The Cost of Financing Insurance with Emphasis on Reinsurance

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					The Cost of Financing Insurance
              with
  Emphasis on Reinsurance
          Glenn Meyers
              ISO
     CAS Ratemaking Seminar
         March 10, 2005
              Fifth Time at CAS
             Ratemaking Seminar
• 2001 – Proof of concept
      http://www.casact.org/pubs/forum/00sforum/meyers/index.htm
• 2002 – Applied to DFA Insurance Company
      http://www.casact.org/pubs/forum/01spforum/meyers/index.htm
• 2003 – Additional realistic examples
  – Primary insurer
    http://www.casact.org/pubs/forum/03sforum/03sf015.pdf
  – Reinsurer http://www.casact.org/pubs/forum/03spforum/03spf069.pdf
• 2004 – No new papers
• 2005 – Emphasis on Reinsurance
       Underlying Themes
• The insurer's risk, as measured by its
  stochastic distribution of outcomes,
  provides a meaningful yardstick that can
  be used to set capital requirements.
• Risk  Capital  Costs money.
• Develop strategy to make most efficient
  use of capital.
    Strategy – Diversification
• Examples
  – Increase volume / Law of large numbers
  – Manage concentrations in property insurance
  – Decide where to grow and/or shrink
• Costs money to diversify

                     $$$
                                             Cost
                                             Benefit
  At some point,
  it doesn’t pay
  to diversify.
                           Diversification
        Strategy – Reinsurance
• Examples – Excess of Loss
  – Coinsurance provisions
  – Treatment of ALAE
  – Stacked contracts with various inuring provisions
• Reinsurance costs money

  •You can buy too much             Pretty Good
  reinsurance.                $$$                 Cost
  •There are often a lot of                       Benefit
  messy details to be
  worked out.

                                    Reinsurance
 Outline of Insurance Strategy
• Grow in lines of business where risk is
  adequate rewarded.
• Shrink in lines of business where risk is
  not adequately rewarded.
• Diversify when cost effective.
• Buy reinsurance when cost effective.
Volatility Determines Capital Needs
                 Low Volatility
                      Chart 3.1
  Size of Loss




                                  Random Loss
                                  Needed Assets
                                  Expected Loss
Volatility Determines Capital Needs
                 High Volatility
                      Chart 3.1
  Size of Loss




                                   Random Loss
                                   Needed Assets
                                   Expected Loss
   Additional Considerations
• Correlation
  – If bad things can happen at the same time,
    you need more capital.
• We will come back to this shortly.
The Negative Binomial Distribution
• Select  at random from a gamma
  distribution with mean 1 and variance c.
• Select the claim count K at random from
  a Poisson distribution with mean .
• K has a negative binomial distribution
  with:

     E K    and Var K     c   2
Multiple Line Parameter Uncertainty

• Select b from a distribution with E[b] =
  1 and Var[b] = b.
• For each line h, multiply each loss by
  b.
     Multiple Line Parameter Uncertainty

        A simple, but nontrivial example


         1  1  3b ,  2  1, 3  1  3b

Pr   1  Pr   3   1/ 6 and Pr   2   2 / 3

              E[b] = 1 and Var[b] = b
        Low Volatility
     b = 0.01 r = 0.50

                             Chart 3.3

             4,000
             3,500
             3,000
Y 2 = X 2




             2,500
             2,000
             1,500
             1,000
              500
                0
                     0   1,000     2,000      3,000   4,000
                                 Y 1 = X 1
        Low Volatility
     b = 0.03 r = 0.75

                             Chart 3.3

             4,000
             3,500
             3,000
Y 2 = X 2




             2,500
             2,000
             1,500
             1,000
              500
                0
                     0   1,000     2,000      3,000   4,000
                                 Y 1 = X 1
        High Volatility
     b = 0.01 r = 0.25

