WPS4765
Policy ReseaRch WoRking PaPeR 4765
Catastrophe Risk Pricing
An Empirical Analysis
Morton Lane
Olivier Mahul
The World Bank
Finance and Private Sector Development Vice Presidency
Global Capital Market Non Banking Financial Institutions Division
November 2008
Policy ReseaRch WoRking PaPeR 4765
Abstract
The price of catastrophe risks is viewed by many are a function of the underlying peril, the expected loss,
to be too high and/or too volatile. Catastrophe risk the wider capital market cycle, and the risk profile of
practitioners point out that, contrary to standard the transaction. The market-based catastrophe risk price
insurance, such as automobile insurance, catastrophe is estimated to be 2.69 times the expected loss over the
re-insurance is exposed to infrequent but potentially very long term, that is, the long-term average multiple is 2.69.
large losses. It thus requires keeping a large amount of When adjusted from the market cycle, the multiple is
capital in hand, generating a cost of capital to be added estimated at 2.33. Peak perils like US Wind are shown to
to the long-term expected loss. This paper pulls together have a much higher multiple than that of non-peak perils
data from about 250 catastrophe bonds issued on the like Japan Wind, revealing the diversification of credit
capital markets to investigate how catastrophe risks are from the market.
priced. The analysis reveals that catastrophe risk prices
This paper--a product of the Global Capital Market Non Banking Financial Institutions (GCMNB) Division, Finance and
Private Sector Development Vice Presidency--is part of a larger effort in the department to support clients in the financial
management of disaster risks. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org.
The author may be contacted at omahul@worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
Catastrophe Risk Pricing: An Empirical Analysis
Morton Lane and Olivier Mahul1
1 Morton Lane is President, Lane Financial LLC. Olivier Mahul is Program Coordinator, Insurance for the Poor,
Finance and Private Sector Development, World Bank. Contact author: omahul@worldbank.org. The authors thank
peer reviewers John Pollner and Rodney Lester for their helpful suggestions. Remaining errors are our own
responsibility.
Introduction
The price of catastrophe reinsurance is viewed by many to be too high and/or too volatile. Practitioners
of catastrophe reinsurance point out that the cost of reinsurance is high because providing protection
against large but infrequent events requires keeping large amounts of capital on hand. The return
necessary to reward this extra capital is a cost of doing business and that cost is reflected in their
protection product. Primary insurance, which serves small but more frequent events of loss, is priced on
the basis of expected, or anticipated, loss plus a safety provision. That safety provision, for unanticipated
loss, is generally small because actual losses tend to be close to expected losses due to the Law of Large
Numbers. Thus the price of primary insurance is seen, with competition, as driving towards expected loss.
Reinsurance on the other hand, because it is concerned with infrequent but large losses, would expect the
capital component of price to be much larger. It is likely to be multiples of expected loss instead of
fractions of expected loss for primary insurance. Added to the concern about high price is the fact that the
traditional reinsurance industry is somewhat opaque. Actual prices are difficult to obtain and analyze.
Further, premiums are not typically decomposed into expected loss and capital load so it is hard to say
whether the high price is justified or not.
The introduction of Insurance Linked Securities (ILS), popularly known as "cat bonds", in the mid-1990s
has now opened up some of the traditional market practices for closer scrutiny. An ILS issued by an
insurer (or reinsurer) represents a reinsurance of a specific risk, transformed into bond form, so that it can
be accepted in the broader capital markets. What was reinsurance is now a security and as such that
security has a price that can be inspected and analyzed. To put this in perspective, about US$22 billion of
ILS have been issued since the mid-1990s and as of Q1 2008 some US$13.5 billion in catastrophe
securities was outstanding (i.e., still on-risk). The premiums associated with those outstandings were
approximately US$840 million. This new market is not a threat to major reinsurers whose premiums are
several orders of magnitude larger, but it could represent a respectable syndicate at Lloyds and is larger
than some of the Bermuda reinsurers, so it is not without consequence.
Figure 1 shows the distribution of perils for all the ILS issued since inception. Each peril is a
classification of geographical zone and natural hazard windstorm or earthquake plus an all "other"
category. The graphs show the distribution of perils by expected loss and by limit, respectively.
