Mixed-asset portfolio allocation under mean-reverting asset returns

Abstract

Standard results about portfolio optimization suggest that the allocation to real estate in a mixed-asset portfolio should be around 15–20%. However, the institutional investors share in real estate is significantly smaller, around 7–9%. Many researches have addressed this point even if as of today no consensus has emerged. In this paper, we built-up an allocation model that can explain the empirical observed weights. For this purpose, we account for the term structure of all standard financial assets and also of real estate asset class (expected returns, volatilities and correlations depending on the time to maturity). We propose a dynamic portfolio optimization model that allows analyzing portfolio weights with respect to the whole term structure modelling, due to its tractability and its good fit when being adequately calibrated. In this framework, we provide explicit and operational solutions to the dynamic mixed-asset portfolio allocation (cash, real estate, stock and bond). The results show that accounting for investment horizon and mean-reverting dynamics allows to better examine how portfolio allocations depend on both risk aversion and investment horizon.

Appendices

Appendix A: Properties of the numeraire portfolio

Recall that the Radon–Nikodym density of the risk-neutral probability is given by:

$$\begin{aligned} \eta _{t}=\exp \left[ -\mathbf {M}_{t}-\int _{0}^{t}A(s)ds\right] \end{aligned}$$

where

$$\begin{aligned} \mathbf {M}_{t}=\int _{0}^{t}\lambda _{r,s}dW_{s}^{r}+\int _{0}^{t}\lambda _{P,s}dW_{s}^{P}+\int _{0}^{t}\lambda _{S,s}dW_{s}^{S}, \end{aligned}$$

and

$$\begin{aligned} A(t)=\frac{1}{2}\left( \lambda _{r,t}^{2}+\lambda _{P,t}^{2}+\lambda _{S,t}^{2}\right) +\lambda _{r,t}\lambda _{P,t}\rho ^{r,P}+\lambda _{r,t}\lambda _{S,t}\rho ^{r,S}+\lambda _{P,t}\lambda _{S,t}\rho ^{P,S}. \end{aligned}$$

The numeraire portfolio H is equal to:

$$\begin{aligned} H_{t}=\frac{\text {exp}\left( \int _{0}^{t}r_{s}ds\right) }{\eta _{t}}. \end{aligned}$$

Consequently, we get:

$$\begin{aligned} \ln H_{t}=\int _{0}^{t}r_{s}\,ds+\mathbf {M}_{t}+\int _{0}^{t}A(s)ds. \end{aligned}$$

Since $\left( W^{r},W^{P},W^{S}\right)$ is a Gaussian process, we can deduce that $\ln H_{t}$ is Gaussian itself. We have to determine its expectation and variance. More precisely, we determine the conditional expectations for powers z of $\mathbb {E}_{t}\left[ \frac{ H_{T}^{z}}{H_{t}^{z}}\right]$. The ratio $\frac{H_{T}^{z}}{H_{t}^{z}}$ satisfies:

$$\begin{aligned} \frac{H_{T}^{z}}{H_{t}^{z}}=\exp \left[ N_{z}(t,T)\right] , \end{aligned}$$

with

$$\begin{aligned} N_{z}(t,T)=z\left[ \int _{t}^{T}r_{s}ds+\mathbf {M}_{T}\mathbf {-M} _{t}+\int _{t}^{T}A(s)ds\right] . \end{aligned}$$

The process $N_{z}(t,T)$ is Gaussian distributed. Thus, to determine $\mathbb {E}_{t}\left[ \frac{H_{T}^{z}}{H_{t}^{z}}\right]$, we have just to compute $\mathbb {E}_{t}\left[ N_{z}(t,T)\right]$ and ${ Var}_{t}\left[ N_{z}(t,T)\right]$, since we have:

$$\begin{aligned} \mathbb {E}_{t}\left[ \frac{H_{T}^{z}}{H_{t}^{z}}\right] =\exp \left[ \mathbb { E}_{t}\left[ N_{z}(t,T)\right] +\frac{1}{2}{} { Var}_{t}\left[ N_{z}(t,T)\right] \right] . \end{aligned}$$

