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A closed-form solution of the multi-period portfolio choice problem for a quadratic utility function

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Abstract

In the present paper, we derive a closed-form solution of the multi-period portfolio choice problem for a quadratic utility function with and without a riskless asset. All results are derived under weak conditions on the asset returns. No assumption on the correlation structure between different time points is needed and no assumption on the distribution is imposed. All expressions are presented in terms of the conditional mean vectors and the conditional covariance matrices. If the multivariate process of the asset returns is independent, it is shown that in the case without a riskless asset the solution is presented as a sequence of optimal portfolio weights obtained by solving the single-period Markowitz optimization problem. The process dynamics are included only in the shape parameter of the utility function. If a riskless asset is present, then the multi-period optimal portfolio weights are proportional to the single-period solutions multiplied by time-varying constants which are dependent on the process dynamics. Remarkably, in the case of a portfolio selection with the tangency portfolio the multi-period solution coincides with the sequence of the single-period solutions. Finally, we compare the suggested strategies with existing multi-period portfolio allocation methods on real data.

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Notes

  1. It can be easily shown that taking the derivative of \(W_t-\dfrac{\alpha }{2}W_t^2\) with respect to \(W_t\) and setting it equal to zero leads to the maximum attainable wealth level at time point \(t\) which is equal to \(1/\alpha \).

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Acknowledgments

The authors are thankful to Professor Endre Boros, the Associate Editor, and an anonymous Reviewer for careful reading of the paper and for their suggestions which have improved an earlier version of this paper. The first and the third author appreciates the financial support of the German Science Foundation (DFG) via the projects BO 3521/3-1 and SCHM 859/13-1 ”Bayesian Estimation of the Multi-Period Optimal Portfolio Weights and Risk Measures”. We also thank David Bauder for his comments used in the preparation of the revised version of the paper.

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Correspondence to Taras Bodnar.

Appendix

Appendix

In this section the proofs of the theorems are given.

Proof of Theorem 1

First, we note that the expression of the optimal weights at period \(T-1\) is given in (11). The rest of the proof is done by using the mathematical induction on the expressions of the portfolio weights and the value function. Let

$$\begin{aligned} \varvec{A}_{i}=E_{i-1}[V_{i+1}\tilde{\mathbf {X}}_{i}\tilde{\mathbf {X}}^\prime _{i}] \quad \text {for}\quad i=1,\ldots ,T-1\quad \text {and}\quad A_T=\varvec{\varSigma }_T+\tilde{\varvec{\mu }}_T\tilde{\varvec{\mu }}_T^\prime . \end{aligned}$$
(51)

Moreover, let

$$\begin{aligned} \tilde{\varvec{\mu }}^{*}_{i}= \left\{ \begin{array}{ll} \tilde{\varvec{\mu }}_{T} &{}\quad \text {for}\quad i=T\\ E_{i-1}\left[ R_{i+1}\tilde{\mathbf {X}}_{i}\right] &{} \quad \text {for}\quad i=1,\ldots ,T-1, \end{array} \right. \end{aligned}$$
(52)

\(R_{i}=\frac{\mathbf {1}^{\prime }\varvec{A}_{i}^{-1}\tilde{\varvec{\mu }}^*_i}{\mathbf {1}^{\prime }\varvec{A}^{-1}_i\mathbf {1}}\), \(V_{i}=\frac{1}{\mathbf {1}^{\prime }\varvec{A}^{-1}_i\mathbf {1}}\) and \(\tilde{s}_i=\tilde{\varvec{\mu }}_i^{*\;\prime } \tilde{\mathbf {Q}}_i\tilde{\varvec{\mu }}^*_i\) with

$$\begin{aligned} \tilde{\mathbf {Q}}_i=\varvec{A}^{-1}_{i}-\frac{\varvec{A}^{-1}_{i}\mathbf {1}\mathbf {1}^{\prime } \varvec{A}^{-1}_{i}}{\mathbf {1}^{\prime }\varvec{A}^{-1}_{i}\mathbf {1}} \quad \text {for} \quad i=1,\ldots ,T-1. \end{aligned}$$

Note, that

$$\begin{aligned} \mathbf {1}^{\prime }\tilde{\mathbf {Q}}_{i}=\mathbf {0}^{\prime },\quad \tilde{\mathbf {Q}}_{i}\mathbf {1}=\mathbf {0},\quad \tilde{\mathbf {Q}}_{i}\varvec{A}_{i}\tilde{\mathbf {Q}}_{i}=\tilde{\mathbf {Q}}_{i}. \end{aligned}$$

Let \(\mathbf {w}^{*\;\prime }_{T-1}\) be the optimal portfolio weights calculated for period \(T-1\) given in (11). First, we calculate the value function for period \(T-2\). It holds that

$$\begin{aligned}&V(T-2,W_{T-2},\mathcal {F}_{T-2})\\&\quad =\max \limits _{\mathbf {w}_{T-2}:\mathbf {w}^{\prime }_{T-2}\mathbf {1}=1} E_{T-2}\Big [\max \limits _{\mathbf {w}_{T-1}:\mathbf {w}^{\prime }_{T-1}\mathbf {1}=1} \left( W_{T-1}\mathbf {w}^{\prime }_{T-1}\tilde{\varvec{\mu }}_{T} -\frac{\alpha }{2}W^2_{T-1}\mathbf {w}^{\prime }_{T-1}\varvec{A}_{T}\mathbf {w}_{T-1}\right) \Big ]\\&\quad =\max \limits _{\mathbf {w}_{T-2}:\mathbf {w}^{\prime }_{T-2}\mathbf {1}=1}E_{T-2} \Big [W_{T-1}\mathbf {w}^{*\;\prime }_{T-1}\tilde{\varvec{\mu }}_{T}-\frac{\alpha }{2} W^2_{T-1}\mathbf {w}^{*\;\prime }_{T-1}\varvec{A}_{T}\mathbf {w}^*_{T-1}\Big ]\\&\quad =\max \limits _{\mathbf {w}_{T-2}:\mathbf {w}^{\prime }_{T-2}\mathbf {1}=1} E_{T-2}\Big [W_{T-1}\left( \frac{\varvec{A}^{-1}_T\mathbf {1}}{\mathbf {1}^{\prime } \varvec{A}^{-1}_T\mathbf {1}}+\frac{1}{\alpha W_{T-1}}\tilde{\mathbf {Q}}_T\tilde{\varvec{\mu }}_T\right) ^{\prime } \tilde{\varvec{\mu }}_T\\&\qquad -\,\frac{\alpha }{2}W^2_{T-1}\left( \frac{\varvec{A}^{-1}_T\mathbf {1}}{\mathbf {1}^{\prime }\varvec{A}^{-1}_T\mathbf {1}}+\frac{1}{\alpha W_{T-1}} \tilde{\mathbf {Q}}_T\tilde{\varvec{\mu }}_T\right) ^{\prime }\varvec{A}_T\left( \frac{\varvec{A}^{-1}_T\mathbf {1}}{\mathbf {1}^{\prime }\varvec{A}^{-1}_T\mathbf {1}}+\frac{1}{\alpha W_{T-1}} \tilde{\mathbf {Q}}_T\tilde{\varvec{\mu }}_T\right) \Big ]\\&\quad =\max \limits _{\mathbf {w}_{T-2}:\mathbf {w}^{\prime }_{T-2}\mathbf {1}=1} E_{T-2}\Big [W_{T-1}\left( \frac{\mathbf {1}^{\prime }\varvec{A}_{T}^{-1}\tilde{\varvec{\mu }}_T}{\mathbf {1}^{\prime }\varvec{A}^{-1}_{T}\mathbf {1}} -\frac{1}{2}\underbrace{\frac{\tilde{\varvec{\mu }}_T^{\prime } \tilde{\mathbf {Q}}_T\varvec{A}_T\varvec{A}_{T}^{-1}\mathbf {1}}{\mathbf {1}^{\prime }\varvec{A}^{-1}_{T}\mathbf {1}}}_{=0} -\frac{1}{2}\underbrace{\frac{\mathbf {1}^{\prime }\varvec{A}_{T}^{-1} \varvec{A}_{T}\tilde{\mathbf {Q}}_T\tilde{\varvec{\mu }}_T}{\mathbf {1}^{\prime }\varvec{A}^{-1}_{T}\mathbf {1}}}_{=0}\right) \\&\qquad +\,\frac{1}{2\alpha }\tilde{\varvec{\mu }}_T^{\prime } \tilde{\mathbf {Q}}_T\tilde{\varvec{\mu }}_T-\frac{\alpha }{2} \frac{W^2_{T-1}}{\mathbf {1}^{\prime }\varvec{A}^{-1}_T\mathbf {1}}\Big ], \end{aligned}$$

