THEORETICAL FOUNDATIONS

Australian retirees can choose from allocated streams which provide for potentially higher investment returns, but offer no longevity insurance, and may be subject to investment risk; fixed income streams with more or less inflation and longevity insurance, but less investment uncertainty; and the means-tested age pension. Social security and taxation regulations interact with retirement income streams in complex ways, but the key features of the system are the more generous tests for pensioners who sacrifice access to their capital over the duration of their life expectancy, income test relief to capital drawdowns, and quite general tax rebates. The popularity of allocated prod- ucts (and unpopularity of lifetime annuities) suggests that flexibility and investment choice remain important decision variables into retirement.

Now we have sketched the income streams market, our next focus is to find the balance between a certain income stream and exposure to risky assets which will maximize the welfare of retirees. The next section sets up the theoretical foundations of optimal portfolio allocations for an agent who aims to keep consumption above a pre-specified level over his or her remaining lifetime.

provision in terms of ‘required gross income in today’s dollars’ in order to identify the minimum consumption stream on which a person can adequately subsist. To describe such a preference for subsistence consumption one needs a non-zero consumption floor in the utility function.

Correspondingly, a simplified version of a habit persistence model is useful for modelling retirement income streams. The model outlined here includes a fixed consumption floor designed to mimic a well-established habit.

Optimal Allocation and Consumption Paths

Consider the lifetime consumption stream and portfolio allocation for an investor with any concave utility function.⁶Assume the date of death, T, is known with certainty and the investor receives no labour income, consuming only out of wealth and investment returns. The agent can choose between one risk-free and one risky asset. In the final period all remaining wealth is consumed, leaving no bequest.

The investor’s problem is to maximize utility over retirement by choosing a consumption stream and allocating wealth between the assets.

max E[Uˆ (C₀,C₁, . . . C_T–1, W_T)] = max E

[ ^S

^{Tt–0U (C, t)}

]

^(7.1)

The investor knows:

P_i(t) = the price of security i,i= 0,1;

W(t) = current wealth;

and can choose:

C(t) = current consumption;

N_i(t) = number of shares of each asset.

The wealth constraint is given by

W(t) – C(t) ≡I(t) =

Σ

¹

0N_i(t)P_i(t) (7.2) so all wealth not consumed in a given period is allocated between the risky and risk-free assets.

Define the share of wealth allocated to security ias:

N_i(t)P_i(t)

w_i(t) ≡ –––––––– (7.3)

I(t) w₀+ w₁= 1

and define next period’s gross return to asset 1, a random variable:

P˜₁ (t + 1)

z˜ (t) ≡ –––––––– (7.4)

P₁(t)

The agent’s portfolio return is the weighted sum of returns over both assets in the portfolio, defined by:

Z˜(t) ≡w₁(t)[z˜(t) – R] + R (7.5) Using Ras the gross return to the risk-free asset, w₀as the portfolio weight allocated to the risk-free asset, and the full investment condition, the value of wealth available for consumption in period t+ 1 is:

w(t+ 1) = [W(t) – C(t)]{w₁(t)[z˜(t) – R] + R} (7.6) The derived utility of wealth function for period tcan be written as:

J[W(t),t] = max_c,w E_t

[ ^Σ

^{s=tT U(C,S)}

]

_(7.7)

J[W(T),T] = U(W_T,T)

If the agent’s preferences are described by U(C,t) = (d^tC^g)/g, as for the conventional constant relative risk aversion (CRRA) investor (here denoted by superscript P), the derived utility of wealth function, by backward induction, is

d^ta^g–1_t W_t^g J(W,t) = –––––––––

g

–1 (7.8)

––––1 g–1 1–g

a_t≡

[

^{1 + (dEt[Z˜Pgat+1])}

]

optimal consumption is

C^P_t= a_tW_t (7.9)

Finance for Australian annuitants 131

and the optimal portfolio is

E_t[(a_t+1Z˜^P)^g–1(z˜– R)] = 0. (7.10) If returns are independently and identically distributed, so that Z˜^Pand a_t+1 are uncorrelated, then the investor looks only to the next period in decision making. If the investment set is changing stochastically over time (so that the moments of the risky asset returns distribution or the interest rate are not constant), then the portfolio held by a multiperiod investor takes into account the future beyond the next period as well.

Making a transition to a positive consumption floor produces analogous results. Choosing a utility function with a subsistence consumption, Cˆ,

d^t(C– Cˆ)^g

U(C,t) = –––––––––––, g< 1 (7.11) g

yields a derived utility of wealth function d^ta_t ^g–1(W– Wˆ

t)^g

J(W,t) = –––––––––––––– (7.12)

g where

Wˆ

t= Cˆ_t

Σ

^T

s=tR^t–s (7.13)

and a_tis defined as in (7.8). It can be shown that the optimal consumption path (with HARA denoted by superscript H) is given by

(W– Wˆ

t) C^H= Cˆ+ –––––––– a_tW_t

W_t (7.14)

(W– Wˆ

t)

= Cˆ+ ––––––––– C^P_t W_t

The HARA agent always consumes their subsistence amount Cˆ, then consumes proportionately to the CRRA agent, after allowing for the wealth set aside ‘in escrow’ to provide future periods’ subsistence, Wˆ

t.

A similar relationship can be set out for allocations between the risky and risk-free assets:

Wˆ

t+1 W– C^H– Wˆ

t+1/R w_0t^H= ––––––––– + –––––––––––––– w_0t^P

(W– C^H)R W– C^H (7.15)

W– C^H– Wˆ

t+1/R w^H_1t= –––––––––––––– w^P_1t

W– C^H

Again the intuition is straightforward: the HARA investor needs to ensure future subsistence by putting the net present value of all future Cˆs into the risk-free asset. The remainder of his or her wealth is then allocated in the same proportions as for the CRRA investor. Given a value for the curvature para- meter, g, the greater the ‘cushion’ between current wealth and Wˆ

t, the more exposure to risk is optimal.

Here one can see the relationship between utility maximization with a consumption floor and portfolio insurance strategies. In its simplest form port- folio insurance uses actual or synthetic put options to enable an equity investor to avoid losses, but capture gains, at the cost of a fixed premium (Leland 1980). The distinguishing feature of portfolio insurance, according to Perold and Sharpe (1988), is that not only does the agent have a floor level of wealth such as that defined by Wˆ

t, but the curvature of the utility function is suffi- ciently low that the investor responds to rising stock prices by buying more stocks. However, they observe that only ‘buy-and-hold’ strategies can be followed by all investors so, in this sense, portfolio insurance has to be a minority taste.

The constant w^P₁ in (7.15) is analogous to the m of Black and Perold’s (1992) constant proportion portfolio insurance (CPPI). Their CPPI decision rule sets exposure to the risky asset at a constant multiple of the cushion between current and floor wealth, W – Wˆ

t. The optimal level of this m for HARA investors is w₁^P, a point made implicitly in Merton (1971) and explic- itly in Kingston (1989). CPPI investors shift money from the worse-perform- ing to the better-performing asset, buying into rising markets and selling into falling markets, a strategy that protects Cˆ while capturing risky payoffs.

The following section presents simulations of these optimal paths to map out a pattern of insured consumption, loosely calibrated to Australian conditions.

SIMULATION OF CONSUMPTION AND WEALTH