• 1. Determination of $\mathbb {E}_{t}\left[ N_{z}(t,T)\right]$. We have:

  $$\begin{aligned} \mathbb {E}_{t}\left[ N_{z}(t,T)\right] =z\left( \int _{t}^{T}\mathbb {E}_{t} \left[ r_{u}\right] \,du+\int _{t}^{T}A(s)ds\right) . \end{aligned}$$

  We deduce: for $u>t$,

  $$\begin{aligned} \mathbb {E}_{t}\left[ r_{u}\right] =\overline{r}+(r_{t}-\overline{r} )e^{-(u-t)k_{r}}. \end{aligned}$$

  Finally, we get:

  $$\begin{aligned} \mathbb {E}_{t}\left[ N_{z}(t,T)\right] =z(T-t)\left( \overline{r}+ \left( r_{t}- \overline{r}\right) \frac{\beta _{r}(T-t)}{T-t}+\frac{1}{T-t}\int _{t}^{T}A(s)ds \right) , \end{aligned}$$

  which is equivalent to:

  $$\begin{aligned} \mathbb {E}_{t}\left[ N_{z}(t,T)\right] =z\Phi _{(t,T)}, \end{aligned}$$

  with:

  $$\begin{aligned} \Phi _{(t,T)}=(T-t)\left[ \overline{r}+(r_{t}-\overline{r})\frac{\beta _{r}(T-t)}{(T-t)}+\frac{1}{T-t}\int _{t}^{T}A(s)ds\right] . \end{aligned}$$

• 2. Determination of ${ Var}_{t}\left[ N_{z}(t,T)\right]$. Step 1: computation of ${ Var}_{t}\left[ \int _{t}^{T}r_{s}ds\right]$ - To calculate ${ Var}_{t}\left[ \int _{t}^{T}r_{s}ds\right] ,$ we use the Fubini’s property for stochastic integrals. We have:

  $$\begin{aligned} r_{t}= & {} \overline{r}+(r_{s}-\overline{r})e^{-(t-s)k_{r}}+a_{r}e^{-(t-s)k_{r}} \int _{s}^{t}e^{(u-s)k_{r}}dW_{u}^{r}.\\ \int _{t}^{T}r_{u}du= & {} \overline{r}(T-t)+(r_{t}-\overline{r})\beta _{r}(T-t)\\&+\int _{t}^{T}\left( a_{r}e^{-(u-t)k_{r}}\int _{t}^{u}e^{(v-t)k_{r}}dW_{v}^{r}\right) du. \end{aligned}$$

Then:

$$\begin{aligned} \int _{t}^{T}\left[ a_{r}e^{-(u-t)k_{r}}\int _{t}^{u}e^{(s-t)k_{r}}dW_{s}^{r} \right] du= & {} \int _{t}^{T}\left( \int _{s}^{T}a_{r}e^{-(u-t)k_{r}}e^{(s-t)k_{r}}du\right) dW_{s}^{r}, \\= & {} \int _{t}^{T}\left( a_{r}\left[ \frac{1-e^{-(T-s)k_{r}}}{k_{r}}\right] \right) dW_{s}^{r},\\ { Var}_{t}\left[ \int _{t}^{T}r_{s}\,ds\right]= & {} a_{r}^{2}\int _{t}^{T}\left[ \frac{1-e^{-(T-s)k_{r}}}{k_{r}}\right] ^{2}ds,\\ { Var}_{t}\left[ \int _{t}^{T}r_{s}\,ds\right]= & {} \frac{a_{r}^{2}}{k_{r}^{2}} \left[ \left( T-t\right) +\frac{1}{2k_{r}}\left[ 1-\exp (-2(T-t)k_{r})\right] -2\beta _{r}(T-t)\right] . \end{aligned}$$

Step 2: computation of ${ Var}_{t}\left[ \mathbf {M}_{T}\mathbf {-M}_{t}\right]$:

$$\begin{aligned} { Var}_{t}\left[ \mathbf {M}_{T}\mathbf {-M}_{t}\right] =2\int _{t}^{T}A(s)ds. \end{aligned}$$