Using the definitions of \(R_i,\,V_i\) and \(\tilde{s}_i\) we obtain

$$\begin{aligned}&V(T-2,W_{T-2},\mathcal {F}_{T-2})\\&\quad =\max \limits _{\mathbf {w}_{T-2}:\mathbf {w}^{\prime }_{T-2}\mathbf {1}=1}E_{T-2} \Big [W_{T-1}R_T+\frac{1}{2\alpha }\tilde{s}_T-\frac{\alpha }{2}W^2_{T-1}V_T\Big ]\\&\quad =\max \limits _{\mathbf {w}_{T-2}:\mathbf {w}^{\prime }_{T-2}\mathbf {1}=1}E_{T-2}\Big [W_{T-2} \mathbf {w}^{\prime }_{T-2}R_T\tilde{\mathbf {X}}_{T-1}+\frac{1}{2\alpha }\tilde{s}_T -\frac{\alpha }{2}W^2_{T-2}V_T(\mathbf {w}^{\prime }_{T-2}\tilde{\mathbf {X}}_{T-1})^2\Big ]\\&\quad =\max \limits _{\mathbf {w}_{T-2}:\mathbf {w}^{\prime }_{T-2}\mathbf {1}=1}\Big [W_{T-2}\mathbf {w}^{\prime }_{T-2} \tilde{\varvec{\mu }}^*_{T-1}+F(\tilde{s}_T) -\frac{\alpha }{2}W^2_{T-2}\left( \mathbf {w}^{\prime }_{T-2}\varvec{A}_{T-1}\mathbf {w}_{T-2}\right) \Big ], \end{aligned}$$

where

$$\begin{aligned} F(\tilde{s}_T)=\frac{1}{2\alpha }E_{T-2}[\tilde{s}_T]. \end{aligned}$$
(53)

\(F(\tilde{s}_T)\) does not depend on \(\mathbf {w}_{T-2}\).

The last expression is similar to the value function at period \(T-1\) [cf. (9)]. Hence, the optimal weights \(\mathbf {w}_{T-2}^*\) are given by

$$\begin{aligned} \mathbf {w}^*_{T-2}=\frac{\varvec{A}^{-1}_{T-1}\mathbf {1}}{\mathbf {1}^{\prime }\varvec{A}^{-1}_{T-1}\mathbf {1}}+\frac{1}{\alpha W_{T-2}}\tilde{\mathbf {Q}}_{T-1}\tilde{\varvec{\mu }}^*_{T-1}\quad \text {with}\quad \tilde{\mathbf {Q}}_{T-1}=\varvec{A}^{-1}_{T-1}- \frac{\varvec{A}^{-1}_{T-1}\mathbf {1}\mathbf {1}^{\prime } \varvec{A}^{-1}_{T-1}}{\mathbf {1}^{\prime }\varvec{A}^{-1}_{T-1}\mathbf {1}}. \end{aligned}$$
(54)

As a result, the following expressions are the basis of the induction

$$\begin{aligned} V(T-2,W_{T-2},\mathcal {F}_{T-2})&= \max \limits _{\mathbf {w}_{T-2}:\mathbf {w}^{\prime }_{T-2} \mathbf {1}=1}\Big [W_{T-2}\mathbf {w}^{\prime }_{T-2}\tilde{\varvec{\mu }}^*_{T-1}+F(\tilde{s}_T)\\&-\,\frac{\alpha }{2}W^2_{T-2}\mathbf {w}^{\prime }_{T-2}\varvec{A}_{T-1}\mathbf {w}_{T-2}\Big ]\\ \mathbf {w}^*_{T-2}&= \frac{\varvec{A}^{-1}_{T-1}\mathbf {1}}{\mathbf {1}^{\prime }\varvec{A}^{-1}_{T-1}\mathbf {1}}+\frac{1}{\alpha W_{T-2}}\mathbf {Q}_{T-1}\tilde{\varvec{\mu }}^*_{T-1} \end{aligned}$$

with \(F(\tilde{s}_T)\) as defined in (53).

In the induction hypothesis we assume that the statement holds for \(t=n\), i.e.,

$$\begin{aligned} V(T-n,W_{T-n},\mathcal {F}_{T-n})&= \max \limits _{\mathbf {w}_{T-n}:\mathbf {w}^{\prime }_{T-n}\mathbf {1}=1} \Big [W_{T-n}\mathbf {w}^{\prime }_{T-n}\tilde{\varvec{\mu }}^*_{T-n+1}\\&\quad +\,F(\tilde{s}_T,\tilde{s}_{T-1},\ldots ,\tilde{s}_{T-n+2}) -\frac{\alpha }{2}W^2_{T-n}\mathbf {w}^{\prime }_{T-n}\varvec{A}_{T-n+1}\mathbf {w}_{T-n}\Big ]\\ \mathbf {w}^*_{T-n}&= \frac{\varvec{A}^{-1}_{T-n+1}\mathbf {1}}{\mathbf {1}^{\prime }\varvec{A}^{-1}_{T-n+1}\mathbf {1}}+\frac{1}{\alpha W_{T-n}} \tilde{\mathbf {Q}}_{T-n+1}\tilde{\varvec{\mu }}^*_{T-n+1}, \end{aligned}$$

where

$$\begin{aligned} F(\tilde{s}_T,\tilde{s}_{T-1},\ldots ,\tilde{s}_{T-n+2}) =\frac{1}{2\alpha }\left( E_{T-2}[\tilde{s}_T] +\sum \limits _{m=T-n+2}^{T-1}E_{m-2}[\tilde{s}_m]\right) . \end{aligned}$$

Note that the last quantity does not depend on \(\mathbf {w}_{T-n}\).

