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1975-1984

Prediction of Beta from Investment

Fundamentals

Barr Rosenberg and James Guy

PREDICTION CRFrERIA

SY s t em a t i c r i s k , a s m eas u red b y b e t a , cap t u re s

that aspect o f inve s tmen t r i sk that cannot be

el iminated by d ivers i f i ca t ion . Consequent ly , i t

p lays the crucia l ro le in evaluat ing ex post the

degree of r i sk under ta ken in a d ivers i f i ed inves t -

me nt p rogra m, hence in judging the ab i li ty o f tha t

inves tment p rogram to ach ieve a des i rab le r i sk-

re turn pos ture . Again , the pred ic t ion of beta es -

sent ial ly predicts the future risk of a diversified

por t fo l io , hence i t s in f luence on por t fo l io beta i so n e o f t h e k ey co n s i d e ra t i o n s i n an y i n v es t m en t

deci s ion . Therefore , among many poss ib le r i sk

measures beta deserves par t i cu lar a t t en t ion and

wil l be the central topic of this art icle. Beta wil l be

def ined , and then , in our d i scuss ions of the appl i-

cat ions of beta, cri teria for opt ima l predic t ion an d

es t imat ion of beta wi l l emerge.

BEI'A

I f the inves tmen t re tu rn on the marke t por t fo l io in

an y t i m e p e r i o d a s s u m es an y ce r t a i n v a l u e , w h a t

re turn can be expec ted , ,on the average , fo r a

secur i ty in the same t ime per iod? For example , i fthe ma rket re tu rn in tha t per iod wi l l be 10 percen t ,

can the secur i ty re tu rn be expected , on the aver-

age , to be 20 percen t , o r f ive percen t?

Not ice that th i s ques t ion refers to the value of

the secur i ty re tu rn to be expected "on the aver-

age ," a l though i t app l ies to a s ing le secur i ty in a

s ing le per iod . The expecta t ion i s to be t aken in the

fo l lowing sense . Sup pos e that , in v iew of every-

t h i n g w e n o w k n o w a b o u t t h e e c o n o m y a n d t h e

specif i c f i rm n, we imagine repeat ing ma ny t imes

the uncer ta in events tha t may occur in the t ime

per iod wi th eac h repet i t ion hav ing the nature o f an

ex p e r i m en t . E ach ex p e r i m en t y i e l d s s o m e m ark e t

re turn r M and some secur i ty re tu rn r , .

Each pai r o f re tu rns ( rM, r , ) ma y be graph ed ,

wi th the secur i ty re tu rn r , on the ver t i ca l ax i s and

the m arket re tu rn r M on the hor izonta l ax is . The

s lope of a regress ion l ine f i t t ed th rough these

Reprinted rom Financial A nalystsJournal (May~June1976):60-72.

p o i n t s , w h i ch m eas u res t h e d eg ree t o w h i ch

higher market re tu rn l eads to an expecta t ion of

greater secur i ty re tu rn , i s the beta o f the secur i ty

(see Fig ure 1) . Wh en the s tock mark et index r i ses

or fal ls , the securi ty price wil l tend to rise or fal l

a l so, and the r i se wil l t end to be more or l ess than

one. Typical ly, the s lope (i .e. , beta) wil l be greater

than zero bu t l ess than th ree . Ma ny secur i t i es have

betas around one, and they tend to r i se and fa l l in

pr ice roughly by the same percen tage that the

market index rises or fal ls . A securi ty with an eg a t i v e b e t a w o u l d t en d t o m o v e ag a i n s t t h e

market , bu t such secur i t i es are rare .

Wh en eac h repet i t ion i s v ie we d in h inds igh t , a

un ique pai r o f re tu rns ( rM, r , ) wi l l have occurred ,

b u t w e a r e co n ce rn ed w i t h t h e ex p ec t a t i o n t h a t

held looking forward in t ime, before the ac tual

re turns h ave occurred . T he values ac tual ly rea l i zed

wi l l no t o rd inar i ly correspond to expecta t ions : Ex

post (i .e., hindsigh t) obse rvat ion that r M = 10

percen t and r , = 20 percen t does no t imply that the

secur i ty ' s be ta was two. The t rue beta cou ld have

been one wi th the addi t ional 10 percen t in secur i ty

r e t u rn b e i n g cau s ed b y r an d o m fac to r s u n i q u e t o

that secur i ty . Beta g ives an ex pected va lue jus t as a

probabil is t ic predict ion for the profi t in a gamble

does : Ex post, the gamble wi l l have e i ther suc-

ceed ed or fai l ed, bu t the resu l t need no t be equal to

the expected value .

Beta i s o f ten expla ined by p lo t t ing a t ime

ser ies o f pai rs o f re tu rns . This corresp onds to

r ep ea t i n g t h e ab o v e ex p e r i m en t a t a s eq u en ce o f

dates . In th i s way , we are ab le to observe more

than one ou tcome and , therefore , to i l lus t ra te the

re la t ionsh ip . Repet i t ion i s somewhat mis lead ing ,

however , s ince i t sugges t s tha t be ta i s unchangingover the sequence. Actual ly , as we wi l l d i scuss

below, there are reasons to expect tha t be ta

changes . The sequ ence i s ac tual ly tha t o f repeat ing

s imi lar bu t changing exper iments . The essen t ia l

mean ing of beta app l ies d i s t inc t ly a t each po in t in

t ime.

N o t e t h a t , f ro m an eco n o m i c v i ew p o i n t , t h e

market re tu rn does no t cause the secur i ty re tu rn .

F/nancial Analysts Journal / January-February 1995 101 © 1995, AIMR

®



1 9 7 5 - 1 9 8 4

Rgure 1. Possible Secudty Retums Plotted againstCorresponding M arket Retum for theHypotheUcaJSecurity "A"

~2

10 •

8- -

6-- • •

4

2 -- •

0

-2 - - •

- 4

--6

- 8

- 1 0 ¢ I I I I I I [ I

-10 -8 -6 -4 -2 0 2 4 6 8 10

rM (%)

In s t ead , b o t h a r e cau s ed b y eco n o m i c ev en t s . T h i s

p o i n t h as c r ea t ed s o m e co n fu s i o n am o n g an a l y s t s

who in terpre t be ta , which i s a regress ion coeff i -

c ien t, as necessar i ly s ta t ing the causal re la t ionsh ip

o f m ark e t r e t u rn s u p o n t h e s ecu r i ty r e tu rn s: T h a t

i s, i f be ta i s two, a marke t re tu rn of 10 percen t

c a u s e s a secur i ty re tu rn of 20 percen t . The correct

wo rd in g of th i s s t a tement i s tha t , as a conseque nce

o f t h e d ep en d en ce o f b o t h m ark e t r e t u rn an d

s ecu r it y r e t u rn u p o n eco n o m i c ev en t s , i f a m ark e t

re turn of 10 percen t i s observed , then the mos t

l ike ly value for the associa ted secur i ty re tu rn i s 20

percen t . The words "mos t l ike ly" inc lude the fo l -

lowing pa t tern of in ference: I f the mark et re tu rn i s

10 percen t , then the associa ted economic events

mus t be of cer ta in types ; i f fo r each se t o f even ts

t h a t co u l d i n d u ce a m ark e t r e t u rn o f 10 p e rcen t w e

co m p u t e t h e s ecu r i t y r e t u rn t h a t w o u l d r e s u l t ,

t h en o n av e rag e t h e r e t u rn , w e i g h t ed b y t h e p ro b -

abi l i ty of the events , is 20 percent .

BETA AS THE CONSEQ UENCE OFUNDERLY ING ECONOMIC EVENTS

I t i s ins t ruct ive to reach a judgment about beta by

carry ing ou t an imaginary exper iment as fo l lows .

One can imagine a l l the var ious events in the

eco n o m y t h a t m ay o ccu r , an d a t t em p t t o an s w er i n

each cas e t h e t w o q u es t io n s : (1) Wh a t w o u l d b e t h e

secur i ty re tu rn as a resu l t o f tha t even t? and (2)

Wh a t w o u l d b e t h e m ark e t r e t u rn a s a re s u l t o f t h a t

event? Looking forw ard in t ime we can see that the

market wi l l be s ign if i can tly af fec ted by c hanges in

the exp ecte d rate of inflat ion, intere st rates , inst i -

tu t ional regu la t ions of a l t ernat ive inves tm ent m e-

d ia , g ro w t h r a te o f r eal GN P , an d m an y o t h e r

fac tors . Fur ther , there are a num be r of less b roadev en t s t h a t a l s o d es e rv e a t t en t i o n : m o v em en t s i n

in ternat ional o f f and o ther ra w m ater ia l p r ices,

d ev e l o p m en t s i n a l t e rn a t iv e d o m es t i c en e rg y s u p -

p l ies , changes in publ ic a t t i tudes toward po l lu t ion

an d co n s u m er d u rab l e s , an d p o s s i b l e ch an g es i n

tax l aw, am ong o thers . E ach of these e vents i s

impor tan t in con t r ibu t ing to the uncer ta in ty of

fu ture market re tu rns . And for each we can an t ic-

ipate the ef fec t up on any par t i cu lar secur i ty . Co n-

s ider , fo r example , a domes t ic o i l s tock . "Energy

cr i s i s" - re la ted events wi l l have a p ropor t ional ly

greater ef fec t up on such a s tock , in f la tion-re la ted

events p roba bly a re la t ively smal ler ef fec t, than for

the ma rket as a whole . As a resu l t, f f we foresee

t h a t t he m a j o r s o u rce o f u n ce r t a in t y i n fu ~ re

re t u rn s i s f ro m d ev e l o p m en t s i n t h e en e rg y p i c -

tu re , we wi l l an t i c ipate an un usua l ly h igh beta , bu t

i f we foresee that the m ajor sourc e of uncer ta in ty

l i es in in f la t ion-re la ted events , we wi l l an t i c ipate

an u n u s u a l l y l o w b e t a .

