Optimal Portfolio Selection Based on Expected Shortfall Under Generalized Hyperbolic Distribution

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1 Optimal Portfolio Selection Based on Expected Shortfall Under Generalized Hyperbolic Distribution B. A. Surya School of Business and Management Bandung Institute of Technology Jln. Ganesha 10, Bandung Indonesia. R. Kurniawan MSc Quantitative Finance University of Zürich/ETH Zürich Zürich - Switzerland February 10, 2012 Abstract This paper discusses optimal portfolio selection problems under expected shortfall as the risk measure. We employ multivariate Generalized Hyperbolic distribution as the joint distribution for the risk factors of underlying portfolio assets, which include stocks, currencies and bonds. Working under this distribution, we find the optimal portfolio strategy. Keywords: Multivariate Generalized Hyperbolic distribution; Expected shortfall; Portfolio optimization 1 Introduction It s been well known in years that financial data are often not normally distributed. They exhibit properties that normally distributed data do not possess. For example, it has been observed that empirical return distributions almost always exhibit excess kurtosis and heavy tail (Cont, [8]). Mandelbrot [18] also concluded that the logarithm of relative price changes on financial and commodity markets exhibit a heavy-tailed distribution. More recently, Madan and Seneta [17] proposed a Lévy process with Variance Gamma distributed increments to model log price processes. Corresponding address: {budhi.surya}@sbm-itb.ac.id. 1

2 Variance Gamma itself is a special case of Generalized Hyperbolic (GH) distribution which was originally introduced by Barndorff-Nielsen [4]. Other subclasses of the GH distribution were also proven to provide an excellent fit to empirically observed increments of financial log price processes, in particular, log return distributions, such as the Hyperbolic distribution (Eberlein and Keller [9]), the Normal Inverse Gaussian Distribution (Barndorff-Nielsen [5]) and the Generalized Hyperbolic skew Student s t distribution (Aas and Haff [1]. The student s t and normal distributions, in particular, are the limit distributions of GH. These give enough reasons for the popularity of Generalized Hyperbolic family distributions: they provide a good fit to financial return data and are also extensions to the much well-known student s t and normal distributions. Since it is clear that return data are nonnormal with heavy tails and that volatility is not designed to capture extreme large losses, alternative risk measures must be considered. A risk measure called Value-at-Risk (VaR) can satisfy this need. Instead of measuring return deviations from its mean, it determines the point of relative loss level that is exceeded at a specified degree. Consequently, when suitably adjusted, it can measure the behavior of negative return distributions at a point far away from the expected return. Hence, it is able to take into account the extreme movements of assets return. Furthermore, it gives an easy representation of potential losses since it is none other than the quantile of loss distribution, when the distribution is continuous. This is the aspect that has mainly contributed to its popularity. However, it has a serious drawback. Although it is coherent for elliptical distributions, when applied to nonelliptical distributions it can lead to a centralized portfolio (Artzner et. al. [3]), which is against the diversity principle. In the same case, it is also a generally nonconvex function of portfolio weights, hence making portfolio optimization an expensive computational problem. The more recent and popularly used risk measure is the expected shortfall (ES). It was made popular by Artzner et al. [3] in response to VaR s drawback. Unlike VaR, it always leads to a diversified portfolio while also being a coherent risk measure. Furthermore, this measure takes into account the behaviour of return distributions at and beyond a selected point. Like VaR, it shows the behaviour of the distributions tails, but, with a much wider scope. Ultimately, these attributes make it more favourable than its classic counterpart and we decide to focus on this measure only in this paper. The remainder of this paper is organized as follows. In Section 2 we briefly discuss the general properties of GH distribution. More details of GH calibration using EM algorithm is discussed in the Appendix. Section 3 briefly discusses the expected shortfall as a coherent risk measure. The asset structures of portfolio is elaborated in more details in Section 4. Section 5 discusses the profit and loss (P&L) distribution in terms of the multivariate Generalized Hyperbolic distribution. Portfolio optimization problems are formulated in Section 6. Section 7 discusses numerical examples on real data of the theoretical framework discussed in the above sections. Section 8 concludes this paper. 2

3 2 Generalized Hyperbolic Distribution Before presenting the Generalized Hyperbolic (GH) distribution, it is essential to present the underlying distribution upon which it is built. Definition 2.1 Generalized Inverse Gaussian distribution (GIG). The random variable W is said to have a Generalized Inverse Gaussian (GIG) distribution, written by W N (λ, χ, ψ), if its probability density function is ( h(w; λ, χ, ψ) = χ λ χψ ) λ ( ( ) w λ 1 exp 1 ( χw 1 + ψw )), w, χ, ψ > 0, λ R 2K λ χψ 2 (2.1) where K λ (.) is the modified Bessel function of the second kind with index λ. The Generalized Hyperbolic distribution belongs to the normal mean-variance mixture distribution class defined as follows. Definition 2.2 Multivariate Normal Mean-Variance Mixture Distribution (MNMVM). A random variable X R d is MNMVM distributed if it has the following representation X = µ + W γ + W AZ, (2.2) where µ, γ R d and A R d k are the distribution parameters, Z N k (0, I k ) is a standard multivariate normal random variable, and W is a nonnegative mixing random variable. Also note that the representation requires Σ := AA to be positive definite. In univariate version of the model, the notation Σ is replaced by σ 2. This new class of distribution was first defined by Barndorff-Nielsen [4] specifically for multivariate Generalized Hyperbolic distribution, and was further explored in general by Barndorff-Nielsen et. al. [6]. Its representation gives a clear decomposition and understanding of its parameters: µ as the location parameter, γ as the skewness parameter, and Σ as the scale parameter, while W acts as the shock factor for the skewness and scale. It also proves to be very useful in constructing the calibration method (see Appendix A.2). The Generalized Hyperbolic distribution is defined from representation (2.2). Definition 2.3 Generalized Hyperbolic Distribution (GH). A random variable X R d is said to be a GH-distributed random variable, denoted by X GH(λ, χ, ψ, µ, Σ, γ) (2.3) iff it has the representation (2.2) with W N (λ, χ, ψ) is a scalar GIG distributed random variable. Additionally, X is called symmetric iff γ = 0. 3

