Optimal portfolio choice with annuitization

Abstract

We study the optimal consumption and portfolio choice problem over an individual's life-cycle taking into account annuity risk at retirement. Optimally, the investor allocates wealth at retirement to nominal, inflation-linked, and variable annuities and conditions this choice on the state of the economy. We also consider the case in which there are, either for behavioral or institutional reasons, limitations in the types of annuities that are available at retirement. Subsequently, we determine how the investor optimally anticipates annuitization before retirement. We find that i) using information on term structure variables and risk premia significantly improves the optimal annuity choice, ii) restricting the annuity menu to nominal or inflation-linked annuities is costly for both conservative and more aggressive investors, and iii) adjustments in the optimal investment strategy before retirement induced by the annuity demand due to inflation risk and time-varying risk premia are economically significant. This holds as well for sub-optimal annuity choices. The adjustment to hedge real interest rate risk is negligible. We estimate that the welfare costs of not taking these three factors into account at retirement are 9% for an individual with an average risk aversion ( = 5). Not hedging annuity risk before retirement causes an additional welfare costs between 1% and 13%, depending on the annuitization strategy implemented at retirement.

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No. 2006–78

OPTIMAL PORTFOLIO CHOICE WITH ANNUITIZATION

By Ralph S.J. Koijen, Theo E. Nijman, Bas J.M. Werker

August 2006




















ISSN 0924-7815

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Optimal Portfolio Choice with Annuitization∗
Ralph S.J. Koijen†
Theo E. Nijman‡
Bas J.M. Werker§
This version: August 2006
First version: March 2006
Abstract
We study the optimal consumption and portfolio choice problem over an
individual’s life-cycle taking into account annuity risk at retirement. Optimally,
the investor allocates wealth at retirement to nominal, inflation-linked, and
variable annuities and conditions this choice on the state of the economy. We
also consider the case in which there are, either for behavioral or institutional
reasons, limitations in the types of annuities that are available at retirement.
Subsequently, we determine how the investor optimally anticipates annuitization
before retirement. We find that i) using information on term structure variables
and risk premia significantly improves the optimal annuity choice, ii) restricting
the annuity menu to nominal or inflation-linked annuities is costly for both
conservative and more aggressive investors, and iii) adjustments in the optimal
investment strategy before retirement induced by the annuity demand due to
inflation risk and time-varying risk premia are economically significant. This
holds as well for sub-optimal annuity choices. The adjustment to hedge real
interest rate risk is negligible. We estimate that the welfare costs of not taking
these three factors into account at retirement are 9% for an individual with an
average risk aversion (γ = 5). Not hedging annuity risk before retirement causes
an additional welfare costs between 1% and 13%, depending on the annuitization
strategy implemented at retirement.
Keywords: optimal life-cycle portfolio choice, annuity risk
JEL classification: D91, G0, G11, G23
∗We thank Lans Bovenberg, Norma Coe, Jiajia Cui, Frank de Jong, Jules van Binsbergen, and Otto van Hemert
and seminar participants at the Second Empirical Asset Pricing Retreat at the University of Amsterdam and the
10th International Congress on Insurance: Mathematics and Economics for useful comments and suggestions.
†Finance Group, CentER, Tilburg University and Netspar, Tilburg, the Netherlands, 5000 LE. Phone: 31-13-
4663238. Email: R.S.J.Koijen@TilburgUniversity.nl.
‡Finance and Econometrics Group, CentER, Tilburg University and Netspar, Tilburg, the Netherlands, 5000 LE.
Phone: 31-13-4662342. Email: Nyman@TilburgUniversity.nl.
§Finance and Econometrics Group, CentER, Tilburg University and Netspar, Tilburg, the Netherlands, 5000 LE.
Phone: 31-13-4662532. Email: Werker@TilburgUniversity.nl.

