An Examination of Insurance Pricing and
Underwriting Cycles
AFIR Conference, September 2003, Maastricht, NL
Chris K. Madsen,
GE Frankona Re, Copenhagen, Denmark
Hal W. Pedersen,
University of Manitoba, Winnipeg, Canada

Overview
• Introduction
• Risky Cash Flows
• Underwriting Cycle
• Pricing
• Areas for Further Study
• Concluding Comments
4/11/2020 - 2 -

Introduction
• Hypothesis
– One “Price of Risk” permeates all financial transactions
• Definitions
– Price of Risk
– Price for One “Standard Unit of Risk”
– “Standard Unit of Risk”
– volatility of measure relative to the expected measure, that is, the Coefficient of Variation
 
StdDev
Mean
4/11/2020 - 3 -

Risky Cash Flows
• When “Price of Risk” Increases
– prices of equities fall
– prices of corporate bonds fall (spreads widen)
– prices of options rise
– prices of insurance rise
• When “Price of Risk” Decreases
– prices of equities rise
– prices of corporate bonds rise
– prices of options fall
– prices of insurance fall
4/11/2020 - 4 -

Risky Cash Flows
• Present Value of Cash Flows
P
 t



1

1

1 r t

 
[ CF t
*
]
• Basic discounting cannot account for this behavior.
– Discount rate or cost of capital reflect the cost of money, not the cost of risk
– The discount rate used for asset pricing is (most likely) not appropriate for insurance pricing
– Present value calculations do not account for the inherent optionality of insurance
4/11/2020 - 5 -

Risky Cash Flows
• There are two ways to be a buyer of risk
– Pay Certain Amount Today => Get Uncertain Cash
Flows in the Future
IRR=10%
50,000
40,000
30,000
10,000
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
-
(10,000)
(20,000)
(30,000)
(40,000)
(50,000)
Limited Downside
Interpretation: Getting 10% return (Other investments must yield more than 10% to be more worthwhile)
– Get Certain Amount Today => Pay Uncertain Cash
Flows in the Future
20,000
(20,000)
(30,000)
(40,000)
(50,000)
50,000
40,000
30,000
20,000
10,000
Limited Upside
-
(10,000)
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
IRR=10%
Interpretation: Paying 10% return (Must get at least
10% return on premium to break even)
4/11/2020 - 6 -

Risky Cash Flows
• What is Insurance?
– Financial transaction on losses
– Selling (naked) call options on losses
• Since insurance is an option, it would make sense to use option theory to price
4/11/2020 - 7 -

Risky Cash Flows
– Market Price of Risk s t

  t
 
 t
 s t
S

S
& P
& P
500
500
– We can use the S&P 500 and implied volatility to find
 t
 s t
S

S
&
&
P
P
500
500
 s t
S & P 500 s S & P 500

S & P 500
4/11/2020 - 8 -

Risky Cash Flows
• Adding the “Price of Risk” the Black
Scholes option pricing model
C

S

N
 
1

K
 e
 r
 t 
N
 
2 d
1
 ln
S
K


 r
 s
2
2


  t s
  t
 ln
S
K


 r

(
 t
 
)
2
2


  t
 t
    t d
2
 d
1
  t
    t

Loss 

 s
L t
L
0





L t
L

0
L
0




L t
S
L t

L
0


1
 r

S

L t
L
0

L
0
 s
L t
L
0 r
Breaking volatility in two:
•
Price of Risk (fluctuates constantly)
• Coefficient of variation on loss returns (constant for a given loss scenario)
4/11/2020 - 9 -

Risky Cash Flows
• Incorporating the “Price of Risk” into utility theory
– Gerber and Pafumi (using exponential utility)
P
 
Loss
 a
2
 
2
Loss
– Including “Price of Risk”
P t
 
Loss
 a
  t
 
2
Loss
2
– Price of insurance in a portfolio
P
1



Loss
1

  t
 a
  t



2
– Total portfolio premium
2
Loss
1

COV
Loss
1
, OtherLosse s


 t
P
Company



N  i

1

Loss i


  t
 a
  t
2


 i
N N 

1 j

1
COV
Loss i
, Loss j


  t

Underwriting Cycle
• A.M.Best (A.M.Best Report, February,
1999)
– “
A.M. Best believes the property/casualty underwriting cycle has been replaced by a permanent ‘down market’”
•
Irving Fisher (Yale University, September,
1929)
– “Stocks have been replaced by what looks like a permanently high plateau”

Underwriting Cycle
• Price of Risk
Price of Risk
1.90%
1.70%
1.50%
1.30%
1.10%
0.90%
0.70%
0.50%
1/2
/19
86
1/2
/19
87
1/2
/19
88
1/2
/19
89
1/2
/19
90
1/2
/19
91
1/2
/19
92
1/2
/19
93
1/2
/19
94
1/2
/19
95
1/2
/19
96
1/2
/19
97
1/2
/19
98
1/2
/19
99
1/2
/20
00
1/2
/20
01
1/2
/20
02
Date
Price of Risk
One Year Moving Average
Long Term Average

