Casualty Reinsurance Seminar, June 7th, 2004, Boston June 7, 2004 “Cat Bond Pricing Using Probability Transforms” published in Geneva Papers, 2004 Shaun

Casualty Reinsurance Seminar, June 7th, 2004, Boston

June 7, 2004

““Cat Bond Pricing Using Probability Cat Bond Pricing Using Probability

TransformsTransforms””

published in published in Geneva Papers, 2004Geneva Papers, 2004

Shaun Wang, Ph.D., FCAS

2

Shaun Wang, June 2004

What is CAT bond?

A high-yield debt instrument: if the issuer (insurance company) suffers a loss from a particular predefined catastrophe, then the issuer's obligation to pay interest and/or repay the principal is either deferred or forgiven.

Covered events: CA Earthquake, Japan Earthquake, FL Hurricane, EU Winter Storm; Multi-Peril & Multi-territory

Actual-dollar trigger or Reference-index trigger

3


Why CAT bond?

For bond issuers:

Alternative source of capital/capacity for insurance companies with large risk transfer needs

Not subject to the risk of non-collectible reinsurance

For investors:

High yield coupon rate

CAT bond performance is not closely correlated with the stock market or economic conditions.

4


Example of Cat-bond transactions(Data Source: Lane Financial LLC)

Cat bond Transaction

Probability of First $

Loss

Probability of Last $

Loss

Expected Loss given

default

Yields Spread Over

LIBOR

Atlas Re A 0.0019 0.0005 0.5789 2.74%

Atlas Re B 0.0029 0.0019 0.7931 3.75%Atlas Re C 0.0547 0.019 0.5923 14.19%

5


State of the Cat-bond Market

In the past, unfamiliar class of assets to investors, led to limited number of transactions

Phenomenal performance of CAT bond portfolios, led to recent surge of interest by institutional investors

Cat Bond Market Grew 42% in 2003 Total bond issuance $1.73 billion Reduced cost of issuing (coupon interest and

transaction costs)

6


Cat-bond offers a laboratory for reconciliation of pricing approaches

Capital market pricing is forward-looking:

prices incorporate all available information

No-arbitrage pricing (Black-Scholes Theory)

Actuarial pricing is back-forward looking

Using historical data to project future costs

Explicit adjustments for risk

7


Financial World

Black-Schole-Merton theory for pricing options and corporate credit default risks

A common measure for fund performance is the Sharpe ratio: ={ E[R] r }/[R], the excess return per unit of volatility

also called “market price of risk”

How can we relate it to actuarial pricing?

8


Ground-up Loss X has loss exceedence curve:

SX(t) =1 FX(t) = Pr{ X>t }.

Layer X(a, a+h); a=retention; h=limit

Actuarial World

0

)(][ dttSXE X

dttShaaXEha

a X )()],([

9


Loss Exceedence Curve

10


Insurance prices by layer implies a

transformed distribution– layer (t, t+dt) loss: Slayer (t, t+dt) loss: SXX(t) dt (t) dt

– layer (t, t+dt) price: Slayer (t, t+dt) price: SXX*(t) dt*(t) dt

– implied transform: Simplied transform: SXX(t) (t) S SXX*(t)*(t)

Venter 1991 ASTIN Paper

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Insight of Gary Venter (91 ASTIN ):

“Insurance prices by layer imply a transformed distribution”

S(x)=1F(x), or Loss Exceedence Curve

12


Attempt #1 by Morton Lane

(Hachemeister Prize Paper) Morton Lane (2001) “Pricing of Risk Transfer

transactions” proposed a 3-parameter model:

EER = 0.55 (PFL)0.49 (CEL)0.57

PFL: Probability of First Loss

CEL: Conditional Expected Loss (as % of principal)

EER: Expected Excess Return (over LIBOR)

13


Attempt #2: Wang Transform(Sharing 2004 Ferguson Prize with Venter)

Let be standard normal distribution:

(1.645)=0.05, (0)=0.5, (1.645)=0.95

Wang introduces a new transform:

F(x)=0.95, =0.3, F*(x) = 1(1.6450.3) =0.91

Fair Price is derived from the expected value under

the transformed distribution F*(x).

λF(x)F*(x) )(ΦΦ 1

14


WT extends the Sharpe Ratio Concept

If FX is normal(), FX* is normal(+ ):

E*[X] = E[X] + [X]

If FX is lognormal( ), FX* is lognormal(+ )

The transform recovers CAPM & Black-Scholes (ref. Wang, JRI 2000)

extends the Sharpe ratio to skewed distributions

))(()(* 1 xFxF

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1-factor Wang transformlambda=0.3

-

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

1 11 21 31 41 51 61 71 81 91

Uniform Distribution

Adj

uste

d D

ensi

ty

f(x)f*(x)

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Unified Treatment of Asset / Loss

The gain X for one party is the loss for the counter party: Y = X

We should use opposite signs of , and we get the same price for both sides of the transaction

))(()( 1* xFxF XX

))(()(

))(()(1*

1*

ySyS

yFyF

YY

YY

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Baseline Sampling Theory

We have m observations from normal(,2). Not

knowing the true parameters, we have to estimate

and by sample mean & variance.

When assessing the probability of future outcomes,

we effectively need to use Student-t with k=m-2

degrees-of-freedom.

