Guaranteed Minimum Maturity Benefit

Study guide: Exam ALTAM

A guaranteed minimum maturity benefit (GMMB) guarantees that a policyholder who is alive at the end of an equity-linked contract's term $n$ receives at least $G$ , so the maturity benefit is $\max(G, F_n)$ . The insurer pays $(G - F_n)^+$ to each survivor, which is a European put on the fund with strike $G$ and expiry $n$ . Its value at issue is the survival probability ${}_np_x$ times a Black-Scholes put on the fund net of all future charges.

$\text{Payoff at } n = \mathbb{1}\{T_x > n\}\,(G - F_n)^+$

$\pi(0) = {}_np_x\left[Ge^{-rn}N(-d_2) - F_0(1-m)^nN(-d_1)\right]$

$d_1 = \frac{\ln\!\left(F_0(1-m)^n/G\right) + \left(r + \tfrac12\sigma^2\right)n}{\sigma\sqrt{n}}$

$\begin{aligned} \pi(t) = {}_{n-t}p_{x+t}\big[ & Ge^{-r(n-t)}N(-d_2) \\ & - F_t(1-m)^{n-t}N(-d_1)\big] \end{aligned}$

Symbols: $T_x$ is the policyholder's future lifetime, $F_0$ is the premium invested and $m$ is the management charge deducted at the start of each year, so $F_n = F_0(1-m)^nS_n/S_0$ for an index $S$ . $r$ is the continuously compounded risk-free rate, $\sigma$ is the index's volatility and $d_2 = d_1 - \sigma\sqrt{n}$ . At a later time $t$ , $\pi(t)$ is the value per policy still in force, $F_t$ is the fund before that year's charge, and $d_1$ and $d_2$ use $F_t(1-m)^{n-t}$ and the remaining term $n - t$ .
Why it is a put: The fund is a fixed number of units of the index, $F_0(1-m)^n/S_0$ of them by maturity, so the guarantee is a put on those units with strike $G$ . If the charge were taken continuously at rate $m$ , the fund would behave like a stock with dividend yield $m$ , and the put would use $F_0e^{-mn}$ in place of $F_0(1-m)^n$ .
The GMAB: A guaranteed minimum accumulation benefit applies the same guarantee at the end of each term of a contract that renews. If the fund is below the guarantee at a renewal date, the insurer tops it up, and the guarantee for the next term is reset, often to the fund value at renewal. Each term is valued like a GMMB, but the reset makes the strikes of later terms depend on the fund's path.
Lapses: A policy that lapses before $n$ loses the guarantee. With lapses, ${}_np_x$ is replaced by the probability that the policy is still in force at $n$ , which lowers the cost of the guarantee.

Value of a 10-Year GMMB at Issue A single premium of $100,000 from a life aged 60 is invested in an equity index fund, and a management charge of $1.5\%$ of the fund is deducted at the start of each year. A policyholder alive at the end of 10 years receives at least $100,000. Use $r = 0.04$ , $\sigma = 0.20$ and the Standard Ultimate Life Table, for which ${}_{10}p_{60} = 0.942549$ . Find the value of the guarantee at issue.
Answer The put is on $100{,}000(0.985)^{10} = 85{,}973.04$ , with strike $100{,}000$ and expiry 10.
 $\begin{align*} d_1 &= \frac{\ln(0.8597304) + 0.06(10)}{0.20\sqrt{10}} \\ &= \frac{-0.15114 + 0.6}{0.63246} \\ &= 0.70972 \end{align*}$ 
So $d_2 = 0.07726$ , $N(-d_2) = 0.46921$ , $N(-d_1) = 0.23894$ and $Ge^{-0.4} = 67{,}032.00$ .
 $\begin{align*} P &= 67{,}032.00\,N(-d_2) - 85{,}973.04\,N(-d_1) \\ &= 31{,}451.97 - 20{,}542.42 \\ &= 10{,}909.55 \\ \pi(0) &= 0.942549(10{,}909.55) \\ &= 10{,}282.79 \end{align*}$ 
The guarantee is worth about $10.3\%$ of the premium. The insurer has to fund it from the charges, and on this contract it takes most of the $1.5\%$ charge (see Equity-Linked Insurance).

Value of the Guarantee After a Market Fall Three years after issue, the policyholder in the first example is alive at 63, and the fund has fallen to $F_3 = 75{,}000$ before the charge for year 4 is taken. Given ${}_7p_{63} = 0.953399$ , find the value of the guarantee per policy in force.
Answer Seven charges remain, so the put is on $75{,}000(0.985)^7 = 67{,}470.65$ , with 7 years to run.
 $\begin{align*} d_1 &= \frac{\ln(0.6747065) + 0.06(7)}{0.20\sqrt{7}} \\ &= \frac{-0.39348 + 0.42}{0.52915} \\ &= 0.05012 \end{align*}$ 
So $d_2 = -0.47903$ , $N(-d_2) = 0.68404$ , $N(-d_1) = 0.48001$ and $Ge^{-0.28} = 75{,}578.37$ .
 $\begin{align*} P &= 75{,}578.37\,N(-d_2) - 67{,}470.65\,N(-d_1) \\ &= 51{,}698.67 - 32{,}386.74 \\ &= 19{,}311.93 \\ \pi(3) &= 0.953399(19{,}311.93) \\ &= 18{,}411.97 \end{align*}$ 
The fall has raised the guarantee's value from $10,283 to $18,412 per policy. A market-consistent reserve for the guarantee rises by the same amount, and so does the hedge the insurer has to hold.

How it appears on the exam

ALTAM asks candidates to "define and calculate payoffs" under a GMMB, given a fund path with its charges, and then to value the "guaranteed minimum accumulation/maturity benefit using the Black-Scholes model" at issue or at a later date.

Reserving and hedging: The section's aim is to "price, reserve, and hedge" these options, so a question can follow the valuation with the value at a later date, as in the second example, or with the replicating portfolio and its rebalancing cost.

Profit testing: In a deterministic profit test the GMMB appears as the top-up $(G - F_n)^+$ paid to survivors in the final year's cash flow under the stated fund path.

Exam traps

The survival weight: The value is the market price of the put times ${}_np_x$ , because only survivors are paid. Leaving out the survival probability overstates the cost, and using a death probability values a GMDB instead.

The charges: The put is written on $F_0(1-m)^n$ , not on $F_0$ . Leaving out the charges understates the cost, which falls from $10,282.79 to $7,596.23 in the first example.

The rate basis: Black-Scholes needs the continuously compounded rate. Given an effective rate of $4\%$ , use $r = \ln 1.04 = 0.03922$ rather than $0.04$ .

A later valuation: At time $t$ , use the remaining term $n - t$ , the survival probability ${}_{n-t}p_{x+t}$ from the policyholder's current age, and the fund net of the charges still to come.