In the inductive step we prove that the last identities also hold for \(t=n+1\). It is sufficient to derive the value function for period \(T-(n+1)\) which is given by

$$\begin{aligned}&V(T-(n+1),W_{T-(n+1)},\mathcal {F}_{T-(n+1)})\\&\quad =\max \limits _{\mathbf {w}^{\prime }_{T-(n+1)}\mathbf {1}=1} E_{T-(n+1)}\left( W_{T-n}\mathbf {w}^{*\;\prime }_{T-n}\tilde{\varvec{\mu }}^*_{T-n+1} +F(\tilde{s}_T,\tilde{s}_{T-1},\ldots ,\tilde{s}_{T-n+2})\right. \\&\qquad -\,\left. \frac{\alpha }{2}W^2_{T-n}\mathbf {w}^{*\;\prime }_{T-n} \varvec{A}_{T-n+1}\mathbf {w}^*_{T-n}\right) =\max \limits _{\mathbf {w}^{\prime }_{T-(n+1)}\mathbf {1}=1} E_{T-(n+1)}\\&\qquad \times \,\Big [W_{T-n}\left( \underbrace{\frac{\mathbf {1}^{\prime }\varvec{A}^{-1}_{T-n+1}\tilde{\varvec{\mu }}^*_{T-n+1}}{\mathbf {1}^{\prime }\varvec{A}^{-1}_{T-n+1}\mathbf {1}}}_{=R_{T-n+1}} -\frac{1}{2}\underbrace{\frac{\tilde{\varvec{\mu }}_{T-n+1}^{*\;\prime } \tilde{\mathbf {Q}}_{T-n+1}\mathbf {1}}{\mathbf {1}^{\prime }\varvec{A}^{-1}_{T-n+1}\mathbf {1}}}_{=0} -\frac{1}{2}\underbrace{\frac{\mathbf {1}^{\prime }\tilde{\mathbf {Q}}_{T-n+1}\tilde{\varvec{\mu }}^*_{T-n+1}}{\mathbf {1}^{\prime }\varvec{A}^{-1}_{T-n+1}\mathbf {1}}}_{=0}\right) \\&\qquad +\,F(\tilde{s}_T,\tilde{s}_{T-1},\ldots ,\tilde{s}_{T-n+2}) +\frac{1}{2\alpha }\underbrace{\tilde{\varvec{\mu }}^{*\;\prime }_{T-n+1}\tilde{\mathbf {Q}}_{T-n+1}\tilde{\varvec{\mu }}^*_{T-n+1}}_{=\tilde{s}_{T-n+1}} -\frac{\alpha }{2}\underbrace{\frac{1}{\mathbf {1}^{\prime }\varvec{A}^{-1}_{T-n+1}\mathbf {1}}}_{=V_{T-n+1}}W^2_{T-n}\Big ]. \end{aligned}$$

Hence, we obtain

$$\begin{aligned}&V(T-(n+1),W_{T-(n+1)},\mathcal {F}_{T-(n+1)})\\&\quad =\max \limits _{\mathbf {w}^{\prime }_{T-(n+1)}\mathbf {1}=1} E_{T-(n+1)}\Big [W_{T-(n+1)}\mathbf {w}^{\prime }_{T-(n+1)}\tilde{\mathbf {X}}_{T-n}R_{T-n+1}\\&\qquad +\,F(\tilde{s}_T,\tilde{s}_{T-1},\ldots ,\tilde{s}_{T-n+1}) -\frac{\alpha }{2}W^2_{T-(n+1)}V_{T-n+1}(\mathbf {w}^{\prime }_{T-n+1}\tilde{\mathbf {X}}_{T-n})^2 \Big ]\\&\quad =\,\max \limits _{\mathbf {w}^{\prime }_{T-(n+1)}\mathbf {1}=1} \left( W_{T-(n+1)}\mathbf {w}^{\prime }_{T-(n+1)}\tilde{\varvec{\mu }}^*_{T-n} +F(\tilde{s}_T,\tilde{s}_{T-1},\ldots ,\tilde{s}_{T-n+1})\right. \\&\qquad -\,\left. \frac{\alpha }{2}W^2_{T-(n+1)}\mathbf {w}^{\prime }_{T-(n+1)} \varvec{A}_{T-n}\mathbf {w}_{T-(n+1)}\right) , \end{aligned}$$

where

$$\begin{aligned} F(\tilde{s}_T,\tilde{s}_{T-1},\ldots ,\tilde{s}_{T-n+1}) =F(s_T,\tilde{s}_{T-1},\ldots ,\tilde{s}_{T-n+2}) +\frac{1}{2\alpha }E_{T-(n+1)}[\tilde{s}_{T-n+1}]. \end{aligned}$$

It is the desired form of the value function at period \(T-(n+1)\). Because this expression is similar to the value function at period \(T-n\), we get the following formula for the weights at period \(T-(n+1)\)

$$\begin{aligned} \mathbf {w}^*_{T-(n+1)}=\frac{\varvec{A}^{-1}_{T-n}\mathbf {1}}{\mathbf {1}^{\prime } \varvec{A}^{-1}_{T-n}\mathbf {1}}+\frac{1}{\alpha W_{T-(n+1)}}\tilde{\mathbf {Q}}_{T-n}\tilde{\varvec{\mu }}^*_{T-n}. \end{aligned}$$

The theorem is proved.

For proving Corollary 1 we use the result of Proposition 6.1.

Proposition 1

Let \(\mathbf {X}\) be a random vector with mean \(\varvec{\mu }\) and positive definite covariance matrix \(\varvec{\varSigma }\). Let \(\varvec{A}=\varvec{\varSigma }+\tilde{\varvec{\mu }}\tilde{\varvec{\mu }}^\prime \) with \(\tilde{\varvec{\mu }}=\varvec{\mu }+\mathbf {1}\). If

$$\begin{aligned} \mathbf {w}=\frac{\varvec{A}^{-1}\mathbf {1}}{\mathbf {1}^{\prime }\varvec{A}^{-1}\mathbf {1}} +\tilde{\alpha }^{-1}\tilde{\mathbf {Q}}\tilde{\varvec{\mu }}\quad \text {with}\quad \tilde{\mathbf {Q}}=\varvec{A}^{-1}-\frac{\varvec{A}^{-1}\mathbf {1}\mathbf {1}^\prime \varvec{A}^{-1}}{\mathbf {1}^\prime \varvec{A}^{-1}\mathbf {1}} \end{aligned}$$
(55)