On e co u l d eas i l y d ev o t e a s m u ch t i m e t o

pred ic t ing beta as i s usual ly devoted to p red ic t ing

secur i ty re tu rns in convent ional secur i ty analys i s .

This paral lel is , in fact , a valuable one to draw on

in th ink ing about beta . In secur i ty analys i s , i t i scu s t o m ary t o d i s t i n g u i s h b e t w een t h e co m p o n en t

of re tu rn resu l t ing f rom even ts speci f ic to the f i rm

i n q u es t io n , an d t h e co m p o n e n t o f r e t u rn s t em -

m i n g f ro m ev en t s a f f ec t i n g t h e eco n o m y o r t h e

m ark e t a s a w h o l e . W h en t h e s u m o f t h es e t w o i s

expected to be pos i t ive , then the secur i ty i s con-

s i d e red to b e a g o o d b u y . N o w , i n en u m era t i n g t h e

even ts speci f ic to the f i rm in ques t ion , the ana lys t

wi l l fo rmulate a p red ic t ion of the expected impact

on re turn and a l so a fo recas t o f the uncer ta in ty of

rea l i z ing that expecta t ion . The former determines

the expecte d speci fi c re tu rn and the l a t t er the

magni tude of speci f i c r i sk . Thus the t asks o f p re-

d ic t ing expec ted re turn and r i sk of re tu rn are

clearly related in this case.

Simi lar ly , in p red ic t ing the component o f se-

cu r i t y r e t u rn a r i s i n g f ro m eco n o m y w i d e ev en t s

ra ther than f rom events speci f i c to tha t par t i cu lar

f i rm, the analys t es t imates the probabi l i t i es o f the

v a r i o u s p o s s ib l e o u t co m es o f t h e ev en t , a n d t h e

102 FinancialAnalysts Jou m al/ January-February 1995



1 9 7 5 - 1 9 8 4

m a g n i t u d e o f t h e r e s p o n s e o f t h e s e c u r i t y r e t u r n t o

t h a t ev en t . T h e p r o d u c t o f t h e s e t w o i s t h e ex -

p ec t e d e f f ec t o f th e ( : ven t u p o n t h e s e cu r i t y re t u r n .

T h e s e e f fe c ts a r e t h e n s u m m e d o v e r al l e c o n o m y -

w i d e e v e n t s t h a t m a y i m p a c t t h e s t o c k t o o b t a i n

t h e e x p e c t e d s e c u ri t y r e t u r n d u e t o e c o n o m y w i d e

f ac t o r s . H e r e ag a i n , a l l t h a t i s n eed ed i s a j u d g -

m e n t a s t o t h e u n c e r t a i n t y a t t a c h i n g to t h e e c o n o -m y w i d e e v e n t s , a n d w e f i n d a p r e d i c ti o n o f t h e

u n c e r t a i n t y o f t h e s e c u r i ty r e t u r n d u e t o e c o n o m i c

ev en t s . T h e r e t u r n o n t h e m ar k e t p o r t f o l i o i s t h e

w e i g h t e d av e r ag e o f t h e i n d i v i d u a l s ecu r i t y re -

t u r n s , s o th i s s am e ap p r o ac h y i e l d s a p r ed i c t i o n o f

t h e u n c e r t a i n t y o f t h e m a r k e t r e t u r n d u e t o e co n -

o m y w i d e ev en t s . S i n ce t h e ev en t s s p ec i f i c t o i n d i -

v i d u a l f i r m s w i l l t en d t o av e r ag e o u t an d co n t r i b -

u t e l i t t l e t o t h e m a r k e t r e t u r n , t h e e c o n o m y w i d e

ev en t s w i l l a ccou n t : f o r t h e g r ea t b u l k o f m ar k e t

r isk.

T h u s t h e r i s k o f m ar k e t r e t u r n i s l a r g e l y ac -

c o u n t e d f o r b y e c o n o m i c e v e n t s t h a t i m p a c t m a n ys t o ck s . Fo r each s t o ck , w e f i n d t h a t t h e s e ev en t s

a l s o h av e an e f f ec t t h a t c an b e p r ed i c t ed b y s ecu r i t y

an a l y s i s . A s an i l l u s t r a t i o n , co n s i d e r T ab l e 1 ,

w h e r e w e g i v e t w o i m a g i n a r y f u t u r e e v e n t s w i t h

eq u a l p r o b ab i l i ty o f g o o d , b ad , an d n o - ch a n g e

o u t c o m e s , a n d d e s c ri b e t h e r e s u l t i n g p e r c e nt a g e

r e t u r n s o n t h e m ar k e t , s t o ck A an d s t o ck B . R e l a-

t i v e t o th e m ar k e t , s t o ck A r e s p o n d s t w o - t h i r d s a s

m u c h t o t h e e n e r g y e v e n t a n d t w o t i m e s a s m u c h

t o t h e i n f l a t i o n ev en t . R e l a t i v e t o t h e m a r k e t , s t o ck

B r e s p o n d s f o u r - th i r d s a s m u c h t o e n e r g y a n d

r e s p o n d s n i l t o i n f l a t i o n . ( T h es e a r e l a t e r r e f e r r ed

to as r e la t ive r esponse coef f ic ien ts . )

Table 1.

% Contribution to Return

E vent O u t co m e M ar k e t S t o ckA StockB

Energy

Inflation

Good +6 +4 +8No change 0 0 0Bad --6 --4 --8Good +3 +6 0No chang(: 0 0 0Bad -3 -6 0

B ecau s e t h e e f f ec t s o f t h e t w o ev en t s a r e

i n d e p e n d e n t , t h e i n f o r m a t i o n g i v e n in t h is t a bl e

can b e r ep r e s en t ed b y t h e t r ee d i ag r am g i v e n i n

F i g u r e 2 . U s i n g t h i s d i ag r am , i t i s e a s y t o d e r i v e

t h e ex p ec t ed v a l u e a n d v a r i an ce o f r e t u r n s o n t h e

m ar k e t , r M , a s a r e s u l t o f t h e t w o ev en t s :

E(rM) = o

VAR(rM) = 1/9 (92 + 62 + 32 + 32 + 32 + 32 + 62 + 92

= 30.

T h i s v a r ian ce o f f u t u r e m ar k e t r e t u r n s can b e

d e c o m p o s e d i n t o t h e v a r ia n c e s i n d u c e d b y t h e t w o

i n d e p e n d e n t e v e n t s . T h e v a r i a n c e i n m a r k e t r e -

t u r n s c a u s e d b y e n e r g y u n c e r t a i n t y a l o n e i s e q u a l

to 1/3162 + 02 + ( -6) 2] = 24, wh ile t ha t ca use d b y

in f la t ion unc er ta in ty a lone i s equa l to 1 /3132 + 0a +

( -3 ) 2] = 6 . Because the se tw o ev en t s ar e ind epe n-

d e n t , t h e s u m o f t he s e t w o s u b v a r i a n c e s s h o u l d

eq u a l t h e t o t a l v a r i an ce o f m ar k e t r e t u r n s , an d

i n d e e d w e h a v e

24 + 6 = 30.

T h e v a r i a nc e o f th e f u t u r e m a r k e t r e t u r n s t e m s

f r o m u n c e r t a i n t y i n e n e r g y a n d i n f la t i on . E n e r g y i s

t h e g r ea t e r s o u r ce o f f u t u r e v a r i an ce ( ac t u a ll y fo u r -

f i f ths o f the to ta l in th i s exam ple) . S tock B is more

r e s p o n s i v e t o t h e e n e r g y f a c t or t h a n t h e m a r k e t ,

an d i t w i l l s h o w a h i g h v o l a t i l i t y i f t h e en e r g y

s i t u a t i o n ch an g es . S t o ck A w i l l s h o w t h e h i g h e r

v o l a t i li t y i f t h e i n f l a t i o n s i t u a t i o n ch a n g es , s i n ce i ts

r esp ons e coef f ic ien t to in f la t ion is h ighe r . S ince

en e r g y i s t h e g r ea t e r s o u r ce o f u n ce r t a i n t y , i t t u r n s

o u t t h a t s t o ck B h as t h e h i g h e r b e t a .

