Risk Analysis & Modelling

Lecture 2: Measuring Risk

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What we will look in this lecture

• Review of statistics

• The central limit theorem

• Implied statistical properties

• Covariance Matrices

• Correlation Matrices

• Calculation of mean and variance of a portfolio using matrices

Our Thought Experiment

• Imagine you have £100 an account• Infront of you is a machine with a red button on it• Every time you press the button the amount you

have in your account changes• The change seems to vary every time you press• We cannot see inside the machine, just observe the

outcome• We want to know if pressing the button is a good

idea

Machine

You have £100

Note: we will be making this machine in the programming section!

Quantifying The Range of Outcomes

• To asses the risk of a game we need to understand the outcomes that can occur and their respective likely-hood

• From this outcome range we can evaluate the range of payoffs that are likely to occur if we play the game and determine if it is to our liking.

We decide to tabulate the results we observe and their frequency

• After 100 presses we find:

Lose £4 10 10%

Lose £2 25 25%

Gain £1 35 35%

Gain £2 25 25%

Gain £4 5 5%

A probability histogram of outcomes

0%

5%

10%

15%

20%

25%

30%

35%

Lose £4 Lose £2 Gain £1 Gain £2 Gain £4

A cumulative probability histogram

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%100%

Lose £4 Lose £2 Gain £1 Gain £2 Gain £4

Cum

ulat

ive

Pro

babi

lity

Estimation Error

• Our histograms are estimations• They are subject to estimation error• Intuitively we can imagine that the larger our

sample the smaller the error.• The 100% accurate underlying probability

distribution is called the population distribution• Our estimation is called the sample distribution

Quantitative Measure

• Our graph and table help us asses the risk and return of playing the game but what if we want to compare risks of various machines?

• We would like to have a parametric measure based on our personal preferences

• We decide we are interested in 2 things:• The centre of the outcomes (what is most likely to happen)• The spread of outcomes about this centre (the uncertainty

of the expected outcome)• The centre is can be defined as the mean, the spread can be

defined as the variance.

Sample Mean & Variance

• The sample mean is defined as

n

1iiX

n

1 X

• The sample variance is defined as

2n

1ii

2 )X(X1-n

1 s

Mean And Variance

0%

5%

10%

15%

20%

25%

30%

35%

Lose £4 Lose £2 Gain £1 Gain £2 Gain £4

We Expect to Gain £1

There are a range of outcomes other than the expectedWe are there uncertain about the exact outcome

Random Variable Operations

• A variable with a ~ on top denotes a random variable

• You cannot treat them like a normal variable, for example:

X~

2. X~

X~

• There is however a special operation you can perform on a random variable called the expectation

• Expectation (E) is a purely abstract concept that states “what would be the expected value if we had knowledge of the population distribution”:

XP . )X~

f( ) )X~

f( E(

• Where PX is the true probability of observation X

Proof of Unbiased Estimation of Sample Mean

• We have said that:

n

1iiX

~

n

1 X

n

1iiX

~

n

1E )XE(

)X~

E(n

1 )XE(

n

1ii

n

1in

1 )XE(

Where is the population mean

n

n. )XE(

So the expected value of our sample mean is the population mean.

Proof of Unbiased Estimation of Sample Variance

2n

1ii

2 )XX~

(1-n

1 s

2n

1ii

2 ))-X()X~

((1-n

1 s

))-X()X~

((1-n

1 )E(s 2

n

1ii

2

E

))-X()-X).(X~

.(2)X~

((1-n

1 )E(s 2

i2

n

1ii

2

E

))-X()X~

(E(1-n

1 )E(s 22

n

1ii

2 E

n

1i

222

1-n

1 )E(s

n

2i )-X( ))-X).(X

~(E( E

222

1-n

1-n )E(s

where

Where is the population variance:

nE

22 ))-X( (

hence

hence

Discrete Vs Continuous Probability

• The game we played in the last section was an example of a discrete random variable

• The number of outcomes from the game was ‘finite’ or of limited number

• If the game had payoffs like: “You Win £2.13312” or “You Lose £4.5633” we could not use our table and histogram

• The outcomes would represent a continuous random variable.

