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Title: Risk Analysis

1
Risk Analysis Modelling

Lecture 5 The Normal Distribution Value At Risk

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www.angelfire.com/linux/riskanalysisRiskCourseHQ_at_
Hotmail.com
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Making Sense of Quantitative Risk

In an earlier lecture we looked at the
mean-variance framework
We saw how we could calculate the mean and
variance of the return on a portfolio from the
statistical properties of the assets it contains
Expected return was relatively easy to interpret
Variance or standard deviation was abstract and
did not mean much other than to give an idea of
the relative risk

4
Value At Risk Implying Potential Loss

People intuitively try to assess risk in terms of
worst case scenarios
Information on how much you could lose on a
portfolio over the next day, month or year makes
much more sense to most people than an abstract
statistic such a variance
Value at Risk originated in the RiskMetrics group
at the investment bank JP Morgan in the early
1990s
It quantifies the worst case scenario in terms of
the probability of observing outcomes worse than
this worst case scenario (ie the quantile of the
loss)
There are a number of methods by which this worst
case scenario can be located, one of the simplest
and most popular is through the use of the normal
distribution

5
The Normal Distribution

The Normal Distribution or Bell Curve was first
introduced by the mathematician Abraham de Moivre
in 1733
The Normal Distribution is frequently observed in
the real world returns on stock markets,
aggregate levels of claims on certain classes of
insurance, measurement errors in an experiment,
individuals heights etc
Its occurrence in the world about us is
explained by the Central Limit Theorem which
states that the sum of a large number of
independent random variables will be normally
distributed

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Normal Distribution PDF and CDF

Normally distributed random variables are
unbounded and can take on any value between minus
and plus infinity
The PDF (Probability Density Function) for the
Normal Distribution is defined as
Where m is the mean or average of the random
variable and s is the standard deviation
The CDF of the normal distribution does not have
a formula and must be evaluated numerically, but
can be written

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Normal CDF PDF Where m 0 and s 2
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NORMDIST and NORMINV

The PDF and CDF for the normal distribution can
be calculated in Excel using the NORMDIST
function
To calculate the PDF for a value X we use the
formula
NORMDIST(X,m,s,FALSE)
To calculate the CDF for a value X we use the
formula
NORMDIST(X,m,s,TRUE)
For example, if we wanted to calculate the
probability that a normally distributed random
variable with a mean of 2 and a standard
deviation of 4 is less than 3 (CDF at 3)
NORMDIST(3,2,4,TRUE)

The inverse of the Normal CDF is calculated using
the NORMINV
NORMINV(P,m,s)
Where P is the probability of the random variable
being less than the level
For example, if we want to calculate the level
such that a normally distributed random variable
with a mean of 2 and a standard deviation of 4
will be less than or equal to 5 (0.05) of the
time
NORMINV(0.05,2,4)

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NORMSINV AND NORMDIST
0.95
NORMDIST(0.5,0,1,TRUE) 0.691
CDF of Standard Normal (Mean 0 and Std Dev 1)
NORMINV(0.95,0,1) 1.644
0.5
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Normally Distributed Random Numbers m0, s1
NORMINV(rand(),0,1)
We use the inverse of the normal CDF function
NORMINV
The computer generates a uniform random number
0.91 using rand()
1
0.91
0
1
NORMINV(0.91,0,1)1.34
The transformed random variable 1.34 is normally
distributed
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Normal Distribution Scatter m0, s1
Density at its highest about the mean (0)
Density decays quickly as we move away from mean
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Fitting the Normal Distribution

The behaviour of a normally distributed random
variable is entirely determined by its mean and
standard deviation (or variance)
We can estimate the mean and standard deviation
of a normally distributed random variable by
taking the mean and variance of a sample of
observations
We can then use this sample mean and standard
deviation to fit a normal distribution to the
random variable

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Central Limit Theory Experiment

The normal distribution is something that occurs
in nature it is not something that just exists
in statistics text books!
The reason we observe the normal distribution in
the world about us is because of the central
limit theorem (CLT) which states the remarkable
fact that if a random variable is the sum of a
large number of other random variables

Then the distribution of Y will be normal
regardless of the distributions of the individual
Xs
We notice that a portfolio is essentially the sum
of the random assets and liabilities it
contains
We will not prove the central limit theorem
mathematically instead look at an example in
which it occurs
We will create a random variable made up from the
average of 8 highly non-normal, uniformly
distributed random variables

