APPENDIX

Static Asset Allocation Problem

First find the MSR portfolio:
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We multiply by e’ on the left-hand side to obtain
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Finally (note that weights sum up to 1),
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The next step consists of finding the right allocation as a function of the investor’s risk aversion:
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We obtain the solution by writing the first-order condition (L is the Lagrangian for the problem and Lw is its first derivative with respect to portfolio weights):
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Dynamic Asset-Liability Allocation Problem with Constant Opportunity Set

We now consider a dynamic asset allocation problem, with an investor allowed to rebalance portfolio between dates 0 and T. In this intertemporal context, information about asset return distribution over the horizon is not sufficient, and one needs to know the distribution of asset return at all points in time.
In what follows, we assume that the investor has access to N locally risky assets and one risk-free asset paying the constant interest rate, with the following dynamics (where ...

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