A Further Examination of Insurance Pricing
And
Underwriting Cycles

July, 2005

Madsen, Chris K.
GE Insurance SolutionsØstbanegade 135, 2100 Copenhagen Ø, Denmark
Phone: +45 33 97 95 02, Fax: +45 33 97 94 41
E-Mail: chris.madsen@ge.com
Haastrup, Svend
GE Insurance SolutionsØstbanegade 135, 2100 Copenhagen Ø, Denmark
Phone: +45 33 18 95 75, Fax: +45 33 97 94 41
E-Mail: svend.haastrup@ge.com
Pedersen, Hal W.
Warren Centre for Actuarial Studies and Research
Faculty of Management
University of Manitoba
Winnipeg, Manitoba R3X 1J8, Canada
Phone: +1 204 474 9529
E-Mail: pedersn@ms.umanitoba.ca

Abstract
This paper is a follow-up to the paper “An Examination of Insurance Pricing andUnderwriting Cycles”. In that paper, Madsen and Pedersen laid out a theory of a "priceof risk" and suggested that this price links all risky financial transactions. In particular,
the paper detailed insurance pricing in terms of this price of risk and supported this withan analysis of the performance of the property and casualty insurance industry for the
past fifty years.
In this paper, we follow up on some of the questions raised as feedback to the original
paper. In addition, we offer some more specific examples based on actual treaty data. We
also extend the use of this to a broader framework and put forth a plausible explanation
for why our industry has been relatively slow to develop a complete and coherent pricing
theory. We discuss that the lack of a coherent pricing theory may in fact be amplifying
the underwriting cycle due to consistent mis-pricing.
We recognize that some of our hypotheses are controversial, but we believe a healthy
debate – supported with actual data and underlying pricing theory - is necessary to move
our industry and its theory of price forward.
Key Words
Price of risk, risk-adjusted value of insurance, insurance pricing, option pricing,underwriting cycle, property and casualty insurance, general insurance, market cycle,
implied volatility

2

Table of Contents
Introduction......................................................................................................................... 4
Background and Review ..................................................................................................... 4
What is Insurance?.............................................................................................................. 6
Pricing Framework............................................................................................................ 13
Two Real-Case Examples ................................................................................................. 14
The Industry Combined Ratio and General Profitability.............................................. 14
Reviewing Actual Data from a Low-volatility Line of Business ................................. 16
Hypothesis............................................................................................................. 20
Test........................................................................................................................ 20
Why? ................................................................................................................................. 25
Summary and Conclusion ................................................................................................. 27
Appendix........................................................................................................................... 28
The Expected Payoff of a Reinsurance Contract Given an Underlying Log-Normal
Distribution ................................................................................................................... 28
First Moment..................................................................................................... 28
Second Moment ................................................................................................ 28
References......................................................................................................................... 29

3

Introduction
Insurance needs a coherent pricing theory. Current actuarial pricing does not account forthe optionality offered by insurance contracts. It is loss estimation rather than pricing andwill only generate acceptable results under elusive conditions. Promoting a common
understanding of the difference between these two terms is essential in moving ourindustry forward.
The views expressed in this paper are those of the authors. We know they arecontroversial, but we believe that a healthy debate is necessary to accelerate thedevelopment of a coherent pricing theory in insurance.
This paper is a follow-up to the paper by Madsen and Pedersen, 2003. In this paper, wefollow up with some more concrete examples and explore further some of the feedback
we have received in the past two years.
Some of the questions that continually haunt the property casualty industry include thefollowing:






Why are property casualty insurers’ returns on equity continually so dismal?
Why do property casualty insurers keep entertaining so many perspectives on risk
management and pricing?
Why do property casualty insurers have so many actuaries?
Why do many property casualty insurers think their business is more art thanscience?
Why is the underwriting cycle such a mystery and why is it so amplified?

We suggest that the answer to these questions is, at least in part, the lack of a coherent
pricing theory.

Background and Review
As suggested in “An Examination of Insurance Pricing”, selling insurance is like sellingcall options. We suggested then that the underwriting cycle was a direct result of insurers
lack of pricing for the inherent optionality in insurance portfolios.
Insurance is ripe with options. Consider the following:
Simple Example
An insurer, A Insurance, buys reinsurance from a reinsurer, B Re. It is a stop-loss
treaty covering property claims arising during the next year. At the moment ofunderwriting, B Re does not know the amount of premium that A Insurance will4

underwrite nor is there anything other than a general indication of the price that A
Insurance will charge. Thus, Be Re is granting A Insurance several powerful
options:

The very nature of insurance is for the insurer to call (assume) the losses ifthey exceed a given strike level (attachment point). This is the exact
definition of an option.

But the optionality does not end there.


A Insurance can change their pricing thus producing more claims that may
flow through the cover. In other words, knowing that a given cover is inplace may change the behavior of A Insurance, and – as a direct result –the behavior of the reinsured losses.
A Insurance can also, if some calamity pierces the coverage during theyear, write new policies below cost, as A Insurance no longer has any costassociated with the policies.

