Download CONDITIONAL TAIL VARIANCE AND

CONDITIONAL TAIL VARIANCE AND CONDITIONAL TAIL SKEWNESS IN FINANCE AND INSURANCE Liang Hong * , Assistant Professor Bradley University Ahmed Elshahat, Assistant Professor Bradley University ABSTRACT Two risk measures Value­at­Risk (VaR) and Conditional­Value­at­Risk (CVaR) have been widely accepted in both finance and insurance fields. In this paper, we propose two new downside risk measures: Conditional­Tail­Variance (CTV) and Conditional­Tail­ Skewness (CTS). We explain why these measures provide information not embraced by VaR nor CVaR, and how we can apply them to better handle extreme events. For theoretical and pedagogical purposes, we derive the formulas of both CTV and CTS under arbitrary distribution assumption. Thus, any particular distribution such as log­ normal, gamma, Pareto or Weibull is just a special case. In particular, we give the formulas for lognormal stock prices. For the use of empiricists in both finance and insurance fields, we also give the estimates of CTV and CTS, and discuss associated numerical methods such as the bootstrapping method and the Monte Carlo simulation. INTRODUCTION Value­at­Risk (VaR) is one of the most well­known downside risk measurement used in the finance and insurance fields. The VaR measures the predicted maximum loss at a pre­specified probability level, over a certain time horizon. Since the year 2000, many academic researchers have investigated the VaR as a measure of downside risk (Alexander and Baptista (2002); Mittnik, Rachev and Schwartz (2002); Pflug and Gaivoronski (2005); Yamai and Yoshiba (2005). Before defining the VaR, two dimensions need to be decided upon. The first dimension is the time interval used in computing the VaR. This time interval reflects the investment horizon, or the time period before revising the portfolio. The second is the confidence level α, which is the probability that the VaR is not expected to exceed a certain maximum loss. VaR can be defined mathematically in several ways. The following notations are used in standard references:
VaR a = inf {x ³ 0 : P [L £ x ] ³ a }
(1) where VaRα is the portfolio's (security's) Value­at­risk with α confidence level. L is the possible portfolio loss variable. x is an arbitrary small value of the portfolio (security). P is the probability function. α is the confidence level. The above equation states that the VaR for a confidence level α is the value x such that the probability that the possible loss L exceeds x is at most some small number 1­α. Even with the VaR, a number of drawbacks exist. These drawbacks can be summarized as follows (Rau­BredowS (2004)): * Corresponding author email: [email protected]. The authors thanks Dr. Ryan Martin for a fruitful discussion on the bootstrapping method.
subadditivity, non­smooth and non­convex function, and ignoring the magnitude of losses beyond the VaR value. First, the subadditivity problem refers to the fact that the VaR computed for a portfolio may be higher than the weighted average of the VaR computed for the individual securities in the optimal portfolio. Thus, due to the subadditivity, the VaR penalizes the portfolio construction and it negates the portfolio diversification concept. Second, the VaR leads to a non­smooth and non­convex function, which will hinder the optimizer from finding the optimality condition. The VaR leads to multiple stationary points that complicate the optimization process (Grootveld and Hallerbach (2004)) Third, the VaR only computes the probable losses until the VaR value; it does not consider the magnitude of the losses beyond the VaR value. All these drawbacks lead to the creation of a new measure that computes the conditional variant of the VaR, namely Conditional Value­at­Risk (CVaR). The CVaR, also known as Conditional­Tail­ Expectations (CTE), measures the expected losses in the tail of the distribution of possible portfolio losses, beyond the portfolio VaR (Artzner et al. (1999)) Thus, the CVaR measures the market risk conditional on the current information. The CVaR can be mathematically defined using the following formula:
CVaRa = E [ X | X > VaR a ]
(2) Manistre and Hancock (2005) gave a good estimate of the variance of CVaR. Lan et al. (2007) develop a two­level simulation procedure that produces a confidence interval for CVaR. A concept that is defined very closely to the CVaR is the expected shortfall (Acerbi and Tasche (2002)). The expected shortfall measurement result has been shown to be equivalent to the CVaR, even though it is defined in a different way. Engle and Manganelli (2004) used a different methodology to compute a different variant of the CVaR. They called their measure conditional autoregressive Value at Risk (CaViaR). Using an autoregressive process, their model specifies the evolution of the CVaR over time. In this paper we go beyond the CVaR and propose two new downside risk measures, the Conditional­Tail­Variance (CTV) and the Conditional­Tail­Skewness (CTS). These two measures provide further improvements over the CVaR by considering not only the expected mean at and beyond the VaR level, but also consider the return variability and skewness. Ignoring the variability and skewness has the potential of underestimating the risk levels. A significant improvement is expected by integrating the first three moments ­ mean, variance, and skewness ­ into the CVaR. We mention that Valdez (2005) studies CTV for elliptically contoured distributions. But his work focused on mathematical aspect only. We will study the general case as well as the application to finance and insurance. The remaining of the paper is organized as follows: Section 2 gives the definitions of CTV and CTS. Section 3 provides the formulas of both CTV and CTS under any arbitrary distribution. Section 4 shows the application of CTV and CTS to log­normal stock price. Section 5 discusses the estimates of CTV and CTS and some numerical methods. Section 6 concludes the paper with a summary and discussion. DEFINITION OF CTV AND CTS Basically, VaR answers the question: Above which level the probability that loss exceeds that level will be less than or equal to α? This means VaR ignores the extreme events. CVaR tells us if things really go wrong (i.e. the loss exceeds that level), then on average what will be the loss. Thus, CVaR takes account of extreme events. To better handle the extreme events in finance and insurance, we are also interested in knowing on average how far away the extreme events can deviate from CVaR. Is CVaR a safe measure to be used to guard the extreme events? However, neither VaR nor CVaR provides the answers to these questions. To answer these questions, we define two new risk measures as follows: Definition 1. Suppose X is a random variable on a given probability space (Ω, F, P), the Conditional­ Tail­Variance (CTV) of X at a given confidence level α, denoted by CTVα , is defined as
CTVa = Var [ X | X > VaR a ]
(3) where VaRα is Value­at­Risk defined above. Definition 2. Suppose X is a random variable on a given probability space (Ω, F, P), the Conditional­ Tail­Skewness (CTS) of X at a given confidence level α, denoted by CTSα , is defined as
CTS a = [
E X 3 | X > VaR a
]
