Exact Solution for the Portfolio Diversification Problem Based on Maximizing the Risk Adjusted Return
aa r X i v : . [ ec on . T H ] M a r Exact Solution for the Portfolio Diversification ProblemBased on Maximizing the Risk Adjusted Return
Abdulnasser Hatemi-J, Mohamed Ali Hajji and Youssef El-KhatibUAE UniversityMarch 5, 2019
Abstract
The potential benefits of portfolio diversification have been known to investors fora long time. Markowitz (1952) suggested the seminal approach for optimizing theportfolio problem based on finding the weights as budget shares that minimize thevariance of the underlying portfolio. Hatemi-J and El-Khatib (2015) suggested findingthe weights that will result in maximizing the risk adjusted return of the portfolio.This approach seems to be preferred by the rational investors since it combines riskand return when the optimal budget shares are sought for. The current paper providesa general solution for this risk adjusted return problem that can be utilized for anypotential number of assets that are included in the portfolio.
Keywords:
Portfolio Diversification, Optimization, Risk and Return.
JEL Classifications:
G10, G12, C6.
The potential portfolio diversification benefits that can result in reducing risk havebeen known to investors for a long time. However, the idea of finding the optimalweights as budget shares was pioneered by Markowitz (1952). In his seminal paperthe optimal weights can be found by minimizing the variance of the portfolio subjectto the budget constraint. In reality, however, it is also important for the investors topay attention to the amount of return. An alternative approach has been suggestedrecently in the literature for the purpose of combining the risk and the return. Thus,the optimization problem in this setting is to find the optimal weights that will resultin maximizing the risk adjusted return subject to the budget constraint according toHatemi-J and El-Khatib (2015). However, the authors provide a closed form solutionfor a portfolio that consists of only two assets. The objective of the current paperis to extend their work and provide an exact solution for the risk adjusted return ofa portfolio that can consist of any potential number of assets subject to the budgetrestriction. he rest of the paper is organized as follows. In section 2, the optimization problemis formulated and the general solution is derived. Section 3 provides two examples. Aconclusion is given in section 4. Finally, an appendix in the end presents some pertinentderivations. Let A i , i = 1 , , . . . , n , be n assets included in a portfolio. Let also r i , i = 1 , , . . . , n, be the return of asset A i . As in Markowitz (1952), it is assumed that the returns arenormally distributed such as r i ∼ N ( µ i , σ i ). The variance and covariance matrix isdefined as Ω = ( σ i,j ) ≤ i,j ≤ n . Let w i , i = 1 , , . . . , n, be the budget shares investedin asset A i . Let f ( w ) = n P i =1 µ i w i be the expected return of the portfolio and g ( w ) = w t Ω w be the variance, w t denotes the transpose of w . The risk adjusted return of theunderlying portfolio is Q ( w ) = n P i =1 w i µ i √ w t Ω w = f ( w ) p g ( w ) . (2.1)The problem is to determine the optimal weights vector w = ( w i ) ≤ i ≤ n such that therisk adjusted return, Q ( w ), of the underlying portfolio is maximized, subject to thebudget constraint n P i =1 w = 1. Thus, the optimization problem at hand is the following.max w ∈ R n Q ( w ) subject to C ( w ) = n X i =1 w i − . (2.2)The following theorem gives the exact optimal solution of the optimization problem(2.2) in a compact form. Theorem 1
The optimal solution, w ∗ , to the maximization problem (2.2) is given by w ∗ = α − ( µ t β )( α t Ω α ) − ( µ t α )( α t Ω β )( µ t β )( α t Ω β ) − ( µ t α )( β t Ω β ) β, (2.3) where α = Ω − u P (Ω − u ) , β = Ω − v − (cid:16)X (Ω − v ) (cid:17) α, with u = ( u i ) i =1 ,...,n and v = ( v i ) i =1 ,...,n are n × constant vectors given recursively by u i = a i u i +1 + b i u i +2 , u n − = 1 , u n = 0 , i = 1 , . . . , n − ,v i = a i v i +1 + b i v i +2 , v n − = 0 , v n = 1 , i = 1 , . . . , n − , and a i = µ i +2 − µ i µ i +2 − µ i +1 , b i = µ i − µ i +1 µ i +2 − µ i +1 , i = 1 , . . . , n − , For an n × vector x , the notation P (Ω − x ) = n P i =1 (Ω − x ) i . roof of Theorem 1: Define the Lagrange function to problem (2.2) by L ( w ; λ ) = Q ( w ) − λC ( w ) , λ ∈ R. (2.4)A solution to problem (2.2) is solution of the system ∂L∂w i = 0 , i = 1 , . . . , n, (2.5) ∂L∂λ = 0 , (2.6)Using the notation f i = ∂f∂w i and g i = ∂g∂w i , (2.5) and (2.6) reduce to2 f i g − f g i = 2 g / λ, i = 1 , . . . , n, (2.7) n X i =1 w i = 1 , (2.8)From (2.7), we see the right-hand side is independent of i . Thus, we have2 f i +1 g − f g i +1 = 2 f i g − f g i , i = 1 , . . . , n − , or g i +1 − g i f i +1 − f i = 2 gf , i = 1 , . . . , n − . (2.9)Again, the right-hand side of (2.9) is independent of i , and we have g i +2 − g i +1 f i +2 − f i +1 = g i +1 − g i f i +1 − f i , i = 1 , . . . , n − . (2.10)Equation (2.10) means that w satisfies the homogeneous linear system( f i +1 − f i )( g i +2 − g i +1 ) − ( f i +2 − f i +1 )( g i +1 − g i ) = 0 , i = 1 , . . . , n − . (2.11)Since f i = µ i and g i = ((Ω + Ω t ) w ) i = 2(Ω w ) i , the ( n − × n system (2.11) can bewritten in matrix form as B Ω w = 0 , (2.12)with B the upper tridiagonal ( n − × n matrix: ( µ − µ ) ( µ − µ ) ( µ − µ ) 0 0 . . .