then

$$\begin{aligned} \mathbf {w}=\frac{\varvec{\varSigma }^{-1}\mathbf {1}}{\mathbf {1}^{\prime }\varvec{\varSigma }^{-1}\mathbf {1}} +\alpha ^{-1}\mathbf {Q}\tilde{\varvec{\mu }}\quad \text {with}\quad \mathbf {Q}= \varvec{\varSigma }^{-1}-\frac{\varvec{\varSigma }^{-1}\mathbf {1}\mathbf {1}^\prime \varvec{\varSigma }^{-1}}{\mathbf {1}^\prime \varvec{\varSigma }^{-1}\mathbf {1}} \end{aligned}$$
(56)

and

$$\begin{aligned} \alpha ^{-1}=\frac{\tilde{\alpha }^{-1}(\mathbf {1}^\prime \varvec{\varSigma }^{-1}\mathbf {1}) -\mathbf {1}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}}{(1+\tilde{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}) \mathbf {1}^\prime \varvec{\varSigma }^{-1}\mathbf {1}-(\mathbf {1}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }})^2}= \frac{\tilde{\alpha }^{-1}-1-R_{ GMV }}{1+s}, \end{aligned}$$
(57)

where \(R_{ GMV }=\frac{\mathbf {1}^\prime \varvec{\varSigma }^{-1}\varvec{\mu }}{\mathbf {1}^\prime \varvec{\varSigma }^{-1}\mathbf {1}}, \quad s=\tilde{\varvec{\mu }}^{\prime }\mathbf {Q}\tilde{\varvec{\mu }}=\varvec{\mu }^{\prime }\mathbf {Q}\varvec{\mu }\).

Proof of Proposition 1

From (56) we obtain

$$\begin{aligned} \mathbf {w}=\left( \frac{1}{\mathbf {1}^\prime \varvec{\varSigma }^{-1}\mathbf {1}} -\alpha ^{-1}\frac{\mathbf {1}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}}{\mathbf {1}^\prime \varvec{\varSigma }^{-1}\mathbf {1}}\right) \varvec{\varSigma }^{-1}\mathbf {1}+\alpha ^{-1}\varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}=C_1\varvec{\varSigma }^{-1}\mathbf {1}+C_2\varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}, \end{aligned}$$
(58)

where

$$\begin{aligned} C_1=\frac{1}{\mathbf {1}^{\prime }\varvec{\varSigma }^{-1}\mathbf {1}}-C_2 \frac{\mathbf {1}^{\prime }\varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}}{\mathbf {1}^{\prime }\varvec{\varSigma }^{-1}\mathbf {1}} \quad \text {and} \quad C_2=\alpha ^{-1}. \end{aligned}$$
(59)

In order to prove the proposition we need to show that (55) can be expressed in the same way. The application of the Sherman–Morrison formula [Harville (1997, Theorem 18.2.8)], i.e.,

$$\begin{aligned} \varvec{A}^{-1}=(\varvec{\varSigma }+\tilde{\varvec{\mu }}\tilde{\varvec{\mu }}^{\prime })^{-1}= \varvec{\varSigma }^{-1}-\frac{\varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}\tilde{\varvec{\mu }}^{\prime } \varvec{\varSigma }^{-1}}{1+\tilde{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}} \end{aligned}$$

leads to

$$\begin{aligned} \mathbf {w}&= (1-K\tilde{\alpha }^{-1})\frac{1+\tilde{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}}{(1+\tilde{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}) \mathbf {1}^\prime \varvec{\varSigma }^{-1}\mathbf {1}-(\mathbf {1}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }})^2} \varvec{\varSigma }^{-1}\mathbf {1}\nonumber \\&\quad +\,\left( -\frac{\tilde{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\mathbf {1}(1-K\tilde{\alpha }^{-1})}{(1+\tilde{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }})\mathbf {1}^\prime \varvec{\varSigma }^{-1}\mathbf {1}-(\mathbf {1}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }})^2} +\frac{\tilde{\alpha }^{-1}}{1+\tilde{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}}\right) \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}, \end{aligned}$$
(60)

where

$$\begin{aligned} K=\mathbf {1}^\prime \varvec{A}^{-1}\tilde{\varvec{\mu }}=\frac{\mathbf {1}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}}{1+\tilde{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}}. \end{aligned}$$
(61)

From the structure of (58) and (60) we get

$$\begin{aligned} \alpha ^{-1}&= C_2=\left( -\frac{\tilde{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1} \mathbf {1}(1-K\tilde{\alpha }^{-1})}{(1+\tilde{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}) \mathbf {1}^\prime \varvec{\varSigma }^{-1}\mathbf {1}-(\mathbf {1}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }})^2}\right) +\frac{\tilde{\alpha }^{-1}}{1+\tilde{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}}\\&= \frac{\tilde{\alpha }^{-1}(\mathbf {1}^\prime \varvec{\varSigma }^{-1}\mathbf {1}) -\mathbf {1}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}}{(1+\tilde{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}) \mathbf {1}^\prime \varvec{\varSigma }^{-1}\mathbf {1}-(\mathbf {1}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }})^2} =\frac{\tilde{\alpha }^{-1}-1-R_{ GMV }}{1+s}. \end{aligned}$$

For proving the proposition we only need to show the equality of the coefficients in front of \(\varvec{\varSigma }^{-1}\mathbf {1}\) in (58) and (60). It holds that

$$\begin{aligned} C_1&= \frac{1}{\mathbf {1}^{\prime }\varvec{\varSigma }^{-1}\mathbf {1}}-C_2\frac{\mathbf {1}^{\prime } \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}}{\mathbf {1}^{\prime }\varvec{\varSigma }^{-1}\mathbf {1}}= \frac{1}{\mathbf {1}^\prime \varvec{\varSigma }^{-1}\mathbf {1}}\\&\quad -\,\left( \frac{\tilde{\alpha }^{-1}}{1+\tilde{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}} -\frac{\tilde{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\mathbf {1}(1-K\tilde{\alpha }^{-1})}{(1+\tilde{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }})\mathbf {1}^\prime \varvec{\varSigma }^{-1} \mathbf {1}-(\mathbf {1}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }})^2}\right) \frac{\mathbf {1}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }}}{\mathbf {1}^\prime \varvec{\varSigma }^{-1}\mathbf {1}}\\&= \frac{(1-K\tilde{\alpha }^{-1})}{\mathbf {1}^\prime \varvec{\varSigma }^{-1}\mathbf {1}} +\frac{(\mathbf {1}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }})^2(1-K\tilde{\alpha }^{-1})}{\mathbf {1}^\prime \varvec{\varSigma }^{-1}\mathbf {1}\left( (1+\tilde{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }})\mathbf {1}^\prime \varvec{\varSigma }^{-1} \mathbf {1}-(\mathbf {1}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }})^2\right) }\\&= \frac{(1-K\tilde{\alpha }^{-1})}{\mathbf {1}^\prime \varvec{\varSigma }^{-1}\mathbf {1}} \left( \frac{(1+\tilde{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }})\mathbf {1}^\prime \varvec{\varSigma }^{-1}\mathbf {1}}{(1+\tilde{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }})\mathbf {1}^\prime \varvec{\varSigma }^{-1}\mathbf {1}-(\mathbf {1}^\prime \varvec{\varSigma }^{-1}\tilde{\varvec{\mu }})^2}\right) . \end{aligned}$$

The last identity completes the proof.