3 / 2 (CTV a ) (4) (provided CTVα >0.) where VaRα is the Value­at­Risk of X at α. Remark. 1. CTV and CTS defined in equations (3) and (4) are fairly general. They are well­defined for discrete, continuous and mixed distributions. 2. Both CTV and CTS depend on α . Usually 1­α is a small number such as 5% or 2.5%. In application, X is often interpreted as the stock price or insurance loss. We see that CTVα is the conditional variance of X when X exceeds VaRα . (i.e. The distribution of X left­truncated by VaRα .) It tells us on average how far away the extreme events deviate from CVaRα. If CTVα is fairly small, one can simply use CVaRα as a benchmark to manage the risks associated with extreme events, because on average each extreme event won't go too far away from CVaRα However, if CTVα is quite large, one does not know whether or not we can use CVaRα as a safe measure for dealing with the risks associated with the extreme events because things can go either way. For a more accurate assessment of risks in this case, we need to look at CTVα and CTS α together. For example, if we know CTVα is fairly large, then Case I: CTSα <0, the tail of the truncated distribution of X at VaRα is left­tailed. Therefore, the extreme events, on average, tend to be less extreme than CVaRα. In this case, using CVaRα as an estimate of average risks is already an overestimate. Case II: CTS α >0, the tail of the truncated distribution of X at VaRα is right­tailed. Therefore the extreme events, on average, tend to be more extreme than CVaRα. We need to have a more conservative estimate than CVaRα . FORMULAS OF CTV AND CTS UNDER ARBITRARY DISTRIBUTION In this section we give the most general formulas of CTV and CTS. Since in most cases, the underlying random variable X we are interested is continuous, we will also give the formulas of CTV and CTS when X is continuous. General Formulas Suppose X is a random variable with cumulative distribution function (CDF) F(x), then from definition 1 and the moments of truncated distributions, we have
¥
æ ¥
ö
x dF ( x ) ç ò xdF ( x ) ÷
ò
ç VaR a
÷
VaR a
CTV a =
-ç
÷
P [ X > VaR a ] ç P [ X > VaR a ] ÷
ç
÷
è
ø
2 2 (5) provided P [ X > VaR a ] > 0 and the integrals exists. Using equations (1) and (2) we can simplify equation (5) to
¥ 2 ò x dF ( x ) CTV a =
VaRa 2 1 - a
- (CVaR a ) (6) Analogously, the formula of CTS can be derived. To avoid involved expression, we express it in terms of CTSα as follows:
¥ 3 ò x dF ( x ) CTS a =
VaRa (7) (1 - a )(CTV a ) 3 / 2 provided CTVα is nonzero and the integral exists. Remark. The integrals in equations (6) and (7) are Lebesuge­Stieltjes integrals. (Interested readers can consult Billingsley (1995), Chow and Teicher (1997), Chung (2000) or Shiryaev (1984) for technical details.) Thus, equations (6) and (7) are applicable, no matter the distribution of X is discrete, continuous or mixed Formulas for Continuous Distribution In both theory and practice, most of the time we are interested in continuous random variables. In other words, we are interested in the case when the underlying random variable X has probability density function (pdf) fX(x). In this case, the measure generated by the CDF F(x) is absolutely continuous with respect to the Lebesgue measure on the real line and the probability density function fX(x) is the Radon­ Nikodym derivative. i.e. dF(x)=fX(x)dx. Thus, equations (6) and (7) can be specialized to
CTV a
ò
=
¥ VaR a x 2 f X ( x ) dx 1 - a
- (CVaR a ) 2 (8) and
¥ 3 ò x f X
CTS a =
( x ) dx VaR a (9) (1 - a )(CTV a ) 3 / 2 APPLICATION TO LOGNORMAL STOCK PRICE In both financial and insurance sectors, lognormal distribution is widely used. (Hardy (2003), Hull (2009) or Shreve (2005)). In this section we show how one can use the results in the previous section when the stock price is assumed to be lognormally distributed. Let's first recall the definition of lognormal random variable and fix some notation. Definition 3. Let X be random variable, then X is lognormally distributed with parameters μ and σ if Y=lnX is normally distributed with mean μ and variance σ which is usually denoted as lnX~N (μ, σ 2 ). In other word, the pdf of Y is given by
