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 otman.gordillo09@imperial.ac.uk

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