Optimal Decumulation of Assets in General Equilibrium James Feigenbaum (Utah State)
Annuities An annuity is an investment that insures against mortality risk by paying an income stream until the investor dies. Optimal annuities: On top of the normal return paid to a risk-free investment, optimal annuities earn a premium equal to mortality risk. If the investor dies, his entire investment is surrendered to pay for the premiums of surviving investors. 1
Yaari (1965) A consumer with no bequest motive will not care what happens to investment after he dies. Optimal annuities earn higher returns than uninsured bonds, so households without heirs will invest exclusively in annuities. With a bequest motive, household will divide wealth between own consumption and bequest. Wealth for own consumption is annuitized. Wealth for bequest goes to uninsured bonds. 2
Realistic Annuities More realistic annuities are irreversible investments. For every $1 of annuity that you purchase, you receive a constant stream of income for the remainder of your life. The return on an annuity is higher than the normal interest rate. Rational households will still annuitize when only realistic annuities are available. 3
Annuities Puzzle Private annuity investments account for only 1% of total wealth of households over age 65 in the US. 2000 Health and Retirement Study (Johnson, Burman, and Kobes (2004)) Pashchenko (2011): Only 6% of households over 65 participate in any annuitization scheme. Social Security and defined benefit pensions are suboptimal annuities not accounted for here. Guo et al (2012): because Social Security does not pay a higher return, it is not a substitute for the annuities discussed previously. 4
Is It Really Good to Annuitize? Much of literature is motivated by presumption consumers hurt themselves by failing to annuitize. What can we do to fix the annuities market? Above results are strictly partial equilibrium. In general equilibrium, Yaari annuities have ambiguous effect with rational households. Feigenbaum et al (FGT) (2013b), Heijdra et al (2010) 5
Motivating Question Economists are almost unanimous in their opinion that people should annuitize. The expectation is that policymakers will eventually nudge people into annuitizing. For example, make it a default practice that 401(k)s roll over into annuities at retirement. What would the consequence of this be with realistic annuities? 6
Pecuniary Externality Why is general equilibrium (GE) so different from partial equilibrium for annuities? There is a two-way causal connection between prices and consumer behavior in GE. McKean (1958), Prest and Turvey (1965) Economists typically focus on how prices affect behavior and ignore reverse relationship. Rational individuals ignore it because their choice is too small to affect aggregate. 7
Coordination Across Generations In an infinite-horizon, representative agent model, ignoring pecuniary externality is innocuous. Decisions now affect future prices, not current prices. In OLG model, different cohorts can coordinate their behavior to exploit the pecuniary externality. If households save more at a given age, this can improve factor prices for later cohorts when they attain the same age. 8
Pecuniary Externality and Bequests The bequest a household receives will be viewed as exogenous by the household. The bequest the household leaves is endogenous. A general equilibrium is a fixed point of the mapping from the prices and bequest the household expects to the prices and bequest generated by its behavior. The pecuniary externality also works through the bequest. 9
Accidental Bequests are Good Annuities mostly transfer the wealth of deceased households to old people. Bequests transfer deceased wealth to young and old people. A bequest received when young permits a steeper declining consumption path. For a fixed K, the best consumption path has marginal utility grow at the discount rate, instead of r. 10
This Paper Previous GE studies compared equilibria where everyone employs Yaari annuities to equilibria where no one annuitizes. Here households have the choice at age 65 to purchase a realistic, irreversible annuity. We consider both a traditional annuity and a bond annuity. The latter does not absorb assets upon death, so there is something left for heirs to inherit. 11
The Punch Line What makes traditional annuities bad is not the annuitization per se but the absorption of the assets when the household dies. In general equilibrium, households earn higher utility if they have a constant consumption stream after retirement. Households should be encouraged to put their assets into bond annuities. 12
Observed Decumulation Behavior Another consumption and saving puzzle. Households do not purchase annuities, yet they also do not spend down their assets as quickly as they rationally should without longevity insurance and no apparent bequest motive. Davies (1981) Here we find this anomalous behavior can increase lifetime utility in general equilibrium. 13
The Model Generalizes Hansen and Imrohoroglu (2008) GE discrete-time OLG model with realistic annuities Probability of surviving to age t {0,..., T} is Q t [0, 1]. Conditional probability Q s t = Q s /Q t. Household works for t T a < T. At T a, the household decides to invest A 0in an irreversible traditional annuity and A b 0 in a bond annuity. 14
Payout Rates Let T r = T a + 1 be the age of retirement. The traditional annuity pays out The bond annuity pays out 15
Preferences The household s utility function is The discount factor >0. Period utility is CRRA with intertemporal elasticity 1 > 0. 16
Constraints For t [0, T a ), For t [T r, T], Boundary conditions: b 0 = b T+1 = 0 Borrowing constraints: A, A b,,..., b T 0. 17
Equilibrium Conditions Production function: Capital and labor: 18
Equilibrium Conditions II Factor prices: where [0, 1] is the depreciation rate. Bequest balance equation: 19
Choice of Annuity A rational household will only invest in the traditional annuity since > b. In fact the internal rate of return of both the bond and the bond annuity is r, which is less than the return on the traditional annuity. Borrowing constraints must bind to avoid arbitrage. The solution with borrowing constraints is pending. 20
Restricted Rational Regime The problem becomes much simpler if we suppose that the government requires households to put all assets into some type of annuity at retirement. We can also solve for the case where there are perfect Yaari annuities and no annuities. 21
Calibration Endowment profile: Gourinchas and Parker (2002). Survivor function: Feigenbaum (2008) Share of capital: = 0.3375 Capital-output ratio: K/Y = 3.0 We vary the intertemporal elasticity 1 and set discount factor to match this target. Consumption ratio: C/Y = 0.70 = 0.10 22
K/Y by Model 23
Compensating Variation in General Equilibrium 24
Consumption Profiles ( = 1) 25
Consumption Profiles ( = 0.25) 26
Consumption Profiles ( = 3) 27
Compensating Variation in Partial Equilibrium 28
Compensating Variation in Quasi- Partial Equilibrium 29
Conclusions 1. Contrary to popular wisdom, traditional annuities that absorb assets upon death lead to suboptimal equilibria in general equilibrium. 2. Unless intertemporal elasticity is very high, rational households do better with bond annuities than other types of annuities. 30