Appendix A: Fixed Housing Demand
This appendix derives the optimal portfolios of a household with fixed housing demand starting from the following form of the optimization problem:
$$ \mathop{{\max }}\limits_{{\mathbf{w}}} \,\,\tfrac{1}{{1 - \gamma }}\left[ {{E_t}\left( {r_{{t + 1}}^U} \right) + \tfrac{1}{2}Va{r_t}\left( {r_{{t + 1}}^U} \right)} \right] $$
It is convenient to define \( {r_p}_{{,t + {1}}} \equiv { \ln }\left( {{1 + }{R_p}_{{,t + {1}}}} \right) \) and \( r_{{i,t + 1}}^W{{ = ln}} \left( {{1 + }R_{{i,t + 1}}^W} \right) \) .We can apply Campbell and Viceira’s lognormal approximation to \( r_{{t + 1}}^U \):
$$ \begin{gathered} r_{{t + 1}}^U = \left( {1 - \theta } \right)\left( {1 - \gamma } \right)\sum\limits_{{i = 1}}^m {{q_i}r_{{i,t + 1}}^W} + \tfrac{{{{\left( {1 - \theta } \right)}^2}{{\left( {1 - \gamma } \right)}^2}}}{2}\sum\limits_{{i = 1}}^m {{q_i}Va{r_t}\left( {r_{{i,t + 1}}^W} \right) - } \tfrac{{{{\left( {1 - \theta } \right)}^2}{{\left( {1 - \gamma } \right)}^2}}}{2}Va{r_t}\left( {\sum\limits_{{i = 1}}^m {{q_i}r_{{i,t + 1}}^W} } \right) \hfill \\ {E_t}\left( {r_{{t + 1}}^U} \right) = \left( {1 - \theta } \right)\left( {1 - \gamma } \right)\sum\limits_{{i = 1}}^m {{q_i}{E_t}\left( {r_{{i,t + 1}}^W} \right)} + \tfrac{{{{\left( {1 - \theta } \right)}^2}{{\left( {1 - \gamma } \right)}^2}}}{2}\sum\limits_{{i = 1}}^m {{q_i}Va{r_t}\left( {r_{{i,t + 1}}^W} \right) - } \tfrac{{{{\left( {1 - \theta } \right)}^2}{{\left( {1 - \gamma } \right)}^2}}}{2}Va{r_t}\left( {\sum\limits_{{i = 1}}^m {{q_i}r_{{i,t + 1}}^W} } \right) \hfill \\ Va{r_t}\left( {r_{{t + 1}}^U} \right) = {\left( {1 - \theta } \right)^2}{\left( {1 - \gamma } \right)^2}\sum\limits_{{i = 1}}^m {Va{r_t}\left( {{q_i}r_{{i,t + 1}}^W} \right)} \hfill \\ \end{gathered} $$
These values are substituted in the optimization problem:
$$ \mathop{{\max }}\limits_{{\mathbf{w}}} \,\,\sum\limits_{{i = 1}}^m {{q_i}{E_t}\left( {r_{{i,t + 1}}^W} \right)} + \tfrac{{\left( {1 - \theta } \right)\left( {1 - \gamma } \right)}}{2}\sum\limits_{{i = 1}}^m {{q_i}Va{r_t}\left( {r_{{i,t + 1}}^W} \right)} $$
Next, we apply Campbell and Viceira’s lognormal approximation to \( r_{{i,t + 1}}^W \):
$$ \begin{gathered} r_{{i,t + 1}}^W = {u_i}{r_{{p,t + 1}}} + {v_i}r_{{i,t + 1}}^L + {x_i}r_{{o,t + 1}}^a + {x_i}{b_{{o,t + 1}}} - {y_i}r_{{i,t + 1}}^a - {y_i}{b_{{i,t + 1}}} + \hfill \\ + \tfrac{1}{2}{u_i}\sigma_p^2 + \tfrac{1}{2}{v_i}{\left( {\sigma_{{i,t + 1}}^L} \right)^2} + \tfrac{1}{2}{x_i}{\left( {\sigma_{{o,t + 1}}^a} \right)^2} + \tfrac{1}{2}{x_i}{\left( {\sigma_{{o,t + 1}}^b} \right)^2} - \tfrac{1}{2}{y_i}{\left( {\sigma_{{i,t + 