Lecture 5: Value At Risk

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What We Will Learn In This Lecture

• We will look at the idea of a stochastic process

• We will look at how our ideas of mean and variance of proportional change relate to the concept of a stochastic process

• We will look at the core concepts of Value At Risk and how they relate to the principles of stochastic processes

Random Variables In Sequence

• So far we have been thinking of random variables as ‘singular’ events.

• We have viewed random variables as single events that occur in isolation and not as part of an accumulative process across time.

• Movements in stock market prices or insurance company claims are not unique events but are a random processes that accumulate on a daily or hourly basis.

• We need to deal with random variables that behave as a sequences.

Our Thought Experiment

• We have a coin that we flip• If it is heads we win £1 and if it is tails we lose £1• If we play this game once we can simply describe the outcomes

in terms the 2 possible outcomes• We could even describe the risks interms of the mean and

variance of the outcomes.• But what if we want to discuss the risk for people who play the

game 10 times, 20 times, 1000 times?• The amount the person stands to loose is obviously the

accumulation of a series of individual random events (coin flips)

• We will call this accumulated sequence a random processes

Graph Of A Possible Game

£0

Total Winnings

Total Losses

NumberOf

GamesPlayed

-£10

+£10

The Expected Payoff And Variance Of Payoff

• Let us say we play with games 100’s of times for a sequence of 5 flips of the coin and from this sample calculate the mean and variance of Payoff for games involving 5 flips

• We will find that on average our payoff is zero and the standard deviation of payoff is 2.23

• If we were to look at the mean and standard deviation of outcomes for various numbers of games played we would find the following:

Games Played Expected Payoff Std Dev Payoff

2 0 1.414

3 0 1.731

4 0 2

5 0 2.236

Standard Deviation Of Payoff

0

0.5

1

1.5

2

2.5

3

3.5

4

1 2 3 4 5 6 7 8 9 10 11 12 13

Number Games Played

Sta

nd

ard

Dev

iati

on

Pay

off

As the number of games played increase the standard deviation increases, but it increases at a decreasing rate.

Intuitive Explanation

• Our simple stochastic process made up of flipping a coin has produced an interesting result.

• As the number of steps in the stochastic process increases the standard deviation of the process increases but increases at a decreasing rate.

• By drawing out a tree of the outcomes we can see the as the number of steps increases the range of outcomes increases but at the same time the number of paths leading to the “centre” increases.

• These two offsetting results lead to the increase of variance at a decreasing rate.

• This is very similar to the result we observed for diversification in a portfolio, you can think of it as diversification across time!

Density of Outcomes1

1 1

21 1

1 3 3 1

1 4 6 4 1

HT

T

T

T

T

T

T

T T T

H

H

H

H

H H

H H

Note: Also known as Pascal’s Triangle

Steps

H

A Markov Chain• Our simple stochastic process involving flipping a coin

and adding the payoffs is an example of a Markov Chain• A Markov Chain is where the probability of the various

future outcomes is only dependent upon the current position of the sequence and not the path that led to that position

• In general path dependant processes need to be analysed using Monte Carlo simulations

• An example of a path dependant process is the modelling of an insurance company cash flows since bankruptcy introduces a ‘barrier’

What We Observe• We normally observe the process and from that need to

derive the stochastic behaviour of the change• In the previous case would could derive a probability

distribution for the payoffs of a single flip by differencing the process of Total Winnings and drawing an associated histogram

• If we were to do this we would see that we have a 50% chance of winning £1 and 50% chance of loosing £1

• If the process was not a Markov chain we could not do this! Why?

Stock Price Stochastic Process

• The stochastic process we will use to model stock prices (and other assets/liabilities) is based on Brownian Motion or Wiener Process

• A Wiener Process is a stationary Markov Chain• The proportional change is at each step is a random

number sampled from a normal distribution• These proportional changes ‘compound’ over time to

produce the movements in stock prices• This ‘compounding’ is different from the

accumulation we saw in the coin flipping example

Stochastic Process

Time

Price

?