Step 3: computation of ${ Cov}_{t}\left[ \int _{t}^{T}r_{s}\,ds;\mathbf {M} _{T}\mathbf {-M}_{t}\right]$:

$$\begin{aligned} { Cov}_{t}\left[ \int _{t}^{T}r_{s}\,ds;\mathbf {M}_{T}\mathbf {-M}_{t} \right] =\int _{t}^{T}a_{r}\left( \lambda _{r,s}+\lambda _{P,s}\rho ^{r,P}+\lambda _{S,s}\rho ^{r,S}\right) \left[ \frac{1-e^{-(T-s)k_{r}}}{k_{r} }\right] ds \end{aligned}$$

To conclude, recall that:

$$\begin{aligned} { Var}_{t}\left[ N_{z}(t,T)\right] = z^{2}\left( { Var}_{t}\left[ \int _{t}^{T}r_{s}\,ds\right] \,{+}\,{ Var}_{t}\left[ \mathbf {M}_{T}\mathbf {-M}_{t}\right] \,{+}\,2{ Cov}_{t}\left[ \int _{t}^{T}r_{s}\,ds,\,\mathbf {M}_{T}\mathbf {-M}_{t}\right] \right) . \end{aligned}$$

Therefore, we get:

$$\begin{aligned} { Var}_{t}\left[ N_{z}(t,T)\right] = z^{2}\left( \begin{array}{c} \frac{a_{r}^{2}}{k_{r}^{2}}\left[ \left( T-t\right) +\frac{1}{2k_{r}}\left[ 1-\exp (-2(T-t)k_{r})\right] -2\beta _{r}(T-t)\right] \\ +2\int _{t}^{T}A(s)ds+2\int _{t}^{T}a_{r}\left( \lambda _{r,,s}+\lambda _{P,s}\rho ^{r,P}+\lambda _{S,s}\rho ^{r,S}\right) \left[ \frac{ 1-e^{-(T-s)k_{r}}}{k_{r}}\right] ds \end{array} \right) . \end{aligned}$$

Appendix B: Optimal weights

In what follows, we consider the CRRA case. At time t, the optimal CRRA portfolio value is given by:

$$\begin{aligned} V_{T}^{{ CRRA}}=\left( \frac{V_{0}}{\mathbb {E}\left[ H_{T}^{\left( \frac{1-\gamma }{ \gamma }\right) }\right] }\right) H_{T}^{\left( \frac{1}{\gamma }\right) }. \end{aligned}$$

(34)

Denote

$$\begin{aligned} \widetilde{A}=\left( \frac{V_{0}}{\mathbb {E}\left[ H_{T}^{\left( \frac{1-\gamma }{\gamma }\right) }\right] }\right) . \end{aligned}$$

To compute the optimal weights $x_{C},x_{B},x_{P},$ and $x_{_{S}}$, first we calculate $V_{t}^{{ CRRA}}$. We have:

$$\begin{aligned} V_{t}^{{ CRRA}}\frac{1}{H_{t}}=\mathbb {E}_{t}\left[ \frac{1}{H_{T}}V_{T}^{{ CRRA}} \right] =\widetilde{A}H_{t}^{\left( \frac{1}{\gamma }\right) }\mathbb {E}_{t}\left[ \left( H_{T}/H_{t}\right) ^{\left( \frac{1}{\gamma }-1\right) }\right] . \end{aligned}$$

which implies

$$\begin{aligned} V_{t}^{{ CRRA}}=\widetilde{A}H_{t}^{\left( \frac{1}{\gamma }\right) }\mathbb {E}_{t}\left[ \left( H_{T}/H_{t}\right) ^{z}\right] \quad \text {with}\quad z=\frac{1}{\gamma }-1. \end{aligned}$$

Using relation (25), we get:

$$\begin{aligned} V_{t}^{{ CRRA}}=\widetilde{A}H_{t}^{\left( \frac{1}{\gamma }\right) }\exp \left[ z\Phi _{(t,\,T)}+\frac{1}{2}z^{2}\Psi _{(0,\,T)}\right] \quad \text {with}\quad z=\frac{1}{ \gamma }-1. \end{aligned}$$

