Incremental VaR and VaR with background risk: traps and ... - CiteSeerX

Incremental VaR and VaR with background risk: traps and misinterpretations Luisa Tibiletti Presented at 27th Seminar of the European Group of Risk and Insurance Economists, Rome, Italy September 18-20, 2000 Abstract. In recent …nancial literature the Incremental Value-at-Risk (IVaR), i.e., the incremental e¤ect on VaR of adding a new instrument to the existing portfolio, has become a standard tool for making portfolio-hedging decisions. Since, calculating the exact IVaR value could be computationally very costly, approximate formulas have been developed. According to the most commonly used formula, IVaR is approximately equal to the current VaR multiplied by the beta coe¢cient of the candidate asset. A spontaneous question arises: could the beta sign be a qualitative indicator of a pro…table (non-pro…table) trade? Fallacy of this conjecture is proved. These results seem to cast shadows on the above approximate formula reliability even for small changes in portfolio composition. Then, VaR sensitivity to an exogenous undiversi…able zero-mean random perturbation (background-risk) is tested and hedging decision rules set out. Finally, a new multidimensional notion of VaR is proposed. Keywords: Value-at-Risk, Incremental VaR, Background-risk, Multidimensional Quantile, Copula JEL classi…cation: C13, D81, G10, G11

1

Introduction

In recent …nancial literature Value-at-Risk (VaR) has become a standard risk tool for designing portfolio strategies. The Incremental VaR (IVaR), i.e., the incremental e¤ect on VaR of adding a new instrument to the existing portfolio, is the gear for making hedging decisions. Since calculating exact IVaR could be cumbersome and computationally very costly, approximate formulas suitable for small portfolio composition changes have been suggested in the literature (Garman, 1996; Dowd, 1998; Jorion, 1997; Culp et al., 1999; Hallerbach, 1999). According to the most commonly used formula, IVaR is approximately equal to the current VaR multiplied by the beta coe¢cient (with respect to the current portfolio) of the candidate asset. This result seems to support the intuitive idea that the beta coe¢cient sign could be a qualitative indicator of a pro…table new entry trade. The opposite strategy should be suggested, if the beta coe¢cient is positive.

1

The aim of this paper is just to test the ground of this conjecture. Through a number of counter-examples, the fallacy of this misleading argument is demonstrated. In order to highlight the real causes of these traps, two cases are analysed. Firstly, the current portfolio and the candidate asset are assumed to have the same elliptically contoured distribution. The hedging decision crucially depends on: (1) the beta coe¢cient, (2) the relative di¤erence between the variances and (3) the relative weight of the candidate asset in the re-calibrated portfolio. For example, it can be shown that properly balancing above gears, the introduction of a zero-mean asset even with a positive beta coe¢cient and a variance greater than that of the current portfolio, may reduce the overall VaR. Secondly, no assumptions on the underlying statistical distributions are made. Now, the impact of a new asset may produce unforeseeable modi…cations on all the high order moments of the augmented portfolio. It is noteworthy, that the sum of random variables does not even preserve the skewness and the kurtosis direction of the addenda (see Tibiletti-Volpe, 1997). Moreover, some examples (even counter-intuitive) cast shadows on the possibility to set out general clear-cut decision procedures. The second part of the work is focused on a slightly di¤erent problem. Suppose a non-tradable zero-mean background risk is added to the existing portfolio. What are the e¤ects on VaR? The issue is relevant in applications, since it enables us to get a feel of VaR sensitivity to exogenous undiversi…able zero-mean random perturbations (background risks). Finally, a new alternative approach, called bivariate VaR, based on an intuitive multidimensional notion of quantile (Tibiletti, 1993) is introduced. By using the concept of copula for describing the dependence structure between the two risk sources involved, the bivariate VaR seems to marry an intuitive theoretical appeal with computational e¢ciency. The paper is organized as follows. Section 2 presents the notion of IVaR. In Section 3 a common approximating formula is discussed. Some common misinterpretations in evaluating IVaR sign is discussed in Section 4. Section 5 focuses on VaR with an additive zero-mean background risk. In Section 6 a new bivariate notion of VaR is proposed. Section 7 concludes the note.

2

The Incremental Value-at-Risk

Value-at-Risk (VaR) is a probability-based metric for quantifying the market risk of assets and portfolios. Its theoretical origins can be traced back long time both in Finance and in Non-life Actuarial Science. In last decade, its adoption is continuing to become widespread not only among securities houses and investment banks, but also among commercial banks, pension funds and in every …nancial institution. Its success is principally imputable to the fact that it is an easy-to-understand measure of risk. VaR refers to the maximum amount we are likely to lose at some speci…c con…dence level q%. Let X be the random current portfolio return with distribution function Fx . VaR can be formally de…ned as the q-quantile of the future random pro…t/loss.

