Approximating the Sum of Correlated Lognormals: An Implementation
August 27, 2015 © Approximating the Sum of Correlated Lognormals: An Implementation † By CHRISTOPHER J. ROOK and MITCHELL C. KERMAN ABSTRACT
Lognormal random variables appear naturally in many engineering disciplines, including wireless communications, reliability theory, and finance. So, too, does the sum of (correlated) lognormal random variables. Unfortunately, no closed form probability distribution exists for such a sum, and it requires approximation. Some approximation methods date back over 80 years and most take one of two approaches, either: 1) an approximate probability distribution is derived mathematically, or 2) the sum is approximated by a single lognormal random variable. In this research, we take the latter approach and review a fairly recent approximation procedure proposed by Mehta, Wu, Molisch, and Zhang (2007), then implement it using C++. The result is applied to a discrete time model commonly encountered within the field of financial economics. † This research originated as an independent study project under the SYS-800 course in the Department of Systems Engineering at Stevens Institute of Technology, School of Systems and Enterprises, Hoboken, NJ 07030. Christopher J. Rook works as a consultant statistical programmer and is finishing a degree in Systems Engineering from Stevens Institute of Technology. Dr. Mitchell C. Kerman is the Director of Program Development and Transition for the Systems Engineering Research Center, a Department of Defense (DoD) University Affiliated Research Center (UARC) led by Stevens Institute of Technology. I. Introduction
Practical problems involving sums of random variables (RVs), say, Z = X + Y, are unavoidable within many disciplines. When X and Y are independent, the probability density function (PDF) of Z can be expressed using the convolution operator. Much theory on RV sums has been developed and closed form expressions for the PDF of Z exist for some X and Y. In general, analytical solutions involving convolutions for sums of independent RVs are often difficult to obtain. When X and Y are correlated, the complexity increases. Sums of n independent and identically distributed ( iid ) RVs, say, Z n = X + X + … + X n , can be approximated by the normal distribution using the central limit theorem (CLT), but only when n ≥
30. Therefore, if n < 30, or if the RVs are not iid , then the CLT does not apply. When n < 30 and the RVs are independent, we may be able to derive the PDF of Z n by applying the convolution operator iteratively. That is, we first determine the PDF of Z = X + X , then recognize that Z = Z + X is also a 2-term sum of independent RVs, with the PDF of Z and X perhaps known. While theoretically sound, it may be unlikely that this technique will produce successive closed form PDFs for each sum. Lognormal RVs appear in many disciplines including finance, fiber optics, inventory management, telecommunications, and reliability theory. By definition, a variable is said to be lognormally distributed when its logarithm is normally distributed. Lognormal RVs tend to appear naturally when a phenomenon involves the product of iid RVs. To see why, we can take the logarithm of the product, which becomes a sum of iid logged RVs that tends to the normal distribution as the number of RVs multiplied increases (via the CLT). Exponentiating the sum then yields a lognormal RV. For example, let P n = X X … * X n , where the X i ’s are iid RVs, so that ln(P n ) = ln(X X … * X n ) = ln(X ) + ln(X ) + … + ln(X n ) ~(cid:4662) N( μ , σ ), for n ≥
30. By exponentiating both sides, P n ~(cid:4662) e (cid:2898)(cid:4666)(cid:2972),(cid:2978) (cid:3118) (cid:4667) , that is, P n is approximately lognormal since its logarithm is approximately normal. Since lognormal RVs appear naturally in such settings, so too will their sum. Unfortunately, the convolution for the sum of two independent lognormal RVs does not have a closed form, and, therefore, neither does the PDF. Mehta, Wu, Molisch, and Zhang (2007) propose a novel and flexible approach to approximate the distribution of a sum of (correlated) lognormal RVs and in this research we review, then implement it, using C++. 2 The remainder of this research is organized as follows. In Section II, we review the literature on lognormal sum approximations, and, in particular, the two methods that motivate the technique proposed by Mehta et al. (2007). In Section III, we present a detailed review of the theoretical concepts required to apply the technique. This section may be skipped by readers already familiar with these concepts. In Section IV, we implement the technique for a 2-term sum, and, in Section V, we present an application to finance. In Section VI, we discuss the extension to a sum of more than two terms. Section VII concludes with our recommendations. Fully documented source code for a C++ implementation is provided in Appendices A and B. II. Literature Review
Fenton (1960) proposed a moment-matching technique to approximate the distribution of a sum of independent lognormal RVs with a single lognormal RV. Probabilities in the center of the distribution can be approximated effectively using this method by matching the 1 st and 2 nd central moments (i.e., the mean and variance). If interest is in upper tail probabilities, then the approximation is derived by matching the 2 nd and 3 rd central moments. For values further in the tail, the approximation can be based on the 3 rd and 4 th central moments. This technique is not customizable for probabilities in the head portion (i.e., near zero) because a practical formula for deriving negative moments of the sum is not known. This approach was used by R.I. Wilkinson at Bell Labs in the 1930’s and is therefore often referred to as the Fenton-Wilkinson (F-W) procedure. Schwartz and Yeh (1982) proposed a similar moment-matching technique but on the log scale. The procedure is iterative, handling two terms at a time and it assumes that each successive sum is lognormal, thus the log is normally distributed. Analytical expressions are provided for the 2-term case, which is sufficient to implement the procedure. Schwartz and Yeh (S-Y) report an improvement over the F-W method and show how the procedure can be applied to correlated lognormal RVs. Mehta et al. (2007) note that F-W is more accurate in the tail while S-Y is more accurate in the head of the distribution and propose a method that is customizable, allowing the user to parameterize the procedure to meet their needs. Instead of matching moments, Mehta et al. (2007) propose matching the moment-generating function (MGF) directly, and, similar to F-W and S-Y, approximate a sum of lognormals with a single lognormal RV. The software is being distributed under the open source MIT license (See: http://opensource.org/licenses/MIT). It depends on the Eigen © and Boost © libraries, and the user shall assume responsibility for code validation. III. Preliminaries
In this section, we provide a general foundation for the techniques that will be involved in approximating the distribution of the sum of (correlated) lognormal RVs with a single lognormal RV. Any reader who is familiar with these topics may skip this section.
A. The Moment-Generating Function
The MGF for a continuous RV, X, with PDF, f(x) , is defined by the following function of both t and x (Freund [1992]): M (cid:2908) (cid:4666)(cid:1872)(cid:4667) (cid:3404) E (cid:2908) (cid:4670)e (cid:3047)(cid:3051) (cid:4671) (cid:3404) (cid:3505) e (cid:3047)(cid:3051) ∗ (cid:1858)(cid:4666)(cid:1876)(cid:4667)(cid:1856)(cid:1876) (cid:2998)(cid:2879)(cid:2998) . In words, it is the expected value of (cid:1857) (cid:3047)(cid:3051) with respect to the RV X. For a discrete RV, we replace the integral by a sum. We refer to this as the MGF because it can be used to derive the moments of X. The n th moment of X is the expected value of X n , denoted E[X n ]. If μ and σ are the mean and variance of X, they are related to the 1 st and 2 nd moments as follows: μ (cid:3404) E(cid:4670)X (cid:2869) (cid:4671) and σ (cid:2870) (cid:3404) E(cid:4670)X (cid:2870) (cid:4671) (cid:3398) (cid:4666)E(cid:4670)X (cid:2869) (cid:4671)(cid:4667) (cid:2870) . The n th moment, as defined here, is sometimes referred to as a moment about the origin to distinguish it from moments about the mean, which are E[(X- μ ) n ]. To derive the n th moment of X using its MGF, we differentiate M (cid:2908) (cid:4666)(cid:1872)(cid:4667) n times with respect to t, then set t equal to zero (Freund [1992]). That is, E(cid:4670)X (cid:2924) (cid:4671) (cid:3404) (cid:1856) (cid:2924) (cid:1856)(cid:1872) (cid:2924) (cid:4670)M (cid:2908) (cid:4666)(cid:1872)(cid:4667)(cid:4671)(cid:3628) (cid:3047)(cid:2880)(cid:2868) . Clearly, from this definition, it immediately follows that the MGF for the sum of two indpendent RVs X and X , namely Z = X + X , is the product of their respective MGFs. In this research, however, we do not assume that X and X are independent. Therefore, this simplified expression is of limited use. (2)(1)(3) 4 B. An Overview of Gaussian Quadrature
We can approximate the area under a curve (i.e., an integral) by dividing the area into rectangles of equal width and summing their areas as depicted in Figure 1 below. In place of rectangles, a more accurate estimate may be derived using trapezoids or polynomials at the top. These methods are known as the trapezoidal rule and
Simpson’s rule , respectively (Anton [1988]).
Figure 1 Approximating the Area Under a Curve using Rectangles
If a total of n rectangles are used in the approximation, then the width can be fixed at w (cid:2928) (cid:3404) (cid:2912)(cid:2879)(cid:2911)(cid:2924) , for r = 1, 2, …, n. Using a fixed width, the midpoint of the r th rectangle is m (cid:2928)∗ (cid:3404) a (cid:3397) (cid:4666)r (cid:3398) 1(cid:4667) ∗w (cid:2928) (cid:3397) (cid:4672) (cid:2933) (cid:3176) (cid:2870) (cid:4673) (cid:3404) a (cid:3397) w (cid:2928) ∗ (cid:4666)r (cid:3398) (cid:2869)(cid:2870) (cid:4667) , for r = 1, 2, …, n. The height of each rectangle is the function evaluated at m (cid:2928)∗ , namely, f( m (cid:2928)∗ ). Therefore, the area under f(x) between points a and b is estimated as: Area Under Curve (cid:3404) (cid:3505) f(cid:4666)x(cid:4667) dx (cid:3406) (cid:3533) w (cid:2928) ∗ f(cid:4666)m (cid:2928)∗ (cid:4667) (cid:2924)(cid:2928)(cid:2880)(cid:2869)(cid:2912)(cid:2911) , and, lim (cid:2924)→(cid:2998) (cid:3437)(cid:3533) w (cid:2928) ∗ f(cid:4666)m (cid:2928)∗ (cid:4667) (cid:2924)(cid:2928)(cid:2880)(cid:2869) (cid:3441) (cid:3404) (cid:3505) f(cid:4666)x(cid:4667) dx (cid:2912)(cid:2911) . (4)(5) 5 The term in (5) is a Riemann sum and merely reflects the fact that using an infinite number of rectangles will yield the area exactly without it being an approximation (Anton [1988]). Depending on the function being integrated, we may need thousands of rectangles (or more) to obtain a good estimate. Clearly, if this integration appears within a larger iterative routine, then it will suffer runtime inefficiencies. A faster approximation can be achieved using numerical quadrature. This technique often requires only a small number of areas be summed, where both the rectangle “widths” and “midpoints” are determined mathematically. To estimate the area under f(x) between a and b using numerical quadrature, we first express f(x) = w(x) * g(x) where w(x) ≥
0 is referred to as a weight function (Golub and Welsch [1969]). Note that, in general, g(x) = [w(x)] -1* f(x). Then: (cid:3505) f(cid:4666)x(cid:4667) dx (cid:3404) (cid:3505) w(cid:4666)x(cid:4667) ∗ g(cid:4666)x(cid:4667) dx (cid:2912)(cid:2911) (cid:3406) (cid:3533) w (cid:2920) ∗ g(cid:3435)t (cid:2920) (cid:3439) (cid:2924)(cid:2920)(cid:2880)(cid:2869) , (cid:2912)(cid:2911) where the pairs (w j , t j ) for j=1, 2, …, n have been specifically derived for w(x) with respect to a given set of orthogonal polynomials. A set of polynomials P j (x) of degree j for j = 1, 2, …, n+1 are said to be orthogonal with respect to w(x) if the following condition holds (Golub and Welsch [1969]): (cid:3505) w(cid:4666)x(cid:4667) ∗ P (cid:2918) (cid:4666)x(cid:4667) ∗ P (cid:2921) (cid:4666)x(cid:4667) dx (cid:3404) 0, for h (cid:3405) k (cid:2912)(cid:2911) . This strict condition requires that the set of polynomials, P j (x), be carefully constructed to satisfy it. For certain weight functions the polynomials have already been derived. The sequence of orthogonal polynomials will take the form (Golub and Welsch [1969]): P (x) = k (x – t ) P (x) = k (x – t ) * (x – t ) ⋮ P n (x) = k n* (x – t ) * (x – t ) * (x – t ) * … * (x – t n ) P n+1 (x) = k n+1* (x – t ) * (x – t ) * (x – t ) * … * (x – t n+1 ) , which reveals an important quality. Namely, the values t j alluded to in (6) are the roots of P n (x) and satisfy a < t j < b, ∀ j = 1, 2, …, n. The weights, w j , are calculated as functions of the polynomials and their derivatives evaluated at the roots. Once the weights, w j , and roots, t j , are (6)(7)(8) 6 known, the sum in (6) can be calculated to estimate the integral. This technique is referred to as numerical quadrature. In Gaussian quadrature, the formula to calculate the weights is given by (Golub and Welsch [1969]): (cid:1875) (cid:2920) (cid:3404) (cid:3398) k (cid:2924)(cid:2878)(cid:2869) k (cid:2924) ∗ 1P (cid:2924)(cid:2878)(cid:2869) (cid:3435)t (cid:2920) (cid:3439) ∗ P (cid:2924)(cid:4593) (cid:3435)t (cid:2920) (cid:3439) . We cannot provide details about the roots, t j , because the orthogonal polynomials will be specific to the weight function, w(x), and our only requirement at this stage is that it be greater than or equal to zero for all x ∈ (a, b). C. Gauss-Hermite Quadrature
When the weight function is w(x) = e (cid:2879)(cid:2934) (cid:3118) and (a, b) = (- ∞ , + ∞ ), the Hermite polynomials are orthogonal. That is, they satisfy (7) (Abramowitz and Stegun [1964]). Weight and root pairs (w j , t j ) have already been calculated for several n and, in this research, we use n=12 “rectangles” to approximate the necessary integrals as suggested by Mehta et al. (2007). The corresponding weights and roots are shown below in Table 1 (Abramowitz and Stegun [1964]). Table 1 Gauss-Hermite Quadrature Weights and Roots for n=12 Roots (t j ) Weights (W j ) (+/-) 0.314240376254 0.570135236262500000 (+/-) 0.947788391240 0.260492310264200000 (+/-) 1.597682635153 0.051607985615880000 (+/-) 2.279507080501 0.003905390584629000 (+/-) 3.020637025121 0.000085736870435880 (+/-) 3.889724897870 0.000000265855168436 D. Overview of Lognormal RVs
In this section we present various results for lognormal RVs. Once finished, we will apply the routine suggested by Mehta et al. (2007) to approximate the sum of correlated lognormal RVs with a single new lognormal RV. (9) 7
D.1 Standard Univariate Form
Let X ~ N( μ x , σ x2 ) be a normally distributed RV. The PDF of X, f(x) , is given by (Casella and Berger [1990]): (cid:1858)(cid:4666)(cid:1876)(cid:4667) (cid:3404) 1√2(cid:2024)σ (cid:3051) e (cid:2879)(cid:2869)(cid:2870)(cid:4672)(cid:3051)(cid:2879) (cid:2972) (cid:3299) (cid:2978) (cid:3299) (cid:4673) (cid:3118) , (cid:1858)(cid:1867)(cid:1870) (cid:3398) ∞ (cid:3407) (cid:1876) (cid:3407) ∞ . The RV Y = e X is then said to be lognormally distributed. We derive the PDF of Y by making the substitution X = ln(Y) in (10) above. The Jacobian of this transformation is (cid:3031)(cid:2908)(cid:3031)(cid:2909) (cid:3404) (cid:2869)(cid:2909) and Y > 0. Therefore, the PDF of the lognormal RV Y is given by f(y) , where: (cid:1858)(cid:4666)(cid:1877)(cid:4667) (cid:3404) 1(cid:1877)√2(cid:2024)σ (cid:3051) e (cid:2879)(cid:2869)(cid:2870)(cid:3436)(cid:2922)(cid:2924)(cid:4666)(cid:3052)(cid:4667)(cid:2879) (cid:2972) (cid:3299) (cid:2978) (cid:3299) (cid:3440) (cid:3118) , (cid:1858)(cid:1867)(cid:1870) (cid:1877) (cid:3410) 0 . The mean and variance of Y are E(Y) = e (cid:2972) (cid:3299) (cid:2878) (cid:3226)(cid:3299)(cid:3118)(cid:3118) and V(Y) = (cid:3435)e (cid:2870)(cid:2972) (cid:3299) (cid:2878)(cid:2978) (cid:3299)(cid:3118) (cid:3439)(cid:3435)e (cid:2978) (cid:3299)(cid:3118) (cid:3398) 1(cid:3439) , respectively (Walpole, Myers, Myers, and Ye [2002]). D.2 Alternative Univariate Form
Using the natural log to define a lognormal RV is not required and any base logarithm will suffice. In this research, base 10 will be used with a constant factor to align with Mehta et al. (2007). Namely, we will define Ϋ = (cid:2908)/(cid:2869)(cid:2868) . Let θ = (cid:4672) (cid:2922)(cid:2924)(cid:4666)(cid:2869)(cid:2868)(cid:4667) (cid:2869)(cid:2868) (cid:4673), then ln( Ϋ ) = θX → Ϋ = e (cid:2968)(cid:2908) . Clearly, θX ~ N(cid:4666)θμ (cid:3051) , θ (cid:2870) σ (cid:3051)(cid:2870) (cid:4667) , therefore Ϋ has the standard form lognormal distribution (see Section III.D.1) based on the underlying normally distributed RV θX. Using (11) above, it immediately follows that the mean and variance of Ϋ are, respectively: E(cid:3427)Y(cid:4663) (cid:3431) (cid:3404) e (cid:3436)(cid:2968)(cid:2972) (cid:3299) (cid:2878) (cid:2968) (cid:3118) (cid:2978) (cid:3299)(cid:3118) (cid:2870) (cid:3440) and
V(cid:3427)Y(cid:4663) (cid:3431) (cid:3404) (cid:3435)e (cid:2870)(cid:2968)(cid:2972) (cid:3299) (cid:2878) (cid:2968) (cid:3118) (cid:2978) (cid:3299)(cid:3118) (cid:3439)(cid:3435)e (cid:2968) (cid:3118) (cid:2978) (cid:3299)(cid:3118) (cid:3398) 1(cid:3439) .
