International Journal of Analysis and Applications Volume 19, Number 3 (2021), 319-340 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-19-2021-319 THE DOWNSIDE AND UPSIDE BETA VALUATION IN THE VARIANCE-GAMMA MODEL ROMAN V. IVANOV∗ Laboratory of Control under Incomplete Information, V.A. Trapeznikov Institute of Control Sciences of RAS, Moscow, Russian Federation ∗Corresponding author: roivanov@yahoo.com Abstract. The paper is aimed to assess the risks and gains of investment portfolio which relate to the impact of a particular asset. We consider the investment portfolios which consist of assets with variance- gamma, gamma distributed and deterministic returns. The returns are assumed to be dependent. We derive analytical formulas for the downside and upside betas in the discussed framework. The established formulas depend on the values of a number of special mathematical functions including the values of the generalized hypergeometric ones. 1. Introduction The basic monetary risk measures value at risk (see, for example, Berkowitz et al. [6], Chen and Tang [8], Ivanov [20], Stoyanov et al. [42]) and conditional value at risk (Kalinchenko et al. [22], Mafusalov and Uryasev [29], Rockafellar and Uryasev [37]) serve to assess the downward risk of the investment portfolio. But if we want to rate the influence of a specific asset on the return of the portfolio, we exploit the market beta. When we form the investment portfolio, it is necessary to estimate how the share increase or decrease for a particular asset impacts the risks and the expected profit of the portfolio. The downside beta serves to evaluate the risk size, the upside beta is used to outlay the profit. Received February 22nd, 2021; accepted March 17th, 2021; published April 1st, 2021. 2010 Mathematics Subject Classification. 60E07, 60E08, 65C20, 91G10, 91G60, 33C20, 33C90. Key words and phrases. downside and upside betas; variance-gamma distribution; investment portfolio; dependence; Appell function. ©2021 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 319 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-319 Int. J. Anal. Appl. 19 (3) (2021) 320 The ideas of use the downside and upside betas go back to the paper by Roy [39] and the monograph by Markowitz [30], where it was argued that investors more care about downside losses and upside gains. In this context, Markowitz [30] suggested to use the semivariance as the basic risk measure. The semivariance beta was introduced in Hogan and Warren [19]. The advantage of downside and upside betas over the traditional ones is proposed in Ang et al. [2] and Tehir et al. [43]. The work by Estrada [13] suggests a capital asset pricing model based on the downside beta. Guy [18] presents a portfolio construction based on the assessment of the values of the upside and downside betas. Rutkowska-Ziarko and Pyke [38] introduce the downside accounting beta suggesting to use it for the measurement of the systemic risk. Altigan et al. [1] claim that the downside beta valuation is not sufficient for the asset pricing on international markets contrary to the results for the US equity market. And therefore it is required to take into account the upside one also if we want to create a general model. Advantages of downside beta-based capital asset pricing model over the traditional one are presented in Ayub et al. [4] and Post and Van Vliet [35]. In the context of the general theory, the downside beta relates to the class of loss-based risk measures