Underwriting Cycle
• Price of Risk
Price of Risk
1.90%
1.70%
1.50%
1.30%
1.10%
0.90%
0.70%
0.50%
1/2/8
6
1/2/8
7
1/2/8
8
1/2/8
9
1/2/9
0
1/2/9
1
1/2/9
2
1/2/9
3
1/2/9
4
1/2/9
5
1/2/9
6
1/2/9
7
1/2/9
8
1/2/9
9
1/2/0
0
1/2/0
1
1/2/0
2
1/2/0
3
Date
Price of Risk
One Year Moving Average
Long Term Average

Underwriting Cycle
• “Fair” Price of Insurance
"Fair" Price of Insurance
100,000
95,000
90,000
85,000
80,000
75,000
70,000
1/3
/19
86
1/3
/19
87
1/3
/19
88
1/3
/19
89
1/3
/19
90
1/3
/19
91
1/3
/19
92
1/3
/19
93
1/3
/19
94
1/3
/19
95
1/3
/19
96
1/3
/19
97
1/3
/19
98
1/3
/19
99
1/3
/20
00
1/3
/20
01
Date
Market Price
One Year Moving Average
Long-Term Average

Underwriting Cycle
• “Fair” Price of Insurance
"Fair" Price of Insurance
100,000
95,000
90,000
85,000
80,000
75,000
70,000
1/3/8
6
1/3/8
7
1/3/8
8
1/3/8
9
1/3/9
0
1/3/9
1
1/3/9
2
1/3/9
3
1/3/9
4
1/3/9
5
1/3/9
6
1/3/9
7
1/3/9
8
1/3/9
9
1/3/0
0
1/3/0
1
1/3/0
2
1/3/0
3
Date
Market Price
One Year Moving Average
Long-Term Average

Underwriting Cycle
• Historical Combined Ratios
Calendar Year Combined Ratios
120.0
115.0
110.0
105.0
High Equity and Bond
Returns
100.0
Actual
95.0
90.0
Low inflation, Inflation soars low interest
85.0
rates
80.0
194
9
195
1
195
3
195
5
195
7
195
9
196
1
196
3
196
5
196
7
196
9
197
1
197
3
197
5
197
7
197
9
198
1
198
3
198
5
198
7
198
9
199
1
199
3
199
5
199
7
199
9
200
1
Year
Source: A.M.Best

Underwriting Cycle
• Combined Ratio Model
CR t

.
2027

.
08836

CR t

1

.
3979

CR t

2

.
4258

C t

1
• Note: C is a significant indicator with a pvalue of 1.3%, but it has the “wrong” sign.
When option prices go up, so does the combined ratio!

Pricing
• Traditional Insurance Pricing
TP t

1 r t
 r t


1
1
 r t

N
• Insurance Option Pricing
C t
• Creating two indexes
TP t
*

TP
TP
0 t , OP t
*

C
C
0 t

Pricing
• Systematic Over and Under Pricing
Systematic Over and Under Pricing
118.0
1.20
In Sample
116.0
1.15
114.0
1.10
112.0
1.05
110.0
Bear
108.0
Bear
Bull
1.00
Combined Ratio
Standard Pricing
Option Pricing
0.95
106.0
0.90
104.0
0.85
102.0
100.0
1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Year
0.80
Source: A.M.Best

Pricing
• Systematic Over and Under Pricing
Systematic Over and Under Pricing
118.0
1.20
116.0
1.15
Out of
Sample
114.0
1.10
112.0
1.05
110.0
108.0
1.00
Combined Ratio
Standard Pricing
Option Pricing
0.95
106.0
0.90
104.0
0.85
102.0
100.0
1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Year
0.80
Source: A.M.Best

Pricing
• Systematic Over and Under Pricing
Systematic Over and Under Pricing
118.0
1.20
Forecast
116.0
1.15
114.0
1.10
112.0
1.05
110.0
108.0
1.00
Combined Ratio
Standard Pricing
Option Pricing
0.95
106.0
0.90
104.0
0.85
102.0
100.0
0.80
1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004
Year
Source: A.M.Best

Areas for Further Study
• Framework
Expected
Earnings
Market
Price of
Risk
Interest
Rates
Equity
Prices
Bond
Prices
Option
Prices
Insurance
Prices
– For example Equity Price = f(interest rates, earnings growth, price of risk)

Areas for Further Study
• Equity Index Prices (S&P 500)
– Theoretical level (no earnings growth): risk-adjusted perpetuity
P t

TrailingEa rnings
RFR
 t

1
2

TrailingDi
   t
 
2
– With earnings growth vidends t
P t

TrailingEa rnings t


 t
TrailingDi
  
2 
EG t vidends t
RFR
 1
2

– Solve for Earnings Growth to get “Implied Earnings
Growth”

Framework/ Areas for Further Study
• Difference between actual and implied earnings growth has 28% correlation with weekly equity returns since 1986
Implied S&P 500 Earnings Growth
10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
1/3/86
-2.0%
1/3/87 1/3/88 1/3/89 1/3/90 1/3/91 1/3/92 1/3/93 1/3/94 1/3/95 1/3/96 1/3/97 1/3/98 1/3/99 1/3/00 1/3/01 1/3/02 1/3/03
-4.0%
Date

Concluding Comments
• Examination of “Price of Risk” to bridge asset pricing and insurance pricing
• Development of insurance pricing option model
• Review of underwriting cycle based on traditional pricing indexes and option pricing indexes
• We have to get a handle on this!

Thanks!