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FINANCE
Distribution des rentabilités
Professeurr André Farber
Solvay Business School
Université Libre de Bruxelles
Le problème
• Comment généraliser le modèle simple (2 états) présenté précédemment?
• Rappel:
• Valeur d’une action = valeur actuelle du cash flow futur.
• Taux d’actualisation = rentabilité attendue
• Aujourd’hui: les leçons du passé.
• Que nous apprend l’historique des rentabilités concernant leurs
distributions de probabilité.
• Trois leçons principales:
• Les rentabilités sont des variables aléatoire distribuées (en première
approximation) selon la loi normale.
• Les rentabilités futures sont imprévisibles.
• La rentabilité attendue d’un actif financier est une fonction de son
risque systématique (le risque après diversification)
February 4, 2011 DESG |2
Capital Asset Pricing Model (CAPM alias MEDAF)
r rF (rM rF )
Taux d’intérêt Prime de risque Beta
Rentabilité attendue sans risque du marché
Beta (equity)
Nov. 27, 2006
Source: finance.yahoo.com (in key statistics
rM Ticker Company Beta
WMT Wal-Mart 0.06
r BUD Budweiser 0.32
KO Coca-Cola 0.76
MSFT Microsoft 0.79
rF ^SPX S&P 500 Index 1.00
SBUX Starbucks 1.17
INTC Intel 1.66
ADBE Adobe 1.81
AAPL Apple 2.03
F Ford 2.27
β 1 Beta
February 4, 2011 DESG |3
Rentabilités (Returns)
• Définition:
divt Pt Pt 1
Rt
Pt 1 Pt 1
• Composantes:
– rendement (dividend yield)
– plus ou moins value (capital gain)
• Période: 1 jour, 1 mois, 1 an,…
February 4, 2011 DESG |4
Ex post and ex ante returns
• Ex post returns are calculated using realized prices and dividends
• Ex ante, returns are random variables
– several values are possible
– each having a given probability of occurence
• The frequency distribution of past returns gives some indications on
the probability distribution of future returns
February 4, 2011 DESG |5
Annual returns (in percent)
19
10
30
50
70
-50
-30
-10
26
19
29
19
32
19
35
February 4, 2011
19
38
19
41
19
44
19
47
19
50
19
53
19
56
19
59
19
62
19
65
19
68
Year-end
19
71
19
74
19
77
19
80
19
DESG
83
19
86
Year-by-Year Total Returns on Large-Company Common Stock
19
89
19
92
19
95
19
98
20
01
20
|6
04
Annual returns ( in percent)
19
10
30
50
70
-50
-30
-10
26
19
29
19
32
19
35
February 4, 2011
19
38
19
41
19
44
19
47
19
50
19
53
19
56
19
59
19
62
19
65
19
68
Year-end
19
71
19
74
19
77
19
80
19
DESG
83
Year-by-Year Total Returns on Government Bonds
19
86
19
89
19
92
19
95
19
98
20
01
20
|7
04
Annual returns (in percent)
19
10
30
50
70
-50
-30
-10
26
19
29
19
32
19
35
February 4, 2011
19
38
19
41
19
44
19
47
19
50
19
53
19
56
19
59
19
62
19
65
19
68
Year-end
19
71
19
74
19
77
Year-by-Year Total Returns on T.Bills
19
80
19
DESG
83
19
86
19
89
19
92
19
95
19
98
20
01
20
|8
04
Frequency distribution
• Supposons que nous observions les rentabilités annuelles pendant 50 ans
.
Realized Return Absolute Relative
frequency frequency
-20% 2 4%
-10% 5 10%
0% 8 16%
+10% 20 40%
+20% 10 20%
+30% 5 10%
50 100%
February 4, 2011 DESG |9
Rentabilité moyenne / rentabilité attendue
• Rentabilité moyenne (arithmétique)
R1 R2 ... RN
Moyenne R
N
• Rentabilité attendue (espérance mathématique):
– Moyenne pondérée par les probabilités des rentabilités
possibles.
r E ( R) p1R1 p2 R2 ... pn Rn
avec pi probabilité de la rentabilité Ri
p1 p2 ... pn 1
February 4, 2011 DESG |10
Variance –Ecart type (volatilité)
• Mesure la dispersion des rentabilités
• Variance
• Ex post: moyenne du carré des écarts par rapport à la moyenne
(R1 R) 2 (R 2 R) 2 ...(R T R) 2
Var 2
T 1
• Ex ante: espérance mathématique du carré des écarts par rapport à la
rentabilité attendue
Var ( RA ) A E( RA RA ) 2
2
Var (R A ) A p1(R A,1 R A ) 2 p2 (R A,2 R A ) 2 ... p N (R A,N R A ) 2
2
• Unité de mesure : carré des écarts, peu intuitif
• Ecart type : Racine carrée de la variance A Var ( RA )
• Unité : rentabilité
February 4, 2011 DESG |11
Statistiques de rentabilité - exemple
Return Proba Squared Dev
-20% 4% 0.08526
-10% 10% 0.03686
0% 16% 0.00846
10% 40% 0.00006
20% 20% 0.01166
30% 10% 0.04326
Exp.Return 9.20%
Variance 0.01514
Standard deviation 12.30%
February 4, 2011 DESG |12
Normal distribution
• Realized returns can take many, many different values (in fact, any
real number > -100%)
• Specifying the probability distribution by listing:
– all possible values
– with associated probabilities
• as we did before wouldn't be simple.
• We will, instead, rely on a theoretical distribution function (the
Normal distribution) that is widely used in many applications.
• The frequency distribution for a normal distribution is a bellshaped
curve.
• It is a symetric distribution entirely defined by two parameters
• – the expected value (mean)
• – the standard deviation
February 4, 2011 DESG |13
Belgium - Monthly returns 1951 - 1999
Bourse de Bruxelles 1951-1999
180.00
160.00
140.00
120.00
Fréquence
100.00
80.00
60.00
40.00
20.00
0.00
00
00
00
00
00
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
00
00
00
00
00
00
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
0.
2.
4.
6.
8.
0.
8.
6.
4.
2.
0.
10
12
14
16
18
20
22
24
26
28
30
-8
-6
-4
-2
-2
-1
-1
-1
-1
-1
Rentabilité mensuelle
February 4, 2011 DESG |14
-8
0
50
100
150
200
250
300
350
400
450
.0
0
-7 %
.5
0
-7 %
.0
0
-6 %
.5
0
February 4, 2011
-6 %
.0
0
-5 %
.5
0
-5 %
.0
0
-4 %
.5
0
-4 %
.0
0
-3 %
.5
0
-3 %
.0
0
-2 %
.5
0
-2 %
.0
S&P 500
0
-1 %
.5
0
-1 %
.0
0
-0 %
.5
0%
0.
00
%
0.
50
%
1.
00
%
1.
50
%
2.
00
%
2.
50
%
DESG
3.
00
%
3.
50
%
4.
00
%
4.
S&P 500 Daily returns (June 96 - Nov 04) StDev = 1.23% n=2,122
50
%
5.
00
%
5.
50
%
6.
00
%
6.
50
%
7.
|15
00
%
7.
50
%
8.
00
%
0
20
40
60
80
100
120
140
160
180
200
-10.0%
-9.5%
-9.0%
February 4, 2011
-8.5%
-8.0%
-7.5%
-7.0%
-6.5%
-6.0%
-5.5%
-5.0%
-4.5%
-4.0%
-3.5%
-3.0%
Microsoft
-2.5%
-2.0%
-1.5%
-1.0%
-0.5%
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
3.5%
DESG
Microsoft Daily 1996-2003 StDev=2.58% (n=1,850)
4.0%
4.5%
5.0%
5.5%
6.0%
6.5%
7.0%
7.5%
8.0%
8.5%
9.0%
|16
9.5%
10.0%
Normal distribution illustrated
Normal distribution
0.0250
0.0200
0.0150
68.26%
0.0100
0.0050
95.44%
0.0000
0
4
8
2
6
0
4
8
2
6
0
.0
.6
.2
.8
.4
.0
.6
.2
.8
.4
0.
0.
0.
1.
1.
2.
2.
2.
3.
3.
4.
-4
-3
-3
-2
-2
-2
-1
-1
-0
-0
Standard deviation from mean
February 4, 2011 DESG |17
Risk premium on a risky asset
• The excess return earned by investing in a risky asset as opposed
to a risk-free asset
•
• U.S.Treasury bills, which are a short-term, default-free asset, will be
used a the proxy for a risk-free asset.
• The ex post (after the fact) or realized risk premium is calculated by
substracting the average risk-free return from the average risk
return.
• Risk-free return = return on 1-year Treasury bills
• Risk premium = Average excess return on a risky asset
February 4, 2011 DESG |18
Total returns US 1926-2002
Arithmetic Standard Risk Premium
Mean Deviation
Common Stocks 12.2% 20.5% 8.4%
Small Company Stocks 16.9 33.2 13.1
Long-term Corporate Bonds 6.2 8.7 2.4
Long-term government bonds 5.8 9.4 2.0
Intermediate-term government 5.4 5.8 1.6
bond (1926-1999)
U.S. Treasury bills 3.8 3.2
Inflation 3.1 4.4
Source: Ross, Westerfield, Jaffee (2005) Table 9.2
February 4, 2011 DESG |19
100 d’histoire boursière internationale
Table 1 Summary Statistics for Annual Real Equity Returns:
16 Markets and a World Index, 1900-2002
Geometric Arithmetic Standard
Country Mean Mean Deviation Autocorrelation
Belgium 1.8 4.0 22.1 0.23
Italy 2.1 6.2 29.4 0.03
Germany 2.8 8.1 32.4 -0.17
France 3.1 5.5 22.7 0.19
Spain 3.2 5.4 22.0 0.33
Japan 4.1 8.8 30.2 0.20
Switzerland 4.1 5.9 19.8 0.20
Ireland 4.3 6.6 22.2 -0.04
Denmark 4.6 6.2 20.1 -0.14
Netherlands 5.0 7.0 21.5 0.09
United Kingdom 5.2 7.1 20.2 -0.05
World 5.4 6.8 17.2 0.13
Canada 5.9 7.2 16.9 0.17
United States 6.3 8.3 20.3 0.01
South Africa 6.7 8.9 22.6 0.04
Sweden 7.3 9.5 22.7 0.13
Australia 7.4 8.9 17.8 -0.02
Average( ex world) 4.6 7.1 22.7 0.08
Source: Dimson, Marsh and Staunton, Irrational Optimism
Financial Analysts Journal, 60, 1 (2004) pp.15-25
February 4, 2011 DESG |20
Market Risk Premium: The Very Long Run
The equity premium puzzle:
1802-1870 1871-1925 1926-1999 1802-2002
Common Stock 6.8 8.5 12.2 9.7
Treasury Bills 5.4 4.1 3.8 4.3
Risk premium 1.4 4.4 8.4 5.4
Source: Ross, Westerfield, Jaffee (2005) Table 9A.1
Was the 20th century an anomaly?
February 4, 2011 DESG |21
Diversification
BERK, J., DE MARZO, P., Corporate Finance, Pearson; 2007
February 4, 2011 DESG |23