T h e b e t a s o f co m p an i e s A an d B can b e ea s i l y

ca l cu l a t ed u s i n g t h i s t r ee d i ag r am . C o n s i d e r , f o r

e x a m p l e , t h e b e t a o f c o m p a n y A , w h i c h i d e f i n e das I

COV(ra, rM) E[(ra - E[ra])(rM -- E[rM])]]3a~ =

VAR(rM) E [ ( r M - - E[rM])2]

W e kn ow tha t VAR(rM) = 30 , so tha t a l l tha t

r em ains i s to ca lcu la te COV(r a , rM). Rem em be r ing

t h a t E ( r a ) = 0 , we have

CO V(ra, rM) = 1/91 9.10 + 6.4 + 3(--2) + 3.6 + 0.0

+ ( -3)( -6) + ( -3) .2

+ ( -6)( -4) + ( -9)( -10)]

= 28, substituting this result in theformula for/3, , we have,

/3a = 28/30 = 14/15.

T h i s b e t a f o r c o m p a n y A c a n b e d e c o m p o s e d i n t o

t h e c o m p o n e n t b e t a s d u e t o t h e t w o e v e n t s. L e t u s

. def ine raM, r~ , an d r~ as th e r e tu rns on the ma rket ,

s t o ck A an d s t o ck B d u e t o th e en e r g y ev en t a l o n e ,

Financial Analysts Journa l January-February 1995 1 0 3



1975 -1984

Rgure 2.

Effect Of EnergyEvent Alone on

rM ra r b

Effect of InflationEvent Alone on

r M ra rb

3 6 0

6 4 8

~ N p = 1 /3 I 0 0 0

NxNNxx]/-

--6 -4 -8

o o o

- 3 - 6 0

6 0

0

-3

0 0

-6 0

6 0

0

-3

0 0

- 6 0

Total Effect ofEnergy + Inflation

Event Alone on

rM ra rb I]PROB

19 10 8 [11/91

16 4 8111/9 I

13 2 8111/9

13 6 0 111/9

I0 0 0 111/9

I-3 6 0 111/9

I-3 2 811 1/9

I-6 -4 -8111/9

19 -10 -8111/9

a n d r h , r~ , a n d r~ a s th e c o r r e s p o n d i n g r e t u r n s d u e

t o t h e i n f l a t i o n e v e n t a l o n e . T h e n , 2

COV(ra, rM) = CO V(r~, reM) + CO V(r~ , rh)

= 1/314.6 + 0.0 + (- 4) (- 6) ]

+ 1/316.3 + 0.0 + ( -6 )( -3 )]

= 2/3{1/316.6 + 0.0 + (- 6) (- 6) ]}

+ 2{1/313.3 + 0.0 + (-3 )(- 3) ]}

= 2/3 VAR(r~) + 2 VAR(r•)

VAR(r~4) VAR(r~I)

fla = 2/3 VAR(rM--------)+ 2 VAR(rM-------)"

Subst i tut ing in the values for VAR(reM), VAR(r/M),

a n d V A R ( r M ) , w e o b t a i n

/3a = 2 /3 24/30 + 2 6/30 = 2/3 4/5

+ 2 1/5 = 14/15.

T h e f i r s t c o m p o n e n t o f / S a r e f l e c t s t h e b e h a v i o r o f

t h e s e c u r i t y r el a t iv e t o e n e r g y , a n d t h e s e c o n d

c o n s i d e r s t h e e f f e c t o f in f l a t io n . A s i n d i c a t e d i n t h e

d e r i v a t i o n , 4 / 5 a n d 1 /5 a r e t h e p r o p o r t i o n a l c o n t r i -

b u t i o n s o f t h e e n e r g y a n d i n f l a t io n e v e n t s t o m a r -

k e t v a r i a n c e , a n d 2 /3 a n d 2 m e a s u r e t h e r e l a t i v e

r e s p o n s e c o e f f i c i e n t s o f st o c k A t o t h e s e e v e n t s .

S i m i l a rl y , i t i s p o s s i b l e t o s h o w t h a t , f o r s e c u -

r i t y B w e h a v e

fib = 4/5 4/3 + 1/5 0 = 16/15.

T h e f o r e g o i n g d i s c u s s i o n i l l u st r a te s t h e p r o p -

o s i ti o n t h a t t h e l e v el o f b e t a i s d e t e r m i n e d b y t w o

k i n d s o f p a r a m e t e r s : (1 ) T h e d e g r e e o f u n c e r t a i n t y

a t t a c h e d t o V a r i o u s c a t e g o r i e s o f e c o n o m i c e v e n t s

( t h e p r o p o r t i o n a l c o n t r i b u t i o n s o f t h e e v e n t s t o

m a r k e t v a r i a n c e ) , a n d ( 2) t h e r e s p o n s e o f t h e

s e c u r i ty r e t u r n s t o t h e s e e v e n t s ( r el a ti v e r e s p o n s e

coef f i c i en t s ) .

I n g e n e r a l, i f w e a s s u m e , f o r e x p o s i t o r y p u r -

p o s es , t h a t ec o n o m i c e v e n t s a re i n d e p e n d e n t o f

e a c h o t h e r , t h e n t h e b e t a o f t h e s e c u r i t y n w i l l b e

I

j= l~=I~vjj= l

w h e r e V / i s t h e c o n t ri b u t io n o f e c o n o m y w i d e e v e n t

j t o m a r k e t v a r i a n c e i n a n y p e r i o d , a n d w h e r e ] ,j , i s

t h e r a t io o f t h e r e s p o n s e s o f th e n t h s e c u r i ty a n d

t h e m a r k e t t o t h e j t h e v e n t O r t h e " r e l a t i v e r e -

104 Financial Analysts Jo um al/ January-February 1995



1975-1984

spon se coef f ic ient ."3 This express ion can be rewr i t -

t e n a s

/

.kj=l /

w hic h c l e a r ly show s t ha t t he be t a f o r a ny onesecur i ty is the w ei gh te d average of i t s re la t ive

r e sponse c oe f f i c i e n t s , e a c h w e igh t e d by t he p r o -

por t i on o f t o t a l va r i a nc e i n ma r ke t r e tu r n due t o

the e ve n t .

T h i s i n s igh t i n t o t he f und a me n ta l de t e r mi -

na n t s o f bet a w i ll be e xp lo i t e d a t ma ny po in t s i n

this a rt icle . For the m om en t i t provid es a grasp o n

the beha vior of a secur i ty ' s be ta ov er t ime . I s be ta

l ike ly to be constant over t ime , to dr i f t r andomly,

o r t o c ha nge i n some p r e d i c t a b l e o r unde r s t a nd-

able way? The answer i s t l ia t be ta wi l l change

wh en e i ther the re la t ive respo nse coef fic ients or

the re la t ive var iances of economic events change .T o t he de gr e e t ha t t he se c ha ng e s c a n be p r e d i c t ed

or e xp l a ine d , c ha nge s i n be t a c a n be p r e d i c t e d o r

e xp l a ine d . For e xa mple , t he mon th ly da t e s on

which the Bureau of Labor S ta t i s t ics announces

inf la t ion ra tes wi ll be da tes u po n whi ch the inf la -

t ion-or iente d events w i l l expla in a la rg er propor -

t ion of mark e t var iance , and wi ll the re fore be da tes

When f i rms wi th h igh re la t ive response to inf la t ion

wi l l have high e r tha n usua l be tas . For anot her

examp le , i f a f i rm chan ges i ts capi tal s t ruc ture ,

thereby increas ing i t s leverage ; i t s r e la t ive re -

spon se coef f ic ient to vi r tua l ly a l l econo mic even ts

will increase, and so as a result wil l i ts beta .

Be c a use be ta ne e d no t be c ons t a n t ove r t ime ,

i t fo l lows tha t es t imat ing th e averag e va lue of be ta

for a secur i ty in som e pas t per iod i s not the same

pr ob l e m a s p r e d i c t ing t he va lue o f bet a i n some

future per iod. This i s the f i r s t d is t inc t ion be tween

his tor ica l es t imat ion and future predic t ion. A sec-

ond equa l ly impor tant d is t inc t ion a r i ses f rom theuse of beta .