Note that (cid:3031)(cid:2908)(cid:3031)(cid:2909)(cid:4663) (cid:3404) (cid:2869)(cid:2968) (cid:2869)(cid:2909)(cid:4663) and Ϋ > 0. If we are given E[ Ϋ ] and V[ Ϋ ], then E[X] = μ (cid:3051) and V[X] = σ (cid:3051)(cid:2870) can be calculated and vice-versa (see Section III.D.4 below). Finally, using (11) with the updated underlying normal distribution, the PDF of Ϋ , f( (cid:1877)(cid:4663) ) , is given by: (10)(13)(12)(11) 8 (cid:1858)(cid:4666)(cid:1877)(cid:4663) (cid:4667) (cid:3404) 1θ(cid:1877)(cid:4663) √2(cid:2024)σ (cid:3051) e (cid:2879) (cid:2869)(cid:2870)(cid:4684)(cid:4672)(cid:2869)(cid:2968)(cid:4673) (cid:2922)(cid:2924)(cid:4666)(cid:3052)(cid:4663) (cid:4667)(cid:2879) (cid:2972) (cid:3299) (cid:2978) (cid:3299) (cid:4685) (cid:3118) , (cid:1858)(cid:1867)(cid:1870) (cid:1877)(cid:4663) (cid:3410) 0 . D.3 Standard Bivariate Form
Let X ~ N( μ (cid:3051) (cid:3117) , σ (cid:3051) (cid:3117) (cid:2870) ) and X ~ N( μ (cid:3051) (cid:3118) , σ (cid:3051) (cid:3118) (cid:2870) ) be joint normal RVs with Cov(X , X ) = σ (cid:4666)(cid:3051) (cid:3117) , (cid:3051) (cid:3118) (cid:4667) . The bivariate PDF of X and X , f(x ,x ) , is given by (Casella and Berger [1990]): (cid:1858)(cid:4666)(cid:1876) (cid:2869) , (cid:1876) (cid:2870) (cid:4667) (cid:3404) 12(cid:2024)σ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) (cid:3493)1 (cid:3398) ρ (cid:2870) e (cid:2879) (cid:2869)(cid:2870)(cid:4666)(cid:2869)(cid:2879)(cid:2977) (cid:3118) (cid:4667)(cid:4680)(cid:3436)(cid:3051) (cid:3117) (cid:2879)(cid:2972) (cid:3299)(cid:3117) (cid:2978) (cid:3299)(cid:3117) (cid:3440) (cid:3118) (cid:2879) (cid:2870)(cid:2977)(cid:3436)(cid:3051) (cid:3117) (cid:2879)(cid:2972) (cid:3299)(cid:3117) (cid:2978) (cid:3299)(cid:3117) (cid:3440)(cid:3436)(cid:3051) (cid:3118) (cid:2879)(cid:2972) (cid:3299)(cid:3118) (cid:2978) (cid:3299)(cid:3118) (cid:3440) (cid:2878) (cid:3436)(cid:3051) (cid:3118) (cid:2879)(cid:2972) (cid:3299)(cid:3118) (cid:2978) (cid:3299)(cid:3118) (cid:3440) (cid:3118) (cid:4681) , where - ∞ < x i < ∞ for i =1, 2 and ρ = (cid:2978) (cid:4666)(cid:3299)(cid:3117), (cid:3299)(cid:3118)(cid:4667) (cid:2978) (cid:3299)(cid:3117) (cid:2978) (cid:3299)(cid:3118) is the correlation between X and X . Now, define new RVs Y (cid:2869) (cid:3404) e (cid:2908) (cid:3117) and Y (cid:2870) (cid:3404) e (cid:2908) (cid:3118) such that X (cid:2869) (cid:3404) ln (cid:4666)Y (cid:2869) (cid:4667) and X (cid:2870) (cid:3404) ln (cid:4666)Y (cid:2870) (cid:4667) . The Jacobian of this transformation is (cid:4672) (cid:2869)(cid:2909) (cid:3117) (cid:2869)(cid:2909) (cid:3118) (cid:4673) and the joint lognormal PDF of Y and Y is defined using (15) as : (cid:1858)(cid:4666)(cid:1877) (cid:2869) , (cid:1877) (cid:2870) (cid:4667) (cid:3404) (cid:2869) (cid:1877) (cid:2870) σ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) (cid:3493)1 (cid:3398) ρ (cid:2870) e (cid:2879) (cid:2869)(cid:2870)(cid:4666)(cid:2869)(cid:2879)(cid:2977) (cid:3118) (cid:4667)(cid:3429)(cid:4678)(cid:2922)(cid:2924) (cid:4666)(cid:3052) (cid:3117) (cid:4667)(cid:2879)(cid:2972) (cid:3299)(cid:3117) (cid:2978) (cid:3299)(cid:3117) (cid:4679) (cid:3118) (cid:2879) (cid:2870)(cid:2977)(cid:4678)(cid:2922)(cid:2924) (cid:4666)(cid:3052) (cid:3117) (cid:4667)(cid:2879)(cid:2972) (cid:3299)(cid:3117) (cid:2978) (cid:3299)(cid:3117) (cid:4679)(cid:4678)(cid:2922)(cid:2924) (cid:4666)(cid:3052) (cid:3118) (cid:4667)(cid:2879)(cid:2972) (cid:3299)(cid:3118) (cid:2978) (cid:3299)(cid:3118) (cid:4679) (cid:2878) (cid:4678)(cid:2922)(cid:2924) (cid:4666)(cid:3052) (cid:3118) (cid:4667)(cid:2879)(cid:2972) (cid:3299)(cid:3118) (cid:2978) (cid:3299)(cid:3118) (cid:4679) (cid:3118) (cid:3433) , where y i ≥ i =1, 2. The means and variances for the lognormal RVs Y and Y are given by E[Y i ] = e (cid:2972) (cid:3299)(cid:3284) (cid:2878) (cid:3226)(cid:3299)(cid:3284)(cid:3118)(cid:3118) and V[Y i ] = (cid:4672)e (cid:2870)(cid:2972) (cid:3299)(cid:3284) (cid:2878) (cid:2978) (cid:3299)(cid:3284)(cid:3118) (cid:4673) (cid:4672)e (cid:2978) (cid:3299)(cid:3284)(cid:3118) (cid:3398) 1(cid:4673) for i =1, 2. The covariance is derived as (Law and Kelton [2000]): σ (cid:4666)(cid:3052) (cid:3117) ,(cid:3052) (cid:3118) (cid:4667) (cid:3404) (cid:4666)e (cid:2978) (cid:4666)(cid:3299)(cid:3117), (cid:3299)(cid:3118)(cid:4667) (cid:3398) 1(cid:4667) ∗ e (cid:4678)(cid:2972) (cid:3299)(cid:3117) (cid:2878)(cid:2972) (cid:3299)(cid:3118) (cid:2878) (cid:2978) (cid:3299)(cid:3117)(cid:3118) (cid:2878) (cid:2978) (cid:3299)(cid:3118)(cid:3118) (cid:2870) (cid:4679) . The transformed joint PDF of Y and Y , (cid:1858)(cid:4666)(cid:1877) (cid:2869) , (cid:1877) (cid:2870) (cid:4667), is derived by substituting X and X with their expressions in terms of Y and Y in the original joint PDF, (cid:1858)(cid:4666)(cid:1876) (cid:2869) , (cid:1876) (cid:2870) (cid:4667) (as shown in (15)), and multiplying by the absolute value of the Jacobian (Freund [1992]). (14)(15)(16)(17) 9 D.4 Alternative Bivariate Form
As in the univariate case, natural logarithms are not required when defining bivariate lognormal RVs; any base is acceptable. In this research, we will use base 10 along with a constant factor. Assume X and X are joint normal RVs as defined in Section III.D.3. Let Ϋ = (cid:2908) (cid:3117) /(cid:2869)(cid:2868) and Ϋ = (cid:2908) (cid:3118) /(cid:2869)(cid:2868) , which implies that ln( Ϋ ) = θX (cid:2869) → Ϋ = e (cid:2968)(cid:2908) (cid:3117) and ln( Ϋ ) = θX (cid:2870) → Ϋ = e (cid:2968)(cid:2908) (cid:3118) . But, θX (cid:2869) ~ N(cid:3435)θμ (cid:3051) (cid:3117) , θ (cid:2870) σ (cid:3051) (cid:3117) (cid:2870) (cid:3439) , and θX (cid:2870) ~ N(cid:3435)θμ (cid:3051) (cid:3118) , θ (cid:2870) σ (cid:3051) (cid:3118) (cid:2870) (cid:3439) with covariance and correlation as (Ross [2009]): Cov(cid:4666)θX (cid:2869) , θX (cid:2870) (cid:4667) (cid:3404) θ (cid:2870)
Cov(cid:4666)X (cid:2869) , X (cid:2870) (cid:4667) (cid:3404) θ (cid:2870) σ (cid:4666)(cid:3051) (cid:3117) , (cid:3051) (cid:3118) (cid:4667) , and, Corr(cid:4666)θX (cid:2869) , θX (cid:2870) (cid:4667) (cid:3404) θ (cid:2870) σ (cid:4666)(cid:3051) (cid:3117) , (cid:3051) (cid:3118) (cid:4667) θ (cid:2870) σ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) (cid:3404) σ (cid:4666)(cid:3051) (cid:3117) , (cid:3051) (cid:3118) (cid:4667) σ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) (cid:3404) ρ , as before. The correlation between the underlying normal RVs does not change form under the alternative representation. The joint lognormal PDF of Ϋ and Ϋ is therefore defined using (16) with the scaled means and variances as: (cid:1858)(cid:4666)(cid:1877)(cid:4663) (cid:2869) , (cid:1877)(cid:4663) (cid:2870) (cid:4667) (cid:3404) (cid:4672)1θ(cid:4673) (cid:2870) (cid:2869) (cid:1877)(cid:4663) (cid:2870) σ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) (cid:3493)1 (cid:3398) ρ (cid:2870) ∗ e (cid:2879) (cid:2869)(cid:2870)(cid:4666)(cid:2869)(cid:2879)(cid:2977) (cid:3118) (cid:4667)(cid:3430)(cid:4684)(cid:4672)(cid:2869)(cid:2968)(cid:4673)(cid:2922)(cid:2924) (cid:4666)(cid:3052)(cid:4663) (cid:3117) (cid:4667)(cid:2879)(cid:2972) (cid:3299)(cid:3117) (cid:2978) (cid:3299)(cid:3117) (cid:4685) (cid:3118) (cid:2879) (cid:2870)(cid:2977)(cid:4684)(cid:4672)(cid:2869)(cid:2968)(cid:4673)(cid:2922)(cid:2924) (cid:4666)(cid:3052)(cid:4663) (cid:3117) (cid:4667)(cid:2879)(cid:2972) (cid:3299)(cid:3117) (cid:2978) (cid:3299)(cid:3117) (cid:4685)(cid:4684)(cid:4672)(cid:2869)(cid:2968)(cid:4673)(cid:2922)(cid:2924) (cid:4666)(cid:3052)(cid:4663) (cid:3118) (cid:4667)(cid:2879)(cid:2972) (cid:3299)(cid:3118) (cid:2978) (cid:3299)(cid:3118) (cid:4685) (cid:2878) (cid:4684)(cid:4672)(cid:2869)(cid:2968)(cid:4673)(cid:2922)(cid:2924) (cid:4666)(cid:3052)(cid:4663) (cid:3118) (cid:4667)(cid:2879)(cid:2972) (cid:3299)(cid:3118) (cid:2978) (cid:3299)(cid:3118) (cid:4685) (cid:3118) (cid:3434) , where (cid:1877)(cid:4663) (cid:3036) ≥ i =1, 2. The means and variances for Ϋ and Ϋ are given by: E(cid:4670)Y(cid:4663) (cid:3036) (cid:4671) (cid:3404) e (cid:2968)(cid:2972) (cid:3299)(cid:3284) (cid:2878) (cid:2968) (cid:3118) (cid:2978) (cid:3299)(cid:3284)(cid:3118) (cid:2870) , and, V(cid:4670)Y(cid:4663) (cid:3036) (cid:4671) (cid:3404) (cid:4672)e (cid:2870)(cid:2968)(cid:2972) (cid:3299)(cid:3284) (cid:2878)(cid:2968) (cid:3118) (cid:2978) (cid:3299)(cid:3284)(cid:3118) (cid:4673) ∗ (cid:4672)e (cid:2968) (cid:3118) (cid:2978) (cid:3299)(cid:3284)(cid:3118) (cid:3398) 1(cid:4673) , for (cid:1861) (cid:3404) 1, 2.