• For a continuous random variable it does not make sense to talk about a specific outcome only a range of outcomes

Cumulative Distribution And Probability Density

• The Cumulative Distribution Function (cdf) gives the probability that an outcome will be less than or equal to a given value

• The Probability Density Function (pdf) describes a function which has the property that the area under it describes the cdf.

Continuous CDF and PDF graphs

+£10

Probability

Outcome-£10 +£10

Outcome-£10

ProbabilityDensity

1.0

0.0

0.4

0.0

-£3-£3

CDF PDF

Area Under Curve

0.3

£1

0.6

£1

Difference is Probability of outcomes between –£3 and £1

Special Distributions• Up until now probability distributions have been of arbitrary

shapes describing the likelihood of a range of outcomes• There are a number of special distributions that frequently

occur in the real world and that are described by well defined functions

• The most important of these is the Normal Distribution or Bell Curve

• The Normal Distribution is observed throughout nature and finance and described by mean and variance

• We can explain why normal distribution occurs using the Central Limit Theorem

The Central Limit Theorem

• The Central Limit Theorem is a precise formulation of the “Law of Large Numbers”

• Imagine we have a variable Y which equal to the average of the observed values for three independent random variable A,B and C:

3

C~

B~

A~

Y~

• The expected value of Y is:

33

)C~

E()B~

E()A~

E()Y

~E( CBA

Y

• The Variance of Y:

9

)- C~

(E)-B~

(E) -A~

E(

3

-- - C~

B~

A~

)- Y~

E(222

2

2 CBACBAY E

0)-C~

).(-B~

(E,0)-C~

).( -A~

E(,0)-B~

).( -A~

E( BBBABA

Since A,B,C are independent the covariances are zero

• The Skew of Y

27

)- C~

(E)-B~

(E) -A~

E(

3

-- - C~

B~

A~

)- Y~

E(333

3

3 CBACBAY E

Covariance

• While looking at the central limit we introduced the concept of covariance which measures the way 2 random variable vary together

• The unbiased sample covariance is

n

ixy n 1

ii )Y).(YXX(1

1s

))μY~

).(μX~

((E YX xy

• Covariance is a product moment

Correlation• Correlation is a normalized measure of covariance• Correlation must be between –1 and +1 due to the

Cauchy-Schwarz inequality.• A strong positive correlation suggests that 2

random variables move about their mean value in unison

• A strong negative correlation suggests that 2 random variable move in opposite directions about their mean value

• Correlation is defined as:

yx

xyxy s.s

sp

Part 2: Portfolio Mean-Variance

Portfolio Risk/Return

• Imagine you have a portfolio of assets.• You have estimates for the means, variances and

covariances of the returns for the various assets• You wish to calculate the mean and variance of

the return for your portfolio• We wish to derive the mean and variance of a

portfolio’s return from the mean, variance and covariance of returns of the assets it contains.

2 Asset Portfolio

• The return on the portfolio (P) is a weighted average of the return on asset A and asset B

B~

.wA~

.wP~

BA

• Where wA is the proportion invested in asset A and wB is the proportion invested in asset B. Investment proportions are often called ‘weights’.

• The weights normally add up to 100%.• For any value for the return in A and B we can

evaluate the return on the portfolio.

• We can use our expectation operator to show that the expected return on the portfolio is a simple weighted average of the return on the assets

)B~

(E.w)A~

(.Ew)B~

.wA~

.w(E)P~

E( BABA

• The variance of the portfolio has a more complex relationship with the assets A and B:

2BBBAAA

2P )μ.w B

~.w.μw-A

~.w(E)μ-P

~E(

BBAAP μ.w.μwμ

)μ.w B~

.(w)..μw-A~

.w(E.2)μ.w B~

.E(w).μw-A~

.w(E BBBAAA2

BBB2

AAA2

P

ABBA2

B2

B2

A2

A2

P ..ww.w.w

3 Asset Portfolio

• We will now examine the case of a 3 Asset Portfolio

C~

.wB~

.wA~

.wP~

CBA

• The expected portfolio return:

CCBBAAP μ.wμ.w.μwμ

• The portfolio variance:

BCCBACCAABBA2

C2

C2

B2

B2

A2

A2

P ww2ww2ww2www

Larger Portfolios

• As we see the equation relating the portfolio variance to the covariance and variance of its assets is messy for 3 assets

• For A modest portfolio of 30 assets it would contain 450 terms!

