Annuity Contract

Study guides: Exam ALTAM, Exam CP-341, Exam ILA-101, Exam ILA-201I, Exam ILA-201U

An Annuity Contract is an insurance contract that pays a stream of income for as long as a life (or lives) survive, or for a period, in exchange for a single premium or a series of premiums. It may be immediate (income starts at once) or deferred (an accumulation phase first), and fixed (guaranteed amounts) or variable (amounts linked to a fund, often with guarantees). It is the product built on the life annuity functions, and the insurer's main exposure is to longevity and investment risk.

$G = \text{EPV(benefits)} + \text{EPV(expenses)}$

$\text{Deferred: } \text{EPV} = B\;{}_nE_x\;\ddot{a}_{x+n}$

$G$ is the single premium set by the equivalence principle; ${}_nE_x = v^n\,{}_np_x$ is the pure endowment factor and $\ddot{a}_{x+n}$ the annuity-due at the start of payout (use $\ddot{a}^{(12)}$ for monthly income). A deferred annuity-certain is the non-life analogue
Product forms: single premium immediate annuity (SPIA); deferred annuity with a cash value during accumulation and annuitization at the end; joint and last-survivor annuity on two lives; guaranteed-period annuity; pension annuities bought in a pension risk transfer; variable annuities with GMDB, GMMB, GMIB or GMWB guarantees whose payoffs are valued with the Black-Scholes Model
A return-of-premium death benefit during deferral, or a guaranteed period, adds an insurance element and raises the price; the insurer profit tests the contract on its own interest, mortality and expense basis
Annuity blocks are long-dated and asset-intensive, which is why they are reinsured by asset-intensive coinsurance, funds withheld or longevity swaps (Reinsurance Transaction)

Pricing a Single Premium Immediate Annuity A life aged 65 buys an annuity of $12{,}000$ a year payable annually in advance. On the pricing basis $\ddot{a}_{65} = 13.0$ . Expenses are $\$200$ plus $3\%$ of the single premium, both at issue. Find the single premium.
Answer
 $\begin{align*} G &= 12{,}000(13.0) + 200 + 0.03G \\ G &= \frac{156{,}000 + 200}{0.97} \\ &= 161{,}030.93 \end{align*}$ 
The expected benefit cost is $\$156{,}000$ ; the remaining $\$5{,}031$ covers expenses.

Deferred Annuity With Return of Premium on Death A life aged 55 pays single premium $G$ for an annuity of $10{,}000$ a year, annually in advance from age 65. If the life dies before 65, $G$ (without interest) is returned at the end of the year of death. Mortality before 65 has a constant force $\mu = 0.01$ , $i = 5\%$ and $\ddot{a}_{65} = 13.0$ . Find $G$ , and the cost of the refund feature.
Answer $p = e^{-0.01} = 0.990050$ , $q = 0.009950$ , so ${}_{10}E_{55} = v^{10}p^{10} = 0.613913(0.904837) = 0.555492$ .
 $\begin{align*} A^{1}_{55:\overline{10}|} &= \sum_{k=0}^{9} v^{k+1}p^{k}q \\ &= \frac{qv\left(1 - (vp)^{10}\right)}{1 - vp} \\ &= 0.073777 \end{align*}$ 
The equivalence principle gives $G = 10{,}000\,{}_{10}E_{55}\,\ddot{a}_{65} + G\,A^{1}_{55:\overline{10}|}$ :
 $\begin{align*} G &= \frac{10{,}000(0.555492)(13.0)}{1 - 0.073777} \\ &= \frac{72{,}213.92}{0.926223} \\ &= 77{,}966.00 \end{align*}$ 
Without the refund the premium would be $72{,}213.92$ , so the return-of-premium death benefit costs $\$5{,}752$ , about $8\%$ more.

How it appears on the exam

ALTAM values annuity contracts on single and joint lives and in multiple state settings (for example CCRC contracts), and profit tests them: the profit vector and signature, NPV, IRR, profit margin and the premium or reserves for a given profit objective.

ALTAM's embedded-options objectives treat the variable annuity as an annuity contract carrying a guarantee: payoffs of GMDB, GMMB, GMIB and GMWB, their Black-Scholes values and the delta hedge.

CP-341 asks to describe and evaluate reinsurance transactions for annuity contracts: asset-intensive coinsurance, funds withheld, pension risk transfer and longevity swaps, and why insurers cede annuity blocks offshore.

Exam traps

Pricing a return-of-premium benefit as if $G$ were known: the refund is a fraction of the premium being solved for, so $G$ appears on both sides and the answer divides by $1 - A^1$ .

Using $a_{65}$ when payments are in advance, which drops the first payment, or an annual $\ddot{a}_{65}$ for monthly income in advance, which overstates the EPV by about $11/24$ of a year's income ( $\ddot{a}^{(12)}_{65} \approx \ddot{a}_{65} - 11/24$ under UDD).

Applying the deferral survival probability twice. ${}_nE_x$ already includes ${}_np_x$ , and $\ddot{a}_{x+n}$ starts from age $x+n$ , so no further survival factor is needed.