The covariance is derived as (Law and Kelton [2000]): σ (cid:4666)(cid:3052)(cid:4663) (cid:3117) ,(cid:3052)(cid:4663) (cid:3118) (cid:4667) (cid:3404) (cid:3435)e (cid:2968) (cid:3118) (cid:2978) (cid:4666)(cid:3299)(cid:3117), (cid:3299)(cid:3118)(cid:4667) (cid:3398) 1(cid:3439) ∗ e (cid:4684)(cid:2968)(cid:3435)(cid:2972) (cid:3299)(cid:3117) (cid:2878)(cid:2972) (cid:3299)(cid:3118) (cid:3439) (cid:3126) (cid:2968) (cid:3118) (cid:4678)(cid:2978) (cid:3299)(cid:3117)(cid:3118) (cid:2878) (cid:2978) (cid:3299)(cid:3118)(cid:3118) (cid:2870) (cid:4679)(cid:4685) . (18)(19)(20)(23)(22)(21) 10 Lastly, if given E[ Ϋ i ] = μ (cid:3052)(cid:4663) (cid:3284) and V[ Ϋ i ] = σ (cid:3052)(cid:4663) (cid:3284) (cid:2870) , i =1,
2, we can derive the underlying normal parameters by solving equations (21) and (22) for μ (cid:3051) (cid:3284) and σ (cid:3051) (cid:3284) (cid:2870) , for i =1, 2. Doing so yields (Law and Kelton [2000]): μ (cid:3051) (cid:3284) (cid:3404) (cid:3436)1θ(cid:3440) (cid:4678)ln(cid:3435)μ (cid:3052)(cid:4663) (cid:3284) (cid:3439) (cid:3398) 12 ln (cid:4680)1 (cid:3397) σ (cid:3052)(cid:4663) (cid:3284) (cid:2870) μ (cid:3052)(cid:4663) (cid:3284) (cid:2870) (cid:4681)(cid:4679) , and, σ (cid:3051) (cid:3284) (cid:2870) (cid:3404) (cid:3436)1θ(cid:3440) (cid:2870) ln (cid:4680)1 (cid:3397) σ (cid:3052)(cid:4663) (cid:3284) (cid:2870) μ (cid:3052)(cid:4663) (cid:3284) (cid:2870) (cid:4681) , for i =1, 2. Finally, if Cov( Ϋ , Ϋ ) = σ (cid:4666)(cid:3052)(cid:4663) (cid:3117) ,(cid:3052)(cid:4663) (cid:3118) (cid:4667) is known the covariance of the underlying normal RVs is given by (Law and Kelton [2000]): σ (cid:4666)(cid:3051) (cid:3117) , (cid:3051) (cid:3118) (cid:4667) (cid:3404) (cid:3436)1θ(cid:3440) (cid:2870) ln (cid:4680)1 (cid:3397) σ (cid:4666)(cid:3052)(cid:4663) (cid:3117) ,(cid:3052)(cid:4663) (cid:3118) (cid:4667) (cid:3627)μ (cid:3052)(cid:4663) (cid:3117) μ (cid:3052)(cid:4663) (cid:3118) (cid:3627)(cid:4681) . D.5 Moment Generating Function for the Lognormal Distribution
Let X ~ N( μ x , σ x2 ) and Y = e (cid:2908) be the corresponding lognormal RV defined in standard form. Using (1) and (11) the MGF for Y is: M (cid:2909) (cid:4666)(cid:1872)(cid:4667) (cid:3404) E (cid:2909) (cid:4670)e (cid:3047)(cid:3052) (cid:4671) (cid:3404) (cid:3505) e (cid:3047)(cid:3052) ∗ (cid:1858)(cid:4666)(cid:1877)(cid:4667) (cid:1856)(cid:1877) (cid:2998)(cid:2868) (cid:3404) 1√2(cid:2024)σ (cid:3051) (cid:3505) e (cid:4680)(cid:3047)(cid:3052)(cid:2879) (cid:2869)(cid:2870)(cid:3436)(cid:2922)(cid:2924)(cid:4666)(cid:3052)(cid:4667)(cid:2879) (cid:2972) (cid:3299) (cid:2978) (cid:3299) (cid:3440) (cid:3118) (cid:4681) (cid:1877) (cid:1856)(cid:1877) (cid:2998)(cid:2868) . The kernel of (27) has an indeterminate form of (cid:4672) (cid:2998)(cid:2998) (cid:4673) as y → ∞ (for t > 0), since the power in the numerator can be expressed as ty – ( (cid:2869)(cid:2870)(cid:2978) (cid:3299)(cid:3118) [ln( y )] - (cid:2972) (cid:3299) (cid:2978) (cid:3299)(cid:3118) ln( y ) + (cid:2972) (cid:3299)(cid:3118) (cid:2870)(cid:2978) (cid:3299)(cid:3118) ). Note that t y – (cid:2869)(cid:2870)(cid:2978) (cid:3299)(cid:3118) [ln( y )] increases without bound (for t > 0) as a function of y since applying L’Hôpital’s Rule twice to lim (cid:3052)→(cid:2998) (cid:4672) (cid:2870)(cid:2978) (cid:3299)(cid:3118) (cid:2930)(cid:3052)(cid:4670)(cid:2922)(cid:2924) (cid:4666)(cid:3052)(cid:4667)(cid:4671) (cid:3118) (cid:4673) shows it to be infinite. This implies that ty increases faster than (cid:2869)(cid:2870)(cid:2978) (cid:3299)(cid:3118) [ln( y )] . To find which value the kernel in (27) approaches as y increases, it suffices to consider the limit: \ lim (cid:3052)→(cid:2998) (cid:3438)e (cid:4680)(cid:3047)(cid:3052)(cid:2879)(cid:4670)(cid:2922)(cid:2924) (cid:4666)(cid:3052)(cid:4667)(cid:4671) (cid:3118) (cid:2870)(cid:2978) (cid:3299)(cid:3118) (cid:4681) (cid:1877) (cid:3442) , which, after one application of L’Hôpital’s Rule (for t > 0), becomes: (24)(25)(26)(27)(28) 11 lim (cid:3052)→(cid:2998) (cid:4678)e (cid:3428)(cid:3047)(cid:3052)(cid:2879) (cid:4670)(cid:3170)(cid:3172) (cid:4666)(cid:3300)(cid:4667)(cid:4671)(cid:3118)(cid:3118)(cid:3226)(cid:3299)(cid:3118) (cid:3432) ∗ (cid:4674)(cid:1872) (cid:3398) (cid:2869)(cid:2870)(cid:2978) (cid:3299)(cid:3118) (cid:2870)(cid:2922)(cid:2924) (cid:4666)(cid:3052)(cid:4667)(cid:3052) (cid:4675)(cid:4679) (cid:3404) e (cid:2998) ∗ (cid:4670)(cid:1872) (cid:3398) 0(cid:4671) (cid:3404) ∞ . Since the kernel of (27) is ≥ ∞ as y increases (for t > 0), the area under it must also approach ∞ which implies that there does not exist an ε > 0 such the MGF is defined ∀ t ∈ (- ε , ε ). This implies that the MGF of a lognormal RV does not exist. Casella and Berger (1990) refer to this as an “interesting property,” namely that all moments for a lognormal RV exist and are finite, but, despite this fact, the MGF does not exist. D.6 Approximating the Lognormal Moment-Generating Function
Let X ~ N( μ x , σ x2 ) and Ϋ = (cid:2908)/(cid:2869)(cid:2868) be the alternative form lognormal RV as defined in Section III.D.2. Using the PDF from (14), the MGF for Ϋ is: M (cid:2909)(cid:4663) (cid:4666)(cid:1872)(cid:4667) (cid:3404) E (cid:2909)(cid:4663) (cid:3427)e (cid:3047)(cid:3052)(cid:4663) (cid:3431) (cid:3404) (cid:3505) e (cid:3047)(cid:3052)(cid:4663) ∗ (cid:1858)(cid:4666)(cid:1877)(cid:4663) (cid:4667) (cid:1856)(cid:1877)(cid:4663) (cid:2998)(cid:2868) (cid:3404) (cid:3505) e (cid:3047)(cid:3052)(cid:4663) ∗ 1θ(cid:1877)(cid:4663) √2(cid:2024)σ (cid:3051) e (cid:2879) (cid:2869)(cid:2870)(cid:4684)(cid:4672)(cid:2869)(cid:2968)(cid:4673) (cid:2922)(cid:2924)(cid:4666)(cid:3052)(cid:4663) (cid:4667)(cid:2879) (cid:2972) (cid:3299) (cid:2978) (cid:3299) (cid:4685) (cid:3118) (cid:1856)(cid:1877)(cid:4663) . (cid:2998)(cid:2868) In the RHS integral from (30), we make the following U-substitution:
Let (cid:1873) (cid:3404) (cid:4672)1θ(cid:4673) ln(cid:4666)(cid:1877)(cid:4663) (cid:4667) (cid:3398) μ (cid:3051) √2σ (cid:3051) → (cid:1856)(cid:1873) (cid:3404) (cid:4672)1θ(cid:4673)√2σ (cid:3051) ∗ 1(cid:1877)(cid:4663) ∗ (cid:1856)(cid:1877)(cid:4663) , and, (cid:1877)(cid:4663) (cid:3404) e (cid:2968)(cid:3435)(cid:3048)√(cid:2870)(cid:2978) (cid:3299) (cid:2878)(cid:2972) (cid:3299) (cid:3439) . As (cid:1877)(cid:4663) →
0, u → - ∞ , and as (cid:1877)(cid:4663) → ∞ , u → ∞ . Writing the integral from (30) in terms of u yields the following expression for the MGF of Ϋ : M (cid:2909)(cid:4663) (cid:4666)(cid:1872)(cid:4667) (cid:3404) (cid:3505) 1√(cid:2024) e (cid:3047)∗(cid:2915) (cid:3216)(cid:3435)(cid:3296)√(cid:3118)(cid:3226)(cid:3299)(cid:3126)(cid:3220)(cid:3299)(cid:3439) e (cid:2879)(cid:3048) (cid:3118) (cid:1856)(cid:1873) (cid:3404) (cid:3505) g(cid:4666)(cid:1873)(cid:4667) ∗ w(cid:4666)(cid:1873)(cid:4667) d(cid:1873) (cid:2998)(cid:2879)(cid:2998) (cid:2998)(cid:2879)(cid:2998) . Here, the weight function w(cid:4666)(cid:1873)(cid:4667) (cid:3404) e (cid:2879)(cid:3048) (cid:3118) is of the form required by Gauss-Hermite quadrature, with appropriate integration limits, thus the MGF from (30) can be approximated (for t < 0) using the weights and roots provided in Table 1 with n=12. Namely, (29)(30)(31)(32)(33)(34) 12 M (cid:2909)(cid:4663) (cid:4666)(cid:1872)(cid:4667) (cid:3404) (cid:3505) 1√(cid:2024) e (cid:3047)(cid:2915) (cid:3216)(cid:3435)(cid:3296)√(cid:3118)(cid:3226)(cid:3299)(cid:3126)(cid:3220)(cid:3299)(cid:3439) e (cid:2879)(cid:3048) (cid:3118) (cid:1856)(cid:1873) (cid:3406) (cid:3533) w (cid:2920) ∗ 1√(cid:2024) e (cid:3047)(cid:2915) (cid:3216)(cid:4672)(cid:3295)(cid:3168)√(cid:3118)(cid:3226)(cid:3299)(cid:3126)(cid:3220)(cid:3299)(cid:4673) (cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869)(cid:2998)(cid:2879)(cid:2998) . Recall that the MGF for a lognormal RV does not exist, something we showed in Section III.D.5, despite (35). The point is to approximate the sum of correlated lognormal RVs with a single lognormal RV which will be accomplished by equating their approximated MGFs for given t < 0, where it does exist. The justification for this is provided by Mitchell (1968) who showed that a single lognormal RV yields a better approximation to the sum than any other distribution examined, a property referred to as “permanence.”
D.7 Approximating the Moment-Generating Function of a Sum
Let S = αΫ + βΫ be a weighted sum of two correlated lognormal RVs with bivariate PDF, (cid:1858)(cid:4666)(cid:1877)(cid:4663) (cid:2869) , (cid:1877)(cid:4663) (cid:2870) (cid:4667) , as shown in (20). The MGF for S is given by: M (cid:2903) (cid:4666)(cid:1872)(cid:4667) (cid:3404) E (cid:2903) (cid:4670)e (cid:3047)(cid:3046) (cid:4671) (cid:3404) E(cid:3427)e (cid:3047)(cid:4666)(cid:2961)(cid:3052)(cid:4663) (cid:3117) (cid:2878)(cid:2962)(cid:3052)(cid:4663) (cid:3118) (cid:4667) (cid:3431) (cid:3404) (cid:3505) (cid:3505) e (cid:3047)(cid:4666)(cid:2961)(cid:3052)(cid:4663) (cid:3117) (cid:2878)(cid:2962)(cid:3052)(cid:4663) (cid:3118) (cid:4667) (cid:1858)(cid:4666)(cid:1877)(cid:4663) (cid:2869) , (cid:1877)(cid:4663) (cid:2870) (cid:4667) (cid:1856)(cid:1877)(cid:4663) (cid:2869) (cid:1856)(cid:1877)(cid:4663) (cid:2870)(cid:2998)(cid:2868)(cid:2998)(cid:2868) . We will evaluate M (cid:2903) (cid:4666)(cid:1872)(cid:4667) in 2 steps. First, we make the following U-substitutions: Let (cid:1873) (cid:3036) (cid:3404) (cid:3436)1θ(cid:3440) ln(cid:4666)(cid:1877)(cid:4663) (cid:3036) (cid:4667) , (cid:1858)(cid:1867)(cid:1870) (cid:1861) (cid:3404) 1, 2 → (cid:1856)(cid:1873) (cid:3036) (cid:3404) (cid:3436)1θ(cid:3440) 1(cid:1877)(cid:4663) (cid:3036) (cid:1856)(cid:1877)(cid:4663) (cid:3036) , (cid:1858)(cid:1867)(cid:1870) (cid:1861) (cid:3404) 1, 2 , and, (cid:1877)(cid:4663) (cid:3036) (cid:3404) e (cid:2968)(cid:3048) (cid:3284) . As (cid:1877)(cid:4663) (cid:3036) → (cid:1873) (cid:3036) → - ∞ , and as (cid:1877)(cid:4663) (cid:3036) → ∞ , (cid:1873) (cid:3036) → ∞ . The Jacobian matrix for this transformation has zeros on the off-diagonal, and terms θe (cid:2968)(cid:3048) (cid:3284) on the i th diagonal, therefore the Jacobian determinant is equal to θ (cid:2870) e (cid:2968)(cid:3048) (cid:3117) e (cid:2968)(cid:3048) (cid:3118) , and the right-side integral in (36) can be expressed using (20) as: (35)(36)(37)(38)(39) 13 M (cid:3020) (cid:4666)(cid:1872)(cid:4667) (cid:3404) (cid:3505) (cid:3505) e (cid:3047)(cid:3435)(cid:2961)(cid:2915) (cid:3216)(cid:3296)(cid:3117) (cid:2878)(cid:2962)(cid:2915) (cid:3216)(cid:3296)(cid:3118) (cid:3439) (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) (cid:3493)1 (cid:3398) ρ (cid:2870) ∗ (cid:2998)(cid:2879)(cid:2998)(cid:2998)(cid:2879)(cid:2998) e (cid:2879) (cid:2869)(cid:2870)(cid:4666)(cid:2869)(cid:2879)(cid:2977) (cid:3118) (cid:4667)(cid:4680)(cid:3436)(cid:3048) (cid:3117) (cid:2879)(cid:2972) (cid:3299)(cid:3117) (cid:2978) (cid:3299)(cid:3117) (cid:3440) (cid:3118) (cid:2879) (cid:2870)(cid:2977)(cid:3436)(cid:3048) (cid:3117) (cid:2879)(cid:2972) (cid:3299)(cid:3117) (cid:2978) (cid:3299)(cid:3117) (cid:3440)(cid:3436)(cid:3048) (cid:3118) (cid:2879)(cid:2972) (cid:3299)(cid:3118) (cid:2978) (cid:3299)(cid:3118) (cid:3440) (cid:2878) (cid:3436)(cid:3048) (cid:3118) (cid:2879)(cid:2972) (cid:3299)(cid:3118) (cid:2978) (cid:3299)(cid:3118) (cid:3440) (cid:3118) (cid:4681) (cid:1856)(cid:1873) (cid:2869) (cid:1856)(cid:1873) (cid:2870) (cid:3404) (cid:3505) (cid:3505) e (cid:3047)(cid:3435)(cid:2961)(cid:2915) (cid:3216)(cid:3296)(cid:3117) (cid:2878)(cid:2962)(cid:2915) (cid:3216)(cid:3296)(cid:3118) (cid:3439) (cid:1858)(cid:4666)(cid:1873) (cid:2869) , (cid:1873) (cid:2870) (cid:4667) (cid:1856)(cid:1873) (cid:2869) (cid:1856)(cid:1873) (cid:2870) (cid:2998)(cid:2879)(cid:2998)(cid:2998)(cid:2879)(cid:2998) . We have written M (cid:3020) (cid:4666)(cid:1872)(cid:4667) as the expected value of a function with respect to U (cid:2869) ~ N( μ (cid:3051) (cid:3117) , σ (cid:3051) (cid:3117) (cid:2870) ) and U (cid:2870) ~ N( μ (cid:3051) (cid:3118) , σ (cid:3051) (cid:3118) (cid:2870) ) with Corr( U (cid:2869) , U (cid:2870) ) = ρ . In (41), (cid:1858)(cid:4666)(cid:1873) (cid:2869) , (cid:1873) (cid:2870) (cid:4667) is a joint bivariate normal PDF with form as in (15). In matrix notation, this PDF can be expressed as (Guttman [1982]): (cid:1858)(cid:4666)(cid:1873) (cid:2869) , (cid:1873) (cid:2870) (cid:4667) (cid:3404) 12(cid:2024)|∑| (cid:2869) (cid:2870)⁄ e (cid:2879)(cid:2869)(cid:2870)(cid:4666)(cid:2203)(cid:4652)(cid:4652)(cid:1318) (cid:2879) (cid:2246)(cid:4652)(cid:4652)(cid:1318)(cid:4667) (cid:4594) ∑ (cid:3127)(cid:3117) (cid:4666)(cid:2203)(cid:4652)(cid:4652)(cid:1318) (cid:2879) (cid:2246)(cid:4652)(cid:4652)(cid:1318)(cid:4667) , (cid:1858)(cid:1867)(cid:1870) (cid:2203)(cid:4652)(cid:4652)(cid:1318) ∈ (cid:1336) (cid:2870) , where, (cid:2203)(cid:4652)(cid:4652)(cid:1318) (cid:3404) (cid:4672)(cid:1873) (cid:2869) (cid:1873) (cid:2870) (cid:4673) , (cid:2246)(cid:4652)(cid:4652)(cid:1318) (cid:3404) (cid:4672)μ (cid:3051) (cid:3117) μ (cid:3051) (cid:3118) (cid:4673) , (cid:1853)(cid:1866)(cid:1856), (cid:2737) (cid:3404) (cid:4680) σ (cid:3051) (cid:3117) (cid:2870) σ (cid:3051) (cid:3117) (cid:3051) (cid:3118) σ (cid:3051) (cid:3117) (cid:3051) (cid:3118) σ (cid:3051) (cid:3118) (cid:2870) (cid:4681) (cid:3404) (cid:4680) σ (cid:3051) (cid:3117) (cid:2870) ρσ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) ρσ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) σ (cid:3051) (cid:3118) (cid:2870) (cid:4681). Here, Σ is referred to as the variance-covariance matrix of (cid:2203)(cid:4652)(cid:4652)(cid:1318) which we assume is symmetric and positive definite. Therefore, its inverse, Σ -1 , exists. Further, Σ -1 will be symmetric and positive definite which implies there exist matrices L and D (where L is lower-triangular with 1’s on the diagonal and D is diagonal with positive real pivots) such that (Meyer [2000]): (cid:2737) (cid:2879)(cid:2869) (cid:3404) (cid:1786)(cid:1778)(cid:1786) (cid:4593) (cid:3404) (cid:3428)1 0(cid:2011) 1(cid:3432) ∗ (cid:3428)(cid:1856) (cid:2869)(cid:2869)
00 (cid:1856) (cid:2870)(cid:2870) (cid:3432) ∗ (cid:4674)1 (cid:2011)0 1(cid:4675) .