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1Introduction
In a seminal paper on annuity choice, Yaari (1965) shows that, in the absence of a bequest motive,
it is optimal to transfer retirement wealth fully into annuities at retirement. Davidoff, Brown, and
Diamond (2005) extend this result and show that full annuitization is optimal if every asset is
available in annuitized form. Annuities allow investors to shift wealth from states of the world in
which no utility is derived to states in which the investor is still alive. Even if annuity markets are
incomplete, it is often optimal to annuitize a sizeable part of retirement benefits, see Davidoff et
al. (2005). A reasonably complete annuity menu comprises nominal, inflation-linked, and variable
annuities linked to a broad equity index. Using these life-contingent instruments, investors are able
to manage exposures to interest rate, inflation, and equity risk, including the corresponding risk
premia, at retirement. Annuitization of retirement wealth exposes the investor to annuity risk: the
utility derived from the annuity payoffs may disappoint if financial market conditions turn out to
be unfavorable at retirement.1This paper addresses the welfare gains of optimal annuity choice at
retirement. Moreover, we explore how investors optimally anticipate annuity risk, for both optimal
and sub-optimal annuity choices, before retirement.
Annuity risk is important because insurance companies generally do not repurchase annuity
products. Irreversibility is a direct consequence of the adverse selection problem in annuity markets
as annuitants generally possess better information concerning their health status, in particular
when they are in bad health. It is therefore hard, if not impossible, to dynamically rebalance
the annuity portfolio in response to changes in financial markets after retirement. Due to the
illiquidity of annuity products after retirement, the investor can essentially manage annuity risk
along two lines.2First, the optimal annuity portfolio at retirement can incorporate financial market
conditions at this date. Second, by trading equity and bonds in the period before retirement, the
investor can construct hedging portfolios which pay off in unfavorable states of the economy at
retirement. Obviously, the optimal investment strategy in the period before retirement depends
on the annuitization strategy followed. Even if, either for behavioral or institutional reasons,
investors restrict attention to only part of the annuity menu mentioned above, see for instance
Diamond (1997) and Brown, Mitchell, and Poterba (2001), hedging risks before retirement is welfare
improving.
Optimal annuity choice has been addressed before in Charupat and Milevsky (2002), Brown et
al. (2001), and Blake, Cairns, and Dowd (2003). Charupat and Milevsky (2002) assume interest
rates and risk premia to be constant and abstract from inflation risk. They show that the optimal
allocation to fixed (i.e., nominal or inflation-linked) and variable annuities coincides with the
1Soares and Warshawsky (2002) illustrate annuity risk in value terms by determining the historical initial payouts
of both nominal and inflation-linked annuities.
2Browne, Milevsky, and Salisbury (2003) determine a liquidity premium required by investors to compensate for
the illiquidity of annuities.
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optimal allocation to stocks and bonds in the period before retirement. Brown et al. (2001) are
mainly interested in the welfare effects of having access to inflation-linked and variable annuities.
They restrict attention to full annuitization using a single product or wealth is split evenly between
inflation-linked and either nominal or variable annuities. Brown et al. (2001) find that both variable
annuities and inflation-linked annuities can be welfare enhancing, depending on the risk preferences
of the annuitant. Blake et al. (2003) consider various distribution programs of retirement wealth in
a model similar to the one in Charupat and Milevsky (2002). They consider portfolios containing
equity and fixed annuities and show that the ability to invest in equities during retirement can
improve welfare significantly. However, both Brown et al. (2001) and Blake et al. (2003) assume
risk premia to be constant and do not explore the possibility to tailor the annuity portfolio to
the state of the economy at retirement. We relax both assumptions below and find significant
additional gains in welfare terms.
Before retirement, the investor can use equity and bond markets to engage in (dynamic)
trading strategies to hedge annuity risk. The recent long-term asset allocation literature does
not (explicitly) account for the state dependence of the value function at retirement as a result
of annuity risk. Notable exceptions are Boulier, Huang, and Taillard (2001), Deelstra, Grasselli,
and Koehl (2003), and Cairns, Blake, and Dowd (2006) in which the investor respectively hedges
a minimal guarantee or the interest rate risk induced by an annuity-like product at retirement.
These papers, however, restrict attention to a single annuity product at retirement and abstract
from risks relevant for long-term investors like inflation and changes in risk premia. In fact, recent
