1. Economic Impact of Capital Level Kevin Zhang, Ph.D., FCAS CNA Insurance Companies

2. 2 Capital level fluctuates Retained earnings, dividends, share buyback, new shares Who cares? Policyholders Shareholders Company management Pricing actuaries

3. 3 A simple model One-period model Company starts at t = 0, with capital c; issues policies Pays all claims at t = 1; returns net assets What will change when c varies? Assume book of business does not change Liability L PV(L) = l

4. 4 How does premium change with C Case 1: If c is large and company is default-free, fair premium = full premium: p = l Surplus = a-l p c l Asset a=p+c

5. 5 Impact on premium (cont’d) Case 2: If c is not high enough, possibility of default exists L is not entirely covered, so policyholders demand premium credit Policyholder deficit = unrecoverable claims ≡ D, and PV(D) = d Covered claims = L – D, present value = l – d

6. 6 Impact on premium, Case 2 (cont’d) Policyholders require more premium credit than d Unrecoverable claims highly correlated with their own loss Expenses in recovering process p c l Surplus < cc < p - d Default-freeNot default-free l - d

7. 7 Insurance profit Default-free: Y = p(1+R) – L (R – investment return) PV(Y) = p – l = 0 Not default-free: Y’ = p’(1+R) – (L-D) PV(Y’) = p’ – ( l - d) < p – l = 0 Default-able companies are less profitable than default-free companies The lower capital adequacy, the lower profit

8. 8 How does shareholder return vary with c? Modigliani-Miller (MM): capital level is irrelevant If an investor has $100, he may (1) invest $100 to the firm; or (2) invest $80 to the firm and $20 in the capital market to earn the same return as the asset 100 (1) 80 20 (2) capital market

9. 9 MM is usually violated 100 Better! 80 20 capital market MM holds if company is default free (also, no frictional cost) If company is not default free: Inadequate c → premium falls more than PV of claims → Insurance profit declines Shareholders prefer higher capital level

10. 10 Return on capital Insurance profit: Y = p(1+R) – L Value of firm at the end of period: S = p(1+R) – L + c(1+R) Return on capital: ROC = (S-c)/c = R + Y/c Capital market return Insurance return on capital

11. 11 ROC ROC = R + Y/c If shareholders invest in capital market, return R If they invest in insurance company, return R + Y/c The only reason for investing in insurance firms is to receive insurance profit Y = p(1+R) - L PV(Y) = p – PV(L)

12. 12 Excess ROC ROC - R = Y/c ROC in excess of market rate R varies in reverse proportion to c : lower c → higher excess return However, this does not mean the less capital the better off the shareholders On the contrary, lower c → lower Y, and shareholders are worse off –reducing c by half doubles the amount of risk; but since Y decrease, ROC – R less than doubles –similar to the CAPM

13. 13 Frictional costs So far, have ignored frictional costs Frictional costs include double taxation, agency costs…. Not all covered by expense provisions, and more volatile than normal business expenses Higher c → higher tax, higher some agency costs

14. 14 Frictional costs (cont’d) Different from the cost of capital – usually means the shareholder required return –Cost of capital  shareholders –Frictional costs  government, employees, agents, etc –From shareholders’ point of view, the frictional cost is the only true cost, and should be minimized

15. 15 Costs of financial distress c too high → tax and some frictional costs too high c too low also hurts –reduce premium adequacy –increases costs of financial distress dealing with auditors and regulators, defending lawsuits keeping up employee morale maintaining customer relationship, retaining business obtaining external funding upon default, direct bankruptcy costs (legal, accounting, filing, administrative)

16. 16 Total costs Frictional cost is an increasing function of c Cost of financial distress is a decreasing function of c Total cost: the sum of the two is minimized at certain level of c

17. 17 Optimal capital level cost cc* Total cost Frictional cost Cost of financial distress