Quar terly Jour nal of Economcs 108, 1993, pp. 219- 43.

Camp bell, J. Y., Let tau, M., Malk iel, B., and Xu, Y. Have In di vid ual Stocks Be come More

Vola tile? An Em piri cal Ex plo ra tion of Idio syn cratic Risk. Jour nal of Fi nance 56, 2001,

pp. 1-43.

Chen, Z. and Knez, P. J. Port fo lio Meas ure ment: The ory and Ap pli ca tions. Re view of Fi -

nan cial Stud ies 9, 1996, pp. 511- 55.

Clements, M. P. and Hen dry, D. F. On the Limi ta tions of Com par ing Mean Fore cast Er -

rors. Jour nal of Fore cast ing 12, 1993, pp. 617- 37.

____________________________. A Re ply to Arm strong and Fil des. Jour nal of Fore -

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de Silva, H., Sa pra, S., and Thor ley, S. Re turn Dis per sion and Ac tive Man age ment. Fi -

nan cial Ana lysts Jour nal 57, 2001, pp. 29- 42.

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Gal ton, F. Re gres sion To wards Me di oc rity in He redi tary Stat ure. Jour nal of the An thro -

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Gla zier, J. and Wilkens, K. Skill and Turn over: Re quire ments for In vest ment Per form -

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namic Econo mies. Jour nal of Po liti cal Econ omy 99, 1991, pp. 225- 62.

Hen riks son, R. D. and Mer ton, R. C. On Mar ket Tim ing and In vest ment Per form ance. II.

Sta tis ti cal Pro ce dures for Evalu at ing Fore cast ing Skills. Jour nal of Busi ness 54, 1981, pp.

513- 33.

Jensen, M. C. Op ti mal Utili za tion of Mar ket Fore casts and the Evalua tion of In vest ment

Per form ance. In G. P. Szego and K. Shell (eds.), Mathe mati cal Meth ods of In vest ment and

Fi nance. Am ster dam: North- Holland, 1972.

Kon, S. J. and Jen, F. C. An Es ti ma tion of Time- Varying Sys tem atic Risk and Per form -

ance for Mu tual Fund Port fo lios: An Ap pli ca tion of Switch ing Re gres sion. Jour nal of Fi -

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Volume 30 Number 8 2005 73

Appendix

The in tui tion as so ci ated with and the steps for cal cu lat ing the beta of as sets with an av er -

age re turn of RO and the beta of as sets with an av er age re turn of RU is pre sented above in

the body of the pa per. Re call that these be tas are de fined, re spec tively, as

bRO = wRO -b- + wRO+ b+ (6)

bRU = wRU -b- + wRU+ b+. (7)

Re fer ring to the ar eas il lus trated in Ex hib its 3a and 3b, the pro por tion of all as sets with a

beta less than one that un der per form the bench mark, wRU- and the pro por tion of all as sets

with a beta greater than one that un der per form the bench mark, wRU+, may be pre sented as

wRU- = Area(E+F)/Area(E+F+G+H) (A.1)

wRU+ = Area(G+H)/Area(E+F+G+H). (A.2)

Based on sym me try con di tions, wRO- = wRU+ and wRO+= wRU-. This im plies that the weights

can be de fined solely in terms of the Ar eas (B) and (D), as shown be low

wRO-= wRU+ = .25– Area(B) + Area(D) (A.3)

wRO+= wRU- = .25+ Area(B) – Area(D). (A.4)

To sim plify the cal cu la tions, the cen ter of the el lipse is moved to the ori gin (0,0) and the

el lipse is ro tated in the beta- return space so that the line q-v is com pletely ver ti cal and the

SML has a slope of zero. See Ex hibit A. The equa tion of the el lipse is then sim ply

2 1

b + = (A. 5)

where a is given by the dis tance q-o and b is given by the dis tance o-s. The calculations are

fur ther sim pli fied by flip ping Area (B) about the R- axis and Area (D) about the beta- axis

and work ing in the upper- right quad rant of the el lipse, which is given by

R b

a = - 1

b . (A. 6)

The in defi nite in te gral of this curve is

Sin

a = - +

ÁÁ

˜˜ Ú - b

2 2

1 . (A. 7)

The only re main ing quan ti ties to be de fined are a and b, which are equal to half of the mi -

nor and ma jor axes of the el lipse, re spec tively.

Managerial Finance 74

To solve for the Area (D), the point (b*, R*), where the line o-w in ter sects the el lipse, must

first be found. The flipped line o-w has the same slope as the SML, which is equal to the

risk pre mium and de noted by RP. Set ting the equa tion of the line o-w equal to the curve in

the fist quad rant, the fol low ing are ob tained

RP RP

R b

1 1

1 1 2

2 . (A. 8)

The area of D is the sum of the tri an gle DT and the defi nite in te gral of the curve (ab) from

point b* to b, as given by

Area D R

Sin

( ) * * = + - +

ÁÁ

˜˜

ÿ - 1

2 2 2

2 2

1 b

(A. 9)

To solve for the Area (B), the point (b**, R**), where the line o-r in ter sects the el lipse, must

be found. In find ing this point, it can be noted that the an gle q- o-r is equal to the an gle w-

o-s. The slope of the line o-r is de noted as q

= -

ÎÍ

°ÿ

- Tan Tan RP

1 ( ) . (A. 10)

The point (b**, R**) can be solved for by set ting the curve (ab) equal to the line R = qb

q q

** ** , . =

= -

a b

R b

a b

1 2 2 2

2 2 2 (A. 11)

The area of B is then the sum of the defi nite in te gral of the curve from the ori gin to b** less

the area of the tri an gle o-r-b**

Volume 30 Number 8 2005 75

s = b

w = (b*,R*)

q = a r =(b**, R**)

DT DE

Slope = RP

Ex hibit A

Ro tated Quar ter of Re gres sion El lipse

Area B

Sin

( )

* *

= - +

ÁÁ

˜˜

ÿ -

b b

2 2

2 b** ** . R (A. 12)

Let ting Area (D) = d and Area (B) = c, then

wRO-=wRU+ = .25 – c+d (A. 13)

wRO+ = wRU-= .25 + c-d. (A. 14)

Managerial Finance 76

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