Capital Modeling
March 22, 2016
Daniel Linton, FCAS, MAAA
Surplus
•
Surplus = Total Assets minus Liabilities
•
Surplus = Capital = Shareholder Equity =
Net Assets
•
Important not to have too little or too much
•
•
Appropriate amount varies from one company to
the next
Board should set benchmarks that properly reflect
overall risk portfolio in conjunction with risk
appetite
Why have surplus?
•
Underwriting results worse than the premiums or
the carried reserves
•
Adverse investment results
•
Other types of losses
•
•
•
Operational
Strategic
Reputational
Difference between Capital Modeling and
Reserving/Pricing
$140,000,000
Size of Loss
$120,000,000
$100,000,000
Expected = $70 Million
$80,000,000
$60,000,000
$40,000,000
$20,000,000
$0
0
mu:
sigma:
1
18.044
0.200
Cumulative Density Function
How much surplus is enough?
Depends on the company’s risk appetite:
•
What probability of insolvency is appropriate to the
company’s business plan?
• This is ultimately a decision that can only be made by
the Board
•
Changes as the company matures
How much surplus is enough?
•
Solvency II (Europe) requires that companies carry
surplus so that the (modeled) probability of ruin in the
next 12 months is less than 0.5% (the 99.5%ile), or
rather 1 in every 200 modeled outcomes
•
Rating Agencies set probabilities of default to go with
various securities:
•
•
BBB about 99.75% (1 in 400), A 99.94% (1 in 1,667),
AA 99.99% (1 in 10,000)
AAA at 99.995% ( 1 in 20,000)
How much surplus is enough?
For example, let’s assume a risk averse company
•
•
Target surplus at the AAA rating level
Probability of insolvency at 0.005%, or one in 20,000
The result is called “economic capital”
•
This is what you need
Anything extra is “free capital”
•
You can do other things with this
•
•
•
Introduce new lines of business
Expand existing coverage
Pay dividends
Risk Exposures
•
Economic capital is comprised of a variety of
risks:
•
•
•
•
•
•
Underwriting Risk
Asset Risk
Operational Risk
Credit Risk
Liquidity Risk
Other (We’ll talk more about this later)
Risk Exposures
Underwriting Risk
•
Risk that premiums are insufficient to cover losses
Asset Risk (market risk)
•
Risk that assets will reduce in value
•
Higher interest rates
•
•
•
(Reduces the value of fixed income securities)
Higher than expected default rates
Poorly performing equities
Example: Underwriting Risk
Assume a company charges $100 million to write a
book of business with a 70% loss ratio.
Percentile
Losses
60%
70%
80%
90%
95%
99%
99.5%
99.75%
99.99%
99.995%
$72,180,000
$76,201,000
$81,192,000
$88,660,000
$95,342,000
$109,264,000
$114,854,000
$120,290,000
$144,358,000
$149,398,000
Expected
$70,000,000
mu:
sigma:
18.044
0.200
Example: Underwriting Risk
Note the tail of the distribution with a small
adjustment to the sigma parameter
Percentile
Losses
60%
70%
80%
90%
95%
99%
99.5%
99.75%
99.99%
99.995%
$72,254,000
$76,798,000
$82,479,000
$91,061,000
$98,818,000
$115,193,000
$121,844,000
$128,351,000
$157,584,000
$163,787,000
Expected
$70,000,000
mu:
sigma:
18.039
0.225
Risk Exposures
Operational Risk
•
Risk of loss due to people, process or systems, and
outside event risk
•
•
•
•
Includes errors, natural disasters, fraud, etc.
Very, very hard to measure, but surplus should
include a provision for this risk
No upside for assuming this risk
Can’t be diversified away
•
Estimated from the methodology prescribed by Basel II
(international banking) regulations
Example – Operational Risk
James Scott
Risk Exposures
Credit Risk
•
Risk of default (e.g. bond default). To some extent,
this is included in the provision for market risk
Liquidity Risk
•
•
Stems from lack of marketability
Asset can’t be bought or sold quickly enough to
prevent or minimize a loss.