Then, we determine $\frac{dV_{t}^{{ CRRA}}}{V_{t}^{{ CRRA}}}$ by using Relation (34). Applying Ito’s formula, we get the martingale part of $\frac{ dV_{t}^{{ CRRA}}}{V_{t}^{{ CRRA}}}$:

$$\begin{aligned} dV_{t}^{{ CRRA}}= & {} (\ldots )dt \\&+\,\frac{1}{\gamma }V_{t}^{{ CRRA}}\left( \lambda _{r,t}dW_{t}^{r}+\lambda _{P,t}dW_{t}^{P}+\lambda _{S,t}dW_{t}^{S}\right) \\&+\,V_{t}^{{ CRRA}}z\left[ \beta _{r}a_{r}dW_{t}^{r}\right] \end{aligned}$$

We obtain the following system:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\lambda _{r,t}}{\gamma }+za_{r}\beta _{r}(T-t) =\delta _{r}(t)= -x_{B}(t)\beta _{r}(D_{B})a_{r} \\ \frac{\lambda _{P,t}}{\gamma } =\delta _{P}(t)= x_{P}(t)\sigma _{t}^{P} \\ \frac{\lambda _{S,t}}{\gamma } =\delta _{S}(t)= x_{S}(t)\sigma _{t}^{S} \end{array} \right. \end{aligned}$$

Therefore, we deduce that, at any time t, the weights are given by:

$$\begin{aligned} \left\{ \begin{array}{l} x_{B}^{{ CRRA}}(t) = -\frac{\delta _{i}(t)}{\beta _{i}(D)a_{i}} \\ x_{P}^{{ CRRA}}(t) = \frac{\delta _{P}(t)}{\sigma _{t}^{P}} \\ x_{S}^{{ CRRA}}(t) = \frac{\delta _{S}(t)}{\sigma _{t}^{S}} \\ x_{C}^{{ CRRA}}(t) = 1-x_{B}^{{ CRRA}}-x_{P}^{{ CRRA}}-x_{S}^{{ CRRA}} \end{array} \right. \end{aligned}$$

Recall that $\Sigma$ is the matrix:

$$\begin{aligned} \Sigma =\left[ \begin{array}{c@{\quad }c@{\quad }c} -\beta _{r}(D_{B})a_{r} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad \sigma _{t}^{P} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \sigma _{t}^{S} \end{array} \right] . \end{aligned}$$

We have:

$$\begin{aligned} \left[ \begin{array}{c} \frac{\lambda _{r,t}}{\gamma }+za_{r}\beta _{r}(T-t) \\ \frac{\lambda _{P}}{\gamma } \\ \frac{\lambda _{S}}{\gamma } \end{array} \right] ={^{t}\Sigma }\cdot \left[ \begin{array}{c} x_{B}^{{ CRRA}}(t) \\ x_{P}^{{ CRRA}}(t) \\ x_{S}^{{ CRRA}}(t) \end{array} \right] , \end{aligned}$$

thus:

$$\begin{aligned} \left[ \begin{array}{c} x_{B}^{{ CRRA}}(t) \\ x_{P}^{{ CRRA}}(t) \\ x_{S}^{{ CRRA}}(t) \end{array} \right] ={^{t}\Sigma ^{-1}}\cdot \left( \left[ \begin{array}{c} \frac{\lambda _{r,t}}{\gamma }+za_{r}\beta _{r}(T-t) \\ \frac{\lambda _{P,t}}{\gamma } \\ \frac{\lambda _{S,t}}{\gamma } \end{array} \right] \right) . \end{aligned}$$

Recall that $z=\frac{1}{\gamma }-1$ and that the optimal portfolio for the logarithm case $X^{{ Log}}$ is given by:

$$\begin{aligned} \left[ \begin{array}{c} x_{B}^{{ Log}}(t) \\ x_{P}^{{ Log}}(t) \\ x_{S}^{{ Log}}(t) \end{array} \right] ={^{t}\Sigma ^{-1}}\cdot \left( \left[ \begin{array}{c} \lambda _{r,t} \\ \lambda _{P,t} \\ \lambda _{S,t} \end{array} \right] \right) . \end{aligned}$$

