Mean-variance portfolio selection with tracking error penalization
MMean-variance portfolio selection with tracking errorpenalization
William Lefebvre ∗ Gr´egoire Loeper † Huyˆen Pham ‡ September 21, 2020
Abstract
This paper studies a variation of the continuous-time mean-variance portfolio selec-tion where a tracking-error penalization is added to the mean-variance criterion. Thetracking error term penalizes the distance between the allocation controls and a refe-rence portfolio with same wealth and fixed weights. Such consideration is motivatedas follows: (i) On the one hand, it is a way to robustify the mean-variance allocationin case of misspecified parameters, by “fitting” it to a reference portfolio that can beagnostic to market parameters; (ii) On the other hand, it is a procedure to track abenchmark and improve the Sharpe ratio of the resulting portfolio by considering amean-variance criterion in the objective function. This problem is formulated as aMcKean-Vlasov control problem. We provide explicit solutions for the optimal portfo-lio strategy and asymptotic expansions of the portfolio strategy and efficient frontierfor small values of the tracking error parameter. Finally, we compare the Sharpe ratiosobtained by the standard mean-variance allocation and the penalized one for four dif-ferent reference portfolios: equal-weights, minimum-variance, equal risk contributionsand shrinking portfolio. This comparison is done on a simulated misspecified model,and on a backtest performed with historical data. Our results show that in mostcases, the penalized portfolio outperforms in terms of Sharpe ratio both the standardmean-variance and the reference portfolio.
Keywords:
Continuous-time mean-variance problem, tracking error, robustified alloca-tion, parameter misspecification. ∗ BNP Paribas Global Markets, Universit´e de Paris and Sorbonne Universit´e, Laboratoire de Probabilit´es,Statistique et Mod´elisation (LPSM, UMR CNRS 8001), Building Sophie Germain, Avenue de France, 75013Paris, wlefebvre at lpsm.paris † BNP Paribas Global Markets, School of Mathematics, Monash University, Clayton Campus, VIC, 3800,Australia, gregoire.loeper at monash.edu ‡ Universit´e de Paris and Sorbonne Universit´e, Laboratoire de Probabilit´es, Statistique et Mod´elisation(LPSM, UMR CNRS 8001), Building Sophie Germain, Avenue de France, 75013 Paris, pham at lpsm.paris a r X i v : . [ q -f i n . C P ] S e p Introduction
The Markowitz mean-variance portfolio selection problem has been initially considered inMarkowitz (1952) in a single-period model. In this framework, investement decision rulesare made according to the objective of maximizing the expected return of the portfolio fora given financial risk quantified by its variance. The Markowitz portfolio is widely usedin the financial industry due to its intuitive formulation and the fact that it produces,by construction, portfolios with high Sharpe ratios (defined as the ratio of the average ofportfolio returns over their volatility), which is a key metric used to compare investmentstrategies.The mean-variance criterion involves the expected terminal wealth in a nonlinear waydue to the presence of the variance term. In a continuous-time dynamic setting, this in-duces the so-called time inconsistency problem and prevents the direct use of the dynamicprogramming technique. A first approach, from Zhou and Li (2000), consists in embed-ding the mean-variance problem into an auxiliary standard control problem that can besolved by using stochastic linear-quadratic theory. Some more recent approaches rely onthe development of stochastic control techniques for control problems of McKean-Vlasov(MKV) type. MKV control problems are problems in which the equation of the stateprocess and the cost function involve the law of this process and/or the law of the control,possibly in a non-linear way. The mean-variance portfolio problem in continuous-time isa McKean-Vlasov control problem of the linear-quadratic type. The state diffusion, whichrepresents the wealth of the portfolio, involves the state process and the control in a linearway while the cost involves the terminal value of the state and the square of its expectationdue to the variance criterion. In Andersson and Djehiche (2011), the authors solved themean-variance problem as a McKean-Vlasov control problem by deriving a version of thePontryagin maximum principle. More recently, Pham and Wei (2017) have developed ageneral dynamic programming approach for the control of MKV dynamics and applied itfor the resolution of the mean-variance portfolio selection problem. In Fischer and Livieri(2016), the mean-variance problem is viewed as the MKV limit of a family of controlledmany-component weakly interacting systems. These prelimit problems are solved by stan-dard dynamic programming, and the solution to the original problem is obtained by passageto the limit.A frequent criticism addressed to the mean-variance allocation is its sensitivity to theestimation of expected returns and covariance of the stocks and the risk of a poor out-of-sample performance. Several solutions to these issues have been considered. An approachconsists in using a more sophisticated model than the Black-Scholes model, in which theparameters are stochastic or ambiguous and to take decisions under the worst-case scenarioover all conceivable models. Robust mean-variance problems have thus been considered inthe economic and engineering literature, mostly on single-period or multi- period models;see, e.g., Fabozzi et al. (2010), Pinar (2016), and Liu and Zeng (2016). In a continuous-timesetting, Ismail and Pham (2019) have developed a robust approach by studying the mean-2ariance allocation with a market model where the model uncertainty affects the covariancematrix of multiple risky assets. In Guo et al. (2020), the authors study the problem ofutility maximization under uncertain parameters in a model where the parameters of themodel do not evolve freely within a given range, but are constrained via a penalty function.Let us also mention uncertain volatility models in Matoussi et al. (2012) and Lin and Riedel(2014) for robust portfolio optimization with expected utility criterion. Another approachis to rely on the shrinking of the portfolio weights or of the wealth invested in each riskyasset in order to obtain a more sparse or more stable portfolio. In DeMiguel et al. (2009),the authors find single-period portfolios that perform well out-of-sample in the presence ofestimation error. Their framework deals with the resolution of the traditional minimum-variance problem with the additional constraint that the norm of the portfolio-weight vectormust be smaller than a given threshold. In Ho et al. (2015), the authors study a one-periodmean-variance problem in which the mean-variance objective function is regularized witha weighted elastic net penalty. They show that the use of this penalty can be justified bya robust reformulation of the mean-variance criterion that directly accounts for parameteruncertainty. In the same spirit, in Chen et al. (2013), l p -norm regularized models are usedto seek near-optimal sparse portfolios.In this paper, we investigate the mean-variance portfolio selection in continuous timewith a tracking error penalization. This penalization represents the distance between theoptimized portfolio composition and the composition of a reference portfolio with the samewealth but fixed weights that have been chosen in advance. Typical reference portfolioswidely used in the financial industry are the equal weights, the minimum variance andthe equal risk contribution (ERC) portfolios. The equal weights portfolio studied, e.g. inDuchin and Levy (2009), is a portfolio where all the wealth of the investor is invested in riskyassets and divided equally between the different assets. The minimum variance portfolio is aportfolio where all the wealth is invested