1 Acknowledgement Research with Zinoviy Landsman, Depart ment of - - PowerPoint PPT Presentation

1 Acknowledgement

• Research with Zinoviy Landsman, Depart

ment of Statistics, Actuarial Re- search Center, University of Haifa, Haifa, ISRAEL

• The authors gratefully acknowledge

financial support of the UNSW Ac- tuarial Foundation of the Institute of Actuaries of Australia, A Research Council Discovery Grant DP0345036, Caesarea Rothschild Insti- tute and Zimmerman Foundation.

2 Aims of Research

• Develop a Pricing Model for a Multi-Line Insurer including friction

and default option value for speci fied model for risks.

• Implement a recently developed dependent Gamma model for lines of busi-

ness.

• Develop approximations and closed form expressions for implementation.

3 Introduction - Insurance Pricing

• Insurance pricing

PV (losses,assuming no default )+PV (surplus costs )−PV (default option )

• Myers and Read (2001), Phillips, Cummins and Allen (1998), Zanjani

(2002), Sherris (2003).

• PV (losses, assuming no default

) not impacted by capital structure or fric- tional costs but re flects the insurance loss price of risk; PV (surplus costs ) are the frictional costs - re flects capital structure - how to determine and how to allocate to line of business/policy?; PV (default option ) re flects capital structure, analogy to corporate bonds, Q probabilities not P prob- abilities, how to allocate to line of business?

4 Model for Insurer

• Insurer with portfolio of

n distinct insurable risks written at the beginn

• f the period.
• PremiumPi for line of business
• i. Pi allows for the expected losses, the

risk loading, the risk of claims not being met due to the insolvency insurer and any costs of capital to be allocated to policyholders. premiums collected at the start of the period are P = Pn

i=1 Pi.

• End of period claims for the

ith risk denoted by Li and total claims at the end of the period L = Pn

i=1 Li.

5 Model for Insurer

• Arbitrage-free model where there exists a probability measure

Q such that the current values of the assets and liabilities are the discounted value of end of period random payments using a risk free discount rate.

• There exists a risk free asset such that an investment of

1 now will return er at the end of the period for certain.

6 Model for Insurer - Insurer Losses/Claims

• L0i denotes the time

0 price or fair value for line of business i ignoring default option given by L0i = EQ h e−rLi

i

for all i = 1, . . . , n and total value of the initial liabilities by L0 = Pn

i=1 L0i.

• V0 denotes the time

0 price or fair value for the assets given by V0 = EQ h e−rV

i

where r is the risk-free continuous compounding rate of interest.

• Complete markets so that we observe

L0i for all i = 1, . . . , n, from the insurance market and V0 from an asset market.

7 Model for Insurer - Default Option Value

• Assuming equal priority for losses by line of business in the event

vency actual loss payments will be

Li L V

if L > V (or

V L ≤ 1)

Li if L ≤ V (or

V L > 1)

• r

Li

"

1 −

µ

1 − V L

¶+#

8 Model for Insurer - Default Option Value

• Premium for line of business

i allowing for default option value Pi = EQ

"

e−rLi

"

1 −

µ

1 − V L

¶+##

= L0i − e−rEQ

"

Li

µ

1 − V L

¶+#

= L0i − D0i where the line of business default value for line i is denoted D0i.

9 Model for Insurer - Capital Structure

• Denote the insurer default option value by

D0 so that D0 = e−r

n

X

i=1

EQ

"

Li

µ

1 − V L

¶+#

=

n

X

i=1

D0i

• Capital of the insurer based on a target solvency ratio of

s will be equal to C = V0 − P = (1 + s) L0 − P = sL0 + D0

• Beginning of period assets are

V0 = (1 + s) L0 = sL0 + D0 + L0 − D0 = C + P

10 Model for Insurer - Liability Assumptions

• Dependent Gamma model: Mathai (1982

), Moschopoulos (1985), Alouini, Abdi, and Kaveh (2001), Furman and Landsman (2004)

• (X0, ...., Xn, Xn+1) aren + 2 independent gamma distributed random

variables with shape parameters γi and common rate parameter α, denoted byG(γi, α), i = 0, ..., n + 1.