4 This representation is consistent with the definition of GH distribution first proposed by Barndorff-Nielsen [4] with pdf K λ d 2 f(x) = c ( (χ + (x µ) Σ 1 (x µ) ) (ψ + γ Σ 1 γ)) e (x µ) Σ 1 γ ( (χ + (x µ) Σ 1 (x µ) ) (ψ + γ Σ 1 γ)) d 2 λ, χ, ψ > 0, λ R (2.4) with the normalizing constant c = ( χψ ) λ ψ λ (ψ + γ Σ 1 γ) d 2 λ (2π) d 2 Σ 1 2 Kλ ( χψ ). (2.5) Now we ll observe the effects of the parameters not present in normal distribution to the shape of the GH distribution. For this purpose, we compare univariate standard normal distribution to GH distribution with varying parameters. The parameters not mentioned will be fixed at level: (λ, χ, ψ, µ, σ, γ) = (1, 1, 1, 0, 1, 0). Figure 1: Effect of γ parameter on GH distribution s shape and comparison with standard normal distribution. 4

5 Figure 2: Effect of χ parameter on GH distribution s shape and comparison with standard normal distribution. Figure 3: Effect of ψ parameter on GH distribution s shape and comparison with standard normal distribution. Figures 1-4 show the effect of changing GH shape paramaters to the shape of the distribution. Figure 1 shows that γ is the main drive to the skewness of the distribution. The more positive the value of γ, the more positively skewed the distribution and the heavier its right tail, and vice versa. Figure 2 and 4 shows that as χ and λ increases, the distribution s peak becomes lower and less acute, but the tails becomes heavier. On the other hand, as ψ increases, the peak gets taller and more acute, but the tails becomes lighter. Although on some cases normal distribution has taller peak, it has considerably lighter tails than the others. 5

6 Figure 4: Effect of λ parameter on GH distribution s shape and comparison with standard normal distribution. Representation (2.2) gives a significant contribution in showing the linearity property of a Generalized Hyperbolic distribution. The following theorem plays a central role in solving optimal portfolio selection problems, discussed in more details in Section 6. Theorem 2.4 If X R d is a Generalized Hyperbolic random variable, i.e, X GH(λ, χ, ψ, µ, Σ, γ), then where B R l d and b R d. BX + b GH(λ, χ, ψ, Bµ + b, BΣB, Bγ), (2.6) 3 Expected Shortfall This section briefly discusses the general properties of expected shortfall associated with uncertain loss in a portfolio due to the volatility of financial market. Hence, a measurement of risk must take into account the randomness of loss and be used to determine the capital reserve to anticipate future loss. In this paper we employ expected shortfall as risk measure, which we shall now discuss. Definition 3.1 Expected Shortfall(ES). For a given β (0, 1), ES is defined as the expectation of a portfolio loss L conditional on itself being at least its valueat-risk V ar β (L): ES β (L) := E[L L > VaR β (L)]. (3.1) Acerbi and Tasche [2] first proved that ES is a coherent risk measure. 6

7 Theorem 3.2 For every β (0, 1), the expected shortfall of a portfolio loss L, ES β (L), is a coherent risk measure: If l is constant, ES β (L + l) = ES β (L) + ES β (l) (translation-invariance criterion) If λ is a positive constant, ES β (λl) = λes β (L) (positive homogeneity criterion) If L is another portfolio loss, ES β (L + L) ES β (L) + ES β ( L) (subadditivity criterion) If L L, ES β (L) ES β ( L) (monotonicity criterion) The subadditivity and positive homogeneity criteria together imply the convexity of the expected shortfall, a property which is very useful in dealing with portfolio optimization problems in later sections. 4 Portfolio Structure The most vital aspect in portfolio optimization problems is the modeling of portfolio risk. As risk comes from portfolio loss value over the holding period, it is important to define the loss function of a portfolio. Denote by V (s) the portfolio value at calendar time s. For a given time horizon, the loss of the portfolio over the period [s, s + ] is defined by L [s,s+ ] := (V (s + ) V (s)). (4.1) In establishing the portfolio theory, is assumed to be a fixed constant. In this case, it is more convenient to use the following definition L t+1 := L [t,(t+1) ] = (V t+1 V t ). (4.2) Here, time series notation is adopted where V t := V (t ). Any random variables with t as the subscript are assumed to be defined in similar way from here on. In the context of risk management where the calendar time t is measured in years, if daily losses are being considered, can be set to 1/250. In this case, t is measured in days, while t acts as a time unit conversion from days to years. Hence, V t and V t+1 represent the portfolio value on days t and t + 1, respectively, and, L t+1 is the loss from day t to day t + 1. From here on equation (4.2) will be used to define portfolio loss with t measured in the time horizon specified by. In standard practice, V t is modelled as a function of time t and a d-dimensional random vector Z t = (Z t,1,..., Z t,d ) of risk factors. In this work, they are assumed to follow some discrete stochastic process. Therefore, V t can be represented as V t = f(t, Z t ) (4.3) 7