• We need to find a more practical way of calculating the portfolio’s statistical properties

• To do this we need to introduce 3 new concepts: weight vector, expected return vector and covariance matrix.

The Weight Vector

• The weight vector is a n by 1 vector containing the proportions invested in the various assets

• For the 3 asset case the weight vector would look like the following:

WA

WB

WC

A

B

C

The Expected Return Vector• The Expected Return Vector is a n by 1 vector

containing the expected returns of the various assets.

• It is important for the Expected Return Vector to maintain the same order as the weight vector.

• For the 3 asset case the Expected Return Vector would look like the following:

E(RA)

E(RB)

E(RC)

A

B

C

The Covariance Matrix

• The covariance matrix is a square matrix which describes the covariances and variances between a set of random variables

• It is a symmetric matrix• It is a positive definite matrix (ie XTCX >0

for every non-zero column vector)• The covariance matrix should maintain the

same order as the weight matrix.

Var(RA) Cov(RA,RB)

Cov(RB,RA) Var(RB)

Var(RA) Cov(RA,RB) Cov(RA,RC)

Cov(RB,RA) Var(RB) Cov(RB,RC)

Cov(RC,RA) Cov(RC,RB) Var(RC)

Covariance Matrix for 2 Assets

Covariance Matrix for 3 Assets

IdenticalTerms

A

A B

B

A

A

B

B

C

C

Expressing Risk & Return with Matrices

• Let C be the covariance matrix, E the expected return vector and W the weight vector for a Portfolio P

• E(RP) = WT.E

• Var(RP) = WT.C.W

2 Asset Portfolio Return With Matrices

E(RP) = WA WB

E(RA)

E(RB)X

E(RP) = WA.E(RA) + WB.E(RB)

E(RP) = WT.E

2 Asset Portfolio Variance With Matrices

Var(RP) = WT.C.W

Var(RP) = WA WB

V(RA) C(RA,RB)

C(RB,RA) V(RB)

WA

WB

X X

Var(RP) = Wa2.Var(Ra) + Wb

2.Var(Rb) +

2.Wa.Wb.Cov(Ra,Rb)

The Correlation Matrix

• The correlation matrix is closely related to the covariance matrix

• It measures the correlation between assets rather than covariances

• Correlation matrices can be converted to covariance matrices using the standard deviation vector

Transformation Between Correlation and Covariance Matrix

• Let D be a square matrix with the standard deviations along the diagonal and zeros everywhere else (a diagonal matrix), let P be the correlation matrix and C be the respective covariance matrix. Then the following relationship is true:

D.P.D = C

2 Asset Correlation to Covariance Matrix Example

1 P(RA,RB)

P(RB,RA) 1

Sd(RA) 0

0 Sd(RB)

Sd(RA) 0

0 Sd(RB)* *

=Sd(RA)*Sd(RA) P(RA,RB)* Sd(RA)*Sd(RB)

P(RB,RA) *Sd(RA)*Sd(RB) Sd(RB)*Sd(RB)

V(RA) C(RA,RB)

C(RB,RA) V(RB)=

Transformation between Covariance and Correlation Matrix

• From our initial relationship we can state:

P= D-1.C .D-1

• Where D-1 is the inverse of the square standard deviation matrix

• Because D is a diagonal matrix its inverse is simply the reciprocal of the elements along the diagonal.

2 Asset Covariance to Correlation Matrix Example

1/ Sd(RA) 0

0 1/ Sd(RB)

1/Sd(RA) 0

0 1/ Sd(RB)* *

=V(RA) / (Sd(RA)*Sd(RA)) C(RA,RB)/(Sd(RA)*Sd(RB))

C(RB,RA)/(Sd(RB)*Sd(RA)) V(RB)/(Sd(RB)*Sd(RB))

=

V(RA) C(RA,RB)

C(RB,RA) V(RB)

1 P(RA,RB)

P(RB,RA) 1