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
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Under other assumptions being equal, the liquid annuity would provide greater utility to the consumer. In order to compensate for this, the illiquid annuity must provide an enhanced rate of return which is on the FIA. The unit price of the FIA become a rx( +λ), where λ is the demanded liquidity premium.
Suppose that the consumer’s personal utility function of consumption is
(1 )
denotes the optimal allocation to the risky asset.
In the dynamic liquid case, the maximal level of expected utility is
1
In the static case, an initial allocation of α to the risky asset will result in a utility
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In this paper, we discuss two types of payment for old-aged benefit under Labor Insurance. We suppose the retirees choose pension benefit is equivalent to hold a 100% of FIA and choose one-time benefit is equivalent to purchase a 100% of VIA, just like participating in the financial market. Hence, the old-age pension benefit should provide additional liquidity premium λ and let the expected utility of wealth of two are equal.Under the one-time benefit, the expected utility is
2 1 2 Under the old-age pension benefit, the expected utility is
1
Let these two functions be equal, and then we can derive the additional liquidity premium λ provided by the pension benefit.
3.2 Implied longevity yield
In the preceding section, we compute the theoretical yield needed to compensate for the retirees select pension benefit. This theoretical yield is liquidity premium.
However, we want to measure the actual rate of return the pension benefit provide.
Following the Milevsky (2005), we apply the concept of the implied longevity yield that could be used to measure the actual rate of return of pension benefit.
Milevsky (2005) measures the return from commercial life annuities which is called implied longevity yield (ILY) is defined equal to the internal rate of return
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(IRR). Assume that there is an insured aged x purchases a life annuity that unit price is a1, and at age x+ the unit price of annuity is τ a2, a2< a1. If the insured decided to forgo the purchase of a life annuity and instead invested a1 in portfolio and withdrew the same amount annum, this strategy is called self-annuitiazation. The required investment portfolio return should replicate the income payout from the annuity and still be able to let insured purchase the annuity priced a2 at x+ . The τ required investment portfolio return is the implied longevity yield δ .
Suppose that the retiree aged x has W0 =w dollars in marketable wealth. If this retiree converted w into a lifetime flow, i.e. purchased annuity, he or she would be entitled to w a/ 1 per annum for life. If, in contrast, the retiree decided to forgo purchasing the life annuity and instead self-annuitized by investing w in risky asset, the wealth dynamics would satisfy the following process:
Milevsky (2005) derives the solution to Eq.(6) is:
This investment portfolio is to contain enough funds to purchase the annuity at age
x+ , so following relationship must hold: τ 2
the right-hand side describes the evolution of wealth under a consumption rate of / 1
Taking the data of commercial annuities into Eq. (9), Milevsky (2005) solves for
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the implied longevity yield δ using numerical techniques.
We consider the Labor Insurance and assume that the insured retire at age x and has wealth W0 =w which is equal to the one-time benefit they deserve to obtain.
Following Milevsky (2005), we accumulate this fund each period by a rate δ , and subtract the consumption. It can be describe as following expression
, (11)
where Ct denotes the withdraw for consumption purpose, is equal to annuitized payment. When the retiree dies, the fund value should be zero. Then we can solve the Eq. (10) and get the δ .
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To measure the welfare loss from the old-age pension benefit since a lack of liquidity, we calculate the liquidity premium λ . We consider the following scenario to compare and analyze the results. The insured with the coefficient of relative risk aversion－1.5, 2, 2.5, and 3 retire at age 60 or 65 separately. As for the capital market parameters in our example, the risky asset is assumed to have drift µ =10% and volatility σ =20%, and set µ =8%, σ =25% to compare. The risk-free rate is assumed to be 2.5%. In our study, we use the 9th Period Taiwan Life Table to carry off the mortality rates, and set the highest age is 100 year-old. Take all the parameters into Eq.(4) and Eq.(5), and let these two functions be equal. We use the interior-reflective Newton method based on the non-linear least-squares algorithms to solve the non-linear equations, and then we could obtain the value of the liquidity premium. The numerical results show in the following Table 4-1.
Table54-1: Liquidity Premium λ (basic point)