• Probability density of the

Xi is fXi (xi) = αγi Γ (γi)e−αXixγi−1

i

, xi > 0

11 Model for Insurer - Liability Assumptions

• End of period claims for the

n lines of business and the value of the assets V are modelled as Li = α0 α X0 + Xi, i = 1, ..., n V = α0 α X0 + Xn+1

• Intuition: common factor

X0 impacts the values of each line of business claims as well as the assets (in flation)

• Each line of business and the assets have a separate independent f

impacting them denoted by Xi,i = 1, ..., n for the line of business i, and Xn+1 for the assets.

12 Model for Insurer - Liability Assumptions

• Claims for each line of business

L1, ..., Ln are gamma distributed random variables, G(λi, α), i = 1, ..., n, where λi = γ0 + γi

• Assets

V are gamma distributed as G(γ0+γn+1, α), since the sum of two independent gamma random variables with the same rate parameter is also gamma with shape parameter equal to the sum of the shape parameters.

13 Model for Insurer - Liability Assumptions

• Total claims liability at the end of the period is

L =

n

X

i=1

Li = nα0 α X0 + X·, where X· =

n

X

i=1

Xi

• X· is distributed

G(γ·, α), with γ· = Pn

i=1 γi. However, the total claims

liability is a sum of 2 gamma random variables,

nα0 α X0 andX·, each with

di fferent rates and so the sum does not have a gamma distribution.

14 Model for Insurer - Liability Assumptions

• Furman and Landsman (2004): can represent

L as a mixed gamma distri- bution with mixed shape parameter L v G(γ0 + γ· + ν, α), where ν is a non negative integer random variable with probabilities pk = Cδk, k ≥ 0, where C = 1 nγ0 : δk = k−1γ0

k

X

i=1

(n − 1 n )iδk−i, k > 0 : δ0 = 1.

15 Model for Insurer - Liability Assumptions

• For the model of claims and assets that we have assumed, the

Q measure probability density of claims for line of business i is GammaG(λi, α), i = 1, ..., n so that fLi (yi) = αλi Γ (λi)e−αyiyλi−1

i

, yi > 0, with EQ [Li] = λi α and V arQ [Li] = λi α2

16 Model for Insurer - Liability Assumptions

• Note that

Cov

³

Li, Lj

´

= γ0 α2 i 6= j Cov (Li, V ) = γ0 α2, i = 1, .., n and ρ

³

Li, Lj

´

= γ0

q

λiλj , i, j = 1, ..., n

17 Model for Insurer - Default Option Value

• Default option value for each line

D0i = e−rEQ

"

Li

µ

1 − V L

¶+#

= e−rEQ

⎡ ⎣ µα0

α X0 + Xi

¶ Ã

1 −

α0 α X0 + Xn+1 nα0 α X0 + X·

!+⎤ ⎦

= e−rα0 α EQ

⎡ ⎣X0 Ã

1 −

α0 α X0 + Xn+1 nα0 α X0 + X·

!+⎤ ⎦

+e−rEQ

⎡ ⎣Xi Ã

1 −

α0 α X0 + Xn+1 nα0 α X0 + X·

!+⎤ ⎦ , i = 1, . . . , n

where Q ∼

Yn+1

i=0 G(γi, α).

18 Model for Insurer - Default Option Value

• Expectation in the

first term (see paper for proofs) EQ

⎡ ⎣X0 Ã

1 −

α0 α X0 + Xn+1 nα0 α X0 + X·

!+⎤ ⎦ = γ0

α EQ0

⎡ ⎣ Ã

1 −

α0 α X0 + Xn+1 nα0 α X0 + X·

!+⎤ ⎦

where dQ0 =

Ã

αγ0+1 Γ (γ0 + 1)e−αx0xγ0

! n+1 Y

i=1

Ã

αγi Γ (γi)e−αxixγi−1

i

αγi

!

dx0 . . . dxn+1

19 Model for Insurer - Default Option Value

• Expectation in the second term

EQ

⎡ ⎣Xi Ã

1 −

α0 α X0 + Xn+1 nα0 α X0 + X·

!+⎤ ⎦ = γi

α EQi

⎡ ⎣ Ã

1 −

α0 α X0 + Xn+1 nα0 α X0 + X·

!+⎤ ⎦

where dQi =

Ã

αγi+1 Γ (γi + 1)e−αxixγi

i αγi+1

!

n+1

Y

j=0,j6=i

⎛ ⎝ αγj

Γ

³

γj

´e−αxjx

γj−1 j

⎞ ⎠ dx0 . . . dxn+1

i = 1, ..., n. ThusQi ∼ G(γi + 1, α)

Yn+1

j=0,,j6=i G(γi, α),i = 1, ..., n.