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Empirical CDF vs Normal CDF Fit
The Empirical CDF for the sum is a very close
match for the normal distribution
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Our Experiment

!
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Simulating Portfolio Asset Behaviour

From an early lecture we discussed how we could
use the mean and variance of the proportional
change in the value of a portfolio (or asset) to
assess the risk and return
If we assume the proportional changes are
Normally Distributed we can simulate the
behaviour of a portfolios value by sampling
random returns from a Normal Distribution with
the appropriate mean and variance and applying
the formula

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Simulating the Portfolio Value
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Multi-Period Simulation
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VaR the Mean-Variance Framework

We can express the relationship between the value
of a portfolio of assets today (V0) and the
random value at some time in the future (Vt) as
Where r is the random return or proportional
change in the portfolios value over the period
of time
We can express the profit or loss as

If we know the mean and variance of the random
return on the portfolio r and we assume it is
normally distributed then can located values r
using the inverse normal CDF such that
Where X is the probability of observing a return
less than or equal to r
We can then take this lower boundary r and
calculate the loss
The probability of observing a loss worse than L
is X
L will be the VaR measure with confidence level X

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Example VaR Calculation

The average return on the portfolio over the next
year is 4 and the standard deviation is 7
If we assume returns are normally distributed, we
can use NORMINV we can calculate the value (r)
such that the annual return will only be less
than this 5 of the time
NORMINV(0.05,0.04,0.07)
This tells us that r is -0.07514 (-7.514)
If the initial value of the portfolio was 10000
then the loss would be -751.4 (-0.07514 10000)
We will only lose more than this 5 of the time,
this is the 5 VaR on the portfolio

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5 VaR Calculation Diagram
Only observe returns less than this 5 of the time
CDF of returns with m 0.04 and s 0.07
0.05
-0.07514
5 VaR -0.07514 10000 -751.4
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VaR Review Question

The annual return on the portfolio is normally
distributed with a mean of 6 and a standard
deviation of 10
The initial value of the portfolio is 250,000
Calculate the 1 VaR

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Locating Quantiles for the Normal Distribution

One useful feature of the normal distribution is
its quantiles can be located by simply taking a
number of standard deviations from the mean
For example, the 5 quantile for a normally
distributed random variable is located 1.645
standard deviations below the mean
The 1 quantile for a normally distributed random
variable is located 2.326 standard deviations
below the mean

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Probability Quantiles On Normal Distributions
PDF(X)
s
m -1.645s
m
Lower 5 tail
PDF(X)
s
m -2.326s
m
Lower 1 tail
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5 Quantile for Normal Disrtibution with m 0.04
and s 0.07
The location of the 5 quantile is 1.645 standard
deviation below the mean
0.05
0.04 1.645 0.07 -0.07515
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VaR Formula

To calculate the 5 VaR over some time horizon we
can simply apply
VaR5V0.(m-1.645s)
Where V0 is the initial value of the portfolio or
asset, m is the mean of random return over the
period and s is the standard deviation over the
period
The 1 VaR formula is equal to
VaR1V0.(m-2.326s)

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Example VaR Formula Calculation

Let us apply these formula to our earlier example
where the average return on the portfolio over
the next year is 4 and the standard deviation is
7 over the year and the initial value of the
portfolio is 10000
Applying the 5 VaR fomula
VaR510000(0.04-1.6450.07)
VaR510000-0.07515 -751.5
There is a slight difference because this is an
approximation (1.645 should be 1.644853.)
This measures the maximum loss over one year
because the mean and standard deviation are
measured over one year

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Positive and Negative VaR

So far we have been calculating VaR as a negative
value (signifying loss)
In practice it is often quoted as a positive
number
This is achieved by multiplying the negative VaR
measure by minus one
We can also adjust our formula to take this into
account
VaRXV0.(cs-m)
Where c is the desired confidence interval (c
1.645 for 5 VaR)
For the rest of the lecture we will use this VaR
measure since this in the convention

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Absolute vs Relative VaR Formula

So far our VaR calculation has measuring the
worst outcome by taking a number of standard
deviations away from the mean, this is known as
Relative VaR
VaR V0.(c.sr-mr)
The formula for Absolute VaR simply makes the
assumption that mr is zero
VaR V0c.sr
Where c is the number of standard deviations from
the mean for our required confidence interval

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Advantages of Absolute VaR

It is obviously conceptually more accurate to use
the actual expected return, as in relative VaR
However, in practice the expected return is very
difficult to predict accurately and errors in the
form of overestimation can lead to
underestimation of risk
This is particularly true for longer time
horizons where small differences compound to
large errors
Over long time horizons the expected return can
over-power the volatility and lead to situations
where the worst probable outcome is a profit!