•
e.g. slow paying reinsurer
Methodology
•
Use Monte Carlo simulation to model results
•
•
Generate 65,000 iterations of asset returns and
underwriting results over a two-year horizon
•
•
Allows you to see all the possible outcomes to
assess the impact of risk
Produces 65,000 balance sheets and income
statements for each of the two years
From these results, we can examine the varying
levels of surplus as well as the changes in
surplus from one year to the next
Diversification
•
•
Important to incorporate the benefits of
diversification throughout the operations of the
company
Ignoring this would distort results
•
•
•
Good years would be very good
Bad years would be quite horrible
For example:
•
•
No plausible link (besides inflation) between public
officials liability and property coverage
No plausible link between asset performance and
the number of large automobile claims
What types of events are included in Base
Economic Model?
•
Allows for extreme and rare events
•
•
•
•
Additional development on runoff years in excess
of two-thirds of the actuarial central estimate
Total losses over the two-year forecast period in
excess of twice the premium.
Investment losses totaling 20% or more
Long term interest rate levels as high as 7%
Scenario Testing
•
Refers to a series of independent tests that
evaluate the impact of a given event on the
company
•
Modeled within the context of the company’s
risk appetite
•
Important: can’t anticipate every possible
scenario (Black Swans)
Scenario Testing
Additional Risks?
What keeps you up at night?
Additional Risks?
•
•
•
•
Material rate of default
Dramatic rise in interest rates
High degree of tail dependency
Reinsurance risk
•
•
•
Slow paying reinsurer
Reinsurer insolvency
Loss of key customer/member
Case Study
Healthy Public Risk Pool
•
•
•
Board wants to evaluate overall capital levels
High level of risk aversion
Surplus = $23.6 million
Probability
Two-Year Underwriting Income
Current Underwriting Program
1.4%
1.2%
1.0%
0.8%
0.6%
0.4%
0.2%
0.0%
-$20
-$15
-$10
-$5
$0
$5
$10
Underwriting Income ($1,000,000)
$15
$20
Two-Year Total Investment Income
Current Underwriting Program
25.0%
Probability
20.0%
15.0%
10.0%
5.0%
0.0%
$0.0
$0.5
$1.0
$1.5
$2.0
Total Investment Income ($1,000,000)
Including Unrealized Gains/Losses
$2.5
$3.0
$3.5
Two-Year Surplus Range of Outcomes
$35
($1,000,000)
$30
$25
$20
$15
$10
$5
$0
0%
20%
40%
60%
Outcome Percentile
80%
100%
Two-Year Surplus Range of Outcomes
Confidence Level
Net Assets
5%
$20,000,000
10%
$21,200,000
20%
$22,800,000
27%
$23,600,000
40%
$24,800,000
60%
$26,500,000
80%
$28,300,000
90%
$29,700,000
95%
$30,700,000
There is a 27% chance that surplus will be at most $23,600,000
in the next two years.
