Lecture 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
a. Fundamentals of Probability
b. Fundamentals of Stats

7

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

Part 1

FUNDAMENTALS OF
PROBABILITY

a. Random Variables and cumulative density function
b. Moments
c. Distribution Functions

9

1. Random Variables
Definition – Univariate distribution functions
•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
Part 1. Fundamentals of probability

(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.
F ( x)  P( X  x)
Discrete F ( x)  x f (x j )
j x


Lecture 1

Continuous F ( x)  f (u )du
• Properties of CDF
•Monotonically, non-decreasing function
•Min value of 0 and maximum value of 1  f (u)du  1

1. Random Variables PDF CDF

Example
•Throw 2 dices
Lecture 1

11

1. Random Variables
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:


  [ x]   xf ( x)dx

Outcome * Probability
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:
[a]  a
[aX  b]  a[ X ]  b
Lecture 1

[ X  Y ]  [ X ]  [Y ]
[ X  Y ]  [ X ]  [Y ]  CovX , Y 
12

1. Random Variables
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:

 2  Var[ X ]  ( X   ) 2    ( x   ) 2 f ( x)dx


Properties:
• Describes the way the distribution is spread out . IT IS A MEASURE OF RISK

• For a, b constants: Var[a]  0
Var[aX  b]  a 2Var[ X ]
Var[ X  Y ]  Var[ X ]  Var[Y ]  2Cov( X , Y )

•Simplified expression of variance:
 2  Var[ X ]  ( X   )2   [ X 2 ]   2
Lecture 1

• Standard Deviation: Square root of Var(X)
SD( X )    Var[ X ]
13

1. Random Variables
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:
 X   3   ( X   )3  
Skew( X )        
 3   
     

    
It is a measure of the asymmetry of the probability density function

Left Right
Long tail Long tail
Lecture 1

14

1. Random Variables
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:
 X    4   ( X   ) 4  
Kurt ( X )        
 
 
     4
 

    
•It is a measure of fatness of the tails of the probability density function.
Lecture 1

15

1. Random Variables
Portfolios can have more than one variable (or risk factors)
Definition – Multivariate distribution functions
Let X and Y be random variables belonging to Ω. The joint (cumulative)
distribution function of (X,Y) is defined as:

FX ,Y ( x, y)  ( X  x, Y  y)...x, y  R

a b


More
FX ,Y (a, b)  ( X  a, Y  b)  f X ,Y ( x, y )dxdy complicated?
 
When RV are independent:
It is just the product of
FX ,Y (a, b)  Fx (a)  Fy (b) densities.
Covariance Two dices: P(1)+P(1)= 1/36* 1/36
Defined as the co-movement between variables
Cov[ x, y]  [( x  [ x])( y  [ y])]  [( x   x )( y   y )]
 
Cov[ x, y ]   ( x   x )( y   y )] f ( x, y )dxdy
Lecture 1

  

• Simplified expression: Cov(X,Y)=E[XY]-E[X]E[Y]
• Cov (X,X)= Var (X) 16

1. Random Variables
Correlation
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:
Cov( X , Y ) Cov( X , Y )
 X ,Y  Corr ( X , Y )  

Var ( X )Var (Y )  XY

Properties:
• -1 ≤ρ≤ 1
Lecture 1

How could be the ρ=-1? 17
17

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
Lecture 1

Distribution Functions
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
Lecture 1

5. Binomial
6. Poisson 19

Part 2

FUNDAMENTALS OF
STATISTICS
a. Returns and properties
b. Portfolio aggregation
c. Regression analysis

20

•Inference about population
•Relation among risk factors:
•i.g. relation between INDU and OIL
Part 2. Fundamentals of statistics
Lecture 1

1. Returns
Definition
• Main input: real data
S0,S1,S2,S3...St,St+1...

• Measuring Returns
St 1  St
rt 

St
rt  Rt Why?
Check the real-life
 St 1 
Rt  Ln example
 St  

•Including dividends or coupons. When horizon is really short the income
return is small.
St 1  D  St
rt 
St
Lecture 1

1. Returns
Properties
• 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 N (  ,  )
2
Lecture 1

1. Returns
Aggregation of returns, more than one period
Time aggregation
Given R0, R1 and R2
 S 2 S1  The return of T periods
R02  Ln    R12  R01 is the aggregate of each
 S1 S0  period

Moments
[ R0, 2 ]  [ R1, 2 ]  [ R0,1 ]
Given that events are independent, they have identical distributions across obs
[ R0, 2 ]  2[ R0,1 ]
[ RT ]  T[ R1 ]

Var[ R0, 2 ]  Var[ R1, 2 ]  Var[ R0,1 ] Events are RV
Var[ R0, 2 ]  2Var[ R0,1 ]
Lecture 1

Var[ RT ]  TVar[ R1 ]

SD[ RT ]  T  SD[ R1 ] Volatility increments
following sqrt T

1. Returns
Time aggregation (Cont)
Variance can be added up from different periods. In this case there is non-
zero correlation between periods.