in risky assets and portfolio weights are optimizedin order to attain the minimal portfolio volatility. The ERC portfolio, presented in Maillardet al. (2010) and in the monography Roncalli (2013), is totally invested in risky assets andoptimized such that the contributions of each asset to the total volatility of the portfolio areequal. The mix of the mean-variance and of this tracking error criterion can be interpretedin two different ways: (i) From a first viewpoint, it is a procedure to regularize and robustifythe mean-variance allocation. By choosing reference portfolio weights which are not basedon the estimation of market parameters, or which are less sensible to estimation error,the allocation obtained is more robust to parameters estimation error than the standardmean-variance one. (ii) From a second viewpoint, this optimization permits to mimic anallocation corresponding to the reference portfolio weights while improving its Sharpe ratiovia the consideration of the mean-variance criterion.We tackle this problem as a McKean-Vlasov linear-quadratic control problem and adoptthe approach developed in Basei and Pham (2019), where the authors give a general methodto solve this type of problems by means of a weak martingale optimality principle. Weobtain explicit solutions for the optimal portfolio strategy and value function, and provide3symptotic expansions of the portfolio strategy and efficient frontier for small values ofthe portfolio tracking error penalization parameter. We then compare the Sharpe ratiosobtained by the standard mean-variance portfolio, the penalized one and the reference port-folio in two different ways. First, we compare these performances on simulated market datawith misspecified market parameters. Different magnitudes of parameter misspecificationsare used to illustrate the impact of the parameter estimation error on the performance ofthe different portfolios. In a second time, we compare the performances of these portfolioson a backtest based on historical market data. In these tests, we shall consider three ref-erence portfolios cited above: the equal weights, the minimum variance and the equal riskcontribution (ERC) portfolios. Finally, we will also consider the case where the referenceportfolio weights are all equal to zero. This case corresponds to a shrinking of the wealthinvested in the different risky assets along the investment horizon.The rest of the paper is organized as follows. Section 2 formulates the mean-varianceproblem with tracking error. In Section 3 we derive explicit solutions for this control prob-lem and provide expansion of this solution for small values of the tracking error penalizationparameter. Section 4 is devoted to the applications of those results and to the comparisonof the mean-variance, penalized and reference portfolio for the different reference port-folios presented above. We show the benefit of the penalized portfolio compared to thestandard mean-variance portfolio and the different reference portfolios on simulated andhistorical data in terms of Sharpe ratio and the lower sensitivity of the penalized portfolioto parameter estimation error. Throughout this paper, we fix a finite horizon T ∈ (0 , ∞ ), and a complete probabilityspace (cid:0) Ω , F , P , F = {F t } ≤ t ≤ T (cid:1) on which a standard F -adapted d -dimensional Brownianmotion W = ( W , ..., W d ) is defined. We denote by L F (0 , T ; R d ) the set of all R d -valued,measurable stochastic processes ( f t ) t ∈ [0 ,T ] adapted to F such that E (cid:2) (cid:82) T | f t | dt (cid:3) < ∞ . Weconsider a financial market with price process P := ( P t ) t ∈ [0 ,T ] , composed of one risk-freeasset, assumed to be constant equal to one, i.e., P ≡
1, and d risky assets on a finiteinvestment horizon [0 , T ]. These assets price processes P it , i = 1 , ..., d satisfy the followingstochastic differential equation: (cid:40) dP it = P it (cid:16) b i dt + (cid:80) nj =1 σ ij dW jt (cid:17) , t ∈ [0 , T ] P i > b i > σ := ( σ ij ) i,j =1 ,...,d ∈ R d × d is the volatility matrixof the d stocks. We denote by Σ := σσ (cid:62) the covariance matrix. Throughout this paper,we will assume that the following nondegeneracy condition holdsΣ ≥ δ I d , δ >
0, where I d is the d × d identity matrix.Let us consider an investor with total wealth at time t ≥ X t , startingfrom some initial capital x >
0. It is assumed that the trading of shares takes placecontinuously and transaction cost and consumptions are not considered. We define the setof admissible portfolio strategies α = ( α , . . . , α d ) as A := (cid:26) α : Ω × [0 , T ] → R d s . t α is F − adapted and (cid:90) T E [ | α t | ] dt < ∞ (cid:27) , where α it , i = 1 , ..., d represents the total market value of the investor’s wealth investedin the i th asset at time t . The dynamics of the self-financed wealth process X = X α associated to a portfolio strategy α ∈ A is then driven by dX t = α (cid:62) t b dt + α (cid:62) t σdW t . (2.1)Given a risk aversion parameter µ >
0, and a reference weight w r ∈ R d , the objective ofthe investor is to minimize over admissible portfolio strategies a mean-variance functionalto which is added a running cost: J ( α ) = µ Var( X T ) − E [ X T ] + E (cid:104) (cid:90) T ( α t − w r X t ) (cid:62) Γ ( α t − w r X t ) dt (cid:105) . (2.2)This running cost represents a running tracking error between the portfolio composition α t of the investor and the reference composition w r X t of a portfolio of same wealth X t and constant weights w r . The matrix Γ ∈ R d × d is symmetric positive definite and is usedto introduce an anisotropy in the portfolio composition penalization. The penalization (cid:82) T ( α t − w r X t ) (cid:62) Γ ( α t − w r X t ), which we will call “tracking error penalization”, is intro-duced in order to ensure that the portfolio of the investor does not move away too muchfrom this reference portfolio with respect to the distance | M | := M (cid:62) Γ M, M ∈ R d .The mean-variance portfolio selection with tracking error is then formulated as V := inf α ∈A J ( α ) , (2.3)and an optimal allocation given the cost J ( α ) will be given by α ∗ t ∈ arg min α ∈A J ( α ) . We complete this section by recalling the solution to the mean-variance problem whenthere is no tracking error running cost, and which will serve later as benchmark for com-parison when studying the effect of the tracking error with several reference portfolios.
Remark 2.1 ( Case of no tracking error ) . When
Γ = 0 , it is known, see e.g. Zhou andLi (2000) that the optimal mean-variance strategy is given by α ∗ t = Σ − b (cid:20) µ e b (cid:62) Σ − b T + x − X ∗ t (cid:21) , ≤ t ≤ T, (2.4)5 here X ∗ t is the wealth process associated to α ∗ . The vector Σ − b , which depends only onthe model parameters of the risky assets, determines the allocation in the risky assets. In the sequel, we study the quantitative impact of the tracking error running cost onthe optimal mean-variance strategy.
Our main theoretical result provides an analytic characterization of the optimal control tothe mean-variance problem with tracking error.
Theorem 3.1.
There exist a unique pair ( K, Λ) ∈ C (cid:0) [0 , T ] , R ∗ + (cid:1) × C ([0 , T ] , R + ) solutionto the system of ODEs dK t = (cid:110) ( K t b − Γ w r ) (cid:62) S − t ( K t b − Γ w r ) − w (cid:62) r Γ w r (cid:111) dt, K T = µd Λ t = (cid:110) (Λ t b − Γ w r ) (cid:62) S − t (Λ t b − Γ w r ) − w (cid:62) r Γ w r (cid:111) dt, Λ T = 0 (3.1) where S t := K t Σ + Γ . The optimal control for problem (2.3) is then given by α Γ t = S − t Γ w r X t − S − t b (cid:2) K t X t + Y t − ( K t − Λ t ) E [ X t ] (cid:3) , (3.2) with Y t = − e − (cid:82) Tt b (cid:62) S − s (Λ s b − Γ w r ) ds R t = 12 (cid:90) Tt b (cid:62) S − s b e − (cid:82) Ts b (cid:62) S − u (Λ u b − Γ w r ) du ds, and X = X α Γ is the wealth process associated to α Γ . Moreover, we have V = J ( α Γ ) = Λ X + 2 Y X + R . Proof.