8 for some measurable function f : R + R d R. The choice of f depends on the assets contained in the considered portfolio, while the risk factors are usually chosen to be the logarithmic price of financial assets, yields or logarithmic exchange rates. In this paper, the risk factors are chosen to take one of these forms since the distribution models of their time increments have been empirically known as have been made clear in the introduction section. Define the increment process (X t ) by X t := Z t Z t 1. Using the mapping (4.3) the portfolio loss can be written as L t+1 = (f(t + 1, Z t + X t+1 ) f(t, Z t )). (4.4) If f is differentiable, a first-order approximation L t+1 of (4.4) can be considered, L t+1 := (f t (t, Z t ) + d f Zt,i (t, Z t )X t+1,i ), (4.5) where the subscripts to f denote partial derivatives. The first-order approximation gives a convenient computation of loss since it represents loss as the linear combination of risk-factor changes. However, it is best used when the risk-factor changes are likely to be small (i.e. if the risk is measured in small time horizon) and when the portfolio value is almost linear in the risk factors (i.e. if the function f has small second derivatives). 4.1 Stock Portfolio Consider a fixed portfolio of d s stocks and denote by λ s i the number of shares of stock i in the potfolio at time t. Denote the stock i price process by (S t,i ). To fit with the Generalized Hyperbolic framework, we choose Z t,i := ln S t,i to be the risk factor since one of the GH subclass, the Variance Gamma distribution, have been chosen to model the log price process of financial assets in the past by Madan and Seneta [17]. The increment ( of the risk factor then assumes the form of stock s log St+1,i return, i.e. X t+1,i = ln S t,i ). Then, and d s V t = λ s i exp(z t,i ) (4.6) d s L t+1 = λ s i S t,i (exp (X t+1,i ) 1). (4.7) Using first order approximation as in equation (4.5) on (4.7), the loss function can be linearized as d s d s ( ) L t+1 = λ s i S t,i X t+1,i = V t wt,i s St+1,i ln, (4.8) 8 S t,i

9 where w s t,i := λ s i S t,i /V t denotes the stock portfolio weight of stock i at time t. Equation (4.8) gives the linearized risk mapping for stock portfolio. The error from such linearization is small as long as the stock log return is generally small. 4.2 Zero Coupon Bond Portfolio Definition 4.1 (Zero Coupon Bond Portfolio) A zero coupon bond with maturity date T and face value K, also known as a zero, is a contract which promises its holder a payment of amount K to be paid at date T. Denote its price at time t, where 0 t T by p z (t, T ). Definition 4.2 (Continuously Compounded Yield) A continuously compounded yield at time t with 0 t T, denoted by y z (t, T ) for a zero coupon bond with maturity T and face value K is defined as the factor y that satisfies p z (t, T ) = Ke y(t t), (4.9) or equivalently as ( y z (t, T ) := 1/ (T t) ln ( pz (t, T ) K )). (4.10) Since y z (t, T ) takes the form of log price of a financial asset, in this case, a zero coupon bond, it is natural to assume y z (t, T ) as the risk factor in zero coupon bond portfolio. Now, consider a fixed portfolio with d z zero coupon bonds, each has maturity T i and face value K i for i = 1, 2,..., d z. Let the current time t be such that t < min 1 i d z T i. Denote by λ z i the number of bonds with maturity T i in the portfolio. Let Z t,i := y z (t, T i ) be the risk factor. Hence, X t+1,i = y z ((t+1), T i ) y z (t, T i ), the increment of the risk factors. The value of the portfolio at time t is then d z d z V t = λ z i p z (t, T i ) = λ z i K exp ( (T i t ) y z (t, T i )). (4.11) Using first order approximation in similar fashion as with the stock portfolio, the linearized loss can be obtained as d z L t+1 = λ z i p z (t, T i ) (y z (t, T i ) (T i t ) X t+1,i ) d z = V t wt,i z (y z (t, T i ) (T i t ) X t+1,i ), (4.12) where w z t,i := λ z i p z (t, T i )/V t denotes the zero coupon bond portfolio weight of bond i at time t. 9

10 4.3 Fixed Coupon Bond Portfolio Definition 4.3 (Fixed Coupon Bond) A fixed coupon bond is a contract that guarantees its holder a sequence of deterministic payments C 1, C 2,..., C n, called the coupons, at time T 1, T 2,..., T n which is arranged in ascending order. For simplicity, let C n includes its face value. Denote its price at time t as p c (t, T ) where T = T n. Definition 4.4 (Yield to Maturity) A yield to maturity at time t < T 1, denoted by y c (t, T ) for a fixed coupon bond with payments C 1, C 2,..., C n at time T 1, T 2,..., T n, where T = T n, the maturity date, is defined as the value y which satisfies n p c (t, T ) = C i e y(ti t). (4.13) Hence, yield to maturity is defined implicitly and can be solved by a numerical root finding method. Now, consider a fixed portfolio with d c fixed coupon bonds, where the i-th bond provides payments C (i) 1,..., C n (i) i at T (i) 1,..., T n (i) i for i = 1, 2,..., d c. Let the current time t be such that t < min 1 i d c T (i) 1. Denote by λ c i the number of i-th bond in the portfolio. Let Z t,i := y c (t, T (i) ) be the risk factor. Hence, X t+1,i = y c ((t + 1), T (i) ) y c (t, T (i) ), the increment of the risk factors. The value of the portfolio at time t is then d c d z V t = λ c ip c (t, T (i) ) = λ c i ( ni j=1 ( C (i) j e T (i) j ) t y c(t,t (i) ) ). (4.14) Using first order approximation in similar fashion as with the stock portfolio, the linearized loss can be obtained as d c L t+1 = λ c ip c (t, T (i) ) ( ( y ) ) c t, T (i) D i X t+1,i d c ( ( = V t w ) ) t,i c yc t, T (i) D i X t+1,i, (4.15) where w c t,i := λ c ip c (t, T (i) )/V t denotes the coupon bond portfolio weight of bond i at time t, and D i := ni j=1 C(i) j ( e T (i) j ) t y c(t,t (i) ) ( T (i) j p c (t, T (i) ) ) t (4.16) denotes the i-th bond s duration, which is a measure of the sensitivity of the bond price with respect to yield changes, since D i = pc(t,t (i) ) y c(t,t (i) ) /p c(t, T (i) ). 10

