A Further Examination of Insurance
PricingandUnderwriting Cycles
AFIR Conference, September 2005, Zurich, Switzerland
Chris K. Madsen, GE Insurance Solutions, Copenhagen, Denmark
Svend Haastrup, GE Insurance Solutions, Copenhagen, Denmark
Hal W. Pedersen, University of Manitoba, Winnipeg, Canada

Problem Statement
Property and casualty insurers’ profitability has long been depressed
relative to the risk they assume. To resolve this problem, one must
understand the “underwriting cycle”. Yet, it is not well understood bythe industry itself.

The underwriting cycle is usually shown as the “combined ratio” overtime. This is total losses plus expenses relative to premium charged.
Thus, any ratio over 100% suggests that more was paid out in losses
Lt + Et
CRt = charged:
and expensed that wasPt

If we can understand the underwriting cycle, we may be able to change
our behavior and make the industry a more attractive place for capital.

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The Time Series
PActual ,t , LActual ,t , E Actual ,tCRt =LActual ,t + E Actual ,tPActual ,tActual premium, losses and expenses
Actual combined ratio+⎛⎞()−XkP⎜⎟Pt = Ε ⎜⎟⎝ 1 + rt ⎠Discounted expected value of future cash
flows under P-measure⎛ ( X − k )+ ⎞⎟Qt = Ε ⎜⎜⎟⎝ 1 + rt ⎠Discounted expected value of future cash
flows under Q-measure

QVIXt

Q=Ε

QVIX t⎛ ( X − k )+ ⎞⎜⎟⎜ 1+ r ⎟t⎝⎠

Discounted expected value of future cash
flows under Q-measure with volatility adjusted
according to Vix index
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Review:The Property & Casualty Insurance Underwriting
Cycle
Madsen & Pedersen in 2003 showed that basic
option pricing can help us understand theunderwriting cycle
In particular, we found that the industry has atendency to under-price relative to the risk-neutral
price and that this explains much of the uw cycle
CRt = .2027 + .08836 ⋅ CRt −1 + .3979 ⋅ CRt − 2 + .4258 ⋅ Ct −1
Insurance is a call option on losses – yet when the value of that
option increases, so does the combined ratio. Insurers are notreflecting the increased price of risk essentially allowing consumers toarbitrage
Option price index
Combined Ratio
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Review:
Comparing “Standard” Pricing with Option Pricing
Systematic Over and Under Pricing

1.20

118.0

116.0

QtVIX tPt

1.15

Forecastfrom 2003

114.0
1.10

1.05
110.0
1.00
108.0

Pricing Index

Combined Ratio (%)

112.0

Combined Ratio
Standard Pricing
Option Pricing

0.95
106.0
0.90
104.0
0.85

102.0

100.0

Key turning points
explained0.80
1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004
YearCRt

Source for CR: A.M.Best

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Review
Feedback to 2003 Paper Industry combined ratio is on reportedresults. As such it contains reserving biases
and will lag actual performance
 To further test our hypothesis, it would beinteresting to look at a single homogeneousline of business with real reserving data
without the inherent lag

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1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
19971998
1999
2000
2001
2002
2003
2004
2005

Combined Ratio
200%

180%
Combined Ratio
Actual Price

160%
160%

PActual ,t

140%
140%

120%
120%

100%
100%

80%
80%

60%
60%

Price relative to Actual Premium

Combined Ratio Versus Pricing Methods

200%

CRt
180%

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Combined Ratio Versus Pricing Methods

200%

200%
Combined Ratio

CRt

180%

180%
Option Price Relative to Premium
160%

PActual ,t
140%

140%

Qt
120%

120%

100%

100%

80%

80%

60%

60%

1970
1971
1972
1973
1974
1975
1976
1977
1978
19791980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005

Combined Ratio

160%

Price relative to Actual Premium

Actual Price

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Combined Ratio Versus Pricing Methods

200%

200%
Combined Ratio

CRtOption Price Relative to Premium
Option Price adjusted for Price of Risk
Actual Price

180%

QtVIX t160%

PActual ,t
140%

140%

Qt
120%

120%

100%

100%

80%

80%

60%

60%

1970
1971
1972
1973
197419751976
1977
1978
1979
1980
19811982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005