Proof of Corollary 1

Under the assumption of independence it holds that

$$\begin{aligned} \varvec{A}_{T-t+1}= \left\{ \begin{array}{ll} \varvec{\varSigma }_T+\tilde{\varvec{\mu }}_T\tilde{\varvec{\mu }}^\prime _T &{}\quad \text {for}\quad t=1\\ V_{T-t+2}(\varvec{\varSigma }_{T-t+1}+\tilde{\varvec{\mu }}_{T-t+1}\tilde{\varvec{\mu }}^\prime _{T-t+1}) &{} \quad \text {for}\quad t=2,\ldots ,T, \end{array} \right. \end{aligned}$$
(62)

and

$$\begin{aligned} \tilde{\varvec{\mu }}^*_{T-t+1}= \left\{ \begin{array}{ll} \tilde{\varvec{\mu }}_T &{}\quad \text {for}\quad t=1\\ R_{T-t+2} \tilde{\varvec{\mu }}_{T-t+1} &{} \quad \text {for}\quad t=2,\ldots ,T\\ \end{array} \right. . \end{aligned}$$
(63)

Let \(R_{T+1}=V_{T+1}=1\). Then,

$$\begin{aligned} \mathbf {w}^*_{T-t}&= \frac{(V_{T-t+2}(\varvec{\varSigma }_{T-t+1} +\tilde{\varvec{\mu }}_{T-t+1}\tilde{\varvec{\mu }}^\prime ))^{-1}\mathbf {1}}{\mathbf {1}^{\prime }(V_{T-t+2} (\varvec{\varSigma }_{T-t+1}+\tilde{\varvec{\mu }}_{T-t+1}\tilde{\varvec{\mu }}^\prime _{T-t+1}))^{-1}\mathbf {1}} +\frac{1}{\alpha W_{T-t}}\tilde{\mathbf {Q}}_{T-t+1} (R_{T-t+2}\tilde{\varvec{\mu }}_{T-t+1})\\&= \frac{(\varvec{\varSigma }_{T-t+1}+\tilde{\varvec{\mu }}_{T-t+1}\tilde{\varvec{\mu }}^\prime _{T-t+1})^{-1}\mathbf {1}}{\mathbf {1}^{\prime }(\varvec{\varSigma }_{T-t+1}+\tilde{\varvec{\mu }}_{T-t+1}\tilde{\varvec{\mu }}^\prime _{T-t+1})^{-1}\mathbf {1}} +\frac{R_{T-t+2}}{\alpha W_{T-t}V_{T-t+2}} \tilde{\tilde{\mathbf {Q}}}_{T-t+1} \tilde{\varvec{\mu }}_{T-t+1} \end{aligned}$$

with

$$\begin{aligned} \tilde{\tilde{\mathbf {Q}}}_{T-t+1}&= \left( \varvec{\varSigma }_{T-t+1}+\tilde{\varvec{\mu }}_{T-t+1} \tilde{\varvec{\mu }}^\prime _{T-t+1}\right) ^{-1}\\&\quad -\,\frac{\left( \varvec{\varSigma }_{T-t+1} +\tilde{\varvec{\mu }}_{T-t+1}\tilde{\varvec{\mu }}^\prime _{T-t+1}\right) ^{-1}\mathbf {1}\mathbf {1}' \left( \varvec{\varSigma }_{T-t+1}+\tilde{\varvec{\mu }}_{T-t+1}\tilde{\varvec{\mu }}_{T-t+1}^\prime \right) ^{-1}}{\mathbf {1}^\prime \left( \varvec{\varSigma }_{T-t+1}+\tilde{\varvec{\mu }}_{T-t+1}\tilde{\varvec{\mu }}^\prime _{T-t+1}\right) ^{-1}\mathbf {1}} \end{aligned}$$

and

$$\begin{aligned} \frac{R_{T-t+2}}{V_{T-t+2}}= \prod \limits _{i=T-t+2}^{T} \frac{\frac{\mathbf {1}^{\prime }(\varvec{\varSigma }_i+\tilde{\varvec{\mu }}_i\tilde{\varvec{\mu }}_i^\prime )^{-1}\tilde{\varvec{\mu }}_i}{\mathbf {1}^{\prime }(\varvec{\varSigma }_i+\tilde{\varvec{\mu }}_i\tilde{\varvec{\mu }}_i^\prime )^{-1}\mathbf {1}}}{\frac{1}{\mathbf {1}^{\prime }(\varvec{\varSigma }_i+\tilde{\varvec{\mu }}_i\tilde{\varvec{\mu }}_i^\prime )^{-1}\mathbf {1}}} = \prod \limits _{i=T-t+2}^{T} \mathbf {1}^{\prime }(\varvec{\varSigma }_i+\tilde{\varvec{\mu }}_i\tilde{\varvec{\mu }}_i^\prime ) ^{-1}\tilde{\varvec{\mu }}_i, \end{aligned}$$

where the last identity follows from the definition of \(R_{T-t+2}\) and \(V_{T-t+2}\) given in (16).

The rest of the proof follows from Proposition 1 if \(\varvec{\varSigma }\) is replaced by \(\varvec{\varSigma }_{T-t+1}\), \(\tilde{\varvec{\mu }}\) by \(\tilde{\varvec{\mu }}_{T-t+1}\) and

$$\begin{aligned} \tilde{\alpha }^{-1}\!=\!\frac{1}{\alpha W_{T-t}}\frac{R_{T-t+2}}{V_{T-t+2}} \!=\!\frac{1}{\alpha W_{T-t}}\left( \prod \limits _{i=T-t+2}^{T} \mathbf {1}^{\prime }(\varvec{\varSigma }_i\!+\!\tilde{\varvec{\mu }}_i\tilde{\varvec{\mu }}_i^\prime )^{-1}\tilde{\varvec{\mu }}_i\right) \!=\!\frac{1}{\alpha W_{T-t}}\left( \prod \limits _{i=T-t+2}^{T}a_i\right) , \end{aligned}$$

where

$$\begin{aligned} a_i=\mathbf {1}^{\prime }(\varvec{\varSigma }_i+\tilde{\varvec{\mu }}_i\tilde{\varvec{\mu }}_i^\prime )^{-1}\tilde{\varvec{\mu }}_i =\frac{1+R_{ GMV ,i}}{(1+R_{ GMV ,i})^2+(1+s_i)V_{ GMV ,i}}. \end{aligned}$$

The last expression is obtained by applying the Sherman–Morrison formula. At last, we recall \(\mathbf {Q}_{T-t+1}\mathbf {1}=\mathbf {0}\) and get (18). Thus the corollary is proved.