2

Let R¤ be the q-quantile of Fx , i.e. Pr (X · R¤ ) = Fx (R¤ ) = q% Suppose X has …nite second order moment. Let ¹x be the mean and ¾2x the variance of X. Let Z = X¡¹ ¾ X , so µ ¶ (R¤ ¡ ¹x ) Pr (X · R¤ ) = Pr Z · = q% ¾x Let ® be the con…dence level of the normalized random variable Z (for example, if Z is normally distributed ®=-1.65 for q=5%). Then R¤ = ¹x + ®¾ x two di¤erent de…nitions can be set out V aRabsolute = (X; q%) = ¡ (¹x + ®¾X ) V aRrelative = (X; q%) = ¡®¾ X In practice, the only relative VaR is used. If we are using a parametric approach to VaR, the relative one is easier to handle because it does not require the evaluation of the mean return. In the following, we will only deal with V aRrelative (for notational simplicity, no subscription will be reported in what follows). In conclusion, we will deal with the following fundamental formula Pr (X ¡ ¹x · ¡V aR (X)) = q%

(1)

Clearly, the smaller its VaR, the more preferable the portfolio. In order to design pro…table portfolio strategies, a logical step is to calculate the VaR of the portfolio either including the candidate asset or not. The Incremental VaR (IVaR) associated with a position in asset Y is, for a …xed value q% IV aR = V aR (portfolio with the asset Y ) - V aR(portfolio without the asset Y ) If IVaR is positive, the candidate asset entry upgrades the portfolio risk, vice versa if it is negative. Unfortunately, any method of calculating IVaR that involves before and after calculations of portfolio VaRs is open to serious practical obstacles: if the assets involved are large, even a fast computer would take time to carry out the necessary matrix operations. So, easy-to-use approximating formulas for a rapid evaluation of IVaR are often used.

3

3

Approximate Solutions for IVaRs: shortcomings and traps

A sketchy notation follows X return of the current portfolio Y return of the candidate new entry asset Y a size of the new entry asset S = X+aY …nal re-balanced portfolio with marginal position in asset Y 1+a ¾ yx covariance between Y and the old portfolio X ¾ 2(:) variance of the random variable (:) ¾ ¯ yx = ¾yx 2 beta coe¢cient of Y with respect to the current portfolio X x

Garman (1996, 1997) was one of the …rst to identify the relevance of calculating IVaR for discriminating pro…table investments. Recently a simpler formula providing the same numerical answer has been proposed by Down (1998 page 49) see also Down (1999) and Jorion (1997 pages 153-155). For a …xed security threshold q% and for a ”small” portion a of the new entry: IVaR=VaR(S)-VaR(X) » = a¯ yx VaR(X)

(2)

where ¯ yx is also called the systematic risk of asset Y vis-à-vis portfolio X (see Jorion, 1997 pages 154 for further formulas to decompose a portfolio’s VaR into incremental VaR). The above shows the possible (with probability q%) variation in monetary terms due to the introduction of Y . By (2), the impact of the candidate new entry, seems to be promptly calculated with negligible computer and human-time wasting. In spite of the advantages stemming from the computational simpli…cations, two spontaneous questions arise: 1. From the quantitative point of view, is (2) a satisfactorily approximating formula? 2. May the sign of the marginal e¤ect of the candidate asset be ascertained by the sign of the right-hand side of (2)? In other words, will the signs of betas provide an acceptable tool for discriminating pro…table assets? With reference to the former question, above mentionated Authors have already highlighted that the size a of the candidate trade should be insigni…cant with respect to the size of the current portfolio, due to the fact that Taylor approximations are used. The latter question seems to be less plain. Intuition seems to suggest a positive answer. The aim of this paper is just to test the foundation of this conjecture and prove its fallacy.

4

4

How to sign IVaR: traps and misinterpretations

As it has been already discussed, formula (2) should be con…ned to very small portfolio adjustments. But the operative attractiveness seems to be another. A preliminary rough screening among candidate pro…table/non-pro…table new entry assets through VaR, can be carried out by signing the correspondent IVaR. According to (2) IVaR seems to have the same sign of the beta coe¢cient of the candidate asset Y . Is this discriminating rule correct? Unfortunately, the answer is no in nearly all the cases. The stressed points are: 1. The sign of betas does not signal the sign of IVaR, even in the case of elliptically contoured distributions. 2. A decreasing in variance does not produce a decreasing in VaR: is that surprising? 3. Stable and non-stable distributions: di¤erent results.

4.1

Old/new portfolios: VaR relations

A common idea throughout the literature concerns the possibility of obtaining information about the V aR (S) only on the basis of: 1) the V aR (X) and 2) the relative standard deviation ¾¾XS . In the following we will show how this conjecture may be highly misleading. For a …xed level of con…dence q% we have Pr (X · ¡V aR (X)) = q% = Pr (S · ¡V aR (S)) Normalizing the variables involved, under the hypothesis of null-mean, we obtain ´ ³ ´ ³ ¡V aR(S) S = q% = Pr · Pr ¾XX · ¡V ¾aR(X) ¾S ¾S X 4.1.1

Normal case

Let ® be the con…dence level of the normalized random variable X . If both variables X and S are normally distributed, it follows ¡V aR(X) ¾X

=®=

¡V aR(S) ¾S

so V aR (S) =

¾S ¾X V

then

5

aR (X)

IV aR = V aR (X)

·

¸ ¾s ¡1 ¾x

(3)

it is worthwhile remarking that (3) is the exact value of IVaR. In conclusion, under normality assumptions, the introduction of Y is pro…table

IV aR < 0 if f ¾s < ¾ x

(4)

i.e., VaR decreases i¤ the variance decreases, as well. Some spontaneous questions arise: 1. could the sign of beta give a clear-cut tool for screening candidate new entries? 2. If the new asset variance is greater than that of the current portfolio, does its entry always produce a favorable contraction in VaR? 3. If variables are no longer normally distributed, but only belong to the same family of stable distributions, is formula (4) still valid? In contrast with our intuition would suggest, all above questions are negatively answered.

