Quantitative Risk Management
MTH5510
Lecture 2
Hasan Fallahgoul
hasan.fallahgoul@monash.edu
Master of Financial Mathematics
Monash University
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Last Lecture
Course details and outline
Defined different types of financial risks
Very brief history of the growth of risk management
Basic concepts
Portfolio value V
Loss L(t, t + ∆) = −(V (t + ∆)− V (t))
Risk factors Vt = f (t,Zt) and risk factor changes
Xt+∆ = Zt+∆ − Zt
Linearized loss
Lδ(t, t + ∆) = −
(
∂t f (t,Zt)∆ +
∑d
i=1 ∂Zi f (t,Zt)Xi,t+∆
)
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This Lecture
Discuss Risk Measures
Introduce common and important examples, VaR and ES
Describe some standard methods for computing these in a
market risk context
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Risk Measures
Risk measures are used for:
Determining risk capital and capital adequacy
Institution needs to hold capital as a buffer against unexpected
future losses
Required by regulators over concerns of solvency
Management tool
Risk measurements can be used to put constraints on units
within a firm
Insurance premiums
Premiums are designed to compensate a company for bearing
the risk of the claim
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Types of Risk Measures
Notional-amount approach
Sum of notional values of each security in the portfolio
Possibly weighted by the asset’s class depending on riskiness
Still used in some standard approaches
Very simple to use
Does not distinguish between long and short positions, does
not use netting
Does not respond well to diversification
Does not handle portfolios of derivatives easily
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Types of Risk Measures
Factor-sensitivity measures
Change in portfolio value for a given change in a risk factor
Typically represented by a derivative: ∂f∂Z
The Greeks are a common example
Can not measure overall riskiness of a portfolio
Not possible to aggregate sensitivities of different risk factors
Can not be aggregated over different types of markets
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Types of Risk Measures
Risk measures based on loss distributions
Statistical quantities describing the distribution of the portfolio
over some horizon ∆
Examples:
Variance
Value-at-Risk
Expected shortfall
Aggregation makes sense
Netting and diversification can be handled
Distributions often estimated from historical data
Difficult task to estimate the distribution
Should be complemented by information from hypothetical
scenarios
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Types of Risk Measures
Scenario-based risk measures
Considers several specific future risk-factor change scenarios
The loss is measured according to each scenario
Risk is set equal to the maximum possible weighted loss
max{w1L(X1), . . . ,wnL(Xn)}
Can be interpreted in a different way:
max{EP[L(X)] : P ∈ P}
This type of quantity is called a Generalized Scenario
Useful when the set of risk factors is small
Provide useful complementary information to measures based
on loss distribution
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Value-at-Risk
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Value-at-Risk
Most widely used risk measure
Let FL(x) = P(L ≤ x) and let α ∈ (0, 1). Definition of
Value-at-Risk:
VaRα = inf{x ∈ R : P(L > x) ≤ 1− α}
= inf{x ∈ R : FL(x) ≥ α}
α is the confidence level
VaRα is the smallest number, x , such that the probability of a
loss greater than x is no larger than 1− α
VaRα is a quantile of the loss distribution
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Value-at-Risk
Typical values of α are 0.95 or 0.99
In market risk contexts, typical values of ∆ are 1 or 10 days
In credit and operational risk contexts, typical value of ∆ is 1
year
VaRα gives no information about the magnitude of losses that
occur with probability less than 1− α
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Visualizing VaRα - Example 1
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Visualizing VaRα - Example 1
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Visualizing VaRα - Example 2
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Visualizing VaRα - Example 2
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Visualizing VaRα - Example 3
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Visualizing VaRα - Example 3
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Quantile Function
For an increasing function T : R→ R, the generalized inverse of
T is
T←(y) = inf{x ∈ R : T (x) ≥ y}
If F is a cumulative distribution function, then F← is called the
quantile function of F . For α ∈ (0, 1), the α-quantile of F is
qα(F ) = F
←(α) = inf{x ∈ R : F (x) ≥ α}
If F is continuous and strictly increasing, then F← = F−1.
In general:
x0 = qα(F ) ⇐⇒ F (x0) ≥ α and F (x) < α for all x < x0
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mean-VaR
Let µ be the mean of the loss distribution
mean-VaR is defined as
VaRmeanα = VaRα − µ
The difference between the two is small when dealing with
market risk
More relevant in credit risk when horizons are generally longer
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VaR for normal distribution
Suppose L ∼ N (µ, σ2), and fix α ∈ (0, 1), then
VaRα = µ+ σΦ
−1(α)
VaRmeanα = σΦ
−1(α)
Proof: since FL is strictly increasing and continuous, we need
FL(VaRα) = α:
P(L ≤ VaRα) = P
(
L− µ
σ
≤ Φ−1(α)
)
= Φ(Φ−1(α))
Typical values:
Φ−1(0.95) = 1.64
Φ−1(0.99) = 2.33
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VaR for scaled Student t distribution
Suppose L ∼ t(ν, µ, σ). This means L−µσ has the Student t
distribution with ν degrees of freedom.
E[L] = µ
V[L] = νσ2ν−2
VaRα = µ+ σt
−1
ν (α)
VaRmeanα = σt
−1
ν (α)
Here, tν is the cumulative distribution function of the standard
Student t distribution with ν degrees of freedom.
Such an expression for VaR holds for any location-scale family.
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Comments on VaR
VaR is not subadditive:
If L = L1 + L2, it is not always true that
qα(FL) ≤ qα(FL1) + qα(FL2)
Contradicts the notion that diversification should reduce risk
The computation of VaR is subject to model risk:
Much more pronounced at high confidence levels (α = 0.9999)
VaR neglects problems with market liquidity. Illiquidity is
considered by risk managers as one of the most important sources
of model risk.