These are powerful and valuable options related to volume and price, and B Re
ignores the value of these at its own peril.
The ladder two points are completely disregarded by B Re in its pricing, and we
can argue about how well B Re is handling the first point.
In the investment world, this disregard simply would not be possible. If A
Insurance were hedging currency risk rather than insurance risk, A Insurance
would be forced to estimate how much currency it would have to hedge at the endof the year. If A Insurance wanted to be completely hedged, it would have tocontinuously add or subtract from its coverage. In other words, A Insurance canhedge, but retains the risk that it has more or less to hedge than it predicted. In theinsurance world, B Re takes on this risk, but does not charge for it.
In the simple example above, the disregard for optionality means that B Re rarely charges
enough. One might conclude that this drains profitability somewhat. In fact, it is a bit
worse than this. When B Re charges too little, B Re sells more because pricing is lower
than the true market equilibrium at the time and B Re is allowing A Insurance toarbitrage. This causes severe adverse losses over time, so B Re raises prices to another
unnatural state – one that is too high. Now, A Insurance does not renew and leaves toplace its business with an (often new) competitor. B Re’s volume drops and B Re needs
two good years with the new price for every bad one to break even. But, as people in theindustry will attest to, one is fortunate to get one good year for each bad. Thus, the poor
results will outweigh the good depressing long-term returns on equity.
In the following sections, we will explore the first of the three bullets mentioned above.
The other two are related and no less important, but we believe these should be addressed
directly through contract terms and conditions.

5What is Insurance?
Insurance is a financial transaction on losses. At the time a policy is written, the insurer
has a certain estimate of what the expected loss is (ultimate estimate or reserve). Overtime, this estimate develops as more information becomes available. Thus, the expected
loss (ultimate estimate or reserve) develops over time much like the price of a stock
develops over time.
Buying insurance is equivalent to buying a call option on losses i . Conversely, sellinginsurance is the equivalent of selling a call option on losses. The insured has an option onlosses that pays only if losses exceed a certain specified threshold. The insurer isgenerally short naked calls (naked because the insurer does not have an underlying
position that offsets the cash flow of the call).
The value of the call option is the present value of the payoff using the risk-free rate andrisk-neutral probabilities as shown in Equation 1. For multi-period examples, the latticein Figure 1 can be extended, and the calculation in Equation 1 is then completed at every
node ii .
Equation 1: Price of Call Option

C=

1[qCu + (1 − q )C d ],
R

where

R = 1 + r , where r is the risk-free rate,
q is the risk-neutral probability and is calculated as q =C u = max(u ⋅ L − K ,0) ,

R−d
,
u−d

C d = max(d ⋅ L − K ,0) ,u is the “up-move” factor,
d is the “down-move” factor, andK is the attachment point1, the ultimate lattice produces a log-normalddistribution and the Black-Scholes result iii . The volatility of this distribution is describedln(u )as σ =. As such, we can define the risk-neutral probability directly in terms ofΔtvolatility, which will be more familiar to people used to work with Black-Scholes, and topeople more familiar with fitting loss distributions. As such, we adopt the following:Note, that in the case, where u =6

u = e σ Δtd = e −σ Δt
As a result, our price becomes a function of volatility and the rate of interest – notsurprising given the fact that this produces the Black-Scholes result and that these are theBlack-Scholes parameters.
Viewed graphically, we can construct a basic binomial lattice similar to what we do withstock options.
Figure 1: Insurance
Lu= uLL
Ld= dL

Cu=Max(uL-K,0)C
Cd=Max(dL-K,0)

In Figure 1, L is the insurer’s estimate of the ultimate underlying loss and C is the priceof the call option (insurance) of a stop-loss treaty with attachment point K. The ultimate
underlying loss estimate develops over time much like the price of a stock develops as
more information becomes available. In fact, we can make analogies to stock options asin the following table:

Table 1: Comparing Equity Options with Loss Options
Stock/Equity Options
Parameter
Interpretation
Risk-free rate ofrinterestσVolatility of equity
returns (note this isequivalent to thevolatility of equity

Loss Options (Insurance)
Parameter
Interpretation
Risk-free rate ofrinterestσVolatility ofultimate loss
relative to initial
estimate7

prices divided by
the beginning equity
price)
Equity price at time0

S0

Equity price at timetStrike price

St
K

L0
Lt
K

Discounted ultimate
loss estimate at time0
Ultimate loss
estimate at time t
Attachment point

We suggest that basic options theory can be used favorably to price insurance policies. In
fact, the fact that insurers do not do this explains much of what we have observed in the
re-insurance market since they were deregulated.
Consider the following very simple Example 1. In Example 1, we are looking at a line ofbusiness with fairly low overall volatility of 9.5% iv . This is the standard deviation ofultimate loss estimate relative to the initial estimate and represents the uncertainty weassociate with this estimate. Note, that the option price produces a price that is
substantially higher than the insurance price. The latter is generally based on the
discounted expected payoff of the reinsurance contract.
Example 1: Option Pricing versus Traditional Insurance Pricing

Option pricing[] R1 (q ⋅ C + (1 − q) ⋅ C )C = Ε Q e − rt (Lt − K ) =+udTraditional insurance pricing
1+
P = Ε P e − rt (Lt − K ) = ( p ⋅ C u + (1 − p) ⋅ C d )R[]

The notation is described below. Note, that u is a function of lambda as well as sigma.
We will explain this later. For now, we introduce that there is a term lambda(t) that scales
the volatility. In the following example, we have set this term to 1, so it has noconsequence for the purpose of this example.

