[PPT] - An Empirical-based Approach for Optimal Reinsurance C HENGGUO W ENG PowerPoint Presentation

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An Empirical-based Approach for Optimal Reinsurance CHENGGUO WENG Department of Statistics & Actuarial Science University of Waterloo This is a joint work with Ken Seng Tan August 31, 2009

C. Weng (c2weng@uwaterloo.ca)

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Outline of Today’s Presentation

Background Motivation Empirical-based Approach Conclusion

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Effect of Reinsurance

Policyholders (Insureds) Insurance Company (Insurer or Cedent)

Premium π0 Loss X

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Effect of Reinsurance

Policyholders (Insureds) Insurance Company (Insurer or Cedent)

Premium π0 Loss X

Reinsurance Company (Reinsurer)

f(X): Ceded Claims

e.g. stop loss:

f(X) = (X − d)+, d ≥ 0

e.g. quota-share:

f(X) = cX, 0 ≤ c ≤ 1

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Effect of Reinsurance

Policyholders (Insureds) Insurance Company (Insurer or Cedent)

Premium π0 Loss X

Reinsurance Company (Reinsurer)

f(X): Ceded Claims

e.g. stop loss:

f(X) = (X − d)+, d ≥ 0

e.g. quota-share:

f(X) = cX, 0 ≤ c ≤ 1 Π(f): Reinsurance Premium

e.g. Expectation premium principle:

Π(f) = (1 + θ)E[f(X)]

Π(f)

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Effect of Reinsurance

Policyholders (Insureds) Insurance Company (Insurer or Cedent)

Premium π0 Loss X

Reinsurance Company (Reinsurer)

f(X): Ceded Claims

e.g. stop loss:

f(X) = (X − d)+, d ≥ 0

e.g. quota-share:

f(X) = cX, 0 ≤ c ≤ 1 Π(f): Reinsurance Premium

e.g. Expectation premium principle:

Π(f) = (1 + θ)E[f(X)]

Π(f)

Insurer’s retained risk: Rf(X) = X − f(X)
Insurer’s total risk: Tf(X) = Rf(X) + Π(f) = X − f(X) + Π(f)
tradeoff between the amount of loss retained and the reinsurance premium

payable to a reinsurer

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Risk Measure Minimization Reinsurance Models

A plausible optimal reinsurance model: minf ρ(Tf(X)) = ρ

X − f(X) + Π(f(X))
s.t.

Π(f(X)) ≤ π 0 ≤ f(x) ≤ x for all x ≥ 0 where ρ is a chosen risk measure, such as variance, VaR and CTE.

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Risk Measure Minimization Reinsurance Models

A plausible optimal reinsurance model: minf ρ(Tf(X)) = ρ

X − f(X) + Π(f(X))
s.t.

Π(f(X)) ≤ π 0 ≤ f(x) ≤ x for all x ≥ 0 where ρ is a chosen risk measure, such as variance, VaR and CTE. Complexity of solving the above risk measure minimization models: if the form of f is specified: technically tractable. stop-loss f(x) = (x − d)+ quota-share f(x) = cx if a general f is considered: infinite dimensional problem, very challenging to obtain the explicit solutions. relies on the premium principle and the risk measure ρ.

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Empirical Approach: Motivation

The general reinsurance model can be formulated as:        minf ρ(X, f) s.t. 0 ≤ f(x) ≤ x for all x ≥ 0, Π(f(X)) ≤ π. Often difficult to solve (due to infinite dimension and nonlinearity) In practice, the distribution of the underlying risk X is estimated from the

bserved data {x1, · · · , xN}.

Empirical-based reinsurance model:

exploits the observed data directly

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Empirical-based Model: Formulation

Collect samples x := (x1, x2, · · · , xN) corresponding to the underlying risk X Introduce decision vector f := (f1, f2, · · · , fN) where each fi represents the reinsurance indemnification for the loss amount xi, i = 1, 2, · · · , N. Define empirical-based estimates as: ρ(X, f) →

ρ(x, f)

0 ≤ f(x) ≤ x, for all x ≥ 0 → 0 ≤ fi ≤ xi, i = 1, 2, · · · , N, Π(f) ≤ π →

Π(f) ≤ π
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Empirical-based Model: Formulation (Cont’d)

General reinsurance model        minf ρ(x, f) s.t. 0 ≤ f(x) ≤ x, for all x ≥ 0, Π(f) ≤ π. Empirical reinsurance model:        minf

ρ(x, f)

s.t. 0 ≤ fi ≤ xi, i = 1, 2, · · · , N,

Π(f) ≤ π.

