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Chapter 3Value at Risk
This chapter deals with risk measures and financial data, including quantilerisk measures and value at risk (VaR).
Contents3.1 Financial Data with R . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3 Quantile Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . 473.4 Value at Risk (VaR) . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1 Financial Data with R
Package quantmod for financial data
The R package quantmod can be installed and run via the following com-mand:
install.packages("quantmod")library(quantmod)getSymbols("1800.HK",from="2007-01-03",to="2011-12-02",src="yahoo")stock=Ad(`1800.HK`)chartSeries(stock,up.col="blue",theme="white")stock.rtn=diff(log(Ad(`1800.HK`)))chartSeries(stock.rtn,up.col="blue",theme="white")n = sum(!is.na(stock.rtn))
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5
10
15
20
stock [2007−01−03/2011−12−02]
Last 5.565
Jan 032007
Jan 022008
Jan 012009
Jan 012010
Jan 032011
Dec 022011
Fig. 3.1: Cumulative stock returns.
−0.2
−0.1
0.0
0.1
0.2
stock.rtn [2007−01−04/2011−12−02]
Last 0.0130224303670248
Jan 042007
Jan 022008
Jan 012009
Jan 012010
Jan 032011
Dec 022011
Fig. 3.2: Stock returns.
3.2 Risk Measures
Types of risk [22] include market risk, liquidity risk, credit risk, counterpartyrisk, model risk, estimation risk. For insurance businesses, a more detailedclassification can be set as follows.
1. Financial risk
a. Investment riski. Credit riskii. Market risk (e.g. depreciation)
b. Liability riski. Catastrophe riskii. Non-catastrophe risk (e.g. claim volatility)
2. Operational risk
a. Business risk (e.g. lower production)b. Event risk (e.g. system failure).
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Value at Risk
The potential loss asssociated to any of the above risks will will be modelledvia a random variable X.
Definition 3.1. A risk measure is a mapping that assigns a value VX to agiven random variable X.
For insurance companies, which need to hold a capital in order to meet futureliabilities, the capital CX required to face the risk induced by a potential lossX can be defined as
CX := VX − LX ,
where
a) LX represents the (positive or negative) liabilities of the company, andb) VX stands for a upper “reasonable” estimate of the potential loss associ-
ated to X.
Estimating the liabilities of the company by IE[X], the required capital isgiven by
CX = VX − IE[X].
getSymbols("^HSI",from="2013-06-01",to="2014-10-01",src="yahoo")returns <- as.vector(diff(log(Ad(`HSI`))))times=index(Ad(`HSI`))m=mean(returns[returns<0],na.rm=TRUE)plot(times,returns,pch=19,cex=0.4,col="blue", ylab="", xlab="", main = '')segments(x0 = times, x1 = times, y0 = 0, y1 = returns,col="blue")abline(h=m,col="red",lwd=3)sum(!is.na(returns))
2014
−0.03
−0.02
−0.01
0.000.01
0.02
Fig. 3.3: Estimating liabilities by the mean for 346 returns.
Example: Guaranteed Maturity Benefits
Variable annuity benefits offered by insurance companies are usually pro-tected via different mechanisms such as Guaranteed Minimum Maturity Ben-
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efits (GMMBs) or Guaranteed Minimum Death Benefits (GMDBs). The com-putation of the corresponding risk measures is an important issue for thepractitioner in risk management.
Given a fund value process (Ft)t∈R+ , an insurer is continuously chargingannualized mortality and expense fees at the rate m from the account ofvariable annuities, resulting into a margin offset income Mt given by
Mt := mFt t ∈ R+.
Denoting by τx the future lifetime of a policyholder at the age x, the futurepayment made by the insurer at maturity T is
(G− FT )+1τx>T
where G is the guarantee level expressed as a percentage of the initial fundvalue F0, δ is a roll-up rate according to which the guarantee increases up tothe payment time. In this case, the random variable X is taken equal to
X := e−rT (G− FT )+1τx>T −
w min(T,τx)
0e−rsMsds.
Coherent risk measures
Definition 3.2. A risk measure V is said to be coherent if it satisfies thefollowing four properties, for any two random variables X, Y :
i) Monotonicity:X 6 Y =⇒ VX 6 VY ,
ii) (Positive) homogeneity:
VλX = λVX , λ > 0,
iii) Translation invariance:
VX+µ = µ+ VX , µ > 0,
iv) Subadditivity:VX+Y 6 VX + VY .
