國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
II. Literature Review
Due to the future life expectancy of retirees is much longer and hard to estimate precisely, the retirees have difficulty to find the investment assets match the duration.
Therefore, many researchers discuss how the retirees use their wealth to get stable payments each period after they retire. The simplest solution is to purchase the commercial annuity products. If they want to hold the annuity, the timing to purchase the annuity would be decided. Besides, they should construct the individual investment strategy to maximize the utility before the time receiving annual benefits.
Most individuals concern that how much of their wealth should be annuitized at the time they retire. Yarri (1965) and Ficher (1973) proposed that the retirees should fully annuitize all their savings under market completeness assumption. However, in reality, as Modigliani (1986), Friedman and Warshawsky (1990), Mirer (1994), and Milevsky and Young (2007a) have pointed out very few people would choose to annuitize their wealth. That is, the economics theories suggest that retirees should enhance their welfare through buying annuities, but the empirical evidence find that retirees do not place their wealth on annuities. This is the so-called annuity puzzle.
The causes of annuity puzzle problem are bequest motives, the expensive loading of the annuity, and liquidity consideration. (Friedman and Warshawsky, 1990;
Vidal-Melia and Lejarraga-Garcia, 2006; Mitchell et al., 1999; Brown and Poterba, 2000; Brown, 2001a, b; Albrecht and Maurer, 2002; Browne et al., 2003)
Life annuities are financial instruments allowing individuals to exchange a lump sum of wealth for a guaranteed income stream to last for the rest of the insured’s life.
First, if individuals want to leave an inheritance to their children; they will find it’s not optimal to annuitize all of their wealth. Second, the insurance company is exposed
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
9
to the longevity risk, thus the inevitable fees will be charged by insurance company and the loading of annuity product is large. Due to these reasons, retirees prefer to construct their own investment strategies, getting constant stream of payments by themselves. (Kotlikoff and Summers, 1981；Kotlikoff and Spivak, 1981；Friedman and Warshawsky, 1990；Mitchell et al., 1999；Brown and Poterba, 2000；Brown, 2001a, b)
When an individual purchases life annuities, he/she exchanges a lump sum of wealth for a guaranteed stream of payments that continue so long as the individual is alive. If an individual doesn’t hold annuity, he/she can invest all their wealth to the financial market. That is, they can do self-annuitization. Hence, the retiree holds the annuity products will loss the liquidity of his/her own wealth. Liquidity problem would reduce annuity demand (Albrecht and Maurer, 2002; Browne et al., 2003).
The main approach to deal with the annuity puzzle problem are self-annuitization (Milevsky and Robinson, 2000; Albrecht and Maurer, 2002; Kingston and Thorp, 2005) and delaying the timing of annuitization (Milevsky, 1998; Stabile, 2006;
Milevsky and Young, 2007a, b; Horneff et al., 2008). Self-annuitization means that retiree do not purchase annuity products but manage his/her own account and withdraw constant amount each period until the account amount is zero or he/she dies.
Therefore, the retiree should find the investment assets that can provide the equivalent amount each period in the financial market.
In the part of delaying the timing of annuitization, we know that the older the retirees buy the annuity product with the same insurance amount, the lower price they spend on. Hence, the concept of delaying the time of annuitization is that retirees can invest their wealth in the market by themselves after they retire for a period, and then choose the perfect timing to buy the annuity products. Milevsky and Young (2007b)
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
find that the appropriate time to annuitize is between age 65-70 in the United State.
The annuity holder transfers the longevity risk to the life insurers, the life insurers should pay annuity to the insured each period. If the retiree wants to do individual strategic asset allocation among the various investment classes, he/she would ensure their investment strategic will offer the same constant amount of pension to maintain his/her future consumption.
Milevsky and Young (2007a) proposed that females wish to annuitize at older age compared to males because the mortality rate of females is lower at each given age. Also, the more risk averse individuals wish to annuitize sooner. And assuming annuitization can take place in small portions at any time; they find the utility-maximizing retirees should acquire a base amount of annuity income and then annuitize additional amounts. Moreover, the retirees will purchase more annuities as they become wealthier and more risk averse.
Under the Labor Insurance, the retirees can’t select the timing of the annuitization since they should decide the type of payment at the time they retire prescribed by Labor Insurance Act. In this study, we focus on the liquidity problem then investigate the difference between the old-age pension benefit and the one-time old-age benefit.
The amount of pension benefit the retiree acquires each period is related to the rate of mortality, assumed interest rate. When the market performed well, the rate of return is superior to the assumed interest of annuity. The retiree is unable to terminate getting payments and gets the rest of amount back to do individual investment.
However, the retiree who selects the one-time old-age benefit could place the lump sum into the financial market to obtain the higher return. Therefore, the pension
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
11
benefit is illiquidity, so that the retiree loses opportunity to participate in the market.
Because of this, the pension benefit should provide the additional premium to compensate the investor caused by its illiquidity feature. Browne et al. (2003) apply the utility function to measure the liquidity premium of the commercial annuity products.
The liquidity premium is the additional premium provided by the annuity pension benefit theoretically. On the other hand, we apply the concept of implied longevity yield which is proposed by Milevsky (2005). The implied longevity yield is used to measure the actual rate of return of pension benefit. Under the social insurance program in Taiwan, in order to offer the retiree long term living care, the annuity pension payment is better than the one-time payment in Labor Insurance.
In this study, we estimate the liquidity premium and the implied longevity yield of the annuity pension benefit then we could prove the annuity benefit is much better for the retiree. Hence, it would encourage the retiree to select annuity pension payment.
‧
3.1 Liquidity premium
To compare the one-time old-age benefit and old-age pension benefit, we should measure the welfare loss from the pension benefits since a lack of liquidity. We refer to Browne et al. (2003) that apply the utility function to estimate the liquidity premium demanded by the retiree holding an illiquidity annuity pension benefit.
We assume that there are a fixed immediate annuity and a variable immediate annuity in the market. A fixed immediate annuity (FIA) provides a fixed payment per unit time, and a variable immediate annuity (VIA), which provides a payment per unit time that varies depending on the value of some market asset Vt. If the retiree selected one-time benefit, he could put all amounts into purchasing a variable immediate annuity. On the other hand, the annuitized payment is equivalent to hold a fixed immediate annuity. Under these assumptions, we may evaluate the utility functions respectively.
If w dollars of the FIA are purchased, the consumer is entitled to continuous payment stream of
Here r denotes the risk-free interest rate, and tpx is the probability that the individual will survive to time t , conditional on being alive at the annuity purchase agex. Similarly, If w dollars of the VIA are purchased, at time t , payment accumulate at the rate of V
( )
asset at time t discount to time 0 then annuitized it in future period, where h is the