8Known at the time of pricing
Ultimate loss estimate at time 0
Cover Terms

1,000
Attachment point
xol

1,000
1,000

sigma (st. dev. of changes in ultimate loss estimate)r (risk-free rate of interest)
lambda(t)

9.5%
5.0%
1.00

Pricingu = exp(sigma(t)*sqrt(dt)) = exp(sigma*lambda(t)*sqrt(dt))
d = 1/u

1.100
0.909

q = (1+r-d)/(u-d)

73.8%
Value

Probability

Payoff
1,100 Lu

Loss value

L

1,000
909 LdOption value

Insurance Pricing

C

P

73.8%

100 Cu

26.2%

-

50.0%

100 Cu

50.0%

-

70
Cd

48
Cd

It is generally the case that the option price will exceed the traditional insurance price. To
demonstrate this and the magnitude of it, we show how the prices behave relative to eachother as we change some of the key underlying parameters. In the graphic below, we holdthe rate of interest constant at 5% and vary only the volatility. The graphic shows theratio of option pricing divided by insurance pricing as shown in Example 1.

9

Figure 2: Comparing the Price/Volatility Trade-offs Under Different Pricing
Methods
Option Pricing divided by Traditional Pricing
7

Option Price Relative to Traditional Price

6

5

4

3

2

1

10
.4
%
12
.2
%
14
.0
%
15
.7
%
17
.4
%
19
.1
%
20
.7
%
22
.3
%
23
.9
%
25
.5
%
27
.0
%
28
.5
%
30
.0
%
31
.5
%
32
.9
%
34
.4
%
35
.8
%
37
.2
%
38
.5
%
39
.9
%

8.
6%

6.
8%

4.
9%

3.
0%

1.
0%

0

Standard Deviation

It is important to note, that this result comes from the fact that ultimate losses tend todevelop with factors u and d=1/u (similar to stock prices). If the underlying losses
developed exactly as expected at the time of pricing, we would observe factors u andd=2(1+r)-u v , and in this case the risk-neutral probabilities would equal the realprobabilities and thus, the option price would equal the discounted expected payoff. But,
the latter is not what we observe, and this simple example suggests that in general,reinsurers tend to underprice low-medium volatility lines and price high volatility lines
more fairly. This is consistent with available market data, where the cat lines, thoughvolatile, are generally fairly profitable, whereas lower volatility lines are more marginal.
In fact, return on risk-adjusted capital for low-volatility lines has historically been
substantially below the level one would expect based on the underlying risk, whereas it
has been much more attractive on catastrophe insurance, which is highly volatile.
In Figure 3, we have varied the interest rate and picked a level of volatility consistent
with that of an average reinsurer with a traditional mix of business.

10Figure 3: Comparing Price/Interest Rate Trade-offs Under Different Pricing
Methods
Option Pricing divided by Traditional Pricing
3.5

Option Price Relative to Traditional Price

3.0

2.5

2.0

1.5

1.0

0.5

0.0
0.0% 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% 7.0% 8.0% 9.0% 10.0% 11.0% 12.0% 13.0% 14.0% 15.0% 16.0% 17.0% 18.0% 19.0% 20.0%
Interest Rate

This suggests that the lower the rate of interest the closer the option price is to thediscounted expected price. As such, we should not be surprised that recent lower rates ofinterest are coinciding with better reinsurer results.
Simple Example (Continued)

B Re’s pricing approach (discounting expected payoffs) will work as long as B
Re’s actuaries have correctly assessed the final distribution. Even small
fluctuations from the true distribution can cost B Re its profit. Consider thefollowing case with an interest rate of 5%:
S

Su

1.10

Sd

0.91

0.9567Here, the underlying insurance (ground-up loss) can develop over time to two
scenarios: The upper case and the lower case. A Insurance’s actuaries havecorrectly assessed this and discounted the losses, and B Re is aware of this.
Now, Be Re’s actuaries go to work. They study the payout of an attachment pointof 1.00 and discount the expected payout associated payout associated with this.

11This is referred to as the price under the “P” measure, as they use realprobabilities.
Under P Measure
C

Cu

0.10

Cd

-

0.0476

Thus, B Re’s actuaries arrive at a price of 0.0476, which is correct in this case. It
is identical to the risk-neutral “Q” measure, since the final distribution wascorrectly assessed. Stated differently, the d=2(1+r)-u, which we mentioned earlier.
Under Q Measure
C

Cu

0.10

Cd

-

0.0476The correct price will always be the one under the Q measure. Otherwise, A
Insurance will have an arbitrage opportunity. If B Re’s price is too low, A
Insurance will sell more insurance, buy a risk-free asset and reinsurance and havean arbitrage opportunity.
Sell
Net Cash
Insurance Buy Bond Buy Call
Outlay0.5011(0.4535) (0.0476)-