Therefore, we have:

$$\begin{aligned} X^{{ CRRA}}=\frac{1}{\gamma }X^{{ Log}}+\left( 1-\frac{1}{\gamma }\right) X^{{ Conservative}}, \end{aligned}$$

where $X^{{ Log}}$ is the optimal portfolio for the logarithm case, and $X^{{ Conservative}}$ corresponds to the optimal portfolio for an infinite relative risk aversion $\gamma$. This latter portfolio is given by:

$$\begin{aligned} \left[ \begin{array}{c} x_{B}^{{ Conservative}}(t) \\ x_{P}^{{ Conservative}}(t) \\ x_{S}^{{ Conservative}}(t) \end{array} \right] =\, ^{t}\Sigma ^{-1}\cdot \left( \left[ \begin{array}{c} -a_{r}\beta _{r}(T-t) \\ 0 \\ 0 \end{array} \right] \right) . \end{aligned}$$

Appendix C (Calibration)

In what follows, we detail how we can calibrate drifts, volatilities and instantaneous correlations of the Brownian motions to respectively expected returns per year, standard deviations per year and assets correlations.¹⁵ Indeed, most of the empirical results about mean reverting asset returns are based on cumulated returns on given time periods [0, T] which are further annualized (see e.g. Campbell and Viceira 2002, 2005; Rehring 2012; Pagliari 2017). Using a continuous-time approach, we have to identify the parameters corresponding to instantaneous variations using those which correspond to annualized characteristics of cumulated returns.

In what follows, we consider two financial assets X and Y defined by:

$$\begin{aligned} dX_{t}= & {} X_{t}\left( \mu _{t}^{X}dt+\sigma _{t}^{X}dW_{t}^{X}\right) , \end{aligned}$$

(35)

$$\begin{aligned} dY_{t}= & {} Y_{t}\left( \mu _{t}^{Y}dt+\sigma _{t}^{Y}dW_{t}^{Y}\right) , \end{aligned}$$

(36)

where $W_{t}=\left( W_{t}^{X},W_{t}^{Y}\right) _{1\le i\le d}$ is a 2-dimensional Brownian motion with correlation matrix $\Sigma _{c,T}$ given by

$$\begin{aligned} \Sigma _{c,T}=\left[ \begin{array}{c@{\quad }c} 1 &{} \rho _{T}^{W^{X},W^{Y}} \\ \rho _{T}^{W^{X},W^{Y}} &{} 1 \end{array} \right] , \end{aligned}$$

where $\rho _{T}^{W^{X},W^{Y}}=\left\langle W_{t}^{X},W_{t}^{Y}\right\rangle _{t}$ is a constant calibrated for a fixed management period [0, T]. Denote:

$$\begin{aligned} \Sigma _{t}=\left[ \begin{array}{c@{\quad }c} \sigma _{t}^{X} &{} 0 \\ 0 &{} \sigma _{t}^{Y} \end{array} \right] . \end{aligned}$$

1.1 Computation of expected returns and variances

Denote $\theta _{t}^{X}=\mu _{t}^{X}-r_{t}$. The random variable $X_{T}$ can be expressed as $X_{T}=\exp \left[ N_{T}^{X}\right]$ where $N_{T}^{X}$ has a Gaussian distribution with

$$\begin{aligned} \mathbb {E}\left[ N_{T}^{X}\right] =\Phi _{T}^{X}, \end{aligned}$$

where

$$\begin{aligned} \Phi _{T}^{X}=\left[ \overline{r}T+(r_{0}-\overline{r})\beta _{r}(T)+\int _{0}^{T}\left( \theta _{s}^{X}-\frac{1}{2}\left[ \sigma _{s}^{X} \right] ^{2}\right) ds\right] , \end{aligned}$$

and

$$\begin{aligned} { Var}\left[ N_{T}^{X}\right] =\Psi _{T}^{X}, \end{aligned}$$

with

$$\begin{aligned} \Psi _{T}^{X}= & {} \frac{a_{r}^{2}}{k_{r}^{2}}\left[ T+\frac{1}{2k_{r}}\left[ 1-\exp (-2Tk_{r})\right] -2\beta _{r}(T)\right] \\&+\int _{0}^{T}\left[ \sigma _{s}^{X}\right] ^{2}ds+2a_{r}\rho _{T}^{Wr,W^{X}}\int _{0}^{T}\sigma _{s}^{X}\left[ \frac{1-e^{-(t-s)k_{r}}}{ k_{r}}\right] ds \end{aligned}$$