1}}^a} \right)^2} - \tfrac{1}{2}{y_i}{\left( {\sigma_{{i,t + 1}}^b} \right)^2} - \hfill \\ - \tfrac{1}{2}Va{r_t}\left[ {{u_i}{r_{{p,t + 1}}} + {v_i}r_{{i,t + 1}}^L + {x_i}r_{{o,t + 1}}^a + {x_i}{b_{{o,t + 1}}} - {y_i}r_{{i,t + 1}}^a - {y_i}{b_{{i,t + 1}}}} \right] \hfill \\ {E_t}\left( {r_{{i,t + 1}}^W} \right) = {u_i}{E_t}\left( {{r_{{p,t + 1}}}} \right) + {v_i}{E_t}\left( {r_{{i,t + 1}}^L} \right) + {x_i}{E_t}\left( {{b_{{o,t + 1}}}} \right) - {y_i}{E_t}\left( {r_{{i,t + 1}}^a} \right) - {y_i}{E_t}\left( {{b_{{i,t + 1}}}} \right) + \hfill \\ + \tfrac{1}{2}{u_i}\sigma_p^2 + \tfrac{1}{2}{v_i}{\left( {\sigma_{{i,t + 1}}^L} \right)^2} + \tfrac{1}{2}{x_i}{\left( {\sigma_{{o,t + 1}}^a} \right)^2} + \tfrac{1}{2}{x_i}{\left( {\sigma_{{o,t + 1}}^b} \right)^2} - \tfrac{1}{2}{y_i}{\left( {\sigma_{{i,t + 1}}^a} \right)^2} - \tfrac{1}{2}{y_i}{\left( {\sigma_{{i,t + 1}}^b} \right)^2} - \hfill \\ - \tfrac{1}{2}Va{r_t}\left[ {{u_i}{r_{{p,t + 1}}} + {v_i}r_{{i,t + 1}}^L + {x_i}r_{{o,t + 1}}^a + {x_i}{b_{{o,t + 1}}} - {y_i}r_{{i,t + 1}}^a - {y_i}{b_{{i,t + 1}}}} \right] \hfill \\ Va{r_t}\left( {r_{{i,t + 1}}^W} \right) = Va{r_t}\left[ {{u_i}{r_{{p,t + 1}}} + {v_i}r_{{i,t + 1}}^L + {x_i}r_{{o,t + 1}}^a + {x_i}{b_{{o,t + 1}}} - {y_i}r_{{i,t + 1}}^a - {y_i}{b_{{i,t + 1}}}} \right] \hfill \\ \end{gathered} $$
After substituting in the optimization problem and dropping constants, we obtain:
$$ \mathop{{\max }}\limits_{{\mathbf{w}}} \,\,\left( {\sum\limits_{{i = 1}}^m {{q_i}{u_i}} } \right)\left[ {{E_t}\left( {{r_{{p,t + 1}}}} \right) + \tfrac{1}{2}\sigma_p^2} \right] + \tfrac{{\left( {1 - \theta } \right)\left( {1 - \gamma } \right) - 1}}{2}\sum\limits_{{i = 1}}^m {{q_i}Va{r_t}\left( {r_{{i,t + 1}}^W} \right)} $$
We can then expand Var
t
(r
i,t+1
W) and drop constant terms:
$$ \begin{gathered} \mathop{{\max }}\limits_{{\mathbf{w}}} \,\,\left( {\sum\limits_{{i = 1}}^m {{q_i}{u_i}} } \right)\left[ {{E_t}\left( {{r_{{p,t + 1}}}} \right) + \tfrac{1}{2}\sigma_p^2} \right] + \left[ {\left( {1 - \theta } \right)\left( {1 - \gamma } \right) - 1} \right] \times \hfill \\ \times \left\{ {\tfrac{1}{2}\sigma_p^2\sum\limits_{{i = 1}}^m {{q_i}u_i^2} + \sum\limits_{{i = 1}}^m {{q_i}{u_i}{v_i}Co{v_t}\left( {{r_{{p,t + 1}}},r_{{i,t + 1}}^L} \right)} + \sum\limits_{{i = 1}}^m {{q_i}{u_i}{x_i}Co{v_t}\left( {{r_{{p,t + 1}}},r_{{o,t + 1}}^a} \right)} - \sum\limits_{{i = 1}}^m {{q_i}{u_i}{y_i}Co{v_t}\left( {{r_{{p,t + 1}}},r_{{i,t + 1}}^a} \right)} } \right\} \hfill \\ \end{gathered} $$
Next, we apply the lognormal approximation formula to \( {r_p}_{{,t + {1}}}{{ = ln}} \left( {{1 + }{R_p}_{{,t + {1}}}} \right) \).