This is the LDU factorization of (cid:2737) (cid:2879)(cid:2869) . We find L and D by equating terms as shown below (using the standard formula for a 2x2 matrix inverse) and solving for (cid:1856) (cid:2869)(cid:2869) , (cid:1856) (cid:2870)(cid:2870) , and (cid:2011) : (cid:2737) (cid:2879)(cid:2869) (cid:3404) 1σ (cid:3051) (cid:3117) (cid:2870) σ (cid:3051) (cid:3118) (cid:2870) (cid:4666)1 (cid:3398) ρ (cid:2870) (cid:4667) (cid:4680) σ (cid:3051) (cid:3118) (cid:2870) (cid:3398)ρσ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) (cid:3398)ρσ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) σ (cid:3051) (cid:3117) (cid:2870) (cid:4681) (cid:3404) (cid:1786)(cid:1778)(cid:1786) (cid:4593) (cid:3404) (cid:3428) (cid:1856) (cid:2869)(cid:2869) (cid:2011)(cid:1856) (cid:2869)(cid:2869) (cid:2011)(cid:1856) (cid:2869)(cid:2869) (cid:2011) (cid:2870) (cid:1856) (cid:2869)(cid:2869) (cid:3397) (cid:1856) (cid:2870)(cid:2870) (cid:3432) → (cid:1856) (cid:2869)(cid:2869) (cid:3404) 1σ (cid:3051) (cid:3117) (cid:2870) (cid:4666)1 (cid:3398) ρ (cid:2870) (cid:4667) , (cid:2011) (cid:3404) (cid:3398)ρσ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) , (cid:1853)(cid:1866)(cid:1856), (cid:1856) (cid:2870)(cid:2870) (cid:3404) 1σ (cid:3051) (cid:3118) (cid:2870) . The LDU factorization therefore yields: (40)(41)(46)(45)(44)(43)(42) 14 (cid:2737) (cid:2879)(cid:2869) (cid:3404) (cid:1786)(cid:1778)(cid:1786) (cid:4593) (cid:3404) (cid:3429) 1 0(cid:3398)ρσ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) (cid:3051) (cid:3117) (cid:2870) (cid:4666)1 (cid:3398) ρ (cid:2870) (cid:4667) 00 1σ (cid:3051) (cid:3118) (cid:2870) (cid:1746)(cid:1745)(cid:1745)(cid:1745)(cid:1744) ∗ (cid:3429)1 (cid:3398)ρσ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) (cid:3404) (cid:1786)(cid:1778) (cid:2868).(cid:2873) (cid:1778) (cid:2868).(cid:2873) (cid:1786) (cid:4593) (cid:3404) (cid:3429) 1 0(cid:3398)ρσ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) (cid:3051) (cid:3117) (cid:3493)(cid:4666)1 (cid:3398) ρ (cid:2870) (cid:4667) 00 1σ (cid:3051) (cid:3118) (cid:1746)(cid:1745)(cid:1745)(cid:1745)(cid:1744) ∗ (cid:1743)(cid:1742)(cid:1742)(cid:1742)(cid:1741) 1σ (cid:3051) (cid:3117) (cid:3493)(cid:4666)1 (cid:3398) ρ (cid:2870) (cid:4667) 00 1(cid:2026) (cid:3051) (cid:3118) (cid:1746)(cid:1745)(cid:1745)(cid:1745)(cid:1744) ∗ (cid:3429)1 (cid:3398)ρσ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) The variance-covariance matrix (cid:2737) can be expressed similarly as: (cid:2737) (cid:3404) (cid:4666)(cid:1786)(cid:1778)(cid:1786) (cid:4593) (cid:4667) (cid:2879)(cid:2778) (cid:3404) (cid:4666)(cid:1786) (cid:4593) (cid:4667) (cid:2879)(cid:2778) (cid:1778) (cid:2879)(cid:2778) (cid:1786) (cid:2879)(cid:2778) (cid:3404) (cid:3429)1 ρσ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) (cid:3051) (cid:3117) (cid:2870) (cid:4666)1 (cid:3398) ρ (cid:2870) (cid:4667) 00 σ (cid:3051) (cid:3118) (cid:2870) (cid:4681) ∗ (cid:3429) 1 0ρσ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) (cid:3404) (cid:3429)1 ρσ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) (cid:3051) (cid:3117) (cid:3493)(cid:4666)1 (cid:3398) ρ (cid:2870) (cid:4667) 00 σ (cid:3051) (cid:3118) (cid:4681) ∗ (cid:4680)σ (cid:3051) (cid:3117) (cid:3493)(cid:4666)1 (cid:3398) ρ (cid:2870) (cid:4667) 00 σ (cid:3051) (cid:3118) (cid:4681) ∗ (cid:3429) 1 0ρσ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) The correlation between U (cid:2869) and U (cid:2870) originates from the correlation between Ϋ and Ϋ and prevents direct application of Gauss-Hermite quadrature to the MGF of S = αΫ + βΫ as shown in (40). To address this, another transformation is made which decorrelates U (cid:2869) and U (cid:2870) (Mehta et al. [2007]) using the decomposition (cid:2737) (cid:2879)(cid:2869) (cid:3404) (cid:1786)(cid:1778)(cid:1786) (cid:4593) shown in (47). A decorrelating transformation based on the LDU factorization of (cid:2737) (cid:2879)(cid:2869) is: Let (cid:2208)(cid:4652)(cid:1318) (cid:3404) 1√2 (cid:1778) (cid:2868).(cid:2873) (cid:1786) (cid:4593) (cid:4666)(cid:2203)(cid:4652)(cid:4652)(cid:1318) (cid:3398) (cid:2246)(cid:4652)(cid:4652)(cid:1318)(cid:4667) , where (cid:2208)(cid:4652)(cid:1318) (cid:3404) (cid:4672)(cid:1878) (cid:2869) (cid:1878) (cid:2870) (cid:4673) .
The variance-covariance matrix for the random vector (cid:2208)(cid:4652)(cid:1318) is:
V(cid:4666)(cid:2208)(cid:4652)(cid:1318)(cid:4667) (cid:3404) 12 (cid:1778) (cid:2868).(cid:2873) (cid:1786) (cid:4593)
V(cid:4666)(cid:2203)(cid:4652)(cid:4652)(cid:1318) (cid:3398) (cid:2246)(cid:4652)(cid:4652)(cid:1318)(cid:4667)(cid:1786)(cid:1778) (cid:2868).(cid:2873) (cid:3404) 12 (cid:1778) (cid:2868).(cid:2873) (cid:1786) (cid:4593) (cid:2737) (cid:1786)(cid:1778) (cid:2868).(cid:2873) (cid:3404) 12 (cid:1778) (cid:2868).(cid:2873) (cid:1786) (cid:4593) (cid:4670)(cid:4666)(cid:1786) (cid:4593) (cid:4667) (cid:2879)(cid:2778) (cid:1778) (cid:2879)(cid:2778) (cid:1786) (cid:2879)(cid:2778) (cid:4671) (cid:1786)(cid:1778) (cid:2868).(cid:2873) (cid:3404) 12 (cid:1778) (cid:2868).(cid:2873) (cid:1778) (cid:2879)(cid:2869) (cid:1778) (cid:2868).(cid:2873) (cid:3404) 12 (cid:1783) .
Since the variance-covariance matrix,
V(cid:4666)(cid:2208)(cid:4652)(cid:1318)(cid:4667) , of the transformed variables Z and Z is diagonal We have based this transformation on the LDU factorization of (cid:2737) (cid:2879)(cid:2869) . The matrix of this transformation is upper triangular, see (57).
A similar decorrelating transformation can be obtained using the Cholesky decomposition of (cid:2737) (cid:3404) (cid:1786)(cid:1786)′ , namely, let (cid:2208)(cid:4652)(cid:1318) (cid:3404) (cid:2869)√(cid:2870) (cid:1786) (cid:2879)(cid:2869) (cid:4666)(cid:2203)(cid:4652)(cid:4652)(cid:1318) (cid:3398) (cid:2246)(cid:4652)(cid:4652)(cid:1318)(cid:4667) . The matrix of this transformation, (cid:1786) (cid:2879)(cid:2869) , would then be lower triangular. (48)(47)(50)(49)(54)(53)(52)(51) 15 (i.e., (½) I ), Z and Z are uncorrelated. Lastly, to compute the Jacobian of this transformation, we first express (cid:2203)(cid:4652)(cid:4652)(cid:1318) in terms of (cid:2208)(cid:4652)(cid:1318) using (51), namely: (cid:2203)(cid:4652)(cid:4652)(cid:1318) (cid:3404) (cid:3435)√2(cid:3439)(cid:4666)(cid:1786) (cid:4593) (cid:4667) (cid:2879)(cid:2869) (cid:4670)(cid:1778) (cid:2868).(cid:2873) (cid:4671) (cid:2879)(cid:2869) (cid:2208)(cid:4652)(cid:1318) (cid:3397) (cid:2246)(cid:4652)(cid:4652)(cid:1318) → (cid:2203)(cid:4652)(cid:4652)(cid:1318) (cid:3404) (cid:3435)√2(cid:3439) (cid:3429)1 ρσ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) (cid:3051) (cid:3117) (cid:3493)(cid:4666)1 (cid:3398) ρ (cid:2870) (cid:4667) 00 σ (cid:3051) (cid:3118) (cid:4681) ∗ (cid:2208)(cid:4652)(cid:1318) (cid:3397) (cid:2246)(cid:4652)(cid:4652)(cid:1318) , → (cid:2203)(cid:4652)(cid:4652)(cid:1318) (cid:3404) (cid:3435)√2(cid:3439) (cid:4680)σ (cid:3051) (cid:3117) (cid:3493)(cid:4666)1 (cid:3398) ρ (cid:2870) (cid:4667) ρσ (cid:3051) (cid:3117) (cid:3051) (cid:3118) (cid:4681) ∗ (cid:2208)(cid:4652)(cid:1318) (cid:3397) (cid:2246)(cid:4652)(cid:4652)(cid:1318) , so that, (cid:1873) (cid:2869) (cid:3404) √2 (cid:4672)σ (cid:3051) (cid:3117) (cid:3493)(cid:4666)1 (cid:3398) ρ (cid:2870) (cid:4667) (cid:1878) (cid:2869) (cid:3397) ρσ (cid:3051) (cid:3117) (cid:1878) (cid:2870) (cid:4673) (cid:3397) μ (cid:3051) (cid:3117) , and, (cid:1873) (cid:2870) (cid:3404) √2σ (cid:3051) (cid:3118) (cid:1878) (cid:2870) (cid:3397) μ (cid:3051) (cid:3118) . The absolute value of the Jacobian determinant for this transformation is: |J| (cid:3404) (cid:3629)Det (cid:3429)√2σ (cid:3051) (cid:3117) (cid:3493)(cid:4666)1 (cid:3398) ρ (cid:2870) (cid:4667) √2ρσ (cid:3051) (cid:3117) (cid:3051) (cid:3118) (cid:3433)(cid:3629) (cid:3404) 2σ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) (cid:3493)(cid:4666)1 (cid:3398) ρ (cid:2870) (cid:4667) . Applying the decorrelating transformation we express the MGF of S from (40) in terms of Z (cid:2869) and Z (cid:2870) as: M (cid:2903) (cid:4666)(cid:1872)(cid:4667) (cid:3404) (cid:3505) (cid:3505) e (cid:3047)∗(cid:3080)∗(cid:2915) (cid:3216)(cid:4680)√(cid:3118)(cid:4678)(cid:3226)(cid:3299)(cid:3117)(cid:3495)(cid:3435)(cid:3117)(cid:3127)(cid:3225)(cid:3118)(cid:3439) (cid:3301)(cid:3117)(cid:3126) (cid:3225)(cid:3226)(cid:3299)(cid:3117)(cid:3301)(cid:3118)(cid:4679)(cid:3126)(cid:3220)(cid:3299)(cid:3117)(cid:4681) e (cid:3047)∗(cid:3081)∗(cid:2915) (cid:3216)(cid:3427)√(cid:3118)(cid:3226)(cid:3299)(cid:3118)(cid:3301)(cid:3118)(cid:3126)(cid:3220)(cid:3299)(cid:3118)(cid:3431) (cid:1858)(cid:4666)(cid:1878) (cid:2869) , (cid:1878) (cid:2870) (cid:4667)(cid:1856)(cid:1878) (cid:2869) (cid:1856)(cid:1878) (cid:2870) , (cid:2998)(cid:2879)(cid:2998)(cid:2998)(cid:2879)(cid:2998) where, (cid:1858)(cid:4666)(cid:1878) (cid:2869) , (cid:1878) (cid:2870) (cid:4667) (cid:3404) σ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) (cid:3493)(cid:4666)1 (cid:3398) ρ (cid:2870) (cid:4667)(cid:2024)|(cid:4666)(cid:1786) (cid:4593) (cid:4667) (cid:2879)(cid:2778) (cid:1778) (cid:2879)(cid:2778) (cid:1786) (cid:2879)(cid:2778) | (cid:2869) (cid:2870)⁄ e (cid:2879)(cid:2208)(cid:4652)(cid:1318) (cid:4594) (cid:2208)(cid:4652)(cid:1318) (cid:3404) σ (cid:3051) (cid:3117) σ (cid:3051) (cid:3118) (cid:3493)(cid:4666)1 (cid:3398) ρ (cid:2870) (cid:4667)(cid:2024)|(cid:1778) (cid:2879)(cid:2778) | (cid:2869) (cid:2870)⁄ e (cid:2879)(cid:2208)(cid:4652)(cid:1318) (cid:4594) (cid:2208)(cid:4652)(cid:1318) (cid:3404) 1(cid:2024) e (cid:2879)(cid:2208)(cid:4652)(cid:1318) (cid:4594) (cid:2208)(cid:4652)(cid:1318) (cid:3404) 1(cid:2024) e (cid:2879)(cid:3053) (cid:3117)(cid:3118) e (cid:2879)(cid:3053) (cid:3118)(cid:3118) , (cid:1858)(cid:1867)(cid:1870) (cid:2208)(cid:4652)(cid:1318) ∈ (cid:1336) (cid:2870) . The MGF of S = αΫ + βΫ from (40) can therefore be expressed in terms of Z (cid:2869) and Z (cid:2870) as: M (cid:2903) (cid:4666)(cid:1872)(cid:4667) (cid:3404) (cid:3505) (cid:3505) 1(cid:2024) e (cid:3047)∗(cid:3080)∗(cid:2915) (cid:3216)(cid:4680)√(cid:3118)(cid:4678)(cid:3226)(cid:3299)(cid:3117)(cid:3495)(cid:3435)(cid:3117)(cid:3127)(cid:3225)(cid:3118)(cid:3439) (cid:3301)(cid:3117)(cid:3126)(cid:3225)(cid:3226)(cid:3299)(cid:3117)(cid:3301)(cid:3118)(cid:4679)(cid:3126)(cid:3220)(cid:3299)(cid:3117)(cid:4681) e (cid:3047)∗(cid:3081)∗(cid:2915) (cid:3216)(cid:3427)√(cid:3118)(cid:3226)(cid:3299)(cid:3118)(cid:3301)(cid:3118)(cid:3126)(cid:3220)(cid:3299)(cid:3118)(cid:3431) e (cid:2879)(cid:3053) (cid:3117)(cid:3118) e (cid:2879)(cid:3053) (cid:3118)(cid:3118) (cid:1856)(cid:1878) (cid:2869) (cid:1856)(cid:1878) (cid:2870) (cid:2998)(cid:2879)(cid:2998)(cid:2998)(cid:2879)(cid:2998) (55)(56)(57)(58)(59)(60)(61)(62)(63)(64) 16 → M (cid:2903) (cid:4666)(cid:1872)(cid:4667) (cid:3404) (cid:3505) (cid:3505) 1(cid:2024) (cid:1860)(cid:4666)(cid:1878) (cid:2869) , (cid:1878) (cid:2870) (cid:4667) e (cid:2879)(cid:3053) (cid:3117)(cid:3118) e (cid:2879)(cid:3053) (cid:3118)(cid:3118) (cid:1856)(cid:1878) (cid:2869) (cid:1856)(cid:1878) (cid:2870) (cid:2998)(cid:2879)(cid:2998) , (cid:2998)(cid:2879)(cid:2998) where, (cid:1860)(cid:4666)(cid:1878) (cid:2869) , (cid:1878) (cid:2870) (cid:4667) (cid:3404) e (cid:3047)∗(cid:3080)∗(cid:2915) (cid:3335)(cid:4680)√(cid:3118)(cid:4678)(cid:3226)(cid:3299)(cid:3117)(cid:3495)(cid:3435)(cid:3117)(cid:3127)(cid:3225)(cid:3118)(cid:3439) (cid:3301)(cid:3117)(cid:3126) (cid:3225)(cid:3226)(cid:3299)(cid:3117)(cid:3301)(cid:3118)(cid:4679)(cid:3126)(cid:3220)(cid:3299)(cid:3117)(cid:4681) e (cid:3047)∗(cid:3081)∗(cid:2915) (cid:3216)(cid:3427)√(cid:3118)(cid:3226)(cid:3299)(cid:3118)(cid:3301)(cid:3118)(cid:3126)(cid:3220)(cid:3299)(cid:3118)(cid:3431) . The