f Y ( y ) = ì ( y - m )2 ü
exp íý
2 s 2 þ
2 p s
î
1 (10) Suppose Xt denote the stock price at time t>0, then under the lognormal stock price assumption, we have 1. Yt=(Xt/Xt­1) is lognormally distributed with some parameters μ and σ. 2. The process (Yt)t has independent increments. Without loss of generality, we may consider the case the observation intervals are of unit length. Thus, ln(Xt/Xt­1) are independently and identically distributed (iid) as a normal distribution with mean μ and variance σ 2 . Hence, it's sufficient to compute the CTV and CTS of a normal distribution of mean μ and variance σ 2. Here we assume the parameters μ and σ 2 are known. Otherwise, standard statistical techniques such as method of moments, maximum likelihood estimation and Bayesian estimation can be employed to estimate μ and σ 2 provided suitable data set is available. (Dudewics and Mishra (1988 or Hogg et al (2005)) A good discussion of these techniques from the financial point of view is provided in Hardy (2003). An alternative of return process is mentioned in Resnick (2007). From equations (8) and (10), we have I æ I ö
CTV a = 2 - ç 1 ÷
1 - a è 1 - a ø
2 where
¥
I 1 =
ò
VaR a
¥
I 2 =
ò
VaR a
ì ( x - m )2 ü
exp íýdx 2 s 2 þ
2p s
î
x ì ( x - m )2 ü
exp íýdx 2 s 2 þ
2 p s
î
x 2 Analogously, from equations (9) and (10), we have
CTVa = where
I 3 (1 - a )(CTV a ) 3 / 2 ¥
I 3 =
ò
VaR a
ì ( x - m )2 ü
exp íýdx 2 s 2 þ
2 p s
î
x 3 Thus, the problem is reduced to evaluating the integrals I1, I2 and I3. To evaluate I1, we first rewrite I1 as
¥
I 1 =
ò
ì ( x - m )2 ü
é
æ VaR a - m öù
exp ídx m
1 N ç
÷ú
ý
ê
s
2 s 2 þ
2 p s
è
øû
ë
î
x - m
VaR a where N (× ) denotes the cumulative distribution function (CDF) of the standard normal distribution. Then we do the change of variable by letting Z=(X­μ) 2 . The invariance property of differentiation form yields
2 ìï (VaR a - m )
s 2 I 1 =
exp í2 p
2 s 2 ïî
üï
é
æ VaR a - m öù
÷ú
ý - m ê1 - N ç
s
ïþ
è
øû
ë
In general, integrals like I2 and I3 have no closed forms and have to be evaluated numerically. Most mathematical software such as Maple and Matlab can provide a fairly accurate approximation of I2 and I3. ESTIMATES, BOOTSTRAPPING METHOD AND MONTE­CARLO SIMULATION Estimates of CTV and CTS Suppose we are given a set of financial data X={x1, x2, ..., xn}, we are interested in estimating CTV and CTS. Since we do not know the underlying distribution behind the data, a practical way is to use the sample CTV and CTS as estimates of CTS and CTS. We will use SCTVα and SCTSα to denote the sample CTV and CTS at a confidence level α respectively. To compute SCTVα and SCTSα, we do the following steps: 1. Order the initial data set X={x1, x2, ..., xn}, and let X’={x(1), x(2), ..., x(n)}to denoted the ordered data set. 2. Compute SCTVα as
n æ n ö
ç å x ( i ) ÷
[1 + n a ]
ç [1 + n a ] ÷
SCTV a =
-ç
n (1 - a ) n ( 1 - a ) ÷
çç
÷÷
è
ø
2 å x ( 2 i ) where [x] denotes the closest integer less or equal than x. 3. Then we compute SCTSα as
n 3 ( i ) SCTS a =
å x [ a]
1 + n 3 / 2 n (1 - a ) (SCTV a ) Bootstrapping Method To get an idea of the accuracy of the above estimates, one often needs to know the variances of these two estimates. The variance of the estimates of CTV (or CTS) can be estimated using the powerful bootstrapping method introduced by Efron (1979). Interested readers can consult DasGupta (2008) for details. For brevity, we only give the algorithm for CTV as the algorithm for CTS is completely
