Pure Endowment

Study guide: Exam FAM

A Pure Endowment pays a sum insured at the end of a term of $n$ years if the insured, aged $x$ at issue, is still alive then, and pays nothing if the insured dies first. Its expected present value (EPV) per unit, ${}_nE_x$ , discounts for interest and survival together, so it serves both as a contract in its own right and as the actuarial discount factor that moves life-contingent values through time. It is the survival part of an endowment insurance.

$Z = \begin{cases} 0 & K_x \le n-1 \\ v^n & K_x \ge n \end{cases}$

${}_nE_x = A^{\phantom{x:}1}_{x:\overline{n}|} = v^n\,{}_np_x$

$\text{Var}(Z) = v^{2n}\,{}_np_x\,{}_nq_x$

${}_{m+n}E_x = {}_mE_x\;{}_nE_{x+m}$

Symbols: $Z = v^n I$ , where $I$ indicates survival of $(x)$ to age $x+n$ and is a Bernoulli variable with probability ${}_np_x$ . The mean and variance follow, and the second moment is ${}^2_{\,n}E_x = v^{2n}\,{}_np_x = v^n\,{}_nE_x$ . The 1 above $\overline{n}|$ in the second block shows that the term must end before the life fails. The payment time is fixed, so there are no annual, $1/m$ -thly or continuous versions to convert between.
Actuarial discount factor: ${}_nE_x$ moves a value at age $x+n$ back to age $x$ for a life who must survive to collect it. That is how a deferred annuity is valued, ${}_{n|}\ddot{a}_x = {}_nE_x\,\ddot{a}_{x+n}$ , and how a whole life value splits, $A_x = A^{1}_{x:\overline{n}|} + {}_nE_x\,A_{x+n}$ . Dividing by ${}_nE_x$ accumulates instead, so the actuarial accumulated value of an annuity is $\ddot{s}_{x:\overline{n}|} = \ddot{a}_{x:\overline{n}|}/{}_nE_x$ .
Part of an endowment: An endowment insurance is a term insurance plus a pure endowment, $A_{x:\overline{n}|} = A^{1}_{x:\overline{n}|} + {}_nE_x$ . The two parts can never both pay, so their present values have covariance $-A^{1}_{x:\overline{n}|}\,{}_nE_x$ . In FAM's objectives, "endowment" covers both this survival benefit and the endowment insurance.
Assumptions: ${}_nE_x$ falls when the interest rate rises and rises when mortality is lighter. That is the opposite of a death benefit's response to mortality, which is why survival benefits carry longevity risk.

Twenty-Year Pure Endowment on the SULT A pure endowment of $100,000 is issued to $(45)$ for 20 years. On the SULT, $\ell_{45} = 99{,}033.9$ and $\ell_{65} = 94{,}579.7$ , and $i = 5\%$ . Find the EPV and the standard deviation of the present value.
Answer ${}_{20}p_{45} = 94{,}579.7/99{,}033.9 = 0.95502$ and $v^{20} = 0.376889$ .
 $\begin{align*} {}_{20}E_{45} &= 0.376889(0.95502) \\ &= 0.359937 \\ \text{EPV} &= 100{,}000(0.359937) \\ &= 35{,}993.70 \\ \text{SD} &= 100{,}000(0.376889)\sqrt{0.95502(0.04498)} \\ &= 37{,}688.9(0.207260) \\ &= 7{,}811.4 \end{align*}$ 
The standard deviation is about $22\%$ of the EPV. Only $4.5\%$ of lives forfeit the benefit, so the value of the contract depends mainly on the interest rate.

The Pure Endowment Bought by Annual Premiums A life aged 55 pays $10,000 at the start of each year for 10 years, while alive, to buy a pure endowment payable at age 65. On the SULT at $5\%$ , $\ddot{a}_{55:\overline{10}|} = 8.0192$ and ${}_{10}E_{55} = 0.59342$ . Find the pure endowment and compare it with the same deposits saved at $5\%$ interest.
Answer By the equivalence principle, $X\,{}_{10}E_{55} = 10{,}000\,\ddot{a}_{55:\overline{10}|}$ .
 $\begin{align*} X &= \frac{10{,}000(8.0192)}{0.59342} \\ &= 135{,}135.3 \\ 10{,}000\,\ddot{s}_{\overline{10}|} &= 10{,}000\,\frac{1.05^{10} - 1}{0.05/1.05} \\ &= 132{,}067.9 \end{align*}$ 
The survivors receive about $3,067 more than a bank account would give. That is their share of the deposits of the $3.3\%$ of lives who die before 65 and forfeit them, which is the survivorship benefit that the division by ${}_{10}E_{55}$ builds in.

How it appears on the exam

FAM asks candidates to identify the present value random variable of a pure endowment and to find its mean, variance or a probability, for example as one part of an endowment insurance or a deferred annuity.

Relationships: FAM's objective on the relationships between insurance, endowment and annuity present values uses ${}_nE_x$ to split whole life and annuity values at age $x+n$ , and to find the covariance between the term and pure endowment parts of an endowment insurance.

Assumption changes: A question may ask how a value that includes ${}_nE_x$ moves when interest or mortality changes, which tests the opposite response of survival and death benefits.

Exam traps

Second moment versus squared mean: The second moment is $v^{2n}\,{}_np_x$ , while the squared mean is $v^{2n}({}_np_x)^2$ . Their difference is the variance, so using one for the other gives a variance of zero.

Accumulating at interest only: Dividing by $v^n$ ignores the forfeited deposits of those who die. The actuarial accumulation divides by ${}_nE_x$ , and the gap was about $3,067 in the second example.

Converting to continuous: A UDD factor such as $i/\delta$ applies only to death benefits. Applying it to ${}_nE_x$ inside an endowment insurance overstates the value, because the maturity payment is already made at a fixed time.

Endowment versus pure endowment: ${}_nE_x$ values a payment on survival only, while $A_{x:\overline{n}|}$ also values the payment on death within the term. Reading "endowment" as one when the question means the other changes the answer by the whole term insurance.