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Capital Consumption

Don MangoAmerican Re-Insurance2003 CARe Seminar

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Goals for Today

1. Get you to admit this is a valid alternative framework (albeit orthogonal) to capital allocation / release / IRR

2. Demonstrate how it can be practically implemented as a means of pricing reinsurance

3. Demonstrate connections to leading edge thinking in financial science

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Problem Statements Capital allocation is a de facto

paradigm a requirement or necessity

Therefore we force-fit our business into a manufacturing-based capital investment framework

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Problem Statements But insurance capital usage is

fundamentally different than it is for manufacturing, being in fact the mirror-image in time

For these decision evaluation processes, capital allocation is sufficient but not necessary

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Problem Statements Even worse, the resulting insurance

IRR framework is now completely fictional (“imputed”), since no capital is actually transferred or returned

However, insurance capital is actually consumed when results are worse than planned

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Actually…

This IS capital allocation for insurance, done right

But I needed new terminology to shake loose the old thought processes

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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

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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

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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

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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

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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 Natural capacity constraint = your

budget

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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

Capacity constraints = ??? Perception

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Two Bets?

Bet #1 = the manufacturing investment decisionSpend then receive

Bet #2 = the insurance investment decisionReceive then spend

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Allocation vs Consumption

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

segments Developing indicated prices for

reinsurance Nearly orthogonal

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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?

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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

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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

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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

The difference between having your own

kiddie pool and joining a swim club This is THE CRITICAL SLIDE!

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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

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

using a “quasi~utility” approach

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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?

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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

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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$

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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$

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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

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Scenario Capital Consumption This is more realistic than imputed

capital flows. (Imputed = fictional) The capital does flow, but in the future. When a segment’s results deteriorate,

the company’s capital is consumed as it is turned into additional reserves.

This is what actually happens, so why don’t we model it? Why pretend?

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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$

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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?

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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.

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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.

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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 judgments made by an insurance company) are put into mathematical form.

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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

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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.

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This links up with: Utility theory in actuarial pricing – from Longley-

Cook, Halliwell, Heyer and Schnapp Probability measure change – from financial

mathematics The Wang Transform – from Shaun Wang Additive Co-Measures – from Rodney Kreps Conditional Risk Charges – from David Ruhm

and Don Mango, 2003 Bowles Symposium

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Risk Charge

Both Expected Utility and Distorted Probability determine a risk charge by:

Risk Charge = Expected Value – Modified Expected Value

How do we calculate the Modified Expected Value?

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Expected Utility

Value x1 x2 … xn

Prob p1 p2 … pn:X

Value U(x1) U(x2) … U(xn)

Prob p1 p2 … pn:)(XU

Modified Expected Value = Sumproduct of Modified Values and Probabilities

Utility function is the modifier

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Distorted Probability

Value x1 x2 … xn

Prob p1 p2 … pn:X

Value x1 x2 … xn

Prob q1 q2 … qn:*X

Modified Expected Value = Sumproduct of Values and Modified Probabilities

Probability Distortion Function is the modifier (changes p q; impress your friends by discussing the “q-measure”)

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Distorted Probability A.k.a. “Measure Change” (change in the

probability measure) In the Black-Scholes world…

Constant interest rate, complete market, no transaction costs, instantaneous perfect hedging, …

…the q-measure is unique. As soon as a few of those constraints are

relaxed, there are infinite q-measures, all of which work.

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))(()(* 1 xFxF

Every value is standard deviations worse If the asset return R has a normal distribution

F(x), transformed F*(x) is also normal with E*[R] = E[R] – [R] = r (risk-free rate) = { E[R] – r }/[R] = the “market price of risk”,

also called the Sharpe ratio It recovers CAPM for assets, and Black-Scholes

formula for Options

Wang Transform

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Risk load R(X) is a probability-weighted average of “riskiness” r(x) over outcomes of the total net loss

g(x) can be thought of as the “riskiness leverage ratio” that multiplies the actual dollar excess that an outcome would entail to get the riskiness.

It reflects that not all dollars are equal, especially dollars that trigger analyst or regulatory tests.

R X dx f x r x

where r x x g x

Kreps’ Co-Measures

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Conditional Risk Charge David Ruhm and Don Mango, 2003

Bowles Symposium paper www.casact.org/coneduc/specsem/sp2003/pap

ers/ruhm-mango.doc

Main principle of conditional risk charge: Each risk receives a charge that represents how much it contributes to undesirable portfolio outcomes.

Generalization of Appendix B of my paper

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Advantages of Method Additive prices. Extends aggregate risk valuation to

any individual risk, including layers of risks.

Handles any underlying dependence structure.

Really works well for Property Cat.

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Goals for Today

1. Get you to admit this is a valid alternative framework (albeit orthogonal) to capital allocation / release / IRR

2. Demonstrate how it can be practically implemented as a means of pricing reinsurance

3. Demonstrate connections to leading edge thinking in financial science