US ES OF BETA

I t i s imp or tant to examin e the uses of be ta , no t

on ly a s a n a id i n unde r s t a nd ing i t , bu t a l so be c a use

the c r i te r ia for predic t ion and es t imat ion probably

a r is e f r om the r e qu i r e me n t s o f usa ge . I n o the r

words , in each appl ica t ion, tha t es t imator or pre -

dic tor should be u sed tha t wi l l func t io n bes t in tha t

appl ica t ion. I f d i f fe rent appl ica t ions im pose di f fe r -

e n t r e qu i r e me n t s , t he n d i f f er e n t e s t ima to r s shou ld

be used. Reca l l tha t we never observe the " t rue"

be t a bu t r a the r o u t c ome s t ha t a r e r a ndom ly d i s tr ib -

uted about an expec ted va lue tha t i s equa l to be ta .

A s a c onse que nc e , w e mus t e s t ima te f r om the

obse r ve d ou t c ome s t he unde r ly ing va lue o f be t a

tha t ge ne r a t e d t he m. S imi l a r l y , w e mus t p r e d i c t

f rom this sam e da ta the va lue Of be ta to be ex-

pec ted in the futu re , as d is t inc t from the t rue va lu e

of be ta in th e pas t .

Performance EvaluationT he mo s t w ide ly r e c ogn iz e d use o f be t a, a t

th is wr i ting, i s in the eva lua t ion of pas t inve s tm ent

pe r f o r ma nc e . For r e a sons r e pe a t e d ly d i sc us se d i n

the l i te ra ture , th is u se of be ta is s t rongly su gges te d

by t he t he or y o f c a p it a l ma r ke ts ; t he w i sdom of

th is c our se ha s b e e n c on f i r m e d by t he e x t r a o rd i -

na r y i nc r e a se i n t he c l ar i ty w i th w hic h i nv e s tme n t

pe r f o r ma nc e !s now be ing a s se s se d a nd pe r c e ive d .

For t h i s pu r pose , t he po r t f o l i o a s a w ho le i s

t he a ppr opr i a t e e n t it y : O ne i s in t e r e s t e d i n t h e

deg ree O por t fol io r i sk ( the be ta of the por t fol io) .

There i s only a der iva t ive in te res t in th e r i sks of theind iv idua l s e cur it i es , t o t he de gr e e t ha t know le dge

of these can be he lpful in assess ing r i sk for the

overall Portfolio.

Inve le,',t StrategyWe n ow turn to the Use of be ta in the se lec t ion

of an inv es tm ent pol icy, tha t i s , to dec is ion mak ing

a s o p p o s e d t o ex post eva lua t ion.

Be c a use t he va lue o f be t a me a sur e s t he e x -

pe c t e d r e sponse t o ma r ke t r e tu r ns a nd be c a use t he

vas t major i ty of r e turns in divers i f ied por t fol ios

c a n be e xp l a ine d by t he h : r e sponse t o t he m a r ke t ,

an accura te predic t ion of be ta i s the most impor -

t a n t s i ng l e e l e me n t i n p r e d i c t i ng t he f u tu r e be ha v-

ior of a por t fol io . To the degree tha t one be l ieves

tha t on e can forecas t the futu re di rec t ion of mark e t

mov e me n t , a f o r ec a s t of be t a , by p r e d i c t ing t he

de gr e e o f r e sponse t o t ha t mov e me n t , p r ov ide s a

predic t ion of the resul tant por t fol io re turn . To th e

de gr e e t ha t on e i s unc e r t a in a bou t t he f u tu r e

mo ve m e nt o f t he ma r ke t , t he f o r e c a s t o f be t a, b y

de t e r min ing one ' s e xposur e t o t ha t unc e r t a in ty ,

provides a predic t ion of por t fol io r i sk . For a less

wel l d ivers i f ied por t fol io , the res idua l r e turns as-

soc i a t e d w i th t he c ompone n t i nve s tme n t s a s sumegr e a t e r p ropor t i ona l impor t a nc e , bu t t he i n f l ue nc e

of t he ove r a l l ma r ke t f a c to r r e ma ins impor t a n t

even in a por t fol io conta ining only one secur i ty .

Thus there i s l i t t le doubt tha t , i f one could

make an accura te predic t ion of future be ta for the

por t f o l i o , i t w ou ld be a n impor t a n t i ng r e d i e n t i n

h i s i nve s tme n t de c i s ion ma k ing . A n d e qua l ly , i f he

could make accura te predic t ions of the be tas for

Financial Analysts Journal / January-February 1995 105



1975 -1984

ind iv idual secur i t i es , these would be impor tan t

ingred ien t s o f h i s por t fo l io rev i s ion deci s ions . For

ins tance , i f the ma nage r d ecides to increase the

por t fo l io beta , then he wi l l seek to exchange cur-

r en t h o l d i n g s w i t h l o w b e t a fo r n ew p u rch as es

wi th h igh beta , and the success o f th i s exchange

wi l l de pe nd on h i s ab i l i ty to fo recas t the d i f ference

in beta. 4In th i s same contex t i t mus t a l so be no ted that

the deci s ion to rev i se the por t fo l io cannot be sep-

ara ted f rom an impl ic i t t ime hor izon . I f the asse t i s

to be held for a four-year per iod , perhaps the

average dura t ion in l arge por t fo l ios , then the ap-

propr ia te hor izon for the forecas t o f beta wi l l be

fo u r y ea rs . H o w e v er , i f t h e a s s e t is p u rch as ed w i t h

a v iew to explo i t ing an an t ic ipated market move-

me nt in the shor t t e rm, say the nex t hal f year , then

t h e b e t a fo recas t s h o u l d b e m ad e w i t h a h o r i zo n o f

s ix months .

Thu s far , two k inds of uses o f beta in the

d ec i s i o n -m ak i n g a s p ec t s o f p o r tfo l io m a n ag e m en thave bee n del ineated : (a) By forecas t ing the re-

s p o n s e t o m ark e t m o v em e n t , i t a l lo w s a fo recas t o f

s ecu r it y r e t u rn w h e n a fo recas t o f m ark e t m o v e-

me nt i s made; and (b) to the degree that the m arket

m o v em e n t i s u n ce r t ai n , b e t a , i n d e t e rm i n i n g t h e

re s p o n s e , d e t e rm i n es t h e ex p ec t ed u n ce r t a in t y o f

secur i ty o r por t fo l io re tu rn . To deve lop cr i t er i a fo r

pred ic tors o f beta , i t i s convenien t to refer to a

typ ical inves tment deci s ion s t ra tegy ( in the sp i r i t

of Tre yno r and Black) that rel ies , in part , on beta.

This wil l be referred to as a "typical control s t rat-

egy . ' s We ass ume that the s t ra tegy includes a

target fo r the por t fo l io beta , which changes over

t ime in response to (a) changing forecas t s o f the

d i rec t ion of mark et mo vem ent , o r (b ) changing

assessments o f the permiss ib le l evel o f sys temat ic

r i sk to be assumed. Transact ions are mot ivated in

par t by cons idera t ions of secur i ty analys i s , in the

sense that secur i t i es regarded as overvalued are

s o l d an d s ecu r i t i e s r eg a rd ed a s u n d e rv a l u ed a r e

purcha sed . Transact ions are a l so in f luenced by a

des i re to main ta in an ap propr ia te l evel o f d ivers i-

f i ca t ion . Also , each t ime that the beta t arget i s

chang ed , a se t o f t ransact ions i s unde r take n w i th

the in ten t ion of reach ing the ne w target . To reacht h e n ew t a rg et w i t h a m i n i m u m o f t r an sac t io n s

(hence a min imum of t ransact ion cos t s ) , there i s a

preference for the purchase of secur i t i es wi th val -

ues o f beta tha t d i f fer f rom the ex i s t ing por t fo l io in

the d i rec t ion of the ne w target , and for the sa le o f

secur i t ies tha t d i f fer in the opp os i te d i rec tion . T hus

t r an s ac t i o n s a r e u n d e r t ak en w i t h t h e m u l t i p l e

goal s o f (1 ) reach ing an appropr ia te por t fo l io beta

wi th a min imal n um ber of t ransact ions ; (2) increas -

ing expected re turn ; and (3) re ta in ing an appropr i -

a te de gree of por t fo l io d ivers i f i ca tion .

D u r i n g p e r i o d s w h en t h e t a rg e t b e t a fo r t h e

por t fo l io i s no t chang ing , there wi l l be t ransact ions

m o t i v a t ed b y t h e d es i re t o i n c reas e ex p ec t ed r e t u rn

and to cont ro l d ivers i f i ca t ion . Beta wi l l remain an

impor tan t cons idera t ion in these t ransact ions , be-cause the need to keep the por t fo l io beta near the

target wi l l serve as an ind i rec t cons t ra in t on pur-

chases and sa les . Transact ions involv ing s tocks

wi th betas d i f fer ing f rom the t arget wi l l requ i re

offse t t ing ad jus tments in o ther t ransact ions . And,

recal l ing that the beta o f the por t fo l io , jus t as the

beta o f a secur ity , ma y change over t ime, t ransac-

t i on s m ay s o m e t i m es b e r eq u i r ed s i m p l y t o ad ju s t

for an undes i rab le d r i f t in the por t fo l io beta .