Fallacy 1 The sign of beta enables us to discriminate between pro…table and non-pro…table candidates new entries. Above conjecture turns out to be false even for normal distributions. Firstly, suppose that the current portfolio X and the candidate new entry Y are normally, or more in general, elliptically distributed. From (4) a fall in VaR goes with a contraction in variance. By straightforward calculations, we have ¾ 2 +a2 ¾ 2 +2a¾x ¾ y rxy

y ¾2s = x (1+a)2 X and Y: So

where rxy is the Pearson correlation coe¢cient between

¢ ¡ a ¾ 2x ¡ ¾ 2y + 2¾ 2x ¾ s < ¾x iff rxy < ; a 6= 0 (5) 2¾x ¾y ³ ´ ¾2 or equivalently, ¯ yx < 1 + a2 1 ¡ ¾y2 : So, IVaR depends on three factors: x

² rxy correlation coe¢cient ² a the portion of Y ² ¾ x and ¾y

It is trivial to remark that no matter what the sign of rxy and the quantity a are, if ¾2y · ¾2x the introduction of any zero-mean asset is always advisable. 6

Remark 1 If ¾2y ¸ ¾ 2x ; the introduction of any zero-mean asset depends on the ful…llment of formula (5). Due to the fact that rxy and ¯ yx have the same sign, the knowledge of the only beta sign is not su¢cient for discriminating favorable investments.

Fallacy 2 If

rxy > 0 and ¾2y > ¾2x the candidate new entry Y increases VaR and variance, so its entry is always to be rejected. Note that the right-hand side of (5) may be positive, even if ¾2y > ¾2x : A necessary and su¢cient condition for its positiveness is 1
0 A possible contraction in the portfolio variance can be obtained when a < 4 1 p : For example, let a = 10 for any rxy < 0:796, corresponding and rxy < a(¡0:05)+2 2 1:5 to ¯ yx = ¾ xy < 0:975: It is worthwhile noting that even a great portion of Y characterized by a high variance and a strong positive correlation, may have a desirable diversi…cation e¤ect on the portfolio. Remark 3 In order to design favorable investment plan, a through analysis is required for taking into account the favorable e¤ect of diversi…cation, which is not captured by the beta coe¢cients. A …nal remark. Suppose a candidate Y has been selected to enter. What is the quantity of Y which minimizes the variance ¡ ¢of the augmented portfolio? Simple calculations are needed. Let M in ¾2S : 1 Let µ = 1+a the relative weight of the ”old” portfolio into the ”new” one, 2 so ¾ 2s = µ ¾ 2x + (1 ¡ µ)2 ¾2y + 2µ (1 ¡ µ) ¾x ¾ y rxy if rxy · 0 the ”best” relative portion of Y is µ ¤ = ¾ 2X ¡¾ X ¾ Y ¾ 2Y ¡¾ X ¾ Y

r

¾2Y ¡¾ X ¾ Y rXY ¾2X +¾2Y ¡2¾ X ¾ Y rXY

or

XY equivalently a¤ = rXY if rxy > 0 and ¾x > ¾y the ”best” relative portion of Y is q ± = Max (0; µ¤ ),vice versa if ¾x < ¾y the ”best” relative portion of Y is µ± = M in (q ¤ ; 1) :

4.1.2

Non-normal case

Whenever the distributions involved are no longer normal, the discussion needed to be stressed along three di¤erent lines: Both the variables X and Y belong to the same elliptically contoured family. In such a case two peculiarities still hold: ² the augmented portfolio distribution still belongs to the same family 7

² the distribution function is fully known when the …rst two moments are given. so analogous conclusions straighten in the normal case can be set out. Formula (3) is still valid: VaR degrades i¤ the variance of the overall portfolio drives to the same direction. Both the variables X and Y have the same stable distribution It is known that the only stable distributions with …nite second order moment belong in the elliptically countured family (see for example Samorodnitsky and Taqqu, 1994). So, this case collapses in the previous one The variables X and Y have di¤erent distributions. In this case no information is available over the …nal distribution S. A complete re-calculation is required.

Fallacy 3 For non-elliptically contoured variables, a lower variance induces a lower VaR. Above is an idea that is still deeply-rooted in the literature. Outside the elliptically contoured world, the strategy of minimizing the variance for reducing VaR, may be absolutely misleading. An intuitive explanation is sketched in the following. Remark 4 The strategy of minimizing the variance is addressed to reducing the ”small” risks (those in the neighbourhood of the mean) and vice versa the slight importance of the ”catastrophical” risks is deserved. According to the strategy of variance minimization, the more data are concentrated on the mean the more the distribution is appreciated. On the other hand, VaR minimization is devoted to shrinking the left-tail below the q-quantile of the distribution to zero, no attention is paid to the risk distribution above the q-quantile. Clearly, no advantages in VaR terms, derive from a possible data concentration around the mean. Finally, we should not be surprised at all if such di¤erently aimed strategies lead to di¤erent solution paths. The paths may coincide in very special circumstances, such as for elliptically countered distributions, where fatness of the tails goes hand-in-hand with the variance, but this must be considered only an exception! Whenever we leave behind the ”arti…cial” elliptical world, many desirable properties fall short. Example 5 Symmetrical distributions In order not to spread out our attention in cumbersome calculations, we will deal with very elementary discrete random variables. A fortiori, analogous counter-examples can be worked out for continuous, non-symmetrical random variables. Let 8

