Indexed Annuity

Study guides: Exam CP-312, Exam INV-201

An indexed annuity, or fixed indexed annuity (FIA), is a deferred annuity whose account value is credited each period with an index-linked credit: a formula applied to the return of an equity index, typically with a participation rate, a cap or a spread on the upside and a floor of $0\%$ on the downside. The principal and past credits are guaranteed, so the policyholder never loses money to the index, and the credit formula is an embedded option, usually a call spread, that the insurer has sold and funds from the yield on its general account.

$c = \max\Big(0,\ \min\big(C,\ p\,R\big)\Big)$

$c_{\text{spread}} = \max(0,\ R - s)$

$c_{\text{monthly}} = \max\left(0,\ \sum_{m=1}^{12}\min\left(C_m,\ R_m\right)\right)$

$AV_t = AV_{t-1}\,(1 + c_t)$

$B = y - \pi$

Symbols: $R = I_1/I_0 - 1$ is the index's price return over the crediting period, which excludes dividends. $p$ is the participation rate, $C$ the annual cap, $s$ the spread or margin, and $C_m$ the monthly cap applied to each month's return $R_m$ . $AV_t$ is the account value after the credit $c_t$ . In the last block, $B$ is the option budget as a share of the account value, $y$ the insurer's expected general account yield and $\pi$ the pricing spread it keeps for expenses, capital and profit.
Crediting strategies: Annual point-to-point compares the index at the start and end of the year. A monthly cap sums the twelve monthly returns, capping each gain but not each loss, and floors the total at zero, so it is path-dependent. A participation rate gives a share of the gain, and a spread subtracts a fixed margin. Credits are locked in each year, the annual reset, so a fall in one year does not have to be made back before the next credit.
The option the insurer is short: With $p = 1$ , the payoff $\max(0, \min(C, R))$ is a long at-the-money call less a call struck at $1 + C$ , a bull call spread. A participation rate with no cap is $p$ at-the-money calls, and a spread is one call struck at $1 + s$ . A monthly cap is a cliquet, an exotic option priced by simulation.
The option budget: The insurer invests the premium in bonds and spends $B$ of each year's account value on options that pay the credit. The cap, participation rate or spread is then set at renewal so the option costs exactly the budget. When rates fall or implied volatility rises, the same budget buys a lower cap, and the insurer can reset only down to the guaranteed minimum cap in the contract.
Hedging: Because the option is reset each year, the insurer's main hedge is to buy the year's call spreads on the account value, a static hedge for the year. Mismatches come from lapses and deaths during the year, from policyholder behaviour at renewal, from basis between the index and the hedge, and from the forward-starting options in future years, whose prices are unknown until each reset.
Modelling index credits: In a projection model the credit can be generated from simulated index scenarios, which shows the distribution of credits and of the hedge's gains and losses, or approximated by assuming that the hedge payoff funds the credit, so the cost of crediting is the option budget. The second approach is common in deterministic profit tests and reserves, and the first is needed to test the hedge, the guaranteed minimums and policyholder behaviour.
RILAs: A registered index-linked annuity gives up the $0\%$ floor for a higher cap. A buffer has the insurer absorb the first losses, such as $10\%$ , and a floor limits the policyholder's loss to a set level, such as $10\%$ . Because the policyholder can lose principal, it is a security registered with the SEC. Its crediting option is the call spread less a put under a buffer, or less a put spread under a floor, as described under Structured Product.

Annual Credits Under a 6% Cap An indexed annuity with an account value of $100,000 credits $100\%$ of the annual point-to-point index return, capped at $6\%$ , with a $0\%$ floor. The index returns $+12\%$ , $+4\%$ , $-9\%$ and $+6.5\%$ in four successive years. Find the credits and the account value after four years. Then find the credits under two alternative strategies on the same returns: a $2\%$ spread with no cap, and a $40\%$ participation rate with no cap.
Answer Under the cap the credits are $6\%$ , $4\%$ , $0\%$ and $6\%$ .
 $\begin{align*} AV_4 &= 100{,}000(1.06)(1.04)(1.00)(1.06) \\ &= 116{,}854.40 \end{align*}$ 
The spread strategy credits $\max(0, R - 0.02)$ , which is $10\%$ , $2\%$ , $0\%$ and $4.5\%$ . The participation strategy credits $0.40R$ when $R > 0$ , which is $4.8\%$ , $1.6\%$ , $0\%$ and $2.6\%$ . The capped strategy does best in moderate years, the spread in strong ones, and none of them loses anything in the $-9\%$ year, which is what the policyholder pays for by giving up dividends and most of the upside.