                             Chart 3.3

             4,000
             3,500
             3,000
Y 2 = X 2




             2,500
             2,000
             1,500
             1,000
              500
                0
                     0   1,000     2,000      3,000   4,000
                                 Y 1 = X 1
        High Volatility
     b = 0.03 r = 0.45

                             Chart 3.3

             4,000
             3,500
             3,000
Y 2 = X 2




             2,500
             2,000
             1,500
             1,000
              500
                0
                     0   1,000     2,000      3,000   4,000
                                 Y 1 = X 1
        About Correlation
• There is no direct connection between r
  and b.
• Small insurers have large process risk
• Larger insurers will have larger
  correlations.
• Pay attention to the process that
  generates correlations.
                                             Correlation and Capital
                                                     b = 0.00
                                                                                     Chart 3.4
                                                                                 Correlated Losses
                       7,000
Sum of Random Losses




                       6,000



                       5,000



                       4,000



                       3,000



                       2,000



                       1,000



                          0
                               1.0   1.0   1.0   1.0   1.0   1.0   1.0   1.0   1.0   1.0   1.0   1.0   1.0   1.0   1.0   1.0   1.0   1.0   1.0   1.0   1.0   1.0   1.0   1.0   1.0

                                                                                       Random Multiplier
                                             Correlation and Capital
                                                     b = 0.03
                                                                                     Chart 3.4
                                                                                 Correlated Losses
                       7,000
Sum of Random Losses




                       6,000



                       5,000



                       4,000



                       3,000



                       2,000



                       1,000



                          0
                               0.7   1.3   1.3   1.0   1.0   0.7   1.0   0.7   1.3   1.3   0.7   1.3   1.3   1.0   0.7   0.7   1.0   1.3   0.7   1.0   1.3   1.0   0.7   0.7   1.0

                                                                                       Random Multiplier
       Calculating an Insurer’s
         Underwriting Risk
• Use the collective risk model.
   – Separate claim frequency and severity analysis
• For each line of insurance:
   – Select a random claim count.
   – Select random claim size for each claim.
• The aggregate loss for all lines = sum of all the
  random claim amounts for all lines.
   – Reflect the correlation between lines of insurance.
Consider the Time Dimension
• How long must insurer hold capital?
  – The longer one holds capital to support a
    line of insurance, the greater the cost of
    writing the insurance.
  – Capital can be released over time as risk is
    reduced.
• Investment income generated by the
  insurance operation
  – Investment income on loss reserves
  – Investment income on capital
The Cost of Financing Insurance
• Includes
   – Cost of capital
   – Net cost of reinsurance

• Net Cost of Reinsurance =
    Total Cost – Expected Recovery
           The To Do List
• Allocate the Cost of Financing back
  each underwriting division.
• Calculate the cost of financing for each
  reinsurance strategy.
• Which reinsurance strategy is the most
  cost effective?
       Doing it - The Steps
• Determine the amount of capital
• Allocate the capital
  – To support losses in this accident year
  – To support outstanding losses from prior
    accident years
• Include reinsurance
• Calculate the cost of financing.
            Step 1
Determine the Amount of Capital
• Decide on a measure of risk
  – Tail Value at Risk
     • Average of the top 1% of aggregate losses
     • Example of a “Coherent Measure of Risk
  – Standard Deviation of Aggregate Losses
     • Expected Loss + K  Standard Deviation
  – Both measures of risk are subadditive
     • (X+Y) ≤ (X) + (Y)
     • i.e. diversification reduces total risk.
            Step 1
Determine the Amount of Capital
• Note that the measure of risk is applied to
  the insurer’s entire portfolio of losses.
         (X) = Total Required Assets
• Capital determined by the risk measure.
                C = r(X) - E[X]
                   Step 2
              Allocate Capital
 • How are you going to use allocated capital?
   – Use it to set profitability targets.
Expected Profit for Line     Total Expected Profit
                           =
Allocated Capital for Line       Total Capital

 • How do you allocate capital?
   – Any way that leads to correct economic
     decisions, i.e. the insurer is better off if
     you get your expected profit.
                 Better Off?
• Let P = Profit and C = Capital. Then the
  insurer is better off by adding a line/policy if:
        P  P P
                
        C  C C
         P  C  C  P  C  P  P  C
           P P
              
           C C
         Marginal return on new business 
          return on existing business.
  OK - Set targets so that marginal return on
  capital equal to insurer return on Capital?