Subadditivity means that the combined risk of several portfolios is lower thanthe sum of risks of those portfolios, as should happen in portfolio diversifica-tion..
The expectation of random variables is an example of coherent risk measure(also called pure premium risk measure)
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Value at Risk
VX := IE[X],
which is additive. More generally, any risk measure of the form
MX = IE[XfX(X)],
cf. e.g. (4.2) below, where fX is a non-decreasing function satisfying fX(λx) =fX(x), and fX(µ+ x) = fX(x), x ∈ R, λ, µ > 0, will be monotone, homoge-neous, and translation invariant.
3.3 Quantile Risk Measures
The cumulative distribution function of a random variable X is the function
FX : R −→ [0, 1]
defined byFX(x) := P(X 6 x), x ∈ R.
A random variableX with cumulative distribution given in Figure 3.4 satisfies
P(X = 0) = 0.25 > 0.
The cumulative distribution function satisfies the following properties:
i) x −→ FX(x) is non-decreasing,ii) x −→ FX(x) is right-continuous,iii) limx→∞ FX(x) = 1,iv) limx→−∞ FX(x) = 0.
Fig. 3.4: Cumulative distribution function with jumps.∗
Definition 3.3. Given X a random variable with cumulative distributionfunction FX : R −→ [0, 1] and a level p ∈ (0, 1), the p-quantile of X defined∗ Picture taken from http://www.probabilitycourse.com/.
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byqpX := infx ∈ R : P(X 6 x) > p.
Performance analytics in R - quantiles of known distributions
The quantiles of various distributions can be obtained in R. For example,
qnorm(.95, mean=.5, sd=1)
shows that the 95%-quantile of a N (0.5, 1) Gaussian random variable is2.144854. On the other hand, the instruction
qt(.90, df=5)
shows the 90%-quantile of a Student t-distributed random variable with 5degrees of freedom is 1.475884.
Performance analytics in R - empirical CDF
The empirical cumulative distribution function can be estimated as
FN (x) := 1N
N∑i=1
1xi6x, x ∈ R.
getSymbols("^STI",from="1990-01-03",to="2015-02-01",src="yahoo")stock.rtn <- as.vector(diff(log(Ad(`STI`))))stock.ecdf=ecdf(as.vector(stock.rtn))plot(stock.ecdf, xlab = 'Sample Quantiles', ylab = '',main = 'Empirical Cumulative Distribution
',col='blue')
−0.3 −0.2 −0.1 0.0 0.1 0.2 0.3
0.00.2
0.40.6
0.81.0
Empirical Cumulative Distribution
Sample Quantiles
Fig. 3.5: Empirical cumulative distribution function.
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Value at Risk
Note that the empirical distribution function has a visible discontinuity at 0,whose height 0.03967611 is given by
sum(!is.na(stock.rtn[stock.rtn==0]))/sum(!is.na(stock.rtn))
3.4 Value at Risk (VaR)
The Value at Risk has two objectives:
i) to provide a measure for risk, andii) to determine an adequate level of capital reserves that matches the cur-
rent level of risk.
In other words, managing risk means here determining a level VX of provisionor capital requirement that will not be “too much” exceeded by X.
In this respect, the probability P(X > V ) that X exceeds the level V is ofa capital importance. Setting V such that for example
P(X 6 V ) > 0.95, i.e. P(X > V ) 6 0.05,
means that insolvency will occur with probability less that 5%.
The 95%-quantile risk measure is the smallest value of V such that
P(X 6 V ) > 0.95, i.e. P(X > V ) 6 0.05.
More precisely, we have the following definition.
Definition 3.4. The Value at Risk V pX at the level p ∈ (0, 1) is the p-quantileof X as defined by
V pX := infx ∈ R : P(X 6 x) > p.
Note that V pX may also be negative, in which case it identifies to a potentialprofit rather than to a liability. On the other hand, the Value at Risk V pXdoes not contain any information on how large losses can be beyond V pX .
By definition of V pX , With probability at least p, the value of X is alwayslower than V pX , i.e. we have
FX(V pX) = P(X 6 V pX) > p.
The function p 7−→ V pX is a non-decreasing function of p ∈ [0, 1], and it is thegeneralized inverse of the cumulative distribution function of X, defined as
x 7−→ FX(x) := P(X 6 x), x ∈ R.