8 ½ < ¡1; p = 0:1 ¡1; p = 0:5 0; p = 0:8 and Y = X= 1; p = 0:5 : 1; p = 0:1 where ¹x = ¹y = 0 and ¾2x = 0:2 , ¾ 2y = 1: Clearly ¾2y > ¾2x: . For q% = 0:1% , it results V aR (X) = 1 in two special cases. Let consider S = X+aY 1+a X Y independently distributed 1. and 8 ¡1; p = 0:05 > > > ¡1+a > ; p = 0:05 > > < 1+a ¡a ; p = 0:4 1+a S0 = so rxy = 0 and ¯ yx = 0 +a ; p = 0:4 > 1+a > > +1¡a > > ; p = 0:05 > : 1+a +1; p = 0:05 For every portion a > 0 and at con…dence level q% = 0:1%; VaR decreases, in fact V aR (S0 ) = ¡ ¡1+a 1+a < V aR (X) : 2. X8and Y positively correlated ¡1; p = 0:08 > > > ¡1+a > ; p = 0:02 > > < 1+a ¡a ; p = 0:4 0:12 1+a > 0 and ¯ yx = 6 S+ = so, rxy = p +a 0:2 ; p = 0:4 > 1+a > > +1¡a > > ; p = 0:02 > : 1+a +1; p = 0:08 Again, even if rxy > 0; for every portion a > 0 and at con…dence level q% = 0:1%; VaR is decreasing, in fact V aR (S+ ) = ¡ ¡1+a 1+a < V aR (X) : Finally, the introduction of any portion a of the asset Y always produces a favorable VaR contraction. So, in contrast with the approximating formula 2 For any level a > 0; IV aR and ¯ yx are always opposite in sign!

Besides, it is worthwhile noting that the introduction of Y is always advisable even if ² ¾ 2y > ¾ 2x ; i.e. the variance of Y is greater than that of current portfolio X ² the decreasing of V aR does not go with a decreasing of variance, in fact correlation and amount a play a crucial role. For example, in the nullcorrelation case ¾ 2x > ¾2S0 if a < 0:5 and, vice versa ¾ 2x · ¾ 2S0 it may happen that

if a ¸ 0:5: In conclusion,

V aR decreases even if the variance increases

9

In conclusion, variance and VaR minimizing strategy may lead to a completely di¤erent results: portfolios minimizing the variance, i.e. the relatively ”small” risks, often increase larger risks as measured by VaR (a seminal idea of this undesirable e¤ect has been already guessed by Andersen et al. (1999), where some numerical explorations are worked out) An explanation of previous examples comes from the fact that VaR is aimed at yielding a ”safety position”: the left-hand side of the risk distribution should be shrunk as ‡at as possible. A more suitable tool for measuring the tail ”fatness” are no longer the central moments of ”low” order, such as the variance, but, vice versa, those of ”high” order, which emphasize the weight of data lying ”far” from the mean. Among the central moments, we will cast a glance over those of third and fourth order, providing a readable interpretation. Skewness 3

: It characterizes Skewness coe¢cient of X is de…ned as Sk (X) = E(X¡¹) ¾ 3X the degree of asymmetry of the distribution around its mean. That is possible, since it is a metric-free coe¢cient. Unlike in the normal case, the con…dence interval is no longer symmetrical around the mean value, but it is tilted toward the direction of skewness. If the skewness is positive, the con…dence interval covers more on the right-hand side of the mean rather than the left one. Clearly, as more right-skewed distributions are as more they are preferred by prudent investors. Figures 1 and 2 show the density function of two variables with the same mean and variance, but opposite in skewness. According to the variance criterium, above variables should be evaluated indi¤erently, but that is no longer true whenever the VaR-criterium is employed: for low value q% , the positively skewed variable is preferred.

u

u

Figure 1. Positively skewed

Figure 2. Negatively-skewed

Kurtosis 4

¡3: It measures the Standard kurtosis of X is de…ned as Kurt (X) = E(X¡¹) ¾ 4X relative peaked, thinness or ‡atness of the distribution compared to a normal distribution. High kurtosis indicates there are more far away from the mean than predict by a standard normal distribution. The lower kurtosis the more desirable the distribution is, in fact tails are thinner than those of a normally distributed variable, with the same mean and variance. 10

The main shortcoming of this index is that no di¤erence between the two tails is made. So, variables displaying the same mean, variance and kurtosis, are evaluated ”indi¤erent” according to the mean-variance-kurtosis criterium,but may receive a di¤erent ranking according to the VaR criterium. An empirical test on how the VaR is a¤ected by the standard deviation as well as skewness and kurtosis has been recently stressed by Li (1999), using ten years of daily observations on twelve di¤erent foreign exchange spot rates.