Therefore, we have:

$$\begin{aligned} \mathbb {E}\left[ X_{T}\right] =X_{0}\exp \left[ \Phi _{T}^{X}+\frac{1}{2} \Psi _{T}^{X}\right] , \end{aligned}$$

and the variance is given by:

$$\begin{aligned} { Variance}\left[ X_{T}\right] =X_{0}^{2}\exp \left[ 2\Phi _{T}^{X}+\Psi _{T}^{X}\right] \left( \exp \left[ \Psi _{T}^{X}\right] -1\right) . \end{aligned}$$

1.2 Calibration to the given return per year g(T) and to the given standard deviation per year $h\left( T\right)$

We must have:

$$\begin{aligned}&\displaystyle \frac{1}{T}\exp \left[ \Phi _{T}^{X}+\frac{1}{2}\Psi _{T}^{X}\right] =g(T), \\&\displaystyle \frac{1}{T}\exp \left[ 2\Phi _{T}^{X}+\Psi _{T}^{X}\right] \left( \exp \left[ \Psi _{T}^{X}\right] -1\right) =h\left( T\right) , \end{aligned}$$

from which, we deduce:

$$\begin{aligned}&\displaystyle \Phi _{T}^{X} ={ Log}\left[ Tg(T)/\sqrt{1+\frac{h\left( T\right) }{Tg^{2}(T)}} \right] , \\&\displaystyle \Psi _{T}^{X} ={ Log}\left[ 1+\frac{h\left( T\right) }{Tg^{2}(T)}\right] . \end{aligned}$$

Therefore, we have:

$$\begin{aligned} \text {For all }\quad T,\,\frac{\partial \Phi _{T}^{X}}{\partial T}= & {} \frac{ \partial }{\partial T}{} { Log}\left[ Tg(T)/\sqrt{1+\frac{h\left( T\right) }{ Tg^{2}(T)}}\right] , \\ \frac{\partial \Psi _{T}^{X}}{\partial T}= & {} \frac{\partial }{\partial T}{} { Log} \left[ 1+\frac{h\left( T\right) }{Tg^{2}(T)}\right] . \end{aligned}$$

Consequently, we get a first relation:

$$\begin{aligned} \frac{\partial \Phi _{T}^{X}}{\partial T}= & {} \frac{\partial }{\partial T} \left[ \overline{r}T+(r_{0}-\overline{r})\beta _{r}(T)\right] +\left( \theta _{T}^{X}-\frac{1}{2}\left[ \sigma _{T}^{X}\right] ^{2}\right) , \\= & {} \frac{\partial }{\partial T}{} { Log}\left[ Tg(T)/\sqrt{1+\frac{h\left( T\right) }{Tg^{2}(T)}}\right] , \end{aligned}$$

which yields to:

$$\begin{aligned} \left( \theta _{T}^{X}-\frac{1}{2}\left[ \sigma _{T}^{X}\right] ^{2}\right) = \frac{\partial }{\partial T}{} { Log}\left[ Tg(T)/\sqrt{1+\frac{h\left( T\right) }{ Tg^{2}(T)}}\right] -\frac{\partial }{\partial T}\left[ \overline{r}T+(r_{0}- \overline{r})\beta _{r}(T)\right] \end{aligned}$$

To get the standard deviation, we use the following second relation:

$$\begin{aligned} \frac{\partial \Psi _{T}^{X}}{\partial T}= & {} \frac{\partial }{\partial T} \left( \frac{a_{r}^{2}}{k_{r}^{2}}\left[ T+\frac{1}{2k_{r}}\left[ 1-\exp (-2Tk_{r})\right] -2\beta _{r}(T)\right] \right) \\&+\left[ \sigma _{T}^{X}\right] ^{2}+2a_{r}\frac{\partial }{\partial T} \left( \rho _{T}^{Wr,W^{X}}\int _{0}^{T}\sigma _{s}^{X}\left[ \frac{ 1-e^{-(t-s)k_{r}}}{k_{r}}\right] ds\right) \\= & {} \frac{\partial }{\partial T}{} { Log}\left[ 1+\frac{h\left( T\right) }{Tg^{2}(T) }\right] \end{aligned}$$

