Insurance Cash Flows

Study guide: Exam 9

Insurance Cash Flows are the amounts and timing of money moving through an insurance contract: premium in, up front or in instalments; commissions and other expenses out, mostly at the start; claim and LAE payments out over the settlement period; reinsurance premiums and recoveries; taxes; and investment income on the funds held in between. From the owners' side they also include the capital committed at inception and released as the liabilities run off.

$\text{PV(losses)} = L \sum_{t} p_t\, v^{t}$

$L$ is the nominal (undiscounted) expected loss, $p_t$ the payout pattern — the share of $L$ paid at time $t$ , summing to $1$ — and $v = 1/(1+r)$ the discount factor at a risk-free rate $r$ . The sum is the discount factor for the line.
Premium is mostly collected before losses are paid. The gap earns investment income, and its length separates short-tail property, paid within about a year, from long-tail liability and workers compensation, paid over many years.
Capital is a cash flow too. It must be held from inception until the liability is settled, so a long-tail line pays the cost of capital for more years. The investment-income credit and the longer capital commitment pull in opposite directions, and pricing has to reflect both — see Economic Value.
The same patterns drive Loss Reserve Discounting, asset–liability duration matching and the management of Interest Rate Risk.
Pricing theory built on a one-period model, as in Mildenhall and Major, collapses the multi-year flows into one period by working with losses discounted to inception; the premium, margin and capital it produces are then present values.

Present Value of Losses: Property Versus Liability Two lines each have nominal expected losses of $70$ per $100$ of premium. Property pays $90\%$ at the end of year 1 and $10\%$ at the end of year 2. Liability pays $10\%$ , $20\%$ , $25\%$ , $25\%$ and $20\%$ at the ends of years 1 to 5. Discount at $4\%$ .
Answer
 $\begin{align*} \text{Property factor} &= \frac{0.90}{1.04} + \frac{0.10}{1.04^2} \\ &= 0.9578 \\ \text{PV}_{\text{prop}} &= 70(0.9578) \\ &= 67.05 \\ \text{Liability factor} &= \frac{0.10}{1.04} + \frac{0.20}{1.04^2} + \frac{0.25}{1.04^3} \\ &\quad + \frac{0.25}{1.04^4} + \frac{0.20}{1.04^5} \\ &= 0.8814 \\ \text{PV}_{\text{liab}} &= 70(0.8814) \\ &= 61.70 \end{align*}$ 
The same nominal $70\%$ loss ratio is a $67\%$ discounted loss ratio for property and $62\%$ for liability, whose average payment is made after $3.25$ years rather than $1.1$ .

Premium Needed for the Same Economic Margin For the two lines above, expenses are $25\%$ of premium, paid at inception, and the insurer wants a margin of $5\%$ of premium in present-value terms. What premium does each line need, compared with a nominal calculation that ignores timing?
Answer The premium must fund expenses, the margin and the discounted losses, so $P(1 - 0.25 - 0.05) = \text{PV(losses)}$ :
 $\begin{align*} P_{\text{nominal}} &= \frac{70}{0.70} \\ &= 100.00 \\ P_{\text{prop}} &= \frac{67.05}{0.70} \\ &= 95.78 \\ P_{\text{liab}} &= \frac{61.70}{0.70} \\ &= 88.14 \end{align*}$ 
Timing is worth about $4\%$ of premium on property and $12\%$ on liability. The liability line holds capital for years longer, though, so the $5\%$ margin is a smaller return per year of capital; a full analysis charges the cost of capital for each year it is held, which recovers part of the apparent discount.