Fallacy 4 Skewness and kurtosis versus are preserved under linear combination: the sum of two positively (negatively) skewed variables is still positively (negatively) skewed. Example 6 Linear combinations of equally skewed variables may turn out to be negatively, positively or null skewed. distributed variables. Let Fallacy even for identically ½ ½ 4 will be proved ¡1; p = 14 ¡1; p = 14 X= and Y = 1 1 3 p = 34 3; ¡3 ; 3 ¢ p = 4¡ 3 ¢ 1 2 2 where ¹x = ¹y = 0, ¾x = ¾ y = 3 ; E X = E Y = ¡ 29 ; so Sk (X) = Sk (Y ) = ¡ p23 < 0 Let construct the following the new variable Sa = X+aY 1+a : 8 > > >
> > :

² a=

1 3

¡1+ 13 a 1+a ; + 13 ;

8 2 < ¡9; p = 0; p = then S 13 = : 1 +3; p =

² a = 1 then S1 =

½

¡ 13 ; p = + 13 ; p =

8 2 < ¡3; p = ² a = 3 then S3 = 0; p = : 1 +3; p =

1 4 1 4 1 2

so Sk(S 13 ) < 0

1 2 1 2

so Sk(S1 ) = 0

1 4 1 4 1 2

so Sk(S3 ) > 0

It is worthwhile noting that a switching in skewness can be shown up even if the portion a of the new entry asset is negligible. For example, let be a = 3:10¡6 . Let the newly-entered be 11

½

¡10¡6 ; p = 1 ¡6 p= 3 10 ; and Sk (Y ) = ¡ p23 < 0: Y =

1 4 3 4

¡ ¢ with ¹y = 0, ¾ 2y = 13 10¡12 ; E Y 3 = ¡ 29 10¡18 ;

In such case, the re-calibrated portfolio coincides with that previously found S3 . So, skewness switches from negative to positive. In conclusion, Even a linear combination of identical variables may not preserve direction in skewness Remark 7 Skewness and kurtosis non-preservation in versus: the relevance of the mixed moments. The mathematical explanation of non-preservation in versus is immediate, whenever a more through insight into the formula of higher order moments is given. By de…nition, the n-th order moment of the sum X + Y is given by n n n E (X + Y ) = E (X) + E (Y ) +

n¡1 Xµ k=1

¶ ¢ ¡ n E X n¡k Y k k

(6)

Above displays the fact that the n-th order moment of the sum depends on both the n-th order pure moments of the addenda, but also on the mixed moments! Consequently, even if the correlation coe¢cient is rXY = 0; i.e. E (XY ) = 0; the mixed moments may be quite far from zero, so they can play the key-role in the determination of skewness and kurtosis (and, in general, of all higher moments) of the sum. How skewness versus may be strongly driven by the mixed moments, instead of the pure moments (i.e., the skewness and kurtosis of the addenda) has been proved by empirical investigations over the daily distribution of twenty assets along a stretch of …ve years (see Peccati-Tibiletti, 1993). Moreover, it is worthwhile noting that the condition rxy = 0, does not guarantee because the mixed moments ¡ ¢ ¡in skewness ¡ ¢ ¢ no-switching ¡ ¢ ¡ and ¢kurtosis, E X 2 Y ; E XY 2 , E X 3 Y ; E XY 3 ; E X 2 Y 2 in general di¤er from zero and they strongly outweighed the pure moments (for a deeper investigation on how skewness and kurtosis1 depend on the type of dependence relation between addenda see Tibiletti-Volpe,1997). Since the n-th order moment of the re-balanced portfolio is given by 1 Tibiletti-Volpe (1997) have shown that the mixed moments display the type of relationship between the addenda. For example, E (XY ) measures the strength of linear dependence between X and Y. Vice versa, the other higher order mixed moments display other type of non-linear dependence. For example, Tibiletti-Volpe (1997) have proved that, once the marginals of X and Y ¡are …xed, ¢ S¡= X + ¢ Y reached its maximum in skewness (i.e. the sum of the mixed moments E X 2 Y +E XY 2 is maxima), when X and Y are positively dependent and the data are clustered around a hyperbole.

12

E

³

X+aY 1+a

´n

=

1 (1+a)n

n ¢o ¡ Pn¡1 ¡ ¢ n n E (X) + an E (Y ) + k=1 nk ak E X n¡k Y k

the skewness and kurtosis of the augmented portfolio can’t be predict without a previous re-calculation of the mixed moments, as well. Unfortunately, easy-to-use formulas for this purpose seem not to exist. In conclusion, the overall skewness and kurtosis vary greatly according to: ² skewness and kurtosis of X and Y ² mixed higher moments, which are not revealed by the correlation between X and Y (or, equivalently, by ¯ yx ) ² quantity a of the new asset. Note that the size of a and the dimension n n in‡uence the relative importance the pure moment an E (Y ) ¢ ¡ n¡kbetween k k Y : In fact, if a is small and n is and the mixed moments a E X large, the former becomes negligible with respect to the latter. Remark 8 Shortcomings of VaR as a measure of risk has been already highlighted in recent literature. Unfortunately, VaR does not turn out to be a ”coherent measure of risk” (for an ample treatment on the VaR shortcoming see Artzner (1999), Artzner et al.(1999), Embrechts et al. (1999), Wirch (1999)). And more precisely, the subadditivity property falls short. In light of the previous remark, this last undesirable feature appears not surprising at all. VaR ”measures” the fatness of the distribution left-tail, which is strongly in‡uenced by the skewness and kurtosis. Unfortunately, under linear combination, skewness and kurtosis are not preserved (even in sign), so the left tail of the …nal distribution may change drastically and VaR as well. No property in marginal changes seem possible to be settled out in a non-elliptically contoured world.

