A C T A U N I V E R S I T A T I S L O D Z I E N S I S
F O L IA O E C O N O M IC A 194, 2005
A d a m D e p t a *
T H E U SE OK BLUM E A N D VASICEK M E T H O D S
IN T H E ESTIM A TIO N OF BETA CO EFFICIENT
IN TH E SING LE-IND EX M O D EL
Abstract
T h is p a p e r will present a lte rn ativ e m eth o d s o f v alu atio n o f coefficients beta. T h e estim ation o f fu tu re coefficients b e ta can be received by d elim itatio n the coefficients b e ta fro m p a st d a ta and use these coefficients as the estim atio n o f fu tu re coefficients beta. A t the beginning we will p re sen t Illum e m eth o d and in the second section V asicek m eth o d .
Key words: coefficient beta in the single-index m odel, B lum e m eth o d , V asicek m eth o d .
I. IN T R O D U C T IO N
On th e capital m ark e t the shares rate o f re tu rn are d eterm in ated by
the fa cto r which reflects the changes o n this m ark e t. T h e ob serv atio n o f
prices o f ran dom ly chosen shares shows, th a t in the p eriod o f good econom ic
situ atio n a t the exchange (m easured with som e o f share indices) th e m ajority
o f share prices grow s, how ever w hen the co n d itio n o f th e m a rk e t w orsens,
the prices o f m ajority o f shares fall. E m pirical o bserv atio ns confirm th at
on m an y capital m ark ets the shares ra te o f re tu rn are to a large extent
related w ith rate o f retu rn o f index m ark e t, w hich reflects the general
situ atio n o n the m a rk e t (Levy, 1971).
Let R m it be the ra te o f retu rn from the m a rk e t index, at, ß t - the
coefficients o f eq u atio n , e. - the random e rro r term , then the ra te o f return
from an i-th share R t can be w ritten with th e help o f the e q u a tio n regress:
T h is eq uatio n defines the linear dependence o f return o r ra te share from
rate o f re tu rn the m ark ets index. In practice th e e q u a tio n o f regress is
estim ated and as a result the follow ing m odel is received:
R i = al + ß l R m
(2)
which is security ch aracteristic line. In this eq u atio n beta coefficient plays
the basic p art. It show s how m any percent ap p ro xim ately will the rate o f
re tu rn sh are grow , when the rate o f retu rn from the m a rk e t index grows
by 1% . By estim ation o f security characteristic line it com plies the d a ta
from p ast, which relate realized the ra te o f re tu rn shares and m ark e t index.
O n th e basis o f ordin ary least squares m ethod the form ula o f p aram eter
ß t is as follows:
.
c o v ( R , . ^ )
Pi
— 2 “ it w /t= t
where:
n - the nu m b er o f periods o f which in fo rm atio n com es from ,
R it - the rate o f re tu rn an i-th share in a i-th perio d,
R Mt - the rate o f re tu rn m ark e t index in a г-th period,
R,. - the arithm etical m ean the rates o f return from an i-th share,
R M - the arithm etical m ean the rates o f re tu rn m a rk e t index,
o j , - the variance o f m ark et index.
II. B L U M E M E T H O D
B lum e explored concerning o f relate between the b eta coefficients in
next periods (Blume, 1971). H e divided the period from July 1926 to June
1968 in to seven-year-periods. N ext, he calculated the beta coefficients by
using to regress the m o n th ly d ata. T hen, he m arked the b eta coefficients
for po rtfo lio s consisting o f one share, to p ortfolio s w ith fifty shares. As
a result o f these studies, he affirm ed th a t the beta coefficients o f large
p o rtfo lio s delivered considerable info rm atio n a b o u t them th e fu tu re beta
coefficients. T h e reasons for differences am ong the beta coefficients with
tw o different periods is firstly the fact, th a t the risk (the beta) can change
the stock o r portfolio, secondly th e beta coefficient in every period be
appointive with random erro r, and the greater this m istak e is, th en the less
ac cu rate p ro g n o ses’ o f coefficients will be for fu tu re period. T h e changes
o f beta coefficient in p ortfolio take dow n m u tu ally, in re la tio n sh ip from
w hat it is observed the sm aller hesitations o f th e real b eta coefficients in
case o f p o rtfo lio th an the individual shares. T h e m istakes o f estim ation the
b eta coefficients fo r individual stocks take dow n m u tu ally , when it will join
in p o rtfo lio these shares, therefore the m istake o f estim atio n the portfolio
coefficient will be lower. L et’s notice th a t the beta coefficients o f po rtfolios
are laden with sm aller m istake and change in sm aller g rade th a n the beta
coefficients o f individual shares, then the historical respects are m ore exact
th an in case o f individual shares.