Given the existence of a pair ( K, Λ) ∈ C (cid:0) [0 , T ] , R ∗ + (cid:1) × C ([0 , T ] , R + ) solution to (3.1),the optimality of the control process in (3.2) follows by the weak version of the martingaleoptimality principle as developed in Basei and Pham (2019). The arguments are recalledin appendix A.1.Here, let us verify the existence and uniqueness of a solution to the system (3.1).(i) We first consider the equation for K , which is a scalar Riccati equation. The equationfor K is associated to the standard linear-quadratic stochastic control problem:˜ v ( t, x ) := inf α ∈A E (cid:20)(cid:90) Tt (cid:16) w (cid:62) r Γ w r ( ˜ X t,x,αs ) − α (cid:62) s Γ w r ˜ X t,x,αs + α (cid:62) s Γ α s (cid:17) ds (cid:21) X t,x,αs is the controlled linear dynamics solution to d ˜ X s = α (cid:62) s b ds + α (cid:62) s σdW s , t ≤ s ≤ T, ˜ X t = x. By a standard result in control theory (Yong and Zhou, 1999, Ch. 6, Thm. 6.1,7.1, 7.2), there exists a unique solution K ∈ C ([0 , T ] , R + ) to the first equation ofsystem (3.1) (more, K ∈ C ([0 , T ] , R ∗ + ) if w r is nonzero). In this case, we have˜ v ( t, x ) = x (cid:62) K t x .(ii) Given K , we consider the equation for Λ. This is also a scalar Riccati equation.By the same arguments as for the K equation, there exists a unique solution Λ ∈ C ([0 , T ] , R + ) to the second equation of (3.1), provided thatΛ T ≥ , w (cid:62) r Γ w r − w (cid:62) r Γ ( K t Σ + Γ) − Γ w r ≥ , K t Σ + Γ ≥ δ I d , ≤ t ≤ T for some δ >
0. We already have that Λ T = 0. From the fact that K >
0, togetherwith the nondegeneracy condition on the matrix Σ, we have that K t Σ + Γ ≥ Γ ≥ δ I d .Since Γ >
0, and under the nondegeneracy condition of matrix Σ, we can use theWoodbury matrix identity to obtain( K t Σ + Γ) − = Γ − − Γ − (cid:18) Γ − + Σ − K t (cid:19) − Γ − . We then get w (cid:62) r Γ w r − w (cid:62) r Γ ( K t Σ + Γ) − Γ w r = w (cid:62) r (cid:18) Γ − + Σ − K t (cid:19) − w r ≥ . (iii) Given ( K, Λ), the equation for Y is a linear ODE, whose unique continuous solutionis explicitly given by Y t = − e − (cid:82) Tt b (cid:62) S − s (Λ s b − Γ w r ) ds . (iv) Given ( K, Λ , Y ), R can be directly integrated into R t = 12 (cid:90) Tt b (cid:62) S − s b e − (cid:82) Ts b (cid:62) S − u (Λ u b − Γ w r ) du ds. (cid:4) We can see from the expression of the optimal control (3.2) that the allocation in therisky assets has two components. One component is determined by the vector S − t Γ w r =( K t Σ + Γ) − Γ w r with leverage X t , and the second one by the vector S − t b = ( K t Σ + Γ) − b K t X t + Y t − ( K t − Λ t ) E [ X t ]]. Computing the average wealth X = E [ X ]associated to α Γ , we can express the control α Γ as a function of the initial wealth of theinvestor x and the current wealth X t α Γ t = S − t Γ w r X t − Λ t S − t b (cid:18) X C ,t + 12 H t (cid:19) (3.3)+ S − t b (cid:20) K t (cid:18) X C ,t + 12 H t − X t (cid:19) − Y t (cid:21) where we set C s,t := e − (cid:82) ts b (cid:62) S − u (Λ u b − Γ w r ) du and H t := C t,T (cid:82) t C s,t b (cid:62) S − s b ds . Remark 3.2.
In the case when Γ is the null matrix, Γ = , we see that the first componentof the optimal control (3.3) vanishes, Y t = − , R t = 14 µ (cid:16) − e b (cid:62) Σ − b ( T − t ) (cid:17) , and the system of ODES (3.1) of ( K, Λ) becomes dK t = K t b (cid:62) Σ − b dt, K T = µd Λ t = Λ t K t b (cid:62) Σ − b dt, Λ T = 0 , which yields the explicit forms K t = µe − b (cid:62) Σ − b ( T − t ) , Λ t = 0 . We get S − t = Σ − K t = Σ − e b (cid:62) Σ − b ( T − t ) µ , C · , · = 1 and H t = µ (cid:82) t b (cid:62) Σ − b e b (cid:62) Σ − b ( T − s ) ds . Thefirst line of the optimal control α Γ equation vanishes and the second line can be rewrittenas α Γ t = Σ − b (cid:20) µ (cid:18) e b (cid:62) Σ − b ( T − t ) + (cid:90) t b (cid:62) Σ − b e b (cid:62) Σ − b ( T − s ) ds (cid:19) + X − X t (cid:21) . Computing the integral in this expression, we recover the optimal control of the classicalmean-variance problem (2.4) . Remark 3.3 ( Limit of α γt for Γ = γ I d → ∞ ) . If we consider Γ in the form Γ = γ I d , theoptimal control can be rewritten as α γt = (cid:18) I d + K t γ Σ (cid:19) − w r X t − γ (cid:18) I d + K t γ Σ (cid:19) − b (cid:2) K t X t + Y t − ( K t − Λ t ) X t (cid:3) . (3.4)8 e show in appendix A.4 that K t and Λ t are bounded functions of the penalization param-eter γ , thus K t γ , Λ t γ −→ γ →∞ .We rewrite Y t as Y t = − e b (cid:62) w r ( T − t ) e − (cid:82) Tt γ b (cid:62) (cid:16) I d + Ksγ Σ (cid:17) − (Λ s b + K s Σ w r ) ds and we get that Y t −→ γ →∞ − e b (cid:62) w r ( T − t ) . Thus the second term of (3.4) vanishes and we get α γt −→ γ →∞ w r X t which corresponds to the reference portfolio. Remark 3.4 ( Expansion for
Γ = γ I d → . We take
Γ = γ I d . Since the covariancematrix Σ is symmetric, there exists an invertible matrix Q ∈ R d × d and a diagonal matrix D ∈ R d × d such that Σ = Q · D · Q − . We can then rewrite the matrix S − t := ( K t Σ + γ I d ) − as S − t = Q · ( K t D + γ I d ) − Q − with (cid:16) ( K t D + γ I d ) − (cid:17) ij = (cid:40) K t d i + γ if i = j if i (cid:54) = j where d i is the i -th diagonal value of the diagonal matrix D . From the nondegeneracycondition of the covariance matrix, we have d i > , ∀ i ∈ (cid:74) , n (cid:75) . As γ −→ , we wantto write the Taylor expansion of the diagonal elements of the inverse matrix ( D + γ I d ) − equal to K t d i (cid:16) γK t d i (cid:17) − . We have that K t −→ γ → µe − ρ ( T − t ) , thus γK t −→ γ → . We can thenwrite the Taylor expansion of the matrix S − t as S − t = Σ − K t − γ (cid:0) Σ − (cid:1) K t + O ( γ ) keeping only the terms up to the linear term in γ .Putting this expression in the differential equation of K , and keeping only the terms up tothe linear term in γ , we get the differential equation dK t dt = K t ρ − γ (cid:107) w r + Σ − b (cid:107) + O ( γ ) , (3.5) where we set ρ := b (cid:62) Σ − b . We look for a solution to this equation of the form K