Risk Measures
Here is where the presentation begins
FOCUS OF THIS CHAPTER
Defining and measuring risk
• Volatility
• VaR (Value–at–Risk)
• ES (Expected Shortfall)
• Holding periods
• Scaling and the square–root–of–time
Notations
p Probability
Q Profit and loss
q Observed profit and loss
w Vector of portfolio weights
X and Y Refer to two different assets
'(.) Risk measure
# Portfolio valued
dividends
Defining Risk
General Definition
• No
universal definition of what constitutes risk
• On a very general level, financial risk could be defined as
“the chance of losing a part or all of an investment”
• Large number of such statements could equally be made,
many of which would be contradictory
Stock indices
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Which asset do you prefer?
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Standard mean variance analysis indicates that all three assets are equally
risky and preferable
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Since we have the same mean
E(A) = E(B ) = E(C ) = 0
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And the same volatility
σA = σB = σC = 1
If one uses mean variance analysis one is indifferent between all three
Which asset is preferred by MV?
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If, however, one asks anybody which asset they would prefer,
they probably would have a personal preference for one
The most popular speculative financial asset in the world is
inverted C — lottery tickets
So what happened?
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The model — mean variance — comes with a set of
assumptions that are beneficial for creating a practical
investment model
But at the same time inconsistent with people’s risk
preference
Oftentimes that is not important
But sometimes it is
Recall “all models are wrong, some models are useful”
Three investment choices
Suppose Tom, Peter and Mary all have the same amount of money to invest
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All have access to the same investment technology
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All are contemplating putting $1 million into Amazon
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Tom is a day trader, aiming to buy and sell within a week
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Peter is a fund manager, and his bonuses depend on quarterly performance
Mary is 22 years old, planning to retire in 40 years and expects to die in 70.
She is saving for her pension that needs to be available when she’s 90 years
old and far from able to manage her own money
Their choices
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While faced with same technology, their preferences are
different
And consequently they will evaluate the three investment
choices A, B , C
differently
Mary will unequivocally pick C
It’s quite possible Tom does as well because she would not be
a day trader if she didn’t like risk
Peter would pick A or B , probably the latter
Which asset is “better”?
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a.
b.
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There is no obvious way to discriminate between the assets
One can try to model the underlying distribution of market
prices and returns of assets, but it is generally unknown.
can identify by maximum likelihood methods
or test the distribution against other other distributions by
using methods such as the Kolmogorov-Smirnov test
Practically, it is impossible to accurately identify the
distribution of financial returns
Risk is a latent variable
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Financial risk is cannot be measured directly
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Risk has to be inferred from the behavior of observed
market prices
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e.g. at the end of a trading day, the return of the day is
known while the risk is unknown
Risk measure and risk measurement
Present in most financial returns
Risk measure a mathematical concept of risk
Risk measurement a number that captures risk, obtained by
applying data to a risk measure
Volatility
Volatility
The standard deviation/error of returns
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a.
b.
Volatility is the standard deviation of returns
Main measure of risk in most financial analysis
It is a sufficient measure of risk when returns are normally
distributed
For this reason, in mean-variance analysis the efficient
frontier shows the best investment decision
If returns are not normally distributed, solutions on the
efficient frontier may be inefficient
Volatility
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The assumption of normality of return is violated for
most if not all financial returns
- See Chapter 1 on the non-normality of returns
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For most applications in financial risk, volatility is likely
to systematically underestimate risk
Value–at–Risk (VaR)
HISTORY
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Until 1994, the only risk measure was volatility
Then the JP Morgan bank proposed a risk measure called
Value-at-Risk and a method to measure it, called Riskmetrics.
Why would JP Morgan do that — to be able to reduce its level
of capital
It used to be called the 415 report because it was created
because the chairman of the bank wanted a single
measurement of the bank’s risk in time for the treasury
meeting at 4:15 PM
VaR
Value-at-risk is a statistical measure of the riskiness of financial
entities or portfolios of assets. It is defined as the maximum
dollar amount expected to be lost over a given time horizon, at
a pre-defined confidence level.
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The most common risk measure after volatility
VAR is determined by three variables: a specific time period,
a confidence level, and the size of the possible loss.
VaR
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Suppose you hold a $1 million portfolio of stocks tracking
the S&P 500 index. For the purpose of our discussion we
may assume that
To address the question of how much this portfolio could
lose on a “bad day,” one could specify a particular bad day
in history – say the October 1987 stock market crash during
which the market declined 22 percent in one day.
Which probability should we use?
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VaR levels of 1% – 5% are very common in practice
Regulators (Basel II) demand 1%
But less extreme numbers, such as 10% are often used in risk
management on the trading floor
More extreme lower numbers, such as 0.1%, may be used for
applications like economic capital, survival analysis or longrun risk analysis for pension funds
Computing VaR for a single asset
For a single asset using daily returns data at confidence level c, the
VaR is computed as:
𝑉0 𝛼𝜎
Where;
𝑉0 is the initial value of the asset
𝛼 the number of the standard deviation below the mean
corresponding to (1-c ) quantile of the normal distribution
𝜎 is the standard deviation of the asset’s return
Computing VaR for a single ass asset
Prob(X<z)
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α
0.1%
0.5%
-3.090 -2.576
1.0%
2.5%
5%
10%
-2.326
-1.960
-1.645
-1.282
For 95% confidence level, c is equal to 0.95
(1-c) =(1-0.95)=0.05 which can be expressed as 5%
The corresponding value of α is 1.645
Computing VaR for a single asset
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Suppose that an investor’s portfolio consists of $10,000 worth
of IBM stock
Assume the standard deviation of the stocks’ returns are
0.0189(1.89%)
If the investor wants to know his portfolio’s VaR over the
coming trading day at a 95% confidence level, how do yoy
think it will be calculated?