which is considered in Cont et al. [9]. The variance-gamma distribution was proposed as a model for market stock returns in Madan and Seneta [28]. An utility-based option pricing theory which exploits the variance-gamma distribution was suggested in Madan and Milne [27]. The price of European call option in the variance-gamma model was derived analytically in the paper by Madan et al. [26]. There is a number of modern research papers which confirms statistically the idea of use the variance- gamma distribution for the financial index modeling. Daal and Madan [12] and Finlay and Seneta [14] approve the variance-gamma model for the exchange rate simulation. Linders and Stassen [23], Moos- brucker [31] and Rathgeber et al. [36] model with the variance-gamma distribution the Dow Jones index returns. Mozumder et al. [32] consider the S&P500 index options in the variance-gamma model. Luciano and Schoutens [25] model the S&P500, the Nikkei225 and the Eurostoxx50 financial indexes by the variance- gamma process. Luciano et al. [24] and Wallmeier and Diethelm [44] confirm the use of the variance-gamma distribution for the modeling of the US and the Swiss stock markets, respectively. Groups of various finan- cial indices are modeled by the multivariate variance-gamma distribution in Nitithumbundit and Chan [34]. Flora and Vargiolu [15] find that the variance-gamma process is the best fit for the carbon price dynamics. Göncü et al. [16] show that the variance-gamma model fits well with the financial data of developed markets. This paper is set to compute the downside and upside betas for the investment portfolio with the variance- gamma, gamma distributed and deterministic asset returns. The gamma and deterministic returns relate to the modeling of credit risk, see Ivanov [21] and My [33]. As usually, the variance-gamma random variables are modeled as the normal mean-variance mixtures and it is supposed that the normal distributions are correlated. The paper develops the direction of research of Madan et al. [26], Ano and Ivanov [3] and Int. J. Anal. Appl. 19 (3) (2021) 321 Ivanov [20], where closed form expressions in the variance-gamma framework are derived for various targets of mathematical finance. 2. Main notations We denote by γ = γ(a,b) the gamma random variable with parameters a,b > 0. The gamma distribution has the probability density function f(γ,x) = baxa−1e−bx Γ (a) , x > 0,(2.1) where Γ(χ) is the gamma function. It has the characteristic function ψ(γ,u) = ( 1 − iu b )−a ,(2.2) the mean and the variance a b and a b2 , respectively. By definition, the variance-gamma distribution is the mean-variance normal mixture, where the mixing density is the gamma distribution. That is, the variance-gamma random variable H is defined as H = r + θγ + σ √ γN,(2.3) where r,θ ∈ R, σ > 0, N is the standard normally distributed random variable and the gamma random variable γ is independent with N. Throughout this work, we do not assume that γ has necessary the mean 1, that is the identity a = b is not required. Next, we set sg(χ) :=   1 if χ > 0, 0 if χ = 0, −1 if χ < 0, and use notations N(χ),χ ∈ R, B(χ1,χ2),χ1 > 0,χ2 > 0, Kχ1 (χ2),χ1 ∈ R,χ2 > 0 for the normal distribution function, the beta function and the MacDonald function (the modified Bessel function of the second kind), respectively. The hypergeometric Gauss function is denoted as F(χ1,χ2,χ3; χ4), χ1,χ2,χ3 ∈ R,χ4 < 1. Int. J. Anal. Appl. 19 (3) (2021) 322 Also, we discuss one of the degenerate Appell functions (or the Humbert series) which is the double sum Φ(χ1,χ2,χ3; χ4,χ5) = ∞∑ m=0 ∞∑ n=0 (χ1)m+n(χ2)m m!n!