analogous. Suppose we are given the financial data set X={x1, x2, ..., xn} in the previous the section, the variance of SCTVα given in equation (14) can be estimated using the following bootstrapping algorithm: 1. Take a sample X * ={x * 1, x*2, ..., x * n} from the given sample X={x1, x2, ..., xn}with replacement. 2. Compute SCTVα using X * ={x * 1, x*2, ..., x * n} and use SCTV * α to denote this value. 3. Repeat the above two steps for a large number of time, say, N times. We will have N different SCTV * α’s. 4. Compute the sample variance of these N different SCTV * α’s and then use that value as the estimate of the variance of SCTVα. Remark. The bootstrapping method may not be applicable for estimating CTV and CTS if the distribution under consideration is heavy­tail (which is often the case for loss distribution in casualty insurance field.) For technical aspects, the reader can consult Resnick (2007). Monte Carlo Simulation If the real­world data are not available, the algorithm just described in Section 5.1 can be used with Monte Carlo simulation to obtain simulated CTV and CTS. For this purpose, one can use the following algorithm: (For illustration purpose, we assume the random variable X follows lognormal distribution with parameter μ and σ 2 . The algorithms for other distributions are analogous.) 1. Generate a large number, say N, of lognormal random numbers and denote them as X (1) ={x (1) 1, x (1) 2, ..., x (1) n}. (The meaning of the superscript will be explained shortly.) 2. Order X (1) ={x (1) 1, x (1) 2, ..., x (1) n} to get the ordered data set {x (1) (1), x (1) (2), ..., x (1) (n)}. 3. Compute SCTV (1) α as
n æ n ( 1 ) ö
x ç å x ( i ) ÷
å
[1 + n a ]
ç [1 + n a ] ÷
( 1 ) SCTV a =
-ç
n (1 - a ) n ( 1 - a ) ÷
çç
÷÷
è
ø
2 ( )
( 1 ) 2 ( i ) 4. Then compute SCTS (1) α as
n SCTS a( 1 ) =
å (x )
[ a]
( 1 ) 3 ( i ) 1 + n (
n (1 - a ) SCTV a( 1 ) ) 3 / 2 5. Now repeat the whole process from step 1 to step 4 for a large number, say M, of times and use superscript k to denote the k­th experiment. Thus, we will obtain M values of SCTVα : SCTV (1) α , SCTV (2) α , …, SCTV (M) α and M values of SCTSα : SCTS (1) α , SCTS (2) α , …, SCTS (M) α . 6. Finally, we get the estimated values of CTV and CTS as M
( k ) å SCTV a
SCTV a = k =1 and
M M
( k ) å SCTS a
SCTS a = k =1 M Remark. This Monte Carlo simulation algorithm can also be used to back­test the assumption taken for the underlying distribution. Suppose we have a data set C of stock price in past several years and we assume the stock price follows lognormal distribution. Then we may compare the estimates of CTV and CTS computed from this data set with the CTV and CTS computed using Monte Carlo simulation. If their difference is large, we may have reasonable concern about the lognormal assumption. SUMMARY In this paper, we study two new downside risk measures that are useful for managing risks in both finance and insurance fields. We derive closed­form formulas for both measures under arbitrary distribution assumption, and show how one can apply these two risk­measures to lognormal stock price. In addition, we give estimate for both CTV and CTS for a given sample data set. The paper also explores the numerical methods such as Monte­Carlo simulation and bootstrapping for CTV and CTS. We should mention that CTV and CTS, like any other risk measures, have their own drawbacks. For example, neither CTV nor CTS is coherent. For detailed discussion of coherence, the reader may consult Artzner et al (1999), Gerber (1979) or Hardy (2003).