Computing VaR for a single asset
𝑉0 𝛼𝜎 = (10,000)(0.0189)(1.645)
= 310.905
This means that there is a 5% chance that the
individual will lose $310.91. Or that the
portfolio would be valued at $9,689.09.
Computing VaR for a single
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VaR can be extended to different time horizons by applying
the square root of time rule
The square–root–of–time rule is commonly assumed when
financial risk is time aggregated whereby high frequency risk
estimates are scaled to a lower frequency T by the
multiplication of √ T.
According to this, the standard deviation increases in
proportion to the square root of time
• 𝜎𝑡−𝑝𝑒𝑟𝑖𝑜𝑑 = √𝑡𝜎1−𝑝𝑒𝑟𝑖𝑜𝑑
Computing VaR for a single asset
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If an investor wants to know his portfolio’s VaR for the
coming month at a 95% confidence interval based on the
assumption that there are 22 days in a month, this would be
calculated as;
• 𝑉0 𝛼𝜎 =(10,000)(0.0189√22)(1.645)
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= 1,458.27
Computing VaR for a single asset
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Similarly, if the investor wants to know what his portfolio’s
VaR is over the coming year, assuming that there are 252
trading days in a year, the calculations would be:
• 𝑉0 𝛼𝜎 =(10,000)(0.0189√252)(1.645)
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= 4,935.46
Computing VaR for a portfolio
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In order to compute VaR for a portfolio of two or more
assets, the correlation between the assets must be
explicitly considered
The lower the correlation, the lower the resulting VaR
Computing VaR for a portfolio
The Value at Risk of a portfolio calculated by determining;
• Weight(proportion of the total invested) of each asset in the
portfolio
• Standard deviation of each asset’s rate of return in the
portfolio
• Correlations among the assets’ rate of returns in the
portfolio
Computing VaR for a portfolio
Once the confidence level and time horizon have been chosen,
the weights, volatilities and correlations can be combined using
Markowitz’s approach to derive the portfolio’s VaR.
𝑇𝑜𝑡𝑎𝑙 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎𝑠𝑠𝑒𝑡
𝑇𝑜𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜
Weights=
∑( 𝑅𝑒𝑡𝑢𝑟𝑛𝑠1 −𝐴𝑣𝑒𝑟𝑎𝑔𝑒1 ) ∗ (𝑅𝑒𝑡𝑢𝑟𝑛𝑠2 −𝐴𝑣𝑒𝑟𝑎𝑔𝑒2 )
Covariance=
𝑆𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒−1
Computing VaR for a portfolio
𝐶𝑜𝑣1,2
Correlations=
𝜎1 𝜎2
Volatilities= √ (𝑤12 𝜎12 + 𝑤22 𝜎22 + 2𝑤1 𝑤2 𝐶𝑜𝑣1,2 )
Computing VaR for a portfolio
Assume that a $100,000 portfolio contains $60,000 worth of
Stock X and $40,000 of stock Y
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Given the following data compute the VaR of this portfolio
with a 95% confidence level over the coming;
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Day
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Month
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Year
Computing VaR for a portfolio
Data
𝑤1 = 0.60
𝑤2 = 0.40
σ1 = 0.0163
σ2 = 0.0154
ρ1,2 = −0.191
Computing VaR for a portfolio
Exercise
Find the VaR at 95% confidence level
Expected Shortfall (ES)
Expected Shortfalls
It is known by several names, including
1. ES
2. Expected tail loss
3. Tail VaR
Expected Shortfall
Expected shortfall is a risk measure sensitive to the shape of
the tail of the distribution of returns on a portfolio, unlike
the more commonly used value-at-risk (VAR). Expected
shortfall is calculated by averaging all of the returns in the
distribution that are worse than the VAR of the portfolio at
a given level of confidence. For instance, for a 95%
confidence level, the expected shortfall is calculated by
taking the average of returns in the worst 5% of cases.
Expected Shortfall
ES = E(Loss|Loss >VaR)
Expected value of the loss given that the loss is greater than
VaR
“a probabilty weighted average of tail losses”
Pros and cons
Pros
• ES in coherent and VaR is not
• It is harder to manipulate ES than VaR
Cons
To calculate ES we first have to know VaR and then integrate over the
tail from VaR to minus infinity
• That means in practice that we need more calculations for ES than
VaR
Length of holding periods
• In
practice, the most common holding period is daily
• Shorter holding periods are common for risk management
on the trading floor
• where risk managers use hourly, 20-minute and even 10
minute holding periods
• this is technically difficult because intraday data has
complicated patterns
Longer holding periods
• Holding
periods exceeding one day are also demanding
• the effective date the sample becomes much smaller
• one could use scaling laws
• Most VaR forecasts require at least a few hundred observations to
estimate risk accurately
• For a 10-day holding period will need at least 3,000 trading days, or
about 12 years
• In most cases data from 12 years ago are fairly useless