Date post: | 03-Dec-2014 |
Category: |
Documents |
Upload: | otmanjavier |
View: | 326 times |
Download: | 4 times |
RISK MANAGEMENT
1
RISK MANAGEMENT
In the last three decades, the world has seen some dramatic economic
failures that were caused not just by wrong financial decisions but by
lack of preparation when there are unexpected changes in market, the
industry or in the company itself.
• LTCM, Bearn Stearns, Lehman Brothers or even Fukushima Daiichi are
perfect examples that sometimes companies are not well prepared to
manage risky events. Those companies were incapable of survive when
faced particular events that at that particular time seemed improbable.
2
RISK MANAGEMENT
Risk management the process designed to reduce or eliminate the
risk created by certain kinds of “improbable” events happening
or having an impact on the business.
Many business risk management plans may focus on keeping the company
viable and reducing financial risks. However, risk management is also
designed to protect employees, customers, and general public
from negative events.
This course addresses the field by three different spheres:
1. Bringing the theoretical tools to understand the foundations of
risk management
2. Illustrating with real-life cases particular situations
3. Applying concepts to workshops
3
COURSE MATERIAL
Slides.
Jorion, P., (2009) Financial Risk Manager Handbook, John Wiley & Sons.
ISBN-10: 0470479612, ISBN-13: 978-0470479612
Each lecture has different materials
CONTENT 1. Quantitative Analysis (bond fundamentals, fundamentals of probability, fundamentals
of statistics)
2. Capital Markets (derivatives, options, fixed-income securities, fixed-income
derivatives, equity currency and commodity markets)
3. Market Risk Management (introduction to market risk, sources of market risk,
hedging linear risk, nonlinear risk options, modelling risk factors, VAR methods)
4. Investment Risk Management (portfolio management, hedge fund risk
management)
5. Credit Risk Management (measuring actuarial default risk, measuring default risk
from market prices, credit exposure, credit derivatives and structured products,
managing credit risk)
6. Legal, Operational and Integrated Risk Management (operational risk, liquidity risk,
firm-wide risk management, legal issues)
7. Regulation and compliance (regulation of financial institutions, the Basel Accord,
the Basel market risk charge)
COMMENTS OTMAN GORDILLO [email protected]
RISK MANAGEMENT LECTURE 1
7
a. Fundamentals of Probability
b. Fundamentals of Stats
LECTURE MATERIAL
Jorion, P., (2009) Financial Risk Manager Handbook, John Wiley & Sons.
ISBN-10: 0470479612, ISBN-13: 978-0470479612
• Introduction to Econometrics. G. S. Maddala, Kajal Lahiri
• An Introduction to the Mathematics of Financial Derivatives.
Ali Hirsa, Salih N. Neftci
• Stochastic Calculus and Financial Applications (Stochastic
Modelling and Applied Probability) J. Michael Steele
FUNDAMENTALS OF PROBABILITY
Part 1
9
a. Random Variables and cumulative density function
b. Moments
c. Distribution Functions
Le
ctu
re 1
Pa
rt 1
. F
un
da
me
nta
ls o
f p
rob
ab
ilit
y
1. Random Variables
•Prices are considered R.V.
• Also known as Stochastic Variable is a variable whose value is subject to
random movements (stocks are examples of this)
•Efficient Market Hyp
• The a SV does not have a fixed value. It can take a set of possible values
(sample space Ω,) each associated to a specific probability
• Can be discrete (specific values) or continuous (any value)
Coin toss H,T x € [0,1]
Die 1,2,3,4,5,6 Price of a stock € [0,+∞]
• Cumulative distribution function (CDF): Describe R.V.
probability that an outcome will be less than, or equal to a specific value.