representation of M (cid:2903) (cid:4666)(cid:1872)(cid:4667) in (65) is of the form required for Gauss-Hermite quadrature using the weights and roots from Table 1. The MGF of S = αΫ + βΫ can therefore be approximated in 2 steps. In Step 1, we apply the quadrature rules to Z (cid:2869) and replace the integral by a sum, then we repeat the process for Z (cid:2870) in Step 2. Step 1: Apply Gauss-Hermite Quadrature Rules (n=12) to Z (cid:2869) → M (cid:2903) (cid:4666)(cid:1872)(cid:4667) ≅ 1(cid:2024) (cid:3505) (cid:4684)(cid:3533) w (cid:2920) ∗ (cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869) (cid:3427)(cid:1860)(cid:3435)t (cid:2920) , (cid:1878) (cid:2870) (cid:3439)(cid:3431)(cid:4685) e (cid:2879)(cid:3053) (cid:3118)(cid:3118) (cid:2998)(cid:2879)(cid:2998) (cid:1856)(cid:1878) (cid:2870) Step 2: Apply Gauss-Hermite Quadrature Rules (n=12) to Z (cid:2870) → M (cid:2903) (cid:4666)(cid:1872)(cid:4667) ≅ 1(cid:2024) (cid:3533) w (cid:3036) ∗ (cid:4684)(cid:3533) w (cid:2920) ∗ (cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869) (cid:3427)(cid:1860)(cid:3435)t (cid:2920) , t (cid:3036) (cid:3439)(cid:3431)(cid:4685) (cid:2869)(cid:2870)(cid:3036)(cid:2880)(cid:2869) → M (cid:2903) (cid:4666)(cid:1872)(cid:4667) ≅ 1(cid:2024) (cid:3533) (cid:3533) w (cid:3036) ∗ w (cid:2920) ∗ (cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869) (cid:1860)(cid:3435)t (cid:2920) , t (cid:3036) (cid:3439) (cid:2869)(cid:2870)(cid:3036)(cid:2880)(cid:2869) The function (cid:1860)(cid:4666)(cid:1878) (cid:2869) , (cid:1878) (cid:2870) (cid:4667) is given in (66) and the final step is to equate M (cid:2903) (cid:4666)(cid:1872)(cid:4667) from (68b) to the approximated MGF for a univariate lognormal RV M (cid:2909)(cid:4663) (cid:4666)(cid:1872)(cid:4667) , shown in the RHS of (35), and solve for the 2 unknowns (cid:2020) (cid:3051) and (cid:2026) (cid:3051) . It is assumed that μ (cid:3051) (cid:3117) , μ (cid:3051) (cid:3118) , σ (cid:3051) (cid:3117) , σ (cid:3051) (cid:3118) , and ρ are all known constants. Since 2 equations are needed to obtain a solution for the two unknowns, we generate these equations using different real values for t < 0 (Mehta et al. [2007]). (65)(66)(68a)(67)(68b) 17 IV. Approximation of Lognormal Sum with Lognormal RV
In this research, we have correlated lognormal RVs Ϋ and Ϋ with Cov( Ϋ , Ϋ ) = σ (cid:4666)(cid:3052)(cid:4663) (cid:3117) ,(cid:3052)(cid:4663) (cid:3118) (cid:4667) . The means and variances are known and given by E[ Ϋ ] = μ (cid:3052)(cid:4663) (cid:3117) , E[ Ϋ ] = μ (cid:3052)(cid:4663) (cid:3118) , V[ Ϋ ] = σ (cid:3052)(cid:4663) (cid:3117) (cid:2870) , and V[ Ϋ ] = σ (cid:3052)(cid:4663) (cid:3118) (cid:2870) . Both Ϋ and Ϋ are defined in non-standard form, so there exist normally distributed RVs X ~ N( μ (cid:3051) (cid:3117) , σ (cid:3051) (cid:3117) (cid:2870) ) and X ~ N( μ (cid:3051) (cid:3118) , σ (cid:3051) (cid:3118) (cid:2870) ) such that Ϋ = (cid:2908) (cid:3117) /(cid:2869)(cid:2868) and Ϋ = (cid:2908) (cid:3118) /(cid:2869)(cid:2868) where Corr(X , X ) = ρ . We will not be provided the means and variances of these underlying normal RVs, but they will be calculated using the expressions in (24), (25), and (26). Lastly, we will be given constants α and β and have interest in approximating the probability distribution of S = αΫ + βΫ . Based on Mitchell (1968), using a univariate lognormal RV to approximate the distribution of S is desirable. We will first calculate the parameters for the underlying normal RVs and then set M (cid:2903) (cid:4666)(cid:1872)(cid:4667) from (68b) equal to M (cid:2909)(cid:4663) (cid:4666)(cid:1872)(cid:4667) from (35) and solve for μ (cid:3051) and σ (cid:3051) . These are the mean and standard deviation, respectively, from the underlying normal distribution that forms the base distribution of our lognormal approximation. The final step is to calculate the corresponding mean and variance for the approximating univariate lognormal distribution using the expressions in (12) and (13). A. An Example
Let Ϋ and Ϋ be (non-standard form) lognormal RVs with μ (cid:3052)(cid:4663) (cid:3117) = 1.0, μ (cid:3052)(cid:4663) (cid:3118) = 2.0, σ (cid:3052)(cid:4663) (cid:3117) (cid:2870) = 3.0, σ (cid:3052)(cid:4663) (cid:3118) (cid:2870) = 4.0, and σ (cid:4666)(cid:3052)(cid:4663) (cid:3117) ,(cid:3052)(cid:4663) (cid:3118) (cid:4667) = 1.73 so that Corr( Ϋ , Ϋ ) = σ (cid:4666)(cid:3052)(cid:4663) (cid:3117) ,(cid:3052)(cid:4663) (cid:3118) (cid:4667) (cid:3435)σ (cid:3052)(cid:4663) (cid:3117) σ (cid:3052)(cid:4663) (cid:3118) (cid:3439) ⁄ = 0.5. Further, define constants α = 1.5 and β = 2.5 with interest in approximating the probability distribution of S = αΫ + βΫ = (1.5) Ϋ + (2.5) Ϋ . Using (24), (25), and (26) from Section
III.D.4, the parameters of the underlying normal distributions are given by: → From Equation (cid:4666)24(cid:4667): (cid:2020) (cid:3051) (cid:3117) (cid:3404) (cid:3436)1θ(cid:3440) (cid:3436)ln(cid:4666)1.0(cid:4667) (cid:3398) 12 ln (cid:3428)1 (cid:3397) 3.0(cid:4666)1.0(cid:4667) (cid:2870) (cid:3432)(cid:3440) (cid:3404) (cid:3398)3.0103 (cid:2020) (cid:3051) (cid:3118) (cid:3404) (cid:3436)1θ(cid:3440) (cid:3436)ln(cid:4666)2.0(cid:4667) (cid:3398) 12 ln (cid:3428)1 (cid:3397) 4.0(cid:4666)2.0(cid:4667) (cid:2870) (cid:3432)(cid:3440) (cid:3404) 1.5051 → From Equation (cid:4666)25(cid:4667): (cid:2026) (cid:3051) (cid:3117) (cid:2870) (cid:3404) (cid:3436)1θ(cid:3440) (cid:2870) ln (cid:3428)1 (cid:3397) 3.0(cid:4666)1.0(cid:4667) (cid:2870) (cid:3432) (cid:3404) 26.1471 (69)(70)(71) 18 (cid:2026) (cid:3051) (cid:3118) (cid:2870) (cid:3404) (cid:3436)1θ(cid:3440) (cid:2870) ln (cid:3428)1 (cid:3397) 4.0(cid:4666)2.0(cid:4667) (cid:2870) (cid:3432) (cid:3404) 13.0736 → From Equation (cid:4666)26(cid:4667): (cid:2026) (cid:4666)(cid:3051) (cid:3117) , (cid:3051) (cid:3118) (cid:4667) (cid:3404) (cid:3436)1θ(cid:3440) (cid:2870) ln (cid:3428)1 (cid:3397) 1.73|1.0 ∗ 2.0|(cid:3432) (cid:3404) 11.7554
Therefore, Corr(X , X ) = ρ (cid:3404) (cid:2869)(cid:2869).(cid:2875)(cid:2873)(cid:2873)(cid:2872)√(cid:2870)(cid:2874).(cid:2869)(cid:2872)(cid:2875)(cid:2869)∗(cid:2869)(cid:2871).(cid:2868)(cid:2875)(cid:2871)(cid:2874) = 0.635811, and (cid:2870) = 0.595744. Using these quantities the function (cid:1860)(cid:4666)(cid:1878) (cid:2869) , (cid:1878) (cid:2870) (cid:4667) from (66) becomes: (cid:1860)(cid:4666)(cid:1878) (cid:2869) , (cid:1878) (cid:2870) (cid:4667) (cid:3404) e (cid:4666)(cid:2869).(cid:2873)(cid:4667)(cid:3047)(cid:2915) (cid:3216)(cid:4674)√(cid:3118)(cid:4672)(cid:4666)(cid:3121).(cid:3117)(cid:3117)(cid:3119)(cid:3120)(cid:4667)√(cid:3116).(cid:3121)(cid:3125)(cid:3121)(cid:3123)(cid:3120)(cid:3120) (cid:3301)(cid:3117)(cid:3126)(cid:4666)(cid:3116).(cid:3122)(cid:3119)(cid:3121)(cid:3124)(cid:3117)(cid:3117)(cid:4667)(cid:4666)(cid:3121).(cid:3117)(cid:3117)(cid:3119)(cid:3120)(cid:4667)(cid:3301)(cid:3118)(cid:4673)(cid:3126)(cid:4666)(cid:3127)(cid:3119).(cid:3116)(cid:3117)(cid:3116)(cid:3119)(cid:4667)(cid:4675) e (cid:4666)(cid:2870).(cid:2873)(cid:4667)(cid:3047)(cid:2915) (cid:3216)(cid:3427)√(cid:3118)(cid:4666)(cid:3119).(cid:3122)(cid:3117)(cid:3121)(cid:3123)(cid:4667)(cid:3301)(cid:3118)(cid:3126)(cid:4666)(cid:3117).(cid:3121)(cid:3116)(cid:3121)(cid:3117)(cid:4667)(cid:3431) When CDF values of the sum S are desired, Mehta et al. (2007) find good results using constants t = -1.0 and t = -0.2 for t to generate the needed equations. Using these values, and setting M (cid:2903) (cid:4666)(cid:1872)(cid:4667) from (68) equal to M (cid:2909)(cid:4663) (cid:4666)(cid:1872)(cid:4667) from (35), we solve the following two non-linear equations for a single μ (cid:3051) and σ (cid:3051) , which are the only two unknown quantities in equations (75) and (76). The quadrature weights and roots, w i , w j , t i , and t j are provided in Table 1. Equation (cid:3036) ∗ w (cid:2920) ∗ e (cid:4666)(cid:2879)(cid:2869).(cid:2873)(cid:4667)∗(cid:2915) (cid:3216)(cid:4674)√(cid:3118)(cid:3436)(cid:4666)(cid:3121).(cid:3117)(cid:3117)(cid:3119)(cid:3120)(cid:4667)√(cid:3116).(cid:3121)(cid:3125)(cid:3121)(cid:3123)(cid:3120)(cid:3120)(cid:3178)(cid:3168)(cid:3126)(cid:4666)(cid:3116).(cid:3122)(cid:3119)(cid:3121)(cid:3124)(cid:3117)(cid:3117)(cid:4667)(cid:4666)(cid:3121).(cid:3117)(cid:3117)(cid:3119)(cid:3120)(cid:4667)(cid:3178)(cid:3284)(cid:3440)(cid:3126)(cid:4666)(cid:3127)(cid:3119).(cid:3116)(cid:3117)(cid:3116)(cid:3119)(cid:4667)(cid:4675) ∗ (cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869) (cid:2869)(cid:2870)(cid:3036)(cid:2880)(cid:2869) e (cid:4666)(cid:2879)(cid:2870).(cid:2873)(cid:4667)∗(cid:2915) (cid:3216)(cid:3427)√(cid:3118)(cid:4666)(cid:3119).(cid:3122)(cid:3117)(cid:3121)(cid:3123)(cid:4667)(cid:3178)(cid:3284)(cid:3126)(cid:4666)(cid:3117).(cid:3121)(cid:3116)(cid:3121)(cid:3117)(cid:4667)(cid:3431) (cid:4675) (cid:3404) 1√(cid:2024) (cid:3533) w (cid:2920) ∗ e (cid:4666)(cid:2879)(cid:2869).(cid:2868)(cid:4667)∗(cid:2915) (cid:3216)(cid:4672)√(cid:3118)(cid:3226)(cid:3299)(cid:3295)(cid:3168)(cid:3126)(cid:3220)(cid:3299)(cid:4673) (cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869) Equation (cid:3036) ∗ w (cid:2920) ∗ e (cid:4666)(cid:2879)(cid:2868).(cid:2871)(cid:4667)∗(cid:2915) (cid:3216)(cid:4674)√(cid:3118)(cid:3436)(cid:4666)(cid:3121).(cid:3117)(cid:3117)(cid:3119)(cid:3120)(cid:4667)√(cid:3116).(cid:3121)(cid:3125)(cid:3121)(cid:3123)(cid:3120)(cid:3120)(cid:3178)(cid:3168)(cid:3126)(cid:4666)(cid:3116).(cid:3122)(cid:3119)(cid:3121)(cid:3124)(cid:3117)(cid:3117)(cid:4667)(cid:4666)(cid:3121).(cid:3117)(cid:3117)(cid:3119)(cid:3120)(cid:4667)(cid:3178)(cid:3284)(cid:3440)(cid:3126)(cid:4666)(cid:3127)(cid:3119).(cid:3116)(cid:3117)(cid:3116)(cid:3119)(cid:4667)(cid:4675) ∗ (cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869) (cid:2869)(cid:2870)(cid:3036)(cid:2880)(cid:2869) e (cid:4666)(cid:2879)(cid:2868).(cid:2873)(cid:4667)∗(cid:2915) (cid:3216)(cid:3427)√(cid:3118)(cid:4666)(cid:3119).(cid:3122)(cid:3117)(cid:3121)(cid:3123)(cid:4667)(cid:3178)(cid:3284)(cid:3126)(cid:4666)(cid:3117).(cid:3121)(cid:3116)(cid:3121)(cid:3117)(cid:4667)(cid:3431) (cid:4675) (cid:3404) 1√(cid:2024) (cid:3533) w (cid:2920) ∗ e (cid:4666)(cid:2879)(cid:2868).(cid:2870)(cid:4667)∗(cid:2915) (cid:3216)(cid:4672)√(cid:3118)(cid:3226)(cid:3299)(cid:3295)(cid:3168)(cid:3126)(cid:3220)(cid:3299)(cid:4673) (cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869) These values are for the quantity t as defined from the MGF which is different from the t i and t j values used to denote the roots for Gauss-Hermite quadrature which are provided in Table 1. (72)(73)(74)(75)(76) 19 B. Implementation Details
Equations 1 and 2 from (75) and (76) must be simultaneously solved for μ (cid:3051) and σ (cid:3051) but note that the left-hand sides of both equations are known constants. Each is a 12 = 144 term sum with no unknown quantities involved. The left-hand sides can thus be subtracted and the equations expressed as simultaneous non-linear equations equal to zero. Further, the equations were generated using t ∈ {-1.0,-0.2} as suggested by Mehta et al. (2007). In general terms let t ∈ { τ , τ }, and let C and C be the LHS of (75) and (76), respectively. It follows that these equations can be expressed as: Equation (cid:2920) ∗ e (cid:4666)(cid:2980) (cid:3117) (cid:4667)(cid:2915) (cid:3216)(cid:4672)√(cid:3118)(cid:3226)(cid:3182)(cid:3178)(cid:3168)(cid:3126)(cid:3220)(cid:3299)(cid:4673) (cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869) (cid:3398) C (cid:2869) (cid:3404) 0