Blume m eth o d depends on it division o b servation on tw o p a rts I and
II, and fo r each these p arts with the help o f o rd in ary least squ ares m ethod
it m ak es estim ation the beta coefficients. In next stage, it tak es place to
regress the b eta coefficient o f second period in re latio n to the coefficient
o f the prev iou s period:
ß u , i = a + b x ß i , i + £j (
4
)T h en , again with th e help o f o rd in ary least squares m eth od for unknow n
p aram eters o f regress the estim ators ä and fi are received. F inally, m odified
the beta coefficient has the follow ing aspect:
ßBlume, i — Ó + S x ß n j
(5)
T h e use o f eq u a tio n (5) leads to low ering o f high values o f beta
coefficients, and th e enlargem ent low.
In the aim o f im age this m eth o d m ark s the beta coefficients fo r the
com panies q u o ted on New Y o rk Stock E xchange in the su p p o rt o n m o nthly
the rates o f re tu rn in period July 1982 to Ju n e 1996. T h e share index D ow
Jones Industrial Average was accepted as the explanatory variable the changes
o f rates o f return shares. These audits were divided into tw o periods: July 1982
to June 1989 and July 1989 to June 1996. T h e results are presented in Table 1:
it was n o t possible to conduct this au d it on the W arsaw Stock Exchange
bccausc the q u a n tity o f observation was to o small.
T able
1.
T h e b e ta coefficients a p p o in ted w ith the use B lum e m eth o dN o. N am e
B eta coefficients B lum e beta coefficients I p eriod 11 p eriod
1 3M C o m p an y 1.0118 0.8193 0.9109
2 A etn a, Inc. 0.7832 1.2464 1.0985
3 A m erican E lectric Pow er C o m p an y Inc. 0.4372 0.5068 0.7737
T ab le 1. (contd.)
N o. N am e
Beta coefficients B lum e beta coefficients 1 period 11 perio d
5 A p p le ra C o rp o ra tio n 1.3893 1.0955 1.0322
6 A v n et Inc. 1.3976 0.8014 0.9031
7 B ausch & L o m b Inc. 0.8037 1.0875 1.0287
8 B axter In te rn a tio n a l Inc. 1.0352 1.0341 1.0053
9 B oeing C o rp o ra tio n 1.2052 1.0693 1.0207
10 Boise C ascad e C o rp o ra tio n 1.5295 1.1970 1.0768
11 C igna C o rp 0.8438 1.2189 1.0864
12 C igna In v estm en t Securities, Inc. 0.2485 0.2619 0.6661
13 C itig ro u p Inc. 0.9983 1.8291 1.3544
14 C o lg ate-P alm o liv e C o. 0.7998 0.9203 0.9553
15 C o m p u te r Sciences C o rp o ra tio n 1.2424 1.0357 1.0060 16 C o n so lid ated E d iso n Inc. 0.2288 0.4621 0.7540
17 C o m in g Inc. 1.1365 0.6385 0.8315
18 C S X C o rp o ra tio n 1.3345 1.2467 1.0986
19 D o w C hem ical Co. 1.2734 1.1425 1.0529
20 D u p o n t E 1 N e m o u rs & Co. 1.2055 1.2006 1.0784
21 E astm a n K o d a k Co. 0.9302 0.5430 0.7896
22 E d iso n In tern a tio n a l 0.2916 0.3799 0.7179
23 Eli Lilly an d C o m p an y 0.9432 1.0964 1.0326
24 E m erso n E lectric C o. 1.1533 1.0983 1.0334
25 E n g elh ard C o rp o ra tio n 1.0243 0.6918 0.8549
26 E xelon C o rp o ra tio n 0.3737 0.4825 0.7630
27 F a n n ie M ae 1.5136 1.4274 1.1780