(χ3)m+n χm4 χ n 5 with χ1,χ2,χ3,χ5 ∈ R and |χ4| < 1, where (χ)l, l ∈ N ∪ {0}, is the Pochhammer’s symbol. For more information on the special mathematical functions above and relations between them, see the monographs by Bateman and Erdélyi [5] and Srivastava and Karlsson [40], the handbook by Gradshteyn and Ryzhik [17] and the papers by Chaudhry et al. [7] and Srivastava et al. [41]. 3. Setup and results Let Aj,t, j = 1, 2, ...,n, be the values of n assets at time moments t = 0, 1. It is assumed that Aj,0 are constant but Aj,1 are random with Law (Aj,1 −Aj,0) = Hj, where Hj = rj + θjγj + σj √ γjNj(3.1) are constant, gamma or variance-gamma random variables in dependence with the values of the parameters rj,θj ∈ R, σj ≥ 0. We suggest that σj > 0 for at least one j ∈ {1, 2, ...,n}. It is supposed that the normal random variables Nj and Nl are correlated with coefficients ρjl, j, l ≤ n. All the gamma random variables are assumed to be independent with the normal ones. We suggest that γj = κjγ, where γ = γ(a,b) is gamma distributed, κj ≥ 0, j = 1, 2, ...,n. Also, we set for the simplicity of formulas below that ρlm = 0 if σl = 0 or σm = 0. This model is a particular case of a more general model which is discussed in Ivanov [20]. Briefly, we assume here that the asset returns are highly dependent with each other. It agrees with the last investigations on the financial market structure, see Cont and Sirignano [10]. Together with it, the strong dependence between stocks originates to the decisions of a large investor (Cont and Wagalath [11]). The value It at time moments t = 0, 1 of the investment portfolio x = (x1,x2, ...,xn) ∈ Rn is defined as It = n∑ j=1 Ij,t, where Ij,t = xjXj,t, j = 1, 2, ...,n. Int. J. Anal. Appl. 19 (3) (2021) 323 Since all Xj,0 are constant, it is enough for the aim to evaluate the portfolio risks to discuss the random increment X = I1 − I0 = n∑ j=1 xjHj(3.2) of the portfolio and the random increments of the investments Xj = Ij,1 − Ij,0 = xj (Xj,1 −Xj,0) = xjHj, j = 1, 2, ...,n.(3.3) Throughout this work we consider the downside beta β−, the investment portfolio characteristics which is defined as β− = E [ (Xj − EXj)(X − EX)I{X≤u} ] E [ (X − EX)2I{X≤u} ] , u ∈ R. Taking into account the value of β−, one could analyze is it expedient to put the size xj into the asset j or not. For example, if β− � 0, it means that the investment xjXj,0 accelerates the downside risk of the portfolio substantially. Together with the downside beta, we discuss the upside beta β+ = E [ (Xj − EXj)(X − EX)I{X≥u} ] E [ (X − EX)2I{X≥u} ] , u ∈ R. The upside beta shows the impact of the investment xj on the potential earnings of the investment portfolio. To introduce the results, now we suggest some auxiliary abbreviations. Let hj = xj ( rj + θjaj bj ) , h = n∑ l=1 hl, ŝ = n∑ l=1 xlrl û = u− ŝ, s1 = n∑ l=1 xlθlκl, s2 = n∑ l=1 ρjlxlσl √ κl, s3 = √√√√ n∑ m,l=1 ρmlxmσmxlσl √ κmκl, s = û √ s21 + 2bs 2 3 s3|s3| , q = − sg(s3)s1√ s21 + 2bs 2 3 . Next, we set Yl = xlσl √ γlNl, xl,σl 6= 0. The lemma below computes the value of the expectation f1 = E ( Yj n∑ l=1 YlI{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ) . Lemma 3.1. If xj,σj 6= 0, the expectation f1 = xjσjs2b a√κj s3Γ(a) ( s3Λ(û) − exp ( ûs1 s23 ) √ 2π ( ûΘ(û, 0) −s1Θ(û, 1) )) ,(3.4) Int. J. Anal. Appl. 19 (3) (2021) 324 where Λ(û) = Γ ( a + 3 2 ) ba+1 √ 2π ( B ( 1 2 ,a + 1 ) √ 2 − s1 s3 √ b F ( a + 3 2 , 