Capital Consumption

Don MangoAmerican Re-Insurance2003 CAS Ratemaking Seminar

Goals for Today Get you to admit this is a valid

alternative framework (albeit orthogonal) to capital allocation / release / IRR

Given it’s a possibility, demonstrate how it can be practically implemented as a means of pricing reinsurance

Two Bets Bet #1

You pay me $10 now I might pay you $50 later

Bet #2 I pay you $10 now You might have to pay me $50 later

Payoff DiagramsBet #1

-20

-10

0

10

20

30

40

50

60

Now Later

Bet #2

-60

-50

-40

-30

-20

-10

0

10

20

Now Later

Bet #1Spend then Maybe Receive You spend now, hope to receive

later You spend NOW, voluntarily With the odds I give you, you can

compute an expected value and decide if you want to make the bet

Bet #2Receive then Maybe Spend You receive now, hope you don’t

have to spend later You MAY spend LATER, involuntarily With the odds I give you, you can

compute an expected value and decide if you want to make the bet

Capital? Bet #1 = $10

You spend $10 capital NOW no matter what

The capital investment is current and certain – i.e., not contingent

Allocated = spent

Capital? Bet #2 = $???

I should be sure you have $40 available LATER, but you don’t spend anything NOW

If Bet #2 hits, you spend $40 capital LATER

Capital expenditure (= allocation) is contingent and in the future

Allocation vs Consumption

Two different but equally valid frameworks for Treating capital Evaluating insurance business

segments Developing indicated prices for

reinsurance Nearly orthogonal

Allocation vs Consumption Four questions:1. What do you do with the total capital?2. How do you evaluate business

segments?3. What does it mean to be in a portfolio?4. How is relative risk contribution

reflected?

Allocation vs ConsumptionQuestion 1: What happens to the total capital?

Allocation Consumption Divided up among the

segments. Either by explicit

allocation, or assignment of the marginal change in the total capital requirement from adding the segment to the remaining portfolio

Left intact Each segment has the right

to “call” upon the total capital to pay its operating deficits or shortfalls

Allocation vs ConsumptionQuestion 2: How are the segments evaluated?

Allocation Consumption Give the allocations to

each segment Evaluate each segment’s

return on their allocated capital

Must clear their hurdle rate

Give each segment “access rights” to the entire capital

Evaluate each segment’s potential calls (both likelihood and magnitude) on the total capital

Must pay for the likelihood and magnitude of their potential calls

Allocation vs ConsumptionQuestion 3: What does being in a portfolio mean?

Allocation Consumption Being standalone with

less capital But still having access to

all the capital if necessary, although it is unclear how this is reflected

Being standalone with potential access to all the capital

But all other segments have similar access rights

This is THE CRITICAL SLIDE!

Allocation vs ConsumptionQuestion 4: How is relative risk contribution reflected?

Allocation Consumption Use a single risk measure

to determine required capital

Select a dependence structure for the aggregation of segment distributions into a portfolio aggregate distribution

The marginal impact of adding a segment to the remaining portfolio is that segment’s risk contribution

Use scenario-level detail generated by stochastic modeling

Use explicit risk-return evaluation via utility function

Segment’s risk contribution is determined at the scenario level, then aggregated over all scenarios

Details of the Framework Scenario analysis Default-free discounting Scenario-level capital consumption Evaluation of capital consumption

using a “quasi~utility” approach

Default-Free Discounting Conditional on its occurrence, a given

scenario’s outcome is certain discount at the default-free rate

Risk-adjusted discounting is too clumsy Overloaded operator Try splitting out default probability from price of risk

in risky debt spreads Reflect uncertainty between scenarios, not

within What is uncertainty within a scenario anyway?

Do you believe the scenario is possible or not?

Scenario Capital Consumption Experience fund

From Finite Reinsurance Fund into which goes all revenue, from

which comes all payments Bakes in investment income

When it drops below zero, and further payments need to be made, gotta “call the parents” for some capital

That capital is spent CONSUMED

Experience Fund Long-Tailed LOB

Example 1Experience Fund for Long-tailed Contract120% Loss Ratio Scenario

Probability 10.0%

Investment Rate 8.0% Loss Ratio 116.2%Ultimate

Loss 120,000

1 2 3 4 5 6 7 8 9

Time

Beginning Fund

Balance Premiums ExpensesPayment

PatternPaid

LossesInvestment

IncomeEnding Fund

BalanceCapital

Call0 -$ 103,305$ 15,000$ 0.0% -$ -$ 88,305$ -$ 1 88,305$ -$ -$ 50.0% 60,000$ 2,264$ 30,570$ -$ 2 30,570$ -$ -$ 25.0% 30,000$ 46$ 615$ -$ 3 615$ -$ -$ 12.0% 14,400$ -$ (13,785)$ 13,785$ 4 -$ -$ -$ 6.0% 7,200$ -$ (7,200)$ 7,200$ 5 -$ -$ -$ 4.0% 4,800$ -$ (4,800)$ 4,800$ 6 -$ -$ -$ 2.0% 2,400$ -$ (2,400)$ 2,400$ 7 -$ -$ -$ 1.0% 1,200$ -$ (1,200)$ 1,200$ 8 -$ -$ -$ 0.0% -$ -$ -$ -$ 9 -$ -$ -$ 0.0% -$ -$ -$ -$