11 4.4 Currency Portfolio The risk mapping for stock and currency portfolios are similar in nature. To see this, consider a currency portfolio with d e number of foreign currencies and denote by λ e i the value of currency i in the corresponding currency denomination, in the portfolio at time t. Denote currency i exchange rate process by (e t,i ) in foreign/domestic value. The risk factor is assumed to be Z t,i := ln e t,i with reasons similar to the ones in stock portfolio case. So, the portfolio value at time t is d e V t = λ e i exp(z t,i ). (4.17) Hence, the currency portfolio is similar to the stock portfolio. With notation replacements, equations (4.6)-(4.8) can be used to derive the risk mapping for linearized currency portfolio loss, which is d e ( ) L t+1 = V t wt,i e et+1,i ln, (4.18) where w e t,i := λ e i e t,i /V t denotes the currency portfolio weight of currency i at time t. It is also appropriate to consider a portfolio consisting of assets valued in foreign currency. Consider a fixed portfolio consisting of d noncurrency assets, each valued in foreign currency, i.e., let asset i be valued in currency i. Let p(t, Z t,i ) be the price of asset i which depends on risk factor Z t,i, λ i be the amount of asset i and e t,i be the exchange rate of currency i with respect to the base currency. Then, this portfolio value at time t in the base currency is d d V t = λ i p(t, Z t,i )e t,i = λ i p(t, Z t,i ) exp(ln e t,i ). (4.19) Hence, each asset of the portfolio contains two risk factors: its intrinsic risk factor and the log exchange rate. It follows that the linearized portfolio loss can be obtained by equation (4.5) as d ( )) L et+1,i t+1 = λ i (p Zt,i (t, Z t,i )e t,i X t+1,i + p(t, Z t,i )e t,i ln (4.20) = V t d e t,i ( ) pzt,i (t, Z t,i ) w t,i p(t, Z t,i ) X t+1,i + Xt+1,i e, (4.21) where w t,i := λ i p(t, Z t,i )e t,i /V t is the asset s weight, while X t+1,i and X e t+1,i are the risk-factor increments for the asset and the foreign currency, respectively. Note that the weight for assets valued in foreign currency differs slightly from the weight of other domestic assets. This weight can be regarded as the usual weight multiplied by the exchange rate of the currency of which it is denominated. This interpretation is consistent with the conversion process of its foreign value to its base value. 11 e t,i

12 5 Profit and Loss Distribution Since the risk mapping for each of the portfolio s assets have been obtained, it is time to determine the distribution of the portfolio loss by employing the linearity property of Generalized Hyperbolic distribution presented by theorem 2.4 in the previous section. Due to this linearity property, to make future calculations tractable, only the linearized portfolio loss function is considered. Using notations from the preceding sections, consider a fixed portfolio containing d s stocks, d e currencies, d z zeros and d c fixed coupon bonds. First, assume that all of the assets are denominated in the base currency. Using the formula of linearized loss from preceding section, the linearized loss of this portfolio can be obtained as where ( Xt+1,i s St+1,i := ln ( ds L t+1 = V t d z d c w s t,ix s t+1,i+ w z t,i ( yz (t, T i ) (T i t ) X z t+1,i) + w c t,i ( yc ( t, T (i) ) D i X c t+1,i) + d e wt,ix e t+1,i e ), (5.1) ( et+1,i e t,i ), X z t+1,i := (y z ((t + 1), T i ) y z (t, T i )) S t,i ), Xt+1,i e := ln and Xt+1,i c := ( y c ((t + 1), T (i) ) y c (t, T (i) ) ) are the risk factors for stocks, currencies, zeros and fixed coupon bonds, respectively. Next, as has been assumed, let X := (Xt+1,1, s..., Xt+1,d s s, Xt+1,1, z..., Xt+1,d z z, Xt+1,1, c..., Xt+1,d c c, Xt+1,1, e..., Xt+1,d e e ) GH(λ, χ, ψ, µ, Σ, γ). Note that X R d, where d := d s + d e + d z + d c. By (5.1), it is clear that L t+1 = V t w (b + BX) (5.2) where w is the weight vector corresponding to each portfolio assets (arranged in similar fashion as X), and b R d and B R d d are constant vector and diagonal matrix, respectively, such that y z (t, T i ), i = d s + 1,..., d s + d z b i = y c (t, T i ), i = d s + d z + 1,..., d s + d z + d c (5.3) 0, otherwise and (T i t ), i = d s + 1,..., d s + d z D B ii = i, i = d s + d z + 1,..., d s + d z + d c 0, i = d d e + 1,..., d and if currency i is not held 1, otherwise 12. (5.4)

13 Now, consider the same case, with the exception that some of the non-currency assets are denominated in foreign currency. By the arguments within section 4.4, equation (5.2) can still be used to represent the linearized loss of this portfolio with the exception that each components of the weight vector that corresponds with assets valued in foreign currency must be multiplied by the foreign exchange rate at time t, and that the diagonal matrix B has to be modified into a sparse matrix with same diagonal entries as before, but with nondiagonal entries { 1, if asset i is valued in currency j B ij = 0, otherwise (5.5) where i = 1,..., d d e and j = d d e + 1,..., d. By evoking theorem 2.4, the preceding representation can be simplified into L t+1 = V t w X, (5.6) where X GH(λ, χ, ψ, b + Bµ, BΣB, Bγ). Also note that by the same theorem, L t+1 GH(λ, χ, ψ, V t w (b + Bµ), (V t ) 2 w BΣB w, V t w Bγ). This is one of the advantages of modeling risk-factor increments with Generalized Hyperbolic distribution, as the linearized portfolio is also Generalized Hyperbolic distributed. Note also that for optimization purposes, equation (5.6) can be used to represent portfolio losses. This concludes that a portfolio loss function can be represented by its weight vector and the vector of risk-factor increments through equation (5.6). 6 Portfolio Optimization As is argued by the linearity property of Generalized Hyperbolic distribution, a portfolio loss function is approximated by its linearized counterpart, and so is the portfolio profit function. So, from here on, the notation L will be used as the linearized portfolio function, replacing the role of L. To set up the portfolio optimization problem, let X GH(λ, χ, ψ, µ, Σ, γ) (6.1) be a d-dimensional Generalized Hyperbolic random variable. This X can be regarded as the variable X in equation (5.6) by adjusting the parameters of X because that equation can be used to represent portfolio losses. Next, let M := { L : L = l + λ X, λ R d, l R } be the set of portfolio losses and the domain for the expected shortfall ES β. Note that by the discussion from the previous section, the distribution of L itself is a function of portfolio weights. Hence, so are its expectation and its risk measure. The followings are the optimization versions which will be discussed. Definition 6.1 (Markowitz Optimization Problem) min ς(w) subject to R(w) = λ and w 1 = 1, (P1) 13