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Diversified Undiversified VaR

In the event of a crash all assets tend to move
down together ie high correlation
When this occurs the effects of diversification
are negated and the volatility of the portfolio
is greater
For this reason it is suggested that when
calculating the variance on a portfolio for a VaR
calculation (worse case scenario) it should
incorporate high positive correlations, not
day-to-day correlations

This can be achieved by setting the correlation
terms in the correlations between assets to 1
(perfect positive correlation)
The effects of this will be to increase the
variance of the portfolio and thus increase the
maximum loss by removing the effect of
diversification from the portfolio
When we calculate VaR on this basis we are
calculating Undiversified VaR
If we use normal day-to-day correlations we
calculate Diversified VaR

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Perfect Correlation Simplification

You may recall from lecture 2 that we discussed
that the relationship between the standard
deviation of a portfolio and the assets contained
in that portfolio simplifies to a linear equation
when there is perfect correlation between the
assets
Where sP is the standard deviation of return on
the portfolio and s1, s2.. are the standard
deviations on the assets in the portfolios
This means that when calculating undiversified
VaR we do not need the quadratic form and the
covariance matrix in calculating sP

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Diversified VaR

Covariance Matrix
The variance used in the value at risk formula
captures diversification through the covariance
matrix
VaRV0.(csP - m)
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Undiversified VaR

w1s1 w2s2 sP

We assume the assets are perfectly correlated (ie
no diversification) in the calculation of the
portfolio variance
VaR V0.(csP - m)
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Monte Carlo Simulation Mixing Actuarial and
Financial Models

In last weeks lecture we built a Monte Carlo
simulation that modelled the underwriting
profitability of the insurance company
Where X is the random underwriting profit or
loss, P the premium income and C the random level
of aggregate claims
We will extend this now to include the effect of
the random profits or losses on investments (I)

Where R is the total profit across both the
underwriting and investment portfolio
We will simulate the random value for I by using
the equation
Where r is the random return on the insurance
companys investment portfolio over the year
randomly sampled from a normal distribution with
an appropriate mean and standard deviation
V0 is the initial value of the insurance
companys investment portfolio at the start of
the year we will assume this is equal to the
initial solvency capital

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Combining Models from Finance Actuarial Science
Using Monte Carlo
Premium Income
Aggregate Claims Model
Model of Profit on Investment
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A Problem With The Model

So far our VaR model has been based about the
equation
Where r is a normally distributed random variable
which can take on any value from plus to minus
infinity
Although widely used, this model has a serious
flaw it can allow the value of the portfolio to
become negative when r is less than -1
We will look at how a different definition of
returns or proportional change can solve this
problem.

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A Better Solution Continuously Compounded Returns

Instead of defining returns or proportional
changes 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 the bank divide the 10 into 2
semi-annual interest payments of 5

Notice that it is slightly larger, 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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What happens as we increase the rate at which the
interest is compounded?
There is a limit of 100e0.1
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The New Equation

We replace our earlier equation its continuously
compounded equivalent
The first thing we notice is that as r approaches
minus infinity Vt approaches 0, (we do not have
to worry about negative portfolio values)
A second more subtle difference is that this
model is now based on the Log-Normal distribution

The VaR using continuously compounded returns is
Where m is the mean of continuously compounded
returns, s is the standard deviation of the
continuously compounded return and c is the
number of standard deviations for the desired
confidence level X (1.645 for 5 confidence etc)

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Important Result!
Log-Normal Distribution
Normal Distribution
If r is a normally distributed then er is
log-normally distributed. The log-normal
distribution is never below zero why is that?
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The Log-Normal Distribution