28 F leetw o o d E n terp rises Inc. 1.4844 1.2076 1.0815
29 F o rd M o to r C o m p an y 1.3505 1.1268 1.0460
30 G e n era l D y n am ics C o rp o ra tio n 1.0810 0.4036 0.7283 31 G o o d y e ar T ire & R u b b e r Co. 1.3295 1.2940 1.1194
32 H ercules In c o rp o ra ted 1.1763 0.9783 0.9807
33 H oneyw ell In te rn a tio n a l, Inc. 0.8061 1.1216 1.0437
34 H u m a n a Inc. 0.9487 1.4542 1.1898
35 In te rn a tio n a l B usiness M achines C o rp o ra tio n 0.8293 0.8677 0.9322 36 In tern a tio n a l F la v o rs & F rag ran c es Inc. 0.9798 1.0333 1.0049 37 In te rn a tio n a l P ap er Co. 1.3278 1.2312 1.0918 38 J o h n s o n & Jo h n so n . 0.9168 1.0672 1.0198 39 K ro g e r C o. 0.5399 1.4330 1.1805 40 M a tte l Inc. 1.4429 0.9846 0.9835 41 M c d o n a ld s C o rp o ra tio n 0.9612 1.0782 1.0246 42 M e d tro n ic Inc. 0.8468 0.9320 0.9604
43 M erck & C o Inc. 0.7891 1.0643 1.0185
44 M o to ro la Inc. 1.5493 1.0187 0.9985
45 N a tio n a l S e m ico n d u cto r C o rp o ra tio n 1.4816 1.2048 1.0802
46 N o b le E nergy, Inc. 1.0625 0.5314 0.7845
47 N o rfo lk S o u th ern C o rp o ra tio n 1.1059 0.9591 0.9723 48 N o rte l N e tw o rk C o rp . 1.3456 1.0255 1.0015 49 N o rth r o p G ru m m a n C o rp o ra tio n 0.7824 0.9235 0.9567
T ab le 1. (contd.)
N o. N am e
B eta coefficients Blum e beta coefficients I period 11 period
50 O ccidental P etro leu m C o rp o ra tio n 0.7192 0.9895 0.9857
51 Pfizer Inc. 0.8654 1.2007 1.0784
52 P helps D o d g e C o rp o ra tio n . 1.4449 1.2104 1.0827
53 Pitney Bowes Inc. 1.3377 1.1970 1.0768
54 P ro c ter & G a m b le C o. 0.7840 1.0464 1.0107 55 Public Service E n terp rise G ro u p Inc. 0.3862 0.4603 0.7532 56 R ad io S h ack C o rp o ra tio n 1.3168 1.2977 1.1210 57 R ockw ell A u to m a tio n , Inc. 1.1370 0.6407 0.8325 58 R oyal D u tc h Petroleum C o m p an y 0.5549 0.7415 0.8768
59 R y d e r System , Inc. 1.3573 1.3505 1.1442
60 S ears R o eb u ck & C o 1.4491 1.2032 1.0795 61 S tew art In fo rm atio n Services C orp. 1.2673 1.0299 1.0034 62 S to rag eT ek C o rp o ra tio n 1.6031 1.0681 1.0202 63 T en n e co A u to m o tiv e Inc. 0.8095 1.1134 1.0401
64 T exas In d u stries 1.0938 1.0292 1.0031
65 T exas In stru m en ts Inc. 1.3893 1.3144 1.1284
66 T h e C o c a-C o la C o m p an y 0.6876 0.8903 0.9421 67 T h e W alt D isn ey C o m p an y 1.1601 1.2595 1.1043 68 U n ited A u to G ro u p Inc. 1.3799 0.9662 0.9755 69 U n io n Pacific C o rp o ra tio n 1.1926 0.9710 0.9775
70 U N IS Y S C O R P 0.9777 1.9968 1.4280
71 U n ited T echnologies C o rp o ra tio n . 1.3953 1.2048 1.0802 72 V alero E nergy C o rp o ra tio n 1.3523 0.7841 0.8955 73 V an K a m p en B ond F u n d 0.2261 0.3488 0.7043
74 V iacom Inc. 1.0843 1.0702 1.0211
75 V iad C o rp o ra tio n . 1.0659 0.9216 0.9558
76 W ach o v ia C o rp o ra tio n . 0.6905 1.0787 1.0249 77 W illiam s C o m p an ies Inc. 1.1622 0.8525 0.9255
78 X ero x C o rp o ra tio n 1.1853 1.0741 1.0228
Source: O w n calculations.
