FIN 685: Risk Management
Topic 6: VaR
Larry Schrenk, Instructor

Types of Risks

Value-at-Risk

Expected Shortfall
Types of Risk

Market Risk

Credit Risk

Liquidity Risk

Operational Risk
VaR

J. P. Morgan Chairman, Dennis
Weatherstone and the 4:14 Report

1993 Group of Thirty

1994 RiskMetrics

Probable Loss Measure

Multiple Methods

Comprehensive Measurement

Interactions between Risks

There is an x percent chance that the firm
will loss more than y over the next z time
period.”

Correlation

Historical Simulation

Monte Carlo Simulation

Historical Prices
– Various periods

Values Portfolio in Next Period

Generate Future Distributions of
Outcomes

Variance-covariance
– Assume distribution, use theoretical to calculate
– Bad – assumes normal, stable correlation

Historical simulation
– Good – data available
– Bad – past may not represent future
– Bad – lots of data if many instruments (correlated)

Monte Carlo simulation
– Good – flexible (can use any distribution in theory)
– Bad – depends on model calibration
Finland 2010

Basel Capital Accord
– Banks encouraged to use internal models to
measure VaR
– Use to ensure capital adequacy (liquidity)
– Compute daily at 99th percentile
• Can use others
– Minimum price shock equivalent to 10 trading
days (holding period)
– Historical observation period ≥1 year
– Capital charge ≥ 3 x average daily VaR of last 60
business days
Finland 2010




At 99% level, will exceed 3-4 times per year
Distributions have fat tails
Only considers probability of loss – not
magnitude
Conditional Value-At-Risk
– Weighted average between VaR & losses
exceeding VaR
– Aim to reduce probability a portfolio will incur
large losses
Finland 2010
E.G. RiskMetrics
 Steps

1. Means, Variances and Correlations from
Historical Data
•
Assume Normal Distribution
2. Assign Portfolio Weights
3. Portfolio Formulae
4. Plot Distribution
n
 
w
r
k k
k 1
n

2

n
w w
i
i 1 j 1


j
i
j
Assuming normal distribution
 95% Confidence Interval

– VaR -1.65 standard deviations from the
mean

99% Confidence Interval
– VaR -2.33 standard deviations from the
mean

Asset
Return
Var
Two Asset
Portfolio
20%
0.04
Weight
Cov
50%
0.02
B
50%
A
12%
0.03
  0 .5  0 .2   0 .5  0 .1 2   0 .1 6

2
 0 .5
2
 0 .0 4   0 .5  0 .0 3 
2
 2  0 .5   0 .5  0 .0 2  0 .0 2 7 5

 = 0.1658

5% tail is 1.65*0.1658 = 0.2736 from mean

Var = 0.16 - 0.2736 =-0.1136

There is a 5% chance the firm will loss more
than 11.35% in the time period

 = 0.1658

1% tail is 2.33*0.1658 = 0.3863 from mean

Var = 0.16 - 0 0.3863 =-0.2263

There is a 1% chance the firm will loss more
than 22.63% in the time period

Steps
1. Get Market Data for Determined Period
2. Measure Daily, Historical Percentage
Change in Risk Factors
3. Value Portfolio for Each Percentage
Change and Subtract from Current
Portfolio Value

Steps
6. Rank Changes
7. Choose percentile loss
•
95% Confidence
–
–
5th Worst of 100
50th Worst of 1000
1.
Model changes in risk factors
– Distributions
– E.g. rt+1 = rt + a + brt + et
2.
3.
Simulate Behavior of Risk Factors Next
Period
Ranks and Choose VaR as in Historical
Simulation

One Number

Sub-Additive

Historical Data

No Measure of Maximum Loss

Holding period
–
–

Risk environment
Portfolio constancy/liquidity
Confidence level
–
–
–
How far into the tail?
VaR use
Data quantity

Benchmark comparison
– Interested in relative comparisons across
units or trading desks

Potential loss measure
– Horizon related to liquidity and portfolio
turnover

Set capital cushion levels
– Confidence level critical here
Uninformative about extreme tails
 Bad portfolio decisions

–
–
–
Might add high expected return, but high loss
with low probability securities
VaR/Expected return, calculations still not well
understood
VaR is not Sub-additive

A sub-additive risk measure is
Risk ( A  B )  Risk ( A )  Risk (B )
Sum of risks is conservative
(overestimate)
 VaR not sub-additive


– Temptation to split up accounts or firms