developments in the dynamic asset allocation literature emphasize the importance for long-term
asset allocation of time variation in interest rates, see Brennan and Xia (2000), and Wachter (2003),
inflation rates, see Campbell and Viceira (2001) and Brennan and Xia (2002), and risk premia, see
Brandt (1999), Campbell and Viceira (1999), Wachter (2002), Campbell, Chan, and Viceira (2003),
Sangvinatsos and Wachter (2005), and Brennan and Xia (2005). It is likely that the same risk
factors play a role in the optimal (conditional) demand for annuities at retirement and the induced
hedging strategies in the period before retirement.3
In this paper, we study the optimal portfolio, consumption, and annuity choice over the
investor’s life-cycle. Our financial market model allows for time-varying interest rates, inflation
rates, and risk premia. In order to keep the problem tractable, we exogenously fix the date at which
the individual converts wealth into annuities. Milevsky and Young (2003), Neuberger (2003), and
Blake et al. (2003) relax this assumption by allowing the investor to select the optimal moment to
invest wealth in annuities. At retirement, we determine the optimal allocation to all three annuity
products, conditionally upon the state of the economy, thus extending the class of annuitization
strategies considered so far in the literature. In addition, we estimate the welfare loss of ignoring
3Bodie and Pesando (1983) show that annuity products inherit the risk-return characteristics from the asset
underlying the annuity.
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conditioning information or, as often observed in real-world annuity markets, restricting attention to
only part of the annuity menu. We solve subsequently for the optimal investment and consumption
strategy in the period before retirement. During this stage of the investor’s life-cycle, the investor
receives a stream of labor income of which a fixed fraction is saved for retirement purposes. The
savings are allocated dynamically to stocks, nominal and inflation-linked bonds, and a nominal
cash account. We derive optimal investment strategies using equity and bonds to hedge annuity
risk induced by both optimal and sub-optimal annuitization strategies. Again, it is important to
explore the optimal responses to sub-optimal annuitization strategies as well. Finally, we calculate
the welfare costs of ignoring annuity risk all together in the investment strategy in the period before
retirement.
This paper provides three main contributions to the extant literature. First of all, we show that
conditioning the annuity choice on financial market conditions improves welfare significantly. The
welfare costs of ignoring information on the term structure and risk premia at retirement range
from 7%-9% of total wealth, depending on the investor’s risk preferences. The optimal conditional
annuity strategy turns out to be a complex function of the state variables, which may limit its
practical use.In fact, we show that 75%-95% of the gains due to incorporating conditioning
information can be obtained by following a simple linear portfolio rule. Secondly, restricting the
annuity menu to nominal or inflation-linked annuities alone can induce significant welfare costs.
Restricting access to nominal annuities is mainly costly for conservative investors, while only having
access to inflation-linked annuities is most harmful for aggressive investors. For an individual with
an average risk aversion (γ = 5), the costs of investing retirement wealth fully in nominal annuities
is estimated to be 28% and likewise 14% for inflation-linked annuities. This corroborates the results
of Brown et al. (2001) and Blake et al. (2003) who confine attention to a few possible retirement
strategies. Thirdly, we determine the optimal hedging strategy in the period before retirement for
four different annuitization strategies. The optimal conditional annuitization strategy invests in all
three annuities and the weights are state dependent. The optimal unconditional strategy may also
use all annuity products, but its allocation is independent of the state of the economy at retirement.
The third and fourth annuitization strategies invest all wealth accumulated in nominal or inflation-
linked annuities, respectively. We find that the welfare costs of not hedging annuity risk before
retirement equal 9% for the first two annuitization strategies and 1% for nominal annuitization for
an individual with an average risk aversion (γ = 5). However, the welfare costs of not hedging the
annuitization strategy which invests all wealth in inflation-linked annuities is negligible. This leads
to the conclusion that hedging annuity risk induced by time variation in inflation and risk premia is
welfare improving, while the annuity risk caused by real interest rates is only of minor importance.
The limited impact of real interest rate risk is a consequence of the relatively strong mean-reversion
implied by our estimates. We find a half-life for the real interest rate of only 8 months. Obviously,
our results depend on the investor’s preferences and the assumptions we have made concerning the
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