00 ( µ − µ ) ( µ − µ ) ( µ − µ ) 0 . . . . . . µ n − µ n − ) ( µ n − − µ n ) ( µ n − − µ n − ) . The general solution of (2.12) can be written as w = Ω − z, here z = ( z i ) i =1 ,...,n is the general solution of the system Bz = 0. If we assume that rank ( B ) = n −
2, i.e., µ i = µ i − , ≤ i ≤ n , then z = ( z i ) i =1 ,...,n is given recursivelyby z i = a i z i +1 + b i z i +2 , i = 1 , . . . , n − , (2.13)with z n − = s, z n = t , s and t are free parameters, and a i = µ i +2 − µ i µ i +2 − µ i +1 , b i = µ i − µ i +1 µ i +2 − µ i +1 , i = 1 , . . . , n − . Since every z i is a linear combination of the free parameters s and t , we can write z = su + tv , where u = ( u i ) i =1 ,...,n and v i =1 ,...,n are n × u i = a i u i +1 + b i u i +2 , v i = a i v i +1 + b i v i +2 , i = 1 , . . . , n − , (2.14)with u n − = 1 , u n = 0 and v n − = 0 , v n = 1. It follows that the general solution of(2.12) is w = Ω − ( su + tv ) = s Ω − u + t Ω − v. Imposing the condition (2.8), n P i =1 w i = 1, we get s = 1 − t P (Ω − v ) i P (Ω − u ) i . Thus, we have w as w = α + tβ, where α = Ω − u P (Ω − u ) i , β = Ω − v − (cid:16)X (Ω − v ) i (cid:17) α. (2.15)From the above derivations, we conclude that the optimal solution w ∗ is of the form w ∗ = α + t ∗ β for some optimal value t ∗ , with α and β as defined above. The optimal value t ∗ is theone that maximizes the risk adjust return Q ( α + tβ ), i.e., ∂Q ( α + tβ ) ∂t = 0. Derivations(see Apprendix A) reveal that t ∗ = − ( µ t β )( α t Ω α ) − ( µ t α )( α t Ω β )( µ t β )( α t Ω β ) − ( µ t α )( β t Ω β ) . (2.16)Therefore, the optimal weights vector is given w ∗ = α − ( µ t β )( α t Ω α ) − ( µ t α )( α t Ω β )( µ t β )( α t Ω β ) − ( µ t α )( β t Ω β ) β, (2.17)which proves the Theorem 1. Examples
In this section, we validate the formula (2.17) by applying it to the case of a portfolioconsisting of two assets, and extend it to the 3 dimensional case.
Example 1.
In the case n = 2, we have µ = [ µ µ ] t , Ω = (cid:20) σ σ σ σ (cid:21) , σ = σ .From equation (2.14), we have u = [ u u ] t = [1 0] t , v = [ v v ] t = [0 1] t and fromequation (2.15), we have α = (cid:18) σ , σ , − σ , , − σ , σ , − σ , (cid:19) , β = (cid:18) − σ , − σ , , σ , − σ , (cid:19) . Substituting these into (2.17), we obtain w = µ σ , − µ σ , µ ( σ , − σ , ) + µ ( σ , − σ , ) ,w = µ σ − µ σ µ ( σ , − σ , ) + µ ( σ , − σ , ) , as found in Hatemi-J and El-Khatib. Example 2.