$$ \begin{gathered} {r_{{p,t + 1}}} = {r_{{f,t + 1}}} + {\mathbf{w\prime}}\left( {{\mathbf{r}}_{{t + 1}}^a - {r_{{f,t + 1}}}1} \right) + \tfrac{1}{2}{\mathbf{w\prime}}diag\left( \Sigma \right) - \tfrac{1}{2}{\mathbf{w\prime}}\Sigma {\mathbf{w}} \hfill \\ {E_t}\left( {{r_{{p,t + 1}}}} \right) = {r_{{f,t + 1}}} + {\mathbf{w\prime}}\left[ {{E_t}\left( {{\mathbf{r}}_{{t + 1}}^a} \right) - {r_{{f,t + 1}}}1} \right] + \tfrac{1}{2}{\mathbf{w\prime}}diag\left( \Sigma \right) - \tfrac{1}{2}{\mathbf{w\prime}}\Sigma {\mathbf{w}} \hfill \\ Va{r_t}\left( {{r_{{p,t + 1}}}} \right) = {\mathbf{w\prime}}\Sigma {\mathbf{w}} \hfill \\ \end{gathered} $$
We again substitute in the optimization problem and drop constant terms:
$$ \begin{gathered} \mathop{{\max }}\limits_{{\mathbf{w}}} \,\,\left( {\sum\limits_{{i = 1}}^m {{q_i}{u_i}} } \right){\mathbf{w\prime}}\left[ {{E_t}\left( {{\mathbf{r}}_{{t + 1}}^a} \right) - {r_{{f,t + 1}}}1 + \tfrac{1}{2}diag\left( \Sigma \right)} \right] + \left[ {\left( {1 - \theta } \right)\left( {1 - \gamma } \right) - 1} \right] \times \left\{ {\tfrac{1}{2}{\mathbf{w\prime}}\Sigma {\mathbf{w}}\sum\limits_{{i = 1}}^m {{q_i}u_i^2} } \right. + \hfill \\ + \left. {{\mathbf{w\prime}}\sum\limits_{{i = 1}}^m {Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,r_{{i,t + 1}}^L} \right)} {q_i}{u_i}{v_i} + {\mathbf{w\prime}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,r_{{o,t + 1}}^a} \right)\sum\limits_{{i = 1}}^m {{q_i}{u_i}{x_i}} - {\mathbf{w\prime}}\sum\limits_{{i = 1}}^m {Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,r_{{i,t + 1}}^a} \right){q_i}{u_i}{y_i}} } \right\} \hfill \\ \end{gathered} $$
The first order condition is:
$$ \begin{gathered} \,\left( {\sum\limits_{{i = 1}}^m {{q_i}{u_i}} } \right)\left[ {{E_t}\left( {{\mathbf{r}}_{{t + 1}}^a} \right) - {r_{{f,t + 1}}}1 + \tfrac{1}{2}diag\left( \Sigma \right)} \right] + \left[ {\left( {1 - \theta } \right)\left( {1 - \gamma } \right) - 1} \right] \times \hfill \\ \times \left\{ {\Sigma {\mathbf{w}}\sum\limits_{{i = 1}}^m {{q_i}u_i^2} } \right. + \left. {\sum\limits_{{i = 1}}^m {Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,r_{{i,t + 1}}^L} \right)} {q_i}{u_i}{v_i} + Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,r_{{o,t + 1}}^a} \right)\sum\limits_{{i = 1}}^m {{q_i}{u_i}{x_i}} - \sum\limits_{{i = 1}}^m {Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,r_{{i,t + 1}}^a} \right){q_i}{u_i}{y_i}} } \right\} = 0 \hfill \\ \end{gathered} $$
This leads to the optimal portfolio weights:
$$ \begin{gathered} \,{\mathbf{w}} = \tfrac{A}{{\gamma + \theta \left( {1 - \gamma } \right)}}{\Sigma^{{ - 1}}}\left[ {{E_t}\left( {{\mathbf{r}}_{{t + 1}}^a} \right) - {r_{{f,t + 1}}}1 + \tfrac{1}{2}diag\left( \Sigma \right)} \right] - \hfill \\ - {\Sigma^{{ - 1}}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,{\mathbf{r\prime}}_{{t + 1}}^L} \right){\mathbf{z}} - \tfrac{{{H_t}{F_{{o,t}}}}}{{{W_t}}}{\Sigma^{{ - 1}}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,r_{{o,t + 1}}^a} \right) + {\Sigma^{{ - 1}}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,{\mathbf{r\prime}}_{{t + 1}}^{{fwd}}} \right){\mathbf{s}} = 0 \hfill \\ \end{gathered} $$
where
$$ \begin{gathered} A \equiv \sum\limits_{{i = 1}}^m {\tfrac{{{W_{{i,net}}}}}{{{W_t}}}} \tfrac{{{p_i}W_{{i,net}}^{{\left( {1 - \theta } \right)\left( {1 - \gamma } \right) - 2}}}}{{\sum {{p_i}W_{{i,net}}^{{\left( {1 - \theta } \right)\left( {1 - \gamma } \right) - 2}}} }} \hfill \\ {s_i} \equiv \tfrac{{{{\bar{H}}_{{t + 1}}}{F_{{i,t}}}}}{{{W_t}}}\tfrac{{{p_i}W_{{i,net}}^{{\left( {1 - \theta } \right)\left( {1 - \gamma } \right) - 2}}}}{{\sum {{p_j}W_{{j,net}}^{{\left( {1 - \theta } \right)\left( {1 - \gamma } \right) - 2}}} }} \hfill \\ {z_i} \equiv \tfrac{{V_{{i,t}}^L}}{{{W_t}}}\tfrac{{{p_i}W_{{i,net}}^{{\left( {1 - \theta } \right)\left( {1 - \gamma } \right) - 2}}}}{{\sum {{p_j}W_{{j,net}}^{{\left( {1 - \theta } \right)\left( {1 - \gamma } \right) - 2}}} }} \hfill \\ {W_{{i,net}}} \equiv {W_t} + V_{{i,t}}^L + {H_t}{F_{{o,t}}} - {{\bar{H}}_{{t + 1}}}{F_{{i,t}}} \hfill \\ \end{gathered} $$
The expression \( {\Sigma^{{ - {1}}}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a, r_{{o,t + 1}}^a} \right) \) is equal to oth column of the n × n identity matrix. The expression \( {\Sigma^{{ - {1}}}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,{\mathbf{r\prime}}_{{t + 1}}^{{fwd}}} \right) \) is equal to the first m columns of the n × n identity matrix. The solution reduces to:
$$ \begin{gathered} {w_o} = \tfrac{A}{{\gamma + \theta \left( {1 - \gamma } \right)}}\Sigma_{{o\bullet }}^{{ - 1}}\left[ {\ln E_t^{{\mathbf{r}}}\left( {1 + {\mathbf{R}}_{{t + 1}}^a} \right) - {r_{{f,t + 1}}}1} \right] - \Sigma_{{o\bullet }}^{{ - 1}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,{\mathbf{r\prime}}_{{t + 1}}^L} \right){\mathbf{z}} + \left[ {{s_o} - \tfrac{{{H_t}{F_{{o,t}}}}}{{{W_t}}}} \right] \hfill \\ {w_i} = \tfrac{A}{{\gamma + \theta \left( {1 - \gamma } \right)}}\Sigma_{{i\bullet }}^{{ - 1}}\left[ {\ln E_t^{{\mathbf{r}}}\left( {1 + {\mathbf{R}}_{{t + 1}}^a} \right) - {r_{{f,t + 1}}}1} \right] - \Sigma_{{i\bullet }}^{{ - 1}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,{\mathbf{r\prime}}_{{t + 1}}^L} \right){\mathbf{z}} + \left[ {{s_i}} \right]{, }i \in \left\{ {1,2,..,m} \right\} - \left\{ o \right\} \hfill \\ {w_j} = \tfrac{A}{{\gamma + \theta \left( {1 - \gamma } \right)}}\Sigma_{{j\bullet }}^{{ - 1}}\left[ {\ln E_t^{{\mathbf{r}}}\left( {1 + {\mathbf{R}}_{{t + 1}}^a} \right) - {r_{{f,t + 1}}}1} \right] - \Sigma_{{j\bullet }}^{{ - 1}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,{\mathbf{r\prime}}_{{t + 1}}^L} \right){\mathbf{z}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,i \in \left\{ {m + 1,..,n} \right\} \hfill \\ \end{gathered} $$
Appendix B: Flexible Housing Demand
This appendix derives the optimal portfolios of a household with flexible housing demand. The optimization problem is:
$$ \mathop{{\max }}\limits_{{\mathbf{w}}} \,\,\tfrac{1}{{1 - \gamma }}\left[ {{E_t}\left( {r_{{t + 1}}^U} \right) + \tfrac{1}{2}Va{r_t}\left( {r_{{t + 1}}^U} \right)} \right] $$
We apply the lognormal approximation formula to \( r_{{t + 1}}^U{{ = ln}} \left( {{1 + }R_{{t + 1}}^U} \right) \):
$$ \begin{gathered} r_{{t + 1}}^U = \sum\limits_{{i = 1}}^m {{q_i}\left( {1 - \gamma } \right)r_{{i,t + 1}}^W} - \sum\limits_{{i = 1}}^m {{q_i}\left( {1 - \gamma } \right)\theta r_{{i,t + 1}}^a} - \sum\limits_{{i = 1}}^m {{q_i}\left( {1 - \gamma } \right)\theta {b_{{i,t + 1}}}} + \hfill \\ + \tfrac{1}{2}\sum\limits_{{i = 1}}^m {{q_i}Va{r_t}\left[ {\left( {1 - \gamma } \right)r_{{i,t + 1}}^W - \left( {1 - \gamma } \right)\theta r_{{i,t + 1}}^a - \left( {1 - \gamma } \right)\theta {b_{{i,t + 1}}}} \right]} - \hfill \\ - \tfrac{1}{2}Va{r_t}\left\{ {\sum\limits_{{i = 1}}^m {{q_i}\left[ {\left( {1 - \gamma } \right)r_{{i,t + 1}}^W - \left( {1 - \gamma } \right)\theta r_{{i,t + 1}}^a - \left( {1 - \gamma } \right)\theta {b_{{i,t + 1}}}} \right]} } \right\} \hfill \\ {E_t}\left( {r_{{t + 1}}^U} \right) + \tfrac{1}{2}Va{r_t}\left( {r_{{t + 1}}^U} \right) = \sum\limits_{{i = 1}}^m {{q_i}\left( {1 - \gamma } \right){E_t}\left( {r_{{i,t + 1}}^W} \right)} - \sum\limits_{{i = 1}}^m {{q_i}\left( {1 - \gamma } \right)\theta {E_t}\left( {r_{{i,t + 1}}^a} \right)} - \sum\limits_{{i = 1}}^m {{q_i}\left( {1 - \gamma } \right)\theta {E_t}\left( {{b_{{i,t + 1}}}} \right)} + \hfill \\ + \tfrac{1}{2}\sum\limits_{{i = 1}}^m {{q_i}Va{r_t}\left[ {\left( {1 - \gamma } \right)r_{{i,t + 1}}^W - \left( {1 - \gamma } \right)\theta r_{{i,t + 1}}^a - \left( {1 - \gamma } \right)\theta {b_{{i,t + 1}}}} \right]} \hfill \\ \end{gathered} $$
We substitute in the optimization problem and drop the constant terms:
$$ \mathop{{\max }}\limits_{{\mathbf{w}}} \,\,\sum\limits_{{i = 1}}^m {{q_i}{E_t}\left( {r_{{i,t + 1}}^W} \right)} - \tfrac{{\gamma - 1}}{2}\sum\limits_{{i = 1}}^m {{q_i}Va{r_t}\left( {r_{{i,t + 1}}^W} \right)} + \theta \left( {\gamma - 1} \right)\sum\limits_{{i = 1}}^m {{q_i}Co{v_t}\left( {r_{{i,t + 1}}^W,r_{{i,t + 1}}^a} \right)} $$