γt = K t + γK t + O ( γ ) . utting this expression in the differential equation (3.5) , we get two differential equations,for the leading order and the linear order in γ respectively (cid:40) dK t dt = K t ρ, K T = µ dK t dt = K t ρ − (cid:107) w r + Σ − b (cid:107) , K T = 0 which yield the explicit solution K γt = K t + γ (cid:107) w r + Σ − b (cid:107) − e − ρ ( T − t ) ρ + O ( γ ) where K t = µe − ρ ( T − t ) is the solution to the differential equation in the unpenalized case.From the expansion for K , we can write the expansion of the differential equation for Λ upto the linear term in γ . We use the expansion K γt = 1 K t (cid:32) − γ (cid:107) w r + Σ − b (cid:107) − e − ρ ( T − t ) K t ρ (cid:33) + O ( γ ) and we get the following expansion of the differential equation of Λ d Λ t dt = Λ t K t ρ (cid:32) − γ (cid:107) w r + Σ − b (cid:107) − e − ρ ( T − t ) K t ρ (cid:33) (3.6) − γ (cid:32) t K t b (cid:62) Σ − w r − (cid:18) Λ t K t (cid:19) b (cid:62) Σ − b − (cid:107) w r (cid:107) (cid:33) + O ( γ ) . As before, we look for a solution of this differential equation of the form Λ γt = Λ t + γ Λ t + O ( γ ) . Plugging this expression into the equation (3.6) , we get the two following differential equa-tions d Λ t dt = ( Λ t ) K t ρ, Λ T = 0 d Λ t dt = 2 Λ t Λ t K t ρ − (cid:16) Λ t K t (cid:17) ρ (cid:107) w r + Σ − b (cid:107) − e − ρ ( T − t ) ρ − (cid:18) Λ t K t b (cid:62) Σ − w r + (cid:16) Λ t K t (cid:17) b (cid:62) Σ − b + (cid:107) w r (cid:107) (cid:19) , Λ T = 0 . The first differential equation yields the solution Λ t = 0 , ∀ t ∈ [0 , T ] . Replacing Λ t by thisvalue in the second differential equation, we get the equation d Λ t dt = −(cid:107) w r (cid:107) nd obtain the solution Λ γt = γ (cid:107) w r (cid:107) ( T − t ) + O ( γ ) . We can also compute the first order expansion of C · , · C γs,t =1 − γ (cid:90) ts ρK u (cid:18) (cid:107) w r (cid:107) ( T − u ) − b (cid:62) Σ − w r ρ (cid:19) du + O ( γ )=1 − γC s,t + O ( γ ) where we set C s,t := e ρ ( T − t ) µρ (cid:110) ρ (cid:107) w r (cid:107) ( t − s ) + (cid:16) e ρ ( t − s ) − (cid:17) (cid:16) (cid:107) w r (cid:107) ( ρT − − b (cid:62) Σ − w r (cid:17)(cid:111) , and we have Y γt = −
12 + γ C t,T . The last expansion we need to compute before rewritting the optimal control is the expansionof H t . We can rewrite H t = e ρT µ (cid:0) − e − ρt (cid:1) − γH t + O ( γ ) with H t := (cid:90) t (cid:0) C s,t + C t,T (cid:1) b (cid:62) Σ − bK s ds + (cid:90) t b (cid:62) Σ − ( K s ) (cid:0) K s I d + Σ − (cid:1) b ds. As shown in appendix A.2, we can rewrite the optimal control α γt = Σ − b α t + γ (cid:16) Σ − w r α , t − Σ − b α , t − Σ − b α , t (cid:17) + O ( γ ) (3.7) where we set Σ − := (Σ − ) , and with α t = µ e ρT + X − X t α , t = (cid:107) w r (cid:107) K t ( T − t ) (cid:16) X + e ρT µ (cid:0) − e − ρt (cid:1)(cid:17) + X C ,t + H t + K t ( K t ) + C t,T K t α , t = e ρT K t µ (cid:0) − e − ρt (cid:1) + ( K t ) α , t = X t K t . (3.8) We see that for γ = 0 , we recover the classical mean-variance optimal control. For non-zero values of γ , we see that a mix of three different portfolio allocations is obtained. Theweight of the allocation Σ − b is modified and two allocations Σ − b and Σ − w r appear withweights γα , t and γα , t . rom this expansion of the control α γ , we can compute the first order asymptotic ex-pansion in γ of the equation giving the relation between the variance of the terminal wealthof the portfolio and its expectation. In the classical mean-variance case, this equation iscalled the efficient frontier formula. As shown in appendix A.3, with the tracking errorpenalization, the first order asymptotic expansion in γ gives V ar ( X T ) = e − ρT − e − ρT (cid:16) X T − X (cid:17) + γ (cid:26) b (cid:62) Σ − w r µ (cid:20) X T − µ e ρT (cid:18) T − − e − ρT ρ (cid:19)(cid:21) − (cid:90) T (cid:18) ρµ α , s + b (cid:62) Σ − bµ α , s (cid:19) e − ρ ( T − s ) ds (cid:27) + O ( γ ) . The leading order term corresponds to the efficient frontier equation of the classical mean-variance allocation computed in Zhou and Li (2000), and thus for γ = 0 , we recoverthis classical result. The linear term in γ contains contributions of the three perturbativeallocations. A modification of ”leverage” of the original mean-variance allocation Σ − b andtwo different allocations Σ − b and Σ − w r . In this section, we apply the results of the previous section and study the allocation obtainedby considering four different static portfolios as reference. First, we shall study theseallocations on simulated data, in the case of misspecified parameters. The misspecificationof parameters means that the market parameters used to compute the portfolio allocationsare different from the ones driving the stocks prices. This study allows us to estimatethe impact of the estimation error on the portfolio performance. In a second time, weperform a backtest and run the different portfolios on real market data. To simplify thepresentation, we will assume now that the tracking error penalization matrix is in the formΓ = γ I d with γ ∈ R ∗ + . With this simplification, we have S − t = ( K t Σ + γ I d ) − and we canrewrite the system of ODEs (3.1) and the optimal control (3.2) as dK t = (cid:110) ( K t b − γw r ) (cid:62) S − t ( K t b − γw r ) − γ ( w r ) (cid:62) w r (cid:111) dt, K T = µd Λ t = (cid:110) (Λ t b − γw r ) (cid:62) S − t (Λ t b − γw r ) − γ ( w r ) (cid:62) w r (cid:111) dt, Λ T = 012nd α γt = γS − t w r X t − Λ t S − t b (cid:18) X C ,t + 12 H t (cid:19) + S − t b (cid:20) K t (cid:18) X C ,t + 12 H t − X t (cid:19) − Y t (cid:21) where S t = K t Σ + γ I d , C s,t := e − (cid:82) ts b (cid:62) S − u (Λ u b − γw r ) du , Y t = − C t,T . We will consider three different classical allocations as reference portfolio.(i)
