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INVESTING RETIREMENT WEALTH: A LIFE-CYCLE MODEL - Retirement and liquid wealthWe model a system of mandatory saving for retirement in the following simple way. During working life the individual must save a fraction, W, of current labor income as retirement wealth. Under this assumption disposable labor income, Yt, is given by:
Formula 6
The amount WYt is added to retirement wealth, denoted by. During working life retirement wealth is illiquid; the individual cannot consume it or borrow against it. At age К retirement wealth is rolled into a riskless annuity, so that the individual receives in each of the retirement years the annuity value corresponding to W^. This assumption of riskless annuitization affects the portfolio choices of older investors. An interesting extension of our work would be to allow investors to choose between riskless and variable annuities.
We will consider several alternative systems governing the investment of retirement wealth during working life. In the first system the individual is forced to hold retirement wealth in riskless assets. This implies that W^ = B|?, where B|? is the dollar amount of retirement wealth investor i has in riskless assets. In alternative systems retirement wealth is partially or fully invested in risky assets, but the allocation remains constant over time and is not controlled by the investor. We interpret this as the social security administration managing the individual’s retirement account. For this reason the fixed cost of investing in stocks applies only to investments outside the retirement account.
Investors also have liquid wealth outside their retirement accounts. We denote liquid wealth of investor i at date t by W£, and liquid holdings of bills and stocks by B%. and Stf, respectively. We assume that the investor faces the following borrowing and short-sales constraints:
Formula 7-8
The borrowing constraint (7) ensures that the investor’s allocation to bills in both the liquid and retirement accounts is non-negative at all dates. It prevents the investor from borrowing against future labor income or retirement wealth. The short-sales constraint (8) ensures that the investor’s allocation to equities is non-negative at all dates.
In each period of a household’s working life (t $ К) the timing of events is as follows. The investor starts the period with liquid wealth W and retirement wealth W.
Then labor income Yu is realized. Following Deaton (1991) we denote cash on hand in period t by
Formula 9
The investor must decide how much to consume, С, whether to pay the fixed cost of entering the stock market (if he has not done so before), and how to allocate the remaining cash on hand between stocks and bills. We denote the proportion of liquid wealth invested in stocks by a^t. The proportion of retirement wealth invested in stocks,, is given exogenously by the retirement system and does not vary over time, so = af for all t. We will consider different values for af.