At each step the proportional change in the stock price is a random variable from a normal distribution

Distribution For Tomorrows Price

• The stock price today is P0 and we know that the daily returns are taken from a normal distribution with mean and standard deviation then we can say that the price tomorrow P1 is:

)~1.(~

001 rPP

• Where r0 the random variable representing daily returns

• We can see that the distribution for P1 is also normal

• We can use the normal distribution to describe the various outcomes for P1

• Note that this is a Markov process, why?

Distribution Of Future Prices

• Let us extend this out to the probability distribution for the price the day after tomorrow:

)~1.(~~

112 rPP

)~~~.~1.()~1).(~1.(~

101001002 rrrrPrrPP

• P2 is not normally distributed! It will be a Chi-Squared distribution because of the product of r0 and r1

We Need A Different Definition of Returns!

• The standard definition of returns makes the probability distribution of prices beyond one step in the future complex

• One solution would be to ignore the compounding effect of returns which would get arid of the nasty cross product term:

)~~1.()~1).(~1.(~

10101002 rrPrrPP

• This would mean that P2 would be normally distributed but will lead to other problems…

Continuously Compounded Returns

• Instead of defining returns like this:

0

010 P

PPr

100 ).1( PPr

• We will see that the continuously compounded definition is better:

0

10 ln

P

Pr 0.01

rePP

Where Do Continuously Compounded Returns Come From?• Imagine you have £100 in your bank and you earn a 10%

annual interest on that amount, at the end of the year you will have 110 in you account: 100 *(1+0.1)

• Let us say your bank now pays interest semi-annually, what rate would they have to pay you to give you the same £110 at the end of the year?

2

21.100110

r

%75.91100

110.2 2

r

• Notice that it is slightly smaller, why is that?

What Happens As We Compound Over Very Short Periods?

• In general we can define the compounding rate as:

n

n

rPP

1.01

• As n approaches infinity the value converges to a non-infinite value:

rn

en

r

1

• Where e is a special number like and is equal to 2.718282..

General Equations

• The relationship between P1 and P0 for a given continuously compounded return r is:

rePP .01

• And by taking natural logs of both side we can see that we can calculate the continuously compounded return as

0

1lnP

Pr

Why Continuously Compounded Returns Are Good

• Let us say we know that continuously compounded returns are described by a normal distribution

• The relationship between the price today P0 and the price tomorrow P1, where r0 is today’s random proportional change