5

VaR with Background Risk

A growing topic under discussion in the comparative statics literature, concerns the study of the e¤ects of the introduction of an additive zero-mean background risk in the model. The problem is not negligible even in Applied Finance. Investigating how the model reacts in terms of VaR, to the introduction of a background risk, permits to get a feel on the VaR sensitivity to an exogenous undiversi…able zero-mean random perturbations (background risks). Our aim is to investigate into the variation between the ex-post and ex-ante the background perturbation, i.e. V aR-variation = V aR (X + Y ) ¡ V aR (X)

13

where Y is a zero-mean background risk. In almost all of the literature, the background risk Y and X are also assumed to be independently distributed. From the technical point of view, VaR-variation looks very similar to IVaR. The main di¤erence relies on the fact that, now, no re-calibration of the portfolio is set out. Although the purposes di¤er, the basic remarks stressed out in the previous session remain valid: ² if X and the background Y belong to the same elliptically contoured family, the ex-post variable X + Y still belongs to the same family. So, the sign of the VaR-variation can be carried out from the new variance. In such a case V aR-variation < 0 if f ¾x+y < ¾x ¾ or equivalently, rxy < ¡ 2¾yx : Clearly, a negative correlation rxy between X and the background risk Y is not su¢cient to produce a favorable e¤ects on the VaR. Note for example, if ¾y = ¾x the favorable diversi…cation e¤ect comes out only i¤ rxy < ¡0:5: Vice versa a positive correlation rxy always unfavorable upgrades the …nal VaR.

² vice versa, if X and Y are no longer in the same elliptically contoured family, the e¤ect of the introduction of the background risk Y may induce an unforeseeable shock over the previous distribution of X. In fact the distribution of X + Y may drastically di¤er from that of X. Let us cast a glance at the central moments of X + Y . Under stochastic independence, formula (6) becomes2 : ¡n¢ ¡ ¢ P n¡k E Yk E ((X ¡ ¹) + Y )n = E (X ¡ ¹)n + E (Y )n + n¡2 k=2 k E (X ¡ ¹) since the latter addenda are non-null in most cases and they upgrade in number and in weight as n increases, the …nal central moment may be distant from the original one. In conclusion, left-tails of X and X + Y may completely di¤er. It is worthwhile noting that not even independence between X and Y can guarantee skewness and kurtosis preservation in versus. For example, with reference to Sk(X + Y ), a switching in the versus may occur if Y is oppositely skewed with respect to X. In fact, the third central moment ex-after is ´ ³ ¢ ¡ 3 3 3 E ((X ¡ ¹) + Y ) = E (X ¡ ¹) +E (Y ) +3E (X ¡ ¹)2 Y +3E (X ¡ ¹) Y 2 = 2 A necessary and su¢cient condition for stochastic independence is the following. The r.v. X and Y are stochastically independent i¤ E (g (X) h (Y )) = E (g (X)) E (h (Y )) for all Borel-measurable functions g and h, provided that the expectations involved exist. Clearly ¢ ¡ are ¢only¡ very ¢ special Borel-measurable functions, so the condition ¡ power functions E X n¡k Y k = E X n¡k E Y k is a necessary, but not a su¢cient condition for independence.

14

´ ³ ¡ ¢ E (X ¡ ¹)3 +E (Y )3 +3E (X ¡ ¹)2 E (Y )+3E (X ¡ ¹) E Y 2 = E (X ¡ ¹)3 +

E (Y )3 3 3 So a skewness direction switching occurs if E (X ¡ ¹) < ¡E (Y ) : Note that if independence hypothesis is weakened in requiring rxy = 0; then a skewness switching in sign may occur even if X and Y are null-skewed or skewed in the same direction3 . With reference to the fourth order moment, under stochastic independence, it holds ¡ ¢ 4 4 4 2 E ((X ¡ ¹) + Y ) = E (X ¡ ¹) + E (Y ) + 6E (X ¡ ¹) E Y 2 which di¤ers from the sum of¡ the¢ of the two pure fourth order moments 2 by the addendum 6E (X ¡ ¹) E Y 2 : So the kurtosis may strongly increase whenever the marginal variances are high. As a consequence, V aR (X + Y ) and V aR (X) may di¤er more than our intuition would suggest. So the only safe way for evaluating V aR-variation seems to require the whole calculation of V aR (X + Y ) :

6

VaR with background risk: a bivariate notion of VaR

Whether the re-calculation of the whole V aR (X + Y ) is too time expensive and only a rough idea about the possible impact of an exogenous factor Y is desired, a di¤erent approach can be followed. From the mathematical point of view, VaR at con…dence level q% is nothing else but the qth-quantile of the distribution. Does a multidimensional de…nition of qth-quantile exist? In spite of the existence of manyfold de…nitions4 of multidimensional median (which is nothing else but the 0:5th-quantile), only one de…nition of multidimensional qth¡quantile for any q% 2 [0; 1] is known to the Author. Tibiletti (1993) proposed a notion stemming from the idea that the quantile should preserve the natural property of segmenting the data into groups of pre-…xed size in percentage (for an application of the multidimensional quantile segmentation for classifying a set of funds according a multi criteria of performance, see Tibiletti, 1994). The investigated case focuses on the probability that two circumstances occur simultaneously: the current portfolio X is below a pre-…xed value and the background risk Y is below/above a threshold. In formulae, when X · x and Y · y

or X · x and Y > y

3 Counter-examples 4 For

are available under request by the Author. a rich and critical survey on the numerous de…nitions known in literature, see Small

(1990).