III. V A SIC E K M E T H O D
O n th e ground the audits Blume (Blume, 1975) and Levy (Levy, 1971)
o f b eta coefficient it noticed, th a t real value o f b e ta coefficient in the
period, w hen we m ak e th e prognosis, it is often closer th e m ean value o f
the beta coefficient, th a n the value estim ated on the basis o f th e historical
data. Vasicek proposed the technique, which relies on fitted th e beta coefficient
in dcpendcncc from grade o f uncertainty the respect o f th e b eta coefficient
(E lton, G ru b e r, 1998). Vasicek procedure relies on calculation from historical
the beta coefficient the weighted average for the share and the m ean o f
the value b eta coefficients in given sam ple shares o f the p ast p erio d where
the weights are added to the variance o f d istrib u tio n historical estim ations
the beta coefficient. T hese weights can be introduced as follows:
fo r ß n -
1T
- 2and for ß l
•
(6)
°>t + °>i
n f i + a j n
T he prognosis o f the beta coefficient for a share:
ß n - j r f r ? i + ^ ~ h t ß n -
Vßl
I<T()l
*+■(Tßi
(7)
where:
ßi
2
- the prognosis o f the beta coefficient fo r an i-th share,
ß i — the average value of the beta coefficients beta in a given sam ple
o f shares from the p ast period,
ß n - the beta coefficient from past for a given share,
стД - the variance o f distribution historical estim ations o f beta coefficient
for given sam ple o f shares,
a flii ~ the variance o f distribution historical estim ations o f beta coefficient
fo r given share.
Vasicek m ethod using the weights corrects observations with large standard
erro rs in larger grade th an observations laden sm all stan d ard errors. In
this m eth o d the weight ascriptitious the beta coefficient o f given share, in
com p ariso n to weight place on average the b eta coefficient in sam ple, is
inversely d ep endent from stan d ard errors o f the beta coefficient given share.
W ith higher values o f the beta coefficients o f concrete shares, the higher
stan d ard erro rs arc connected than in case o f shares with low er the beta
coefficients. T herefore for shares w ith higher coefficients, the b eta coefficients
will reduced in larger grade in relation to difference am o n g their value and
average value for sam ple, than will enlarged the beta coefficients for shares
on low coefficients. F rom this it results th a t, the average assessm ent o f
futu re beta coefficient will be lower from average coefficient in sam ple of
shares, on the ground which it takes place to estim ation.
In the aim of im age this m ethod m ark s the beta coefficients for the
com panies qu o ted on New Y ork Stock Exchange in th e su p p o rt on m onthly
the rates o f re tu rn in period July 1982 to Ju n e 1996. T h e share index D ow
Jones Industrial Average was accepted as the explanatory variable the changes
of rates o f re tu rn shares. These audits were divided into tw o periods: July
1982 to Ju n e 1989 and July 1989 to Ju n e 1996.
T o every from periods with the help o f ord in ary least squ ares m ethod it
was estim ated separately the beta coefficients fo r every with com panies.
I hen on the ground the form ula (7) it was m arked the prognosis o f the
beta coefficient for an i-th share. T he results arc presented in T ab le 2.
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o] — ŕn <4T ab le 2. (co n td .) ■ оi Q \ N o. N am e B e ta coefficients а1 ‘К ß 2 — ■ ■ R V asicek b e ta coefficients I p eriod II p erio d * , л
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2 j "h 27 F a n n ie М ае 1.5136 1.4274 0.1169 1.3448 1.461728 F leetw o o d E n terp rises Inc. 1.4844 1.2076 0.1051 1.3356 1.4407
29 F o rd M o to r C o m p an y 1.3505 1.1268 0.0690 1.2616 1.3306
30 G e n era l D y n am ics C o rp o ra tio n 1.0810 0.4036 0.0535 1.0258 1.0793 31 G o o d y e a r T ire & R u b b e r C o. 1.3295 1.2940 0.0821 1.2253 1.3075