The U.S. market captures about 42% of the catastrophe reinsurance capacity, while it captures almost
60% of the coverage issued through ILS, as shown on Figure 2. European catastrophe risks capture 28%
of the traditional catastrophe reinsurance capacity and only 17% of the ILS coverage.
This paper aims to analyze the pricing of catastrophe reinsurance and to identify key components that
impact catastrophe risk prices, with a particular attention to the amount of capital that is necessary to
protect against extreme risks. The paper examines the pricing of the catastrophe risks assumed by the ILS
market and thereby we can make an inference to the whole catastrophe reinsurance market. Proceeding in
this way, there are some obvious benefits and some pitfalls. The most obvious benefit is that prices are
revealed in ways not available in the traditional reinsurance market. The equivalent of the reinsurance
price or premium is the "spread" over LIBOR associated with an ILS at its issue2. Further, each bond that
is issued comes with a risk analysis that provides the investor with an independent view of that for which
he is being compensated. This allows a linkage of price to risk, a most important capital component. This
2 Note that while prices are available, they are the prices of "Private Placements". The ILS market is still not a
public market.
2
linkage is one that is unique to the ILS market and is usually only available to those few who participate
in the market as broker or underwriter. Another advantage is that a secondary market in ILS trading has
developed that allows insight into changes in price over time. This is unavailable in traditional markets
where the philosophy has been to "write and ride" risk on an annual basis.
Figure 1. Distribution of Perils for Index-Linked Securities since Inception
Expected Loss Exposure (1997 - Q1 2008)
Japan Quake Other CAT
7%
Japan Wind 3%
4%
Euro Quake
1%
US Wind
44%
Euro Wind
22%
US Quake
19%
Exposure by Potential Limit (1997 - Q1 2008)
Other CAT
Japan Quake 2%
10%
Japan Wind
7% US Wind
Euro Quake 34%
1%
Euro Wind
16%
US Quake
30%
Source: Lane Financial LLC (2008).
3
Figure 2. Fraction of Coverage in Catastrophe Reinsurance and ILS Markets, by Geographic Zone
US
Europe
Japan
Other
0% 10% 20% 30% 40% 50% 60%
ILS Limits by zone, all CAT ILS issued in calendar 2006
Catastrophe Reinsurance Protection Purchased 2006
Source: Lane Financial LLC (2008).
Among the pitfalls of the proposed approach is the extrapolation from the ILS market to the catastrophe
reinsurance market as a whole. And while it is true that the risks covered by the ILS market are similar to
those covered in the market as a whole, the ILS market is less rich and deep compared to the traditional
reinsurance market. Finally, of course, the capital standards of the ILS market are quite different from the
traditional market. All ILS are fully collateralized; traditional reinsurance is collateralized to some
acceptable level of certainty (usually 1 in 250 years, i.e., 0.4%). It is acceptable, that is, to cedants,
shareholders and rating agencies. ILS risk assumption is unleveraged; traditional market risk assumption
is leveraged to some degree. Inferences can nevertheless be made concerning the risk profiles of the ILS
market which are comparable to the market as a whole.
The paper proceeds in four parts as follows. Section 2 examines the issue prices of 247 ILS tranches and
their respective risk statistics since 1997. Importantly, the risk statistics are broken down into component
perils since peril diversity can be a determinant of capital use. Prices are then deflated by an index of
general market prices to see if the cyclic effect of prices can be isolated.3
In Section 3, price relationships are examined without cyclic effects. Ideally this would be done by
simultaneously examining all the ILS issued on a particular date. Given the small number of ILS this
simultaneity of issue is a rarity4. However, an insight can be made with simultaneous evaluation, if not
simultaneous issue. This is essentially the information that comes from the secondary market. Two cross
section analyses are conducted at different points in the cycle: the most recent hard part in mid-2006 and
the most recent part at the end of the first quarter of 2008. The ILS market is not the only part of the
catastrophe market that is visible. Since the mid-1990s another vehicle that trades non-traditionally is the
Industry Loss Warranty (ILW) market5. This market resides somewhere between the traditional market
and the securities market in a series of private transactions. ILWs have become sufficiently commonplace
for brokers to exist and provide potential participants with price "indication" sheets for a range of
3 See Annex 1 for a description of the data used in this analysis.
4 The closest the market ever came to a simultaneous issue of many securities at once was the Successor series of
Swiss Re in June 2006.