Setting the Cap from the Option Budget The insurer earns $5.0\%$ on its general account and prices for a spread of $2.0\%$ , so the option budget is $3.0\%$ of the account value. One-year call options on the index trade at these prices per $1 of index: $0.0797$ at the money, $0.0506$ struck at $107\%$ and $0.0472$ struck at $108\%$ . Find the annual point-to-point cap the budget supports with $100\%$ participation, and the uncapped participation rate it supports.
Answer A cap $C$ costs the at-the-money call less the call struck at $1 + C$ .
 $\begin{align*} \text{Cost}(7\%) &= 0.0797 - 0.0506 \\ &= 0.0291 \\ \text{Cost}(8\%) &= 0.0797 - 0.0472 \\ &= 0.0325 \\ C &\approx 7\% + \frac{0.0300 - 0.0291}{0.0325 - 0.0291}(1\%) \\ &= 7.26\% \end{align*}$ 
The insurer would declare a cap of about $7.25\%$ . With no cap, the budget buys $p = 0.0300/0.0797 = 37.6\%$ of an at-the-money call. On $100,000 the insurer spends $3,000 on options. If the index rises $10\%$ , the call spread pays $7,250, which funds the credit exactly, and the insurer's own margin is the $2\%$ it kept from the bond yield.

A Monthly Cap Against Annual Point-to-Point A monthly-cap strategy caps each month's index return at $2.0\%$ , with no limit on monthly losses, and floors the annual sum at $0\%$ . In one year the monthly returns, in percent, are $3.0$ , $1.5$ , $-2.0$ , $2.5$ , $0.8$ , $-5.0$ , $1.2$ , $2.2$ , $-1.0$ , $3.5$ , $1.0$ and $0.5$ . Find the credit, and compare it with a $6\%$ annual point-to-point cap.
Answer The capped gains are $2.0$ , $1.5$ , $2.0$ , $0.8$ , $1.2$ , $2.0$ , $2.0$ , $1.0$ and $0.5$ , which sum to $13.0$ , and the uncapped losses sum to $-8.0$ .
 $\begin{align*} c_{\text{monthly}} &= \max(0,\ 13.0 - 8.0)\% \\ &= 5.0\% \\ R_{\text{annual}} &= (1.030)(1.015)\cdots(1.005) - 1 \\ &= 8.18\% \end{align*}$ 
The point-to-point strategy credits $\min(6\%, 8.18\%) = 6\%$ , more than the monthly cap's $5.0\%$ . The monthly cap loses whenever a few large monthly gains are clipped while the losses count in full, so it pays most in steady, low-volatility rises. That is why it is cheap, and why its credit depends on the path and not just the year's return.

How it appears on the exam

INV-201: The objective asks candidates to "identify and evaluate embedded options in liabilities", naming the indexed annuity. Expect to decompose a crediting formula into calls, price it with Black-Scholes or given option prices, and explain how the insurer hedges it and which risks remain.

CP-312: The modelling objective lists "index-linked credits" among the complex components a cash flow model must handle, read in Cardinal's chapter on fixed and fixed indexed annuities. Expect to project credits under a crediting strategy, explain the choice between scenario-based and budget-based credits, and describe how the hedge and lapse behaviour enter the model.

Renewal questions: Questions also give an option budget and option prices and ask for the cap, participation rate or spread the insurer can offer, or how the cap must change when rates or volatility move.

Exam traps

Price return, not total return: The index credit uses the price index, which excludes dividends. Crediting on total return overstates the credit and the option's value, and the dividend yield lowers the call's price in Black-Scholes.

Order of cap and participation: Check whether the participation rate applies before the cap, $\min(C, pR)$ , or after it. With $p = 80\%$ , a cap of $6\%$ and $R = 10\%$ , the first gives $6\%$ and $\min(C, R)$ times $p$ gives $4.8\%$ .

Monthly losses are not capped: In a monthly cap only the gains are capped. Applying a floor to each month, or capping the losses, overstates the credit, and the $0\%$ floor applies only to the annual sum.

The insurer is short the option: The policyholder is long the call spread, and the insurer funds it from its yield. Treating the option budget as income, or the hedge payoff as profit, double counts the credit.