We then apply the lognormal approximation formula to \( r_{{i,t + 1}}^W{{ = ln}} \left( {{1 + }R_{{i,t + 1}}^W} \right) \):
$$ \begin{gathered} r_{{i,t + 1}}^W = {u_i}{r_{{p,t + 1}}} + {v_i}r_{{i,t + 1}}^L + {x_i}r_{{o,t + 1}}^a + {x_i}{b_{{o,t + 1}}} + \hfill \\ + \tfrac{1}{2}{u_i}\sigma_p^2 + \tfrac{1}{2}{v_i}{\left( {\sigma_{{i,t + 1}}^L} \right)^2} + \tfrac{1}{2}{x_i}{\left( {\sigma_{{o,t + 1}}^a} \right)^2} + \tfrac{1}{2}{x_i}{\left( {\sigma_{{o,t + 1}}^b} \right)^2} - \hfill \\ - \tfrac{1}{2}Va{r_t}\left( {{u_i}{r_{{p,t + 1}}} + {v_i}r_{{i,t + 1}}^L + {x_i}r_{{o,t + 1}}^a + {x_i}{b_{{o,t + 1}}}} \right) \hfill \\ {E_t}\left( {r_{{i,t + 1}}^W} \right) = {u_i}{E_t}\left( {{r_{{p,t + 1}}}} \right) + {v_i}{E_t}\left( {r_{{i,t + 1}}^L} \right) + {x_i}{E_t}\left( {r_{{o,t + 1}}^a} \right) + {x_i}{E_t}\left( {{b_{{o,t + 1}}}} \right) + \hfill \\ + \tfrac{1}{2}{u_i}\sigma_p^2 + \tfrac{1}{2}{v_i}{\left( {\sigma_{{i,t + 1}}^L} \right)^2} + \tfrac{1}{2}{x_i}{\left( {\sigma_{{o,t + 1}}^a} \right)^2} + \tfrac{1}{2}{x_i}{\left( {\sigma_{{o,t + 1}}^b} \right)^2} - \hfill \\ - \tfrac{1}{2}Va{r_t}\left( {{u_i}{r_{{p,t + 1}}} + {v_i}r_{{i,t + 1}}^L + {x_i}r_{{o,t + 1}}^a + {x_i}{b_{{o,t + 1}}}} \right) \hfill \\ Va{r_t}\left( {r_{{i,t + 1}}^W} \right) = Va{r_t}\left( {{u_i}{r_{{p,t + 1}}} + {v_i}r_{{i,t + 1}}^L + {x_i}r_{{o,t + 1}}^a + {x_i}{b_{{o,t + 1}}}} \right) \hfill \\ \end{gathered} $$
After substituting and dropping constant terms, the optimization becomes:
$$ \begin{gathered} \mathop{{\max }}\limits_{{\mathbf{w}}} \,\,\left( {\sum\limits_{{i = 1}}^m {{q_i}{u_i}} } \right)\left[ {{E_t}\left( {{r_{{p,t + 1}}}} \right) + \tfrac{1}{2}\sigma_p^2} \right] + \theta \left( {\gamma - 1} \right)\sum\limits_{{i = 1}}^m {{q_i}{u_i}Co{v_t}\left( {{r_{{p,t + 1}}},r_{{i,t + 1}}^a} \right)} - \hfill \\ - \tfrac{\gamma }{2}\sigma_p^2\sum\limits_{{i = 1}}^m {{q_i}u_i^2} - \gamma \sum\limits_{{i = 1}}^m {{q_i}{u_i}{v_i}Co{v_t}\left( {{r_{{p,t + 1}}},r_{{i,t + 1}}^L} \right)} - \gamma Co{v_t}\left( {{r_{{p,t + 1}}},r_{{o,t + 1}}^a} \right)\sum\limits_{{i = 1}}^m {{q_i}{u_i}{x_i}} \hfill \\ \end{gathered} $$
Next, we apply the lognormal approximation formula to \( {r_p}_{{,t + {1}}}{{ = ln}} \left( {{1 + }{R_p}_{{,t + {1}}}} \right) \):
$$ \begin{gathered} {r_{{p,t + 1}}} = {r_{{f,t + 1}}} + {\mathbf{w\prime}}\left( {{\mathbf{r}}_{{t + 1}}^a - {r_{{f,t + 1}}}1} \right) + \tfrac{1}{2}{\mathbf{w\prime}}diag\left( \Sigma \right) - \tfrac{1}{2}{\mathbf{w\prime}}\Sigma {\mathbf{w}} \hfill \\ {E_t}\left( {{r_{{p,t + 1}}}} \right) = {r_{{f,t + 1}}} + {\mathbf{w\prime}}\left[ {{E_t}\left( {{\mathbf{r}}_{{t + 1}}^a} \right) - {r_{{f,t + 1}}}1} \right] + \tfrac{1}{2}{\mathbf{w\prime}}diag\left( \Sigma \right) - \tfrac{1}{2}{\mathbf{w\prime}}\Sigma {\mathbf{w}} \hfill \\ Va{r_t}\left( {{r_{{p,t + 1}}}} \right) = {\mathbf{w\prime}}\Sigma {\mathbf{w}} \hfill \\ \end{gathered} $$
