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Abstract

Households that contemplate moving to different cities or trading up/down in the future are exposed to substantial housing risk. In order to mitigate this risk, we derive optimal portfolios using CME housing futures. Housing investment risk is hedged by selling housing futures amounting to the full value of the home. Housing consumption risk is hedged by buying housing futures in each city where the household might move. The size of the hedges depends on the probability of moving, on home values, and on labor income in each region. The hedging demands offset each other when the household intends to live in the same home indefinitely.

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Notes

  1. Ignoring housing consumption leads these authors to exaggerate the exposure to housing risk. Englund et al. (2002) conclude, “renters experience almost no losses relative to the unrestricted portfolio.” However, renters are clearly exposed to housing price risk since unexpected increases in real estate would also drive up rent prices, which would make them poorer, everything else constant. A similar argument would be that countries with no oil reserves are still exposed to oil price risk.

  2. Firms such as Fiserv CSW and Moody’s Economy.com provide reliable home price forecasts.

  3. Default risk considerations are ignored.

  4. Futures positions require a small margin account. However, margin requirements could be satisfied with other existing investments without affecting the overall portfolio.

  5. Depreciation can be effortlessly introduced by letting the future home value be \( \left( {{1} - \delta } \right){H_t}{P_o}_{{,t + {1}}} \).

References

  • Bodie, Z., Merton, R. C., & Samuelson, W. F. (1992). Labor supply flexibility and portfolio choice in a life-cycle model. Journal of Economic Dynamics and Control, 16(3–4), 427–449.

    Article  Google Scholar 

  • Campbell, J. Y. (2006). Household finance. Presidential Address to the American Finance Association, January.

  • Campbell, J. Y., and Viceira L. M. (2001). Appendix to strategic asset allocation. Resource Document. http://kuznets.fas.harvard.edu/~campbell/papers.html. Accessed 2 June 2011.

  • Campbell, J. Y., and Viceira, L. M. (2002). Strategic asset allocation: portfolio choice for long-term investors. Oxford University Press.

  • Case, K. E., & Shiller, R. J. (1989). The efficiency of the market for single-family homes. American Economic Review, 79(1), 125–137.

    Google Scholar 

  • Case, K. E., & Shiller, R. J. (2003). Is there a bubble in the housing market? Brookings Papers on Economic Activity, 2, 299–362.

    Article  Google Scholar 

  • Case, K. E., Shiller, R. J., & Weiss, A. N. (1993). Index-based futures and options markets in real-estate. Journal of Portfolio Management, 19(2), 83–92.

    Article  Google Scholar 

  • Cao, M., & Wei, J. (2010). Valuation of housing index derivatives. Journal of Futures Markets, 30(7), 660–688.

    Google Scholar 

  • Cocco, J. F. (2005). Portfolio choice in the presence of housing. Review of Financial Studies, 18(2), 535–567.

    Article  Google Scholar 

  • Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1981). The relation between forward prices and futures prices. Journal of Financial Economics, 9, 321–346.

    Article  Google Scholar 

  • Damgaard, A., Fuglsbjerg, B., & Munk, C. (2003). Optimal consumption and investment strategies with a perishable and an indivisible durable consumption good. Journal of Economic Dynamics and Control, 28, 209–253.

    Article  Google Scholar 

  • Englund, P., Hwang, M., & Quigley, J. M. (2002). Hedging housing risk. Journal of Real Estate Finance and Economics, 24(1–2), 167–200.

    Article  Google Scholar 

  • Hull, J. C. (1998). Introduction to futures and options markets. New Jersey: Prentice Hall.

    Google Scholar 

  • Iacoviello, M., & Ortalo-Magne, F. (2003). Hedging housing risk in London. Journal of Real Estate Finance and Economics, 27(2), 191–209.

    Article  Google Scholar 

  • Shiller, R. J. (2003). The new financial order: risk in the 21 st century. Princeton University Press.

  • Shiller, R. J., & Weiss, A. N. (1999). Home equity insurance. Journal of Real Estate Finance and Economics, 19(1), 21–47.

    Article  Google Scholar 

  • Sinai, T., & Souleles, N. S. (2005). Owner-occupied housing as a hedge against rent risk. Quarterly Journal of Economics, 120(2), 763–789.

    Google Scholar 

Download references

Acknowledgements

We would like to thank John Y. Campbell, Robert C. Merton, Edward Glaeser and Luis Viceira for advice and multiple edits on earlier drafts. All errors remain our own.

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Correspondence to Michael Joseph Seiler.

Appendices

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:

$$ \begin{gathered} {w_o} = \tfrac{A}{\gamma }{\mathbf{\Sigma }}_{{o\bullet }}^{{ - 1}}\left[ {\ln E_t^{{\mathbf{r}}}\left( {1 + {\mathbf{R}}_{{t + 1}}^a} \right) - {r_{{f,t + 1}}}1} \right] - \Sigma_{{o\bullet }}^{{ - 1}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,{\mathbf{r\prime}}_{{t + 1}}^L} \right){\mathbf{z}} + \left[ {\theta \left( {1 - \tfrac{1}{\gamma }} \right){s_o} - \tfrac{{{H_t}{F_{{o,t}}}}}{{{W_t}}}} \right] \hfill \\ {w_i} = \tfrac{A}{\gamma }{\mathbf{\Sigma }}_{{i\bullet }}^{{ - 1}}\left[ {\ln E_t^{{\mathbf{r}}}\left( {1 + {\mathbf{R}}_{{t + 1}}^a} \right) - {r_{{f,t + 1}}}1} \right] - \Sigma_{{i\bullet }}^{{ - 1}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,{\mathbf{r\prime}}_{{t + 1}}^L} \right){\mathbf{z}} + \left[ {\theta \left( {1 - \tfrac{1}{\gamma }} \right){s_i}} \right],i \in \left\{ {1,2,..,m} \right\} - \left\{ o \right\} \hfill \\ {w_j} = \tfrac{A}{\gamma }{\mathbf{\Sigma }}_{{j\bullet }}^{{ - 1}}\left[ {\ln E_t^{{\mathbf{r}}}\left( {1 + {\mathbf{R}}_{{t + 1}}^a} \right) - {r_{{f,t + 1}}}1} \right] - \Sigma_{{j\bullet }}^{{ - 1}}Co{v_t}\left( {{\mathbf{r}}_{{t + 1}}^a,{\mathbf{r\prime}}_{{t + 1}}^L} \right){\mathbf{z}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{ }j \in \left\{ {m + 1,..n} \right\} \hfill \\ \end{gathered} $$

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Voicu, C., Seiler, M.J. Deriving Optimal Portfolios for Hedging Housing Risk. J Real Estate Finan Econ 46, 379–396 (2013). https://doi.org/10.1007/s11146-011-9328-x

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