0~

01 . rePP

• P1 is log normally distributed

• Now the relationship between P0 and P2

1~

12 . rePP

1010~~

0

~~

02 ... rrrr ePeePP

• Now the relationship between P0 and P3

210210~~~

0

~~~

02 .... rrrrrr ePeeePP

• The relationship between P0 and PT

RT ePP

~

0.

1

0

~~ T

iirRwhere

• Because e is a special type of function with a unique one-to-one mapping between the domain and range we can map the probability of observing a given P directly to the probability of observing a given R

Random Returns(domain)

Random Prices(map)

There is a unique one-to-one mapping between a given random return and a given random price, therefore we say that the probability of observing a random price is determined by the probability of observing the random return it relates to!

The Behaviour Of Continuously Compounded Returns Across Time

• We have noted that continuously compounded returns over say a T day period time is simply equal to the sum of the individual random returns observed on each of those T days

• Also we can say that if prices are a Markov Chain then each of those return is sampled from the same distribution

• So we could say:

1210~...~~~~

TrrrrR

.)~(....)~()~()~()~

( 1210 TrErErErERE T 2

1210 .)~(...)~()~()~()~

( TrVarrVarrVarrVarRVar T

.)~

( TRDevStd

• R will be normally distributed since it is the sum of T normally distributed normal variables

• The mean of R’s distribution will be T. and the standard deviation T1/2.

T.

T1/2.

Probability Distribution of R

T. +1.96.T1/2.T. -1.96.T1/2.

Lower 2.5% tail Upper 2.5% tail

Lognormal Probability Distribution of P(T)

P0.e*T

TTeP **96.1*0. TTeP **96.1*

0.

Lower 2.5% tailUpper 2.5% tail

An example• Imagine the price today is 100 and we know that the daily continuously

compounded return follow a normal distribution with a mean of 0.3% and standard deviation of 0.1%

• Calculate the expected value of return in two days, the return which will only expect to see values greater than 2.5% of the time and the expected return we only expect to see values less than 2.5% of the time

006.0003.0*2)( RE

0087.096.1*001.0*2003.0*2 UpperR

0032.096.1*001.0*2003.0*2 LowerR

• Translating these to values to the levels for prices:

6.100*100)( 006.0 ePE

87.100*100 0087.0 eUpperP

32.100*100 0032.0 eLowerP

Price Diffusion Boundaries

Time

Price

100

ExpectedPath

Lower Probabilistic Boundary

Upper Probabilistic Boundary

Value At Risk• Value-At-Risk can be defined as “An estimate, with a given

degree of confidence, of how much one can lose from one’s portfolio over a given time horizon”.

• It is very useful because it tells us exactly what we are interested in: what we could loose on a bad day

• Our previous ideas of mean and variance of return on a portfolio were abstract

• VaR gives us a very concrete definition of risk, such as, we can say with 99% certainty we will not loose more than X on a given day

• “Value at Risk” is literally the value we stand to lose or the value at risk!

The Value Of Risk On A Portfolio

• We are normally interested in describing the value at risk on a portfolio of assets and liabilities

• We know how to describe mean and variance of return on our portfolio interms of the mean, variance and covariance of returns on the assets and liabilities it contains

• We will now use this to describe the stochastic process of the portfolio’s value across time

• From this stochastic process of the portfolio’s value we will estimate the Value At Risk for a given time horizon

Our Method• We can derive the continuously compounded mean and variance of a

portfolio’s continuously compounded return for a portfolio from the expected return and covariance matrix of continually compounded returns for the assets it contains

• Under the assumption that the proportional changes in the portfolio’s value are normally distributed we can translate the mean and variance of these proportional changes to the diffusion of the portfolios value across time

• Using the diffusion process we can put a probabilistic lower bound of the portfolios value across time:

• So for example if we wanted to calculate the value of the portfolio we would only be bellow 2.5% of the time we would use the formula:

)1.(.)( **96.1*0

**96.1*00

TTTT ePePPTVaR

• Where is the mean of returns on the portfolio and is the standard deviation of returns on the portfolio

Portfolio Value Diffusion

PV0

Expected Path For Portfolio

Portfolio Value WillOnly Go Bellow this2.5% of the time

Time

Por

tfol

io V

alue

T

Value At Risk At Time T

Other Confidence Intervals• The number -1.96 is the number of standard deviations

bellow the mean we must go to be sure that only 2.5% of the observation that can be sampled from that normal distribution will be bellow that level

• Sometimes we might want to be even more confident such that say only 1% of the possible outcomes is bellow our value (-2.32 standard deviations bellow the mean)

• We can use the Excel Function NORMSINV to calculate the number of standard deviations bellow the mean we must go for a given level of confidence.

• For Example NORMSINV(0.01) = -2.32.

Zero Drift VaR

• One thing to notice is that the drift in the portfolio value introduced by a positive expected return can mean the Value at Risk is negative (ie we don’t expect to lose money even in the worse case scenario)!

• Sometimes VaR is calculated under the assumption that expected returns on the portfolio are zero:

)1.()( **96.10

TePTVaR

• This is used as an estimate of VaR over short time periods such as days, or where we are uncertain of our estimates of expected return.

Diversified & Undiversified VaR

• Diversified VaR relates to the situation where we use estimates of the covariances of the portfolio’s assets to reflect their actual value

• Undiversified VaR is where we restrict all the correlations between the assets to be 1 (ie perfect correlation). This is a pessimistic calculation and is based on the observation that in a crash correlations between assets are high (ie everything goes down)