Equal-weights portfolio : in this classical equal-weights portfolio, the same capitalis invested in each asset, thus w ew r = 1 d e where d is the number of risky assets considered and e ∈ R d is the vector of ones.(ii) Minimum variance portfolio : the minimum variance portfolio is the portfoliowhich achieves the lowest variance while investing all its wealth in the risky assets.The weight vector of this portfolio is equal to w min-var r = Σ − ee (cid:62) Σ − e . These weights correspond to the one-period Markowitz portfolio when every assetexpected return b i is taken equal to 1. In that case, only the portfolio variance isrelevant and is minimized during the optimization process.(iii) ERC portfolio : the equal risk contributions (ERC) portfolio, presented in Maillardet al. (2010) and in the monograph Roncalli (2013) is constructed by choosing a riskmeasure and computing the risk contribution of each asset to the global risk of theportfolio. When the portfolio volatility is chosen as the risk measure, the principle ofthe ERC portfolio lays in the fact that the volatility function satisfies the hypothesisof Euler’s theorem and can be reduced to the sum of its arguments multiplied bytheir first partial derivatives. The portfolio volatility σ ( w ) = √ w (cid:62) Σ w of a portfoliowith weights vector w ∈ R d can then be rewritten as σ ( w ) = d (cid:88) i =1 w i ∂ i σ ( w ) = d (cid:88) i =1 w i (Σ w ) i σ ( w ) . The term under the sum w i (Σ w ) i σ ( w ) , corresponding to the i -th asset, can be interpretedas the contribution of this risky asset to the total portfolio volatility. The equal risk13ontribution allocation is then defined as the allocation in which these contributionsare equal for all the risky assets of the portfolio, w i (Σ w ) i σ ( w ) = w j (Σ w ) j σ ( w ) for every i, j ∈ (cid:74) , d (cid:75) . The equal risk contribution allocation is thus obtained when the portfolioweights w ∗ are given by w ∗ = (cid:40) w ∈ [0 , d : d (cid:88) i =1 w i = 1 , w i (Σ w ) i = w j (Σ w ) j , ∀ i, j ∈ (cid:74) , d (cid:75) (cid:41) . With this risk measure, the ERC portfolio weights can be expressed in a closed-formonly in the case where the correlations between every couple of stocks are equal, thatis corr( P i , P j ) = c, ∀ i, j ∈ (cid:74) , d (cid:75) , with the additional assumption that c ≥ − d − .Under these assumptions, and with the constaint that (cid:80) di =1 ( w erc r ) i = 1, the weightsof this portfolio are equal to ( w erc r ) i = σ − i (cid:80) dj =1 σ − j where σ i is the volatility of the i -th asset.In the general case, the weights of the ERC portfolio do not have a closed form andmust be computed numerically by solving the following optimization problem w erc r = argmin w ∈ R d d (cid:88) i =1 d (cid:88) j =1 (cid:16) w i (Σ w ) i − w j (Σ w ) j (cid:17) s.t e (cid:62) w = 1 and 0 ≤ w i ≤ , ∀ i ∈ (cid:74) , d (cid:75) . (iv) Control shrinking (zero portfolio) : this is the portfolio where all weights areequal to zero, w ir = 0 for all i . This case corresponds to a shrinking of the controls ofthe penalized allocation, in the same spirit as the shrinking of regression coefficientsin the Ridge regression (or Tikhonov regularization). In this section we compare, for each reference portfolio, the classical dynamic mean-varianceallocation, the reference portfolio and the “tracking error” penalized portfolio. In a realinvestment situation, expected return and covariance estimates are noisy and biased. Thus,in order to compare the three portfolios and observe the impact of adding a tracking errorpenalization in the mean-variance allocation, we will run Monte Carlo simulations, assum-ing that the real-world expected returns b real and covariances σ real are equal to reference14xpected returns b and covariances σ plus some noise: b = . . . . , v = . . . . , C = . . − .
05 0 . .
05 1 . − .
03 0 . − . − .
03 1 . − . .
10 0 . − .
13 1 . , with the volatilties v and correlations C and b real = b + (cid:15) × noise , σ real = σ + (cid:15) × noisewhere the covariance matrix σ is obtained from v and C . The noise follows a standardnormal distribution N (0 ,
1) and (cid:15) is its magnitude. We use Monte Carlo simulations toestimate the expected Sharpe ratio of each portfolio, equal to the average of the portfoliodaily returns R divided by the standard deviation of those returns: E (cid:2) E [ R ]Stdev( R ) (cid:3) .We consider an investment horizon of one year, with 252 business days and a dailyrebalancing of the portfolio. The risk aversion parameter µ is chosen so that the targetedannual return of the classical mean-variance allocation is equal to 20%, thus µ = e b (cid:62) Σ − b x ∗ . according to Zhou and Li (2000). The initial wealth of the investor x is chosen equal to 1and we choose the penalization parameter γ = µ/ µ depends onthe value of the stocks expected return and covariance matrix and on the targeted return,and can be very big, we express γ a function of this µ in order for the penalization to berelevant and non-negligible.For each reference portfolio, we compare the reference portfolio, the classical mean-variance allocation and the penalized one for values of noise amplitude (cid:15) ranging from 0 to1. For each value of (cid:15) , we run 2000 scenarios and we plot the graphs of the average Sharperatio as a function of (cid:15) .On the following graphs, we can see that in the four cases, the mean-variance and thepenalized portfolios are superior to the reference. In the case where the equal weightsportfolio is chosen as reference, the penalized portfolio’s Sharpe ratio is lower than themean-variance one for small values of (cid:15) . For (cid:15) greater than approximately 0.25, the penal-ized portfolio’s Sharpe ratio becomes larger and the gap with the mean-variance’s Sharpetends to increase with (cid:15) . The same phenomenon occurs in the case where the ERC port-folio is chosen as reference, with a smaller gap between the mean-variance and penalizedportfolios’ Sharpe ratios. When the minimum variance portfolio is chosen as reference, thepenalized portfolio’s Sharpe ratio is lower than the one of the mean-variance portfolio forall (cid:15) in the interval [0 , (cid:15) in the interval[0 , Equal-weights reference portfolio A v e r a g e S h a r p e r a t i o Sharpe mean-varianceSharpe penalized, = /100