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Adjust for Parameter Uncertainty

Baseline: For normal distributions, Student-t

properly reflects the parameter uncertainty

Generalization: For arbitrary F(x), we propose the

following adjustment for parameter uncertainty:

))(()( 1* xFQxF

19


A Two-Factor Model

Wang transform with adjustment for parameter uncertainty:

))(()(* 1 yFQyF

where is standard normal CDF, and Q is Student-t CDF with k degrees-of-freedom

20


Student-t Adjustmentk=7

-

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

1 11 21 31 41 51 61 71 81 91


Ad

jus

ted

De

ns

ity

f(x)f*(x)

21


1-factor Wang transformlambda=0.3

-

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

1 11 21 31 41 51 61 71 81 91


Adj

uste

d D

ensi

ty

f(x)f*(x)

22


2-factor transformlambda=0.3, k=7

-

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

1 11 21 31 41 51 61 71 81 91


Ad

jus

ted

De

ns

ity

f(x)f*(x)

23


Insights for the second factor

Explains investor behavior: greed and fear

Investors desire large gains (internet lottery)

Investors fear large losses (market crash)

Consistent with “volatility smile” in option prices

Quantifies increased parameter uncertainty in the tails

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Empirical Studies

16 CAT-bond transactions in 1999

Fit well to the 2-factor Wang transform

Better fit than Morton Lane’s 3-parameter model (in his

2001 Hachmeister Prize Paper)

12 CAT bond transactions in 2000

Use 1999 estimated parameters to price 2000

transactions, remain to be the best-fit

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1999 Cat-bond transactions(Data Source: Lane Financial LLC)

Cat bond Transaction

Probability of First $ Loss

Probability of Last $

Loss

Expected Loss given

defaultModel Yields

Spread

Empirical Yields

Spread

Mosaic 2A 0.0115 0.0012 0.3652 3.88% 4.06%Mosaic 2B 0.0525 0.0115 0.541 10.15% 8.36%Halyard Re 0.0084 0.0045 0.75 4.82% 4.56%Domestic Re 0.0058 0.0044 0.8621 4.36% 3.74%Concentric Re 0.0062 0.0022 0.677 4.01% 3.14%Juno Re 0.006 0.0033 0.75 4.15% 4.26%Residential Re 0.0076 0.0026 0.5789 4.08% 3.71%

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Fit Wang transform to 1999 Cat bondsDate Sources: Lane Financial LLC Publications

Yield Spread for Insurance-Linked Securities

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

16.00%

Transactions

Yie

ld S

prea

d

Model-Spread

Empirical-Spread

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Use 1999 parameters to price 2000 Cat Bonds

Fitted versus Empirical Spread

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

Transactions

Yie

ld S

pre

ad

Model-Spread

Empirical-Spread

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Corporate Bond Default:Historial versus Implied Default Frequency

Corporate Bond Historical Bond default freq

Rating Class p p* Ratio p*/p p** Ratio p**/p

AAA 0.00015 0.00077 5.2 0.00971 64.7AA 0.0004 0.00185 4.6 0.01362 34.1A 0.00075 0.00322 4.3 0.01721 23.0

BBB 0.0017 0.00659 3.9 0.02393 14.1BB 0.0075 0.02372 3.2 0.04735 6.3B 0.02 0.05438 2.7 0.07995 4.0

CCC 0.08 0.16977 2.1 0.18821 2.4

lambda=0.45 lambda=0.45; k=61-factor transform 2-factor transform

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Fit 2-factor model to corporate bonds

Bond Rating and Yield Spread

0

200

400

600

800

1,000

1,200

1,400

AAA AA A BBB BB B CCC

Bond Rating

Sp

rea

d (

ba

sis

po

ints

)

Model Fitted Spread

Actual Spread

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Risk Premium for Corporate Bonds

Use 2-factor Wang transform to fit historical default probability & yield spread by bond rating classes

Compare the fitted parameters for “corporate bond” versus “CAT-bond”

parameters are similar,

“CAT-bond” has lower Student-t degrees-of-freedom,

In 1999, CAT-bond offered more attractive returns for the risk than corporate bonds

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Cat bond vs. Corporate Bond (before)

Before Sept. 11 of 2001 fund managers were less familiar (or comfortable) with the cat bond asset class.

Fund managers were reluctant to expose themselves to potential career risks, since they would have difficulties in explaining losses from investing in cat bonds, instead of conventional corporate bonds.

At that time, because of investors’ weak appetite for cat bonds, cat bonds issuers had to offer more attractive yields than corporate bonds with comparable default frequency & severity.

32


Cat bond vs. Corporate Bond (after)

During 2002-3, fund managers' interest in cat bond has growth significantly, due to superior performance of the cat bond class. They now complaint about not having enough cat bond issues to feed their risk appetite.

In the same time period, the perceived credit risk of corporate bonds increased, in tandem with the general broader market. Investors began to value more the benefit of low correlation between cat bond and other asset classes.

It has been reported that the yields spreads on cat bonds have tightened while the yields spreads on corporate bonds have widened (cross over) – Polyn April 2003.

Documents

Casualty Reinsurance Seminar, June 7th, 2004, Boston June 7, 2004 “Cat Bond Pricing Using Probability Transforms” published in Geneva Papers, 2004 Shaun