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Since the cumulative distribution function of X is non-decreasing, its general-ized inverse p 7−→ V pX is nondecreasing, left-continuous, and it admits limitson the right, see e.g. Proposition 2.3-(2) of [18]. In general we have, for allx ∈ R,
V pX 6 x ⇐⇒ P(X 6 x) > p, (3.1)
and, for all x ∈ R such that P(X = x) = 0,
V pX = x ⇐⇒ p = P(X 6 x).
In particular, if P(X = V pX) = 0 then
p = P(X 6 V pX) and 1− p = P(X > V pX). (3.2)
a) Monotonicity. If X 6 Y then
P(Y 6 x) = P(X 6 Y 6 x) 6 P(X 6 x), x ∈ R,
henceP(Y 6 x) > p =⇒ P(X 6 x) > p, x ∈ R,
which shows thatV pX 6 V
pY ,
hence Value at Risk satisfies the monotonicity property of coherent riskmeasures.
b) Positive homogeneity and translation invariance. Value at Risk also sat-isfies the positive homogeneity and translation invariance properties ofcoherent risk measures, in addition to monotonicity. Indeed, for all b > 0and a ∈ R we have
V paX+b = infx ∈ R : P(aX + b 6 x) > p= infx ∈ R : P(X 6 (x− b)/a) > p= infay + b ∈ R : P(X 6 y) > p= b+ a infy ∈ R : P(X 6 y) > p= b+ aV pX .
c) Subadditivity and coherence. Although Value at Risk satisfies monotonic-ity, positive homogeneity and translation invariance, it is not subadditivein general. Namely, the Value at Risk V pX+Y of X +Y may be larger thanthe sum V pX +V pY . Therefore, Value at Risk is not a coherent risk measure.Proof. We show that Value at Risk is not subadditive by considering twoindependent Bernoulli random variables X,Y ∈ 0, 1 with the distribu-tion P(X = 1) = P(Y = 1) = 2%,
P(X = 0) = P(Y = 0) = 98%.
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Value at Risk
hence V 0.975X = V 0.975
Y = 0. On the other hand, we haveP(X + Y = 2) = (0.02)2 = 0.04%,
P(X + Y = 1) = 2× 0.02× 0.98 = 3.92%,
P(X + Y = 0) = (0.98)2 = 96.04%,
henceV 0.975X+Y = 1 > V 0.975
X + V 0.975Y = 0.
Nevertheless, Value at Risk is sub-additive (and hence coherent) on (notnecessarily independent) Gaussian random variables. Indeed, by (3.5), forany two random variables X and Y we have
σ2X+Y = Var[X + Y ]= IE[(X + Y )2]− (IE[X + Y ])2
= IE[X2] + IE[Y 2] + 2 IE[XY ]− IE[X]2 − IE[Y ]2 − IE[X] IE[Y ]= Var[X] + Var[Y ] + 2(IE[XY ]− IE[X] IE[Y ])= Var[X] + Var[Y ] + 2(IE[(X − IE[X])(Y − IE[Y ])] (3.3)6 Var[X] + Var[Y ] + 2
√IE[(X − IE[X])2]
√IE[(Y − IE[Y ])2] (3.4)
=(√
Var[X] +√
Var[Y ])2,
where from (3.3) to (3.4) we applied the Cauchy-Schwarz inequality.
Hence in particular when X and Y are Gaussian, by (3.5) we get
V pX+Y = µX+Y + σX+Y qpZ
= µX + µY + σX+Y qpZ
6 µX + µY + (σX + σY )qpZ= V pX + V pY .
Examples
i) Discrete distribution.
Consider X ∈ 10, 100, 110 with the distribution
P(X = 10) = 90%, P(X = 100) = 9.5%, P(X = 110) = 0.5%.
Compute V 99%X .
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ii) Gaussian distribution.
Consider X ' N (µ, σ2) written as
X = µ+ σZ
where Z ' N (0, 1) is a standard normal random variable. We have
V pX = µ+ σqpZ = µ+ σV pZ (3.5)
where the normal quantile qpZ at level p satisfies P(Z > qpZ) = p.
iii) Exponential distribution.
For example, if X has an exponential distribution with parameter λ > 0and mean 1/λ, we have
P(X 6 x) = 1− e−λx, x > 0,
and we find
V pX := infx ∈ R : P(X 6 x) > p
= − 1
λlog(1− p) = IE[X] log 1
1− p .
When p = 95% this yields
V pX ' 2.996 IE[X].