The Log-Normal distribution is widely used
throughout finance and actuarial science
It is closely related to the normal distribution
Where Y is a Log-Normally Distributed random
variable and X is normally distributed
M is the minimum value for the Log-Normal (in
finance this is frequently set to zero)
X has a special name the normal counter part

We can transform a log-normally distributed
random number into its normal distributed counter
part simply by applying the following formula
This relationship turns out to be very useful
since it allows us to describe define the CDF and
PDF of the log-normal distribution in terms of
the normal distribution
We can think of the log-normal as the normal
distribution in a different form

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Log-Normal Excel Formula

The PDF of a log-normally distributed at a value
Y is
NORMDIST(LN(Y-M),m,s,FALSE)
Where m is the mean of the normal counterpart and
s is the standard deviation (Note this uses the
density for the normal distribution)
The CDF for a log-normally distributed random
variable is
NORMDIST(LN(Y-M),m,s,TRUE)

The inverse CDF for a log-normally distributed
random variable is
EXP(NORMINV(P,m,s))M
Where P is the probability of the log-normally
distributed random variable being less than or
equal to some level
Notice that we are doing here is finding the
quantile of the normal counterpart and then
implying the quantile of the log-normal from this

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Fitting the Log-Normal

The simplest method of fitting involves
transforming the log-normal dataset into a set of
values for the Normal Counterpart
Where Y is a value from the dataset (lognormal),
X is the associated normal counterpart value and
M is the minimum
One problem with this transformation is it will
not work for values where Y equal M, you can
ignore such values or set M below all the values
in your dataset
You can then simply take the mean and variance of
the normal counterpart dataset to fit the
lognormal

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Product Limit Theory

Like the Normal Distribution, the Log-Normal
distribution also occurs in the world about us
The explanation behind why we see the Log-Normal
distribution is the Product Limit Theory
The Product Limit theory states that if we
multiply any number of independent random
variables we can expect their product to be
Log-Normally Distributed

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Our Experiment

!
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Geometric Brownian Motion

If we simulate the behaviour of the value of an
asset or portfolio using the equation
By selecting normally distributed values for the
continuously compounded return r, we simulating a
discrete form of Geometric Brownian Motion
Geometric Brownian Motion is one of the most
important stochastic processes in Finance and is
widely use in option pricing

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Discrete Geometric Brownian Motion
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Uses of the Log-Normal in Actuarial Science

The Log-Normal distribution is frequently used in
actuarial science as a distribution to describe
both individual claim severities and for the
aggregate claims distribution
The Log-Normal distribution exhibits skew and has
a fairly long tail (allowing it to model large
claims)
It is commonly used to describe the random
severity of claims for fire insurance

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Appendix Risk And Time

Our ability to calculate VaR analytically is
limited to a one day horizon (if we use daily
data)
This is because our estimates of the mean and
variance of return are for a fixed time horizon
If we measure return on a daily basis and then
estimate the mean and variance of daily returns
we can only talk about the risk over a one day
horizon
This is obviously a serious limitation which we
will now address and in the process we will
learn something important about the nature of risk

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Multi-Period Risk

The relationship between risk over a single
period (one day to the next) and risk over a
number of period (such as risk over a month) are
obviously related
Intuitively we expect the risk to increase the
longer the horizon over which we invest
We can think of the multi-period return R as
being related to the single period return r
We would calculated the VaR over a longer time
horizon by placing some boundary on R like we did
r

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What We Want.
Distribution Describing Multi-Day Returns
SD(R)
Worst outcome R
E(R)
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Important Result!
Normal Distribution
Normal Distribution
Normal Distribution

s02 s12
s12
s02

m1
m0
m0 m1
The sum (or convolution) of two independent,
normally distributed random variables is a
normally distributed random variable whose mean
is the sum of the mean of the two random
variables r0 and r1 and whose variance is the sum
of the variances of the two random variable r0
and r1
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From Return To Loss

The relationship between the value of the
portfolio today and in the next period

Since r0 is normally distributed v1 is normally
distributed. The relationship between the value
of the portfolio in the next period and the
period after that

Re-expressing

If we want to perform a Monte Carlo simulation
this formula doesnt cause us any problems we
just generate r0 and r1 from a normal
distribution and generate samples for R and v2
If we want to analyse the risk using mathematics
rather than simulation then this formula causes
us problems.
The sum of 2 normally distributed random numbers
(or the convolution) is a normally distributed
random number
The product between the 2 normally distributed
terms (r0 and r1) causes us problems since it
results in an unpleasant product normal
distribution.