REFERENCES: Acerbi, C., Tasche, D., 2002. On the Coherence of Expected Shortfall, Journal of Banking and Finance Vol. 26, Iss. 7, p. 1487­1503. Alexander, G. J., Baptista, A. M., 2002. Economic Implications of Using a Mean­VaR Model for Portfolio Selection: A Comparison with Mean­Variance Analysis, Journal of Economic Dynamics and Control Vol. 26, No. 7­8, p. 1159­1193. Artzner. P., Delbaen., F., Eber., J.M., Heath., D., 1999. Coherent Measures of Risk, Mathematical Finance, Vol. 9, Iss. 3, p. 203­228. Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A., Nesbitt, C.J., 1997. Actuarial Mathematics, 2nd edition. Society of Actuaries. Brzezniak, Y.S., Zastawniak, T., 2000. Basic Stochastic Processes. Corrected edition. Springer. Bali, T.G, Theodossiou. P., 2007. A Conditional­SGT­VaR Approach with Alternative GARCH Models, Annals of Operations Research, Vol. 151, Iss. 1, p. 241­267. Billingsley, P., 1995. Probability and Measure, 3rd edition. Wiley. Chen, S.X., 2008. Nonparametric Estimation of Expected Shortfall, Journal of Financial Econometrics, Vol. 6, Iss. 1, p. 87­107. Chow, Y.S., Teicher, H., 1997. Probability Theory: Independence, Interchangeability, Martingales, 3rd edition. Springer. Chung, K.L., 2000. A Course in Probability Theory, Revised 2nd edition. Academic Press. DasGupta, A., 2008. Asymptotic Theory of Statistics and Probability, Springer. Dudewicz, D.C., Mishra, S.N., 1988. Modern Mathematical Statistics. Wiley. Efron, B., 1979. Bootstrap Methods: Another Look at the Jackknife, Annals of Statistics, Vol. 7, Iss. 1: 1– 26. Engle, R. F., Manganelli, S., 2004. CAViaR: Conditional Autoregressive Value at Risk by Regression Quantiles, Journal of Business and Economic Statistics, Vol. 22, Iss. 4, p. 367­382. Gerber, H.U., 1979. An Introduction to Mathematical Risk Theory, S.S. Huebner Foundation monograph series, No.8., University of Pennsylvania. Giamouridis, Daniel, Ntoula, I., 2009. A Comparison of Alternative Approaches for Determining the Downside Risk of Hedge Fund Strategies, The Journal of Futures Markets, Vol. 29, Iss. 3, p. 244. Grootveld, H., Hallerbach, W. G., 2004. Upgrading Value­at­Risk from Diagnostic Metric to Decision Variable: A Wise Thing to do?, Risk Measures for 21st Century, Wiley. Hardy, M., 2003. Investment Guarantees: Modeling and Risk Management for Equity­Linked Life Insurance, Wiley. Hogg, R.V., McKean, J.W., Craig, A.T., 2005. Introduction to Mathematical Statistics, 6th edition. Prentice Hall. Hull, J., 2009. Options, Futures and Other Derivatives. 7th edition, Prentice Hall. Lan, H., Nelso, B., Staum, N.J., 2007. A Confidence Interval for Tail Conditional Expectation Via Two­ level Simulation, Proceedings of the 2007 Winter Simulation Conference. Klugman, S.A., Panjer, H.H., Willmot, G.E., 2009. Loss Models: From Data to Decisions, 3rd edition Wiley.
Manistre, B.J., Hancock, G.H., 2005. Variance of CTE Estimator, North American Actuarial Journal, Vol. 9, Iss. 2, p. 129­156, Mittnik, S., Rachev, S., Schwartz, E., 2002. Value­at­Risk and Asset Allocation with Stable Return Distributions, Allgemeines Statistisches Archiv., Vol. 86, Iss. 1, p. 53­67. Pflug, G., Gaivoronski, A.A., 2005, Value­at­Risk in Portfolio Optimization: Properties and Computational Approach, Journal of Risk, Vol. 7, Iss.2, p. 1­31. Rau­Bredow, H., 2004. Value at Risk, Expected Shortfall, and Marginal Risk Contribution, Risk Measures for the 21st Century, Wiley Finance, Chichester p. 61­68. Resnick, S.I., 2007. Heavy­Tailed Phenomena: Probabilistic and Statistical Modeling}, Springer. Ross, S., 2003. Introduction to Probability Models, 8th edition. Academic Press. Ross, S., 2006. Simulation. 4th Edition, Academic Press. Shiryaev, A.N., 1984. Probability, 2nd edition. Springer. Simone, M., 2000. Conditional Autoregressive Value at Risk and Other Essays in Financial Econometrics [Ph.D. dissertation], ProQuest. Shreve, S., 2005. Stochastic Calculus for Finance II: Continuous­Time Model. Springer. Valdez, E.A., Tail conditional variance for elliptically contoured distributions, Belgian Actuarial Bulletin, Vol. 5, Iss. 1, 2005. Yamai, Y., Yoshiba, T., 2005. Value­at­Risk versus Expected Shortfall: A Practical Perspective, Journal of Banking and Finance, Vol. 29, Iss. 4, p. 997­1015.