32 H ercules In c o rp o ra te d 1.1763 0.9783 0.0533 1.1165 1.1698
33 H oneyw ell In te rn a tio n a l, In c . 0.8061 1.1216 0.0457 0.7709 0.8167
34 H u m a n a Inc. 0.9487 1.4542 0.0705 0.8849 0.9554
35 In te rn a tio n a l B usiness M a ch in es C o rp o ra tio n 0.8293 0.8677 0.0342 0.8023 0.8364 36 In te rn a tio n a l F la v o rs & F ra g ra n c e s Inc. 0.9798 1.0333 0.0504 0.9327 0.9831
37 In te rn a tio n a l P a p e r Co. 1.3278 1.2312 0.0623 1.2489 1.3112 38 J o h n s o n & Jo h n s o n . 0.9168 1.0672 0.0398 0.8821 0.9218 39 K ro g e r C o. 0.5399 1.4330 0.0715 0.5031 0.5746 40 M a tte l Inc. 1.4429 0.9846 0.2058 1.1597 1.3655 41 M c d o n a ld s C o rp o ra tio n 0.9612 1.0782 0.0322 0.9316 0.9639 42 M e d tro n ic Inc. 0.8468 0.9320 0.0687 0.7914 0.8600 43 M e rck & C o Inc. 0.7891 1.0643 0.0317 0.7652 0.7970 44 M o to ro la Inc. 1.5493 1.0187 0.0917 1.4138 1.5055
45 N a tio n a l S em ico n d u cto r C o rp o ra tio n 1.4816 1.2048 0.1381 1.2864 1.4246
46 N o b le E nergy, In c. 1.0625 0.5314 0.1008 0.9604 1.0611
47 N o rfo lk S o u th e rn C o rp o ra tio n 1.1059 0.9591 0.0414 1.0623 1.1037
48 N o rte l N e tw o rk C o rp . 1.3456 1.0255 0.0770 1.2468 1.3238
49 N o rth ro p G ru m m a n C o rp o ra tio n 0.7824 0.9235 0.0774 0.7246 0.8020 50 O ccid en tal P e tro le u m C o rp o ra tio n 0.7192 0.9895 0.0521 0.6835 0.7356
51 Pfizer Inc. 0.8654 1.2007 0.0430 0.8299 0.8729 52 P helps D o d g e C o rp o ra tio n . 1.4449 1.2104 0.1516 1.2360 1.3876 53 P itney B ow es Inc. 1.3377 1.1970 0.0668 1.2525 1.3192 54 P ro c ter & G a m b le C o. 0.7840 1.0464 0.0291 0.7622 0.7913
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55 P ublic Service E n te rp rise G ro u p Inc. 0.3862 0.4603 0.0261 0.3766 0.4027
56 R a d io S h a c k C o rp o ra tio n 1.3168 1.2977 0.0967 1.1953 1.2920
57 R ockw ell A u to m a tio n , Inc. 1.1370 0.6407 0.0555 1.0768 1.1323
58 R o y a l D u tc h P etro leu m C o m p a n y 0.5549 0.7415 0.0267 0.5407 0.5674
59 R y d e r System , Inc. 1.3573 1.3505 0.0681 1.2691 1.3372
60 S ears R o e b u c k & C o 1.4491 1.2032 0.0634 1.3615 1.4249
61 S te w a rt In fo rm a tio n Services C o rp . 1.2673 1.0299 0.1023 1.1437 1.2459
62 S to rag eT ek C o rp o ra tio n 1.6031 1.0681 0.2792 1.1763 1.4555
63 T en n eco A u to m o tiv e Inc. 0.8095 1.1134 0.0342 0.7830 0.8173
64 T exas In d u stries 1.0938 1.0292 0.0768 1.0137 1.0905
65 T exas In stru m e n ts Inc. 1.3893 1.3144 0.1086 1.2455 1.3540
66 T h e C o c a-C o la C o m p an y 0.6876 0.8903 0.0288 0.6687 0.6975
67 T h e W alt D isn ey C o m p an y 1.1601 1.2595 0.0859 1.0650 1.1510
68 U n ite d A u to G ro u p Inc. 1.3799 0.9662 0.0870 1.2654 1.3524
69 U n io n Pacific C o rp o ra tio n 1.1926 0.9710 0.0526 1.1328 1.1854
70 U N IS Y S C O R P 0.9777 1.9968 0.0570 0.9246 0.9816
71 U n ited T ech n o lo g ies C o rp o ra tio n . 1.3953 1.2048 0.0615 1.3135 1.3750
72 V alero E n erg y C o rp o ra tio n 1.3523 0.7841 0.1886 1.1091 1.2977
73 V a n K a m p e n B o n d F u n d 0.2261 0.3488 0.0119 0.2235 0.2354 74 V iacom In c. 1.0843 1.0702 0.0968 0.9842 1.0810 75 V iad C o rp o ra tio n . 1.0659 0.9216 0.0655 0.9993 1.0648 76 W a c h o v ia C o rp o ra tio n . 0.6905 1.0787 0.0775 0.6395 0.7169 77 W illiam s C o m p an ies In c 1.1622 0.8525 0.0787 1.0750 1.1537 78 X e ro x C o rp o ra tio n 1.1853 1.0741 0.0496 1.1292 1.1788
<
P
РГK
n
S-Ł
I
o 3 o —5s
Source: O w n calcu la tio n s.
n о
о
W ith the results contained in T able 2 it is possible to notice th a t if the
beta coefficient estim ated on the basis o f historical d a ta is higher th an 1,
then the corrected the sam e beta coefficient with the help o f Vasicek m ethod
is sm aller th an the one which was estim ated on the g ro u n d o f historical
d a ta . In case, when estim ated on the ground o f historical d a ta the beta
coefficient is sm aller th a n 1, then the corrected it will be higher from him.