Proof of Theorem 2

The expression of the optimal weights at period \(T-1\) is given in (28). The rest of the theorem’s statement is proved by using the mathematical induction on the expressions of the portfolio weights and the value function. We use similar notations as in the proof of Theorem 1. Let \(\breve{\varvec{A}}_i=E_{i-1}[(1-\tilde{s}_{i+1})\breve{\mathbf {X}}_{i}\breve{\mathbf {X}}^\prime _{i}]\) for \(i=1,\ldots ,T-1\) and \(\breve{\varvec{A}}_T=\varvec{\varSigma }_T-\breve{\varvec{\mu }}_T\breve{\varvec{\mu }}_T^\prime ,\)

$$\begin{aligned} \breve{\varvec{\mu }}^*_i=\left\{ \begin{array}{ll} \breve{\varvec{\mu }}_{T} &{}\quad \text {for}\quad i=T\\ E_{i-1}[(1-\tilde{s}_{i+1})\tilde{\mathbf {X}}_{i}]&{} \quad \text {for}\quad i=1,\ldots ,T-1, \end{array} \right. \end{aligned}$$
(64)

and \(\breve{s}_i=\breve{\varvec{\mu }}_i^{*\;\prime } \breve{\varvec{A}}^{-1}_i\breve{\varvec{\mu }}^*_i\) for \(i=2,\ldots ,T\).

Let \(\mathbf {w}^{*\;\prime }_{T-1}\) be the optimal portfolio weight calculated at period \(T-1\) in the case of a riskless asset as expressed in (28). First, we calculate the value function at period \(T-2\). It holds that

$$\begin{aligned}&V(T-2,W_{T-2},\mathcal {F}_{T-2})\\&\quad =\max \limits _{\mathbf {w}_{T-2}}E_{T-2}\Big [W_{T-1} \left( R_{f,T}+\mathbf {w}^{*\;\prime }_{T-1}\breve{\varvec{\mu }}_{T}\right) \\&\qquad -\,\frac{\alpha }{2}W^2_{T-1}\left( \mathbf {w}^{*\;\prime }_{T-1} \breve{\varvec{A}}_{T}\mathbf {w}^*_{T-1}+R^2_{f,T}+2R_{f,T}\mathbf {w}^{*\;\prime }_{T-1} \breve{\varvec{\mu }}_{T}\right) \Big ]\\&\quad =\max \limits _{\mathbf {w}_{T-2}}E_{T-2}\Big [W_{T-1}\left( R_{f,T} +\left( \frac{1}{\alpha W_{T-1}}-R_{f,T}\right) \breve{\varvec{\mu }}^\prime _T\breve{\varvec{A}}^{-1}_T \breve{\varvec{\mu }}_{T}\right) \\&\qquad -\,\frac{\alpha }{2}W^2_{T-1}\left( \left( \frac{1}{\alpha W_{T-1}} -R_{f,T}\right) \breve{\varvec{\mu }}^\prime _T\breve{\varvec{A}}^{-1}_T\breve{\varvec{A}}_{T}\left( \frac{1}{\alpha W_{T-1}}-R_{f,T}\right) \breve{\varvec{A}}^{-1}_T\breve{\varvec{\mu }}_T+R^2_{f,T}\right. \\&\qquad +\,\left. 2R_{f,T}\left( \frac{1}{\alpha W_{T-1}}-R_{f,T}\right) \breve{\varvec{\mu }}^\prime _T\breve{\varvec{A}}^{-1}_T\breve{\varvec{\mu }}_{T}\right) \Big ]. \end{aligned}$$

Using the definition of \(\breve{s}_T\) we obtain

$$\begin{aligned}&V(T-2,W_{T-2},\mathcal {F}_{T-2})=\max \limits _{\mathbf {w}_{T-2}}E_{T-2}\Big [W_{T-1}R_{f,T} (1-\breve{s}_T)+\frac{\breve{s}_T}{\alpha }\\&\qquad -\frac{\alpha }{2}W^2_{T-1}\left( \left( \frac{1}{\alpha W_{T-1}}-R_{f,T}\right) ^2\breve{s}_T+R^2_{f,T} +2R_{f,T}\left( \frac{1}{\alpha W_{T-1}}-R_{f,T}\right) \breve{s}_T\right) \\&\qquad =\max \limits _{\mathbf {w}_{T-2}}E_{T-2}\Big [W_{T-1} R_{f,T}(1-\breve{s}_T)+\frac{\breve{s}_T}{2\alpha } -\frac{\alpha }{2}W^2_{T-1}R^2_{f,T}\left( 1-\breve{s}_T\right) \Big ]\\&\quad =\max \limits _{\mathbf {w}_{T-2}}\Big [W_{T-2}R_{f,T}\left( E_{T-2}[1-\breve{s}_T]R_{f,T-1}+\mathbf {w}^{\prime }_{T-2}\breve{\varvec{\mu }}^*_{T-1}\right) \\&\qquad -\,\frac{\alpha }{2}W^2_{T-2}R^2_{f,T} \left( \mathbf {w}^{\prime }_{T-2}\varvec{A}_{T-1}\mathbf {w}_{T-2} +E_{T-2}[1-\breve{s}_T]R^2_{f,T-1}\right. \\&\qquad \left. +\,2R_{f,T-1} \mathbf {w}^{\prime }_{T-2}\breve{\varvec{\mu }}^*_{T-1}\right) +\frac{E_{T-2}[\breve{s}_T]}{2\alpha }\Big ]. \end{aligned}$$

The last expression is similar to the value function at the period \(T-1\). Hence, it is maximized on the weights \(\mathbf {w}_{T-2}^*\) expressed as

$$\begin{aligned} \mathbf {w}^*_{T-2}=\left( \frac{1}{\alpha W_{T-2}}(R_{f,T})^{-1}-R_{f,T-1}\right) \breve{\varvec{A}}^{-1}_{T-1}\breve{\varvec{\mu }}^*_{T-1}. \end{aligned}$$
(65)

Hence, the basis of induction are the following expressions

$$\begin{aligned}&V(T-2,W_{T-2},\mathcal {F}_{T-2})= \max \limits _{\{\mathbf {w}_{T-2}\}}\Big [W_{T-2} R_{f,T}\left( b_TR_{f,T-1}+\mathbf {w}^{\prime }_{T-2}\breve{\varvec{\mu }}^*_{T-1}\right) \\&\quad -\frac{\alpha }{2}W^2_{T-1}R^2_{f,T} \left( \mathbf {w}^{\prime }_{T-2}\varvec{A}_{T-1}\mathbf {w}_{T-2}+b_TR^2_{f,T-1}+2R_{f,T-1}\mathbf {w}^{\prime }_{T-2}\breve{\varvec{\mu }}^*_{T-1}\right) +F(\breve{s}_T)\Big ]\\&\mathbf {w}^*_{T-2}=\left( R_{f,T-1}-\frac{1}{\alpha W_{T-2}}(R_{f,T})^{-1}\right) \breve{\varvec{A}}^{-1}_{T-1}\breve{\varvec{\mu }}^*_{T-1} \end{aligned}$$

with \(F(\breve{s}_T)=\frac{E_{T-2}[\breve{s}_T]}{2\alpha }\) and \(b_T=E_{T-2}[1-\breve{s}_T]\).