10 -$ -$ -$ 0.0% -$ -$ -$ -$

TOTAL 103,305$ 15,000$ 100.0% 120,000$ -$ 29,385$ NPV 103,305$ 15,000$ 86.2% 103,479$ 21,714$

Experience FundShort-Tailed LOB

Example 1AExperience Fund for Short-tailed Contract120% Loss Ratio Scenario

Investment Rate 8.0% Loss Ratio 120.0%Ultimate

Loss 120,000

1 2 3 4 5 6 7 8 9

Time

Beginning Fund

Balance Premiums ExpensesPayment

PatternPaid

LossesInvestment

IncomeEnding Fund

BalanceCapital

Call0 -$ 100,000$ 15,000$ 0.0% -$ -$ 85,000$ -$ 1 85,000$ -$ -$ 80.0% 96,000$ -$ (11,000)$ 11,000$ 2 -$ -$ -$ 15.0% 18,000$ -$ (18,000)$ 18,000$ 3 -$ -$ -$ 5.0% 6,000$ -$ (6,000)$ 6,000$ 4 -$ -$ -$ 0.0% -$ -$ -$ -$ 5 -$ -$ -$ 0.0% -$ -$ -$ -$ 6 -$ -$ -$ 0.0% -$ -$ -$ -$ 7 -$ -$ -$ 0.0% -$ -$ -$ -$ 8 -$ -$ -$ 0.0% -$ -$ -$ -$ 9 -$ -$ -$ 0.0% -$ -$ -$ -$

10 -$ -$ -$ 0.0% -$ -$ -$ -$

TOTAL 100,000$ 15,000$ 100.0% 120,000$ 35,000$ NPV 100,000$ 15,000$ 90.9% 109,084$ 30,380$

Chart 1: Capital Consumption Profile Over TimeShort versus Long Tail with 120% Loss Ratio

$-

$2,000

$4,000

$6,000

$8,000

$10,000

$12,000

$14,000

$16,000

$18,000

$20,000

0 1 2 3 4 5 6 7 8 9 10

Short TailLong Tail

Capital Calls (Philbrick/Painter) The entire surplus is available to every

policy to pay losses in excess of the aggregate loss component.

We can envision an insurance company instituting a charge for the access to the surplus.

This charge should depend, not just on the likelihood that surplus might be needed, but on the amount of such a surplus call.

Capital Calls (Philbrick/Painter) We can think of a capital allocation method as

determining a charge to each line of business that is dependant on the need to access the surplus account.

Conceptually, we might want to allocate a specific cost to each line for the right to access the surplus account.

In practice though, we tend to express it by allocating a portion of surplus to the line, and then requiring that the line earn (on average) an adequate return on surplus.

Capital Call Cost Function Risk-based overhead expense loading Pricing decision variable Application of utility theory Borch (1961):

To introduce a utility function which the company seeks to maximize, means only that such consistency requirements (in the various subjective judgements made by an insurance company) are put into mathematical form.

Capital Call Cost Function Make the implicit explicit Express your preferences explicitly, in

mathematical form, and apply them via a utility function

The mythical Risk Appetite Enforce consistency in the many

judgments being made

Implicit Preferences Preferences buried in Kreps’ “Marginal

Standard Deviation” risk load approach: The marginal impact on the portfolio

standard deviation is our chosen functional form for transforming a given distribution of outcomes to a single risk measure.

Risk is completely reflected, properly measured and valued by this transform.

Upward deviations are treated the same as downward deviations.

Property Cat ExampleExample 4Property Catastrophe Contract

Premium 1,000,000$ Limit 10,000,000$

No Loss Scenario Loss ScenarioProbability 98.0% 2.0%Premiums 1,000,000$ 1,000,000$ Expenses -$ -$

Losses -$ 10,000,000$ Capital Call Amount -$ 9,000,000$

Capital Call Factor 0.0% 400.0%Capital Call Charge -$ 36,000,000$

Expected NPV 800,000$ Expected Capital Call Cost 720,000$

Expected Risk-adjusted NPV 80,000$

Property Cat Example How would you do this with capital allocation? Allocate a percentage of the limit – say 5% --

based on marginal portfolio capital requirements?

What does that mean? What happens if the event occurs? Where does the money to pay the claim come from?

Does the sum of the marginals add up to the company’s total capital? If not, what does it mean?

Building Bridges Pricing via probability measure change –

from voluminous capital markets literature

Utility theory in pricing – from Halliwell, Heyer and Schnapp

The Wang Transform – from Wang The market cost of risk – from Van Slyke Additive Co-Measures – from Kreps

Final Thought:

This actually IS capital allocation for insurance, done right

And now…FIRE AWAY !