Equation (cid:2920) ∗ e (cid:4666)(cid:2980) (cid:3118) (cid:4667)(cid:2915) (cid:3216)(cid:4672)√(cid:3118)(cid:3226)(cid:3182)(cid:3178)(cid:3168)(cid:3126)(cid:3220)(cid:3299)(cid:4673) (cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869) (cid:3398) C (cid:2870) (cid:3404) 0
The constants C and C in (77) and (78) are specific to the problem being addressed, and { τ , τ } may also be application specific. If we denote the LHS of (77) and (78) by M (cid:2909)(cid:4663)(cid:4666)(cid:2869)(cid:4667) (cid:4666)μ (cid:3051) , σ (cid:3051) (cid:4667) and M (cid:2909)(cid:4663)(cid:4666)(cid:2870)(cid:4667) (cid:4666)μ (cid:3051) , σ (cid:3051) (cid:4667) , respectively, then we seek to solve the following non-linear system for μ (cid:3051) and σ (cid:3051) : (cid:1813)(cid:4652)(cid:4652)(cid:4652)(cid:1318) (cid:3404) (cid:3437)M (cid:2909)(cid:4663)(cid:4666)(cid:2869)(cid:4667) (cid:4666)μ (cid:3051) , σ (cid:3051) (cid:4667)M (cid:2909)(cid:4663)(cid:4666)(cid:2870)(cid:4667) (cid:4666)μ (cid:3051) , σ (cid:3051) (cid:4667)(cid:3441) (cid:3404) (cid:4672)00(cid:4673) . Newton’s method is often used in optimization problems to solve a similar set of non-linear equations, namely that of the gradient vector being equal to zero. It applies here as well and operates by approximating the vector (cid:1813)(cid:4652)(cid:4652)(cid:4652)(cid:1318) by its 1 st order Taylor expansion around some given initial starting point (cid:4666)μ (cid:3051)(cid:2868) , σ (cid:3051)(cid:2868) (cid:4667) , namely: (cid:1813)(cid:4652)(cid:4652)(cid:4652)(cid:1318) ≅ (cid:1813)(cid:4652)(cid:4652)(cid:4652)(cid:1318) (cid:2777) (cid:3404) (cid:3437)M (cid:2909)(cid:4663)(cid:4666)(cid:2869)(cid:4667) (cid:4666)μ (cid:3051)(cid:2868) , σ (cid:3051)(cid:2868) (cid:4667)M (cid:2909)(cid:4663)(cid:4666)(cid:2870)(cid:4667) (cid:4666)μ (cid:3051)(cid:2868) , σ (cid:3051)(cid:2868) (cid:4667)(cid:3441) (cid:3397) (cid:1743)(cid:1742)(cid:1742)(cid:1742)(cid:1741) (cid:2034)(cid:2034)μ (cid:3051) (cid:4674)M (cid:2909)(cid:4663)(cid:4666)(cid:2869)(cid:4667) (cid:4675) (cid:3435)(cid:2972) (cid:3299)(cid:3116) ,(cid:2978) (cid:3299)(cid:3116) (cid:3439) (cid:2034)(cid:2034)σ (cid:3051) (cid:4674)M (cid:2909)(cid:4663)(cid:4666)(cid:2869)(cid:4667) (cid:4675) (cid:3435)(cid:2972) (cid:3299)(cid:3116) ,(cid:2978) (cid:3299)(cid:3116) (cid:3439) (cid:2034)(cid:2034)μ (cid:3051) (cid:4674)M (cid:2909)(cid:4663)(cid:4666)(cid:2870)(cid:4667) (cid:4675) (cid:3435)(cid:2972) (cid:3299)(cid:3116) ,(cid:2978) (cid:3299)(cid:3116) (cid:3439) (cid:2034)(cid:2034)σ (cid:3051) (cid:4674)M (cid:2909)(cid:4663)(cid:4666)(cid:2870)(cid:4667) (cid:4675) (cid:3435)(cid:2972) (cid:3299)(cid:3116) ,(cid:2978) (cid:3299)(cid:3116) (cid:3439) (cid:1746)(cid:1745)(cid:1745)(cid:1745)(cid:1744) ∗ (cid:3436)μ (cid:3051) (cid:3398) μ (cid:3051)(cid:2868) σ (cid:3051) (cid:3398) σ (cid:3051)(cid:2868) (cid:3440) . (77)(78)(79)(80) 20 The matrix shown in (80) consists of the corresponding 1 st order partial derivatives of both functions evaluated at the initial starting point, and it is therefore known once the derivatives have been calculated. By setting this 1 st order Taylor expansion of (cid:1813)(cid:4652)(cid:4652)(cid:4652)(cid:1318) at (cid:4666)μ (cid:3051)(cid:2868) , σ (cid:3051)(cid:2868) (cid:4667) (i.e., (cid:1813)(cid:4652)(cid:4652)(cid:4652)(cid:1318) (cid:2777) ), equal to zero and solving for (cid:4666)μ (cid:3051) , σ (cid:3051) ), we arrive at: (cid:1813)(cid:4652)(cid:4652)(cid:4652)(cid:1318) (cid:2777) (cid:3404) (cid:4672)00(cid:4673) ↔ (cid:1743)(cid:1742)(cid:1742)(cid:1742)(cid:1741) (cid:2034)(cid:2034)μ (cid:3051) (cid:4674)M (cid:2909)(cid:4663)(cid:4666)(cid:2869)(cid:4667) (cid:4675) (cid:3435)(cid:2972) (cid:3299)(cid:3116) ,(cid:2978) (cid:3299)(cid:3116) (cid:3439) (cid:2034)(cid:2034)σ (cid:3051) (cid:4674)M (cid:2909)(cid:4663)(cid:4666)(cid:2869)(cid:4667) (cid:4675) (cid:3435)(cid:2972) (cid:3299)(cid:3116) ,(cid:2978) (cid:3299)(cid:3116) (cid:3439) (cid:2034)(cid:2034)μ (cid:3051) (cid:4674)M (cid:2909)(cid:4663)(cid:4666)(cid:2870)(cid:4667) (cid:4675) (cid:3435)(cid:2972) (cid:3299)(cid:3116) ,(cid:2978) (cid:3299)(cid:3116) (cid:3439) (cid:2034)(cid:2034)σ (cid:3051) (cid:4674)M (cid:2909)(cid:4663)(cid:4666)(cid:2870)(cid:4667) (cid:4675) (cid:3435)(cid:2972) (cid:3299)(cid:3116) ,(cid:2978) (cid:3299)(cid:3116) (cid:3439) (cid:1746)(cid:1745)(cid:1745)(cid:1745)(cid:1744) ∗ (cid:3436)μ (cid:3051) (cid:3398) μ (cid:3051)(cid:2868) σ (cid:3051) (cid:3398) σ (cid:3051)(cid:2868) (cid:3440) (cid:3404) (cid:3398) (cid:3437)M (cid:2909)(cid:4663)(cid:4666)(cid:2869)(cid:4667) (cid:4666)μ (cid:3051)(cid:2868) , σ (cid:3051)(cid:2868) (cid:4667)M (cid:2909)(cid:4663)(cid:4666)(cid:2870)(cid:4667) (cid:4666)μ (cid:3051)(cid:2868) , σ (cid:3051)(cid:2868) (cid:4667)(cid:3441) which is a simple linear system of 2 equations and 2 unknowns that we solve for the vector: (cid:3436)μ (cid:3051) (cid:3398) μ (cid:3051)(cid:2868) σ (cid:3051) (cid:3398) σ (cid:3051)(cid:2868) (cid:3440) . The solution immediately yields values for (cid:4666)μ (cid:3051) , σ (cid:3051) ) that we label (cid:4666)μ (cid:3051)(cid:2869) , σ (cid:3051)(cid:2869) (cid:4667) and then the process is repeated using (cid:4666)μ (cid:3051)(cid:2869) , σ (cid:3051)(cid:2869) (cid:4667) as the new starting point which yields (cid:4666)μ (cid:3051)(cid:2870) , σ (cid:3051)(cid:2870) (cid:4667) , and so on. We stop at iteration i when an objective criteria is met, such as when the functions M (cid:2909)(cid:4663)(cid:4666)(cid:2869)(cid:4667) (cid:3435)μ (cid:3051)(cid:3036) , σ (cid:3051)(cid:3036) (cid:3439) and M (cid:2909)(cid:4663)(cid:4666)(cid:2870)(cid:4667) (cid:3435)μ (cid:3051)(cid:3036) , σ (cid:3051)(cid:3036) (cid:3439) both become smaller than some predetermined threshold level ε . The only remaining step for implementing Newton’s method is to derive the elements of the 2x2 matrix of partial derivatives shown in (80). Using the chain rule along with the fact that the derivative of a sum equals the sum of the derivatives, these quantities are given by: (cid:2034)(cid:2034)(cid:2020) (cid:3051) (cid:4674)M (cid:2909)(cid:4663)(cid:4666)(cid:3036)(cid:4667) (cid:4666)μ (cid:3051) , σ (cid:3051) (cid:4667)(cid:4675) (cid:3404) 1√(cid:2024) (cid:3533) w (cid:2920) ∗ (cid:2034)(cid:2034)μ (cid:3051) (cid:3428)e (cid:4666)(cid:2980) (cid:3284) (cid:4667)(cid:2915) (cid:3216)(cid:4672)√(cid:3118)(cid:3226)(cid:3182)(cid:3178)(cid:3168)(cid:3126)(cid:3220)(cid:3299)(cid:4673) (cid:3432) (cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869) (cid:3404) θ ∗ τ (cid:3036) √(cid:2024) (cid:3533) w (cid:2920) ∗ e (cid:4666)(cid:2980) (cid:3284) (cid:4667)(cid:2915) (cid:3216)(cid:4672)√(cid:3118)(cid:3226)(cid:3299)(cid:3178)(cid:3168)(cid:3126)(cid:3220)(cid:3299)(cid:4673) ∗ (cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869) e (cid:2968)(cid:3435)√(cid:2870)(cid:2978) (cid:3182) (cid:2930) (cid:3168) (cid:2878)(cid:2972) (cid:3299) (cid:3439) for (cid:1861) (cid:3404) 1, 2, and, (81)(82)(83)(84a)(84b) 21 (cid:2034)(cid:2034)σ (cid:3051) (cid:4674)M (cid:2909)(cid:4663)(cid:4666)(cid:3036)(cid:4667) (cid:4666)μ (cid:3051) , σ (cid:3051) (cid:4667)(cid:4675) (cid:3404) 1√(cid:2024) (cid:3533) w (cid:2920) ∗ (cid:2034)(cid:2034)σ (cid:3051) (cid:3428)e (cid:4666)(cid:2980) (cid:3284) (cid:4667)(cid:2915) (cid:3216)(cid:4672)√(cid:3118)(cid:3226)(cid:3182)(cid:3178)(cid:3168)(cid:3126)(cid:3220)(cid:3299)(cid:4673) (cid:3432) (cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869) (cid:3404) θ ∗ τ (cid:3036) ∗ √2√(cid:2024) (cid:3533) w (cid:2920) ∗ t (cid:2920) ∗ e (cid:4666)(cid:2980) (cid:3284) (cid:4667)(cid:2915) (cid:3216)(cid:4672)√(cid:3118)(cid:3226)(cid:3299)(cid:3178)(cid:3168)(cid:3126)(cid:3220)(cid:3299)(cid:4673) (cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869) ∗ e (cid:2968)(cid:3435)√(cid:2870)(cid:2978) (cid:3182) (cid:2930) (cid:3168) (cid:2878)(cid:2972) (cid:3299) (cid:3439) for (cid:1861) (cid:3404) 1, 2. The 1 st order partial derivatives above are evaluated at the starting point for each iteration, thus they are completely known and constitute the 2x2 matrix from (80). All quantities in the system of 2 linear equations from (82) are known except (cid:4666)μ (cid:3051) , σ (cid:3051) ) and we will solve for these using the technique described. Solving a linear system of 2 equations with 2 unknowns is trivial. Once found, the mean and variance of the approximating univariate lognormal RV for S = αΫ + βΫ , namely E(cid:4670)S(cid:4671) and V (cid:4670)S(cid:4671) , are derived using (12) and (13).
B.1 Initial Values for Newton’s Method
Newton’s method requires the selection of initial values, namely (cid:4666)μ (cid:3051)(cid:2868) , σ (cid:3051)(cid:2868) (cid:4667) , and choosing values that are closer to the actual solution can reduce the number of iterations needed to achieve convergence. Since we are interested in the sum S = αΫ + βΫ , an obvious choice, motivated by the F-W approximation, is to use the values that correspond to E[S] and V[S], both of which are known. That is, E(cid:4670)S(cid:4671) (cid:3404) E(cid:3427)αY(cid:4663) (cid:2869) (cid:3397) βY(cid:4663) (cid:2870) (cid:3431) (cid:3404) αμ (cid:3052)(cid:4663) (cid:3117) (cid:3397) βμ (cid:3052)(cid:4663) (cid:3118)
V(cid:4670)S(cid:4671) (cid:3404) V (cid:4680)(cid:4666)α β(cid:4667) ∗ (cid:4678)Y(cid:4663) (cid:2869)
Y(cid:4663) (cid:2870) (cid:4679)(cid:4681) (cid:3404) (cid:4666)α β(cid:4667)V (cid:4678)Y(cid:4663) (cid:2869)
Y(cid:4663) (cid:2870) (cid:4679) (cid:4672)αβ(cid:4673) (cid:3404) (cid:4666)α β(cid:4667) (cid:4680) σ (cid:3052)(cid:4663) (cid:3117) (cid:2870) σ (cid:4666)(cid:3052)(cid:4663) (cid:3117) ,(cid:3052)(cid:4663) (cid:3118) (cid:4667) σ (cid:4666)(cid:3052)(cid:4663) (cid:3117) ,(cid:3052)(cid:4663) (cid:3118) (cid:4667) σ (cid:3052)(cid:4663) (cid:3118) (cid:2870) (cid:4681) (cid:4672)αβ(cid:4673) . The initial values are then derived using (24) and (25) as: μ (cid:3051)(cid:2868) (cid:3404) (cid:3436)1θ(cid:3440) (cid:4678)ln(cid:4666)E(cid:4670)S(cid:4671)(cid:4667) (cid:3398) 12 ln (cid:4680)1 (cid:3397) V(cid:4670)S(cid:4671)(cid:4666)E(cid:4670)S(cid:4671)(cid:4667) (cid:2870) (cid:4681)(cid:4679) , σ (cid:3051)(cid:2868) (cid:3404) (cid:3496)(cid:3436)1θ(cid:3440) (cid:2870) ln (cid:4680)1 (cid:3397) V(cid:4670)S(cid:4671)(cid:4666)E(cid:4670)S(cid:4671)(cid:4667) (cid:2870) (cid:4681) . (85a)(85b)(86)(87)(88)(89) 22 V. An Application to Finance
Let R be the total annual return on an arbitrary investment portfolio. For prior years, we can calculate the value of R as:
R (cid:3404) End Balance (cid:3398) Start BalanceStart Balance .
From (90), the portfolio’s ending balance can be derived using R as:
End Balance (cid:3404) (cid:4666)Start Balance(cid:4667) ∗ (cid:4666)1 (cid:3397) R(cid:4667) .
Future unobserved values of R will be taken as RVs following some probability distribution. The quantity R consists of an inflation component, I, and a real return component, r, and it can be decomposed as follows: (cid:4666)1 (cid:3397) R(cid:4667) (cid:3404) (cid:4666)1 (cid:3397) I(cid:4667) ∗ (cid:4666)1 (cid:3397) r(cid:4667) .
Here, both I and r are RVs but the inflation rate can be difficult to model because it possesses a deterministic component. The inflation rate is heavily influenced by central banks via monetary policy, which can make treating it as a pure RV problematic. Further, central banks often have a target inflation rate and the process of keeping it on target can add serial correlation to the observations. To remove it from the model, we divide both sides of (91) by (1 + I), which leaves: (cid:4666)1 (cid:3397) r(cid:4667) (cid:3404) (cid:4666)1 (cid:3397) R(cid:4667)(cid:4666)1 (cid:3397) I(cid:4667) .