PayoffudPay
Get Call
Insurance Sell Bond Payoff(0.5762)0.47620.1000(0.4762)0.4762--A Insurance can replicate the reinsurance payoff by selling .5011 of insurance andbuying .4535 of the risk-free asset and spending .0476 on reinsurance. Thisrequires no cash outlay and A Insurance is now completely hedged, as we can see.
The net payoff is zero in both the upper and lower case.
But B Re is under pressure to write business and while B Re’s actuaries think theyknow the distribution, they cannot be completely sure, so press the price. This canbe illustrated a couple of ways (all of which lead to the same point). We will saythat B Re assumes the total loss will be slightly better than estimated (a little
under 5% to be exact). Now, the Q and P measure prices no longer match and A
Insurance has an arbitrage opportunity.
B Re assumes that the underlying price for A Insurance is given, but that theultimate will develop just a little more favorably than originally estimated.
S

Su

1.05

Sd

0.87

Net
Payoff-0.9567and now

12Under P Measure
C

Cu

0.05

Cd

-

0.0249

A Insurance happily accepts the reinsurance contract at 0.0249 and writes more
direct insurance effectively arbitraging:
Under P Measure
Sell
Net Cash
Insurance Buy Bond Buy Call
Outlay0.5011(0.4535) (0.0249)
0.0227

PayoffudPay
Get Call
Insurance Sell Bond Payoff(0.5762)0.47620.1000(0.4762)0.4762--A Insurance now generates .0227 of risk-free profit per .5011 of direct insurance
A Insurance writes. A Insurance is completely hedged as long as B Re wants toaccept the business.
In reality, A Insurance does not need to worry. B Re has already committed andmade no requirements as to underlying volume. As such, A Insurance canarbitrage as much as they want and B Re is exposed.
The subtle point made here that the consequences of under-pricing can be as dire in theinsurance industry as they are in the financial markets. In other words, pricing below “fairvalue” is allowing arbitrage and since there are no real contractual limits to volume, thearbitrage can be exploited. In the end, this depresses profits and increases the volatility of
the underwriting cycle. We will discuss this further, but first take a look at our suggested
pricing framework, which we will then apply to two real-case examples.

Pricing Framework
The price of a call option is given by the following:[] R1 (q ⋅ C + (1 − q) ⋅ C ) , whereC = Ε Q e − rt (Lt − K ) =
+

u

Net
Payoff-d

R represents the risk-free rate plus one,q the risk-neutral probability,
Cu the call option payoff in the “up” case, andCd the call option payoff in the “down” case.
As we saw in the simple example above, this must necessarily be so. Otherwise, there isan opportunity for arbitrage.
We apply this to the pricing of an excess of loss contract, but make one additionalchange. Namely, we propose that the value of a cash flow is the same whether the source13

is insurance, equities or some other financial asset. In other words, we recognize that
there is a single price of risk that permeates all financial transactions. This price of risk
can be described by the implied option volatility and we have observable market prices
for this through the Chicago Board Options Exchange (CBOE) “Vix”, which trade at the
average implied volatility of several S&P 500 options. When the Vix are high, the priceof risk is high as the markets perceive greater risk. We submit that this translates totransactions beyond the equity markets vi .
R−d
R − e −σ t Δt
R − e −σλt Δt, whereq== σ Δt=**u−d e t− e −σ t Δt e σλt Δt − e −σλt Δt
*

N

λ =
*
t

vix t

∑ vix

, vix = t =1

t

,
N
vix
σ is the long-term standard deviation of the underlying ultimate loss estimate divided by
the initial loss estimate,σ t is the standard deviation at time t reflecting the market price of risk (estimatedimplied volatility) of the underlying ultimate loss estimate divided by the initial loss
estimate,
Δt is the time increment during which the option is in effect, andR = 1 + r , where r is the risk-free rate.
One can rewrite the Black-Scholes formula with this formulation. Black-Scholes will stayintact as scaling the long-term volatility to the volatility of today is equivalent to using
today’s volatility thus producing today’s prices.
Thus, we have made two important changes to insurance pricing:

We price under the Q measure
We create the implied volatility of an insurance contract and use this in
our pricing

Two Real-Case Examples
The Industry Combined Ratio and General Profitability

In “An Examination of Insurance Pricing”, we demonstrated that a “price of risk”
observed through market prices can be adapted to insurance prices. In other words, there
is a single price of risk in the financial markets (including insurance) that permeates all
financial transactions. In an incomplete market such as insurance, prices may differ fromthis true “arbitrage-free” or risk-neutral price, but they do so at the peril of market

14

participants. We will review later why such a discrepancy can exist in an incomplete
market for extended periods of time.
In “An Examination of Insurance Pricing”, we used the “Vix vii ” option prices from the
Chicago Board Options Exchange (CBOE) to get at the price of risk.
We used this information to show that

The underwriting cycle moves exactly the opposite of what one would
expect based on option pricing. This is because actuarial pricing
techniques use basic expected discounting rather than option pricing. In
option pricing, one raises the price when interest rates increase. Most
insurers do exactly the opposite failing to recognize that a higher rate ofinterest increases the probability last a layer will be pierced.
The turning points in the underwriting cycle can be identified by creating
an option price index and a traditional price index. Whenever the two
methods cross, the cycle turns. Stated differently, sometimes insurers
charge too much and sometimes too little. Sometimes they are just right,
but this occurs by sheer chance.