14 where λ is the target expected return, w is the portfolio weights, and ς(w) := ES β (L(w)) (6.2) R(w) := E[ L(w)] (6.3) By representation (5.6), the portfolio loss L has a one-to-one correspondence with w, the portfolio weights. Hence, it follows that ς(w) is convex on R d since ES β (L(w)) is convex. Definition 6.2 (RORC Optimization Problem) R(w) ς(w) max subject to w 1 = 1. (P2) The objective function in this problem can be nonconvex, but, it can easily be seen that the solution of this problem lies on the efficient frontier of the problem (P1). In cases where the maximum of the RORC cannot be found analytically, this fact can be used to find the maximum of the ratio by selecting the point on the efficient frontier that yields the highest value of the ratio. This way, the nonconvex problem is reduced to a fixed number of convex optimization problems which are efficiently solvable. 6.1 Optimization in Symmetric Case In the symmetric case, γ in equation (6.1) has to be set to 0, and so, E[w X] = w E[X] = w µ. (6.4) Hence, the set of feasible weights for the Markowitz optimization problem can be defined as S := {w : w 1 = 1, w µ = µ p }. The problem will first be discussed. Assume β > 0.5. Since w X GH(λ, χ, ψ, w µ, w Σw, 0), then X := ( w X + w µ)/ w Σw GH(λ, χ, ψ, 0, 1, 0) doesn t depend on w. Hence, note that ES β (L) = µ p + q w Σw, (6.5) where q :=ES β ( X) does not depend on w. Since E[ X] = 0, β > 0.5 and X is symmetric, it is clear that q is a nonnegative constant. This implies that We have just proven the following proposition. argmin w ES(L) = argmin w w Σw. (6.6) Proposition 6.3 (Equality of Markowitz-Optimal Weights) In the framework of symmetric Generalized Hyperbolic, the optimal portfolio composition obtained from Markowitz optimization problem using Expected Shortfall at confidence level β 0.5 is equal to that of using volatility as risk measure. 14

15 So, optimization under expected shortfall is equivalent to that under volatility. Hence, by standard Lagrangian method, the optimal weights can be obtained as where K 1 and K 2 are constant vectors w = K 1 + K 2 µ p, (6.7) K 1 = 1 D [A 1Σ 1 1 A 2 Σ 1 µ] (6.8) K 2 = 1 D [A 3Σ 1 µ A 2 Σ 1 1], (6.9) with A 1 := µ Σ 1 µ, A 2 := µ Σ 1 1, A 3 := 1 Σ 1 1 and D := A 1 A 3 A 2 2. Equation (6.5) can be further used to find the expected return level µ p which yields the minimum value of ES. Since ES can be regarded as a function of µ p, define a function f with f(µ p ) := µ p + q w Σw. (6.10) Substituting w from (6.7) to (6.10) yields f(µ p ) = µ p + q K 2 ΣK 2 µ 2 p + 2K 1 ΣK 2 µ p + K 1 ΣK 1. (6.11) By differentiation, it can be obtained that f K 2 ΣK 2 µ p + K 1 ΣK 2 (µ p ) = 1 + q (6.12) K 2 ΣK 2 µ 2p + 2K 1 ΣK 2 µ p + K 1 ΣK 1 f (K 2 ΣK 2 ) (K 1 ΣK 1 ) (K 1 ΣK 2 ) 2 (µ p ) = q ( ) K2 ΣK 2 µ 2 p + 2K 1 ΣK 2 µ p + K 3/2. (6.13) 1 ΣK 1 By Schwarz inequality, (K 2 ΣK 2 ) (K 1 ΣK 1 ) (K 1 ΣK 2 ) 2 0. It follows then that f is nonnegative and so f is convex with respect to µ p. Observe that if f = 0, the minimum of f doesn t exist. Assume it is positive. To see another condition for the existence of the minimum, observe that lim f (µ p ) = 1 ± q K 2 ΣK 2. (6.14) µ p ± By the monotonicity of f, it follows that the minimum of f exists if lim µ p f (µ p ) = 1 + q K 2 ΣK 2 > 0. (6.15) Assume the minimum exists. Define J 1 := K 2 ΣK 2, J 2 := K 1 ΣK 2 and J 3 := K 1 ΣK 1 to simplify matters. Since f is convex, the minimum can be found by solving the root of f. Observe that the root-finding problem of f can be transformed into the problem of solving J 1 (1 q 2 J 1 )µ 2 p + 2J 2 (1 q 2 J 1 )µ p + (J 3 q 2 J 2 2 ) = 0. (6.16) 15