Thus a typ ical con t ro l s t ra tegy wi l l involve a

cons t ra in t on the por t fo l io beta tha t induces a

preference for the p urchas e (or sa le) o f s tocks wi th

par t i cu lar k inds of ind iv idual betas ; in o therw o rd s , t h e b e t a o f each i n d i v i d u a l s t o ck a s s u m es

impor tanc e as a mean s to ach ieve a por t fo lio t arget

value . The por t fo l io beta being the average value

o f t h e i n d i v i d u a l b e t a s , w e i g h t ed b y i n v es t m en t

propor t ion s , the impor tan ce of the ind iv idual be-

t a s w i l l b e d e t e rm i n ed b y t h e i n v es t m en t p ro p o r -

t ions. Since the typical portfol io wil l by defini t ion

involve inves tments in secur i t i es tha t are p ropor-

t ional to their market capi tal izat ions, i t fol lows that

the typ ical weig h t o f an ind iv idual beta , as an

ingred ien t in the cont ro l s t ra tegy , wi l l be in p ro-

port i on to the capi tal izat ion of the fi rm.

ValualionFinal ly, a third class of uses appl ies to the

valuat ion of conver t ib le asse t s. C ons ide r any asse t ,

such as conver t ib le bonds , conver t ib le p refer red

s tock , warran t s and op t ions , tha t p rov ides the

oppor tun i ty to exerc i se a convers ion in to the un-

der ly ing secur ity . An impor tan t d eterm inant o f the

value of any such asse t i s the to ta l r i sk of the

under ly ing secur i ty , fo r the s imple reason that

such asse t s p rov ide one-s ided c la ims on the un-

der ly ing secur i ty . The h igher the under ly ing r i sk ,

the more l ike ly that the secur i ty p r ice wi l l change

significant ly. Since one profi ts ( loses) i f the secu-

r i ty p r ice goes in one d i rec t ion a nd i s unaffec ted i f

the secur i ty goes in the o ther , the greater the

expected r i sk , the greater the expected prof i t

( loss) . Know ledge of the value of beta perm i t s

pred ic t ion of one impor tan t e leme nt o f r i sk . Not ice

that th i s use of beta ar i ses because i t s usefu lness as

a m eas u re o f r is k o f t h e u n d e r l y i n g co m m o n i r a -

106 Financial Analysts Jou m al/ January-February 1995



1975-.-1984

pl ies an es t imate o f the value of the conver t ib le

asset .

CRITERIA FOR PREDIC'RON

Fo r each u s e o f b e t a d es c r i b ed ab o v e , o n e s h o u l d

as k w h a t p ro p e r t i e s an ap p ro p r i a t e m eas u re o f

beta sho uld have. I t i s bey on d the scope of th i s

art icle to discus s cri teria for the est im ator o f beta tobe used in h i s to r ica l per formance evaluat ion . We

m ay n o t e i n p as s i n g t h a t t h e ap p ro p r ia t e m eas u re

re la tes to an average l evel o f r i sk assumed in the

por t fo l io dur ing the eva luat ion per iod , so that i t i s

an es t imator o f a pas t r i sk l evel . The prob lem of

choos ing am ong a l t ernat ive es t imators o f the aver-

age value in the pas t p rov ides a good veh ic le fo r

in t roducing the conc epts o f b ias , var iance , an d

m ean s q u a re e r ro r a s em p l o y ed i n t h e co n t ex t o f

es t imat ion prob lems .

H o w a re w e t o ch o o s e am o n g s ev e ra l a lt e rn a -

t ive es t imates o f the ave rage va lue of the por t fo l io

beta over the h i s to r ical pe r iod? (Recal l tha t be ta i sn o m o re t h an an u n d e r l y i n g t en d e n cy an d t h a t t h e

actual resu l t s observed e x p o s t do no t t e l l us what

t h e ex ac t u n d e r l y i n g t en d en cy w as . ) T h e d i s t r ib u -

t ions of es t imated values fo r four imaginary es t i -

mators are p lo t t ed in Figure 3 .

Rgure 3.

•(c)

-- ~ d ~n ~

Suppose that the t rue average for a por t fo l io

or secur i ty beta was f t ,, and that ~ , i s an es t imator

of th i s and has an ex pected value f in . The qual i ty o f

th i s es t imator can be ju dge d b y th ree cr i ter ia : b ias ,

var iance , and mea n square er ror. 6 I f the es t imator

i s unbiased , i t s expected value equal s the t rue

under ly ing av erage , and the b ias , ~ , - f t ,, is zero .

Est imators (a) and (b) are unbiased in Figure 3.

Freedom from b ias i s obvious ly des i rab le .

Of a g ro u p o f u n b i a s ed e s t im a t o r s , t h e m o s t

des i rab le i s the one that i s the mos t accura te .

A ccu racy m ay b e d e f i n ed b y t h e s m a l l n es s o f t h e

var iance of es t imat ion er ror . Th us the bes t unbi -

ased es t imator i s the unbiased es t imator wi th the

smal les t var iance , i. e . , min imu m E[~, - ~n] 2. In

Figure 3 , a i s the mos t des i rab le unbiased es t ima-

tor. A cr i ter ion for compar ing b ia sed and unbia sed

es t i m a t o r s w h en i t i s n o t i m p o r t an t w h e t h e r t h eerror in ~n i s der ived f rom the b ias o r es t imat ion

error i s the mean square er ror , MSE. Whereas the

var iance of the es t imator i s the expected squared

devia t ion of the es t imated beta f rom i t s mean , the

m ean s q u a re e r ro r is th e m ea n o f t h e s q u a red

devia t ion of the es t imated beta f rom the t rue

value, i .e. , E[~ , - /3,] 2 . O f course, wh en fl~e

es t i m a t o r i s u n b i a s ed t h es e t w o m eas u res a r e

equivalen t . For any es t imator ~ , , the fo rmal re la-

t ions l~ip between b ias , BIAS(~, ) , var iance ,

VAR(fl , ) , and mean square error, MSE(~n), isg i v en b y 7

MSE(~n) = VAR(~,) + [BIAS(~n)]2.

As can be seen , by min imizing the MSE of the

es t imate , w e are in fac t min imizing the su m of the

var iance and the sq uared b ias o f tha t es t imator . As

such , min imizing the MSE imposes an arb i t rary

jud gm ent as to the re lat ive impor tan ce of the b ias

and variance. If i t is thought cri t ical to have an

unbiased es t imator , then min imizing the MSE

would no t au tomat ica l ly p rov ide one. I t i s qu i te

poss ib le tha t a b iased es t imator wi th low var iance

w o u l d b e ch o s en i n p re f e r en ce t o an u n b i a s ed

es t imator wi th h igh var iance . This po in t i s ampl i -fied graphical ly in Figure 3. Est imates (c) and (d)

are bo th b iased to the same ex ten t , bu t (c) i s

super ior to (d) because i t has a lower var iance . C an

(c) be superior to ei ther (a) or (b), even though (c)

i s b iased and (a) and (b) are no t? Us ing the MSE

cri terion, i t is qui te possible that (c) is superior to

(b) as long as

VAR(c) + [BIAS(c)] < VAR(b).

Let us now turn to the main top ic o f th i s

ar ti c le , namely , the pred ic t ion of beta and the

cr i t er i a fo r good pred ic t ion . Cons ider the case

where the cr i t er i a are concerned wi th the manage-me nt o f a por t fo l io o f s tocks and o ther non conver t -

ible assets , as dis t inct from convert ible assets .

Clear ly, the f i r st requ i rem ent i s a p red ic t ion of the

beta of the exis t ing portfol io. This wil l provide an

ind icat ion of the por t fo l io ' s respo nse to an t ic ipated

market movements as wel l as a p red ic t ion of the

por t fo l io ' s exposure to market r i sk . Natura l ly , the

pred ic t ion should re la te to the p lanning hor izon .

Financial Analysts Joumal / January-February 1995 107



1975- 1984

That i s, we are conce rned w i th an es t imate o f beta

for the fu ture per iod for which p lans are being

m a d e .

The por t fo l io beta in the fu ture i s the we igh te d

average of the ind iv idual secu r i ty betas , each

w e i g h t ed b y t h e p ro p o r t i o n a t e i n v es t m en t i n t h a t

secur i ty , s

tip =n

where Wp, i s the propor t ion of the to ta l inves t -

me nt n ow in s tock n , wi th ~ Wpn = 1 . The pre-

d ic ted por t fo l io be ta i s 9

= Z wp.K.tZ

The pred ic t ion er ror wi l l therefore be ~ W~n(fln -

f in )- Thu s the pred ic t ion er ror fo r the por t fo l io beta

i s the we igh t ed ave rage of the pred ic t ion er rors fo r

the ind iv idual secur i t i es , each weigh ted in p ropor-

t ion to the value o f the inv es tm ent in tha t secur i ty .