20 Model for Insurer - Default Option Value

• To evaluate the expression

EQi

⎡ ⎣ Ã

1 −

α0 α X0 + Xn+1 nα0 α X0 + X·

!+⎤ ⎦ , i = 1, ..., n

• Note that under

Q0 V = α0 α X0 + Xn+1 ∼ G(γ0 + 1 + γn+1, α) and under Qi,i = 1, ..., n V = α0 α X0 + Xn+1 ∼ G(γ0 + γn+1, α)

21 Model for Insurer - Default Option Value

• UnderQ0 andQi, i = 1, ..., n, the distribution of

nα0 α X0 + X· will not

be Gamma

• Its distribution can be represented as a mixture of Gamma random v

ables - Furman and Landsman (2004).

22 Model for Insurer - Default Option Value

• Under

Q0 L = nα0 α X0 + X· ∼ G(γ0 + 1 + γ· + ˜ ν, α) where ˜ ν is a non-negative integer random variable de fined in ˜ pk = ˜ C˜ δk, k ≥ 0, ˜ C = 1 nγ0+1 : ˜ δk = k−1(γ0 + 1)

k

X

i=1

(n − 1 n )i˜ δk−i, k > 0 : ˜ δ0 = 1.

23 Model for Insurer - Default Option Value

• Under

Qi, i = 1, ..., n, L = nα0 α X0 + X· ∼ G(γ0 + 1 + γ· + ν, α), where ν is a non-negative integer random variable de fined previously.

24 Model for Insurer - Some Functions

• Define the following functions (evaluate numerically)
• ψ (γ) = Γ0(γ)

Γ(γ) = d dγ ln Γ (γ) is the Psi (Digamma) function

• ψ(γ; a) =

1 Γ(γ)

R ∞

ln(a + x)xγ−1 exp(−x)dx whereψ(γ; 0) is simply the digamma function ψ(γ)

• φ(γ; a) =

1 Γ(γ)

R ∞

1 a+xxγ−1 exp(−x)dx, γ > 1 so that

φ(γ; 0) =

1 γ−1.

25 Model for Insurer - Log-Normal Approximation

• Log normal approximation

D0i = e−rα0γ0 α2

⎡ ⎣1

2 − eµΛ0+1

2σ2 Λ0Φ

⎛ ⎝− ³

µΛ0 + σ2

Λ0

´

σΛ0

⎞ ⎠ ⎤ ⎦

+e−rγi α

⎡ ⎣1

2 − eµΛi+1

2σ2 ΛiΦ

⎛ ⎝− ³

µΛi + σ2

Λi

´

σΛi

⎞ ⎠ ⎤ ⎦ , i = 1, ..., n.

26 Model for Insurer - Moments of Λ

• Under

Q0 µ0

Λ = ψ

³

γ0 + γn+1 + 1

´

−

⎛ ⎝

∞

X

k=0

ψ (γ0 + 1 + γ· + k) ˜ pk

⎞ ⎠

σ2

Λ0

= Γ00 ³ γ0 + γn+1 + 1

´

Γ

³

γ0 + γn+1 + 1

´ − ψ ³

γ0 + γn+1 + 1

´2

+

∞

X

k=0

"

Γ00 (γ0 + 1 + γ· + k) Γ (γ0 + 1 + γ· + k) − ψ (γ0 + 1 + γ· + k)2

#

˜ pk −2CovQ0 [ln V, ln L]

27 Model for Insurer - Moments of Λ (approximation)

• We also have

CovQ0 [ln V, ln L] = E[ψ(γ·; nα0X0)ψ(γn+1; α0X0)] − Eψ(γ·; nα0X0)Eψ(γn+1; α0X0)

• For large

γ· andγn+1 CovQ0 [ln V, ln L] ≈ n(γ0 + 1) (γ· − 1)(γn+1 − 1).