16 Equation (6.16) yields µ 1 = 2J 2(1 q 2 J 1 ) D 2J 1 (1 q 2 J 1 ) and µ 2 = 2J 2(1 q 2 J 1 ) + D, (6.17) 2J 1 (1 q 2 J 1 ) where D := 4( 1 + q 2 J 1 )(J 1 J 3 J2 2 ). If D 0, then, only one between µ 1 and µ 2 can be the solution. To see which one, observe that (J 1 q 1 µ p+j 2 ) 2 f J 1 µ 2 p+2j 2 µ p+j 3 if µ p J 2 J 1 (µ p ) =. (6.18) (J 1 + q 1 µ p+j 2 ) 2 J 1 µ 2 p+2j 2 µ p+j 3 if µ p > J 2 J 1 Hence, it is necessary for µ p to be greater than J 2 J 1 to have f (µ p ) = 0. Since it is clear that µ 1 J 2 J 1, the root is µ 2. This result can be summed up by the following proposition. Proposition 6.4 (Global Minimum of ES) In symmetric GH framework of portfolio optimization, the minimum of ES at confidence level β 0.5 of portfolio loss exists iff (K 2 ΣK 2 ) (K 1 ΣK 1 ) (K 1 ΣK 2 ) 2 > 0 (6.19) and If it exists, the minimum is achieved at 1 + q K 2 ΣK 2 > 0. (6.20) µ p = 2 (K 1 ΣK 2 ) (1 q 2 (K 2 ΣK 2 )) + D 2 (K 2 ΣK 2 ) (1 q 2 (K, (6.21) 2 ΣK 2 )) where D = 4( 1 + q 2 (K 2 ΣK 2 ))((K 2 ΣK 2 ) (K 1 ΣK 1 ) (K 1 ΣK 2 ) 2 ). Next, to find the optimal weights of the RORC optimization problem, the following proposition can be used. Proposition 6.5 (RORC in Symmetric Framework) Let β 0.5 at the framework of symmetric Generalized Hyperbolic. Let RORC σ and RORC ES be the RORCs for return-volatility and return-es optimization problems, respectively, with equal set of weight constraints. Then, if µ = argmaxrorc σ (µ) and the global minimum value of ES β is positive, µ = argmaxrorc ES (µ). Proof By equation (6.5), ES β can be expressed as a function of µ p only, i.e., ES β (µ p ) = µ p + qσ(µ p ), (6.22) where σ(µ p ) is the minimum volatility corresponding to µ p. Let µ = argmax RORC σ (µ), then, for every µ R, µ µ (6.23) σ( µ) σ(µ ) µσ(µ ) µ σ( µ). (6.24) 16

17 Next, note that µ ES β ( µ) µ ES β (µ ) = µes β(µ ) µ ES β ( µ) ES β ( µ)es β (µ ) (6.25) = q ( µσ(µ ) µ σ( µ)) 0, ES β ( µ)es β (µ ) (6.26) where the last equality is due to equation (6.22), while the inequality is due to equation (6.24) and the positivity of q and ES β. The proof is complete. Hence, optimal returns which yield the minimum RORC for optimizations using ES is equal to those using volatility. It is therefore easier to compute the minimum RORC using volatility as risk measure. In the following discussions, the existence conditions of minimum RORC will be derived using this property. Note that given targeted expected return, µ p, the portfolio loss volatility can be expressed as a function of µ p σ(l) = ( ) Kw Σw = K K 1 ΣK 1 (K 1 ΣK 2 ) 2 K, (6.27) 2 ΣK 2 where K := E[W ] is the expectation of the Generalized Inverse Gaussian mixing distribution which depends only on the inner parameters of GH: χ, λ and ψ, and therefore is a constant. So, the RORC with volatility as the risk measure can be expressed as RORC(µ p ) = µ p σ (µ p ). (6.28) The first derivative of RORC will first be analyzed. It can be obtained that RORC (µ p ) = σ ) µp 2.(σ2 σ σ 2 = 2σ2 µ p (σ 2 ) 2σ 3 = K. 2K 1ΣK 2 µ p + K 1ΣK 1 σ 3. (6.29) Set µ p := K 1 ΣK 1. From the third equation of (6.29), there can be three cases. 2K 1 ΣK 2 First, if K 1ΣK 2 < 0, then the RORC function will be strictly increasing over (, µ p) and strictly decreasing over (µ p, ). In this case, the RORC will then be maximized at µ p = µ p. Second, if K 1ΣK 2 0, then the RORC will increase to its asymptote µ p lim µ p σ = 1. (6.30) K.K 2 ΣK 2 17

18 as µ p tends to, where its asymptote is the maximum value. In the case where K 1ΣK 2 > 0, this asymptote is its maximum value since µ p lim µ p σ = 1. (6.31) K.K 2 ΣK 2 The preceding results therein lead to the following proposition. Proposition 6.6 (Conditions of the Existence of Maximum RORC) In the symmetric GH framework, maximum RORC with ES at confidence level β 0.5 as the risk measure exists iff K 1ΣK 2 < 0. If it exists, the maximum is achieved at the expected return on the level 6.2 Optimization in Asymmetric Case µ p := K 1ΣK 1 2K 1ΣK 2. (6.32) In the asymmetric GH framework, ES β (L) = ES β ( w X) cannot be expressed in basic functions like in equation (6.5). It is not a linear transformation of volatility, since the factor q in the equation contains the term w γ, i.e. q 2 = ES β (Y ), Y GH ( λ, χ, ψ, 0, 1, w γ w Σw ). (6.33) But, for the purpose of numerical computations, it can be expressed in integral form as follows ES β ( w X) = E[ w X w X > VaR β ( w X)] (6.34) = E[w X w X VaR 1 β (w X)] (6.35) ( ) 1 F 1 Y (1 β) = yf Y (y)dy, (6.36) 1 β where Y = w X GH (λ, χ, ψ, w µ, w Σw, w γ), and F Y andf Y denote the cdf and pdf of Y, respectively. The computation of F 1 Y (1 β) can be done by numerical root finding methods, while the integral in (6.36) is done by numerical integration methods. 7 Numerical Results For simulation purposes, the following assets are used: 4 stocks, 3 foreign currencies, 2 Indonesian government-issued zero coupon bonds and 1 Indonesian governmentissued international fixed coupon bond. We take IDR as the base currency. The stocks are chosen from international blue-chip companies: 1. ASII (Astra International), denominated in IDR 18

