Finance - Master SDMR Part 1
Olivier Brandouy University of Paris 1, Panth´ eon-Sorbonne, IAE (Sorbonne Graduate Business School)
Olivier Brandouy (IAE - Paris)
Finance - Master SDMR
February 14, 2011
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Preliminary informations
1 2
Webpage : http://brandouy.free.fr/MastFIN Some useful softwares : I
I
R, software environment for statistical computing and graphics and the Rmetrics packages. A list of useful packages F
F I
3
The ”Rmetrics” suite : fOptions, fPortfolio, timeSeries, fExtremes, fImport and their dependencies. performanceAnalytics and car
OpenOffice (an open alternative to micro$oft office); requires some extra-libraries (NLPSolver.oxt and scsolver.uno.oxt for optimization)
Scope : I I
Mainly (ordinary) Stocks Quantitative approach to AM, (option valuation and greek-related stuff – if we have time for that)
Olivier Brandouy (IAE - Paris)
Finance - Master SDMR
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Outline
A 18 hours course. 6 hours around stock market price motions, basic concpets of risk in finance and a first use of ”R” for illustrations. 12 hours around portfolio management and CAPM.
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Asset Returns and Risks
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Data
Some clarifications (if needed) : Prices vs. return Historical vs. Expected returns Implicitly Ex-Ante vs. Ex-post evaluation pb.
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Historical returns(1)
Definition Returns computed using historical data (past prices such as daily closing prices) specific frequency free sources for historical data (see one of the files I sent you): I
Stocks, Bonds http://finance.yahoo.com/
I
Options http://www.liffe-data.com/
can be used as (weak) proxies for expected returns
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Historical returns(2) Algebraic returns Rt =
pt+1 − pt → pt+1 = pt × (1 + Rt ) pt
(1)
suppose discrete capitalization practically not usable with softwares Continuously compounded returns a.k.a. Continuous returns pt+1 = pt × (1 +
Rt n ) n
(2)
Rt n ) = e rt n→+∞ n rt = log (pt+1 ) − log (pt ) lim (1 +
(3) (4)
Additive properties Usual rates of return in finance Olivier Brandouy (IAE - Paris)
Finance - Master SDMR
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Expected Returns
Definition Future is hardly predictable ! This is the role of ”strategists” and ”financial analysts”. Nevertheless Scenarios :
Different scenarios, different outcomes in terms of return (at a specific maturity T ) Probabilities associated with these future economic states Expected returns aggregate these possible outcomes
Models :
CAPM !
Consensus :
I/B/E/S
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Individual Assets Risk
0.4
0.4
Randomness is not a simple concept ! → X a r.v. following a uniform distribution in [1, 6] (a dice : Figure 1(a) presents the distribution of realizations. → Central limit theorem can explain why stock returns are said to be Normally distributed in first approximation Figure 1(a).
:
0.3
: :
density
:
0.0
0.0
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0.2 0.1
density
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X
0
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4
6
X
(a) Uniform distribution function in (0,6)
(b) Normal distributions with various µ and σ
Figure: Deviations from Gaussianity
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Asset (centered)moments The k th -moments of the probability distribution for a given random variable X is:
0.6 0.5 0.4 density
0.3
: : :
0.1 0.0
0.0
0.1
0.2
density
: : :
0.2
0.3
0.4
0.5
0.6
m(X )µ = E [(X − E (X ))k ] Pn ¯ )k It can be estimated using : 1/n i=1 (Xi − X k = 2 → variance, k = 3 → skewness, k = 4 → kurtosis
−4
−2
0
2
4
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X
0
2
4
X
(a) Skewed Normal Distributions Olivier Brandouy (IAE - Paris)
−2
(b) Normal distributions with various kurtosis
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What about ”Expected” volatility ?
→ One can use here : Investors consensus Models that forecast volatility (such as ARCH models) Implied Volatility extracted from options quotations (this will be discussed further in my ”Risk Management” class)
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Finance - Master SDMR
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Punchline
The parametric (Gaussian) approximation is useful but, as shown later, doesn’t provide a correct framework to model price dynamics in Finance Many models are grounded on this assumption (for example, CAPM, Black-Scholes...) Financial engineers usually use more sophisticated laws and models
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A review of stylized facts
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What are stylized facts ?