Figure 1: The highest average Sharpe ratio attained by the equal-weight portfolio is equalto 0.047 for (cid:15) = 0. • Minimum-variance reference portfolio A v e r a g e S h a r p e r a t i o Sharpe mean-varianceSharpe penalized, = /100
Figure 2: The highest average Sharpe ratio attained by the minimum-variance portfolio isequal to 0.057 for (cid:15) = 0. 16
ERC reference portfolio A v e r a g e S h a r p e r a t i o Sharpe mean-varianceSharpe penalized, = /100
Figure 3: The highest average Sharpe ratio attained by the ERC portfolio is equal to 0.051for (cid:15) = 0. • Control shrinking (zero reference) A v e r a g e S h a r p e r a t i o Sharpe mean-varianceSharpe penalized, = /100
Figure 4: In this case the reference weights are equal to zero, and no Sharpe ratio iscomputed for the reference portfolio. 17 .2 Performance comparison on a backtest
We now compare the different allocations on a backtest based on adjusted close daily pricesavailable on Quandl between 2013-09-03 and 2017-12-28 for four stocks: Apple, Microsoft,Boeing and Nike. Here we chose a value of µ which corresponds to an annual expectedreturn of 25%. In our example, we express again γ as a function of µ and we consider twodifferent values, γ = µ and γ = µ/ µ e b (cid:62) Σ − b T + x in the mean-variance control equation (2.4), its leverage decreases and its wealth curve flattens. Thesame phenomenon occurs for the penalized allocation with large penalization parameter γ = µ . In this case, the high value of the penalization parameter keeps the penalizedportfolio controls close to the ones of the mean-variance portfolio. On the contrary, thereference portfolios have constant weights and no target wealth. We can see that in eachcase the reference portfolio’s wealth keeps increasing over the entire horizon. The wealth ofthe penalized portfolio with penalization parameter γ = µ/
100 follows the wealth of thesereference portfolio due to the small value of the tracking error penalization.For these three reference portfolios, we observe that the penalized portfolio with pe-nalization parameter γ = µ outperforms both the mean-variance and the reference portfo-lios in terms of Sharpe ratio whereas the penalized portfolio with penalization parameter γ = µ/
100 outperforms the mean-variance but underperforms the reference portfolio. Thiscan be attributed to the larger weight of the mean-variance criterion with respect to thetracking error in the optimized cost (2.2) with penalization parameter γ = µ .Finally, Figure 8 corresponds to the case of a reference portfolio with weights all equalto zero. This corresponds to a shrinking of the optimal control of the penalized portfolio.In that case, for a better visualization, we plot the total wealth of the mean-varianceand penalized portfolios for penalization parameters γ = µ and γ = µ/
100 normalizedby the standard deviation of their daily returns. On this graph, we can see that thenormalized wealth of the two penalized portfolio is higher than the one of the mean-variance allocation. Similarly to the three precedent reference portfolios, the two penalizedportfolios outperform the mean-variance allocation in terms of Sharpe ratio. As previously,we observe that the Sharpe ratio of the penalized portfolio with penalization parameter γ = µ is greater than the one with γ = µ/ Equal-weights reference portfolio T o t a l p o r t f o li o w e a l t h Wealth mean-varianceWealth penalized, =Wealth penalized, = /100Wealth equal weights
Figure 5: Sharpe ratios:Mean-variance : 0.183Equal weights : 0.258Penalized γ = µ : 0.260Penalized γ = µ/
100 : 0.226 • Minimum variance reference portfolio T o t a l p o r t f o li o w e a l t h Wealth mean-varianceWealth penalized, =Wealth penalized, = /100Wealth minimum variance
Figure 6: Sharpe ratios:Mean-variance : 0.183Minimum variance : 0.255Penalized γ = µ : 0.256Penalized γ = µ/
100 : 0.220 19
ERC portfolio T o t a l p o r t f o li o w e a l t h Wealth mean-varianceWealth penalized, =Wealth penalized, = /100Wealth ERC
Figure 7: Sharpe ratios:Mean-variance : 0.183ERC : 0.258Penalized γ = µ : 0.260Penalized γ = µ/
100 : 0.225 • Zero portfolio (shrinking) T o t a l p o r t f o li o w e a l t h Wealth MarkowitzWealth penalized, =Wealth penalized, = /100
Figure 8: Total wealth of the mean-variance and penalized portfolios for γ = µ and γ = µ/ γ = µ : 0.252Penalized γ = µ/
100 : 0.221 20
Conclusion
In this paper, we propose an allocation method based on a mean-variance criterion plusa tracking error between the optimized portfolio and a reference portfolio of same wealthand fixed weights. We solve this problem as a linear-quadratic McKean-Vlasov stochasticcontrol problem using a weak martingale approach. We then show using simulations thatfor a certain degree of market parameter misspecification and the right choice of referenceportfolio, the mean-variance portfolio with tracking error penalization outperforms thestandard mean-variance and the mean-variance allocations in terms of Sharpe ratio. An-other backtest based on historical market data also shows that the mean-variance portfoliowith tracking error outperforms the traditional mean-variance and the reference portfoliosin terms of Sharpe ratio for the four reference portfolios considered.
A Appendix
A.1 Proof of Theorem 3.1
The proof of Theorem 3.1 is based on the weak optimality principle lemma stated in Baseiand Pham (2019), and formulated in the case of the mean-variance problem (2.3) as:
Lemma A.1 ( Weak optimality principle ) . Let { V αt , t ∈ [0 , T ] , α ∈ A} be a family ofreal-valued processes in the form V αt = v t ( X αt , E [ X αt ]) + (cid:90) t ( α s − w r X αs ) (cid:62) Γ ( α s − w r X αs ) ds, for some measurable functions v t on R × R , t ∈ [0 , T ] , such that:(i) v T ( x, ¯ x ) = µ ( x − ¯ x ) − x , for all x, ¯ x ∈ R ,(ii) the function t ∈ [0 , T ] → E [ V αt ] is nondecreasing for all α ∈ A (iii) the map t ∈ [0 , T ] → E (cid:2) V α ∗ t (cid:3) is constant for some α ∗ ∈ A .Then, α ∗ is an optimal portfolio strategy for the mean-variance problem with tracking error (2.3) , and V = J ( α ∗ ) . We aim to construct a family of processes { V αt , t ∈ [0 , T ] , α ∈ A} as in Lemma (A.1),and given the linear-quadratic structure of our optimization problem, we look for a mea-surable function v t in the form: v t ( x, x ) = K t ( x − x ) + Λ t x + 2 Y t x + R t (A.1)21or some deterministic processes ( K t , Λ t , Y t , R t ) to be determined. Condition ( i ) in Lemma(A.1) fixes the terminal condition K T = µ, Λ T = 0 , Y T = − / , R T = 0 . For any α ∈ A , with associated wealth process X := X α , let us compute the derivative ofthe deterministic function t → E [ V αt ] = E (cid:104) v t ( X t , E [ X t ]) + (cid:82) t ( α s − w r X s ) (cid:62) Γ ( α s − w r X s ) ds (cid:105) with v t as in (A.1). From the dynamics of X = X αt in (2.1) and by applying It’s formula,we obtain d E [ V αt ] dt =Var( X t ) (cid:16) ˙ K t + w (cid:62) r Γ w r (cid:17) + X t (cid:16) ˙Λ t + w (cid:62) r Γ w r (cid:17) + 2 X t ˙ Y t + ˙ R t (A.2)+ E [ G t ( α )]where G t ( α ) := α (cid:62) t S t α t + 2 (cid:110)(cid:0) K t ( X t − X t ) + Y t + Λ t X t (cid:1) b (cid:62) − X t w (cid:62) r Γ (cid:111) α t . By completing the square in α , and setting S t := K t Σ + Γ and ˜ ρ t := b (cid:62) S − t b , we rewrite G t ( α ) as G t ( α ) = E (cid:104)(cid:0) α t − α Γ t (cid:1) (cid:62) S t (cid:0) α t − α Γ t (cid:1)(cid:105) − Var( X t ) (cid:110) K t ˜ ρ t + w (cid:62) r Γ S − t Γ w r − K t b (cid:62) S − t Γ w r (cid:111) − X t (cid:110) Λ t ˜ ρ t + w (cid:62) r Γ S − t Γ w r − t b (cid:62) S − t Γ w r (cid:111) − X t (cid:110) Λ t Y t ˜ ρ t − Y t b (cid:62) S − t Γ w r (cid:111) − Y t ρ t with α Γ t := S − t Γ w r X t − S − t b (cid:2) K t X t + Y t − ( K t − Λ t ) X t (cid:3) . The expression in (A.2) is thenrewritten as d E [ V αt ] dt = E (cid:104)(cid:0) α t − α Γ t (cid:1) (cid:62) S t (cid:0) α t − α Γ t (cid:1)(cid:105) + Var( X t ) (cid:110) ˙ K t − K t ˜ ρ t + w (cid:62) r Γ w r + 2 K t b (cid:62) S − t Γ w r − w (cid:62) r Γ S − t Γ w r (cid:111) + X t (cid:110) ˙Λ t − Λ t ˜ ρ t + w (cid:62) r Γ w r + 2Λ t b (cid:62) S − t Γ w r − w (cid:62) r Γ S − t Γ w r (cid:111) + 2 X t (cid:16) ˙ Y t + Y t b (cid:62) S − t Γ w r − Λ t Y t ˜ ρ t (cid:17) + ˙ R t − Y t ˜ ρ t . Therefore, whenever 22 ˙ K t − K t ˜ ρ t + w (cid:62) r Γ w r + 2 K t b (cid:62) S − t Γ w r − w (cid:62) r Γ S − t Γ w r = 0˙Λ t − Λ t ˜ ρ t + w (cid:62) r Γ w r + 2Λ t b (cid:62) S − t Γ w r − w (cid:62) r Γ S − t Γ w r = 0˙ Y t + Y t b (cid:62) S − t Γ w r − Λ t Y t ˜ ρ t = 0˙ R t − Y t ˜ ρ t = 0holds for all t ∈ [0 , T ], we have d E [ V αt ] dt = E (cid:104)(cid:0) α t − α Γ t (cid:1) (cid:62) S t (cid:0) α t − α Γ t (cid:1)(cid:105) which is nonnegative for all α ∈ A , i.e., the process V αt satisfies the condition ( ii ) of Lemma(A.1). Moreover, we see that V αt = 0 , ≤ t ≤ T if and only if α t = α Γ t , 0 ≤ t ≤ T . X Γ := X α Γ is solution to a linear McKean-Vlasov dynamics and, since K ∈ C ([0 , T ] , R ∗ + ),Λ ∈ C ([0 , T ] , R + ) and Y ∈ C ([0 , T ] , R ), X Γ satisfies the square integrability condition E (cid:34) sup ≤ t ≤ T | X Γ t | (cid:35) < ∞ , which implies that α Γ is F -progressively measurable and (cid:82) T E [ | α Γ t | ] dt < ∞ . Therefore, α Γ ∈ A , and we conclude by the verification lemma A.1 that it is the uniqueoptimal control. (cid:4) .2 Computation linear expansion of α γ for Γ = γ I d → α γt =Σ − b (cid:18) µ e ρT + X − X t (cid:19) + γ (cid:0) Σ − w r + Σ − b (cid:1) X t K t − γ (cid:107) w r (cid:107) ( T − t ) Σ − K t b (cid:18) X + e ρT µ (cid:0) − e − ρt (cid:1)(cid:19) − γ (cid:18) Σ − bX C ,t + Σ − b X K t (cid:19) − γ (cid:18) Σ − b H t − b e ρT K t µ (cid:0) − e − ρt (cid:1)(cid:19) − γ (cid:40) (cid:0) K t (cid:1) Σ − (cid:0) K t + Σ − (cid:1) b + Σ − b C t,T K t (cid:41) + O ( γ )=Σ − b (cid:18) µ e ρT + X − X t (cid:19) + γ Σ − w r X t K t − γ Σ − b (cid:40) (cid:107) w r (cid:107) K t ( T − t ) (cid:18) X + e ρT µ (cid:0) − e − ρt (cid:1)(cid:19) + X C ,t + H t K t (cid:0) K t (cid:1) + C t,T K t (cid:41) − γ Σ − b (cid:40) e ρT K t µ (cid:0) − e − ρt (cid:1) + 12 (cid:0) K t (cid:1) (cid:41) + O ( γ ) . A.3 Computation linear expansion of
V ar ( X T ) for Γ = γ I d → We recall that the linear expansion of the optimal control can be written as α γt = Σ − b α t + γ (cid:16) Σ − w r α , t − Σ − b α , t − Σ − b α , t (cid:17) + O ( γ )where the coefficients α , t , α , t and α , t are given by (3.8). The average total wealth ofthe portfolio constructed by the optimal control at time t is given by the ODE dX t = ρζ − γ (cid:16) ρα , t + b (cid:62) Σ − bα , t (cid:17) + (cid:18) γ b (cid:62) Σ − w r K t − ρ (cid:19) X t + O ( γ ) , X t = X , ζ := X + µ e ρT . We get the solution X T = X e − ρT + ζ (cid:0) − e − ρT (cid:1) + γ (cid:26) b (cid:62) Σ − w r µ (cid:18) T ζ − µ e ρT − e − ρT ρ (cid:19) − (cid:90) T (cid:16) ρα , s + b (cid:62) Σ − bα , s (cid:17) e − ρ ( T − s ) ds (cid:27) + O (cid:0) γ (cid:1) = X T + γX T + O ( γ )with (cid:40) X T := X e − ρT + ζ (cid:0) − e − ρT (cid:1) X T := b (cid:62) Σ − w r µ (cid:16) T ζ − µ e ρT − e − ρT ρ (cid:17) − (cid:82) T (cid:16) ρα , s + b (cid:62) Σ − bα , s (cid:17) e − ρ ( T − s ) ds and X T = (cid:16) X T (cid:17) + 2 γX T X T + O ( γ )The average of the square of the portfolio wealth at time t is given by the ODE dX t = (cid:16) ζ − γα , t (cid:17) ρ − γb (cid:62) Σ − bα , t (cid:16) ζ − γα , t (cid:17) + 2 γ w (cid:62) r Σ − bK t (cid:16) ζ − γα , t (cid:17) X t − ρX t + O ( γ )which gives the solution X T = X e − ρT + ζ (cid:0) − e − ρT (cid:1) − γζ (cid:90) T (cid:110) ρα , s + b (cid:62) Σ − bα , s (cid:111) e − ρ ( T − s ) ds + 2 γ w (cid:62) r Σ − bµ ζ (cid:90) T X s ds + O ( γ ) . We can then compute the variance of the terminal total wealth of the portfolio given bythe control (3.7)