In o rder fo r the pred ic t ion er ror to be smal l , i t i sn eces s a ry t h a t t h e p red i c t i o n e r ro r s fo r th e i n d M d -

ual s tocks be smal l and average ou t to zero . I f the

es t i m a t i o n e r ro r s a r e i n d ep en d en t an d a r e ex -

pected to equal zero (which wi l l be the case i f the

es t imators are unbiased) then the es t imat ion er ror

wi l l t end to average ou t to zero .

The qual i ty o f the forecas t be ta fo r any one

s tock can be judged us ing the same cr i t er i a as was

sugges ted in the evaluat ion of es t imates o f the

historical average beta. If the t rue future beta is f in,

and the forecas t be ta i s f t , and has an expected

valu e of f t , , the n th e fore cast is unb ias ed i f fin - f l~= 0 . From a grou p of such unbiase d forecas t s , the

opt imal es t imate i s tha t wi th the min imum forecas t

var iance . I f , on the o ther hand , we are cons ider ing

b i a s ed an d u n b i a s ed fo recast s o f b e t a w e s h o u l d

ch o o s e t h a t o n e w i t h t h e m i n i m u m m ean s q u a re

forecas t er ror , M SE. Not ice that i t i s the t rue fu ture

v a l u e o f fin, not the presen t value , tha t i s to be

pred ic ted .

I f w e w ere co n ce rn ed w i t h e s t i m a t in g t h e b e t a

for a s ingle s tock n , f in , the p reced ing cons ider-

a t ions would suff ice . But s ince we are es t imat ing

beta fo r a num ber of secur i ti es , n = 1 . . . . N , we

mus t cons ider cr i ter i a fo r a co l lec t ion of es t imatesfin . . . n = 1 . . . N such that the col lect ion wil l

perform opt imal ly in use . Suppose that a p red ic-

t ion ru le i s def ined that p roduces , fo r each n , a

pre dict io n fin" The n a cri terion for this predict i on

ru le might t ake the form of a condi t ion ap ply ing to

a w e i g h t ed av e rag e o f t h e p ro p e r t i e s o f t h e e s t im a-

tor fo r the ind iv idual secur i ti es .

Cons ider , fo r example , the que s t ion of unbi-

asedness. T h e s t ro n g es t r eq u i r em en t o f u n b i a s ed -

n es s w o u l d b e t h a t t h e ex p ec t ed v a l u e o f t h e

es t imator fo r each and every ind iv idual secur i ty

should equal the v alue of beta fo r tha t secur i ty . A

w eak e r r eq u i r em en t w o u l d b e t h a t t h e av e rag e

es t i m a t ed b e t a for each i n d u s t ry s h o u l d eq u a l t h e

t rue average beta fo r tha t indus t ry . Compar ing the

req u i r em en t w i t h t h e p rev i o u s o n e , t h e d i f f er en cehere i s tha t some es t imators wi th in the indus t ry

c o u l d b e u p w a r d b i a s e d a n d o t h e r s d o w n w a r d

biased as long as the average b ias we re zero . A s ti ll

w eak e r s t a t em en t w o u l d b e t h a t t h e av e rag e p re -

d ic ted beta fo r a l l s tocks should equal the t rue

average value . This l as t s t a tement i s equ ivalen t to

asser t ing that the expected value for a p red ic ted

beta o f a s tock se lec ted a t random from the s tock

ex ch an g e s h o u l d eq u a l t h e ex p ec t ed t ru e v a l u e fo r

a secur i ty se lec ted a t random. This condi t ion re-

qu i res on ly that the average b ias , averaged over a l l

securi t ies , is zero.

Each of these pred ic t ion cr i ter i a involve an

av e rag e o v e r m an y s ecu r it i es . O v e r w h a t g ro u p o f

secur i t i es should th i s average be t aken? How

s h o u l d t h e s ecu r i t i e s b e w e i g h t ed ? T h es e t w o

ques t ions can be co l lapsed in to a s ingle ques t ion of

weight ing wi th in the un iverse o f secur i t i es , be-

cause those secur i t i es no t inc luded in the group

o v e r w h i ch t h e av e rag e i s t ak en w o u l d au t o m a t i -

ca l ly have a weigh t o f zero .

T h e an s w er t o t h e w e i g h t i n g p ro b l em fo l l o w s

di rec t ly f rom the cr i t er ion that the er rors in the

p red i c t ed b e t a s s h o u l d av e rag e o u t w h en

w e i g h t ed b y th e fu t u re p ro p o r t i o n a t e i n v es t m en t s

in the por t fo l io . What i s des i red i s unbiasedness ,

w h e n w e i g h t ed b y th e fu t u re i n v es t m en t p ro p o r -

t ions. 1° Thus, ideal ly, a s l ight ly differen t set of

w e i g h t s m u s t b e u s ed t o ev a l u a t e u n b i a s ed n es s fo r

each fu ture inves tment por t fo l io . In p ract i ce , i t i s

s impler and probably suff ic ien t to ach ieve unbi -

a s ed n es s r e l a t i v e t o t h e av e rag e i n v es t m en t

w e i g h t s t o b e ex p ec t ed fo r t h e u s e r o f th e p red ic -

t ion ru le . Since the sum of the inves tme nt weigh ts ,

summed across a l l po ten t ia l ins t i tu t ional users o f

the pred ic t ion ru le , approximates the aggregate

market values , a natura l cr i t er ion i s to def ine

unbiasedness re la t ive to a cap i ta l i za t ion-weightedaverage.

H av i n g s e t tl ed t h e q u es t i o n o f w e i g h ti n g , t h e

next i s sue i s tha t o f the s t r i c tness o f the unbiase d-

n es s co n d i t i o n : M u s t t h e p red i c t i o n b e u n b i a s ed

for every secur i ty , fo r g roups of secur i t i es such as

indus t r i es , o r on ly for the en t i re sample? The

answer i s again that the average expected pred ic-

t ion er ror fo r the gr oup of secur i t ies in any por t fo-

108 Financial Analysts Journal / January-February 1995



1975-1984

l io should be zero . I f a ll por t fo lios wer e iden t ica l to

the ma rket por t fo l io , then the absen ce of b ias for

the cap i ta l i za t ion-weighted market would suff ice .

But in fac t ind iv idual por t fo l ios d i f fer . Some em-

p h as i ze o n e i n d u s t ry g ro u p , s o m e em p h as i ze an -

o ther . Some concent ra te on s tocks wi th a par t i cu-

lar fundamenta l character i s t i c , some on s tocks

with a part icular technical characteris t ic. I t fol lowsthat , f f the averag e ex pected pred ic t ion er ror i s to

be zero for al l portfol ios , i t is desirable that the

p red i c t or b e u n b i a s ed fo r each i n d u s t ry g ro u p an d

for each fundamenta l o r t echnical character i s t i c

that m ay serve as a bas i s fo r por tfo l io se lec tion .

The ques t ion of the appro pr ia te cr i ter ion for

accuracy o f th e e s t i m a t o r s m ay b e ap p ro ach ed i n a

s imi lar fash ion : F rom the p o in t o f v iew of p red ic t -

ing portfol io risk, i t is the s ize of the error in

pred ic t ing the por t fo l io beta tha t i s impor tan t , as

d i s t inc t f rom the betas o f ind iv idual s tocks in the

portfol io. Moreover, i t is the error i tself that mat-

ters , no t the source f rom which i t der ives . Thus i ti s immater ia l whether an er ror resu l t s f rom b ias o r

f rom var iance in the es t imator . I t fo l lows that the

appropr ia te cr i t er ionfor accuracy in the pred ic t ion

of por t fo l io r i sk i s a min im um me an square er ror

pred ic tor . We are no t on ly concerned wi th pred ic-

t ions of beta fo r the pred ic t ion o f port fo l io r isk , bu t

a l so for making deci s ions wi th regard to poss ib le

portfol io revis ions. The respect ive cri teria for pre-

d ic t ion of ind iv idual secur i ty betas and of the

prese n t r i sk of the por t fo l io mus t be suc h as to

y ie ld a good cont ro l o f r i sk for the eventual por t -

fo l io cons t ru cted us in g these pred ic t ions . T hus the

form of these cr i t er i a mus t be der ived f rom the

deci s ion procedure . I f, fo r example , the ma nager

fo l lows a typ ical con t ro l s t ra tegy wi th a des i red

por t fo l io beta o f 1 .3 , then a good beta p red ic tor i s

one such that by re ly ing on the pred ic tor he wi l l

indeed tend to ach ieve a por t fo l io beta o f 1 .3 .