15

In general, background risk Y has the nature of an undiversi…able risk. For example Y could be the prevision error about the in‡ation tax, or about a peculiar …nancial index, or, Y may represent the percentage discard between X and a basket of similar …nancial products. But also some widely known structured …nancial products may be entailed in such frame, as for example, the bivariate contingent claims. A …nancial example is provided by the digital binary options, that are debt instruments promising to pay a coupon if the prices of two assets X and Y are above some prede…ned strike levels at some future date. Another example can be provided by an insurance policy which pays the reimbursement if loss X is below a threshold and an exogenous index Y is above/below a pre…xed value y ¤ . Given the random vector (X; Y ) we denote by Fc (x; y) = Pr (X · x; Y · y) Fd (x; y) = Pr (X > x; Y > y) Fc;d (x; y) = Pr (X · x; Y > y) Fd;c (x; y) = Pr (X > x; Y · y) its partially cumulative-decumulative distribution functions. In order to keep the note self-contained, de…nition of multidimensional quantile is re-called in Appendix. To extend the VaR concept, we have to focus on the left-tail of X. So, we consider the qth-quantile points (x¤ ; y¤ ) lying on the qth-level set of Fc if we are interested in the values of Y · y ¤ or Fc;d if we are interested in the values of Y > y ¤ . That is context depending. In order to keep the main ideas in focus, we will only concern with absolutely continuous random vectors. De…nition 9 Let us …x the con…dence level q%² [0; 1] : For any …xed threshold y¤ for the background-risk Y , there exists a unique x¤ such that Pr (X · x¤ ; Y · y ¤ ) = q%

(7)

Fc (x¤ ; y¤ ) = q%

(8)

in short,

The value V aRy¤ (X) = ¡x¤ is called the bivariate V aR of X under the condition Y · y ¤ : De…nition 10 Let us …x the con…dence level q%² [0; 1] : For any …xed threshold y¤ for the background-risk Y , there exists a unique x¤ such that Pr (X · x¤ ; Y > y ¤ ) = q% or equivalently,

16

(9)

Fc;d (x¤ ; y ¤ ) = q%

(10)

The value V aRy¤ (X) = ¡x¤ is called the bivariate V aR of X under condition Y > y¤: Remark 11 V aRy¤ (X) (or V aRy¤ (X)) is the maximum amount we are likely to lose at con…dence level q%, when simultaneously Y · y ¤ (or Y > y ¤ ): Remark 12 By formula (8) , V aRy¤ (X) (and V aRy¤ (X) as well, by formula (10)) can be implicitly de…ned as function of y ¤ . Then, a 2D plot of V aRy¤ (X) ³ ´ ¡ ¢ (and V aRy¤ (X)) on plane y ¤ ; V aRy¤ (X) (and y ¤ ; V aRy¤ (X) , respectively), may be easily plotted.

6.1

How to calculate bivariate VaRs

The bivariate VaR strongly depends on the dependence structure between X and Y: Let F1 and F2 be their marginal distributions. Let us consider separately two cases. ² X and Y are independently distributed If we are interested in the VaR whenever the background risk is below a …xed threshold, i.e. Y · y¤ ; we look at Pr (X · x¤ ; Y · y ¤ ) = F1 (x¤ ) :F2 (y ¤ ) = q% so, we can easily …nd out q F2 (y ¤ ) %

F1 (x¤ ) = Denote by F1¡1 the inverse of F1

V aRy¤ (X) = ¡F1¡1

³

´

q F2 (y ¤ ) %

so, V aRy¤ (X) is equal to the unidimensional V aR (X) at con…dence level

q F2 (y ¤ ) %:

Vice versa, if we are interested in the VaR for Y > y : Pr (X · x¤ ; Y > y ¤ ) = F1 (x¤ ) : [1 ¡ F2 (y ¤ )] = q% so, F1 (x¤ ) =

q% [1¡F2 (y ¤ )]

then V aRy¤ (X) = ¡F1¡1

17

³

´

q 1¡F2 (y¤ ) %

so V aRy¤ (X) turns out to be equal to the unidimensional V aR (X) at con…dence level 1¡Fq2 (y¤ ) %: ² X and Y are dependently distributed. Copula seems to be well suited for constructing multivariate distributions in the case of stochastic independence. This notion has been seminal introduced by Sklar (1959) and for the subsequent thirty years its use has been con…ned to a restrict circle of researchers (see for a historical review Schweizer, 1993). But in the last ten years its use has been spread out in many di¤erent …elds (see for example Tibiletti (1995), Embrechts et al. (1999), Ceske and Hernàndez (1999) see, also, Bouyé et al.(2000) for a practical guide). The key-idea is that each bivariate distribution function can be re-written in order to separate the di¤erent role played by the dependence structure between X and Y (whose information is captured by the copula) and their marginals F1 and F2 : Fc (x; y) = C (F1 (x) ; F2 (y))

(11)