1 2 , 3 2 ;− s21 2bs23 )) I{û=0}+ + |s|a+ 1 2 es(1 + q)a+1 ba+1 √ 2π (( |s|Ka+ 3 2 (|s|) + sKa+ 1 2 (|s|) ) Φ̂(0)− − (1 + q)sKa+ 1 2 (|s|) Φ̂(1) ) I{û6=0} and Θ(û,j) = Γ ( a + 1 2 + j )( 2s23 s21 + 2bs 2 3 )a+ 1 2 +j I{û=0}+ + 2 ( û2 s21 + 2bs 2 3 )a+j 2 + 1 4 Ka+j+ 1 2 ( |û| √ s21 + 2bs 2 3 s23 ) I{û 6=0} with Φ̂(j) = B(a + 1 + j, 1)Φ ( a + 1 + j,−a,a + 2 + j; 1 + q 2 ,−s(1 + q) ) . Next, we discuss the expectation f2(ζ,α) = E ( γ ζ j ( n∑ l=1 xlθlγl )α YjI{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ) . for ζ,α ∈ N∪{0}. Lemma 3.2. When xj,σj 6= 0, we have that f2(ζ,α) = − s2xjσjs α 1 b aκ ζ+ 1 2 j Γ(a) √ π ( 2ζ+α+as 2(ζ+α+a) 3 Γ ( α + a + 1 2 ) (s21 + 2bs 2 3) ζ+α+a+ 1 2 I{û=0}+(3.5) + exp ( ûs1 s23 )√ 2 s3 ( û2 s21 + 2bs 2 3 )ζ+α+a 2 + 1 4 Kζ+α+a+ 1 2 ( |û| √ s21 + 2bs 2 3 s23 ) I{û6=0} ) . Also, we calculate the function f3(ζ,α) = E ( γ ζ j ( n∑ l=1 xlθlγl )α I{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ) . Int. J. Anal. Appl. 19 (3) (2021) 325 Lemma 3.3. Let ζ,α ∈ N∪{0}. Then f3(ζ,α) = κ ζ js α 1 Γ(ζ + α + a + 1 2 ) Γ(a)bζ+α √ 2π I{û=0} ( B ( 1 2 ,ζ + α + a ) √ 2 − s1 s3 √ b × × F ( ζ + α + a + 1 2 , 1 2 , 3 2 ;− s21 2bs23 )) + κ ζ j|s| ζ+α+a−1 2 (1 + q)ζ+α+aI{û 6=0} Γ(a)bζ+αe−ss−α1 √ 2π × × ( B(ζ + α + a, 1) ( |s|Kζ+α+a+ 1 2 (|s|) + sKζ+α+a−1 2 (|s|) ) Φ̃(0)− − (1 + q)sB(ζ + α + a + 1, 1)Kζ+α+a−1 2 (|s|) Φ̃(1) ) ,(3.6) where Φ̃(j) = =Φ ( ζ + α + a + j, 1 − ζ −α−a,ζ + α + a + 1 + j; 1 + q 2 ,−s(1 + q) ) . Set f4(ζ,α) = E ( γ ζ j ( n∑ l=1 xlθlγl )α n∑ l=1 YlI{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ) for ζ,α ∈ N∪{0}. Lemma 3.4. The expectation f4(ζ,α) = − s3b asα1 κ ζ j Γ(a) √ 2π ( 2e ûs1 s2 3 Kζ+α+a+ 1 2 ( |û| √ s21 + 2bs 2 3 s23 ) I{û=0}×(3.7) ( û2 s21 + 2bs 2 3 )ζ+α+a 2 + 1 4 + Γ ( ζ + α + a + 1 2 )( 2s23 s21 + 2bs 2 3 )ζ+α+a+ 1 2 I{û6=0} ) . Finally, set f5 = E (( n∑ l=1 Yl )2 I{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ) . Lemma 3.5. Let the functions Θ(û,j) and Λ(û) be defined in Lemma 3.1. Then f5 = s3b a Γ(a) ( s1√ 2π Θ(0, 1) + s3Λ(0) ) I{û=0}+(3.8) + s3b a Γ(a) ( s3Λ(û) + exp ( ûs1 s23 ) √ 2π (s1Θ(û, 1) − ûΘ(û, 0)) ) I{û6=0}. The proofs of Lemmas 3.1–3.5 are placed in Section 4. Now can introduce the main results of the paper. The theorem below gives us an analytical expression for the value of the downside beta. Int. J. Anal. Appl. 19 (3) (2021) 326 Theorem 3.1. The downside beta β− = β−n β−d , with β−n = xj [ rj ( ŝf3(0, 0) + f3(0, 1) + f4(0, 0) ) +θj ( ŝf3(1, 0)+(3.9) + f3(1, 1) + f4(1, 0) )] +ŝf2(0, 0) + f2(0, 1) + f1 −hj ( ŝf3(0, 0)+ + f3(0, 1) + f4(0, 0) −hf3(0, 0) ) −h [ xj ( rjf3(0, 0) + θjf3(1, 0) ) +f2(0, 0) ] and β−d = ŝ 2f3(0, 0) + 2ŝf3(0, 1) + f3(0, 2) + 2ŝf4(0, 0) + 2f4(0, 1)+(3.10) + f5 − 2h ( ŝf3(0, 0) + f3(0, 1) + f4(0, 0) ) +h2f3(0, 0), where the expectations f1,f2,f3,f4,f5 are computed in Lemmas 3.1–3.5, respectively. Let ĥj = xj n∑ l=1 xl ( rj ( rl + θlal bl ) +θj ( rlaj bj + + θlκlκj(a + 1)a b2 ) +σj ( rl + σlρlja √ κlκj b )) and ĥ = n∑ l,m=1 xlxm ( rl ( rm + θmam bm ) +θl ( rmal bl + + θmκlκma(a + 1) b2 ) + σlσmρlma √ κlκm b ) . The next theorem derives the size of the upside beta. Theorem 3.2. The upside beta β+ = ĥj −hjh−β−n ĥ−h2 −β−d , where β−n and β − d are defined in (3.9) and (3.10). The proofs of Theorems 3.1 and 3.2 are given in Section 4. The following example considers the case of the investment portfolio which consists of three assets and two of them are risk-free and low risk ones. Int. J. Anal. Appl. 19 (3) (2021) 327 Example 3.1. Assume that n = 3, H1 = r1, H2 = r2 + θ2γ, H3 = r3 + θ3γ + σ3 √ γN3 and j = 2. Then h2 = x2 ( r2 + θ2a2 b2 ) , h = r1 + ∑3 l=2 xl ( rl + θlal bl ) , ŝ = ∑3 l=1 xlrl, û = u − ŝ, s1 = ∑3 l=2 xlθl, s2 = 0, s3 = σ3|x3|, ĥ2 = =x2 ( x1 ( r2r1 + θ2r1a b ) + 3∑ l=2 xl ( r2 ( rl + θla b ) +θ2 (rla b + θl(a + 1)a b2 ))) , ĥ = 3∑ l,m=1 xlxm ( rl ( rm + θmI{m6=1}a b ) + +θlI{l 6=1} ( rma b + θmI{m 6=1}a(a + 1) b2 )) + σ23a b , f1 ≡ 0, f2(ζ,α) ≡ 0, β−n = xj ( rj ( ŝf3(0, 0) + f3(0, 1) + f4(0, 0) ) + +θj ( ŝf3(1, 0) + f3(1, 1) + f4(1, 0) )) −hj ( ŝf3(0, 0) + f3(0, 1)+ +f4(0, 0) −hf3(0, 0) ) −h ( xj ( rjf3(0, 0) + θjf3(1, 0) )) , where f3, f4 are determined by Lemma 3.3, Lemma 3.4 and β − d is calculated in (3.10). Example 3.2 discusses the case when there are n assets in the portfolio and they are the medium dependent between each other. Example 3.2. Let γ1 ≡ γ2 ≡ ... ≡ γn ≡ γ and ρlm = 0, l 6= m. We have that hj = xj ( rj + θja b ) , h = ∑n l=1 hl, ŝ = ∑n l=1 xlrl, û = u− ŝ, s1 = ∑n l=1 xlθl, s2 = xjσj, s3 = √∑n l=1 x 2 l σ 2 l , ĥj = xj ( xjσ 2 ja b + n∑ l=1 xl ( rj ( rl + θla b ) +θj (rla b + θl(a + 1)a b2 ) +σjrl )) , ĥ = n∑ l,m=1 xlxm ( rl ( rm + θma b ) +θl ( rma b + θma(a + 1) b2 )) + a b n∑ l=1 x2l σ 2 l and β−n , β − d are computed with respect to (3.9), (3.10). 4. Proofs Proof of Lemma 3.1. It is easy to notice that ( Yj, ∑n l=1 Yl ∣∣∣γ1,γ2, ...,γn) is a Gaussian vector with the covariance matrix   (xjσj)2γj ∑nl=1 ρjlxjσjxlσl√γjγl∑n l=1 ρjlxjσjxlσl √ γjγl ∑n m,l=1 ρmlxmσmxlσl √ γmγl   . Int. J. Anal. Appl. 19 (3) (2021) 328 Hence Law ( Yj, n∑ l=1 Yl ∣∣∣γ1,γ2, ...,γn)= Law(Yj, Ỹ ),(4.1) where Ỹ = σỸ Ñ with (4.2) σỸ = √√√√ n∑ m,l=1 ρmlxmσmxlσl √ γmγl and the standard normal random variables Nj and Ñ are correlated with the coefficient (4.3) ρjỸ = ∑n l=1 ρjlxjσjxlσl √ γjγl xjσj √ γj √∑n m,l=1 ρmlxmσmxlσl √ γmγl . Set σ̂j = xjσj √ γj(4.4) and (4.5) ũ = u− n∑ l=1 xl(rl + θlγl). Then E ( Yj n∑ l=1 YlI{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn)= =E ( YjỸ I{Ỹ≤ũ} ∣∣∣γ1,γ2, ...,γn)= ∫ ũ −∞ ∫ ∞ −∞ xy 2πσỸ σ̂j √ 1 −ρ2 jỸ × ×exp  − 1 2 ( 1 −ρ2 jỸ ) [ x2 σ2 Ỹ − 2ρjỸ xy σỸ σ̂j + y2 σ̂2j ]dydx Int. J. Anal. Appl. 19 (3) (2021) 329 and since ∫ ∞ −∞ y exp ( − 1 2 ( 1 −ρ2 jỸ ) [y2 σ̂2j − 2ρjỸ xy σỸ σ̂j ]) dy = = exp ( ( xρjỸ )2 2σ2 Ỹ ( 1 −ρ2 jỸ ))∫ ∞ −∞ y exp ( − 1 2 ( 1 −ρ2 jỸ ) [ y σ̂j − xρjỸ σỸ ]2 ) dy = = σ̂2j exp ( ( xρjỸ )2 2σ2 Ỹ ( 1 −ρ2 jỸ ))(∫ ∞ −∞ ( y − xρjỸ σỸ ) × × exp ( − 1 2 ( 1 −ρ2 jỸ ) [y − xρjỸ σỸ ]2 ) dy + xρjỸ σỸ × × ∫ ∞ −∞ exp ( − 1 2 ( 1 −ρ2 jỸ ) [y − xρjỸ σỸ ]2 ) dy ) = = exp ( ( xρjỸ )2 2σ2 Ỹ ( 1 −ρ2 jỸ ))xρjỸ σ̂2j σỸ √ 2π ( 1 −ρ2 jỸ ) , we have that E ( Yj n∑ l=1 YlI{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn)= ∫ ũ −∞ x2ρjỸ σ̂j σ2 Ỹ √ 2π × × exp ( − x2 2σ2 Ỹ ) dx = − ρjỸ σ̂j√ 2π ∫ ũ −∞ xd exp ( − x2 2σ2 Ỹ ) = = − ρjỸ σ̂j√ 2π ũ exp ( − ũ2 2σ2 Ỹ ) + ρjỸ σ̂j√ 2π ∫ ũ −∞ exp ( − x2 2σ2 Ỹ ) dx = = − ρjỸ σ̂j√ 2π ũ exp ( − ũ2 2σ2 Ỹ ) +ρjỸ σỸ σ̂jN ( ũ σỸ ) .(4.6) Next, one can observe that σỸ = √√√√γ n∑ m,l=1 ρmlxmσmxlσl √ κmκl and ũ = u− n∑ l=1 xlrl −γ n∑ l=1 xlθlκl. Moreover, we have that ρjỸ = γ ∑n l=1 ρjlxjσjxlσl √ κjκl xjσj √ κjγ √ γ ∑n m,l=1 ρmlxmσmxlσl √ κmκl Int. J. Anal. Appl. 19 (3) (2021) 330 and σ̂j = xjσj √ κjγ. Set û = u− n∑ l=1 xlrl, s1 = n∑ l=1 xlθlκl, s2 = n∑ l=1 ρjlxlσl √ κl, s3 = √√√√ n∑ m,l=1 ρmlxmσmxlσl √ κmκl. Then σỸ = s3 √ γ, ũ = û−s1γ, ρjỸ = s2 s3 and we get from (4.6) that E ( Yj n∑ l=1 YlI{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ) =(4.7) =xjσj s2 s3 √ κj ( s3 ∫ ∞ 0 gN ( û−s1g s3 √ g ) f(γ,g)dg− − 1 √ 2π ∫ ∞ 0 (û−s1g) √ g exp ( − (û−s1g)2 2s23g ) f(γ,g)dg ) = = xjσjs2b a√κj s3Γ(a) ( s3 ∫ ∞ 0 gaN ( û−s1g s3 √ g ) exp(−bg)dg− − exp ( ûs1 s23 ) √ 2π ∫ ∞ 0 (û−s1g)ga− 1 2 exp ( − û2 + (s1g) 2 2s23g − bg ) dg ) . First, ∫ ∞ 0 ga± 1 2 exp ( − û2 2s23g − s21 + 2bs 2 3 2s23 g ) dg = =2 ( û2 s21 + 2bs 2 3 )a+1 2 ±1 4 Ka+1±1 2 ( |û| √ s21 + 2bs 2 3 s23 ) (4.8) with respect to the formula 3.471.9 from Gradshteyn and Ryzhik [17] if û 6= 0. When û = 0,∫ ∞ 0 ga± 1 2 exp ( − s21 + 2bs 2 3 2s23 g ) dg = = Γ ( a + 1 ± 1 2 )( 2s23 s21 + 2bs 2 3 )a+1±1 2 .