Estimating the liabilities of the company by IE[X], we check that therequired capital CX satisfies
CX = V pX − IE[X] = IE[X] log 11− p − IE[X],
i.e.CX = V 95%
X − IE[X] ' 1.996 IE[X],
which means doubling the estimated amount of liabilities.
iv) Estimating risk probabilities from moments.
We note that in general, by the Chebyshev inequality, for every r > 0we have
xrP(X > x) = xr IE[1X>x
]6 IE
[Xr
1X>x]
6 IE [Xr] ,
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Value at Risk
henceP(X 6 x) > 1− 1
xrIE[Xr], x > 0,
and
V pX = infx ∈ R : P(X 6 x) > p
6 inf
x ∈ R : 1− 1
xrIE[Xr] > p
= inf
x ∈ R : xr >
11− p IE[Xr]
=(IE[Xr]1− p
)1/r
=‖X‖Lr(Ω)
(1− p)1/r .
For example, with p = 95% and r = 1 we get
V 95%X 6
11− p IE[X] = 20 IE[X].
To summarize, a smaller Lr-norm of X tends to make the value at riskVX smaller.
Performance analytics in R - Value at Risk
We are using the PerformanceAnalytics R package, which can be installedvia the commands
install.packages("PerformanceAnalytics")library(PerformanceAnalytics)chart.CumReturns(stock.rtn,main="Cumulative Returns")chartSeries(stock,up.col="blue",theme="white")VaR(stock.rtn, p=.95, method="historical")
The historical 95%-Value at Risk over N samples (xi)i=1,...,N can be es-timated using by inverting the empirical cumulative distribution functionFN (x), x ∈ R. It is found equal to V 95%
X = −0.05121668.
VaR(stock.rtn, p=.95, method="gaussian")
The Gaussian 95%-Value at Risk is estimated from (3.5) by
V 95%X = µ+ σqpZ ,
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where µ = IE[X] and σ2 = Var[X], and is found equal to V 95%X =
−0.05222171. It can be recovered up to approximation as
m=mean(stock.rtn,na.rm=TRUE)s=sd(stock.rtn,na.rm=TRUE)q=qnorm(.95, mean=0, sd=1)m-s*q
which yields −0.05224278.
The next lemma is useful for random simulation purposes, and it will also beused in the proof of Proposition 4.4 below.
Lemma 3.5. Any random variable X can be represented as X = V UX whereU is uniformly distributed on [0, 1].
Proof. It suffices to note that by (3.1) we have
P(V UX 6 x) = P(U 6 P(X 6 x)) = P(X 6 x) = FX(x), x ∈ R.
Exercises
Exercise 3.1 Consider a random variable X having the Pareto distributionwith probability density function
fX(x) = γθγ
(θ + x)γ+1 , x ∈ R+.
a) Compute the cumulative distribution function
FX(x) :=w x
0fX(y)dy, x ∈ R+.
b) Compute the value at risk V pX at the level p for any θ and γ, and then forp = 99%, θ = 40 and γ = 2.
Exercise 3.2 Consider two independent random variables X and Y withsame distribution given by
P(X = 0) = P(Y = 0) = 90% and P(X = 100) = P(Y = 100) = 10%.
a) Plot the cumulative distribution function of X on the following graph:
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Value at Risk
0.88
0.90
0.92
0.94
0.96
0.98
1.00
1.02
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210−10−20x
FX(x)
Fig. 3.6: Cumulative distribution function of X.
b) Plot the cumulative distribution function of X+Y on the following graph:
0.82
0.84
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
1.02
1.04
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210−10−20x
FX+Y (x)
Fig. 3.7: Cumulative distribution function of X + Y .
c) Give the values at risk V 99%X+Y , V 95%
X+Y , V 90%X+Y .
d) Compute the Tail Value at Risk
η90%X := 1
1− pw 1
pV qXdq
at the level p = 90%.e) Compute the Tail Value at Risk
ηpX+Y := 11− p
w 1
pV qX+Y dq
at the levels p = 90% and p = 80%.
Exercise 3.3 Consider a random variable X with the cumulative distributionfunction
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0.96
0.97
0.98
0.99
1.00
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160−10−20x
FX(x)
Fig. 3.8: Cumulative distribution function of X.
a) Give the value of P(X = 100).b) Give the value of V qX for all q in the interval [0.97, 0.99].c) Compute the value of V qX for all q in the interval [0.99, 1].
Hint: We have
FX(x) = P(X 6 x) = 0.99 + 0.01× (x− 100)/50, x ∈ [100, 150].
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