In the induction hypothesis we assume that the statement holds for \(t=n\), i.e.,

$$\begin{aligned}&V(T-n,W_{T-n},\mathcal {F}_{T-n})\\&\quad =\max \limits _{\{\mathbf {w}_{T-n}\}} \Big [W_{T-n}\left( \prod \limits _{i=T-n+2}^{T}R_{f,i}\right) \left( b_{T-n+2}R_{f,T-n+1}+\mathbf {w}^{\prime }_{T-n}\breve{\varvec{\mu }}^*_{T-n+1}\right) \\&\quad -\,\frac{\alpha }{2}W^2_{T-n}\left( \prod \limits _{i=T-n+2}^{T} R^2_{f,i}\right) \left( \mathbf {w}^{\prime }_{T-n}\varvec{A}_{T-n+1}\mathbf {w}_{T-n} +b_{T-n+2}R^2_{f,T-n+1}\right. \\&\quad \left. +\,2R_{f,T-n+1}\mathbf {w}^{\prime }_{T-n}\breve{\varvec{\mu }}^*_{T-n+1}\right) +F(\breve{s}_T,\ldots ,\breve{s}_{T-n+2})\Big ],\\&\mathbf {w}^*_{T-n}=\left( \frac{1}{\alpha W_{T-n}} \left( \prod \limits _{i=T-n+2}^{T}R_{f,i}\right) ^{-1} -R_{f,T-n+1}\right) \breve{\varvec{A}}^{-1}_{T-n+1}\breve{\varvec{\mu }}^*_{T-n+1}. \end{aligned}$$

with \(F(\breve{s}_T,\ldots ,\breve{s}_{T-n+2})=\frac{1}{2\alpha }\left( E_{T-2}[\breve{s}_T]+\sum \nolimits _{m=T-n+2}^{T-1}\prod \nolimits _{i=m}^{T-1}b_iE_{m-2}[\breve{s}_m]\right) \) and \(b_i=E_{i-2}[1-\breve{s}_i]\).

In the inductive step we prove that the last identities also hold for \(t=n+1\). It is sufficient to derive the value function for period \(T-(n+1)\) which is given by

$$\begin{aligned}&V(T-(n+1),W_{T-(n+1)},\mathcal {F}_{T-(n+1)})\\&\quad =\max \limits _{\mathbf {w}_{T-(n+1)}}E_{T-(n+1)} \Big [W_{T-n}\left( \prod \limits _{i=T-n+2}^{T}R_{f,i}\right) \left( b_{T-n+2}R_{f,T-n+1} +\mathbf {w}^{*\;\prime }_{T-n}\breve{\varvec{\mu }}^*_{T-n+1}\right) \\&\qquad -\,\frac{\alpha }{2}W^2_{T-n}\left( \prod \limits _{i=T-n+2}^{T} R^2_{f,i}\right) \left( \mathbf {w}^{*\;\prime }_{T-n}\varvec{A}_{T-n+1}\mathbf {w}^*_{T-n}+b_{T-n+2} R^2_{f,T-n+1}\right. \\&\qquad \left. +\,2R_{f,T-n+1}\mathbf {w}^{*\;\prime }_{T-n}\breve{\varvec{\mu }}^*_{T-n+1}\right) -F(\breve{s}_T,\ldots ,\breve{s}_{T-n+2})\Big ]\\&\quad =\max \limits _{\mathbf {w}_{T-(n+1)}}E_{T-(n+1)}\Big [W_{T-n}\prod \limits _{i=T-n+2}^{T}R_{f,i} \left( b_{T-n+2}R_{f,T-n+1}\right. \\&\qquad \left. +\,\left( \frac{1}{\alpha W_{T-n}}\left( \prod \limits _{i=T-n+2}^{T}R_{f,i}\right) ^{-1}-R_{f,T-n+1}\right) \right. \\&\qquad \times \,\left. \breve{\varvec{\mu }}^{*\;\prime }_{T-n+1} \breve{\varvec{A}}^{-1}_{T-n+1}\breve{\varvec{\mu }}^*_{T-n+1}\right) \\&\qquad -\,\frac{\alpha }{2}W^2_{T-n}\prod \limits _{i=T-n+2}^{T}R^2_{f,i} \left( \left( \frac{1}{\alpha W_{T-n}}\left( \prod \limits _{i=T-n+2}^{T} R_{f,i}\right) ^{-1}\right. \right. \\&\qquad \left. \left. -\,R_{f,T-n+1}\right) \breve{\varvec{\mu }}^{*\;\prime }_{T-n+1} \breve{\varvec{A}}^{-1}_{T-n+1}\varvec{A}_{T-n+1}\right. \\&\qquad \times \,\left. \left( \frac{1}{\alpha W_{T-n}} \left( \prod \limits _{i=T-n+2}^{T}R_{f,i}\right) ^{-1}-R_{f,T-n+1}\right) \breve{\varvec{A}}^{-1}_{T-n+1}\breve{\varvec{\mu }}^*_{T-n+1}+b_{T-n+2}R^2_{f,T-n+1}\right. \\&\qquad +\,\left. 2R_{f,T-n+1}\left( \frac{1}{\alpha W_{T-n}} \left( \prod \limits _{i=T-n+2}^{T}R_{f,i}\right) ^{-1} -R_{f,T-n+1}\right) \breve{\varvec{\mu }}^{*\;\prime }_{T-n+1} \breve{\varvec{A}}^{-1}_{T-n+1}\breve{\varvec{\mu }}^{*}_{T-n+1}\right) \\&\qquad +\,F(\breve{s}_T,\ldots ,\breve{s}_{T-n+2})\Big ]. \end{aligned}$$

Using the definition of \(\breve{s}_i\) and denoting \(\xi =\prod \limits _{i=T-n+2}^{T}R_{f,i}\) we receive

