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Risk
Management
Session 4
Analytics of Risk
Management II:
Statistical Measures of Risk
Lecturer:
Mr. Frank Lee
Overview
Quantitative measures of risk - 3 main types:
1. Sensitivity – derivative based measures
2. Volatility & Statistical measures of risk:
◦
◦
◦
◦
Attitudes to risk
Relation with Finance Theory
Portfolio Theory, CAPM & APT
Post-modern Portfolio Theory
Downside Risk measures
3.
◦
◦
Statistical underpinning
Value at Risk
Value = Expected Net Present Value
n
E(NPV) =
Σ
E(Rt) - E(Ct)
t=1
(1 + R)t
A Set of Definitions
Risk - The outcome in a particular situation
is unknown but the probability distribution
from which the outcome is to drawn is
known
 Uncertainty - The outcome in a particular
situation is unknown as is the probability
distribution from which the outcome is
drawn.

Problems of which situations are
risky and which are uncertain
Risky situations can form the basis for
tractable financial analysis
 Uncertain situations are considerably less
analytically tractable

Examples of Risk and Uncertainty
Risk - Prices in Financial Markets
Share Prices
Interest Rates
Exchange Rates
 Uncertainty Terrorist Attacks
Wars
Financial Crises

Statistics and Estimates
Mean - Measure of Central Tendency, like Median
and Mode.
 Variance - Measure of dispersion round the mean
the squared term ensures that Variance is positive
and gives extra weight to observations furthest from
the mean.
 Standard Deviation - Square Root of Variance. It
is in same metric as Mean

Probability Distributions
We generally focus on normal
distributions
 Normal Distributions are entirely
specified by Mean and Standard Deviation

Normal Distribution
Frequency
34.13%
34.13%
-s
+s
m
Return
Risk and Return
Return - Average or Expected Return
 Risk

◦ For Normal Distributions Standard Deviation is totally
satisfactory
◦ For Non-normal Distributions there may be a diversity of
statistical and psychological measures or risks
Attitudes to Risk
Utility Function - Defined over a probability
distribution of returns.
 Mean and Variance approach
 Higher Moments
 Time

Attitudes to Risk
Risk Averse - Like Return Dislike Risk
 Risk Neutral - Like Return and totally
unconcerned about Risk
 Risk Loving - Like Return and Like Risk

Utility and Risk
Utility
U(B)
U(*)
U(A)
W(A)
W(*)
W(B)
Wealth
Risk Aversion and Utility
W(A) + W(B) = W(*)
2
U(*) > U(A) + U(B)
2
Prefer W(*) to bet with 0.5 probability
of W(A) and 0.5 probability of W(B) which
has the same expected value of wealth of W(*)
Preferences for Risk and Return
Return
B
A
X
C
D
Risk
For Risk Averse Investor
A Preferred to X
 X Preferred to D
 C and X no ordering
 B and X no ordering

Indifference Curves for the Risk
Averse Investor
Return
U1
U2
U3
x
Risk
Indifference Curve Slope is a Measure
of Risk Aversion
Return
a
b
c
d
ab > cd
xy
xy
x
y
Risk
Measures of Risk





Standard Deviation
Combination of Moments
Value at Risk
Expected Tail Loss
Moments Relative to Benchmarks - Risk Free
Rate, Zero Return, Capital Asset Pricing Model,
Arbitrage Pricing Theory
Measuring Risk
Variance - Average value of squared deviations from
mean. A measure of volatility.
n
2
=

pi (ri E (r ))2
i 1
Standard Deviation - Square root of variance
(square root of average value of squared deviations
from mean). A measure of volatility.
=
n
2
 pi (ri E (r ))
i 1
2
Standard Deviation






Square root of variance
Equates risk with uncertainty
Implies symmetric, normal return distribution
Upside volatility penalized same as downside
volatility
Measures risk relative to the mean
Same risk for all goals
Moments
ith Moment around a = E(R - a)i
Measure of Skewness = E(R - E(R))3
(Minus value skewed to left, Positive Value skewed to right)
Measure of Kurtosis = E(R - E(R))4
(Larger value flatter the distribution)
Modern Portfolio Theory
Portfolio Theory Assumptions





Investors Risk Averse
Investors only interested in the Mean and
Standard Deviation of the Distribution of
Returns on an Asset
Investors have knowledge of Mean and
Standard Deviation of Returns
A Range of Risky Assets
At Least One Riskless Asset
Portfolio standard deviation
Measuring Risk
Unique
risk
Market risk
0
5
10
Number of Securities
15
Portfolio Risk
Covariance =
n
Covxy   pi (rxi  rx )( ryi  ry )
i 1
Correlation =
 xy 
Covxy
 x y
(-1 < xy <1)
Variables Trend Together
Y
Figure 1
X
Variables Trend in Opposite
Directions
Y
Figure 2
X
Correlation Values





