Economic Impact of Capital Level

Kevin Zhang, Ph.D., FCAS

CNA Insurance Companies

2

Capital level fluctuates

Retained earnings, dividends, share buyback, new shares

Who cares?

• Policyholders• Shareholders• Company management• Pricing actuaries

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

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

lAsset a=p+c

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

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-free Not default-free

l - d

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

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

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

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

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

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

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

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

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

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

Optimal capital level

cost

cc*

Total cost

Frictional cost

Cost of financial distress