Many empirical reinsurance model can be cast as Second-Order Conic

(SOC) programming: A wide class of optimization problems Efficient softwares are available for solving SOC programming: e.g., CVX (Grant and Boyd, 2008)

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Empirical Model: Variance Minimization

Consider the following variance minimization model:    minf

Var(Tf) = Var(X − f(X) + Π(f))

s.t. 0 ≤ f(x) ≤ x, Π(f) ≤ π. Empirical version of the goal function:

Var(Tf) =

1 N − 1

N

i=1
xi − fi) − (¯

x − ¯ f 2 , where ¯ x denotes the mean of x, and ¯ f denotes the mean of f. Empirical version of the constraints: 0 ≤ fi ≤ xi, i = 1, 2, · · · , N, and Π(f) ≤ π. Empirical variance minimization model:        minf∈RN N

i=1

xi − fi) − (¯

x − ¯ f 2 s.t. 0 ≤ fi ≤ xi, i = 1, 2, · · · , N.

Π(f) ≤ π.
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Reinsurance Premium Budget Constraint

P1. Expectation principle: Π(f) = (1 + θ)E[f] with θ > 0.
Π(f) ≤ π ⇐

⇒ (1 + θ) ¯ f ≤ π,

P2. Standard deviation principle: Π(f) = E[f] + β
Var[f], where β > 0.
Π(f) ≤ π ⇐

⇒ ¯ f + β √ N − 1 N

i=1
fi − ¯

f 2 1/2 ≤ π. The empirical variance minimization model can be cast as a SOC programming problem

for as many as ten principles.

We analyzed the empirical solutions by some numerical examples (under expectation

and std premium principles).

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Empirical Model: CTE minimization

Theoretical CTE minimization model:    minf CTEα(Tf) ≡ CTEα

X − f(X) + Π[f(X)]
s.t.

0 ≤ f(x) ≤ x, Π[f(X)] ≤ π, Technical model:      min(ξ,f) Gα(ξ, f) ≡ ξ + 1 α

E

X − f(X) + Π(f(X)) − ξ

+ . s.t. 0 ≤ f(x) ≤ x, Π(f(X)) ≤ π. We developed the following fact: (ξ∗, f ∗) ∈ arg min(ξ,f)Gα(ξ, f) if and only if f ∗ ∈ arg minfCTEα(Tf), ξ∗ ∈ arg minξGα(ξ, f ∗). We proved that stop-loss is the optimal solution given that Π is the expectation principle.

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Empirical Model: CTE minimization (Cont’d)

Empirical model:        min(ξ,f)

Gα(ξ, f) ≡ ξ +

1 αN

N

i=1
xi − fi +

Π(f) − ξ + , s.t. 0 ≤ fi ≤ xi, i = 1, 2, · · · , N,

Π(f) ≤ π.

Empirical CTE minimization model:                  min(ξ,f,z) ξ + 1 αN

N

i=1

zi, s.t. 0 ≤ fi ≤ xi,

Π(f) ≤ π,

zi ≥ 0, zi ≥ Π(f) − fi − ξ + xi, i = 1, 2, · · · , N. The above empirical model can be cast as Second-order conic programming for as many as ten reinsurance premium principles.

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Example: CTE minimization

Use samples from an exponential loss distribution with mean 1000 Reinsurance premium principle: expectation principle with loading factor θ = 0.2 standard deviation principle with loading factor β = 0.2 Consider different levels of the reinsurance premium budget The solutions f ∗ are illustrated by their scatter plots against sample x

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Soln’s: CTE min & Expectation Principle (1/2)

1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000

1) π = 80

1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000

2) π = 200

1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000

3) π = 400

1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000

4) π = 600

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Soln’s: CTE min & Expectation Principle (2/2)

1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000

5) π = 800

1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000

6) π = 1000

1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000

7) π = 1500

1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000

8) π = 2000

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Soln’s:: CTE min & Std Principle (1/2)

1000 2000 3000 4000 5000 6000 500 1000 1500 2000

1) π = 50

1000 2000 3000 4000 5000 6000 500 1000 1500 2000

2) π = 80

1000 2000 3000 4000 5000 6000 500 1000 1500 2000

3) π = 100

1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000

4) π = 120

1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000

5) π = 150

1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000

6) π = 200

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EA: CTE min & Std Dev Principle (2/2)

1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000

7) π = 400

1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000

8) π = 600

1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000

9) π = 800

1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000

10) π = 1000

1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000

11) π = 1500

1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000

12) π = 1500

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Conclusion: Pros & Cons

Pros:

empirical data based finite dimensional reinsurance models flexibility of the goal function and the reinsurance premium principle empirical solutions are consistent with the theoretical solutions e.g., Variance and CTE minimization with expectation premium principle

Cons:

empirical-based model will turn out to be a large scale programming when the sample size is extremely large = ⇒ issues such as computational time and requirement for a substantially large computer’s memory will arise

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Acknowledgement

SOA/CAS Ph.D. Grant 2008 and 2009 and Travel Grant University of Waterloo Department of Statistics & Actuarial Science

C. Weng (c2weng@uwaterloo.ca)

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