In the case n = 3, we have µ = [ µ µ µ ] t , Ω = σ σ σ σ σ σ σ σ σ ,where σ , = σ , , σ , = σ , , and σ , = σ , . From equation (2.14), the vectors u and v , from u = [ u u u ] t = n µ − µ µ − µ , , o t , v = [ v v v ] t = n µ − µ µ − µ , , o t . Similarly,we obtain the α and β vectors and when Substituted into (2.17), we obtain w = µ ( σ , σ , − σ , σ , ) + µ ( σ , σ , − σ , σ , ) + µ (cid:0) σ , − σ , σ , (cid:1) ∆ ,w = µ ( σ , σ , − σ , σ , ) + µ (cid:0) σ , − σ , σ , (cid:1) + µ ( σ , σ , − σ , σ , )∆ ,w = µ (cid:0) σ , − σ , σ , (cid:1) + µ ( σ , σ , − σ , σ , ) + µ ( σ , σ , − σ , σ , )∆ , where ∆ = µ (cid:0) σ , − ( σ , + σ , ) σ , + ( σ , − σ , ) σ , + σ , σ , (cid:1) + µ ( − σ , ( σ , − σ , + σ , ) + σ , ( σ , − σ , ) + σ , σ , )+ µ ( σ , ( σ , − σ , ) + σ , ( σ , − σ , ) + ( σ , − σ , ) σ , ) Investors regularly make use of portfolios in order to remove or at least reduce theunsystematic risk of their investments. A critical issue within this context is finding he optimal weights as budget shares. The seminal paper of Markowitz (1952) suggestsfinding the optimal weights that result in minimum variance as a measure of risk of theunderlying portfolio. This classical approach might lead to selecting a portfolio thatwould be the safest but not necessarily the one that would give the highest amountof return per unit of risk. Hatemi-J and El-Khatib (2015) suggest finding the optimalweights that combine risk and return based on the maximization of the risk adjustedreturn. They managed to provide solution for the 2 × n × n , where n represents the number of assets.In the future, applications can be provided to show that the risk adjusted returnof a portfolio based on our suggested approach has higher value compared to the riskadjusted return of other alternative portfolios. It should be also mentioned that thismethod can be extended to the financial markets in which short selling is not possible,that is, all weights must be positive. Other extensions could be allowing for highermoments in the data generating process of the underlying assets such as the third andthe fourth moments. A Appendix
In this appendix we present the derivation of the formula for t ∗ given in (2.16). Let α and β be two constant n × h ( t ) = Q ( α + tβ ) = f ( α + tβ ) p g ( α + tβ ) . The optimal t ∗ is the value of t for which h ′ ( t ) = 0, where h ′ ( t ) = ∂Q∂t = 2 f t g − f g t g / . Then t ∗ is the value of t for which 2 f t g − f g t = 0 . (A.1)We have f ( α + tβ ) = n X i =1 µ i ( α i + tβ i ) = ⇒ f t = n X i =1 µ i β i g ( α + tβ ) = X i,j σ i,j α i α j + X i,j σ i,j ( α i β j + α j β i ) t + X i,j σ i,j ( β i β j ) t = ⇒ g t = X i,j σ i,j ( α i β j + α j β i ) + 2 X i,j σ i,j ( β i β j ) t hen (A.1) becomes2 f t g − f g t = 2 n X i =1 µ i β i ! n X i,j =1 σ i,j α i α j + n X i,j =1 σ i,j ( α i β j + α j β i ) t + n X i,j =1 σ i,j ( β i β j ) t − n X i =1 µ i ( α i + tβ i ) ! n X i,j =1 σ i,j ( α i β j + α j β i ) + 2 n X i,j =1 σ i,j ( β i β j ) t = n X i =1 µ i β i ! n X i,j =1 σ i,j α i α j − n X i =1 µ i α i ! n X i,j =1 σ i,j ( α i β j + α j β i ) + t (cid:20) n X i =1 µ i β i ! n X i,j =1 σ i,j ( α i β j + α j β i ) − n X i =1 µ i β i ! n X i,j =1 σ i,j ( α i β j + α j β i ) − n X i =1 µ i α i ! n X i,j =1 σ i,j ( β i β j ) (cid:21) + t (cid:20) n X i =1 µ i β i ! n X i,j =1 σ i,j ( β i β j ) − n X i =1 µ i β i ! n X i,j =1 σ i,j ( β i β j ) | {z } =0 (cid:21) = 2( µ t β )( α t Ω α ) − ( µ t α )( α t Ω β + β t Ω α ) + t (cid:2) ( µ t β )( α t Ω β + β t Ω α ) − µ t α )( β t Ω β ) (cid:3) It follows that 2 f t g − f g t = 0 for t ∗ = − µ t β )( α t Ω α ) − ( µ t α )( α t Ω β + β t Ω α )( µ t β )( α t Ω β + β t Ω α ) − µ t α )( β t Ω β )Since Ω is symmetric, we have β t Ω α = α t Ω β and t ∗ reduces to t ∗ = − ( µ t β )( α t Ω α ) − ( µ t α )( α t Ω β )( µ t β )( α t Ω β ) − ( µ t α )( β t Ω β ) , as in (2.16). References [1] Aase, K.K., 1984. Optimum portfolio diversification in a general continuous-timemodel. Stochastic Process. Appl. 18 (1), 8189.[2] Hatemi-J, A. and El-Khatib Y., Portfolio selection: An alternative approach, Eco-nomics Letters 135 (2015) 141143.[3] Hatemi-J, A., Roca, E., 2006. A re-examination of international portfolio diversifi-cation based on evidence from leveraged bootstrap methods. Econ. Model. 23 (6),9931007.
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