Finally, we get:

$$\begin{aligned}&\left[ \sigma _{T}^{X}\right] ^{2}+2a_{r}\frac{\partial }{\partial T} \left( \rho _{T}^{Wr,W^{X}}\int _{0}^{T}\sigma _{s}^{X}\left[ \frac{ 1-e^{-(T-s)k_{r}}}{k_{r}}\right] ds\right) \\&\quad =\frac{\partial }{\partial T}{} { Log}\left[ 1+\frac{h\left( T\right) }{Tg^{2}(T) }\right] -\frac{\partial }{\partial T}\left( \frac{a_{r}^{2}}{k_{r}^{2}} \left[ T+\frac{1}{2k_{r}}\left[ 1-\exp (-2Tk_{r})\right] -2\beta _{r}(T) \right] \right) , \end{aligned}$$

where we have:

$$\begin{aligned}&\frac{\partial }{\partial T}\left( \rho _{T}^{Wr,W^{X}}\int _{0}^{T}\sigma _{s}^{X}\left[ \frac{1-e^{-(T-s)k_{r}}}{k_{r}}\right] ds\right) \\&\quad =\frac{\partial }{\partial T}\left( \rho _{T}^{Wr,W^{X}}\right) \int _{0}^{T}\sigma _{s}^{X}\left[ \frac{1-e^{-(T-s)k_{r}}}{k_{r}}\right] ds+\rho _{T}^{Wr,W^{X}}\frac{\partial }{\partial T}\left( \int _{0}^{T}\sigma _{s}^{X}\left[ \frac{1-e^{-(T-s)k_{r}}}{k_{r}}\right] ds\right) . \end{aligned}$$

1.3 Computation of covariances and correlations

In what follows, we search to calibrate the correlations of the three Brownian motions, namely the three parameters ${\rho }_{{T}}{^{r,\,P}}$, ${\rho }_{{T}}{^{r,\,S}}$ and ${\rho }_{{T}}{^{P,\,S}}$ to Rehring (2012) data (annualized correlations of cumulated asset returns). For this purpose, let us denote:

$$\begin{aligned} A_{t}= & {} E\left[ \int _{0}^{t}r_{s}ds\right] =\overline{r}t+(r_{0}-\overline{r })\beta _{r}(t); \\ B_{t}= & {} \frac{a_{r}^{2}}{k_{r}^{2}}\left[ t+\frac{1}{2k_{r}}\left[ 1-\exp (-2tk_{r})\right] -2\beta _{r}(t)\right] ; \\ C_{t}= & {} a_{r}\rho _{T}^{Wr,W^{X}}\int _{0}^{t}\sigma _{s}^{X}\left[ \frac{ 1-e^{-(t-s)k_{r}}}{k_{r}}\right] ds; \\ D_{t}= & {} a_{r}\rho _{T}^{Wr,W^{Y}}\int _{0}^{t}\sigma _{s}^{Y}\left[ \frac{ 1-e^{-(t-s)k_{r}}}{k_{r}}\right] ds. \end{aligned}$$

Taking account of the interest rate randomness, the covariance of asset prices X and Y is given by:

$$\begin{aligned} { Cov}_{T}^{X,Y}= & {} X_{0}Y_{0}\exp \left[ \int _{0}^{T}\left( \theta _{s}^{X}+\theta _{s}^{Y}\right) ds+\left( 2A_{T}+B_{T}+C_{T}+D_{T}\right) \right] \\&\times \left( \exp \left[ \int _{0}^{T}\sigma _{s}^{X}\sigma _{s}^{Y}\rho _{T}^{W^{X},W^{Y}}ds+\left( B_{T}+C_{T}+D_{T}\right) \right] -1\right) . \end{aligned}$$