$$\begin{aligned}&V(T-(n+1),W_{T-(n+1)},\mathcal {F}_{T-(n+1)})\\&\quad =\max \limits _{\{\mathbf {w}_{T-(n+1)}\}}E_{T-(n+1)}\Big [W_{T-n}R_{f,T-n+1}\xi b_{T-n+2}(1-\breve{s}_{T-n+1})+\frac{b_{T-n+2}}{\alpha }\breve{s}_{T-n+1}\\&\qquad +\,F(\breve{s}_T,\ldots ,\breve{s}_{T-n+2})-\frac{\alpha }{2}W^2_{T-n}b_{T-n+2}\xi ^2\left( \left( \frac{\xi ^{-1}}{\alpha W_{T-n}}-R_{f,T-n+1}\right) ^2\breve{s}_{T-n+1}+R^2_{f,T-n+1}\right. \\&\qquad +\,\left. 2R_{f,T-n+1}\left( \frac{\xi ^{-1}}{\alpha W_{T-(n+1)}}-R_{f,T-n+1}\right) \breve{s}_{T-n+1}\right) \Big ]\\&\quad =\,\max \limits _{\{\mathbf {w}_{T-(n+1)}\}}E_{T-(n+1)}\Big [W_{T-n}\xi R_{f,T-n+1} b_{T-n+2} (1-\tilde{s}_{T-n+1})\\&\qquad +\,\left( \frac{\breve{s}_{T-n+1}}{2\alpha }b_{T-n+2}+F(\breve{s}_T,\ldots ,\breve{s}_{T-n+2})\right) -\,\frac{\alpha }{2}W^2_{T-n}(\xi R_{f,T-n+1})^{2}b_{T-n+2} (1-\tilde{s}_{T-n+1})\Big ]\\&\quad =\,\max \limits _{\{\mathbf {w}_{T-(n+1)}\}}\Big [W_{T-(n+1)}\xi R_{f,T-n+1} b_{T-n+2}\left( E_{T-(n+1)}(1-\tilde{s}_{T-n+1})R_{f,T-n}\right. \\&\qquad \left. +\,\mathbf {w}^{\prime }_{T-(n+1)}\breve{\varvec{\mu }}^*_{T-n}\right) \\&\qquad -\,\frac{\alpha }{2}W^2_{T-(n+1)}(\xi R_{f,T-n+1})^{2}b_{T-n+2} \left( \mathbf {w}^{\prime }_{T-(n+1)}\varvec{A}_{T-n}\mathbf {w}_{T-(n+1)}\right. \\&\qquad \left. +\,E_{T-(n+1)}(1-\tilde{s}_{T-n+1})R^2_{f,T-n}\right. \\&\qquad +\,\left. 2R_{f,T-n}\mathbf {w}^{\prime }_{T-(n+1)}\breve{\varvec{\mu }}^*_{T-n}+F(\breve{s}_T,\ldots ,\breve{s}_{T-n+1})\right) . \end{aligned}$$

where \(F(\breve{s}_T,\ldots ,\breve{s}_{T-n+1})=F(\breve{s}_T,\ldots ,\breve{s}_{T-n+2})+\frac{1}{2}\frac{E_{T-(n+1)}[\breve{s}_{T-n+1}]}{\alpha }b_{T-n+2}\).

It is a desired form of the value function at period \(T-(n+1)\). Because this expression is similar to the value function at period \(T-n\), we get the following formula for the weights at period \(T-(n+1)\)

$$\begin{aligned} \mathbf {w}^*_{T-(n+1)}=\left( \frac{(\xi R_{f,T-n+1})^{-1}}{\alpha W_{T-n}}-R_{f,T-n}\right) \breve{\varvec{A}}^{-1}_{T-n}\breve{\varvec{\mu }}^*_{T-n}. \end{aligned}$$

Substituting \(\xi =\prod \nolimits _{i=T-n+2}^{T}R_{f,i}\) leads to the expression given in the statement of Theorem 2. The theorem is proved.

Proposition 2

Let \(\mathbf {X}\) be a random vector with mean \(\varvec{\mu }\) and positive definite covariance matrix \(\varvec{\varSigma }\). Let \(\breve{\varvec{A}}=\varvec{\varSigma }+\breve{\varvec{\mu }}\breve{\varvec{\mu }}^\prime \) and \(\breve{\varvec{\mu }}=\varvec{\mu }-r_{f}\mathbf {1}\). If

$$\begin{aligned} \mathbf {w}=\tilde{\gamma }^{-1}\breve{\varvec{A}}^{-1}\breve{\varvec{\mu }}\end{aligned}$$
(66)

then

$$\begin{aligned} \mathbf {w}=\gamma ^{-1}\varvec{\varSigma }^{-1}\breve{\varvec{\mu }}\quad \text {with}\quad \gamma ^{-1}=\frac{\tilde{\gamma }^{-1}}{1+\breve{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\breve{\varvec{\mu }}}. \end{aligned}$$
(67)

Proof of Proposition 2

The application of the Sherman–Morrison formula, i.e.,

$$\begin{aligned} \breve{\varvec{A}}^{-1}=(\varvec{\varSigma }+\breve{\varvec{\mu }}\breve{\varvec{\mu }}^{\prime })^{-1}=\varvec{\varSigma }^{-1} -\frac{\varvec{\varSigma }^{-1}\breve{\varvec{\mu }}\breve{\varvec{\mu }}^{\prime }\varvec{\varSigma }^{-1}}{1+\breve{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\breve{\varvec{\mu }}} \end{aligned}$$

leads to

$$\begin{aligned} \mathbf {w}=\tilde{\gamma }^{-1}\varvec{\varSigma }^{-1}\breve{\varvec{\mu }}-\tilde{\gamma }^{-1} \frac{\varvec{\varSigma }^{-1}\breve{\varvec{\mu }}\breve{\varvec{\mu }}^{\prime }\varvec{\varSigma }^{-1}}{1+\breve{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\breve{\varvec{\mu }}}\breve{\varvec{\mu }}=\frac{\tilde{\gamma }^{-1}}{1+\breve{\varvec{\mu }}^\prime \varvec{\varSigma }^{-1}\breve{\varvec{\mu }}}\varvec{\varSigma }^{-1}\breve{\varvec{\mu }}, \end{aligned}$$

what completes the proof of the proposition.

Proof of Corollary 2

Under the assumption of independence

$$\begin{aligned} \breve{\varvec{A}}_{T-t+1}= \left\{ \begin{array}{ll} \varvec{\varSigma }_{T}+\breve{\varvec{\mu }}_{T}\breve{\varvec{\mu }}_{T}^\prime &{}\quad \text {for}\quad t=1\\ (1-\tilde{s}_{T-t+2}) (\varvec{\varSigma }_{T-t+1}+\breve{\varvec{\mu }}_{T-t+1}\breve{\varvec{\mu }}_{T-t+1}^\prime )&{} \quad \text {for}\quad t=2,\ldots ,T\\ \end{array} \right. \end{aligned}$$
(68)

and

$$\begin{aligned} \breve{\varvec{\mu }}_{T-t+1}^*= \left\{ \begin{array}{ll} \breve{\varvec{\mu }}_{T} &{}\quad \text {for}\quad t=1\\ (1-\tilde{s}_{T-t+2})\breve{\varvec{\mu }}_{T-t+1} &{} \quad \text {for}\quad t=2,\ldots ,T\\ \end{array} \right. . \end{aligned}$$
(69)

Then the statement of the corollary follows from Proposition 2 if \(\varvec{\varSigma }\) is replaced by \(\varvec{\varSigma }_{T-t+1}\) and \(\breve{\varvec{\mu }}\) by \(\breve{\varvec{\mu }}_{T-t+1}\), and

$$\begin{aligned} \tilde{\gamma }^{-1}=\Big [\frac{1}{\alpha W_{T-t}}\left( \prod \limits _{i=T-t+2}^{T}R_{f,i}\right) ^{-1}-R_{f,T-t+1}\Big ]. \end{aligned}$$

Proof of Theorem 3

The results of Theorem 3 follow Theorem 2 and the application of the Sherman–Morrison formula.

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Bodnar, T., Parolya, N. & Schmid, W. A closed-form solution of the multi-period portfolio choice problem for a quadratic utility function. Ann Oper Res 229, 121–158 (2015). https://doi.org/10.1007/s10479-015-1802-z

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