Discrete
Continuous
• Properties of CDF
•Monotonically, non-decreasing function
•Min value of 0 and maximum value of 1
)()( xXPxF
xjx jxfxF )()(
duufxF )()(
1)( duuf
Definition – Univariate distribution functions
11
1. Random Variables Example
Le
ctu
re 1
Pa
rt 1
. F
un
da
me
nta
ls o
f p
rob
ab
ilit
y
PDF CDF
•Throw 2 dices
12
Moments WHY? Because they describe the risk factors
Each factor is a R.V.
Let X be a continuous random variable with density f(x). The expected
value of X (first moment) is denoted by E[X] and is defined by:
Properties:
• It is the “centre of gravity” of the distribution
• When we talk about the Value at Risk (VaR), we assume that more of the
daily looses are “near” µ
• For a, b constants:
dxxxfx )(][
aa ][
bXabaX ][][
][][][ YXYX
Le
ctu
re 1
Pa
rt 1
. F
un
da
me
nta
ls o
f p
rob
ab
ilit
y
Outcome * Probability
1. Random Variables
YXCovYXYX ,][][][
13 13
Moments Let X be a continuous random variable with density f(x). The variance of X
(second moment) is denoted by Var[X] and is defined by:
Properties:
• Describes the way the distribution is spread out . IT IS A MEASURE OF RISK
• For a, b constants:
•Simplified expression of variance:
• Standard Deviation: Square root of Var(X)
dxxfxXXVar )()()(][ 222
0][ aVar
][][ 2 XVarabaXVar
2222 ][)(][ XXXVar
),(2][][][ YXCovYVarXVarYXVar
Le
ctu
re 1
Pa
rt 1
. F
un
da
me
nta
ls o
f p
rob
ab
ilit
y
][)( XVarXSD
1. Random Variables
14 14 14
Moments Let X be a continuous random variable with density f(x). The skewness of X
(third moment) is denoted by Skew[X] and is defined by:
It is a measure of the asymmetry of the probability density function
3
33)(
)(
XXXSkew
Le
ctu
re 1
Pa
rt 1
. F
un
da
me
nta
ls o
f p
rob
ab
ilit
y
Long tail Long tail
Left Right
1. Random Variables
15 15
Moments Let X be a continuous random variable with density f(x). The kurtosis of X
(fourth moment) is denoted by Kurt[X] and is defined by:
•It is a measure of fatness of the tails of the probability density function.
4
44)(
)(
XXXKurt
Le
ctu
re 1
Pa
rt 1
. F
un
da
me
nta
ls o
f p
rob
ab
ilit
y
1. Random Variables
16 16
Let X and Y be random variables belonging to Ω. The joint (cumulative)
distribution function of (X,Y) is defined as:
When RV are independent:
Definition – Multivariate distribution functions
RyxyYxXyxF YX ,)...,(),(,
a b
YXYX dxdyyxfbYaXbaF ),(),(),( ,,
Covariance Defined as the co-movement between variables
• Simplified expression: Cov(X,Y)=E[XY]-E[X]E[Y]
• Cov (X,X)= Var (X)
)])([(])][])([[(],[ yx yxyyxxyxCov
dxdyyxfyxyxCov yx ),()])((],[
Le
ctu
re 1
Pa
rt 1
. F
un
da
me
nta
ls o
f p
rob
ab
ilit
y
Portfolios can have more than one variable (or risk factors)
)()(),(, bFaFbaF yxYX It is just the product of
densities.
Two dices: P(1)+P(1)= 1/36* 1/36
1. Random Variables
More
complicated?
17 17 17
To find out whether the strength of covariance, we need a normalization by
the standard deviations of X and Y. The normalized covariance is called
correlation between X and Y:
Properties:
• -1 ≤ρ≤ 1
How could be the ρ=-1?