2

where C : [0; 1] ! [0; 1] is the copula associated to the cumulative function Fc . It is worth mentioning three special copulas: ² The product copula C (F1 (x) ; F2 (y)) = F1 (x) F2 (y) if and only if X and Y are independently distributed; ² The minimum copula W (F1 (x) ; F2 (y)) = max (F1 (x) + F2 (y) ¡ 1; 0) in the case of minimum association between X and Y ; ² The maximum copula M (F1 (x) ; F2 (y)) = min (F1 (x) ; F2 (y)) in the case of maximum association between X and Y . Many families of copulas have been studied in the literature in the last forty years. A class which seems to be well suited for …nancial applications, both for its width and easiness in elicitation and numerical estimation is the so-called Archimedean class (see the seminal work of Genest and MacKay, 1986 and for an exhaustive list of Archimedean families see Nelsen, 1999). A relevant statistical advantage in using the Archimedean copulas is that the relationship between each member of this family and association indices of common use, such as Kendall’s tau and Spearman’s rho, has been worked out. With an explanatory purpose, an example is carried out. Example 13 Gumbel copula The family of Gumbel copulas is de…ned o as follows n 1 ® ® ® C (u; v) = exp ¡ [(¡ lg u) + (¡ lg v) ] 18

where ® 2 [1; +1) is the generator parameter of the family. Note that ® = 1 gives the product¡copula, i.e. ¢ the independence case. For this family, Kendall’s tau coe¢cient is 1 ¡ ®¡1 : If VaR under the condition Y · y¤ is looked after, it su¢ces to Pr (X · x¤ ; Y · y ¤ ) = Fc (x¤ ; y ¤ ) = C (F1 (x¤ ) ; F2 (y ¤ )) = q% by straightforward calculations, we can easily …nd out n o 1 F1 (x¤ ) = exp ¡ [¡ (¡ lg F2 (y ¤ ))® + (¡ lg q)® ] ® so,

³ n o´ 1 V aRy¤ (X) = ¡F1¡1 exp ¡ [¡ (¡ lg F2 (y ¤ ))® + (¡ lg q)® ] ® and V aRy¤ (X) is equal to the unidimensional V aR (X) at con…dence level n o 1 exp ¡ [¡ (¡ lg F2 (y¤ ))® + (¡ lg q)® ] ® %:

Vice versa, if VaR under the condition Y > y ¤ is looked after, the partially cumulative-decumulative function should be taken Pr (X · x¤ ; Y > y ¤ ) = Fc;d (x¤ ; y ¤ ) = F1 (x¤ ) ¡ C (F1 (x¤ ) ; F2 (y¤ )) = q% by straightforward calculations, we can easily …nd out o n 1 F2 (y ¤ ) = exp ¡ [¡ (¡ lg F1 (x¤ ))® + (¡ lg (q ¡ F1 (x¤ )))® ] ® Then the value x¤ can be implicitly de…ned by ³ n o´ 1 y¤ = F2¡1 exp ¡ [¡ (¡ lg F1 (x¤ ))® + (¡ lg (q ¡ F1 (x¤ )))® ] ®

therefore V aRy¤ (X) = ¡x¤ can be always computed, at least numerically. Remark 14 In conclusion, in both cases of independence or non-independence the values of V aRy¤ (X) and V aRy¤ (X) can be drawn from the unidimensional V aR (X) evaluated at a properly modi…ed con…dence level. So, at con…dence level q% and for every …xed value y ¤ V aR-variation = V aRy¤ (X) ¡ V aR (X) and V aR ¡ variation = V aRy¤ (X) ¡ V aR (X) can be easily obtained comparing the unidimensional V aR (X) at di¤erent con…dence levels. In the case of independence, q V aR-variation = (V aR (X) at con…dence level F2 (y ¤ ) %) - (V aR (X) at con…dence level q%) and 19

V aR ¡ variation = (V aR (X) at con…dence level 1¡Fq2 (y¤ ) %) - (V aR (X) at con…dence level q%) If independence assumption is dropped, the decumulative-cumulative distribution function can be easily elicited by means of the copula and simple formulas for VaR-variation can be provided.

7

Conclusion

VaR is a risk measure focusing on the probability mass spread over the extreme left-tail of the distribution, i.e. that part of distribution devoted to capturing information on catastrophic events. The problem tackled in the article is the following: Is this part of distribution very sensitive to even small portfolio re-calibrations? Can IVaR be satisfactorily approximated by formulas involving only the beta coe¢cients among the assets? These questions seem to be negatively answered outside the ”arti…cial” world of elliptically contoured variables. Unfortunately, the extreme tails do not possess ”intuitive preservation” properties. For evaluating the extreme tails, well suited tools are the metricfree coe¢cients of skewness and kurtosis. We have shown that a re-calibrated portfolio may not even preserve skewness and kurtosis versus of the addenda, even when the variables involved are identical. Thus a new-entry asset may change drastically the extreme left-tail of the …nal distribution and the VaR, as well. Consequently, outside the elliptical world, approximating formulas involving only the …rst two mixed moments of the …nal distribution, are to be discouraged. In the second part of the paper, the impact of the introduction of a background risk Y is discussed. Due to a di¤erent approach to the problem, a bivariate de…nition of VaR is introduced. The elicitation of the bivariate cumulativedecumulative distribution functions of (X; Y ) is easily achieved by means of the notion of copula. Handy formulas for calculating the VaR-variation, in stochastic independence and dependence between the primary risk X and the background risk Y; have been set out.

Appendix In more than one dimension, the relationship between cumulative and decumulative function does no longer hold. In fact Fc (x; y) 6= 1 ¡ Fd (x; y) : The following relations exist Fd (x; y) = 1 ¡ F1 (x) ¡ F2 (y) + Fc (x; y) Fc;d (x; y) = F1 (x) ¡ Fc (x; y) Fd;c (x; y) = F2 (y) ¡ Fc (x; y) where F1 and F2 are the one dimensional marginals of X and Y , respectively (see for example Nelsen, 1999). Whenever the con…dence level q% is …xed, the following level sets can be constructed

20

© ª Ac = ©(x; y) ²