Because the por t fo l io rev i s ion deci s ion en ta i l s the

sale of specific securi t ies within the portfol io and

the purchase of o thers , i t becomes necessary to

pred ic t the betas o f ind iv idual secur i t i es - - -h igh-

l igh t ing another essen t ia l d i s t inc t ion between fu-

tu re p red ic t ion and h i s to r ica l evaluation : In p red ic-

t ion , the r i sk l evel s o f ind iv idual secur i ti es assum epr imary impor tance . Again , any er ror in the pre-

dict ion of r isk for the exis t ing portfol io, regardless

of i ts source or nature , wi l l be equal ly ser ious as

long as we accep t the pred ic ted value as the bas i s

for subsequent por t fo l io rev i s ion .

However , in modi fy ing the por t fo l io , we wi l l

cons ider a l t ernat ive comb inat ions of sa les and pur-

chases , fo l lowing the " ' typ ica l con t ro l s t ra tegy"

o u t l i n ed p rev i o u s l y . Ou r d ec i s i o n w i l l d ep en d i n

some form on the pred ic t ions of the betas fo r the

ind iv idual secur i t i es. P resum ably , we wi l l se lec t a

g ro u p o f s a le s an d p u rch a s es t h a t m o v e i n t h e

d i rec t ion of the des i red beta , whi le a l so ach iev ing

an increase in expected re turn . I t i s l ike ly that

certain "characteris t ics" of the s tocks wil l influence

the choice . Thus we cons ider curren t ly " 'popular"s t o cks fo r p u rch as e , a n d cu r r en t l y " u n p o p u l a r "

s tocks for sa le . Or we cons ider curren t ly h igh P/E

s tocks for purchase , and curren t ly low P/E s tocks

for sa le . An y one of an in f in it e num ber of deci s ion

ru le s m ay b e u s e d i n w h i ch t h e m a j o r i n g red i en t is

a fo recas t o f excess re tu rn on the ind iv idual secu-

r i ty . But i f th i s fo recas t o f excess re tu rn sho ws any

dependence a t a l l across d i f feren t s tocks , i t i s

p robable tha t the dependence wi l l t ake the form of

a bel i ef on the p ar t o f the mana ger tha t s tocks wi th

more of some character i s t i cs o r g roups of charac-

teris t ics are desirable. Another form of this ap-

p ro ach w o u l d b e b as ed o n t h e b e l i e f t h a t s t o ck s in

some sectors wi l l ou tperform o thers .

Ob v i o u s l y , w e w an t t h e p red i c t io n o f b e t a t o

be as accura te as poss ib le fo r each s tock , so that i t s

cont r ibu t ion to the expected change in beta i s as

accura te ly measured as poss ib le . But i t i s a l so

impor tan t tha t the l a w of averages wi l l opera te to

r ed u ce t o w ard ze ro o v e r a n u m b er o f d ec i s io n s t h e

average value of the er rors in the ind iv idual s tocks

selec ted . In o ther words , we want the pred ic t ion

ru le to be unbiased re la t ive to the deci s ion ru le

b e i n g u s ed .

The impor tan ce of th i s po in t can be ind icated

by an i l lus t ra t ion that we shal l develop in some

deta i l . Cons ider a por t fo l io manager who con-

s t ruct s h i s por t fo l io us ing s tocks curren t ly exper i -

encing t rad ing vo lume above thei r h i s to r ica l aver-

age . Then , when rev i s ing h i s por t fo l io , tha t

por t fo l io manager might se l l f rom the ex i s t ing

por t fo l io those s tocks wi th below average vo lume,

and might buy s tocks wi th curren t ly h igh t rad ing

v o l u m e . N o w , s u p p o s e t h a t a t t h e s am e t i m e t h e

por t fo l io manage r a t t empts to con t ro l the por t fo l io

risk and l imit beta to, for examp le, 1.2. If the

pred ic ted b eta value on h i s cur ren t por t fo l io i s 1.3,

he might reasonably se lec t fo r sa le those s tocksfrom the por t fo l io tha t wer e h igh in p red ic te d beta ,

an d r ep l ace t h es e w i t h s t o ck s f ro m am o n g t h e

act ively t raded l i st tha t were low in beta , wh i le a l so

meet ing h i s o ther cr i t er i a fo r h igher expected re-

turn.

Having se t up th i s i l lus t ra t ion , cons ider now

the ef fec t s o f a p red ic t ion ru le tha t i s negat ively

b iased re la t ive to changes in share t rad ing vo lume

Financial Analysts Journal / January-February 1995 10 0



1975 -1984

in compar i son to h i s to r ica l averages . In o ther

wor ds , i f the s tock is curren t ly po pular , the pre-

d ic ted beta wi l l be too low, and , i f the s tock i s

curren t ly unpop ular , the pred ic ted beta wil l be too

high . I t should be apparen t tha t the por t fo l io

manager would no t ach ieve h i s goal o f con t ro l l ing

r i sk by us ing such a ru le . The ave rage pred ic te d

beta fo r the s tocks that t i e so ld would be too h ighso that the sa le would reduce the beta o f h i s

por t fo l io l ess than he expected , and the average

p red i c t ed b e t a fo r t h e s t o ck s h e b o u g h t w o u l d b e

too low, resu l t ing in a g reater increase in beta f rom

the purchase than he expected . These two ef fec t s

combine to resu l t in the t ransact ions reducing beta

less than expected. In fact , i f the bias is large

eno ugh , the t ransact ions might ac tual ly increase

beta desp i te the fac t tha t a reduct ion i s p red ic ted .

T h i s ex am p l e w as d ev e l o p ed a t s o m e l en g t h

b ecau s e t h e co n v en t io n a l m e t h o d s n o w b e i n g u s ed

to pred ic t be ta do show this kind of bias and , as a

resu l t , th i s k ind of er ror i s be ing ma de on aneve ryd ay bas i s . I t is qu i te conceivab le fo r a por t fo-

l io manag er , wi th the bes t in ten t ions , to con t inue

to produce a beta o f 1 .3 on a regular bas i s , a l -

thou gh cont inual ly rev i s ing h i s por t fo l io to ach ieve

an apparen t beta o f 1 .2 , s imply because the pre-

d ic t ion ru le , by being b iased re la t ive to one of the

character i s t i cs employed for s tock se lec t ion , as -

ser t s tha t be ta wi l l be reduced , wh en in fac t i t wi l l

not .

Thus we see that in se lec t ing s tocks i t i s

des i rab le tha t the pred ic t ion ru le fo r ind iv idual

secur i ty betas again be unbiased re la t ive to the

character i st i cs em ploy ed in the deci s ion ru le . 11

Subject to th i s requ i rement , the pred ic t ion ru le

should be as accura te as poss ib le- - - i . e . , should

exhib i t min imum mean square er ror .

Final ly , l e t us tu r n to the th i rd us e of p red ic ted

beta , nam ely , the valuat ion of conver t ib le asse t s .

Cons ider an inves tor in conver t ib le asse t s who wi l l

repeated ly use the pred ic t ion ru le to value a con-ver t ib le asse t p r io r to making a buy or se l l deci -

s ion . For th i s purpos e the imp or tan t po in t i s tha t

he mak e prof i t ab le deci s ions on average . So in th is

case our cr i ter ion for the choice of a p red ic t ion ru le

fo r b e t a i s d e r i v ed f ro m t h e r eq u i r em en t t h a t

"go od" valuat ions of conver t ib le asse t s resul t ,

w h e re " g o o d n es s " i s m eas u red b y t h e p rof i tab i li ty

o f an i n v es t m en t s t r a t eg y b as ed u p o n t h e v a l u a -

t ions . Any er ror in the pred ic ted beta feeds

t h ro u g h t o a co n s eq u en t e r ro r i n t h e v a l u a t i o n o f

the conver t ib le asse t , and the re la t ionsh ip between

the former and the l a t t er i s a compl ica ted one. I t

fo l lows that a s imple cr i t er ion appl ied to the valu-at ion rule for convert ible assets wil l resul t in a

compl ica ted cr i t er ion for the under ly ing pred ic t ion

of ri sk. In par t icu lar the des i re fo r a min im um-

var iance unbiase d pred ic tor o f conver t ib le asse t

value (no t a bad cr i t er ion for a valuat ion ru le) ,

y ie lds a h igh ly complex cr i t er ion for the nature o f

t h e p red i c t o r o f r i sk o n t h e u n d e r l y i n g co m m o n ,

that , among o ther th ings , does no t requ i re tha t the

unde r ly ing pred ic tor b e unbia sed . 12 Thus the cr i-

t er i a fo r beta p red ic t ions to be used for asse t

valuat ion are crucia l ly dependent on the exact

context .

F O O T N O T E S

1 . Fo r an ex p l an a t i o n o f su b sc r i p t n o t a t i o n , s ee J .L . V a l en t i n e

an d E .A . M en n i s , Quantitative Techniques for Financial Anal-

ysis, 1 s t ed . (C h a r l o t t e sv i l le , V a . : C FA R esea rch Fo u n d a -

tion, 1971).