Correlation
YX
YX
YXCov
YVarXVar
YXCovYXCorr
),(
)()(
),(),(,
Le
ctu
re 1
Pa
rt 1
. F
un
da
me
nta
ls o
f p
rob
ab
ilit
y
1. Random Variables
2. Distribution Functions 1. Uniform :
• Same weight on each observation
2. Normal:
• The most important
• Bell like shape
• Symmetrical around mean. Skew around 0
• Mean=mode
• Kurtosis is around 3
Le
ctu
re 1
Pa
rt 1
. F
un
da
me
nta
ls o
f p
rob
ab
ilit
y
19
3. Log-Normal
• Calculate Ln(X). If it is normally distributed, X has a lognormal
distribution
4. T-Student
• Fatter tails than the normal distribution
Distribution Functions L
ec
ture
1
Pa
rt 1
. F
un
da
me
nta
ls o
f p
rob
ab
ilit
y
5. Binomial
6. Poisson
FUNDAMENTALS OF STATISTICS
Part 2
20
a. Returns and properties
b. Portfolio aggregation
c. Regression analysis
Le
ctu
re 1
Pa
rt 2
. F
un
da
me
nta
ls o
f s
tati
sti
cs
•Inference about population
•Relation among risk factors:
•i.g. relation between INDU and OIL
Le
ctu
re 1
Pa
rt 2
. F
un
da
me
nta
ls o
f s
tati
sti
cs
1. Returns
• Main input: real data
S0,S1,S2,S3...St,St+1...
• Measuring Returns
•Including dividends or coupons. When horizon is really short the income
return is small.
Definition
t
ttt
S
SSr
1
t
tt
S
SLnR 1
tt Rr
t
ttt
S
SDSr
1
Why?
Check the real-life
example
Le
ctu
re 1
Pa
rt 2
. F
un
da
me
nta
ls o
f s
tati
sti
cs
1. Returns
• Efficient Market Hypothesis
* Prices reflect all available information. No history affect the present
* No arbitrage possible
* Prices are always fair
Prices are moving randomly Unpredictable
Prices and returns are independent
They are stochastic variables
We can use
Properties
),( 2N
Le
ctu
re 1
Pa
rt 2
. F
un
da
me
nta
ls o
f s
tati
sti
cs
1. Returns
Given R0, R1 and R2
Time aggregation
0112
0
1
1
202 RR
S
S
S
SLnR
The return of T periods
is the aggregate of each
period
][][][ 1,02,12,0 RRR
][][][ 1,02,12,0 RVarRVarRVar Events are RV
Given that events are independent, they have identical distributions across obs
][2][ 1,02,0 RR
][][ 1RTRT
][2][ 1,02,0 RVarRVar
][][ 1RTVarRVar T
][][ 1RSDTRSD T
Aggregation of returns, more than one period
Moments
Volatility increments
following sqrt T
Le
ctu
re 1
Pa
rt 2
. F
un
da
me
nta
ls o
f s
tati
sti
cs
1. Returns
Variance can be added up from different periods. In this case there is non-
zero correlation between periods.
Comparing. Suppose that T=2. Variance is different!
Time aggregation (Cont)
][2][][][ 1112 RVarRVarRVarRVar
1][2][ 12 RVarRVar
Autocorrelation coefficient
> 0 Trend
< 0 Mean reversion
][][ 1RTVarRVar T 1][2][ 12 RVarRVar
> 0 Trend <
][][ 1RTVarRVar T 1][2][ 12 RVarRVar
< 0 Mean reversion >
Le
ctu
re 1
Pa
rt 2
. F
un
da
me
nta
ls o
f s
tati
sti
cs
2. Portfolio aggregation Aggregation of assets
N shares (assets)
S prices
q number of each asset
Value of portfolio
Weight to each asset
Value of portfolio
One Period
N
tiit SqW1
,
t
tii
tiW
Sqw
,
, 1
1
, N
tiw
Period t+1
N
tiit SqW1
1,1
Dollar change
N
tii
N
tiitt SqSqWW1
,
1
1,1
N
titiitt SSqWW1
,1,1
Le
ctu
re 1
Pa
rt 2
. F
un
da
me
nta
ls o
f s
tati
sti
cs
2. Portfolio aggregation
Rate of return
N
ti
titi
t
tt
S
SSwi
W
WW
1 ,
,1,1 We already know wi
N
titp rwir1
1,1,
Portfolio return is a
linear combination of
the return of assets
Le
ctu
re 1
Pa
rt 2
. F
un
da
me
nta
ls o
f s
tati
sti
cs
3. Regression analysis
• How one variable can affect other variable?
