Payoff

Study guides: Exam FAM, Exam ALTAM, Exam INV-201

A Payoff is the amount a derivative or an embedded guarantee pays its holder at the time it pays, written as a function of the underlying price or rate at that time, ignoring what was paid for the contract. Profit is the payoff less the accumulated premium, and the price is the risk-neutral expected present value of the payoff.

$\text{Call: } (S_T - K)^+ = \max(S_T - K,\,0)$

$\text{Put: } (K - S_T)^+ = \max(K - S_T,\,0)$

$\text{Long forward: } S_T - F_{0,T}$

$\text{GMMB: } \mathbb{1}\{T_x > n\}\,(G - F_n)^+$

$S_T$ is the underlying's price at expiry, $K$ the strike, $F_{0,T}$ the forward price, $G$ the guaranteed amount and $F_n$ the policyholder's fund at maturity $n$ ; $(\cdot)^+$ keeps only the positive part, so an option's payoff is never negative while a forward's can be
Interest-rate options (INV-201): a caplet pays $N\,\tau\,(L - K)^+$ and a floorlet $N\,\tau\,(K - L)^+$ , where $N$ is the notional, $\tau$ the accrual fraction and $L$ the reference rate set at the start of the period. The payment is made at the end of the period, in arrears. A cap is a strip of caplets
Embedded guarantees (ALTAM): a GMDB pays $(G_t - F_t)^+$ at death; a GMMB pays $(G - F_n)^+$ at maturity if alive. A GMWB guarantees withdrawals totalling the premium even if the fund runs out; a GMIB guarantees an annuity rate. $G$ may be the premium, a roll-up or a ratchet to past fund highs
Each guarantee is in effect a put option on the fund, which is why its value comes from the Black-Scholes Model and its risk is managed by hedging

Bull Call Spread at Three Prices An investor buys a call with $K = 100$ for $8.00$ and sells a call with $K = 120$ for $2.50$ , both expiring together. Find the payoff and the profit (ignoring interest on the premiums) if $S_T = 90$ , $110$ or $130$ .

Answer Payoff $= (S_T - 100)^+ - (S_T - 120)^+$ ; net premium $= 8.00 - 2.50 = 5.50$ .

$S_T = 90$ : payoff $0$ , profit $-5.50$

$S_T = 110$ : payoff $10$ , profit $4.50$

$S_T = 130$ : payoff $30 - 10 = 20$ , profit $14.50$

The payoff is capped at $120 - 100 = 20$ : the investor gives up gains above 120 to cheapen the bet on a rise.

GMDB and GMMB Payoffs on a Variable Annuity A single premium of $1{,}000$ buys a 3-year variable annuity. At the end of each year the fund earns the index return and then a $2\%$ management charge is deducted. Index returns are $+10\%$ , $-30\%$ and $+5\%$ . On death the contract pays, at the end of the year of death, the larger of the fund and the premium rolled up at $3\%$ a year. At maturity a living policyholder receives at least the premium. Find the guarantee payoff for death in each year and at maturity.

Answer Fund values:

$F_1 = 1{,}000(1.10)(0.98) = 1{,}078.00$

$F_2 = 1{,}078.00(0.70)(0.98) = 739.51$

$F_3 = 739.51(1.05)(0.98) = 760.95$

The GMDB guarantee is $G_t = 1{,}000(1.03)^t$ : $1{,}030.00$ , $1{,}060.90$ , $1{,}092.73$ .

Death in year 1: $(1{,}030.00 - 1{,}078.00)^+ = 0$

Death in year 2: $1{,}060.90 - 739.51 = 321.39$

Death in year 3: $1{,}092.73 - 760.95 = 331.77$

Survival to maturity (GMMB): $1{,}000 - 760.95 = 239.05$

The insurer's cost is the guarantee payoff only; the fund itself belongs to the policyholder.

Caplet Payoffs on a Floating-Rate Loan A borrower holds a cap on a $\$10$ million quarterly floating-rate loan with cap rate $4\%$ . The reference rates set at the start of three quarters are $3.6\%$ , $4.3\%$ and $4.8\%$ . Find each caplet's payoff, taking $\tau = 0.25$ .

Answer

$3.6\%$ : $10{,}000{,}000 \times 0.25 \times (0.036 - 0.04)^+ = 0$

$4.3\%$ : $10{,}000{,}000 \times 0.25 \times 0.003 = \$7{,}500$

$4.8\%$ : $10{,}000{,}000 \times 0.25 \times 0.008 = \$20{,}000$

Each is paid at the end of its quarter, so together with the loan the borrower's interest never exceeds $4\%$ .

How it appears on the exam

INV-201 asks for the payoff of forwards, futures, swaps, calls, puts, caps, floors and swaptions, alone or combined into a strategy (spread, straddle, collar), often as a payoff table or diagram, and asks what view the strategy expresses.

ALTAM gives a fund path (index returns and charges) and asks for the GMDB, GMMB, GMIB or GMWB payoff, as the first step before valuing it with the Black-Scholes Model or profit testing it.

Written parts ask which guarantee behaves like which option, and why a roll-up or ratchet makes it more expensive.

Exam traps

Reporting profit when payoff is asked, or the reverse: a long call that expires out of the money has payoff $0$ but profit equal to minus the accumulated premium.

Projecting the fund without its charges: $1{,}000(1.10)(0.70)(1.05) = 808.50$ instead of $760.95$ in the second example, which understates every guarantee payoff.

Paying the caplet at the reset date instead of in arrears, or forgetting the accrual fraction $\tau$ — a payoff four times too large on a quarterly cap.

Adding the GMMB to the GMDB on a death in the final year: only one benefit is triggered, by the way the contract ends.