We have updated the key graphic below.

Figure 4: Systematic Over- and Under-pricing
Systematic Over and Under Pricing
1.20

118.0

116.0
1.15
114.0
1.10

1.05
110.0

1.00

108.0

Pricing Index

Combined Ratio (%)

112.0

Combined Ratio
Standard Pricing
Option Pricing

106.0
0.95
104.0
0.90
102.0
0.85
100.0

98.0

0.80
1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006
Year

15

As can be seen, the graphic looks one year ahead and suggests that the property andcasualty insurance industry will have at least one more “good” year in terms of financial
performance. This is because traditional pricing (red triangles) suggests a higher pricethan the option price (green squares) thus improving financial results.
This graphic is interesting and thought-provoking, but the generality of it makes it
difficult to demonstrate the value of our proposition without a doubt:

It is based on insurer’s reported results with the inherent lag in reporting andpossible reserving bias.
The indexes do not propose a correct price per se. Rather they are based on a
general model of the industry.

To address these and other questions, we reviewed actual data to develop a specificexample of how an insurer would use this in practice.
Reviewing Actual Data from a Low-volatility Line of BusinessThis example is from a real book of business, so the lags and reporting issues that one
encounters above are not present in this data to the same extent. In addition, the data is ahomogeneous portfolio, which should facilitate the explanation of results.
The line of business is excess of loss treaty business on a low to medium volatility book.
The combined ratio through time is shown below, where we show underwriting years on
the x-axis and the combined ratio on the y-axis.

Figure 5: Actual Segment Combined Ratio

16

Combined Ratio
200%

180%

Combined Ratio

160%

140%

120%

100%

80%

19
70
19
71
19
72
19
73
19
74
19
75
19
76
19
77
19
78
19
79
19
80
19
81
19
82
19
83
19
84
19
85
19
86
19
87
19
88
19
89
19
90
19
91
19
92
19
93
19
94
19
95
19
96
19
97
19
98
19
99
20
00
20
01
20
02
20
03
20
04
20
05

60%

Our first goal is to try to explain this behavior. We need to be sensitive to the fact that thereinsurance market in Europe was heavily regulated until about 1994. Deregulation had
been in progress for about ten years up to this point. As such, we can separate the above
into three segments:Regulated market place (~1983 and prior)o Average combined ratio of 109.5% (excluding 1970)
o Standard deviation of combined ratio of 20.0%
o Prices are set centrally and jointly. Reinsurers are very profitable and there
is little competition on price. At this time, pricing did not matter because
prices were unnaturally high.
Market place in transition (~1984 through ~1993)
o The market begins to change in some locations allowing price to fluctuate.
Pricing begins to matter and more actuaries are hired.
o Average combined ratio of 163.1%
o Standard deviation of combined ratio of 42.2%
Deregulated market place (~1994 and later)o 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 allowing parameters to be easily modified to get the price one
wants.
o Average combined ratio of 120.4%
o Standard deviation of combined ratio of 27.5%

17

Our focus will be on the latter two periods, as they are more indicative of the future.
To set an option price structure for this segment, we need to estimate ground-up losses.
Ideally, the ground-up distribution would be readily available, but in most cases, thecurrent environment does not offer one. To develop a ground-up loss distribution, we
define a few parameters that relate the ground-up loss distribution to the reinsurance
distribution (which is readily available). Based on these parameters, the ground-up
distribution will be entirely described.
a1 =Ε(GL )
Ε(RL )a2 =AP
Ε(GL )

Stdev(GL)f GroundUp =Ε(GL)StDev( RL)
Ε( RL)

Here,

GL
RL
AP

is the ground-up loss,
is the reinsurance loss, andis the attachment point.

In addition, we set
CRi ,0 =Ε(RLi )0

= 90% , where CR is the combined ratio, RL the reinsurance loss and RP
RPi
the reinsurance premium. The subscript i denotes the underwriting year.
Δt = 4.4 years , which is the duration of the payout pattern.

Note, that for simplicity we have included expenses in our ultimate loss estimate. This
simplifies notation and has no further implications.
The last two assumptions can be reviewed directly from the data. The first is the initial
combined ratio estimate. In other words, how was the business viewed at the time it wasunderwritten? This may be directly available if the reinsurance data is complete from thedate of inception. In our case, we have set it to 90% for all underwriting years consistent
with the initial indications, though they will vary from underwriting year to underwriting18

year. If the data is readily available by underwriting year, one can simply use it to get amore accurate result.
The second is the duration of the business based on the claims payment pattern. This canbe estimated directly from the payout pattern. This gives us an indication of how long it
will take to pay out this business on average. We expect to make adjustments to ourultimate loss estimate for the same period of time on average.
Given the mean and standard deviation of the ground-up log-normal distribution as wellas the layer that is reinsured, one can determine the reinsurance expected loss andstandard deviation (see Appendix). We can reverse this, so we find the ground-up meanand standard deviation given the reinsurance information.
For any underwriting year, i, we have