• If risk measure is subadditive then:
       Sum of Marginal Capitals is  Capital
• Will be strictly subadditive without perfect
  correlation.
• If insurer is doing a good job, strict
  subadditivity should be the rule.
OK - Set targets so that marginal return on
capital equal to insurer return on Capital?

If the insurer expects to make a return,
                  e = P/C
then at least some of its operating divisions
must have a return on its marginal capital
that is greater than e.
Proof by contradiction
   DPk P                     P
If    = º e then: P = å DPk = å DCk < P         !
   DCk C              k      C k
  Ways to Allocate Capital #1

• Gross up marginal capital by a factor to
  force allocations to add up.
• Economic justification - Long run result
  of insurers favoring lines with greatest
  return on marginal capital in their
  underwriting.
                   Reference
• The Economics of Capital Allocation
  – By Glenn Meyers
  – Presented at the 2003 Bowles Symposium
    http://www.casact.org/pubs/forum/03fforum/03ff391.pdf
• The paper:
  – Asks what insurer behavior makes
    economic sense?
  – Backs out the capital allocation method
    that corresponds to this behavior.
  Ways to Allocate Capital #2

• Average marginal capital, where
  average is taken over all entry orders.
  – Shapley Value
  – Economic justification - Game theory
• Additive co-measures – Kreps
• Capital consumption – Mango
Remember the time dimension.
     Allocate capital to
   prior years’ reserves.
•   Target Year 2003 - prospective
•   Reserve for 2002 - one year settled
•   Reserve for 2001 - two years settled
•   Reserve for 2000 - three years settled
•   etc
              Step 3
            Reinsurance

• Skip this for now
             Step 4
The Cost of Financing Insurance
The cash flow for underwriting insurance
• Investors provide capital - In return they:
• Receive premium income
• Pay losses and other expenses
• Receive investment income
  – Invested at interest rate i%
• Receive capital as liabilities become
  certain.
             Step 4
The Cost of Financing Insurance
Net out the loss and expense payments
• Investors provide capital - In return they:
• Receive profit provision in the premium
• Receive investment income from capital
  as it is being held.
• Receive capital as liabilities become
  certain.
• We want the present value of the income
  to be equal to the capital invested at the
  rate of return for equivalent risk
             Step 4
The Cost of Financing Insurance
Capital invested in year y+t                         C(t)
Capital needed in year y+t if division k            Ck(t)
is removed
Marginal capital for division k               Ck(t)=C(t)-Ck(t)
Sum of marginal capital                             SM(t)
Allocated capital for division k           Ak(t)=Ck(t)×C(t)/SM(t)
Profit provision for division k                     Pk(t)
Insurer’s return in investment                        i
Insurer’s target return on capital                    e
             Step 4
The Cost of Financing Insurance
Time     Financial Support          Amount Released
         Allocated at time t           at time t
 0              Ak(0)                      0
 1             Ak(1)           Relk(1) = Ak(0)(1+i) – Ak(1)
 ---             ---                            ---
  t             Ak(t)          Relk(t) = Ak(t –1)(1+i) – Ak(t)
 ---             ---                            ---

