Whole Life Insurance

Study guides: Exam MAS-I, Exam DISC-IA

Whole Life Insurance is a life insurance contract with an unlimited term: it pays a benefit of 1 (one unit) on the death of the insured, whenever that occurs. In the life-table model of Exam MAS-I the benefit is paid at the end of the year of death, and its actuarial present value (the net single premium) for a life aged $x$ is $A_x$ .

$A_x = \sum_{k=0}^{\infty} v^{k+1}\,{_k}p_x\,q_{x+k}$

$A_x = v\,q_x + v\,p_x\,A_{x+1}$

$P = \frac{A_x}{\ddot{a}_x}$

$v = 1/(1+i)$ : a death in year $k+1$ (survive $k$ years, probability ${_k}p_x$ , then die, $q_{x+k}$ ) is paid at time $k+1$
Recursion: either $(x)$ dies in the first year and 1 is paid in a year, or $(x)$ survives and holds, one year from now, a whole life policy on $(x+1)$
Level annual premium: a premium $P$ paid at the start of each year while $(x)$ is alive is a life-annuity-due, so $A_x = P\,\ddot{a}_x$ ; since $\ddot{a}_x = (1 - A_x)/d$ , $P$ follows from $A_x$ and $d$ alone
Variance: the second moment ${}^2\!A_x$ of the present value is $A_x$ at discount factor $v^2$ (double the force of interest), and $\text{Var} = {}^2\!A_x - (A_x)^2$ . The exam's Illustrative Life Table lists $1000A_x$ , $1000\,{}^2\!A_x$ and $\ddot{a}_x$ at $i = 0.06$
Paid at the moment of death (continuous variant): $\bar{A}_x = E\!\left[e^{-\delta T_x}\right] = \int_0^\infty e^{-\delta t}\,{_t}p_x\,\mu_{x+t}\,dt$ , with ${}^2\!\bar{A}_x$ at $2\delta$ ; under a constant force of mortality $\mu$ , $\bar{A}_x = \mu/(\mu + \delta)$
As a product: Whole life is a cash value policy: coverage lasts for life as long as premiums are paid, and the premiums are typically level, so they start higher than for the same amount of term insurance and build a cash value. A limited-payment version, such as premiums payable to age 65, has higher premiums. The owner can take a policy loan against the cash value, and any unpaid loan and interest is subtracted from the death benefit or cash value. If premiums stop, state nonforfeiture laws require the policy to give a paid-up benefit and, once premiums have been paid for at least three full years, a cash surrender value. Whole life's premiums follow a set schedule, unlike the flexible premiums of universal life.

Whole Life Value by Recursion The Illustrative Life Table ( $i = 0.06$ ) gives $1000q_{40} = 2.78$ and $1000A_{41} = 168.69$ . Calculate $A_{40}$ .

Answer $A_{40} = \frac{q_{40} + p_{40}\,A_{41}}{1.06} = \frac{0.00278 + 0.99722 \times 0.16869}{1.06} = \frac{0.17100}{1.06} = 0.16132$ This is the table's own $1000A_{40} = 161.32$ .

Level Annual Premium from the Illustrative Life Table Using the Illustrative Life Table ( $A_{40} = 0.16132$ , $\ddot{a}_{40} = 14.8166$ , $i = 0.06$ ), find the level annual premium for a whole life insurance of 1 on $(40)$ .

Answer $P = \frac{A_{40}}{\ddot{a}_{40}} = \frac{0.16132}{14.8166} = 0.010888$ From $A_{40}$ alone, with $d = 0.06/1.06$ : $\ddot{a}_{40} = (1 - 0.16132)(1.06)/0.06 = 14.8167$ , and $P = 0.16132/14.8167 = 0.010888$ again.

Paid at the Moment of Death, Constant Force A life is subject to constant force of mortality $\mu = 0.02$ and the force of interest is $\delta = 0.05$ . Calculate $\bar{A}_x$ and the variance of the present value $e^{-\delta T_x}$ .

Answer $\bar{A}_x = \frac{\mu}{\mu + \delta} = \frac{0.02}{0.07} = 0.2857$ ${}^2\!\bar{A}_x = \frac{0.02}{0.02 + 2(0.05)} = \frac{0.02}{0.12} = 0.1667$ $\text{Var} = {}^2\!\bar{A}_x - (\bar{A}_x)^2 = \frac{1}{6} - \frac{4}{49} = \frac{25}{294} \approx 0.0850$

How it appears on the exam

MAS-I: The objective is to calculate simple whole life or annuity problems. Questions give Illustrative Life Table values and ask for an EPV, a level premium $A_x/\ddot{a}_x$ , or the effect of a change in the table, as in the examples.

DISC-IA: The Life and Health Insurance section asks for the features of whole life against term and universal life: lifetime coverage, level scheduled premiums, cash values, policy loans and the nonforfeiture options if premiums stop.