Ε(RLi ) = CR0 ⋅ RPi
Ε(GLi ) = a1 ⋅ Ε(RLi )APi = a 2 ⋅ Ε(GLi ) = a 2 ⋅ a1 ⋅ Ε(RLi )StDev (RL )StDev (GL )= f GroundUp ⋅Ε(GL )
Ε(RL )
Thus, we now have to estimate 3 parameters to fully describe the value of the option: a1 ,a 2 , and f GroundUp . These can be estimated in a variety of ways.
Any distributional assumption can be made at this point (unless the three parameters are
directly available from existing data in which case one can avoid the restrictions of
distributional assumptions), and the parameters found either through algebraic
manipulation or through simulation.
For our purposes, in order to estimate the parameters, we assume the ground-up loss is
log-normal. This is consistent with the binomial lattice framework and the log-normal iscommonly used in the insurance industry as an estimate of aggregate losses. Once thethree parameters are given in addition to the reinsurance data, our ground-up distribution
is entirely described.
Once the parameters, and through them the ground-up distribution, have been found,pricing the option is a straight-forward exercise as laid out in our previous examples. But,
it is important to recognize our previous work: The price of risk differs over time. As
such, for pricing purposes, there is not a constant volatility. It fluctuates and we can use
our price of risk concept to recognize this.
In our original paper, we defined a price of risk as volatility relative to return. The
required us to estimate a coefficient of variation for insurance, which includes the19estimation of a return (e.g. “drift” in reserve estimates) in insurance. There are two
problems with this:The results are very sensitive to the mean return
We know from options theory that the return does not matter

Accordingly, we will now modify our concept of the price of risk. It is important to note,that is leaves our original work intact albeit with different parameters. In other words, thegraph in the previous section (Figure 4) can be reproduced in the following framework.
To avoid confusion, we introduce our new price of risk as a “Price of Risk Factor” or λ* .
It is simply a scaling factor of volatility measured by the current value of vix relative to
its trailing historical average.

λ*t =

vix t
vix

,
N

∑ vix

where vix = t =1

tNWhatever volatility we get for the ground-up loss, we will rescale it based on our “Priceof Risk Factor”.

σ GroundUp,t = σ GroundUp ⋅ λ*t , where σ GroundUp is the actuarial estimate of volatility of theground-up loss.
To test our theory further, we find the parameters a1 , a 2 , and f GroundUp through aminimum sum of squared errors approach. Stated differently, since we are trying tounderstand the underwriting cycle, we simply let our history of combined ratios tell us
what reasonable assumptions for our three parameters would be. We assume theparameters are the same for each underwriting year. This makes sense since this is afairly homogeneous portfolio. Given complete data and an individual treaty, theparameters would be explicit by treaty by underwriting year.
Hypothesis
Traditional insurance pricing does not properly account for the optionality inherent ininsurance contracts. As such, insurance contracts can be arbitraged thus extending
losses when prices are low relative to the arbitrage-free price. Thus, we should be able toshow that the combined ratio tends to be extended when the arbitrage-free price is highrelative to the traditional (actual) price.
Test
2

⎛
C ⎞
Minimize ∑ ⎜⎜ CRi − i ⎟⎟ , wherePi ⎠i =1 ⎝
I20

We allow losses to be rescaled to get the best fit possible
The adjusted option price is the option price incorporating our Price of Risk
FactorHere,
CRi is the reinsurance combined ratio for underwriting year i,
Ci is the call option (reinsurance) price as laid out in our pricing framework (using
the ground-up log-normal distribution as described by parameters a1 , a 2 , andf GroundUp ), andPi is the actual reinsurance premium for underwriting year i.
In our case, we get the following results:

a1 = 9.5%
a 2 = 90.5%
f GroundUp = 9.6%
Reviewing these for reasonability, they do not seem unreasonable. One could argue thatthey do not even have to be reasonable per se, as they are akin to state variable pricing
factors or simply the parameter values that best describe the available data. As such, they
do not have to be observable.
The graph below summarizes the results.
Figure 6: Combined Ratio versus Pricing Methods

21Combined Ratio Versus Pricing Methods
200%

200%
Actual Combined Ratio
180%

160%

160%

140%

140%

120%

120%

100%

100%

80%

80%

60%

60%

Price relative to Actual Premium

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

19
70
19
71
19
72
19
73
19
74
19
75
19
76
19
77
19
78
19
79
19
80
19
81
19
82
19
83
19
84
19
85
19
86
19
87
19
88
19
89
19
90
19
91
19
92
19
93
19
94
19
95
19
96
19
97
19
98
19
99
20
00
20
01
20
02
20
03
20
04
20
05

Combined Ratio

180%

On the graph, we see the combined ratio as before in the columns. The orange (diamond)line is the actual price relative to premium, which naturally is 100% and the price is theactual premium. The blue (crosses) line is the unadjusted option price using a constantvolatility relative to premium. The purple (stars) line is the adjusted option price using
the Price of Risk Factor relative to premium.
As is visually obvious, the adjusted options price has a correlation coefficient of 79.2%
with the actual combined ratio. The main deviations are 1989 and 1990 as well as 2002
and later.
The “Burns’ Day Storm” hit in late January, 1990, causing severe losses in northern
Europe. It was one of the strongest storms on record. Such deviation from our pricewould be expected. This is why insurance exists and it is the eventuality we price for.
As for 2002 and later, we suggest that the ultimate here is not done developing. In otherwords, we would not be surprised if losses here continue to accumulate and get closer toour price. In fact, the drift term of reserves would suggest this to be true.
There are many interesting observations that extend this result. For example, one cancalculate the associated return on equity assuming a capital factor of 15% of reserves perannum.