19 2. BACH (Bank of China), denominated in CNY 3. INTC (Intel Corporation), denominated in USD 4. RDS (Royal Dutch Shell), denominated in EUR The foreign currencies are chosen to represent the three most developed regions: 1. CNY (Chinese Yuan) 2. EUR (Euro) 3. USD (US dollar) The zero coupon bonds are government issued of series 1. ZC3 with specifications ISIN: IDB BB Number: EH Amount Issued: IDR 1,500, Amount Outstanding: IDR 1,249, Par Amount: IDR 1,000,000 Maturity: 11/20/2012 Denomination: IDR 2. ZC5 wih specifications ISIN: IDB BB Number: EH Amount Issued: IDR 3,150, Amount Outstanding: IDR 1,263, Par Amount: IDR 1,000,000 Maturity: 2/20/2013 Denomination: IDR while the fixed coupon bond is of series 1. INDO38 with specifications Par Amount: USD 1,000,000 Coupon: 7.75% p.a. Coupon payment date: 17 January & 17 July 19

20 Maturity: 11/17/2038 Denomination: USD As has been explained before, the source of uncertainty of a portfolio comes from its risk factors. The increments of those risk factors are assumed to be Generalized Hyperbolic distributed. For stocks and currencies, the increments are their log returns, respectively. While for zero coupon bonds and fixed coupon bonds, they are the continuously compounded yields and yield-to-maturity, respectively. For bonds, the calculations of time to coupon maturities are done under the actual/365 rule. The data obtained for all of the assets are their trading prices from 6 February 2008 until 4 March The stocks and currencies log returns can be calculated directly. For bonds, the prices are quoted as the percentage of their par values. Yields of zero coupon bonds can be calculated using equation (4.10). Their trading prices can be used as the value of pz(t,t ). Yield to maturity of a fixed coupon bond K can be calculated by solving equation (4.13) for y by root finding methods. Since both sides of the equation can be divided by the bond s par value, the trading price can be plugged directly into variable p c (t, T ). Note that for INDO38, there are 57 coupon payments with the last coincide with the redeem payment of the bond, i.e., C i = for i = 1,..., 56 and C 57 = Additionally, the following terms are used on the portfolio weights: unconstrained and constrained. An asset weight is called unconstrained if asset shortings are permitted, and constrained if no shorts are allowed. 7.1 Accuracy and Robustness of First-Order Approximation of Loss In this section, we will address the issue of the accuracy and robustness of the firstorder approximation scheme of portfolio loss function introduced in section 4 via equation (4.5) by means of our assets. Our formal method of assessing the approximation performance will be by testing whether the first-order loss approximation signficantly differs from the true realized loss via the 2-sample Kolmogorov-Smirnov test, where the data is obtained from the historical loss values of the assets during the sampling period. If they are not significantly different (the null hypothesis), then the calibration and optimization results in later sections can be justified in the sense that they represent approximations to the results when true loss is used. Figures 5 gives a graphical view of the accuracy of the first-order approximation in terms of distribution shapes for individual selected assets. In many cases, the tail density between the true and the approximation differs. In the case of INDO38 s loss, the first-order loss has heavier tails than the true loss. In the remaining cases, differences can be seen from the lower tails, while the upper tails only slightly differ. But, overall these differences are not too significant and can be tolerated by the Kolmogorov-Smirnov test as shown in table 1 by the high p-values. As Expected Shortfall strongly dependes on the shape of the tail-distribution rather than the whole, the Kolmogorov-Smirnov test is applied to the conditional distributions on 20

21 that the losses of the assets are below certain quantiles, which we choose to be 95%, 99% and 99.9%. Figure 5: QQplot comparison between empirical true and 1st order loss of a) ZC5; b) INDO38; c) ASII; d) CNY (red lines indicate 45-degree lines) 95%-quantile 99%-quantile 99.9%-quantile Asset KS p-value KS p-value KS p-value ZC ZC INDO ASII BACH INTC RDS CNY EUR USD Table 1: 2-sample Kolmogorov-Smirnov statistics (KS) and p-values of true versus first-order loss for each individual asset s losses conditioned on losses being below certain quantiles. 21

22 To test how robust the linear approximation is, we will compare the true and first-order approximate losses of a sample of generated portfolios with perturbed weights. The procedure is as follows. First, for each asset in the basket, a random weight following continuous uniform distribution from some interval is generated. For the three bonds, each of the weight is drawn from [-100,100]. For the four stocks, each is drawn from [-10000,10000], while for the currencies, each is drawn from [ , ]. Afterwards, the true losses and the approximate first-order losses of a portfolio composed according to the previously simulated weights are calculated using the dataset of the historic prices. This procedure is then repeated 2000 times where in each step the random weights are resampled from the aforementioned uniform distributions, giving 2000 series of pairs of true and approximate losses. For each series of loss pairs, a Kolmogorov-Smirnov test is performed to check if the hypothesis that the conditional distributions of the true and the approximate losses, on that the losses are below the 95th-quantile, coincide can be sustained or not. Figure 6: Dataplot of statistics and p-values of 2-sample Kolmogorov-Smirnov tests applied to 2,000 pairs of portfolio true and first-order losses, conditioned on losses being below the 95th-quantile Figures 6 summarizes the result. The statistics range from to with average of , while the p-values range from to with average of In every case, we have found that the true and first-order loss populations are not significantly different. Thus, we conclude that the first-order approximation of portfolio loss is accurate and robust, and therefore is justified. 7.2 Calibration Results Since our paper revolves around portfolio optimization and that optimization in 22