Definition Empirical properties that can be observed in most of financial data and that characterize, in a qualitative way, the kind of risks one faces when investing in individual assets. Risk in the portfolio context will be reviewed later. A good reference : Rama Cont, 2001, ”Empirical properties of asset returns, stylized facts and statistical issues”, Quantitative Finance, 1, 223-236 (can be downloaded from my webpage).
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Finance - Master SDMR
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A (selected) series of stylized facts
1
Absence of autocorrelations
2
Fat tails
3
Gain loss asymmetry
4
Aggregational Gaussianity
5
Volatility clustering
Remark Notice these stylized facts highlight the weaknesses of the parametric framework to assess risk in Finance.
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Finance - Master SDMR
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10000 points
8000 6000
5000
Krach de 1987 4000
Indice en points
10000
12000
15000
Dow-Jones : ordinary risk ?
2000
Bulle Internet
0
0
Crise de 1929
1900
1920
1940
1960
1980
2000
1940
1960
Date
2000
1980
2000
return
ï0.1
0.0
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0.1 0.0 ï0.1 ï0.2 ï0.3
Rentabilité
1980 Date
1900
1920
1940
1960
1980
2000
1940
Date
1960 Date
(a) Dow-Jones (1928 – 2007)
(b) Simulated Index
Figure: DJ w.r.t. a Simulated i.i.d. index Olivier Brandouy (IAE - Paris)
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44.9 44.8 44.7 44.5
44.6
Cotation en USD
45.0
45.1
Intraday prices : Pfizer
10:36
11:24
12:12
13:00
13:48
14:36
13:48
14:36
0.001 ï0.001 ï0.003
Rentabilité
0.003
01.fev.2001, Heure
10:36
11:24
12:12
13:00
01.fev.2001, Heure
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Fat tails
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30 0
10
20
Fréquence
40
50
60
Distribution of returns, Dow-Jones
ï0.10
ï0.05
0.00
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Rentabilités
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1eï05
1eï03
densité
1eï01
Distribution of returns, Pfizer
ï20
Olivier Brandouy (IAE - Paris)
ï10
0
Finance - Master SDMR
10
20
February 14, 2011
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0.000 ï0.001
Quantiles empiriques
ï0.003
ï0.2
ï0.002
ï0.1
Quantiles empiriques
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Non Normality of Asset Returns, QQ-Plots
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Quantiles théoriques
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Quantiles théoriques
(a) Dow-Jones Index, 1928 – 2007
(b) Pfizer, Feb. 1st 2001
Figure: DJ and Pfizer returns, Q-Q Plot
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1e+00 1eï01 1eï02
Probabilité >= x
1eï02
Rentabilités Négatives
Rentabilités Négatives
Rentabilités positives
Rentabilités positives 1eï03
1eï03
Probabilité >= x
1eï01
1e+00
Fat Tails (1)
Rentabilités positives ajustées à la Gaussienne
0.001
Rentabilités positives ajustées à la Gaussienne
1eï04
Rentabilités négatives ajustées à la Gaussienne
1eï04
Rentabilités négatives ajustées à la Gaussienne
0.010
0.100
1.000
0.001
Rentabilités Absolues
0.010
0.100
1.000
Rentabilités Absolues
(a) Dow-Jones Index, 1928 – 2007
(b) Pfizer, Feb. 1st 2001
Figure: Survival function of empirical returns w.r.t. / theoretical Gaussian distribution
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Autocorrelation
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0.8 0.6
ACF
0.0
0.0
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0.6 0.4 0.2
ACF
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1.0
1.0
Autocorrelation (1)
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40
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Lag
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Lag
x : lag (a) Return Autocorrelations, DJI
(b) Autocorrelations, Absolute value of returns, DJI
returns, y : level of autocorrelation. Hash lines : threshold under which these coefficients are non significant. Figure: Empirical autocorrelations
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0.6
ACF
0.4 0.0
ï0.2
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ACF
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Autocorrelation (2)
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30
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Lag
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Lag
(a) Return Autocorrelations, Pfizer
(b) Autocorrelations, Absolute value of returns, Pfizer
x : lag returns, y : level of autocorrelation. Hash lines : threshold under which these coefficients are non significant. Figure: Empirical autocorrelations
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Volatility clustering
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Non stationary variance processes (1)
