Reinstatements

Study guides: Exam 9, Exam GI-301

Reinstatements restore the limit of an excess of loss layer, typically a property catastrophe cover, after a loss has used it up, so the layer is available again for a later event in the same period. A reinstatement is usually paid for with a reinstatement premium, most often pro rata as to amount and less often also pro rata as to time.

$\text{RP} = P \times r \times \frac{\text{Limit reinstated}}{L}$

$\text{Aggregate limit} = (1 + n)\,L$

$\begin{aligned} &P\left(1 + r\,\frac{E[\min(S,\, nL)]}{L}\right) \\ &\quad = \text{Required total premium} \end{aligned}$

$P$ is the annual (up-front) premium, $r$ the reinstatement rate ( $100\%$ is common; Clark's illustration uses $110\%$ ), $L$ the occurrence limit, $n$ the number of reinstatements and $S$ the year's total loss to the layer before any aggregate cap. Pro rata as to time multiplies RP by the fraction of the term left at the loss date. Clark notes few contracts do this, because catastrophes such as hurricanes are seasonal.
Aggregate limit. With $n$ reinstatements the most the reinsurer pays in a year is $(1+n)L$ . Pro rata as to amount means that, with one reinstatement, four half-limit losses produce the same result as two full-limit losses.
Effect on loss cost. Limiting reinstatements caps expected ceded loss at $E[\min(S, (1+n)L)]$ . Expected reinstatement premium is additional premium, so the up-front premium needed to hit a target is lower by the factor $1 + r\,E[\min(S, nL)]/L$ . A reinsurer that ignored it would overprice the cover. Free reinstatements ( $r = 0$ ) must be paid for in $P$ instead.
Who bears what. The cedant pays the reinstatement premium at the worst moment, straight after a catastrophe. Its net cost of an event is retention, plus loss above the limit, plus RP, and the net PML must include it. For the reinsurer, premium that arrives with the losses makes the result less volatile.
Priced from catastrophe model output (the annual loss distribution to the layer — see Catastrophe Expected Loss Cost). Before models, the payback approach set $P$ so that $L$ would be repaid over a chosen number of years. See Reinsurance Contract Provisions.

Computing a Reinstatement Premium A calendar-year catastrophe cover is $\$20$ M xs $\$30$ M with an annual premium of $\$3$ M and one reinstatement at $100\%$ . A hurricane on September 1 causes a gross loss of $\$42$ M. Find the recovery and reinstatement premium if the reinstatement is (a) pro rata as to amount only, (b) also pro rata as to time. What cover remains?
Answer Layer loss: $\min(\max(42 - 30, 0), 20) = \$12$ M.

(a) As to amount:
 $\begin{align*} \text{RP} &= \$3\text{M} \times 1.00 \times \frac{12}{20} \\ &= \$1.8\text{M} \end{align*}$ 
(b) Also as to time, with four of twelve months left:
 $\begin{align*} \text{RP} &= \$1.8\text{M} \times \frac{4}{12} \\ &= \$0.6\text{M} \end{align*}$ 
In case (a) the cedant recovers $\$12$ M and pays back $\$1.8$ M, so its net cost of the event is $\$30\text{M} + \$1.8\text{M} = \$31.8$ M. The full $\$20$ M occurrence limit is available again for the next event. Of the $\$40$ M aggregate limit, $\$28$ M remains: $\$8$ M of reinstatement capacity is left after the $\$12$ M reinstated.

Pricing a Layer with One Paid Reinstatement A $\$10$ M xs $\$10$ M catastrophe layer has one reinstatement at $100\%$ , pro rata as to amount. The annual loss to the layer, before any cap, is $\$0$ ( $p = 0.80$ ), $\$5$ M ( $0.10$ , one partial event), $\$10$ M ( $0.06$ ), $\$20$ M ( $0.03$ , two full events) or $\$30$ M ( $0.01$ , three full events). The reinsurer targets a $60\%$ loss ratio on total premium. Find the up-front premium.
Answer With aggregate limit $\$20$ M, ceded loss is $0, 5, 10, 20, 20$ :
 $\begin{align*} E[\text{Ceded}] &= 0.10(5) + 0.06(10) + 0.03(20) + 0.01(20) \\ &= 1.9 \end{align*}$ 
Limit reinstated, $\min(S, 10)$ , is $0, 5, 10, 10, 10$ :
 $\begin{align*} E[\min(S, 10)] &= 0.10(5) + 0.06(10) + 0.03(10) + 0.01(10) \\ &= 1.5 \end{align*}$ 
 $\begin{align*} P\left(1 + \frac{1.5}{10}\right) &= \frac{1.9}{0.60} \\ P &= \frac{3.1667}{1.15} \\ &= \$2.754\text{M} \end{align*}$ 
The rate on line is $27.5\%$ . Expected reinstatement premium is $0.15 \times 2.754 = \$0.413$ M, which brings total expected premium to $\$3.167$ M.

Comparisons: with a free reinstatement the whole $\$3.167$ M must be charged up front. With no reinstatement the aggregate limit is $\$10$ M, expected loss falls to $\$1.5$ M, and $P = 1.5/0.60 = \$2.5$ M.