So the return on the portfolio value over 1
period is normally distributed while the
distribution describing the return over 2 periods
is a complicated convolution of a product normal
and normal distribution
This means our assumption that actuarial returns
are normally distributed has an unpleasant side
effect the type of distribution describing risk
changes over the time horizon we measure risk!

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Problem With Actuarial Returns
Returns Compounding
Convolution of Product Normal and Normal! ?
Normal Distributed Portfolio Value
v0
Normal Actuarial Returns
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We Need A Different Definition of Returns!

The standard definition of returns makes
estimating the probability distribution of values
beyond one step in the future complex
One solution would be to ignore the compounding
effect of returns which would get arid of the
tricky cross product term

This would mean that P2 would be normally
distributed but will lead to other problems (like
negative portfolio values)

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2 Day Horizon With Continuously Compounded Returns

Let us say we assume that continuously compounded
returns are described by a normal distribution
The relationship between the portfolio value
today v0 and the value tomorrow v1, where r0 is
todays random proportional change

r0 is normally distributed by assumption
v1 is log normally distributed since er0 is
log-normally distributed

Now the relationship between v0 and v2

R is equal to r0 r1 so it is normally
distributed
v2 is log-normal since er0r1 is log-normally
distributed
Let us say that r0 and r1 are both sampled from
the same normal distribution with a given mean m
and standard deviation s
Then the mean of R is 2.m (m m) and the
variance is 2.s2 (s2 s2)

Now the standard deviation of R over 2 days is

Since R is normally distributed with a mean of
2.m and standard deviation of the lower
boundary can be expressed as

1 tail
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2 Day Horizon Example

Daily continuously compounded returns have a mean
of 0.00108 or (0.108) and a standard deviation
of 0.0102
Calculate the 5 relative VaR over a 2 day time
horizon on a portfolio with value of 10,000
The barrier for returns that we will only observe
a worse return 5 of the time is

The value of the portfolio in this worst case
scenario is

The loss associated with this worst outcome is

Lets verify this with a Monte Carlo simulation in
Excel.

74
Continuously Compounding
Portfolio Value Compounding
Log-Normal Distribution
Log-Normal Distribution
v0
Normal Continuously Compounding Returns
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Further Into The Future

We can extend these results to derive the mean
and standard deviation of return over a 3 day
period interms of the mean and standard deviation
of return over one day

Or over a period of T days to

The distribution describing R or the accumulated
continuously compounded returns on T periods will
be normally distributed
The mean of Rs distribution will be and
the standard deviation

Probability Distribution of R
Lower 1 tail
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Var Equations

The worst return we can expect to observe on our
portfolio over a time horizon T is therefore

Where c is the number of standard deviations away
from the mean for the confidence interval of
interest (such as 1.64 for the 5 level) and R
boundary on the worst return at that confidence
The value of the portfolio when we observe this
worse return scenario is

The loss or Value at Risk in the event of
observing this portfolio is

This formula gives us the worst loss we can
expect to observe on a portfolio after a period
of time T, given that normally distributed,
continuously compounded returns have a mean of
E(R) and standard deviation of SD(R)

79
Over Short Time Horizons

For small values of r we can make the following
approximation (first order Taylor)

So for short time horizons we can often simply
the VaR formula

As the time horizon extends the error of this
approximation decreases

80
Portfolio Diffusion Boundaries
Price
2.5 Upper Probabilistic Boundary
2.5 Lower Probabilistic Boundary
100
Portfolio will be between upper and lower
boundary 95 of the time
Time
81
Risk Time

The result that risk scales with the square root
of time is very important
It tells us that risk or standard deviation
increases over time, but it does so at a
decreasing rate
The intuitive reason behind this is we have
diversification across time!
We have diversification because we assume that
returns from one day to the next are uncorrelated
or independent so there is a tendency for
returns above and below the mean to offset each
other
So heavy losses on one day might be offset by
large gains on another day
This offsetting of gains and losses decreases the
accumulation of risk

82
Risk vs Time
Risk increases at a decreasing rate because the
accumulation in risk is offset by diversification
between different days

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