23 symmetric framework can be solved analytically, for the purpose of current section, we will give focus only on the asymmetric Generalized Hyperbolic model. Additionally, an advantage of modeling with the asymmetric model is that it gives us an additional degrees of freedom as we do not have to fix the parameter γ to 0. We can then observe from the calibration result whether the value of γ and its skewness are close to 0 or not to deem whether the better model is the asymmetric or the symmetric one. The calibration results are presented for the percentage risk-factor increments of the assets used (original risk-factor increments data multiplied by 100). They will be presented for both the univariate and multivariate asymmetric Generalized Hyperbolic distributions. When discussing about the multivariate Generalized Hyperbolic calibration of the data, the following order is used for the order of the elements of GH random vector: 1. ZC3 yield increment 2. ZC5 yield increment 3. INDO38 yield increment 4. ASII log return 5. BACH log return 6. INTC log return 7. RDS log return 8. CNY log return 9. EUR log return 10. USD log return The calibration is performed by employing the EM algorithm detailed in the Appendix. Again, note that the calibration can only be done with fixed λ. For nonnegative λ, χ-algorithm is used, while for negative λ, ψ-algorithm is used. For each λ, the calibration is terminated when the likelihood increment between the current and previous iteration is less than a specified value. More specifically, denote the observed Generalized Hyperbolic random vector by X = (X 1,..., X n ) and the Generalized Hyperbolic estimated parameters at iteration k by ˆθ k = (λ (k), χ (k), ψ (k), µ (k), Σ (k), γ (k) ). Given positive ɛ, the calibration process is terminated at iteration k + 1 if L(ˆθ k+1 ; X) L(ˆθ k ; X) < ɛ. (7.1) Here, L(θ, X) is the likelihood function (see Appendix A.1). The EM algorithm is then combined with a one dimensional optimization method to find the value of λ 23

24 Empirical Asset s k ZC ZC INDO ASII BACH INTC RDS CNY EUR USD Table 2: Empirical skewness and kurtosis of percentage assets returns. that yields the highest likelihood value. MATLAB fminbnd function is used for this purpose, where the search region for λ is from 10 to 10. Table 2 gives a list of the empirical skewness and kurtosis of each individual asset s risk-factor increment. Clearly, the assets are heavily skewed and therefore not symmetric. Therefore, our decision of choosing the asymmetric Generalized Hyperbolic to model the assets distribution is justified. Table 3 shows the calibrated parameters of the univariate Generalized Hyperbolic distribution for all of the assets used. µ γ σ 2 χ ψ λ ZC ZC INDO ASII BACH INTC RDS CNY EUR USD Table 3: Parameters of calibrated univariate Generalized Hyperbolic distribution for portfolio assets. The followings are the parameters of the calibrated multivariate GH distribution: 24

25 λ = χ = ψ = µ = γ = Σ = , and also the expectation, covariance and correlation of X E[X] = Cov(X) =

26 Corr(X) = From the expected value, we see that stocks yield the highest returns amongst other asset classes, with the highest being that of ASII log return. Correlations between assets within the same class are positive, while correlations between zero coupon bonds and INDO38 are negative, although they are mostly positive with the stocks. Their correlations with CNY are negative, but are of varying signs with EUR and USD. Stocks are negatively correlated with CNY and USD, while being positively correlated with EUR. 7.3 Goodness of Fit In this section, the goodness of fit of calibrated parameters of Generalized Hyperbolic on the data are analyzed. To gain some confidence that Generalized Hyperbolic provides a good fit, some univariate examples of the data will first be analyzed. Figure 7: QQplot comparison between normal and GH distribution of daily a) ZC5 yield increment; b) INDO38 yield increment; c) ASII log return; d) CNY log return Figure 7 shows the comparison between the QQplots of normal distribution and the Generalized Hyperbolic distribution. The normal distribution does not provide a 26

27 good fit since its QQplots are not linear. They form the inverted S shape that show that the actual distributions of the data have higher kurtosis than normal and have heavier tail. Meanwhile, the calibrated Generalized Hyperbolic distributions show a significant improvement over normal distribution. In every case, the QQplot of the GH distribution is almost linear, showing a good fit to the empirical distribution. Figure 8: Models for ASII daily log return. Figure 9: Models for CNY/IDR daily log return. To get a better look at how the kurtosis and skewness of the theoretical distributions match the ones of the actual distributions, a comparison between the histogram of the empirical distribution and the pdf of the theoretical distributions are shown by figures The figures include the comparisons between normal 27

28 Figure 10: Models for ZC05 daily yield return. Figure 11: Models for INDO38 daily yield return. and GH distributions, as well as the marginal distributions of the multivariate GH distributions. The empirical log pdf is generated by the Epanechnikov smoothing kernel. It can be seen in all cases that GH distribution gives the better fit over normal distribution. One of the main reason is that it can adapt its shape to have the excess kurtosis feature that all of the assets exhibit. Moreover, as is shown by the log pdf plots, the distribution s tail heaviness can match that of the empirical distribution, unlike the normal one. Between the univariate and the marginal distributions, it can also be seen that the univariate gives the better fit. The main reason is because in the marginal 28

29 case, the shape parameters are fixed as the result of the interdependence structure between the assets, unlike those from the univariate case. This lessens the freedom of individual parameter calibration to fit univariate empirical data. Normal GH Asset log L s k log L ZC ZC INDO ASII BACH INTC RDS CNY EUR USD Marginal GH Asset s k log L ZC ZC INDO ASII BACH INTC RDS CNY EUR USD Table 4: Comparison between calibrated univariate and marginal distributions of percentage assets returns, showing log-likelihood values (log L), skewness and kurtosis. Comparisons between the empirical and theoretical kurtosis and skewness, as well as the log likelihood values of the theoretical distributions can be seen at tables 2 and 4. The likelihood values are the highest in the univariate cases, as is predicted. Unfortunately, we cannot provide any statistical test for the skewness and kurtosis of the GH distribution so we have to settle with the rather intuitive argument. But, it can be seen that for most assets the empirical skewness is significantly different than zero, so the symmetric model is somewhat doubtful. This further justifies our decision to calibrate using the asymmetric GH distribution. The fitted kurtosis of the univariate models also deviate much from the empirical kurtosis for some assets, although not as much as in the marginal cases. Although the deviations are large, it does not necessarily condemn the GH model as a poor fit to our data. 29