After substituting again and dropping constant terms, the problem is:
$$ \begin{gathered} \mathop{{\max }}\limits_{{\mathbf{w}}} \,\,\left( {\sum\limits_{{i = 1}}^m {{q_i}{u_i}} } \right){\mathbf{w\prime}}\left[ {\left( {{E_t}\left( {{\mathbf{r}}_{{t + 1}}^a} \right) - {r_{{f,t + 1}}}1} \right) + \tfrac{1}{2}diag\left( \Sigma \right)} \right] + \theta \left( {\gamma - 1} \right){\mathbf{w\prime}}\sum\limits_{{i = 1}}^m {{q_i}{u_i}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,r_{{i,t + 1}}^a} \right)} - \hfill \\ - \tfrac{\gamma }{2}{\mathbf{w\prime}}\Sigma {\mathbf{w}}\sum\limits_{{i = 1}}^m {{q_i}u_i^2} - \gamma {\mathbf{w\prime}}\sum\limits_{{i = 1}}^m {{q_i}{u_i}{v_i}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,r_{{i,t + 1}}^L} \right)} - \gamma {\mathbf{w\prime}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,r_{{o,t + 1}}^a} \right)\sum\limits_{{i = 1}}^m {{q_i}{u_i}{x_i}} \hfill \\ \end{gathered} $$
The first order condition is:
$$ \begin{gathered} \left( {\sum\limits_{{i = 1}}^m {{q_i}{u_i}} } \right)\left[ {\left( {{E_t}\left( {{\mathbf{r}}_{{t + 1}}^a} \right) - {r_{{f,t + 1}}}1} \right) + \tfrac{1}{2}diag\left( \Sigma \right)} \right] + \theta \left( {\gamma - 1} \right)\sum\limits_{{i = 1}}^m {{q_i}{u_i}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,r_{{i,t + 1}}^a} \right)} - \hfill \\ - \gamma \Sigma {\mathbf{w}}\sum\limits_{{i = 1}}^m {{q_i}u_i^2} - \gamma \sum\limits_{{i = 1}}^m {{q_i}{u_i}{v_i}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,r_{{i,t + 1}}^L} \right)} - \gamma Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,r_{{o,t + 1}}^a} \right)\sum\limits_{{i = 1}}^m {{q_i}{u_i}{x_i}} = 0 \hfill \\ \end{gathered} $$
From the first order condition, the solution becomes:
$$ \begin{gathered} {\mathbf{w}} = \tfrac{A}{\gamma }{\Sigma^{{ - 1}}}\left[ {\left( {{E_t}\left( {{\mathbf{r}}_{{t + 1}}^a} \right) - {r_{{f,t + 1}}}1} \right) + \tfrac{1}{2}diag\left( \Sigma \right)} \right] - \hfill \\ - {\Sigma^{{ - 1}}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,{\mathbf{r\prime}}_{{t + 1}}^L} \right){\mathbf{z}} - \tfrac{{{H_t}{F_{{o,t}}}}}{{{W_t}}}{\Sigma^{{ - 1}}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,r_{{o,t + 1}}^a} \right) + \theta \left( {1 - \tfrac{1}{\gamma }} \right){\Sigma^{{ - 1}}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,{\mathbf{r\prime}}_{{t + 1}}^{{fwd}}} \right){\mathbf{s}} \hfill \\ \end{gathered} $$