Definition AutoRegressive Conditional Heteroskedasticity : variance of the error term is related to the squares of the previous innovations. Roughly speaking, ARCH process can be described using three relations : 1
rt = mean return +noiset (noise is an innovation)
2
noiset = volatilityt × innovationt
3
2 2 volatilityt2 = x + y × volatilityt−1 + z × noiset−1
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Non stationary variance processes (2)
(a) Returns, DJI, from 02/12/1997 to 02/12/2007
(b) Simulated GARCH(1,1)
Figure: Calibrated ”Garch Model” using real data
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Agregationnal Gaussianity
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60 20
40
BNP.PA.Close
80
BNP (2001-01-01, 2010-01-01), various granularity
2003−01−01
2005−10−19
2008−08−06
Time
60
BNP.PA.Close
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60 40
BNP.PA.Close
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80
80
90
(a) Daily prices
2003−01−01
2005−10−18
2008−08−04
2003−01−01
Time
2008−07−13
Time
(b) Weekly prices Olivier Brandouy (IAE - Paris)
2005−10−07
(c) Monthly prices
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0
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Density
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BNP (2001-01-01, 2010-01-01), histograms
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Density
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Density
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(d) Daily prices
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ret2
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(e) Weekly prices Olivier Brandouy (IAE - Paris)
−0.2
(f) Monthly prices
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Gain/Loss Asymeties
DIY ! Using the DJ series How many ups and downs ? Compute the semi-variance w.r.t a return=0. What can you conclude ?
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A brief discussion on technical analysis and chartism
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On the market randomness Pros and cons : EMH hypothesis vs. Chartists Chartists, examples and results : Example Triangles, Rectangles, Moving Averages:
d{k}
d{k+n}
p
=...%
d{k+1} m
s
n d{k+n+1}
(g) Rectangle
(h) Moving Average 10 / 90
Figure: Two Chartists Strategies Olivier Brandouy (IAE - Paris)
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A Simulation
Methodology (use dj.csv): 1
Identify signals
2
initial endowment = first value of DJ
3
Enter the market at the first ”Buy” signal
4
Buy as much trackers as you can (TC apply or not)
5
At each ”Sell” signal, complete liquidation of your position (TC apply or not)
Remark Notice you cannot obtain consecutive ”Buy” signals with MM10,90 which highly facilitates computations.
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Example of chartist strategy (without TC)
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Example of chartist strategy (with TC)
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Data snooping problem (1) Imagine you want to hire ”someone who can predict the market at least in 75% of the cases”. You build a test based on the next 14 closing prices for an Index (such as the DJ) limiting the problem to correct predictions of ”ups” and ”downs”. Candidates are successful if they can predict accurately at least 11 moves among 14. The probability of success can be computed as follows : 14 14 14 14 14 14 14 C14 = 0.02869 11 0.5 + C12 0.5 + C13 0.5 + C14 0.5
One candidate has really few chances to succeed (specifically if he/she has no skills to predict the market)! The situation is far different if you organize a massive recruitment process...
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Data snooping problem (2) 10 candidates : probability of success = 1 − (0.9713)10 = 0.2525 30 candidates : P = 58% 100 candidates : P = 94.5% 300 candidates : P = 99.98% 1000 candidates : P = 100% Remark You will ALWAYS find a rule that performs well in some specific time window (or succesful ”market forecaster”). NOT in the long run or in other time windows. Successful fund manager one year tend to be poorly performing the next years. Active management hardly (if ever) proved to be a good solution on the long term (market is efficient).
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Decades of academic research
Cowles (1930, 1944) ”Can Stock Market Forecasters Forecast ?” Analysis of 45 firms that tried to predict the market. Poor results, even worst that simple random draws. Inertia (passive management) is the best choice. Same conclusions in Working (1934), Kendall (1953), Samuelson (1965) ... and many others. Remark Some discordant voices (Brock, Lakonishok & Le Baron 1992), some intriguing results (hedge funds, Jim Simons), and the ”keep silent” argument.
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