One - Perfect linear relation
Between zero and one variables trend together
Zero - No relation between variables
Between zero and minus one variables trend in
opposite directions
Minus one - variables have negative perfect linear
relation
Portfolio Return & Risk
Expected Portfolio Return  (x1 r1 )  (x 2 r2 )
Portfolio Variance  x12σ 12  x 22σ 22  2(x1x 2ρ 12σ 1σ 2 )
Portfolio Risk
The shaded boxes contain variance terms; the remainder contain
covariance terms.
1
2
3
STOCK
To calculate
portfolio
variance add
up the boxes
4
5
6
N
1
2
3
4
5
6
STOCK
N
Hedging with a Portfolio
Return
A
B
Time
Simple Portfolio Impacts
Return/SD
Asset y
Return/SD
Asset x
Correlation
Case 1
20/10
10/5
-1
Case 2
20/10
10/5
0
Case 3
20/10
10/5
1
Correlation Coefficients Revisited
Chart 1: Portfolio Opportunity Sets: different correlations
30
all investment in N
25
expected return (%)
20
correlation=0
correlation=-1
15
correlation = 1
all investment in M
correlation=0.25
10
5
0
0.0
5.0
10.0
15.0
risk (standard deviation (%))
20.0
25.0
30.0
The Set of Risky Assets
E(R)
A
SD(R)
Optimal Set of Risky Assets
E(R)
y
x
SD(R)
Adding the Riskless Asset
E(R)
Rf
SD(R)
The Capital Market Line
E(R)
G
E
The Capital
Market Line
H
D
SD(R)
Investors Choice
E(R)
y
x
SD(R)
Portfolio Theory Conclusions
All Investors hold the same set of risky assets if
they hold risky assets
 All investors must hold the market portfolio
 Their risk preferences determine whether they
gear up or down by borrowing or lending

Security Market Line
Return
Market Return = rm
.
Efficient Portfolio
Risk Free
Return
=
rf
1.0
BETA
Security Market Line
Return
SML
rf
1.0
BETA
SML Equation = rf + B ( rm - rf )
Beta and Unique Risk
 im
Bi  2
m
Covariance with the
market
Variance of the market
Downside Risk Measures
Expert Opinions
Markowitz (1992): Since an investor worries about
underperformance rather than over-performance,
semi-deviation is a more appropriate measure of
investor's risk than variance.
 Sharpe (1963): Under certain conditions the meanvariance approach leads to unsatisfactory predictions
of investor behavior.

Post-Modern Portfolio Theory
Two Fundamental Advances on MPT:

Downside risk replaces standard deviation

PMPT permits non-normal return distributions
‘PMPT’ v MPT
• Risk measure:
Downside Risk vs. Standard Deviation
• Probability distribution:
Lognormal vs. Normal.
• The same application:
Asset allocation/portfolio optimalisation
and performance measurement
Downside risk measures
Shortfall probability
Average shortfall
Semi-variance
LPM 0 
LPM 1 
LPM 2 

 p   r 
r  
0
r

 p   r 
r  
1
r

 p   r 
r  
2
r
Downside Risk
Defined by below-target semideviation
 Standard deviation of below-target returns
 Differentiates between risk and uncertainty
 Naturally incorporates skewness
 Recognizes that upside volatility is better than downside
volatility
 Combines frequency and magnitude of bad outcomes
 No single riskless asset

Value at Risk (VAR)
Value at Risk





VaR is a potential loss
The ‘maximum’ loss at a present confidence level.
The confidence level is the probability that the
loss exceeds this upper bound.
VaR applies to all risks – market, credit, default…
VaR applies as long as we can build up a
distribution of future values of transactions or
losses
Value at Risk (VAR) - The level of losses
relative to 0 or the Mean which will only
be exceeded in a particular proportion of
instances over a particular time period.
Value at Risk
Frequency
RVAR = |E(R) - x|
AVAR = |0 - x|
a
x
0
E(R)
Risk
Definition of VAR
Absolute VAR = (0 - x)
Relative VAR = (y - x)
Probability Distribution and Value at Risk
Key Choice Parameters
Time Period
 Confidence Level

Time Period

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

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Liquidity of Portfolio
Regulatory Framework (10 Days)
Measurement Technique - Does one Assume
Normality ?
How does one deal with changing composition of
Portfolio
Required Data for Testing
Confidence Level
Risk Management/Capital Requirement
 Regulatory Requirement (1% VAR)
 Testing - Higher so more extreme observations
 Accounting and Comparison

Measuring Value at Risk
Variance/Covariance
 Historical Simulation
 Monte Carlo Simulation
 Stress Testing

Issues in Modelling VaR

Need to move from a ‘standalone’ VaR
(distribution of losses on individual assets) to the
portfolio loss distribution (combines losses from
all individual assets in the portfolio).

Difficult to model the loss distribution of a
portfolio.
Issues in Modelling VaR cont’d
The focus on high losses implies modelling of the
‘fat tail’ of the distribution rather than looking at
the central tendency.
 Expected Tail Loss - ETL is quintile average;
expected loss if we get a loss in excess of VAR.


The normal distribution does a poor job of
modelling distribution tails (e.g. for credit risk the
loss distributions are highly skewed.
Issues with VAR

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How does one deal with Non-normality ?
How does one deal with financial crises ?
How does one deal with shifting parameter
values ?
What types of risks is it best applied to ?
If normal just a multiple of Standard Deviation !
VAR Example
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Portfolio value is £100million, volatility (st.
deviation) is 5%. Assuming normality, what is the
1% VaR of the portfolio over the next 10 days?
VaR= 2.33 x 5% x £100m = £11.65million
VaR 10 days = £11.65 x 100.5=£36.84m
Can adjust for expected return if any
E.g. if E(z) = 0.1 percent per day, daily VaR in the
example becomes £11.55m
Because small, daily E(z) is ignored in practice
Summary of Conclusions on Risk and
Return I
Investors Choose Between Distributions of
Returns
 Return is Mean or Expected Return
 With Normal Distributions of Returns Mean
and Standard Deviation of Returns totally
summarise the information in the Distribution
of Returns

Summary of Conclusions on Risk and
Return II
The appropriate measures depend on investor
preferences
 Investor Preferences seem to concentrate on
returns relative to the market, making losses and
worst case options
 The falling cost of computation and the increasing
risks associated with financial markets have driven
the increasing focus on these issues.

Summary of Conclusions on Risk and
Return III
We mainly consider the standard approaches to
Risk and Return but these have a restricted view
of risk and return.
 Many developments in finance related to the risk
management issues raised have been or will be
discussed further in other courses on the
programme.