28 Model for Insurer - Moments of Λ

• Under

Qi, i = 1, ..., n µi

Λ = ψ

³

γ0 + γn+1

´

−

⎛ ⎝

∞

X

k=0

ψ (γ0 + 1 + γ· + k) pk

⎞ ⎠

σ2

Λi =

Γ00 ³ γ0 + γn+1

´

Γ

³

γ0 + γn+1

´ − ψ ³

γ0 + γn+1

´2

+

∞

X

k=0

"

Γ00 (γ0 + 1 + γ· + k) Γ (γ0 + 1 + γ· + k) − ψ (γ0 + 1 + γ· + k)2

#

pk −2CovQi [ln V, ln L] , i = 1, ..., n.

29 Model for Insurer - Moments of Λ (approximation)

• We also have

CovQi [ln V, ln L] = E[ψ(γ· + 1; nα0X0)ψ(γn+1; α0X0)] − Eψ(γ· + 1; nα0X0)Eψ(γn+1; α0X0)

• Approximation

CovQi [ln V, ln L] ≈ nγ0 γ·(γn+1 − 1), i = 1, ..., n.

30 Model for Insurer - Frictional Costs

• Frictional costs (transactions costs and other costs such as additional

ation and agency costs are assumed to occur as a percentage of the

• f period surplus and are only incurred provided the insurer is solven
• Denote the value of these as

Cc then Cc = EQ h cce−r (V − L)+i =

n

X

i=1

EQ h cce−rLi (Λ − 1)+i where cc is the costs of capital as a percentage of the surplus, provided is positive.

31 Model for Insurer - Frictional Costs

• If we denote the costs of capital allocated to line of business

i ∈ {1, ..., n} byCi

c we then have

Ci

c

= cce−rEQ h Li (Λ − 1)+i = ccD0i − cce−rEQ [Li (1 − Λ)] = ccD0i − ccLΛi

32 Model for Insurer - Frictional Costs

• We have

LΛi = e−rEQ [Li (1 − Λ)] = e−r

∙γ0

α EQ0 (1 − Λ) + γi α EQi (1 − Λ)

¸

= e−r

∙γ0

α (1 − EQ0 [Λ]) + γi α (1 − EQi [Λ])

¸

, i = 1, ..., n.

33 Model for Insurer - Frictional Costs

• Approximation for the expectation of

Λ EQ0 [Λ] ≈ (γ0 + 1 + γn+1)( 1 γ· − 1 − n(γ0 + 1) (γ· − 1)(γ· − 2)), and EQi [Λ] ≈ (γ0 + γn+1)( 1 γ· − n(γ0 + 1) γ·(γ· − 1)), i = 1, ..., n and under the log normal assumption EQi [Λ] = exp(µi

Λ + 1

2σ2

Λi), i = 0, 1, ..., n,

34 Model for Insurer - Frictional Costs

• Following Estrella (2004) we assume that

financial distress costs are a percentage of the absolute value of the shortfall of assets over liabilities in the event of insolvency. Denoting the value of these financial distress costs by Cf we have Cf = EQ h cfe−r (L − V )+i = EQ h cfe−rL (1 − Λ)+i where cf is the financial distress costs as a percentage of the asset shor in the event of insolvency.

35 Model for Insurer - Frictional Costs

• Allocation to line of business

i Ci

f = cfe−rEQ h

Li (1 − Λ)+i = cfD0i where Ci

f is the allocation of the

financial distress costs to line of business i.

36 Model for Insurer - By Line Price

• An expression for the price by line of business, including the frictional
• f capital and default option value, for a multi-line insurer is given

L0i + Ci

c + Ci f − D0i = e−rλi

α − ccLΛi −

³

1 − cf − cc

´

D0i

• Closed form expressions for

LΛi andD0i are derived in this paper under a recently developed dependent gamma distribution model.

37 Conclusions and Summary

• Results for a model for multiline insurer with dependent gamma distrib

allowing for default put and frictional costs.

• Closed form and approximate expressions for default option value b

as well as allocation of frictional costs.

• Issues - Optimal capital structure (see Chandra and Sherris, 2005).
• Issues - Marginal versus full balance sheet allocations.