Lecture 5: Value At Risk

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Lecture 5 Value At Risk
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http//www.angelfire.com/linux/lecturenotes
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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

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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.

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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

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Graph Of A Possible Game
Total Winnings
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0
Number Of Games Played
-10
Total Losses
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The Expected Payoff And Variance Of Payoff

• Let us say we play with games 100s 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
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Standard Deviation Of Payoff
As the number of games played increase the
standard deviation increases, but it increases at
a decreasing rate.
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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!

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Density of Outcomes
Steps
1
H
T
1
1
H
T
H
T
1
1
2
T
H
H
T
T
H
1
1
3
3
T
H
H
T
T
T
H
H
1
4
6
4
1
Note Also known as Pascals Triangle
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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

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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?

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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

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Stochastic Process
Price
At each step the proportional change in the stock
price is a random variable from a normal
distribution
?
Time
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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 m and standard deviation s
  then we can say that the price tomorrow P1 is

• 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?

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Distribution Of Future Prices

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

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

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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

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

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Continuously Compounded Returns

• Instead of defining returns like this

• We will see that the continuously compounded
  definition is better

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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
  (10.1)
• 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?

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

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What Happens As We Compound Over Very Short
Periods?

• In general we can define the compounding rate as

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

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

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General Equations

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

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

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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 todays random
  proportional change

• P1 is log normally distributed

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• Now the relationship between P0 and P2

• Now the relationship between P0 and P3

• The relationship between P0 and PT

where
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• 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 Prices (map)
Random Returns (domain)
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!
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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

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• R will be normally distributed since it is the
  sum of T normally distributed normal variables
• The mean of Rs distribution will be T.m and the
  standard deviation T1/2.s

Probability Distribution of R
T1/2.s
T.m -1.96.T1/2.s
T.m 1.96.T1/2.s
T.m
Lower 2.5 tail
Upper 2.5 tail
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Lognormal Probability Distribution of P(T)
P0.emT
Upper 2.5 tail
Lower 2.5 tail
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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

• Translating these to values to the levels for
  prices

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Price Diffusion Boundaries
Price
Upper Probabilistic Boundary
Expected Path
Lower Probabilistic Boundary
100
Time
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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 ones 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!

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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 portfolios value across time
• From this stochastic process of the portfolios
  value we will estimate the Value At Risk for a
  given time horizon

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Our Method

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

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

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Portfolio Value Diffusion
Portfolio Value
Value At Risk At Time T
Expected Path For Portfolio
PV0
Portfolio Value Will Only Go Bellow this 2.5 of
the time
Time
T
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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.

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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 dont 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

• 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.

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Diversified Undiversified VaR

• Diversified VaR relates to the situation where we
  use estimates of the covariances of the
  portfolios 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)