22

T

ROE = T

∑ PVCF
t =1

t∑ PVCapitalt =1

iThat is, we sum the present value of results for each underwriting year at the time ofunderwriting and divide by the present value of the capital requirement. This gives anindication of the return on equity. Based on this, we develop the following graphic.
Figure 7: Risk/Reward Trade-offs of Pricing Alternatives

Risk/Reward Trade-Offs of Price Alternatives
30.0%

25.0%

20.0%

Expected ROE

15.0%

10.0%

ROE OP Adj
ROE OPROE TP

5.0%

0.0%
0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

35.0%

-5.0%

-10.0%-15.0%
Standard Deviation of ROE

Which business would you prefer to be in? Industry members will recognize that thesituation with the yellow (triangle) marker is more commonly observed than the othertwo. This is the actual case. The “OP” (square) marker is the option price but not yet
adjusted for the price of risk. The “OP Adj” (diamond) marker is based on the option
price of our framework.
Taking a different view on this, we can restate what the market would have looked likeunder our options prices. It is very interesting that the following dynamics exist.

23

Table 2: Combined Ratios and Volatility Under Different Pricing Methods

Expected Value
Standard Error

Actual CR
133.5%
31.1%

Adj CR
119.9%
25.4%

Adj CR incl
PoR
101.4%
20.2%

Here, we have looked at the combined ratio with our implied volatility adjustment as wellas with it. “PoR” denotes price of risk..
We suggest that a third of the volatility underwriting cycle be eliminated if option pricing
was adopted. In fact, it might be more significant than this. This is based on the simple
notion that consumers intuitively know when they are getting a good deal. When prices
are below the fair price, they buy more insurance or press terms and conditions. In otherwords, reinsurers implicitly sell more insurance when they do so at a loss and this iscostly when the attachment point is fixed. Conversely, insurers sell less when they sell at
a premium. The extension of this is that volatility increases. To the extent that one would
charge the fair price at all times, one would expect to remove at least part of thisphenomenon.
The following graph shows the combined ratio as it has been and what it would havebeen under the various prices.
Figure 8: Combined Ratio under Different Pricing Assumptions

Combined Ratio Under Different Pricing Assumptions
250%
Actual CR
Adj CR

Combined Ratio

200%

Adj CR incl PoR

150%

100%

50%

0%
1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

Underwriting Year24

Some may argue that given that the option price is consistently above the actual price,
one would have written nothing if one has stuck to our pricing framework. This isrealistically correct and for most, it would have been a good decision. Why would anyone
want to write a line at prices that generate an ROE of –11% with high volatility?
Especially when arbitrage is readily available for the same reasons.
But there are lines, where the opposite is true. In particular, in the current environment
with low interest rates, any medium to high volatility line has a good chance of providing
segments where the market price is close to the option price. Following our pricing
structure by segment, one would naturally shift into the proper segments at the right timewith favorable results. In other words, following our framework in the current marketplace, one would tend to be closer to the market in the high volatility segments and thus
end up writing a lot of this type of business. Conversely, one would largely exit the low
volatility markets as the market would be lower.
Exiting a business does not imply that the insurer is not willing to write the business. It
simply means that the reinsurer’s price is above the market price and, as such, one would
naturally expect to lose the more opportunistic customers.

Why?
Observers of the industry may question how such a situation can develop. This situation
has perplexed the reinsurance industry for as long as it has been deregulated and there are
hardly any industry conferences where the underwriting cycle is not discussed. It has
always baffled the authors of this paper that an industry can operate with such little
understanding of the fundamentals that drive its business.
But the reason the cycle exists to the extent it does, is that the source is deeply rooted andthat it did not use to matter. Reinsurers “suffer” from a long and profitable history until
the industry was deregulated. In an environment of heavy regulation where everybody
was making money, it did not matter if prices were set right. They were always high.
Now, in a period of deregulation, prices have been forced to unnaturally low levels
because actuarial techniques do not recognize optionality in general, and most actuaries
use the same techniques for pricing. In other words, while companies differ in the prices,
the source of those prices is usually the same (often erroneous) pricing framework.
The reinsurance market is an incomplete market. This means that all risks cannot bereadily hedged with available market derivatives, though as we have seen, insurers canreplicate reinsurance payoffs and generate arbitrage profits. But there is no easy way toarbitrage insurance ad infinitum to eliminate discrepancies, because it is not a continuous
market. Thus, arbitrage can exist, but it does not move prices as it would in a continuousand complete market. This arbitrage is a direct cost to reinsurers and depresses profits.

