Capital Allocation
Capital Consumption
Don Mango
American Re-Insurance
2003 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 Diagrams
Bet #1
60
50
40
30
20
10
0
-10
-20
Now Later
Bet #2
-10
-20
-30
-40
-50
-60
20
10
0
Now Later
Bet #1
Spend 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 #2
Receive 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
1.
2.
3.
4.
Four questions:
What do you do with the total capital?
How do you evaluate business segments?
What does it mean to be in a portfolio?
How is relative risk contribution reflected?
Allocation vs Consumption
Question 1: What happens to the total capital?
Allocation
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
Consumption
Left intact
Each segment has the right to “call” upon the total capital to pay its operating deficits or shortfalls
Allocation vs Consumption
Question 2: How are the segments evaluated?
Allocation
Give the allocations to each segment
Evaluate each segment’s return on their allocated capital
Must clear their hurdle rate
Consumption
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 Consumption
Question 3: What does being in a portfolio mean?
Allocation
Being standalone with less capital
But still having access to all the capital if necessary, although it is unclear how this is reflected
Consumption
Being standalone with potential access to all the capital
But all other segments have similar access rights
This is THE CRITICAL SLIDE !
Allocation vs Consumption
Question 4: How is relative risk contribution reflected?
Allocation
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
Consumption
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 1
Experience Fund for Long-tailed Contract
120% Loss Ratio Scenario
Probability
Investment Rate 8.0% Loss Ratio
5 1
Time
4
5
6
2
3
0
1
9
10
7
8
2
Beginning
3 4
Fund
Balance Premiums Expenses
$ -
$ 88,305
$ 30,570
$ 615
$ -
$ -
$ -
$ -
$ -
$ -
$ -
$ 103,305
$
$
$
$
$
$
$
$
$
$
-
-
-
-
-
-
-
-
-
-
$ 15,000
$
$
$
$
$
$
$
$
$
$
-
-
-
-
-
-
-
-
-
-
10.0%
116.2%
6
Ultimate
7
Loss 120,000
8 9
Payment
Pattern
0.0%
50.0%
25.0%
12.0%
6.0%
4.0%
2.0%
1.0%
0.0%
0.0%
0.0%
Paid
Losses
$ -
$ 60,000
$ 30,000
$ 14,400
$ 7,200
$ 4,800
$ 2,400
$ 1,200
$ -
$ -
$ -
Investment
Income
Ending Fund
Balance
$ -
$ 2,264
$ 46
$ -
$ -
$ -
$ -
$ -
$ -
$ -
$ -
$
$
$
88,305
30,570
$
(13,785)
$
$
$
$
615
(7,200)
(4,800)
(2,400)
(1,200)
$
$
$
-
-
-
Capital
Call
$ -
$ -
$ -
$ 13,785
$ 7,200
$ 4,800
$ 2,400
$ 1,200
$ -
$ -
$ -
TOTAL
NPV
$ 103,305
$ 103,305
$ 15,000
$ 15,000
100.0%
86.2%
$ 120,000
$ 103,479
$ $ 29,385
$ 21,714
Experience Fund
Short-Tailed LOB
Example 1A
Experience Fund for Short-tailed Contract
120% Loss Ratio Scenario
Investment Rate 8.0% Loss Ratio
5 1
Time
4
5
6
2
3
0
1
9
10
7
8
2
Beginning
3 4
Fund
Balance Premiums Expenses
$ -
$ 85,000
$ -
$ -
$ -
$ -
$ -
$ -
$ -
$ -
$ -
$ 100,000
$
$
$
$
$
$
$
$
$
$
-
-
-
-
-
-
-
-
-
-
$ 15,000
$
$
$
$
$
$
$
$
$
$
-
-
-
-
-
-
-
-
-
-
120.0%
6
Ultimate
7
Loss 120,000
8 9
Payment
Pattern
0.0%
80.0%
15.0%
5.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
Paid
Losses
$ -
$ 96,000
$ 18,000
$ 6,000
$ -
$ -
$ -
$ -
$ -
$ -
$ -
Investment
Income
Ending Fund
Balance
$ -
$ -
$ -
$ -
$ -
$ -
$ -
$ -
$ -
$ -
$ -
$
$
$
85,000
(11,000)
(18,000)
$ (6,000)
$
$
$
$
$
$
$
-
-
-
-
-
-
-
Capital
Call
$ -
$ 11,000
$ 18,000
$ 6,000
$ -
$ -
$ -
$ -
$ -
$ -
$ -
TOTAL
NPV
$ 100,000
$ 100,000
$ 15,000
$ 15,000
100.0%
90.9%
$ 120,000
$ 109,084
$ 35,000
$ 30,380
Chart 1: Capital Consumption Profile Over Time
Short versus Long Tail with 120% Loss Ratio
$20,000
$18,000
$16,000
$14,000
$12,000
$10,000
$8,000
$6,000
$4,000
$2,000
$-
0 1 2 3 4 5 6
Short Tail
Long Tail
7 8 9 10
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 Example
Example 4
Property Catastrophe Contract
Premium
Limit
$ 1,000,000
$ 10,000,000
Probability
Premiums
Expenses
Losses
Capital Call Amount
Capital Call Factor
Capital Call Charge
Expected NPV
Expected Capital Call Cost
Expected Risk-adjusted NPV
No Loss Scenario
98.0%
$ 1,000,000
$ -
$ -
$ -
0.0%
$ -
$ 800,000
$ 720,000
$ 80,000
Loss Scenario
2.0%
$ 1,000,000
$ -
$ 10,000,000
$ 9,000,000
400.0%
$ 36,000,000
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:
