Transcript Lecture 5: Value At Risk

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
Total Winnings
+£10
£0
Number
Of
Games
Played
-£10
Total Losses
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
2
3
Expected Payoff
0
0
Std Dev Payoff
1.414
1.731
4
5
0
0
2
2.236
Standard Deviation Of Payoff
4
Standard Deviation Payoff
3.5
3
2.5
2
1.5
1
0.5
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Number Games Played
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 Outcomes
Steps
1
H
T
1
1
T
H
1
H
1
1
1
2
T
T
H
T
H
T
3
H
T
4
T
H
1
3
H
T
H
6
Note: Also known as Pascal’s Triangle
T
4
H
1
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
Price
At each step the proportional
change in the stock price is a
random variable from a
normal distribution
?
Time
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:
~
P1  P0 .(1  ~
r0 )
• 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:
~ ~
P2  P1.(1  ~
r1 )
~
P2  P0 .(1  ~
r0 ).(1  ~
r1 )  P0 .(1  ~
r0 .~
r1  ~
r0  ~
r1 )
• 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:
~
P2  P0 .(1  ~
r0 ).(1  ~
r1 )  P0 .(11  ~
r0  ~
r1 )
• This would mean that P2 would be normally
distributed but will lead to other problems…
Continuously Compounded
Returns
• Instead of defining returns like this:
P1  P 0
r0 
P0
(1  r0 ).P 0  P1
• We will see that the continuously compounded
definition is better:
 P1
r0  ln 
 P0



P1  P0 .e r0
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?
 r
110  100.1  
 2
2
 110 
r  2. 2
 1  9.75%
 100 
• 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:
r

P1  P0 .1  
n

n
• As n approaches infinity the value converges
to a non-infinite value:
n
r

r
1


 e
n

• Where e is a special number like p and is
equal to 2.718282..
General Equations
• The relationship between P1 and P0 for a
given continuously compounded return r is:
P1  P0 .e r
• And by taking natural logs of both side we
can see that we can calculate the continuously
compounded return as
 P1 
r  ln  
 P0 
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
P1  P0 .e
~
r0
• P1 is log normally distributed
• Now the relationship between P0 and P2
P2  P1.e
~
r0
~
r1
~
r1
P2  P0 .e .e  P0 .e
~
r0  ~
r1
• Now the relationship between P0 and P3
~
r0
~
r1
~
r2
P2  P0 .e .e .e  P0 .e
~
r0  ~
r1  ~
r2
• The relationship between P0 and PT
PT  P0 .e
~
R
where
~ T 1 ~
R   ri
i 0
• 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:
~ ~ ~ ~
R  r0  r1  r2  ...  ~rT 1
~
E(R)  E(~r0 )  E(~r1 )  E(~r2 )  ....  E(~rT 1 )  T .m
~
2
~
~
~
~
Var( R)  Var(r0 )  Var(r1 )  Var(r2 )  ...  Var(rT 1 )  T .s
~
Std Dev ( R )  T .s
• 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.m and the
standard deviation T1/2.s
Probability Distribution of R
T1/2.s
T.m -1.96.T1/2.s
Lower 2.5% tail
T.m +1.96.T1/2.s
T.m
Upper 2.5% tail
Lognormal Probability
Distribution of P(T)
P0 .e
m *T 1.96*s * T
P0 .e
P0.em*T
Lower 2.5% tail
m *T 1.96*s * T
Upper 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
E ( R)  2 * 0.003  0.006
UpperR  2 * 0.003  2 * 0.001*1.96  0.0087
LowerR  2 * 0.003  2 * 0.001*1.96  0.0032
• Translating these to values to the levels for prices:
E ( P)  100 * e0.006  100.6
UpperP  100 * e0.0087  100.87
LowerP  100 * e 0.0032  100.32
Price Diffusion Boundaries
Price
100
Upper Probabilistic
Boundary
Expected
Path
Lower Probabilistic
Boundary
Time
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:
VaR(T )  P0  P0 .eT *m 1.96*
T *s
 P0 .(1  eT *m 1.96*
T *s
)
• Where m is the mean of returns on the portfolio and s is the standard
deviation of returns on the portfolio
Portfolio Value
Portfolio Value Diffusion
Expected Path
For Portfolio
Value At Risk At
Time T
PV0
Portfolio Value Will
Only Go Bellow this
2.5% of the time
T
Time
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:
VaR(T )  P0 .(1  e
1.96* T *s
)
• 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)