• How two variables are correlated?
• How the SP500 can affect the performance of AAPL?
•short the income return is small.
Main idea
ttt xy
Errors are independent
from Xt
Errors are independent
Errors follow
0),( XCov
0][
),( 2 N
Obs
Est
Error
Le
ctu
re 1
Pa
rt 2
. F
un
da
me
nta
ls o
f s
tati
sti
cs
3. Regression analysis
• Minimize the square of errors
OLS. Ordinary least squares
22)ˆ(minmin iii yyu
xy ˆˆ x
y
i
ii yx
x
yxCov
xx
yyxx
),(
)var(
),(
)(
))((ˆ
2
2)ˆ(Re ii yyquaressidualSumS
umSqExplainedSTotSumSqSxySyyRSS
TSS
ESSr 2
r=1: perfect fit. Errors will be 0. RSS=0
r=0: no fit
And F.O.C.
But ii xy ˆˆˆ
2)ˆˆ(min ii xy
2)ˆˆ( ii xyQ Then the expression to
minimize is
𝐹𝑖𝑛𝑑 𝛼
0)1)(ˆˆ(2 ii xy
Q
0)ˆˆ( ii xy
Q
0ˆˆ
ii xyQ
0ˆˆii xny
0ˆˆ xy
ˆˆ xy
Le
ctu
re 1
Pa
rt 2
. F
un
da
me
nta
ls o
f s
tati
sti
cs
3. Regression analysis OLS. Ordinary least squares
x
y
i
ii yx
x
yxCov
xx
yyxx
),(
)var(
),(
)(
))((ˆ
2
2)ˆˆ( ii xyQ 𝐹𝑖𝑛𝑑 𝛽
0))(ˆˆ(2 iii xxy
Q
ˆˆ xy
0))(ˆˆ( iii xxy
Q
0)ˆˆ2
iiii xxyx
But
)ˆ(ˆ 2xyxxyx iiii
)ˆ(ˆ 2xynxxyx iii
nxnyxxyx iii
22 ˆˆ
nxx
nyxyx
i
ii
22
2)(
))((ˆ
xx
yyxx
i
ii
Le
ctu
re 1
Pa
rt 2
. F
un
da
me
nta
ls o
f s
tati
sti
cs
3. Regression analysis OLS. Ordinary least squares
2)ˆ(Re ii yyquaressidualSumS
umSqExplainedSTotSumSqRSS
TSS
RSS
TSS
RSSTSS
TSS
ESSr
12
r=1: perfect fit. Errors will be 0. RSS=0
r=0: no fit
Now, the fit of the regression
2
)ˆˆ( ii xyRSS
2))(ˆ(( xxyyRSS ii
))((ˆ2)(ˆ)( 222 xxyyxxyyRSS iiii
𝑆𝑦𝑦 + 𝛽 2𝑆𝑥𝑥 − 2𝛽 𝑆𝑥𝑦
SxxSxy
But
SxySyyRSS
Le
ctu
re 1
Pa
rt 2
. F
un
da
me
nta
ls o
f s
tati
sti
cs
3. Regression analysis
•More than one variable, so we use a matrix approach
Multivariate
Y = X β + e
TNNTTTT
N
T e
e
e
xxxx
x
x
xxxx
y
y
y
...
...
...
...
...
...............
...
...
...
...
...
2 2
1
2
1
,3,2,1,
1,3
1,2
,13,12,11,11
yXXX 1)(12 )()ˆ( XXVar