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Comments on VaR
The horizon parameter ∆ should reflect the period over which the
institution is committed to hold its portfolio.
Affected by contractual and legal constraints, and liquidity
considerations
Typically varies across markets
For overall risk management, it should be chosen to reflect its
main exposures
Also affected by computation method
Linearized loss requires small ∆
If we assume the composition of the portfolio is constant, we
should take small ∆
Statistical models for X t+∆ will be easier to calibrate for small
∆
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Comments on VaR
Different confidence parameters α are appropriate for different
purposes.
The Basel committee proposes α = 0.99 and ∆ = 10 days for
market risk
A bank would typically take α = 0.95 and ∆ = 1 day for
trader limits
Backtesting models should use lower values of α to ensure
more observations that breach VaR
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Expected Shortfall
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Expected Shortfall
Expected shortfall is often preferred to VaR by many risk
managers. Suppose E[|L|] <∞, and let FL be the CDF of L.
ESα =
1
1− α
∫ 1
α
qu(FL)du
=
1
1− α
∫ 1
α
VaRu(L)du
If FL is continuous, we can also write:
ESα =
E[L1L≥qα(L)]
1− α
= E[L|L ≥ VaRα]
which is the expectation of the loss conditioned on the loss being
at least as large as VaR.
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Expected Shortfall for normal distribution
Suppose L ∼ N (µ, σ2) and fix α ∈ (0, 1).
ESα = µ+ σ
φ(Φ−1(α))
1− α
Typical values:
φ(Φ−1(0.99))
1− 0.99 = 2.67
φ(Φ−1(0.95))
1− 0.95 = 2.06
This confirms that ESα ≥ VaRα for normal distributions.
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Expected Shortfall for scaled Student t distribution
Suppose L ∼ t(ν, µ, σ) and fix α ∈ (0, 1).
ESα(L) = µ+ σESα(L˜)
where L˜ = L−µσ has the Student t distribution with ν degrees of
freedom.
ESα(L˜) =
(
gν(t
−1
ν (α)
1− α
)(
ν + (t−1ν (α))2
ν − 1
)
where tν is the CDF and gν is the PDF for the Student t with ν
degrees of freedom.
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Law of Large Numbers for ESα
Let (Li )i∈N be a sequence of iid random variables with CDF FL.
Then
lim
n→∞
∑bn(1−α)c
i=0 Li ,n
bn(1− α)c+ 1 = ESα a.s.
where L0,n ≥ · · · ≥ Ln−1,n are the order statistics of L0, . . . , Ln−1.
In English: if we have a large number of observations of a loss
distribution, we can approximate ESα by averaging the largest
(1− α) fraction of them.
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Standard Methods
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Standard Market Risk Methods
Variance-covariance method
Historical simulation
Monte Carlo
Backtesting
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Variance-Covariance Method
Rick factor changes Xt+∆ ∼ N (µ,Σ). We consider the linearlized
loss and assume it is an accurate representation of the loss. The
general form will be:
Lδt,t+∆ = −(c + b · Xt+∆)
for some constant c and vector b. This implies:
Lδt,t+∆ ∼ N (−c − b · µ,bTΣb)
Both VaRα and ESα can now be easily calculated using formulas
from earlier.
VaRα = −c − b · µ+
√
bTΣbΦ−1(α)
ESα = −c − b · µ+
√
bTΣb
φ(Φ−1(α))
1− α
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Variance-Covariance Method
Strengths
Closed form expression for VaRα and ESα
Fast and easy to implement
Extends to multivariate distributions that are closed under
linear operations
Weaknesses
Linearization may not always be a suitable approximation
Normal distribution of changes not likely to be the true
distribution
Often leads to underestimation of these risk measures
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Historical Simulation
Consider the past n observations of the risk factor changes
X˜t−(n−1)∆, . . . , X˜t . If we apply each of these observed risk factor
changes to our current portfolio, we have a sequence of realizations
of the loss L.
From the sequence of losses (L˜i )
n−1
i=0 , the order statistics are found,
L˜0,n ≥ · · · ≥ L˜n−1,n.
We estimate VaRα as L˜bn(1−α)c,n
This is equivalent to using a distribution for L which is equal to the
empirical distribution of (L˜i )
n−1
i=0 .
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Historical Simulation
Strengths
Easy to implement
No statistical estimation of X is necessary
No assumptions about dependence of risk factor changes
Weaknesses
Depends on sufficient quantity of data for all risk factors
Missing data for certain risk factors can reduce the effective
value of n
Previous market behaviour may not be a good indication of
current market behaviour
Can artificially add extreme scenarios to the set of observed
realizations
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Monte Carlo
Generate m independent realizations of risk-factor changes,
denoted X˜0t+∆, . . . , X˜
m−1
t+∆ .
In a similar fashion to historical simulation, each of these changes
is applied to the current portfolio to generate a sequence of loss
realizations, (L˜i )
m−1
i=0 .
The distribution of Xt+∆ is generally calibrated to historical data
from a finite number of points, X˜t−(n−1)∆, . . . , X˜t .
But then m may be chosen as large as desired, keeping
computational constraints in mind.
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Monte Carlo
Strengths
Easy to implement
Allows for flexibility in distribution choice for Xt+∆
Can achieve any desired level of accuracy within limits of
computation
Weaknesses
The specific choice of distribution is important and can have
significant effects of results
For large portfolios, the computational cost can be significant
This is particularly significant if there are derivatives in the
portfolio that can not be evaluated analytically
Variance reduction techniques can help