In other words, a 27% chance that surplus will decrease over a two
year time period
Base Results
Capital Level
Req. Capital
99.5%ile
$8,400,000
99.75%ile
$9,300,000
99.94%ile
$11,100,000
99.99%ile
$12,600,000
99.995%ile
$12,800,000
Scenario Testing
•
•
•
•
Interest Rate Risk
Default Risk
Loss of Key Members
Slow Paying Reinsurer
Scenario 1: Interest Rate Risk
Considers a dramatic rise in interest rates
•
Not unlike the increases observed in the 1980’s
Increasing interest rates will reduce the market
value of the investment portfolio
• Over 90% of financial instruments mature within
5 years
Scenario 1: Interest Rate Risk
Capital Level
Req. Capital
Capital Level
Req. Capital
99.5%ile
$9,200,000
99.5%ile
$8,400,000
99.75%ile
$10,100,000
99.75%ile
$9,300,000
99.94%ile
$11,700,000
99.94%ile
$11,100,000
99.99%ile
$13,400,000
99.99%ile
$12,600,000
99.995%ile
$14,000,000
99.995%ile
$12,800,000
Scenario 1
Base Results
Scenario 2: Default Risk
•
•
•
Approximately 40% of investment portfolio
comprised of asset backed securities
Played a large role in the economic downturn in
2008
Assumed a 25% rate of default
•
About 60% of securities need to default to run out
of surplus at 99.995%ile; over 90% at 99.5%ile
Scenario 2: Default Risk
Capital Level
Req. Capital
Capital Level
Req. Capital
99.5%ile
$12,500,000
99.5%ile
$8,400,000
99.75%ile
$13,500,000
99.75%ile
$9,300,000
99.94%ile
$15,200,000
99.94%ile
$11,100,000
99.99%ile
$16,700,000
99.99%ile
$12,600,000
99.995%ile
$16,900,000
99.995%ile
$12,800,000
Scenario 2
Base Results
Scenario 3: Interest Rate Risk/Default Risk
•
•
Considers significant increase in interest rates
in conjunction with a 25% rate of default
Contemplated a high degree of tail correlation
between these two events
•
Results in capital needs that are effectively the
sum of scenarios 1 and 2
Scenario 3: Interest Rate Risk/Default Risk
Capital Level
Req. Capital
Capital Level
Req. Capital
99.5%ile
$13,200,000
99.5%ile
$8,400,000
99.75%ile
$14,100,000
99.75%ile
$9,300,000
99.94%ile
$15,800,000
99.94%ile
$11,100,000
99.99%ile
$17,400,000
99.99%ile
$12,600,000
99.995%ile
$18,000,000
99.995%ile
$12,800,000
Scenario 3
Base Results
Scenario 4: Loss of Key Members
•
•
A small number of members account for about 25% of
total premium
Assumes that overall expenses remain unchanged over
the next two years
Scenario 4: Loss of Key Members
Capital Level
Req. Capital
Capital Level
Req. Capital
99.5%ile
$5,100,000
99.5%ile
$8,400,000
99.75%ile
$5,800,000
99.75%ile
$9,300,000
99.94%ile
$7,200,000
99.94%ile
$11,100,000
99.99%ile
$8,800,000
99.99%ile
$12,600,000
99.995%ile
$9,300,000
99.995%ile
$12,800,000
Scenario 4
Base Results
Scenario 5: Slow Paying Reinsurer
Considered a $10,000,000 property loss
$5,000,000 paid in year 1
$5,000,000 paid in year 2
Pertinent question: Does the company have
enough liquid assets to continue meeting its
obligations?
Scenario 5: Slow Paying Reinsurer
Capital Level
Req. Capital
Capital Level
Req. Capital
99.5%ile
$9,400,000
99.5%ile
$8,400,000
99.75%ile
$10,400,000
99.75%ile
$9,300,000
99.94%ile
$12,100,000
99.94%ile
$11,100,000
99.99%ile
$13,600,000
99.99%ile
$12,600,000
99.995%ile
$13,800,000
99.995%ile
$12,800,000
Scenario 5
Base Results
Scenario 6: Slow Paying Reinsurer in
Conjunction with Interest Rate Risk and
Default Risk
Combines Scenarios 1, 2, and 5
•
Interest rates increasing leading to defaults on
securitized assets resulting in a reinsurer’s inability to
meet its obligations
Must consider overall surplus as well as liquid
assets
Scenario 6: Slow Paying Reinsurer in
Conjunction with Interest Rate Risk and
Default Risk
Capital Level
Req. Capital
Capital Level
Req. Capital
99.5%ile
$14,100,000
99.5%ile
$8,400,000
99.75%ile
$15,000,000
99.75%ile
$9,300,000
99.94%ile
$16,700,000
99.94%ile
$11,100,000
99.99%ile
$18,300,000
99.99%ile
$12,600,000
99.995%ile
$19,000,000
99.995%ile
$12,800,000
Scenario 6
Base Results
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Questions? Comments?