Define and calculate payoffs under each of the following options embedded in insurance and annuity contracts: guaranteed minimum death benefit, guaranteed minimum maturity benefit, guaranteed minimum income benefit, and guaranteed minimum withdrawal benefit.

A variable annuity holder watches her subaccount drop 35 percent. Without a rider she eats the loss. With a GMxB rider, the insurer wrote her a put. Pricing that put is the whole game.

Let $F_t$ be the account (subaccount) value at time $t$, which moves with the underlying investments. Let $G_t$ be the guaranteed benefit base, evolved by contract rules. Every rider pays the policyholder $\max(F_t, G_t)$ at its trigger event. The insurer's net cost is $\max(G_t - F_t, 0)$. That is a long put on $F_t$ struck at $G_t$.

KEY: $F_t$ is what the account is worth; $G_t$ is what the contract promises. The guarantee bites only when markets push $F_t$ below $G_t$.

Withdrawals usually reduce $G_t$. Pro-rata adjustments shrink $G$ by the same percentage as $F$, which protects the insurer; dollar-for-dollar adjustments shrink $G$ by the cash withdrawn, which protects the holder. Read the contract before computing.

Pays $\max(F_{T_d}, G_{T_d})$ on death at random time $T_d$. The insurer's shortfall $\max(G_{T_d} - F_{T_d}, 0)$ is a put with a random exercise time driven by mortality.

Common mistakes

• Mixing roll-up and ratchet. Roll-up grows $G$ at fixed rate $g$ regardless of $F$; ratchet only steps $G$ up when $F$ exceeds the prior peak. A 5 percent roll-up on a flat market still raises $G$; a ratchet does not.
• Dropping the survival factor in GMMB. GMMB pays only if the annuitant is alive at maturity. Forgetting ${}_n p_x$ overstates the EPV by 5 to 20 percent depending on age and term.
• Annuitizing $F_T$ at the guaranteed factor. The guaranteed factor applies to $G_T$; the current factor applies to $F_T$. Crossing them flips which side wins.

Bottom line

• Every GMxB payoff is max(account value, guaranteed base); the insurer covers the shortfall $\max(G_t - F_t, 0)$, a put on the fund struck at the guaranteed base $G_t$.
• GMDB pays on death, GMMB at maturity if alive, GMIB at annuitization, GMWB through scheduled withdrawals even after the fund hits zero.
• Four base mechanics: return of premium, roll-up at rate $g$, ratchet (high-water mark), reset to current fund value at fixed intervals.
• Survival matters: GMMB, GMIB, and GMWB EPVs carry a survival factor, while GMDB EPV carries a mortality density (cash flow conditional on death).

Exam shortcut

Write the payoff as $\max(F, G)$ first, then expand to the shortfall $\max(G - F, 0)$. The shortfall view makes the put structure obvious and prevents sign errors. DECISION: Mortality trigger → integrate against ${}_t p_x \mu_{x+t}$ (GMDB). Survival trigger → multiply by ${}_t p_x$ alone (GMMB, GMIB, GMWB). Mnemonic "DMIW" for the four riders in their natural order: Death, Maturity, Income, Withdrawal.

The full lesson (about 1,740 words, 12 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 7a

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