IV. C O N C L U S IO N S
In the lite ra tu re on the subject there is w idespread discussion related
to the usefulness the presented m ethods for estim ation o f the beta coefficients.
E stim ations o f these coefficients can be applied in co n stru ction o f investm ent
po rtfolio s which can protect from risk with the help o f fu tu re contracts.
T h e use o f introduced m ethods can arouse som e d o u b t p articu larly in case
o f developing m ark ets, where large influence o n stock prices is exerted by
beh av io u r o f small investors. D ecisions u n dertaken th ere by investors cause
th a t the price o f qu o ted stocks can change considerably, som etim es from
session to session. W e should also underline the fact th a t except choice of
m ethod estim ation o f beta coefficient, wc should analyse the influence of
the value o f this coefficient on the choicc o f m ark e t p o rtfo lio as level of
reference o f studied period and com partm ent tem porary between observations.
R E F E R E N C E S
B lum e M .E . (1971), O n the assessm ent o f risk, Journal o f Finance, 6, 1, M a rch .
B lum e M .E . (1975), B elas and their regression tendencies, Journal o f Finance, 10, 3, June. E lto n E .J., G ru b e r M .J. (1998), Nowoczesna teoria portfelow a i analiza papierów wartościowych,
W IG P R E S S , W arszaw a.
Ja ju g a K ., Jaju g a Г. (1998), Inwestycje, instrum enty finansow e, ry zy ko fin a n so w e, inżynieria fin a n so w a , W yd. N a u k . P W N , W arszaw a.
Levy R . (1971), O n th e sh o rt-term sta tio n a rity o f b e ta coefficients, Financial A n a lysts Journal, 27, 5, D ecem ber.
Reilly F.K.., B row n K..C. (2001), Analiza inw estycji i zarządzanie portfelem , P W E , W arszaw a. S m ag a E. (1995), R y z y k o i zw rot w inwestycjach, F u n d a c ja R ozw oju R ach u n k o w o ści w Polsce,
W arszaw a.
T arczyński W ., M ojsiew icz M . (2001), Zarządzanie ryzykiem , P W E , W arszaw a.
T arczyński W ., Z w olankow ski M. (1999), Inżynieria finansow a, A gencja W yd. Placet, W arszaw a. W ierzbicki M . (1995), A naliza portfelow a, M O T T E , Ł ódź.
A d a m D e p ta
Z A S T O S O W A N IE M E T O D B IA JM E ’A O R A Z V A SIC K A W S Z A C O W A N IU W S P Ó Ł C Z Y N N IK A В Е Г А
W M O D E L U JE D N O W S K A Z N IK O W Y M Streszczenie
N a ry n k u kap itało w y m k ształto w an ie się stó p zw rotu akcji jest zd eterm in o w an e działaniem czynnika odzw ierciedlającego zm iany n a tym ry n k u . O b serw acja cen losow o w ybranych akcji pokazuje, że w czasie d o b rej k o n iu n k tu ry n a giełdzie (m ierzonej k tó ry m ś z indeksów giełdowych) w iększość cen akcji rośnie, n a to m iast kiedy sytuacja na ry n k u się p o g arsza , ceny większości akcji sp ad ają. O bserw acje em piryczne p o tw ierd zają, że n a wielu ry n k a ch k ap itało w y ch stopy zw rotu większości akcji są w dużym stopniu po w iązan e ze sto p ą zw ro tu z indeksu ry n k u , odzw ierciedlającego o g ó ln ą sytuację n a rynku.
Celem arty k u łu jes t przedstaw ienie alternatyw nych m etod szacow ania w spółczynników beta. O szacow anie przyszłych współczynników beta m ożna otrzym ać przez w yznaczenie w spółczynników b e ta d la d an y ch z przeszłości i w ykorzystanie tych w spółczynników ja k o szacu n k ó w przyszłych w spółczynników beta. P rzedstaw ione zostały' dw ie m etody szacow ania w spółczynnika beta: B lum e’a o ra z V asicka.