The quantity r is referred to as the real (or inflation-adjusted) annual return on the investment portfolio and it can be positive or negative. The value (1 + r) is the compounding real annual return and it must be ≥ iid RVs is approximately lognormal. To find the best fitting lognormal distribution for (1 + r), we examine the historical record. For example, if we invest our portfolio in an S&P 500 Index Fund or in 10-Year Treasury Bonds, then the historical record will reveal the annual total returns R (cid:2930) (90)(91)(92) 23 for each investment, along with the inflation rate I (cid:2930) , and it is a simple matter to construct r (cid:2930) for time points t=1, 2, …, N, as: r (cid:2930) (cid:3404) (cid:4666)1 (cid:3397) R (cid:2930) (cid:4667)(cid:4666)1 (cid:3397) I (cid:2930) (cid:4667) (cid:3398) 1 . The values ( (cid:2930) (cid:4667) can be fit to a lognormal distribution using a hypothesis test such as the Anderson-Darling (A/D) test. The null hypothesis is that the returns originate from a given lognormal distribution and a p-value is generated. For example, using historical data on the S&P 500 Index (stocks) and 10-Year Treasury Bonds (bonds), along with the corresponding inflation rates from 1928 – 2013, the real compounding returns (1 + r s ) and (1 + r b ), for stocks and bonds, respectively, are best fit by the following LogNormal( μ, σ ) distributions : (cid:4666)1 (cid:3397) r (cid:2929) (cid:4667) ~ LogNormal(cid:4666)1.0837, 0.2153(cid:4667) (cid:4666)A/D p (cid:3398) value (cid:3404) 0.000(cid:4667) (cid:4666)1 (cid:3397) r (cid:2912) (cid:4667) ~ LogNormal(cid:4666)1.0214, 0.0825(cid:4667) (cid:4666)A/D p (cid:3398) value (cid:3404) 0.559(cid:4667) As shown, S&P 500 Index real compounding returns, (1 + r s ), have a p-value that leads to rejection of the null hypothesis that they originate from a lognormal distribution, perhaps suggesting that the daily returns are not iid . The corresponding hypothesis for 10-Year Treasury Bond returns cannot be rejected at any reasonable significance level. Note that similar null hypotheses for both (1 + r (cid:2929) (cid:4667) and (cid:4666)1 (cid:3397) r (cid:2912) (cid:4667) with respect to the normal distribution cannot be rejected at any reasonable significance level. Regardless, we will accept this disparity for the benefit of using RVs that have a domain which is consistent with the practical application. Finally, the sample correlation and covariance between these real compounding returns, at a given time point, is measured as: Corr(cid:4670)(cid:4666)1 (cid:3397) r (cid:2929) (cid:4667), (cid:4666)1 (cid:3397) r (cid:2912) (cid:4667)(cid:4671) (cid:3404) 0.04387
Cov(cid:4670)(cid:4666)1 (cid:3397) r (cid:2929) (cid:4667), (cid:4666)1 (cid:3397) r (cid:2912) (cid:4667)(cid:4671) (cid:3404) 0.00078
Let R s and R b be total annual returns for the stock and bond investments detailed above, respectively. A diversified portfolio would invest the proportion α in stocks and (1- α ) in bonds. The total annual return on this portfolio is α R s + (1- α )R b and the corresponding compounding (93)(94)(95)(96)(97) 24 return is (1 + α R s + (1- α )R b ) = α (1 + R s ) + (1- α )(1 + R b ). By decomposing each total return into its inflation and real component the compounding return can be written as α (1 + r s )(1 + I) + (1- α )(1 + r b )(1 + I). To obtain the compounding real return, we divide by (1 + I) which yields α (1 + r s ) + (1- α )(1 + r b ) = (1 + α r s + (1- α )r b ). This is a weighted sum of correlated lognormal RVs, see (94) and (95). Since α is the proportion invested in stocks, it is often referred to as the equity ratio. The CDF of real compounding returns on a diversified stock and bond portfolio can thus be approximated using the techniques presented here. Consider diversified portfolios consisting of equity ratios α ∈ {0.25, 0.50, 0.75}. Probabilities for the compounding return S = (1 + α r s + (1- α )r b ) will be derived using the MGF technique presented and compared with simulated probabilities and probabilities derived from the moment-matching (M-M) lognormal distribution . We will examine probabilities from both the head and tail along with those near the mean. The method presented here will use t ∈ {(-1.0, -0.2), (-0.001, -0.005)} as proposed by Mehta et al. (2007), who note that some t-sets work better in the head portion, and others in the tail of the distribution of S. The results of this analysis are shown below in Table 2. Table 2 Comparison of CDF Probabilities using Various Methods a Method Equity Ratio ( α ) CDF Probabilities P(S ≤ s) 0.01 0.05 0.10 0.30 0.50 0.80 0.90 0.95 0.99 Simulation b M-M 0.25
MGF(1) c MGF(2) c a P robabilities are for S = αΫ + (1- α ) Ϋ where Ϋ ~ LogNormal( ), Ϋ ~ LogNormal( ) and Cov( Ϋ , Ϋ ) = . The cell values represent the lognormal domain values, s, that yield P(S ≤ s). b Simulations were run in C++ using a sample size of N = 200,000,000. c MGF(1) uses t ∈ {-1.0, -0.2} and MGF(2) uses t ∈ {-0.001, -0.005}. The moment-matching lognormal distribution would be the one with mean and variance equal to E[S] and V[S], respectively, and derived in (86) and (87). It is motivated by the F-W approach for sums of independent RVs.
25 The CDF values from Table 2 generated via simulation can be viewed as the best representation of the true probabilities. The first item of note from Table 2 is that the CDF probabilities using the M-M lognormal approximation and using the MGF technique presented here with t ∈ {-0.001, -0.005} are identical. When values of t near zero are used, the equations from (77) and (78) are instantly satisfied without iterating and the procedure converges to the initial values. To see this, note that when t ∈ {0.0, 0.0}, (cid:1860)(cid:4666)(cid:1878) (cid:2869) , (cid:1878) (cid:2870) (cid:4667) from (66) equals 1, and C , C become: C (cid:2869) (cid:3404) C (cid:2870) (cid:3404) 1(cid:2024) (cid:3533) (cid:3533)(cid:3427)w (cid:3036) ∗ w (cid:2920) (cid:3431) (cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869) (cid:2869)(cid:2870)(cid:3036)(cid:2880)(cid:2869) . Further, equations (77) and (78) are identical and both reduce to: (cid:2920)(cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869) (cid:3398) 1(cid:2024) (cid:3533) (cid:3533)(cid:3427)w (cid:3036) ∗ w (cid:2920) (cid:3431) (cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869) (cid:2869)(cid:2870)(cid:3036)(cid:2880)(cid:2869) (cid:3404) 0 .
But since, (cid:3533) w (cid:2920)(cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869) (cid:3406) √(cid:2024) , and, (cid:3533) (cid:3533)(cid:3427)w (cid:3036) ∗ w (cid:2920) (cid:3431) (cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869) (cid:2869)(cid:2870)(cid:3036)(cid:2880)(cid:2869) (cid:3406) (cid:2024) , the equation in (99) is automatically satisfied, thus converges at the initial values. For this reason, using two values of t near zero is not recommended, as any initial values satisfy the equations and the procedure converges instantly to these values. It is straightforward to prove the results from (100). Note that the area under a standard normal RV equals 1 since it is a valid PDF. Let z ~ N(0,1), then: (cid:3505) 1√2(cid:2024) e (cid:2879)(cid:2869)(cid:2870)(cid:3053) (cid:3118) d(cid:1878) (cid:3404) 1 (cid:2878)(cid:2998)(cid:2879)(cid:2998) . Let u = (cid:2869)√(cid:2870) (cid:1878) , then du = (cid:2869)√(cid:2870) d(cid:1878) , and this expression becomes: (cid:2879)(cid:2931) (cid:3118) du (cid:3404) 1 (cid:2878)(cid:2998)(cid:2879)(cid:2998) → (cid:3505) e (cid:2879)(cid:2931) (cid:3118) du (cid:3404) √(cid:2024) (cid:2878)(cid:2998)(cid:2879)(cid:2998)
The integral on right side of the arrow is now of the form required by Gauss-Hermite quadrature with non-weight function g(u) from (6) equal to 1. Therefore, it can be estimated using Gauss-Hermite quadrature by: (98)(99)(100)(101)(102) 26 √(cid:2024) (cid:3404) (cid:3505) e (cid:2879)(cid:2931) (cid:3118) du (cid:3406) (cid:3533) w (cid:2920)(cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869) , (cid:2878)(cid:2998)(cid:2879)(cid:2998) which is the LHS identity from (100). For the 2 nd identity in (100), consider two independent RVs z i ~ N(0,1), i=1, 2. Their joint PDF must also integrate to 1, thus: (cid:3505) (cid:3505) 12(cid:2024) e (cid:4672)(cid:2879)(cid:2869)(cid:2870)(cid:3053) (cid:3117)(cid:3118) (cid:4673) e (cid:4672)(cid:2879)(cid:2869)(cid:2870)(cid:3053) (cid:3118)(cid:3118) (cid:4673) d(cid:1878) (cid:2869) d(cid:1878) (cid:2870) (cid:3404) 1 (cid:2878)(cid:2998)(cid:2879)(cid:2998)(cid:2878)(cid:2998)(cid:2879)(cid:2998) . Let u i = (cid:2869)√(cid:2870) (cid:1878) (cid:2919) , then du i = (cid:2869)√(cid:2870) d(cid:1878) (cid:2919) , for i=1, 2, so that (104) can be written as: (cid:2024) (cid:3404) (cid:3505) (cid:3505) e (cid:3435)(cid:2879)(cid:2931) (cid:3117)(cid:3118) (cid:3439) e (cid:3435)(cid:2879)(cid:2931) (cid:3118)(cid:3118) (cid:3439) du (cid:2869) du (cid:2870) (cid:3404) (cid:2878)(cid:2998)(cid:2879)(cid:2998) (cid:3505) e (cid:2879)(cid:2931) (cid:3117)(cid:3118) du (cid:2869) (cid:3505) e (cid:2879)(cid:2931) (cid:3118)(cid:3118) du (cid:2870) (cid:3406) (cid:2878)(cid:2998)(cid:2879)(cid:2998)(cid:2878)(cid:2998)(cid:2879)(cid:2998)(cid:2878)(cid:2998)(cid:2879)(cid:2998) (cid:3533) (cid:3533) w (cid:2919) w (cid:2920)(cid:2869)(cid:2870)(cid:2920)(cid:2880)(cid:2869)(cid:2869)(cid:2870)(cid:2919)(cid:2880)(cid:2869) , where the identity from (103) was applied twice. As seen in Table 2, using an equity ratio of α =0.25, the best performing method is MGF(1) which uses the technique presented in this research with t ∈ {-1.0, -0.2}. When the equity ratio is α =0.50, the method presented in this research works best with t ∈ {-1.0, -0.2} for probabilities in the head and select upper tails, while t ∈ {-0.001, -0.005} works better for some probabilities in the center and upper tail of the distribution. Thus, if interest is in portfolios with equal weighting of stocks/bonds (i.e., α =0.50), an optimization technique such as that described by Mehta et al. (2007) would be beneficial using various combinations of t ∈ { τ , τ } along with some intuitive criteria or metric to determine the best performing t-set. Finally, with an equity ratio of α =0.75, the MGF(1) approach is generally more accurate in the head portion of the distribution, whereas MGF(2) is more accurate in the tail. These results again demonstrate the need to optimize over the 2-member t-set and determine which values perform best. We provide code to perform this optimization in Appendix B. In terms of implementing these results using the code presented in Appendix A, we enter the lognormal parameters within the function main(), which is the application’s entry point and exists within the code file LnSum.cpp. The following arrangement was used to derive the approximating lognormal mean and variance for MGF(2) with α =0.25 in Table 2. (103)(104)(105) 27 // Declare/initialize local variables. //====================================== vector
Consider the sum S = a Ϋ + a Ϋ + … + a n Ϋ n , where a i is a known constant and Ϋ i ~ LogNormal( μ (cid:3052)(cid:4663) (cid:3284) , σ (cid:3052)(cid:4663) (cid:3284) (cid:2870) ) with Cov( Ϋ i , Ϋ j ) = σ (cid:4666)(cid:3052)(cid:4663) (cid:3284) ,(cid:3052)(cid:4663) (cid:3285) (cid:4667) , for i (cid:3405) j = 1, 2, …, n. As with a 2-term sum, the distribution of S will be approximated with a univariate lognormal RV by solving the simultaneous equations in (77) and (78). Regardless of how many terms constitute the sum, there will be two equations to solve for two unknown parameters. The unknowns are the mean and variance of the normal distribution on which the approximating lognormal RV is based (using the scale factor). In (77) and (78), only the constants C and C will change, and they represent two approximations to the MGF of S at different t < 0 values. In vector notation, S (cid:3404) (cid:4666)a (cid:2869) a (cid:2870) … a (cid:2924) (cid:4667) (cid:1737)(cid:1735)Y(cid:4663) (cid:2869) Y(cid:4663) (cid:2870) ⋮Y(cid:4663) (cid:2924) (cid:1740)(cid:1738) .
The expected value and variance of the sum S are known and given by:
E(cid:4670)S(cid:4671) (cid:3404) (cid:4666)a (cid:2869) a (cid:2870) … a (cid:2924) (cid:4667) (cid:3438)μ (cid:3052)(cid:4663) (cid:3117) μ (cid:3052)(cid:4663) (cid:3118) ⋮μ (cid:3052)(cid:4663) (cid:3172) (cid:3442) , (106)(107) Declare a 2-element vector named uniMuVar to hold the approximating lognormal mean and variance. Set the t-values that define the two non-linear equations from (77) and (78 ) that must be solved. Declare and populate the 2x2 variance-covariance matrix for the lognormal RVs that constitute the sum, along with a 2-element vector holding the means, and a 2-element vector holding the sum constants. Invoke the function and retrieve the mean and variance for the approximating lognormal RV.
28 and,
V(cid:4670)S(cid:4671) (cid:3404) V (cid:1743)(cid:1742)(cid:1742)(cid:1741)(cid:4666)a (cid:2869) a (cid:2870) … a (cid:2924) (cid:4667) (cid:1737)(cid:1735)Y(cid:4663) (cid:2869) Y(cid:4663) (cid:2870) ⋮Y(cid:4663) (cid:2924) (cid:1740)(cid:1738)(cid:1746)(cid:1745)(cid:1745)(cid:1744) (cid:3404) (cid:4666)a (cid:2869) a (cid:2870) … a (cid:2924) (cid:4667)V (cid:1737)(cid:1735)Y(cid:4663) (cid:2869) Y(cid:4663) (cid:2870) ⋮Y(cid:4663) (cid:2924) (cid:1740)(cid:1738) (cid:3438)a (cid:2869) a (cid:2870) ⋮a (cid:2924) (cid:3442) (cid:3404) (cid:4666)a (cid:2869) a (cid:2870) … a (cid:2924) (cid:4667) (cid:1743)(cid:1742)(cid:1742)(cid:1742)(cid:1741) σ (cid:3052)(cid:4663) (cid:3117) (cid:2870) σ (cid:4666)(cid:3052)(cid:4663) (cid:3117) ,(cid:3052)(cid:4663) (cid:3118) (cid:4667) … σ (cid:4666)(cid:3052)(cid:4663) (cid:3117) ,(cid:3052)(cid:4663) (cid:3172) (cid:4667) σ (cid:4666)(cid:3052)(cid:4663) (cid:3117) ,(cid:3052)(cid:4663) (cid:3118) (cid:4667) σ (cid:3052)(cid:4663) (cid:3118) (cid:2870) … σ (cid:4666)(cid:3052)(cid:4663) (cid:3118) ,(cid:3052)(cid:4663) (cid:3172) (cid:4667) ⋮ ⋮ ⋱ ⋮σ (cid:4666)(cid:3052)(cid:4663) (cid:3117) ,(cid:3052)(cid:4663) (cid:3172) (cid:4667) σ (cid:4666)(cid:3052)(cid:4663) (cid:3118) ,(cid:3052)(cid:4663) (cid:3172) (cid:4667) … σ (cid:3052)(cid:4663) (cid:3172) (cid:2870) (cid:1746)(cid:1745)(cid:1745)(cid:1745)(cid:1744) (cid:3438)a (cid:2869) a (cid:2870) ⋮a (cid:2924) (cid:3442) . Here,
E(cid:4670)S(cid:4671) and
V(cid:4670)S(cid:4671) will be used to compute the starting points for Newton’s method as they were in (88) and (89) for a 2-term sum. With n-terms, the constants C and C will consist of sums containing 12 n terms. To derive C and C we proceed exactly as in (36) for a 2-term sum. Here, let (cid:2737) be the variance-covariance matrix of the u i ’s, where, (cid:1873) (cid:3036) (cid:3404) (cid:3436)1θ(cid:3440) ln(cid:4666)(cid:1877)(cid:4663) (cid:3036) (cid:4667) , for (cid:1861) (cid:3404) 1, 2, … , (cid:1866) , are the underlying correlated normal RVs. Further, let (cid:2737) (cid:3404) (cid:1786)(cid:1786) (cid:4593) be its Cholesky decomposition, where L is lower triangular, unique, and has positive real pivots. The transformation that decorrelates the PDF in (40) for a 2-term sum now becomes: Let (cid:2203)(cid:4652)(cid:4652)(cid:1318) (cid:3404) √2(cid:1786)(cid:2208)(cid:4652)(cid:1318) (cid:3397) (cid:2246)(cid:4652)(cid:4652)(cid:1318) , where (cid:2203)(cid:4652)(cid:4652)(cid:1318) (cid:3404) (cid:3438)(cid:1873) (cid:2869) (cid:1873) (cid:2870) ⋮(cid:1873) (cid:2924) (cid:3442), (cid:2208)(cid:4652)(cid:1318) (cid:3404) (cid:3438)(cid:1878) (cid:2869) (cid:1878) (cid:2870) ⋮(cid:1878) (cid:2924) (cid:3442) , (cid:1853)(cid:1866)(cid:1856), (cid:2246)(cid:4652)(cid:4652)(cid:1318) (cid:3404) (cid:3438)μ (cid:3051) (cid:3117) μ (cid:3051) (cid:3118) ⋮μ (cid:3051) (cid:3172) (cid:3442) . Then, (cid:2208)(cid:4652)(cid:1318) (cid:3404) 1√2 (cid:1786) (cid:2879)(cid:2869) (cid:4666)(cid:2203)(cid:4652)(cid:4652)(cid:1318) (cid:3398) (cid:2246)(cid:4652)(cid:4652)(cid:1318)(cid:4667) .
