Optimal retirement planning under partial information
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Nicole Bäuerle
Abstract
The present paper analyzes an optimal consumption and investment problem of a retiree with a constant relative risk aversion (CRRA) who faces parameter uncertainty about the financial market. We solve the optimization problem under partial information by making the market observationally complete and consequently applying the martingale method to obtain closed-form solutions to the optimal consumption and investment strategies. Further, we provide some comparative statics and numerical analyses to deeply understand the consumption and investment behavior under partial information. Bearing partial information has little impact on the optimal consumption level, but it makes retirees with an RRA smaller than one invest more riskily, while it makes retirees with an RRA larger than one invest more conservatively.
A Appendix
In this appendix we collect some proofs and lengthy calculations.
A.1 Proof of Lemma 2.1
Proof.
Note first that (2.5) implies
In order to achieve a more explicit representation of
Defining
which is identical to (A.1). In other words, the solution to (A.1) can be expressed by
A.2 Proof of the optimality of c I *
Proof.
Given
Inserting
This then implies for
Since c is a feasible consumption and
A.3 Detailed calculations of Section 2.2
Note that we obtain
Recall that Y is a standard Brownian motion under
A.4 Detailed calculations of Section 2.3
We obtain here
An easy computation now yields that
A.5 Proof of Theorem 3.1
Proof.
We proceed as before. Let
where
Using the Itô–Doeblin formula, we obtain
Equating this with the wealth equation implies
which concludes the proof. ∎
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