                                              Rel k  t 
       Then Pk  0   Ak  0   
                                               1  e 
                                                          t
                                        t 1
        Back to Step 3
    Reinsurance and Other
     Risk Transfer Costs
• Reinsurance can reduce the amount of,
  and hence the cost of capital.
• When buying reinsurance, the
  transaction cost (i.e. the reinsurance
  premium less the provision for expected
  loss) is substituted for capital.
   Step 4 with Risk Transfer
The Cost of Financing Insurance
Time    Financial Support          Amount Released
        Allocated at time t           at time t
 0          Ak(0)+Rk(0)                   0
 1            Ak(1)           Relk(1) = Ak(0)(1+i) – Ak(1)
 ---            ---                         ---
  t            Ak(t)          Relk(t) = Ak(t –1)(1+i) – Ak(t)
 ---            ---                         ---
                                                   Rel k  t 
 Then Pk  0   Ak  0   Rk  0   
                                                    1  e 
                                                               t
                                             t 1

 The Allocated $$ should be reduced with risk transfer.
 Step 4 Without Risk Transfer
The Cost of Financing Insurance
Time     Financial Support          Amount Released
         Allocated at time t           at time t
 0              Ak(0)                      0
 1             Ak(1)           Relk(1) = Ak(0)(1+i) – Ak(1)
 ---             ---                            ---
  t             Ak(t)          Relk(t) = Ak(t –1)(1+i) – Ak(t)
 ---             ---                            ---

                                              Rel k  t 
       Then Pk  0   Ak  0   
                                               1  e 
                                                          t
                                        t 1
               Examples
• Use ISO Underwriting Risk Model
• Parameterization based on analysis of
  industry data.
• Big and small insurer
  – Big Insurer is 10 x Small Insurer
• Three reinsurance strategies
  Expected Loss
for small insurer is
   10 times less,
Various Risk
 Measures
Various Risk
 Measures
Different measures of risk imply
  different amounts of capital
                         Implied Capital
  Amount




                                                      Capital
                                                      Liabilities




           2xStd. Dev.    VaR@99%          TVaR@99%
   Allocating (Cost of) Capital
• Calculate marginal capital for each profit
  center.
• Calculate the sum of the marginal capitals for
  all capital centers.
• Diversification multiplier equals the total
  capital divided by the sum of the marginal
  capitals.
• Allocated capital for each profit center equals
  the product of the diversification multiplier
  and the marginal capital for the profit
  center.
             Capital for Multiline vs Standalone Insurer




                            Diversification Benefit
Amount




         CMP-M CMP-S HO-M    HO-S   Auto-M Auto-S Cat-M   Cat-S Total-M Total-S
Note capital is
 allocated to
loss reserves
    Optimizing Reinsurance
• User input
  – Target return on capital
  – Return on investments (sensitivity analysis
    on investment income)
  – Corporate income tax rate
  – Cost of reinsurance
  – Insurer expense provisions
List of Reinsurance
     Strategies
     Cost of Financing Insurance =
Cost of Capital + Net Cost of Reinsurance

• Cost of capital = target return x capital
• Net cost of reinsurance
  = Premium – Expected Recovery
• Minimize the cost of financing.
                              
                                    Rel k  t 
Cost of Financing  Ak  0   
                                    1  e 
                                               t
                             t 1
     Big Insurer
Cost of Financing with
  No Reinsurance
    Small Insurer
Cost of Financing with
  No Reinsurance
     Big Insurer
Cost of Financing with
  Cat Reinsurance
    Small Insurer
Cost of Financing with
  Cat Reinsurance
      Big Insurer
 Cost of Financing with
 Cat Reinsurance and
XS of Loss Reinsurance
     Small Insurer
 Cost of Financing with
 Cat Reinsurance and
XS of Loss Reinsurance
   Optimize reinsurance by
minimizing the cost of financing

    Big Insurer           Small Insurer




                                                 Net Reins
                                                 Capital




                                                        Note: Small insurer
                                                        costs multiplied by 10.

   No Re   Cat   All Re   No Re   Cat   All Re
           Re                     Re
 Discussion of Behavioral Issues
• Smooth out earnings – Wall Street punishes
  shock losses.
• Question – Cat limit to capital ratio?
  – Answer – 10 to 15%.
• Impairment issues – Can you raise additional
  capital if you lose 1/3 of capital?
• Silos – Divisional incentives work against
  corporate objectives.