Economic Value

Study guides: Exam 9, Exam ILA-201I, Exam ILA-201U

The Economic Value of a set of insurance cash flows is their expected value at one date — normally policy inception — after discounting each flow for the time value of money and, where it is risky, for its risk. A policy creates value for its owners when the present value of what it brings in exceeds the present value of everything it commits them to, including a fair return on the capital it ties up.

$V_0 = \sum_{t} \frac{E[CF_t]}{(1 + r)^{t}}$

$CF_t$ is the net cash flow at time $t$ : premium in; commissions, expenses, claims, LAE and taxes out. The discount rate $r$ must suit the flow. Claim payments are normally discounted at a risk-free rate (IFRS 17 adds a liquidity premium — IFRS 17 Discount Rates), with their risk priced separately through a risk load; discounting them at the insurer's riskier portfolio yield would credit the policyholder with investment risk the insurer keeps.
Two equivalent views. The policy view values premium less expenses less losses, all in present value, and compares the result with the margin the capital requires. The owners' view values the equity flows — capital paid in at inception and released as the liabilities run off — at the cost of capital, giving a net present value or an internal rate of return to compare with a hurdle rate.
Nominal profit misleads. Underwriting Profit ignores timing: a long-tail line can create value at a combined ratio above 100% if investment income on its reserves is large enough, and a short-tail line can destroy it below 100% if it holds much capital for a thin margin.
Economic value added is the margin actually earned less the margin the capital requires, $M - \iota\, Q$ in Mildenhall and Major's notation; it is positive only when pricing beats the cost of capital (Insurance Margin, Return on Capital).
Mildenhall and Major's one-period pricing framework sets investment income aside by working with present-value losses, so its premium, loss, margin, capital and assets are all discounted amounts. See Insurance Cash Flows for the flows themselves and Loss Reserve Discounting for their reserving counterpart.

Nominal Profit Versus Economic Value A policy's premium of $100$ is collected at inception, where expenses of $25$ are paid. Losses of $72$ are paid $30\%$ , $40\%$ and $30\%$ at the ends of years 1, 2 and 3. The risk-free rate is $4\%$ . Compare the nominal underwriting profit with the present value of the policy's cash flows.
Answer
 $\begin{align*} \text{Nominal profit} &= 100 - 25 - 72 \\ &= 3.00 \\ \text{PV(losses)} &= 72\left(\frac{0.30}{1.04} + \frac{0.40}{1.04^2} + \frac{0.30}{1.04^3}\right) \\ &= 72(0.9250) \\ &= 66.60 \\ V_0 &= 100 - 25 - 66.60 \\ &= 8.40 \end{align*}$ 
Waiting to pay the losses is worth $5.40$ — the investment income earned on the premium before claims are paid. The policy's economic value is nearly three times its accounting profit. But $8.40$ is not yet profit to the owners: the capital that supports the policy for three years must still earn its cost, which is the owners' view in the next example.

NPV and IRR to the Owners Premium of $100$ is received at inception and expenses of $20$ are paid then. A single loss payment of $75$ is made at the end of year 2. Owners contribute capital of $50$ at inception, and all funds are invested at the risk-free rate of $4\%$ until the loss is paid, when everything left is released to the owners. Their cost of capital is $12\%$ . Ignoring tax, does the policy create value?
Answer
 $\begin{align*} \text{Invested at start} &= 100 - 20 + 50 \\ &= 130 \\ \text{Funds after 2 years} &= 130(1.04)^2 \\ &= 140.61 \\ \text{Released after 2 years} &= 140.61 - 75 \\ &= 65.61 \\ \text{NPV} &= -50 + \frac{65.61}{1.12^2} \\ &= 2.30 \\ \text{IRR} &= \sqrt{65.61/50} - 1 \\ &= 14.55\% \end{align*}$ 
The owners earn $14.55\%$ a year against a $12\%$ hurdle, so the policy adds $2.30$ of value at inception. Were the same loss paid at the end of year 4 instead, the release would be $130(1.04)^4 - 75 = 77.08$ and the NPV $-50 + 77.08/1.12^4 = -1.01$ : two more years of investment income do not pay for two more years of capital that earns $4\%$ but costs $12\%$ . The timing of the cash flows, not just their size, decides the value.