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Financial Econometrics

(EF5070)

Week 11

November 13/17, 2023

Financial Econometrics (EF5070) | Dr. Ferenc Horvath 1/14

Contents

• Extreme Values, Quantiles, and Value at Risk • Tsay, Chapter 7, Sections 7.1-7.4.

Financial Econometrics (EF5070) | Dr. Ferenc Horvath 2/14

Value at Risk (VaR)

• We are at time ��, and we are interested in the potential loss in our portfolio value at time �� + l, where l ∈ R>0.

• Let �� l denote the loss in our portfolio value (in dollars) at time �� + l.

• Definition: the Value at Risk (VaR) with tail probability �� is the loss such that the probability that we will suffer a greater loss than this (or a loss exactly equal to this) is ��. I.e.,

�� = Pr �� l ≥ �� .

• Note: the VaR with tail probability �� is the 1 − �� h quantile of �� l . (To be precise, this holds only if �� l is continuous. Otherwise, whether this holds or not depends on whether equality is allowed in the VaR definition and in the definition of quantiles. For the rest of this lecture, we will assume that �� l is continuous.)

Financial Econometrics (EF5070) | Dr. Ferenc Horvath 3/14

Value at Risk (VaR)

• KeyelementsofVaRcalculation:

• the tail probability �� (usually 5% or 1% for risk

management and 0.1% for stress testing);

• the time horizon l (1 day, 1 week, 1 month, 1 year, 5 years, etc.);

• frequency of available data (which might be different from the time horizon l);

• distribution of the loss �� l .

Financial Econometrics (EF5070) | Dr. Ferenc Horvath 4/14

Value at Risk (VaR)

• Some practical considerations regarding VaR calculation:

• When VaR is calculated in terms of continuously-compounded returns (i.e., log returns) ��, the VaR in terms of dollars is often calculated by the approximation �� × �� , instead of the precise way of �� × �� − 1 .

• In principle, the predictive distribution used during the VaR calculation should account for parameter uncertainty. In practice, paramter uncertainty is usually ignored and the conditional predictive distribution is used.

• VaR does not say anything about the loss one might suffer if the loss happens to be greater than the VaR value. Hence, other risk measures also need to be used in addition to VaR, e.g., Expected Shortfall (a.k.a. Conditional Value at Risk (CVaR)).

Financial Econometrics (EF5070) | Dr. Ferenc Horvath 5/14

RiskMetrics

• RiskMetrics is a model of portfolio returns developed by J.P. Morgan, in the framework of which VaR can be determined in an intuitive and simple way.

• Denoting the daily log return by �� and the information available at time �� − 1 by ��

�� , RiskMetrics assumes that �� |�� ~�� 0, ��2 , where ��−1 �� −1 ��

��2=��2+1−��2, ��∈0,1.

�� −1 ��−1

• That is, RiskMetrics assumes that the dynamics of the daily log return can be described by

• only a constant (i.e., ARMA(0,0)) in the mean equation and

• an IGARCH(1,1) (and no constant) in the volatility equation.

• A very elegant property of this model is that ��|��~��0,��2 ∀��∈N ,

where�� denotesthe��-periodlogreturn. ��

�� +1 >0

Financial Econometrics (EF5070) | Dr. Ferenc Horvath 6/14

RiskMetrics

• I.e., in the RiskMetrics model, the conditional variance of the log return is proportional to the investment horizon. (For the derivation of this result, see Tsay, pp. 329.)

• Hence,

�� = ��×��1,

where �� is the VaR corresponding to a ��-period investment horizon. This is also referred to as the square root of time rule in VaR calculation under RiskMetrics.

• When the portfolio consists of several subportfolios with correlated log returns and each log return follows an IGARCH(1,1) model as previously described, the portfolio VaR can be calculated as

��

��= ෍��2+2෍��,

��

where �� is the correlation between the log returns of portfolios �� and ��.

��=1

��<��

��

Financial Econometrics (EF5070) | Dr. Ferenc Horvath 7/14

Expected Shortfall

• Definition: the Expected Shortfall (ES) is the expected value of the loss, conditional on the loss being greater than the VaR.

• For example, in the RiskMetrics model, where �� |�� ~�� 0, ��2 , it ��−1 ��

can be shown that

�� =��1−��, 1−��

where �� ⋅ is the probability density function (PDF) of the standard normal distribution.

Financial Econometrics (EF5070) | Dr. Ferenc Horvath 8/14

Quantile estimation

• Thequantile(andhencetheVaR)canbeestimatedby the quantile of the empirical distribution. I.e.,

• arrangetheobservedlogreturnsinascendingorder,

• then find the log return such that 100 × �� % of the

observed returns are lower than or equal to and

100 × 1 − �� % of the observed returns are higher than that return.

• If necessary, interpolate linearly between two observed log returns.

Financial Econometrics (EF5070) | Dr. Ferenc Horvath 9/14

Quantile estimation

• Advantagesofthisapproach:

• itdoesnotassumeaparticularformofdistribution,and • itisverysimple.

• Disadvantagesofthisapproach:

• it works well only when the sample is representative for

the future;

• forsmalltailprobabilities,itisnotanefficientestimate.

Financial Econometrics (EF5070) | Dr. Ferenc Horvath 10/14

Quantile regression

• An alternative approach to estimate the quantile (and hence the VaR) is called quantile regression.

• This approach assumes that the variable in question (e.g., the log return) is the sum of a deterministic function of observable (explanatory) variables and a zero-mean random variable (i.e., a noise).

• For example,

where �� is the log return at time ��, �� is a vector of explanatory variables

�� = ��′�� + �� , ��

��

observable at time �� − 1, �� is a vector of parameters, and �� is a random

variable the conditional mean of which is zero and which exhibits no autocorrelation.

Financial Econometrics (EF5070) | Dr. Ferenc Horvath 11/14

Quantile regression

• Then, the �� quantile of the conditional distribution of �� is estimated by ��

where

��ො|�� =��, ��−1 ��

��

�� =argmin෍�� −��′�� , ��

and �� ⋅ ��

is defined by

�� =ቊ��,

��

�� − 1 ��,

��≥0 �� < 0.

��=1

Financial Econometrics (EF5070) | Dr. Ferenc Horvath 12/14

Quantile regression

Source of image: https://scikit-learn.org/stable/auto_examples/linear_model/plot_quantile_regression.html

Financial Econometrics (EF5070) | Dr. Ferenc Horvath 13/14

Quantile regression

Source of image: https://en.wikipedia.org/wiki/Quantile_regression#/media/File:Quantilsregression.svg

Financial Econometrics (EF5070) | Dr. Ferenc Horvath 14/14