To prove that this transformation is decorrelating in the z i ’s, (cid:1796)(cid:4666)(cid:2208)(cid:4652)(cid:1318)(cid:4667) (cid:3404) 12 (cid:1786) (cid:2879)(cid:2869) (cid:1796)(cid:4666)(cid:2203)(cid:4652)(cid:4652)(cid:1318) (cid:3398) (cid:2246)(cid:4652)(cid:4652)(cid:1318)(cid:4667)(cid:4666)(cid:1786) (cid:2879)(cid:2869) (cid:4667) (cid:4593) (cid:3404) 12 (cid:1786) (cid:2879)(cid:2869) (cid:2737)(cid:4666)(cid:1786) (cid:4593) (cid:4667) (cid:2879)(cid:2778) (cid:3404) 12 (cid:1786) (cid:2879)(cid:2869) (cid:1786)(cid:1786) (cid:4593) (cid:4666)(cid:1786) (cid:4593) (cid:4667) (cid:2879)(cid:2778) (cid:3404) 12 (cid:1783) . Note that since L is lower-triangular with positive real pivots, it has the following general form: (108)(109)(110)(111)(112)(113) 29 (cid:1786) (cid:3404) (cid:1743)(cid:1742)(cid:1742)(cid:1742)(cid:1741)(cid:1864) (cid:2869)(cid:2869) (cid:2870)(cid:2869) (cid:1864) (cid:2870)(cid:2870) (cid:2871)(cid:2869) (cid:1864) (cid:2871)(cid:2870) (cid:1864) (cid:2871)(cid:2871) … 0⋮ ⋮ ⋮ ⋱ ⋮(cid:1864) (cid:3041)(cid:2869) (cid:1864) (cid:3041)(cid:2870) (cid:1864) (cid:3041)(cid:2871) … (cid:1864) (cid:3041)(cid:3041) (cid:1746)(cid:1745)(cid:1745)(cid:1745)(cid:1744) , where l ii > 0, i=1, 2, …, n. This implies that (cid:2203)(cid:4652)(cid:4652)(cid:1318) from (111) consists of the following elements: (cid:3438)(cid:1873) (cid:2869) (cid:1873) (cid:2870) ⋮(cid:1873) (cid:2924) (cid:3442) (cid:3404) (cid:1737)(cid:1736)(cid:1735) √2(cid:4666)(cid:1864) (cid:2869)(cid:2869) (cid:1878) (cid:2869) (cid:4667) (cid:3397) μ (cid:3051) (cid:3117) √2(cid:4666)(cid:1864) (cid:2870)(cid:2869) (cid:1878) (cid:2869) (cid:3397) (cid:1864) (cid:2870)(cid:2870) (cid:1878) (cid:2870) (cid:4667) (cid:3397) μ (cid:3051) (cid:3118) ⋮√2(cid:4666)(cid:1864) (cid:3041)(cid:2869) (cid:1878) (cid:2869) (cid:3397) (cid:1864) (cid:3041)(cid:2870) (cid:1878) (cid:2870) (cid:3397) ⋯ (cid:3397) (cid:1864) (cid:3041)(cid:3041) (cid:1878) (cid:3041) (cid:4667) (cid:3397) μ (cid:3051) (cid:3289) (cid:1740)(cid:1739)(cid:1738). Here, Ϋ i = (cid:2908) (cid:3167) /(cid:2869)(cid:2868) where X (cid:2919) ~ N(cid:3435)μ (cid:3051) (cid:3167) , σ (cid:3051) (cid:3167) (cid:2870) (cid:3439) , so that Ϋ i follows the standard form lognormal distribution with underlying normal RVs θX (cid:2919) ~ N(cid:3435)θμ (cid:3051) (cid:3167) , θ (cid:2870) σ (cid:3051) (cid:3167) (cid:2870) (cid:3439) , for i = 1, 2, …, n. By making this decorrelating transformation, we express the MGF of S in terms of the z i ’s as was done in (61) for a 2-term sum. The required weight functions appear for each z i , i = 1, 2, …, n and the non-weight function in terms of the Ϋ i ’s from (36) is now given by: (cid:3436)1(cid:2024)(cid:3440) (cid:2924)(cid:2870) e (cid:3047)(cid:4666)(cid:2911) (cid:3117) (cid:3052)(cid:4663) (cid:3117) (cid:2878) (cid:2911) (cid:3118) (cid:3052)(cid:4663) (cid:3118) (cid:2878) … (cid:2878) (cid:2911) (cid:3172) (cid:3052)(cid:4663) (cid:3172) (cid:4667) . In terms of the u i ’s, the non-weight function becomes: (cid:3436)1(cid:2024)(cid:3440) (cid:2924)(cid:2870) e (cid:3047)(cid:4666)(cid:2911) (cid:3117) (cid:2915) (cid:3216)(cid:3296)(cid:3117) (cid:2878) (cid:2911) (cid:3118) (cid:2915) (cid:3216)(cid:3296)(cid:3118) (cid:2878) … (cid:2878) (cid:2911) (cid:3172) (cid:2915) (cid:3216)(cid:3296)(cid:3172) (cid:4667) . Finally, in terms of the z i ’s, the non-weight function is given by: (cid:1860)(cid:4666)(cid:1878) (cid:2869) , (cid:1878) (cid:2870) , … , (cid:1878) (cid:2924) (cid:4667) (cid:3404) (cid:3436)1(cid:2024)(cid:3440) (cid:2924)(cid:2870) e (cid:3047)(cid:4666)(cid:2911) (cid:3117) (cid:2915) (cid:3216)(cid:3435)√(cid:3118)(cid:4666)(cid:3287)(cid:3117)(cid:3117)(cid:3301)(cid:3117)(cid:4667)(cid:3126)(cid:3220)(cid:3299)(cid:3117)(cid:3439) (cid:2878) (cid:2911) (cid:3118) (cid:2915) (cid:3216)(cid:3435)√(cid:3118)(cid:4666)(cid:3287)(cid:3118)(cid:3117)(cid:3301)(cid:3117)(cid:3126)(cid:3287)(cid:3118)(cid:3118)(cid:3301)(cid:3118)(cid:4667)(cid:3126)(cid:3220)(cid:3299)(cid:3118)(cid:3439) (cid:2878) … (cid:2878) (cid:2911) (cid:3172) (cid:2915) (cid:3216)(cid:3435)√(cid:3118)(cid:4666)(cid:3287)(cid:3289)(cid:3117)(cid:3301)(cid:3117)(cid:3126)(cid:3287)(cid:3289)(cid:3118)(cid:3301)(cid:3118)(cid:3126)⋯(cid:3126)(cid:3287)(cid:3289)(cid:3289)(cid:3301)(cid:3289)(cid:4667)(cid:3126)(cid:3220)(cid:3299)(cid:3289)(cid:3439) (cid:4667) . The constants C and C from (77) and (78) for an n-term sum S are then constructed by applying Gauss-Hermite quadrature to the n-dimensional integration of h (·) using two values for t < 0. As noted, the sum for each C i will consist of 12 n terms with each term representing a unique combination of the weight and root pairs across the n-dimensions. For example, the first term in the sum would use the first weight from Table 1 for each of the n-dimensions and each z i would be replaced by the corresponding first root from Table 1. This is repeated until all unique (114)(115)(116)(117)(118) 30 combinations have been represented. The weights are multiplied by each other as done on the left-hand sides of (75) and (76) for the 2-term case. The formal expression for C i , i=1, 2 is: C (cid:2919) (cid:3404) (cid:3436)1(cid:2024)(cid:3440) (cid:2924)(cid:2870) (cid:3533) (cid:3533) … (cid:3533) (cid:4686)(cid:4684)(cid:3537) w (cid:2921) (cid:3168) (cid:2924)(cid:2920)(cid:2880)(cid:2869) (cid:4685) e (cid:3047)(cid:4666)(cid:2911) (cid:3117) (cid:2915) (cid:3216)(cid:4672)√(cid:3118)(cid:4666)(cid:3287)(cid:3117)(cid:3117)(cid:3176)(cid:3169)(cid:3117)(cid:4667)(cid:3126)(cid:3220)(cid:3299)(cid:3117)(cid:4673) (cid:2878) (cid:2911) (cid:3118) (cid:2915) (cid:3216)(cid:4672)√(cid:3118)(cid:4666)(cid:3287)(cid:3118)(cid:3117)(cid:3176)(cid:3169)(cid:3117)(cid:3126) (cid:3287)(cid:3118)(cid:3118)(cid:3176)(cid:3169)(cid:3118)(cid:4667)(cid:3126)(cid:3220)(cid:3299)(cid:3118)(cid:4673) (cid:2878) (cid:2869)(cid:2870)(cid:2921) (cid:3172) (cid:2880)(cid:2869)(cid:2869)(cid:2870)(cid:2921) (cid:3118) (cid:2880)(cid:2869)(cid:2869)(cid:2870)(cid:2921) (cid:3117) (cid:2880)(cid:2869) … (cid:3397) a (cid:2924) e (cid:2968)(cid:3435)√(cid:2870)(cid:4666)(cid:3039) (cid:3289)(cid:3117) (cid:2928) (cid:3169)(cid:3117) (cid:2878) (cid:3039) (cid:3289)(cid:3118) (cid:2928) (cid:3169)(cid:3118) (cid:2878) … (cid:2878) (cid:3039) (cid:3289)(cid:3289) (cid:2928) (cid:3169)(cid:3172) (cid:4667)(cid:2878)(cid:2972) (cid:3299)(cid:3289) (cid:3439) (cid:4667)(cid:4675) . Here, (w , r ) is the 1 st weight/root pair from Table 1, (w , r ) is the 2 nd , and so on. VII. Summary/Conclusion
Sums of lognormal RVs appear naturally in many disciplines and consequently must be modeled accurately within complex systems. Two common modeling procedures are the F-W (Fenton [1960]) and S-Y (Schwartz and Yeh [1982]) methods. Each has their benefits and drawbacks, for example, working well within some regions but not others. Mehta et al. (2007) propose a new and novel approach that is parameterizable, allowing the user to customize the CDF precision in regions of special interest. As is common with academic research, the paper assumes a high prerequisite level of technical expertise that may not be held by all who could benefit from it. We have therefore filled in the gaps and presented the material in a pedagogical fashion. We step the reader through all technical details required to understand the method for a (correlated) 2-term sum, and provide sufficient technical details for a full understanding of sums involving more than two (correlated) terms. To emphasize the importance of such a procedure we provided an application to financial economics, and particularly to approximating CDF probabilities for the compounding return on a diversified portfolio of stocks and bonds, in discrete time. Such models are important within financial economics fields such as retirement planning where critical decisions on asset allocation and withdrawal rates are made periodically (e.g., yearly), not continuously. We have also included original source code from a C++ implementation that solves the required set of non-linear equations using Newton’s method, with starting values motivated by the F-W approximation. Mehta et al. (2007) made use of MATLAB’s built-in non-linear solvers. Such an implementation may suffer run-time inefficiencies within a large financial application being optimized over a planning period that spans several decades. Lower level programming (119) 31 languages can be more appropriate for such implementations. The technique we use converges rapidly, requiring only a small number of iterations. As seen in Table 2, the improvements are modest when using arbitrary MGF values for t to generate the two required equations. Mehta et al. (2007) suggest that the user optimize over the t-set and find values suitable to their application. For example, CDF probabilities for a set of sum values would be generated via simulation over a region of interest, or the entire sum domain, as was done in Table 2. Optimization over the 2-member t-set would then compute the (weighted) sum of absolute %-deviations between the simulated and approximated values, and the best performing t-set would be chosen for that particular application. The best performing t-set would be the one that yields the minimum sum value. To achieve greater accuracy over particular regions of the lognormal sum domain, weights can be introduced for each absolute %-deviation (see, Mehta et al. [2007]). Arguably, the F-W approach can be similarly optimized using various mean/variance combinations, but there is a distinction. Under the Mehta et al. (2007) framework, the optimal t-set may work well for a variety of related sums, whereas, the F-W optimization would need to be repeated whenever the sum changes. We have included C++ source code to simulate sum values and optimize the t-set in Appendix B. The computations are multi-threaded to reduce processing time. With respect to the finance application provided in Section V, we derived optimal t-sets and these are shown in Figure 2 along with the corresponding univariate lognormal CDF approximations. The sum of absolute %-differences was unweighted for this implementation which results in greater absolute precision in the head portion of the distribution, and this is clearly seen as α increases. We can enhance the univariate approximations shown in Figure 2 by weighting the sum of absolute %-differences in a manner that gives more importance to increasing values on the lognormal sum domain, or by partitioning the domain into sections and optimizing the t-set within each section. The univariate approximation would then be conditional on the section that a particular domain value resides in. Figure 3 shows the result of using a simple weighting scheme, namely, values of S < 0.75 receive a weight of 1.0, values between 0.75 and 1.10 receive a weight of 15.0, and values > 1.10 receive a weight of 50.0. 32 Figure 2 CDF Approximations using the Unweighted Optimized t-Set
Figure 3 CDF Approximations for α = 0.75 using the Weighted Optimized t-Set Appendix A: Lognormal Approximation Source Code
Filename: stdafx.h /* / The MIT License (MIT) / / Copyright (c) 2015 Chris Rook / / Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), / to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, / and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: / / The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. / / THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, / FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER / LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER / DEALINGS IN THE SOFTWARE. (License source: http://opensource.org/licenses/MIT) / / Filename: stdafx.h / / Summary: / / This is the header file where we include other header files, define constants, namespaces, inline functions, and function prototypes. / /***************************************************************************************************************************************************/ // Constants. //============= const double pi = 3.141592653589793; const double sf = log(10.00)/10.00; // Scaling factor ln(10)/10 to align with Mehta et al. (2007) // Inline functions to derive the underlying normal mean/variances from the lognormal mean/variances. //===================================================================================================== inline long double NMean(const Eigen::VectorXd inM, const Eigen::MatrixXd inV, const int i) {return (1.0/sf)*(log(inM(i)) - (0.5)*log(1.0 + inV(i,i)/pow(inM(i),2)));} // See (24). inline long double NVar(const Eigen::VectorXd inM, const Eigen::MatrixXd inV, const int i, const int j) {return pow((1.0/sf),2)*log(1.0 + inV(i,j)/abs(inM(i)*inM(j)));} // See (25) & (26). // Function prototypes. //======================= vector
Filename: LnSumOpt.cpp /* / The MIT License (MIT) / / Copyright (c) 2015 Chris Rook / / Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), / to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, / and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: / / The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. / / THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, / FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER / LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER / DEALINGS IN THE SOFTWARE. (License source: http://opensource.org/licenses/MIT) / / Filename: LnSumOpt.cpp / / Function: main() / / Summary: / / The main() function here is the entry point for the application that optimizes the univariate approximation over the 2-member t-set. To find / the best performing t-set we first discretize the domain scale of the sum S = aY1 + bY2 + cY3 + etc ... into k values. The CDF probability for / each value is then derived using simulation. The k domain values and k CDF probabilities are stored in arrays/vectors. We then iterate over / various 2-member combinations of t and, for each combination, we invoke the function LnSumApprox() to find the approximating univariate / lognormal mean and variance for this t-set. Using these parameters we compute the approximated CDF probability and then take the (weighted) / absolute %-difference between the simulated and approximated values for each of the k domain values. These k values are then summed and the / smallest sum of (weighted) absolute %-differences yields the optimal 2-member t-set. As suggested by Mehta et al. (2007), weights can be added / to Taylor the approximation to suit the user's needs. This can be done in the function tSetOpt() and we indicate where to add weights (if / desired) when constructing the sum. The quantities we construct in main() are as follows: / / 1.) The / threaded call is executed by the function SimProb(). To optimize over the 2-member t-set the function ThrdtSetOpt() splits the job by sectioning / variable t1 into set sizes that increase in value. Iteration over the 2-member t-set is then done section-by-section. The results are inspected / and the best performing t-set across all calls to tSetOpt() is selected. The results for the best performing t-set are returned by ThrdtSetOpt() / in a 5-member vector, then printed to the screen. / /***************************************************************************************************************************************************/ // and corresponding divisor. long long int n=20000000000; // Total simulation sample size. double *dvals = new double[k]; // High value on the sum domain, and array to hold sum domain values. long double *probs = new long double[k]; // Array to hold the simulated probabilities for comparison. vector Abramowitz, Milton, and Irene A. Stegun, 1964, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , National Bureau of Standards Applied Mathematics Series, 55, pp. 924. Anton, Howard, 1988, Calculus 3 rd Edition (John Wiley & Sons, New York, NY). Casella, George, and Roger L. Berger, 1990, Statistical Inference (Wadsworth & Brooks/Cole, Pacific Grove, California). Fenton, Lawrence F., 1960, The Sum of Log-Normal Probability Distributions in Scatter Transmission Systems, IRE Transactions on Communications Systems Mathematical Statistics 5 th Edition (Prentice Hall, Englewood Cliffs, NJ). Golub, Gene H., and John H. Welsch, 1969, Calculation of Gauss Quadrature Rules, Mathematics of Computation Linear Models: An Introduction (John Wiley & Sons, New York, NY). Law, Averill M., and W. David Kelton, 2000, Simulation Modeling and Analysis 3 rd Edition (McGraw-Hill International Series, New York, NY). Mehta, Neelesh B., Jingxian Wu, Andreas F. Molisch, and Jin Zhang, 2007, Approximating a Sum of Random Variables with a Lognormal, IEEE Transactions on Wireless Communications Matrix Analysis and Applied Linear Algebra (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA). Mitchell, R. L., 1968, Permanance of the Log-Normal Distribution, Journal of the Optical Society of America Introduction to Probability and Statistics for Engineers and Scientists 4 th Edition (Elsevier Academic Press, New York, NY). Schwartz, S. C., and Y. S. Yeh, 1982, On the Distribution Function and Moments of Power Sums With Log-Normal Components, The Bell Systems Technical Journal