(4.9) Next, the integral I = ∫ ∞ 0 gaN ( û−s1g s3 √ g ) exp(−bg)dg Int. J. Anal. Appl. 19 (3) (2021) 331 is quite similar to the one at the bottom of p.207 of Ivanov and Ano [3]. If û = 0, I = Γ ( a + 3 2 ) ba+1 √ 2π ( B ( 1 2 ,a + 1 ) √ 2 − s1 s3 √ b F ( a + 3 2 , 1 2 , 3 2 ;− s21 2bs23 )) (4.10) due to Case 2.2, p.208 of Ivanov and Ano [3]. When û 6= 0, I = |s|a+ 1 2 es(1 + q)a+1 ba+1 √ 2π ( B(a + 1, 1) ( |s|Ka+ 3 2 (|s|) +(4.11) sKa+ 1 2 (|s|) ) Φ ( a + 1,−a,a + 2; 1 + q 2 ,−s(1 + q) ) − (1 + q)sB(a + 2, 1)Ka+ 1 2 (|s|) Φ ( a + 2,−a,a + 3; 1 + q 2 ,−s(1 + q) )) , where s = û √ s21+2bs 2 3 s3|s3| and q = − sg(s3)s1√ s21+2bs 2 3 , with respect to Case 3.2, p.210 of Ivanov and Ano [3]. Hence we get (3.4) from (4.7)–(4.11). Proof of Lemma 3.2. Keeping in mind (4.1), we get that E ( YjI{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn)= = E ( YjI{Ỹ≤ũ} ∣∣∣γ1,γ2, ...,γn)= ∫ ũ −∞ ∫ ∞ −∞ y 2πσỸ σ̂j √ 1 −ρ2 jỸ × × exp  − 1 2 ( 1 −ρ2 jỸ ) [ x2 σ2 Ỹ − 2ρjỸ xy σỸ σ̂j + y2 σ̂2j ]dydx = = ∫ ũ −∞ xρjỸ σ̂j σ2 Ỹ √ 2π exp ( − x2 2σ2 Ỹ ) dx = − ρjỸ σ̂j√ 2π exp ( − ũ2 2σ2 Ỹ ) , where σỸ , ρjỸ , σ̂j, ũ are defined in (4.2), (4.3), (4.4), (4.5), respectively. Hence E ( γ ζ j ( n∑ l=1 xlθlγl )α YjI{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ) = = − s2xjσj( ∑n l=1 κlxlθl) ακ ζ+ 1 2 j s3 √ 2π E ( γζ+α+ 1 2 exp ( − (û−s1γ)2 2s23γ )) = = − s2xjσj( ∑n l=1 κlxlθl) α exp ( ûs1 s23 ) baκ ζ+ 1 2 j s3Γ(a) √ 2π × × ∫ ∞ 0 gζ+α+a− 1 2 exp ( − û2 2s23g − s21 + 2bs 2 3 2s23 g ) dg = = − s2xjσj( ∑n l=1 κlxlθl) α exp ( ûs1 s23 ) baκ ζ+ 1 2 j √ 2 s3Γ(a) √ π × × ( û2 s21 + 2bs 2 3 )ζ+α+a 2 + 1 4 Kζ+α+a+ 1 2 ( |û| √ s21 + 2bs 2 3 s23 ) (4.12) Int. J. Anal. Appl. 19 (3) (2021) 332 due to the formula 3.471.9 from Gradshteyn and Ryzhik [17] when û 6= 0. If û = 0, E ( γ ζ j ( n∑ l=1 xlθlγl )α YjI{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ) = = − s2xjσj( ∑n l=1 κlxlθl) αbaκ ζ+ 1 2 j s3Γ(a) √ 2π ∫ ∞ 0 gζ+α+a− 1 2 exp ( − s21 + 2bs 2 3 2s23 g ) dg = = − 2ζ+α+as2s 2(ζ+α+a) 3 xjσj( ∑n l=1 κlxlθl) αbaΓ ( α + a + 1 2 ) κ ζ+ 1 2 j Γ(a) (s21 + 2bs 2 3) ζ+α+a+ 1 2 √ π .(4.13) Thus, we get (3.5) from (4.13) and (4.12). Proof of Lemma 3.3. Since P ( n∑ l=1 Yl ≤ u− n∑ l=1 xl(rl + θlγl) ∣∣∣γ1,γ2, ...,γn)= = P ( Ỹ ≤ ũ ∣∣γ1,γ2, ...,γn)= N( ũ σỸ ) , we have that E ( γ ζ j ( n∑ l=1 xlθlγl )α I{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ) = = κ ζ j ( n∑ l=1 xlθlκl )α E ( γζ+αN ( ũ σỸ )) = = baκ ζ j (∑n l=1 xlθlκl )α Γ(a) ∫ ∞ 0 gζ+α+a−1 exp(−bg)N ( û−s1g s3 √ g ) dg. Hence we get similarly to the proof of Lemma 3.1 that E ( γ ζ j ( n∑ l=1 xlθlγl )α I{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ) =(4.14) = baκ ζ j (∑n l=1 xlθlκl )α Γ(ζ + α + a + 1 2 ) Γ(a)bζ+α+a √ 2π ×( B ( 1 2 ,ζ + α + a ) √ 2 − s1 s3 √ b F ( ζ + α + a + 1 2 , 1 2 , 3 2 ;− s21 2bs23 )) Int. J. Anal. Appl. 19 (3) (2021) 333 if û = 0 and E ( γ ζ j ( n∑ l=1 xlθlγl )α I{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ) =(4.15) = baκ ζ j (∑n l=1 xlθlκl )α |s|ζ+α+a− 1 2 es(1 + q)ζ+α+a Γ(a)bζ+α+a √ 2π ×( B(ζ + α + a, 1) ( |s|Kζ+α+a+ 1 2 (|s|) + sKζ+α+a−1 2 (|s|) ) × Φ ( ζ + α + a, 1 − ζ −α−a,ζ + α + a + 1; 1 + q 2 ,−s(1 + q) ) − − (1 + q)sB(ζ + α + a + 1, 1)Kζ+α+a−1 2 (|s|)× Φ ( ζ + α + a + 1, 1 − ζ −α−a,ζ + α + a + 2; 1 + q 2 ,−s(1 + q) )) when û 6= 0. We have (3.6) from (4.14) and (4.15). Proof of Lemma 3.4. We have with respect to (4.1) that E ( γ ζ j ( n∑ l=1 xlθlγl )α n∑ l=1 YlI{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn)= = γ ζ j ( n∑ l=1 xlθlγl )α E ( Ỹ I{Ỹ≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn)= = γ ζ j ( n∑ l=1 xlθlγl )α∫ ũ −∞ x σỸ √ 2π exp ( − x2 2σ2 Ỹ ) dx = = − σỸ γ ζ j (∑n l=1 xlθlγl )α √ 2π exp ( − ũ2 2σ2 Ỹ ) . Hence E ( γ ζ j ( n∑ l=1 xlθlγl )α n∑ l=1 YlI{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ) =(4.16) = − s3b asα1 κ ζ j Γ(a) √ 2π ∫ ∞ 0 gζ+α+a− 1 2 exp ( −bg − (û−s1g)2 2s23g ) dg = = − s3b asα1 κ ζ j exp ( ûs1 s23 )√ 2 Γ(a) √ π ( û2 s21 + 2bs 2 3 )ζ+α+a 2 + 1 4 × × Kζ+α+a+ 1 2 ( |û| √ s21 + 2bs 2 3 s23 ) Int. J. Anal. Appl. 19 (3) (2021) 334 if û 6= 0 and E ( γ ζ j ( n∑ l=1 xlθlγl )α n∑ l=1 YlI{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ) =(4.17) = − s3b asα1 κ ζ j Γ(a) √ 2π Γ ( ζ + α + a + 1 2 )( 2s23 s21 + 2bs 2 3 )ζ+α+a+ 1 2 when û = 0 similarly to (4.8) and (4.9), respectively. We get (3.7) from (4.16) and (4.17). Proof of Lemma 3.5. Conditional expectation E (( n∑ l=1 Yl )2 I{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn)= = E ( Ỹ 2I{Ỹ≤u−∑nl=1 xl(rl+θlγl)}) ∣∣∣γ1,γ2, ...,γn)= = ∫ ũ −∞ x2 σỸ √ 2π exp ( − x2 2σ2 Ỹ ) dx = − σỸ√ 2π ( ũ exp ( − ũ2 2σ2 Ỹ ) − − ∫ ũ −∞ exp ( − x2 2σ2 Ỹ ) dx ) = − ũσỸ√ 2π exp ( − ũ2 2σ2 Ỹ ) +σ2 Ỹ N ( ũ σỸ ) . Therefore E (( n∑ l=1 Yl )2 I{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ) = =s3 ( s1√ 2π E ( γ 3 2 exp ( − s21 2s23 γ )) +s3E ( γN ( − s1 s3 √ γ ))) = = s3b a Γ(a) ( s1√ 2π ∫ ∞ 0 ga+ 1 2 exp ( − s21 + 2bs 2 3 2s23 g ) dg+ +s3 ∫ ∞ 0 gaN ( − s1 s3 √ g ) exp(−bg)dg ) =(4.18) = s3b a Γ(a) ( s1√ 2π Γ ( a + 3 2 )( 2s23 s21 + 2bs 2 3 )a+ 3 2 + + s3Γ ( a + 3 2 ) ba+1 √ 2π [ B ( 1 2 ,a + 1 ) √ 2 − s1 s3 √ b F ( a + 3 2 , 1 2 , 3 2 ;− s21 2bs23 )]) Int. J. Anal. Appl. 19 (3) (2021) 335 if û = 0 as in (4.9) and (4.10). When û 6= 0, E (( n∑ l=1 Yl )2 I{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ) = s3 ( s3E ( γN (û−s1γ s3 √ γ )) + + 1 √ 2π [ s1E ( γ 3 2 exp ( − (û−s1γ)2 2s23γ )) −ûE ( γ 1 2 exp ( − (û−s1γ)2 2s23γ ))]) = = s3b a Γ(a) ( s3 ∫ ∞ 0 gaN (û−s1g s3 √ g ) exp(−bg)dg+ + exp ( ûs1 s23 ) √ 2π [ s1 ∫ ∞ 0 ga+ 1 2 exp ( − û2 2s23g − s21 + 2bs 2 3 2s23 g ) dg− − û ∫ ∞ 0 ga− 1 2 exp ( − û2 2s23g − s21 + 2bs 2 3 2s23 g ) dg ]) and hence E (( n∑ l=1 Yl )2 I{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ) =(4.19) = s3b a Γ(a) ( s3|s|a+ 1 2 es(1 + q)a+1 ba+1 √ 2π [ B(a + 1, 1) ( |s|Ka+ 3 2 (|s|) + + sKa+ 1 2 (|s|) ) Φ ( a + 1,−a,a + 2; 1 + q 2 ,−s(1 + q) ) − (1 + q)sB(a + 2, 1)Ka+ 1 2 (|s|) Φ ( a + 2,−a,a + 3; 1 + q 2 ,−s(1 + q) )] + exp ( ûs1 s23 )√ 2 √ π [ s1 ( û2 s21 + 2bs 2 3 )a 2 + 3 4 Ka+ 3 2 ( |û| √ s21 + 2bs 2 3 s23 ) − − û ( û2 s21 + 2bs 2 3 )a 2 + 1 4 Ka+ 1 2 ( |û| √ s21 + 2bs 2 3 s23 )]) in this case similarly to (4.8) and (4.11). We establish (3.8) from (4.18) and (4.19). Proof of Theorem 3.1. We have that β− = E [ (Xj − EXj)(X − EX)I{X≤u} ] E [ (X − EX)2I{X≤u} ] = = E ( XjXI{X≤u} ) − EXjE ( XI{X≤u} ) E ( X2I{X≤u} ) − 2EXE ( XI{X≤u} ) + (EX) 2 P(X ≤ u) + = EXjEXP(X ≤ u) − EXE ( XjI{X≤u} ) E ( X2I{X≤u} ) − 2EXE ( XI{X≤u} ) + (EX) 2 P(X ≤ u) (4.20) and hence it is needed to compute consequently E ( XjXI{X≤u} ) , E ( XI{X≤u} ) , E ( XjI{X≤u} ) , P(X ≤ u) and E ( X2I{X≤u} ) . Int. J. Anal. Appl. 19 (3) (2021) 336 One can see that E ( XjXI{X≤u}|γ1,γ2, ...,γn ) = xj(rj + θjγj) n∑ l=1 xl(rl + θlγl)× ×P ( n∑ l=1 Yl ≤ u− n∑ l=1 xl(rl + θlγl) ∣∣∣γ1,γ2, ...,γn)+xj(rj + θjγj)× ×E ( n∑ l=1 YlI{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn)+ n∑ l=1 xl(rl + θlγl)× ×E ( YjI{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn)+ +E ( Yj n∑ l=1 YlI{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn). Hence we have that E ( XjXI{X≤u} ) = xj ( rj [ ŝf3(0, 0) + f3(0, 1) ] + + θj [ ŝf3(1, 0) + f3(1, 1) ] +rjf4(0, 0) + θjf4(1, 0) ) + + ŝf2(0, 0) + f2(0, 1) + f1.(4.21) Next, E ( XI{X≤u}|γ1,γ2, ...,γn ) = n∑ l=1 xl(rl + θlγl)× ×P ( n∑ l=1 Yl ≤ u− n∑ l=1 xl(rl + θlγl) ∣∣∣γ1,γ2, ...,γn)+ +E ( n∑ l=1 YlI{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn) and therefore E ( XI{X≤u} ) = ŝf3(0, 0) + f3(0, 1) + f4(0, 0).(4.22) Further, E ( XjI{X≤u}|γ1,γ2, ...,γn ) = xj(rj + θjγj)× ×P ( n∑ l=1 Yl ≤ u− n∑ l=1 xl(rl + θlγl) ∣∣∣γ1,γ2, ...,γn)+ +E ( YjI{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn) and then E ( XjI{X≤u} ) = xj ( rjf3(0, 0) + θjf3(1, 0) ) +f2(0, 0).(4.23) Int. J. Anal. Appl. 19 (3) (2021) 337 Also, P(X ≤ u|γ1,γ2, ...,γn) = P ( n∑ l=1 Yl ≤ u− n∑ l=1 xl(rl + θlγl) ∣∣∣γ1,γ2, ...,γn) and hence P(X ≤ u) = f3(0, 0).(4.24) Moreover, E ( X2I{X≤u}|γ1,γ2, ...,γn ) = ( n∑ l=1 xl(rl + θlγl) )2 × × P ( n∑ l=1 Yl ≤ u− n∑ l=1 xl(rl + θlγl) ∣∣∣γ1,γ2, ...,γn)+ + 2 n∑ l=1 xl(rl + θlγl)E ( n∑ l=1 YlI{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn)+ + E (( n∑ l=1 Yl )2 I{∑nl=1 Yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn) and E ( X2I{X≤u} ) = ŝ2f3(0, 0) + 2ŝf3(0, 1)+ + f3(0, 2) + 2ŝf4(0, 0) + 2f4(0, 1) + f5(0, 0).(4.25) Keeping in mind the identities EXj = hj and EX = h, we get exploiting (4.20)–(4.25) that β− = β−n β−d , where β−n = xj ( rj [ ŝf3(0, 0) + f3(0, 1) ] + + θj [ ŝf3(1, 0) + f3(1, 1) ] +rjf4(0, 0) + θjf4(1, 0) ) + + ŝf2(0, 0) + f2(0, 1) + f1 −hj [ ŝf3(0, 0)+ f3(0, 1) + f4(0, 0) ] +hjhf3(0, 0) −h [ xj ( rjf3(0, 0) + θjf3(1, 0) ) +f2(0, 0) ] Int. J. Anal. Appl. 19 (3) (2021) 338 and β−d = ŝ 2f3(0, 0) + 2ŝf3(0, 1)+ + f3(0, 2) + 2ŝf4(0, 0) + 2f4(0, 1) + f5(0, 0)− − 2h [ ŝf3(0, 0) + f3(0, 1) + f4(0, 0) ] +h2f3(0, 0). Proof of Theorem 3.2. One can observe that β+ = E [ (Xj − EXj)(X − EX)I{X≥u} ] E [ (X − EX)2I{X≥u} ] = = E [(Xj − EXj)(X − EX)] − E [ (Xj − EXj)(X − EX)I{X≤u} ] E [(X − EX)2] − E [ (X − EX)2I{X≤u} ] = = EXjX − EXjEX − E [ (Xj − EXj)(X − EX)I{X≤u} ] EX2 − (EX)2 − E [ (X − EX)2I{X≤u} ] . Since EXjX = E [ xj(rj + θjγj + σj √ γjNj) n∑ l=1 xl(rl + θlγl + σl √ γlNl) ] = = xj [ rj n∑ l=1 xl ( rl + θlal bl ) + θj n∑ l=1 xl ( rlaj bj + θlEγjγl ) + + σj n∑ l=1 xl ( rl + σlE √ γlγjNlNj )] = xj [ rj n∑ l=1 xl ( rl + θlal bl ) + + θj n∑ l=1 xl ( rlaj bj + θlκlκj(a + 1)a b2 ) + σj n∑ l=1 xl ( rl + σlρlja √ κlκj b )] = ĥj and EX2 = E ( n∑ l=1 xl(rl + θlγl + σl √ γlNl) )2 = =E   n∑ l,m=1 xlxm(rl + θlγl + σl √ γlNl)(rm + θmγm + σm √ γmNm)   = = n∑ l,m=1 xlxm ( rl ( rm + θmam bm ) +θl (rmal bl + θmκlκma(a + 1) b2 ) + + σlσmρlma √ κlκm b ) = ĥ, we get that β+ = ĥj −hjh−β−n ĥ−h2 −β−d . Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. Int. J. Anal. Appl. 19 (3) (2021) 339 References [1] Y. Altigan, T.G. Bali, K.O. Demirtas and A.D. Gunaydin, Downside Beta and Equity Returns around the World, J. Portfolio Manage. 44 (7) (2018), 39–54. [2] A. Ang, J. Chen and Y. Xing, Downside risk, Rev. Financ. 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