Taking account of the interest rate randomness, the correlation $\rho _{T}^{X,Y}$ of asset prices X and Y is given by:

$$\begin{aligned} \rho _{T}^{X,Y}={ Cov}_{T}^{X,Y}/\left( \sqrt{{ variance}\left( X_{T}\right) }\sqrt{ { variance}\left( Y_{T}\right) }\right) , \end{aligned}$$

which yields to:

$$\begin{aligned} \rho _{T}^{X,Y}= \frac{\exp \left[ \left( B_{T}+C_{T}+D_{T}\right) \right] \left( \exp \left[ \rho _{T}^{W^{X},W^{Y}}\int _{0}^{T}\sigma _{s}^{X}\sigma _{s}^{Y}ds+\left( B_{T}+C_{T}+D_{T}\right) \right] -1\right) }{\sqrt{\left( \exp \left[ \int _{0}^{T}\left[ \sigma _{s}^{X}\right] ^{2}ds\right] -1\right) }\sqrt{ \left( \exp \left[ \int _{0}^{T}\left[ \sigma _{s}^{Y}\right] ^{2}ds\right] -1\right) }}. \end{aligned}$$

Thus, we get:

$$\begin{aligned} \rho _{T}^{W^{X},W^{Y}}= & {} \frac{1}{\int _{0}^{T}\left( \sigma _{s}^{X}\sigma _{s}^{Y}\right) ds}\\&\times \left( \begin{array}{c} { Log}\left[ \begin{array}{c} 1+\rho _{T}^{X,Y}\exp \left[ -\left( B_{T}+C_{T}+D_{T}\right) \right] \\ \sqrt{\left( \exp \left[ \int _{0}^{T}\left[ \sigma _{s}^{X}\right] ^{2}ds \right] -1\right) }\sqrt{\left( \exp \left[ \int _{0}^{T}\left[ \sigma _{s}^{Y}\right] ^{2}ds\right] -1\right) } \end{array} \right] \\ -\left( B_{T}+C_{T}+D_{T}\right) \end{array} \right) . \end{aligned}$$

Note that, if $r_{t}$ were deterministic, then we would have a special case with $\mu _{s}^{X}=r_{s}+\theta _{s}^{X}$, $\mu _{s}^{Y}=r_{s}+\theta _{s}^{Y}$ and $B_{t}=C_{t}=D_{t}=0$. In such a case, the covariance of asset prices X and Y is given by:

$$\begin{aligned} \sigma _{T}^{X,Y}=X_{0}Y_{0}\exp \left[ \int _{0}^{T}\left( \mu _{s}^{X}+\mu _{s}^{Y}\right) ds\right] \left( \exp \left[ \rho _{T}^{W^{X},W^{Y}}\int _{0}^{T}\sigma _{s}^{X}\sigma _{s}^{Y}ds\right] -1\right) \end{aligned}$$

Therefore, for a fixed management period [0, T] and for a given correlation function $\rho _{T}^{X,Y}$ defined by:

$$\begin{aligned} \rho _{T}^{X,Y}={ Cov}_{T}^{X,Y}/\left( \sqrt{{ variance}\left( X_{T}\right) }\sqrt{ { variance}\left( Y_{T}\right) }\right) , \end{aligned}$$

we find:

$$\begin{aligned} \rho _{T}^{X,Y}= \frac{\left( \exp \left[ \rho _{T}^{W^{X},W^{Y}}\int _{0}^{T}\sigma _{s}^{X}\sigma _{s}^{Y}ds\right] -1\right) }{\sqrt{\left( \exp \left[ \int _{0}^{T}\left[ \sigma _{s}^{X}\right] ^{2}ds\right] -1\right) }\sqrt{ \left( \exp \left[ \int _{0}^{T}\left[ \sigma _{s}^{Y}\right] ^{2}ds\right] -1\right) }}. \end{aligned}$$

Thus, we get:

$$\begin{aligned} \rho _{T}^{W^{X},W^{Y}}= & {} \frac{1}{\int _{0}^{T}\sigma _{s}^{X}\sigma _{s}^{Y}ds }\\&\times \,\, { Log}\left[ 1+\rho _{T}^{X,Y}\sqrt{\left( \exp \left[ \int _{0}^{T}\left[ \sigma _{s}^{X}\right] ^{2}ds\right] {-}1\right) }\sqrt{\left( \exp \left[ \int _{0}^{T}\left[ \sigma _{s}^{Y}\right] ^{2}ds\right] -1\right) }\right] . \end{aligned}$$

The previous formula can be applied not only to the real estate–stock case but also, when the bond is involved. However, as detailed in Sect. 2, the bond modelling is such that the term structure is affine (see Relation 7).