Var[ R2 ]  Var[ R1 ]  Var[ R1 ]  2Var[ R1 ]
Var[ R2 ]  2Var[ R1 ]{1  }

 Autocorrelation coefficient
 > 0 Trend
 < 0 Mean reversion

Comparing. Suppose that T=2. Variance is different!

Var[ RT ]  TVar[ R1 ] Var[ R2 ]  2Var[ R1 ]{1  }
<
 > 0 Trend
Lecture 1

Var[ RT ]  TVar[ R1 ] Var[ R2 ]  2Var[ R1 ]{1  }
>
 < 0 Mean reversion

2. Portfolio aggregation
Aggregation of assets
One Period
N shares (assets)
S prices
q number of each asset
N

Value of portfolio Wt   qi Si ,t
1
N

Weight to each asset wi ,t 
qi Si ,t
Wt
w
1
i ,t 1

Period t+1 N

Value of portfolio Wt 1   qi Si ,t 1
1
N N
Dollar change Wt 1  Wt   qi Si ,t 1   qi Si ,t
Lecture 1

1 1

Wt 1  Wt   qi Si ,t 1  Si ,t 
N

1

2. Portfolio aggregation
Rate of return
Wt 1  Wt N  Si ,t 1  Si ,t 
  wi



We already know wi
Wt 1  S i ,t 

rp ,t 1   wi ri ,t 1 
N

Portfolio return is a
linear combination of
1 the return of assets
Lecture 1

3. Regression analysis
Main idea
• How one variable can affect other variable?
• How two variables are correlated?
• How the SP500 can affect the performance of AAPL?

yt    xt   t

Errors are independent
from Xt
Obs
Cov( X ,  )  0
Error
Est Errors are independent

[ ]  0

Errors follow
  N ( ,  2 )
Lecture 1

•short the income return is small.

OLS. Ordinary least squares
• Minimize the square of errors
min  ui  min  ( yi  yi ) 2
2
ˆ And F.O.C.

But ˆ ˆ ˆ
yi    xi
min  ( yi    xi ) 2

ˆ ˆ
Then the expression to
minimize is Q   ( yi    xi ) 2
ˆ ˆ

𝐹𝑖𝑛𝑑 𝛼
Q ˆ ˆ
y    x  0
 2 ( yi    xi )(1)  0
ˆ ˆ

ˆ
y  x  
ˆ
Q
  ( yi    xi )  0
ˆ ˆ

Lecture 1

Q
  yi       xi  0
ˆ ˆ


y i  n    xi  0
ˆ ˆ

𝐹𝑖𝑛𝑑 𝛽 Q   ( yi    xi ) 2
ˆ ˆ
Q

 2 ( yi    xi )( xi )  0
ˆ ˆ ˆ
  x y  x yn
i i

x x n
2 2
Q i
  ( yi    xi )( xi )  0

ˆ ˆ

ˆ
  ( x  x )( y  y )
i i
 xi yi    xi  ˆ  xi )  0
ˆ 2

 (x  x) i
2

ˆ
But y   x  
ˆ

( x i  x )( yi  y ) Cov( x, y )  ( x, y ) y
ˆ  
 ( xi  x ) 2 x
 xi yi    xi   xi ( y  x )
var( x)
ˆ 2 ˆ

 xi yi    xi  x n( y  x )
ˆ ˆ
2
Lecture 1

 xi yi    xi  x yn  x 2 n
ˆ ˆ
2

Now, the fit of the regression

Re sidualSumSquares   ( yi  yi )
2
ˆ
RSS   ( yi    xi )
ˆ ˆ
2

 ˆ
RSS  (( y  y   ( x  x )) 2
i i

RSS   ( yi  y ) 2   (
ˆ2
xi  x ) 2  2 ( yi  y )( xi  x )
ˆ

𝑆𝑦𝑦 + 𝛽 2 𝑆𝑥𝑥 − 2𝛽 𝑆𝑥𝑦
But   ˆ Sxy
Sxx
ˆ
RSS  Syy  Sxy
RSS  TotSumSq  ExplainedSumSq
ESS TSS  RSS
Lecture 1

RSS
r2    1
TSS TSS TSS
r=1: perfect fit. Errors will be 0. RSS=0
r=0: no fit

Multivariate
•More than one variable, so we use a matrix approach

 y1   x1,1 x1, 2 x1,3 ... x1, N   1   e1 
 y 2  x ...    2   e2 
   2,1    

 ...    x3,1 ...   ...    ... 
      
 ...   ... ... ... ... ...   ...   ... 
 yT   xT ,1
   xT , 2 xT ,3 ... xT , N    N  eT 
   

Y = X β + e

  ( X X ) 1 X y
Var ( )   2 ( X X ) 1
ˆ
Lecture 1

Lecture 1

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