V ar ( X T ) = X T − X t = e − ρT − e − ρT (cid:16) X T − X (cid:17) + γ b (cid:62) Σ − w r µ (cid:18) ζT − µ e ρT − e − ρT ρ (cid:19) − γ (cid:90) T (cid:18) ρµ α , s + b (cid:62) Σ − bµ α , s (cid:19) e − ρ ( T − s ) ds + O ( γ ) . (cid:4) .4 Proof that K t and Λ t are bounded in γ To prove this, we use a theorem from Gronwall (1919) (also in Hairer et al. (1993), Theorem14.1, p93). We rewrite the differential equation of K as dK t dt = f ( t, K t , γ ) , K T = µ with f ( t, K t , γ ) := ( K t b − γw r ) (cid:62) ( K t Σ + γ ) − ( K t b − γw r ) − γ (cid:107) w r (cid:107) , where (cid:107) · (cid:107) denotesthe euclidean norm in R d .For t ∈ [0 , T ], the partial derivatives ∂f /∂K and ∂f /∂γ exist and are continuous in theneighbourhood of the solution K t . Then the partial derivative ∂K t ∂γ = ψ t exists, is continuous, and satisfies the differential equation ψ (cid:48) t = ∂f∂K ( t, K t , γ ) ψ t + ∂f∂γ ( t, K t , γ ) . Recalling that the derivative of the inverse of a nonsingular matrix M whose elements arefunctions of a scalar parameter p w.r.t this parameter is equal to ∂M − ∂ p = − M − ∂M∂ p M − ,we can compute the partial derivatives ∂f /∂K and ∂f /∂γ , and we obtain the followingdifferential equation for ψ (cid:40) ( ψ t ) (cid:48) = (cid:2) −(cid:107) σ (cid:62) S − t ( K t b − γw r ) (cid:107) + 2 b (cid:62) S − t ( K t b − γw r ) (cid:3) ψ t − (cid:107) w r + S − t ( K t b − γw r ) (cid:107) , t ∈ [0 , T ] ψ T = 0 . This ODE has an explicit solution given by ψ t = (cid:90) Tt A s e − (cid:82) st B u du ds with A t ≥ , ∀ t ∈ [0 , T ] equal to A t := K t γ (cid:107) (cid:18) I d + K t γ Σ (cid:19) − ( b + Σ w r ) (cid:107) −→ γ →∞ B t :=2 K t γ ( b + Σ w r ) (cid:62) (cid:18) I d + K t γ Σ (cid:19) − ( b + Σ w r ) − K t γ ( b + Σ w r ) (cid:62) (cid:18) I d + K t γ Σ (cid:19) − Σ (cid:18) I d + K t γ Σ (cid:19) − ( b + Σ w r ) − b (cid:62) w r − (cid:107) σ (cid:62) w r (cid:107) .
26e have B t → γ →∞ − b (cid:62) w r − (cid:107) σ (cid:62) w r (cid:107) , thus ψ t −→ γ →∞ , ∀ t ∈ [0 , T ] and K t is bounded in γ for every t ∈ [0 , T ].In the same spirit, we rewrite the differential equation of Λ t as d Λ t dt = g ( t, Λ t , γ ) , Λ t = 0with g ( t, Λ t , γ ) := (Λ t b − γw r ) (cid:62) S − t (Λ t b − γw r ) − γ (cid:107) w r (cid:107) . The partial derivative ∂ Λ t ∂γ = φ t exists, is continuous and satisfies the differential equation (cid:40) φ (cid:48) t = 2 b (cid:62) S − t (Λ t b − γw r ) φ t − (cid:2) (cid:107) w r + S − t (Λ t b − γw r ) (cid:107) + ψ t (cid:107) σ (cid:62) S − t (Λ t b − γw r ) (cid:107) (cid:3) , t ∈ [0 , T ] φ = 0 . which gives the explicit solution φ t = (cid:90) Tt C s e − (cid:82) st D u du ds with C t ≥ , ∀ t ∈ [0 , T ] equal to C t := (cid:107) γ (cid:18) I d + K t γ Σ (cid:19) − (Λ t b + K t Σ w r ) (cid:107) + ψ t (cid:107) γ σ (cid:62) (cid:18) I d + K t γ Σ (cid:19) − (Λ t b + K t Σ w r ) − σ (cid:62) w r (cid:107) and D t := 2 (cid:34) γ b (cid:62) (cid:18) I d + K t γ Σ (cid:19) − (Λ t b + K t Σ w r ) − b (cid:62) w r (cid:35) . We showed that K t γ , ψ t −→ γ →∞ t ∈ [0 , T ]. Thus C t −→ γ →∞ D t −→ γ →∞ − b (cid:62) w r and φ t −→ γ →∞ ∀ t ∈ [0 , T ]. Λ t is then bounded in γ for every t ∈ [0 , T ].27 eferences Daniel Andersson and Boualem Djehiche. A maximum principle for sdes of mean-fieldtype. Applied Mathematics & Optimization, 63(3):341–356, 2011.Matteo Basei and Huyˆen Pham. A weak martingale approach to linear-quadratic mckean–vlasov stochastic control problems. Journal of Optimization Theory and Applications,181(2):347–382, 2019.Caihua Chen, Xindan Li, Caleb Tolman, Suyang Wang, and Yinyu Ye. Sparse portfolioselection via quasi-norm regularization. arXiv preprint arXiv:1312.6350, 2013.Victor DeMiguel, Lorenzo Garlappi, Francisco J Nogales, and Raman Uppal. A generalizedapproach to portfolio optimization: Improving performance by constraining portfolionorms. Management science, 55(5):798–812, 2009.Ran Duchin and Haim Levy. Markowitz versus the talmudic portfolio diversification strate-gies. The Journal of Portfolio Management, 35(2):71–74, 2009.Frank J Fabozzi, Dashan Huang, and Guofu Zhou. Robust portfolios: contributions fromoperations research and finance. Annals of operations research, 176(1):191–220, 2010.Markus Fischer and Giulia Livieri. Continuous time mean-variance portfolio optimizationthrough the mean field approach. ESAIM: Probability and Statistics, 20:30–44, 2016.Thomas Hakon Gronwall. Note on the derivatives with respect to a parameter of thesolutions of a system of differential equations. Annals of Mathematics, pages 292–296,1919.Ivan Guo, Nicolas Langren´e, Gr´egoire Loeper, and Wei Ning. Robust utility maximizationunder model uncertainty via a penalization approach. Available at SSRN 3612503, 2020.Ernst Hairer, Syvert P Nørsett, and Gerhard Wanner. Solving ordinary differential equa-tions i. nonstiff problems, volume 8 of, 1993.Michael Ho, Zheng Sun, and Jack Xin. Weighted elastic net penalized mean-varianceportfolio design and computation. SIAM Journal on Financial Mathematics, 6(1):1220–1244, 2015.Amine Ismail and Huyˆen Pham. Robust markowitz mean-variance portfolio selection underambiguous covariance matrix. Mathematical Finance, 29(1):174–207, 2019.Qian Lin and Frank Riedel. Optimal consumption and portfolio choice with ambiguity.arXiv preprint arXiv:1401.1639, 2014.Jun Liu and Xudong Zeng. Correlation ambiguity. Available at SSRN 2692692, 2016.28´ebastien Maillard, Thierry Roncalli, and J´erˆome Teiletche. The properties of equallyweighted risk contribution portfolios. The Journal of Portfolio Management, 36(4):60–70, 2010.Harry Markowitz. Portfolio selection. The Journal of Finance, 7(1):77–91, 1952. doi:10.1111/j.1540-6261.1952.tb01525.x. URL https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1540-6261.1952.tb01525.xhttps://onlinelibrary.wiley.com/doi/abs/10.1111/j.1540-6261.1952.tb01525.x