where
$$ \begin{gathered} A \equiv \sum\limits_{{i = 1}}^m {\tfrac{{{W_{{i,total}}}}}{{{W_t}}}} \tfrac{{{p_i}W_{{i,total}}^{{ - \gamma - 1}}F_{{i,t}}^{{ - \theta \left( {1 - \gamma } \right)}}}}{{\sum {{p_j}W_{{j,total}}^{{ - \gamma - 1}}F_{{j,t}}^{{ - \theta \left( {1 - \gamma } \right)}}} }} \hfill \\ {s_i} \equiv \tfrac{{{W_{{i,total}}}}}{{{W_t}}}\tfrac{{{p_i}W_{{i,total}}^{{ - \gamma - 1}}F_{{i,t}}^{{ - \theta \left( {1 - \gamma } \right)}}}}{{\sum {{p_j}W_{{j,total}}^{{ - \gamma - 1}}F_{{j,t}}^{{ - \theta \left( {1 - \gamma } \right)}}} }} \hfill \\ {z_i} \equiv \tfrac{{V_{{i,t}}^L}}{{{W_t}}}\tfrac{{{p_i}W_{{i,total}}^{{ - \gamma - 1}}F_{{i,t}}^{{ - \theta \left( {1 - \gamma } \right)}}}}{{\sum {{p_j}W_{{j,total}}^{{ - \gamma - 1}}F_{{j,t}}^{{ - \theta \left( {1 - \gamma } \right)}}} }} \hfill \\ {W_{{i,total}}} \equiv {W_t} + V_{{i,t}}^L + {H_t}{F_{{o,t}}} \hfill \\ \end{gathered} $$
The expression \( {\Sigma^{{ - {1}}}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,r_{{o,t + 1}}^a} \right) \) is the oth column of the n × n identity matrix and the expression \( {\Sigma^{{ - {1}}}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,{\mathbf{r\prime}}_{{t + 1}}^{{fwd}}} \right) \) is equal to columns 1 through m of the n × n identity matrix. Therefore, the solution can be written as:
$$ \begin{gathered} {w_o} = \tfrac{A}{\gamma }{\mathbf{\Sigma }}_{{o\bullet }}^{{ - 1}}\left[ {\ln E_t^{{\mathbf{r}}}\left( {1 + {\mathbf{R}}_{{t + 1}}^a} \right) - {r_{{f,t + 1}}}1} \right] - \Sigma_{{o\bullet }}^{{ - 1}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,{\mathbf{r\prime}}_{{t + 1}}^L} \right){\mathbf{z}} + \left[ {\theta \left( {1 - \tfrac{1}{\gamma }} \right){s_o} - \tfrac{{{H_t}{F_{{o,t}}}}}{{{W_t}}}} \right] \hfill \\ {w_i} = \tfrac{A}{\gamma }{\mathbf{\Sigma }}_{{i\bullet }}^{{ - 1}}\left[ {\ln E_t^{{\mathbf{r}}}\left( {1 + {\mathbf{R}}_{{t + 1}}^a} \right) - {r_{{f,t + 1}}}1} \right] - \Sigma_{{i\bullet }}^{{ - 1}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,{\mathbf{r\prime}}_{{t + 1}}^L} \right){\mathbf{z}} + \left[ {\theta \left( {1 - \tfrac{1}{\gamma }} \right){s_i}} \right],i \in \left\{ {1,2,..,m} \right\} - \left\{ o \right\} \hfill \\ {w_j} = \tfrac{A}{\gamma }{\mathbf{\Sigma }}_{{j\bullet }}^{{ - 1}}\left[ {\ln E_t^{{\mathbf{r}}}\left( {1 + {\mathbf{R}}_{{t + 1}}^a} \right) - {r_{{f,t + 1}}}1} \right] - \Sigma_{{j\bullet }}^{{ - 1}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,{\mathbf{r\prime}}_{{t + 1}}^L} \right){\mathbf{z}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{ }j \in \left\{ {m + 1,..n} \right\} \hfill \\ \end{gathered} $$