25

How can this exist? In an incomplete market, participants are cursed (or blessed
depending on your perspective) by not knowing the truth. As such, they can make up
their own truth and our survival instinct usually makes such truth one that is favorable tooneself. This is not a new concept.
In an inefficient and incomplete market, participants can delude themselves. A recent
study by McKinsey & Co. found that 80% of insurance CEOs believed that they would
do the right thing in the next down market. But only 20% believed that other participants
would… This is reminiscent of the classic introductory psychology course scenario
where 80% of a given pool of participants think they are above average, though – orrather because - they have nothing to base it on.
Similarly, a behavorial study by Professor Vernon Smith compared aspects of complete
and incomplete markets and found that incomplete markets stay incomplete while
complete markets quickly converge to efficiency (even with few participants). It is
important to note his definitions of a complete and incomplete market:
Complete:
Incomplete:

Continuous two-sided bid-ask system such as electronic exchanges.
Bilateral markets, where buyers and sellers find each other and makedirect contact.

What really was interesting to Prof. Smith was the fact that given the choice, incomplete
market participants chose to stay with the incomplete market. Not knowing any better,they simply had “lots of confidence to just go it alone. We have a long way to go tounderstand this choice.” He calls this “survival economics” rather than “maximization of
profits”. Though he never specifically mentions the reinsurance market, the authors findthe parallels eerie.
So, there is a human comfort in being able to make up the truth such that it suits oneself.
But, it is not very profitable.
Change will necessarily come at some point, but given that we are in an incomplete
market and that this scenario is itself fairly stable, we need a pivotal event. A pricing
theory and framework such as what we are proposing here could go a long way. Also,change of the market structure could facilitate. As more existing (run-off) insurance
losses are traded, it is plausible that a bid-ask exchange could be developed that would
spill over to the current market place.
The parallels here between the insurance market of today and the options market of 30
years ago are interesting. Prior to the emergence of the Chicago Board Options Exchange(CBOE) and the Black-Scholes option theory, options pricing was hardly as organized
and straight-forward as it is today. A similar exchange for insurance risk combined with acoherent theory of pricing could do wonders to the viability of the industry in the longterm.

26

Summary and Conclusion
We have described in detail the underwriting cycle of the industry in general as well as a
specific segment. We have shown that our hypothesis explains more than 60% of thevariation in the underwriting cycle and that the industry would generate returns closer towhat one would expect given the level of risk, if it were to follow our framework.
We have showed that any reinsurer or insurer would be well served to price assumingthat underlying loss estimates will develop like a lattice with factor u and d=1/u. This
would force a price rigor and structure that would at the same time make arbitrage against
the insurer more difficult. The main parameter to estimate is the volatility and the result is
not as sensitive to changes in this parameter as traditional pricing is to changes in other
distributional parameters.
We have also shown that reinsurance can be replicated with direct insurance, and that thismeans that insurance can actually be arbitraged. But the market remains incomplete andwe have explained this.
The reinsurance market needs to be more complete. Further product development isneeded for this to develop, but a coherent pricing theory is missing to set the structurearound the pricing of the financial products. We have shown that option theory can beapplied successfully to accomplish this and hope to see more products introduced withthis in mind.

27

Appendix
The Expected Payoff of a Reinsurance Contract Given an Underlying LogNormal Distribution{

Υ = min (b − a ), (Χ − a )+}First Moment
We have∞b

Ε(Υ ) = ∫ ( x − a ) f ( x )dx + ∫ (b − a ) f (x )dx , henceab

Ε(Υ ) = σ [ f (a ) − f (b )] + (μ − a )[F (b ) − F (a )] + (b − a )[1 − F (b )]2Second Moment
We have( )∞bΕ Υ 2 = ∫ ( x − a ) f ( x )dx + ∫ (b − a ) f ( x )dx , hence
2

2

a

b

Ε(Υ ) =

(μ − a )σ 2 [ f (a ) − f (b )]− (b − a )σ 2 f (b ) + σ 2 [F (b ) − F (a )]22+ (μ − a ) [F (b ) − F (a ) + (b − a ) (1 − F (b ))]Here,f ( x) =1−e( x − μ )2
2σ 2

σ ⋅ 2πF ( x) = Φ ( x) , which is the cumulative standard normal distribution
a is the attachment pointb is the upper limit
X denotes the ground-up loss random variable and Y the reinsurance loss randomvariable.

28

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To see this, graph the payoff function of an excess of loss contract. It is exactly the
payoff of a call option.
ii
In continuous time, this becomes Brownian motion, which can be described asdL= μ ⋅ dt + σ ⋅ dz .
L

30

iii

Note, that many distributions can be explained by changing u’s and d’s in the binomial
lattice.
iv
Volatility of ultimate reserves will generally range from 0% to 60+% depending on lineof business.
⎛L ⎞v
If p=50% and L0 = Ε P ⎜ 1 ⎟ , then d=2R-u.
⎝R⎠vi
For more detail on this, please see Madsen and Pedersen 2003.
vii
The CBOE Volatility Index - more commonly referred to as "VIX" - is an up-to-theminute market estimate of expected volatility that is calculated by using real-time S&P
500 (SPX) index option bid/ask quotes. VIX uses nearby and second nearby options withat least 8 days left to expiration and then weights them to yield a constant, 30-day
measure of 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.

31