2. Not e t hat r M = r~ + ffM an d r a = r~ + r~. Because the e ven ts

a r e i n d e p e n d e n t , E(r ~, ~M) = 0 = E(r~, r~). Fu r t h e r , b ecau se

t h e re i s n o rea so n t o ex p ec t an y d ep en d en ce b e t w e en r~ an d

r/M or r~ and ffM, E( r~, r~) = 0 = E(rcM, r~). C o n s e q u e n t l y ,

COV(r=, rM) =E[r~a + r~] [rim + reM]

= E[r~, rh l + E[t~, dia + E[r~, riM]

+ E[r~, rh]

= E[t~, rh] + E[r~, reMl

= COV(r~, r~) + COV(r~, r~ia).

3 . A fo rma l p ro o f o f t h i s eq u a t i o n i s g i v en a s fo l lo w s : Le t t h e

m a r k e t r e t u r n g e n e r a t e d b y t h e jth fac t o r b e d en o t ed b y fj ,

w i t h r M = ~ j f/ , an d t h e m ark e t v a r i an ce re su l t i n g f ro m t h e

jth fac tor = VAR(~) = Vj . For exposi tory c onven ience , le t us

a s su m e t h a t t h e fac t o rs a re i n d ep en d e n t , so C OV ~ , f i ) = 0

for i ~ j . Wi thou t loss of general i ty , the fac tors are s tan-

d a rd i zed so t h a t t h e mark e t re sp o n se co e f f i c i en t i s 1 .

Th en r , = ~ j ' D ~ + Un, whe re "Dn is the securi ty re t urn

cau sed b y t h e jth fac t o r d i v i d ed b y t h e mark e t re t u rn cau sed

b y t h e same fac t o r , an d u n i s t h e sp ec i f i c co mp o n en t o f

re t u rn fo r s ecu r i t y n , i n d e p en d en t o f t h e fac t ors . Th e re fo re ,

= COV(rn, rM)/VAR(rM)

co v ( ~ ~jn~ + u., Z~)J J

= VAR(E~)J

~ ,j. vj + X Z 0j i j~i

=

j i j~i

110 Financial Analysts Jou m al/ January-February 1995



1975-1984

This equat ion can be viewed in another l ight . ~ . can be

considered to be that component of beta ar is ing from a

specific economic ewm t. C onsequently , to der ive the over-

al l beta , we should weight each one of these components

by the importance of that specif ic event to overal l market

variance.

4 . The discussion in the text indicates that an investor wil l

make use of h is predict ions about the future and his

at t i tude toward r isk to der ive a portfol io with a par t icularbeta value: In th is process , the investor is choosing be-

tween many portfol ios with different beta Values . When

confronted with such a der is ion process , some market

par t idpants s implify the portfol io problem by advocat ing

that an investor has to choose between just two extreme

portfolios . I f he expects the s tock market to r ise , he should

be ful ly invested in com mon s tocks with as hi gh a beta

value as is possible . I f he expects the s tock market to

decl ine , however , he should hold no common s tocks and

should be ful ly invested in some f ixed-interes t assets whos e

va lue does no t depend on mo vemen ts in the s tock marke t .

Such an approach is based on the naive bel ief that we kno w

with cer ta inty whe ther the market wil l r ise or fal l. W e can

never be so cer ta in . To reduce the exposure to th is uncer-

ta inty , i t is prude nt to se lect an in term ediate portfol io that

balances the r isks of an exposed posi t ion against thebenefits f rom the expected movement . Consequently, a t any

point in time, the optimal portfolio ~ be some mixture of

fixed-interest and equity securities, and, de pendi ng on the

uncertainty of our predictions and our risk attitude, the

portfolio could have one of many different beta levels.

5 . See J.L. Treynor and F. Black, "H ow to Us e Securi ty

Analysis to Improve Portfol io Select ion," The Journal ofBusiness (January 1973):66-86.

6. In prindple, none of these criteria is really appropriate.

One should f i rs t consider the investment s t ra tegy and

evaluate the cost of making an error . O nce th is is der ided,

the error is measured in such a way as to maximize the

present value of the contemplated investment s t ra tegy.

7. The derivation of this formula is simple~

MSE[ ~.] = E[~. - /3.12

= ElK - E(K ) + E(f~ .) - ~]2

= E [ A . - k . +/~. - a . ] =

= E[~ . - ~.] 2 + E[~ - /3,112

+ 2E[~, - ~,.] [K - /3.].

Now , E (K - K) 2 -= VAR(K), by def ini t ion

E(~ . - ft.)2 = [BIAS (K)] 2, since ~ . and fin are both

pararneters, whose difference is eclual

to BIAS (/~.), the expectation of the

:BIAs (K) ~ is equza tO BIAS (K) 2.

And E[K - K ] [K - /3 . ] = [~ - /3 . ]E tK - ~ ]

= IK - 8 .1 [ / ~ . - ~

=0.

8. The variance of returns on an individual security, n, is

re la ted to i ts beta , and the var iance of re turns o n the market

by the fol lowing expression:

VAR(r.) = ~.VAR(r,) + VAR(u.),

where VAR(u.) is the unsystematic r isk of the secur i ty n. I f

we combine N secur i t ies in a port fo l io wi th each secur i ty

weighted by W., the expected return and var iance of

returns for the port fo l io are

N

E(rp) = ~ E[Wpn(a, + 18,rM+ u,)]n=l

an d

.N

VAR(rp) = ~ W~./~VAR(rta)n= l

N+ Z W~VAR(u.).

n= l

In a d iversi f ied por t fo l io , the last term is c lose to zero and

N

VAR(rp) = VAR(rM) ~ W.2/~ = ~pVAR(rM).,=1

Also,

COV ( .~ Wnrn, rM)

CO V(rp, rM) ~- ~=1/3pVAR(rM) VAR(rM)

WnCOV(rn, rM)N

n= l

E w.~..VAR(rM) .=1

9. In future per iods , the inves tment proport ions wil l change

as a consequence of Stock price changes, and the portfolio

beta wil l therefore a lso change. Ne vertheless , the expected

weights in the future wil l be c lose to the exis t ing invest-

ment proport ions , so that ~:he predicted portfol io beta using

current investment proport ions is appropria te even when

the uncer ta in future changes in investment proport ions are

taken into account .

10. T here exists a problem of circularity here. The estim ates of

beta are used to determine the inv estmen t propert ies in anyfuture portfol io , but yet these future investment propor-

t ions are needed in order to choose between the var ious

est imates of ~. The choice of " typical investment propor-

t ions" Suggested in the text s idesteps th is problem.

11. As in the predict ion of portfol io beta , there is the quest ion

of appropria te W eights for the def ini t ion of unbiasedness .

Paralleling the previous discussion, a natural Criterion is

to def ine the unbiasedness re la t ive to a capi ta l izat ion

weighte d average. For purposes of portfol io revis ion, how-

ever , th is weight ing is less c lear ly indicated. The problem is

that the ent i re se t of beta predictors for securi t ies being

considered for purchase and sale inf luences the t ransact ion

deris ion, a l though only a f ract ion of the securi t ies under

considerat ion may actual ly be t raded. For ins tance, a mon g

e igh t secu r i t i e s r eg a rded a s cand ida te s fo r above -ave rage

apprec iat ion , the One wi th the h ighes t p red ic ted be ta

may be chosen fo r pu rchase . Whe the r th i s i s a l so the

s tock wi th the h ighes t t rue be ta depen ds on the e r ro r s in

est imating a l l e ight s ta t is t ics , regardless of the capi ta l i -

zat ion of those securi t ies . Nevertheless , i t is a reasonable

approx ima t ion to a s se r t tha t the expec ted in f luence o f an

error in es t imating ~8 is proport iona l to the capi ta l izat ion

of that asset .

12. To see this, note thai the typical valuation rule for the

estima ted v alue 17 of a convertible asset, as a fun ction of the

Financial Analysts Joumal / January-February 1995 111



1975-1984

es t ima ted m ean ~ an d v a r ian ce ~ o f th e re tu rn to th e

u n d e r ly in g co mmo n s to ck , h a s th e p ro p e r t ie s o f th e in te -

gral

V ~ f " X exp {-1/ 2 (X - ~')2/'} dX.

dxo

Th e in teg ra l i s a n o n l in ea r fu n c t io n o f ~ an d ~ , so th a t a

l inear or quadrat ic cr i ter ion on ~ ' (e .g . , E[~] = E[V], or

MINIMIZE VAR[~] ) imp l ie s a n o n q u ad ra t ic c r i t e r io n o n ~ .

In d eed , th e c r i t e r io n can o n ly b e wr i t t e n in th e fo rm o f an

in teg ra l eq u a t io n .

112 Financial Analysts Joum al / January-February 1995

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