1. 1 Cash-Flow Solvency Leigh J. Halliwell, FCAS, MAAA Consulting Actuary leigh@lhalliwell.com Casualty Loss Reserve Seminar Boston, MA September 12, 2005 Leigh J. Halliwell, FCAS, MAAA Consulting Actuary leigh@lhalliwell.com Casualty Loss Reserve Seminar Boston, MA September 12, 2005

2. 2 Financial Solvency Latin solvere – (generally) to solve, to free, to loosen; (financially) to pay off or discharge, especially a debt Solvent (adj) – able to pay one’s debts or liabilities Solvency (n) – ability to pay one’s debts

3. 3 Balance-Sheet Solvency “Balance Sheet, a tabular statement of assets and liabilities” (Oxford Universal Dictionary) With (non-negative) values assigned to each item! What values: GAAP, SAP, market, fair? Surplus as balancing item: A  L + S Balance-sheet solvent  A  L  S  0 BS solvency is the ability to pay, to a high probability, one’s liabilities as they come due by liquidating one’s assets in their respective markets at their current market values.

4. 4 BS Solvency: Aims and Concerns Study historic behavior of asset prices, and how asset prices interact (i.e., returns and correlations). Time the markets, i.e., “Buy low and sell high.” Allocate capital as a solvency cushion against plausible market downturns. How to treat assets and liabilities that have no liquid markets, and hence no market values? Use “fair” value? The greatest threat to a BS-solvent entity is an adverse revaluation of its assets and liabilities. “Just get me to the next accounting period!” Butsic (1994), 667-674.

5. 5 BS Solvency and Duration Bond income and loss outgo most important to insurers Interest-rates are the main determinant of the market value of high-grade bonds. They and underlying inflation can affect loss payments. Butsic (1981, 62f.) AD & PD models Macauley duration theoretically immunizes cash flows from interest-rate fluctuations. Cf. Feldblum (1989) and Halliwell (1999). D’Arcy (1996) gives caveats (504-509). “… cash inflow from … new policies is adequate to pay all losses and expenses” (D’Arcy, 505) Cf. Bustic (1994) on insurers as going concerns (674f).

6. 6 Cash Flows and Market Values Cash flows fundamental; market values derivative. Market participants value stochastic cash flows (Halliwell, 2003). Confusion between the two levels abounds in financial theory and practice. Account values are not cash flows. Example: $100 in a bank account paying 4% per year: Incorrect:$100  $100/1.00 + $4/1.04  $103.85 Correct:$0   $100)/1.00 + ($100 + $4)/1.04 An account is a wrapper with which cash flows in and out. The account merely capitalizes the cash flow. Don’t double count!

7. 7 Cash-Flow Solvency CF solvency is the ability to pay, to a high probability, one’s liabilities as they come due from one’s current pool of cash; mathematically, for all t, CumCashIn(t)  CumCashOut(t). Impervious to market valuation; solvent on a deeper level and to a more stringent standard Shifts attention off of investment and onto underwriting. For too long the insurance industry has been “straining out a gnat and swallowing a camel.” (Mt 23.24) Discourages the writing new business to pay old claims Doable with the present state of actuarial science and DFA

8. 8 The Major Objection to CF Solvency “Providing such a cash pool by a portfolio of high-quality (even risk-free bonds) unduly reduces investment income!” Four answers (unlikely to convince many, especially CFOs): Are alternatives, e.g., liquidity capital, any better? “We make money the old-fashioned way. We underwrite!” Regulators and policyholders will appreciate CF solvency. Ratings agencies might eventually accept it. The noble and aesthetic – It’s right and beautiful!

9. 9 Example of CF Solvency We have 250 equiprobable loss-runoff scenarios over a 10-year horizon. Their derivation considered all plausible economic and social conditions (in good DFA parlance). We desire to schedule cash inflows from US Treasuries that will fund 225 scenarios, i.e., achieve CF solvency to the 90 th percentile. From the many possible schedules we will choose the least expensive one in terms of current Treasury prices.

10. 10 Fundamental Theorem of Funding A schedule (or funding arrangement) cannot discharge runoff scenarios, if its present value is less than theirs; i.e., less “in-PV” cannot fund greater “out-PV”. (However, greater in-PV in does not necessarily fund less out-PV.) Proof: To discharge a scenario, income from x to x+  x must precede outgo. Since v(t) decreases, for all  x, v(t income )  x  v(t outgo )  x. Hence,   PV income    PV outgo. Therefore, the cost of funding 225 scenarios is at least the 225 th order statistic of the present values of the scenarios.

11. 11 Steepest-Descent Algorithm Fund the envelope of the cumulative outflows of the 225 lowest-PV scenarios. Envelope(t) = Max(cumoutflows(t)). So PV[Fund]  PV order statistic 225. Are more than 225 scenarios funded? If not, the envelope is best. If yes, chip away from the envelop the scenario that most reduces the PV. Repeat this until 225 scenarios remain. “Steepest Descent”: Simple, but may not be optimal. Counting down from 250 better for 225, but not for 200. Excel demonstration: Cash-Flow Solvency.xls Extra conservatism: no reinvestment, no mid-interval flows

12. 12 References Butsic, Robert P., “The Effect of Inflation of Losses and Premiums for Property-Liability Insurers,” Inflation Implications for Property- Casualty Insurance, CAS 1981 Discussion Paper Program, 58-109. “Solvency Measurement for Property-Liability Risk- Based Capital Applications,” Journal of Risk and Insurance, Vol 61 (Dec 1994), 656-690. D’Arcy, Stephen P., “Investment Issues in Property-Liability Insurance,” Foundations in Casualty Actuarial Science (Third Ed., 1996), Chapter 8. Feldblum, Sholom, “Asset-Liability Matching for Property/Casualty Insurers,” Valuation Issues, 1989 CAS Special Interest Seminar, 117-154. Halliwell, Leigh J., “Insights into Present Value and Duration,” Spring 1999 Forum, 408-437. “The Valuation of Stochastic Cash Flows,” Spring 2003 Forum, 1-68.