Combined Ratio

160%

Price relative to Actual Premium

180%

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VIX: The CBOE Volatility Index
 The CBOE Volatility Index - more commonly referred to as "VIX" - is an
up-to-the-minute market estimate of expected volatility that iscalculated by using real-time S&P 500 (SPX) index option bid/askquotes.
 VIX uses nearby and second nearby options with at least 8 days left toexpiration and then weights them to yield a constant, 30-day measureof the expected volatility of the S&P 500 Index.
 The underlying for options is an "Increased-Value" Volatility Index(VXB), which is calculated at 10 times the value of VIX. For example,
when the level of VIX is 12.81, VXB would be 128.10.
 The VIX began trading in 1986.
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These should be “good times”, since consumers are willing topay relatively more. Instead, the industry prices too low allowing
consumers to arbitrage, which is a direct drain on industry
capital
VIX Prices
40.00%

VIX

35.00%

30.00%

25.00%

20.00%

15.00%

1/3/2005

1/3/2004

1/3/2003

1/3/2002

1/3/2001

1/3/2000

1/3/1999

1/3/1998

1/3/1997

1/3/1996

1/3/1995

1/3/1994

1/3/1993

1/3/1992

1/3/1991

1/3/1990

1/3/1989

1/3/1988

1/3/1987

10.00%
1/3/1986

Price (Implied Volatility)

10 per. Mov. Avg. (VIX)

Date
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Why Can This Exist?
Classic incomplete market (bilateral, where buyersand sellers find each other and make directcontact)Prof. Vernon Smith’s behavioral economics study: People, when
given a choice, choose to stay in an incomplete market. This is“survival economics” rather than “maximization of profits”
Impetus needed to move from incomplete to complete. Players preferincomplete markets because they feel better.

No recognition that insurance can be arbitraged


Reinsurance coverage can be replicated by selling insurance andbuying bonds. This enables arbitrage, but on a limited scale sincetransaction costs are high (insurance license, staff etc)
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Pricing Framework
We make two important changes to the status quo
 We price under the Q measure We create the implied volatility of an insurance
contract and use this in our pricing

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Pricing Framework
Traditional insurance pricing (status quo)

[P=Ε eP− rt( Lt − K ) ]+1= ( p ⋅ C u + (1 − p ) ⋅ C d )ROption pricing
1+Q− rtC = Ε [e (Lt − K ) ] = (q ⋅ C u + (1 − q ) ⋅ C d )R

R−dR − e −σ t ∆t
R − e −σλt ∆t== *q=*u − d e σ t ∆t − e −σ t ∆t e σλt ∆t − e −σλt ∆t*Nλ =*tvixtvixvix =∑ vixt =1t

N
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What If Industry Pricing Reflected
This
Combined Ratio Under Different Pricing Assumptions
250%
Actual CR

ROE of –12% (31% St. Dev.)

Adj CR

Combined Ratio

200%

Adj CR incl PoR

150%

ROE of 5% (20% St. Dev.)
100%

50%

Expected Value
Standard Error

Actual CR
133.5%
31.1%

Adj CR
119.9%
25.4%

Adj CR incl
PoR
101.4%
20.2%

ROE of 23% (15% St. Dev.)

0%
1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

Underwriting Year

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Conclusion
We have showed the the VIX index can help explain thegeneral underwriting cycle as well as a segment specific cycle
Fascinating area – dire need for more researchMore publicly traded insurance linked securities will makearbitrage easier forcing better pricing and supporting theory
Thanks

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A Further Examination of Insurance
PricingandUnderwriting Cycles
AFIR Conference, September 2005, Zurich, Switzerland
Chris K. Madsen, GE Insurance Solutions, Copenhagen, Denmark
Svend Haastrup, GE Insurance Solutions, Copenhagen, Denmark
Hal W. Pedersen, University of Manitoba, Winnipeg, Canada

Appendix

A Historical Perspective
 Regulated market place (~1983 and prior)

Average combined ratio of 109.5% (excl. 1970)
Standard deviation of combined ratio of 20.0%
Price are set centrally and jointly. Reinsurers are very profitable and there is little competition
on price. Pricing did not matter, as prices were generally high.

 Market place in transition (~1984 through ~1993)



The market begins to change in some locations allowing price to fluctuate. Pricing begins tomatter and more actuaries are hired.
Average combined ratio of 163.1%
Standard deviation of combined ratio of 42.2%

 Deregulated market place (~1994 and later)Market prices fall where they may. Actuaries price deals extensively but no coherent pricing
theory exists and there is always uncertainty about the right price leaving room for parameters
to be modified and pricing biased.
Average combined ratio of 120.4%
Standard deviation of combined ratio of 27.5%

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