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)
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Figure24-1: Liquidity Premium λ (basic point)－Male, age 60
Figure34-2: Liquidity Premium λ (basic point)－Male, age 60 0
100 200 300 400 500 600
1.5 2 2.5 3
=10%, 20%
µ σ =
= 8%, 20%
µ σ =
= 10%, 25%
µ σ =
λ
γ
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According to the Figure 4-1, 4-2, first we observe that a more risk averse insured requires less compensation for the illiquidity annuity for the same return and volatility of risky asset. The reason is the insured with grater aversion to risk is more unwilling to take the additional market risk, thus they will require less liquidity premium.
Second, the numerical results illustrate that the liquidity premium is an increasing function of the market rate of return. Because when the market performs well, it will induce insured to do self investment, the annuity payment should provide higher premium to compensate the insured. Third, we find the negative relationship between the greater volatility and the required liquidity premium. The intuition is the insured receives more risk while the risky asset is having higher volatility, so that the insured would prefer to hold the FIA. Fourth, comparing the liquidity premium between genders, the liquidity premium females require is higher than the males, since the female’s life expectancy is more than the males (Figure 4-3). Last, because the mortality is increasing by age, the latter the age of retire, the lower the liquidity premium.
(2) Labor Insurance in Taiwan
In this part, we use Labor Insurance to illustrate that pension benefit is better 0
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than one-time benefit. Now, we brief introduce the Labor Insurance in Taiwan. The workers above 15 full years and below 60 years of age shall all be insured under this program compulsorily, with their employers, or the organizations to which they belong reckoned as the insured units. The insurance premium of Labor Insurance is calculated using the insured person's monthly insurance salary and insurance premium rate. The premium shall be paid by the insured person, insured unit, and the government in accordance with the percentage prescribed by the Labor Insurance Act (Appendix C). The insurance benefits include the permanent disability benefits, old-age benefits, and medical care benefits, etc. When the insured retire at 60 years old, he/she could apply for the old-age benefit
There are two methods for retirees to get the pension payments: one-time old-age benefit and old-age pension benefit11
Assumed the insured retiree at age 60, the coverage years are 15, 20, 30, and 35, 40. Refer to the Table of Grades of Insurance Salary (Appendix A), we choose the . If the insured person chooses the pension benefit, he/she could select the better one from the following two formulas: (1) average monthly insurance salary × coverage years × 0.775% + 3000; (2) average monthly insurance salary × coverage years × 1.55%. Otherwise, the insured person chooses the one-time benefit then he/she could take the product of average monthly insurance salary product and issued months.
(3) Implied longevity yield
Liquidity premium means the theoretical yield needed to compensate for the retirees select the pension benefit. In this part, we measure the actual rate of return the annuitized pension benefit provides which is called the implied longevity yield.
11 In this paper, we only consider the insured person has insurance coverage before January 1, 2009.
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lowest, middle, the highest grades monthly insurance salary, are 17,800, 27,600, and 43,900 separately. Due to these assumptions, we could understand how the length of coverage years and the amount of monthly insurance salary effect on the implied longevity yield.
We assume the average monthly insurance salary under the pension benefit is identical to it under the one-time benefit. The benefit amount is showed in Table 4-2.
Table64-2: Old-age benefit age 60
coverage years 15 20 30 35 40
low salary (17,880)
one-time benefit 268,200 447,000 804,600 804,600 804,600 pension benefit 5,079 5,771 8,314 9,700 11,086 middle salary
(27,600)
one-time benefit 414,000 690,000 1,242,000 1,242,000 1,242,000 pension benefit 6,417 8,556 12,834 14,973 17,112 high salary
(43,900)
one-time benefit 658,500 1,097,500 1,975,500 1,975,500 1,975,500 pension benefit 10,207 13,609 20,414 23,816 27,218 To simplify the calculation process, we transfer the monthly payment to annuitized payment by let pension benefit multiplied by 12. Table 4-3 shows the implied longevity yield (ILY) we derive from Eq.(10).
Table74-3: Implied Longevity Yield δ (basic point) age 60
coverage years 15 20 30 35 40
Male
lower salary 2,883 1,784 1,364 1,642 1,931 middle salary 2,234 1,699 1,364 1,642 1,931 higher salary 2,234 1,699 1,364 1,642 1,931
Female
lower salary 2,908 1,802 1,378 1,659 1,950 middle salary 2,254 1,716 1,378 1,659 1,950 higher salary 2,254 1,716 1,378 1,659 1,950
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Figure54-4: Implied Longevity Yield δ －Male, Female age 60 (basic point) According to Table 4-3 and Figure 4-4, we find that the insured with lowest average monthly insurance salary earn the highest ILY under the coverage years less than 30, and ILY decreases by the greater monthly salary. However, the ILY values are all the same between different monthly insurance salary if the coverage years which excess 30. Regardless the gender, the 15 coverage years corresponds to the highest ILY value. And we notice the ILY value of over 30 coverage years is greater than ILY of 30 coverage years. The reason is that under the one-time benefit, for every one full year of insurance coverage, one month of average monthly insurance salary would be issued. For the part which is more than 15 years, 2 month of average monthly insurance would be issued for every one extra year of insurance coverage, and the highest limit is 45 months. Hence, the largest coverage years are 30 years under the one-time benefit, but the coverage years could increase over 30 years under the pension benefit. Besides, the female’s ILY values are greater than the male’s, since the mortality of female is lower than the male at any given age.
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Table84-4: Implied Longevity Yield minus risk-free rate (basic point) age 60
coverage years 15 20 30 35 40
Male
Comparing the implied longevity yield to the liquidity premium is the main purpose in this paper. According to Table 4-1 and Table 4-4, we could observe the ILY values minus risk-free rate are greater than the liquidity premium in all situations. On the 15 coverage years and middle monthly insurance salary, the ratio of ILY to liquidity premium varied from 3.88 to 38.9 for males and 3.55 to 36.44 for females by
Figure64-5: Liquidity premium and Implied Longevity Yield (basic point)－Male, age 60, µ=10%,σ =20%
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Figure74-6: Liquidity premium and Implied Longevity Yield (basic point)－Female, age 60, µ=10%,σ =20%
the coefficient of relative risk aversion from 1.5 to 3. On the 20 coverage years, the ratio of ILY to liquidity premium varied from 2.83 to 28.41 for males and 2.6 to 26.65 for females. On the 30 coverage years, the ratio of ILY to liquidity premium varied
Table94-5: The ratio of ILY to liquidity premium
coverage years Male Female
15 3.88 ~ 38.9 3.55 ~ 36.44 20 2.83 ~ 28.41 2.6 ~ 26.65
30 2.18 ~ 21.84 2 ~ 20.51
35 2.72 ~ 27.29 2.5 ~ 25.62 40 3.29 ~ 32.96 3.01 ~ 30.91
from 2.18 to 21.84 for males and 2 to 20.51 for females. On the 35 coverage years, the ratio of ILY to liquidity premium varied from 2.72 to 27.29 for males and 2.5 to 25.62 for females. On the 40 coverage years, the ratio of ILY to liquidity premium varied from 3.29 to 32.96 for males and 3.01 to 30.91 for females (Table 4-5). As the result,
0
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the actual rate of return is greater than theoretical yield that the annuitized pension benefit should provide. Therefore, we propose that the annuitized pension benefit is better than the one-time benefit under Labor Insurance.