5 The ILW market is also known interchangeably as the OLW market, meaning Original Market Loss Warranty.
4
coverages. Price "indications" are not transaction prices. Nevertheless, they convey useful information
and we incorporate them as a further step in the cross section analysis to further isolate price
relationships.
In Section 4, ILW are incorporated into the analysis because we are able to associate an independent
estimate of the risk of each ILW with its price "indication".6 A remodeling exercise is conducted to
homogenize the risk statistics. While a particular ILS prospectus comes with a risk analysis from one of
the major modeling companies (AIR Worldwide, RMS or EQECAT) it often only comes with summary
statistics and they are not always directly comparable. The remodeling of nearly all ILS by one company
allows two important features not previously available, namely a full risk profile for each ILS, and a set of
profiles on the same basis. This set of data is used to compute alternative risk and/or capital measures
such as Value at Risk and Tail Value at Risk and try to relate these to price.
Finally, the paper concludes with several observations about catastrophe risk pricing and capital
allocation, which may be helpful to international financial institutions like the World Bank in structuring
its efforts in this new arena. However, the exercise must be viewed as a best preliminary and the first in
many such studies. The results are empirical and derived from unique sources and perspectives. As the
market matures and progresses it is hoped that more telling insights can be derived as well as better
theories of this important area. Hopefully the results herein will stimulate such activity.
Analysis of the Time Series of Original Issue Statistics
The simple linear model
Our objective is to explain ILS pricing and thereby make inference to the impact of capital on price. The
pure transaction database available is the set of original issue prices for 2477 ILS issued since 1997. For
each issue there is a known spread over LIBOR8 (the equivalent of reinsurance premium) and an
associated expected loss9. So the simple linear model, Model 1, is:
Premium Spread = a + b * (Expected Loss) (1)
If, in this expression, a=0 then Spread is proportional to Expected Loss. If also b>1 then Spread contains
some "load" above Expected Loss. Often rationales for insurance pricing assume that
Premium Spread = Function (Expected Loss, Load, Other Expenses)
In this expression the load is an amount over cost to generate the risk takers profit. More pointedly, load
is to generate return on, or be the cost of, capital. On the ILS market, the investor is typically not
confronted with expenses so our focus is on Expected Loss (EL) and Load. If Load is related to return on
6 The independent measure is provided courtesy of AIR Worldwide, one of the three independent modeling agencies
that regularly provide modeling to ILS prospectuses.
7 The actual number of issues to Q1 2008 is actually higher, 251, but certain issues were excluded due to a lack of
information or regularity of form.
8 The term "Premium Spread" is used to convey the fact that in an ILS the Spread over LIBOR is the equivalent of
what Premium would be if the same risk were in reinsurance form. We also use it interchangeably to mean price.
9 Typically, the probability of attachment and exhaustion will also be available to investors, often with some other
statistics on perils but we do not examine those here.
5
capital then for each individual ILS it is related to how much capital each ILS consumes and that is
related to its riskiness.
The statistical results of the simple model are listed in Table 1 for the transactions since inception (1997-
Q1 2008) and for the recent transactions (2003-Q1 2008). In rounded terms, the typical ILS issued
between 1997 and Q1 2008 was priced according to a rule of 320 basis points plus double the EL. Or,
alternatively, it was 2.7 times the expected loss. In both cases there is a high degree of explanation of
price, as measured by the Multiple Correlation coefficient (84% and 90% respectively). Expected Loss is
an important determinant of price, but the model may not be a good price predictor. It, nevertheless,
shows that there is a load associated with price. In the simplest proportionate case it is the amount over
and above expected loss. In this case, a further 1.7 times expected loss.
The ten year period is better in the sense that it contains two full market cycles, but it suffers from the
disadvantage of containing a period of experimentation and infrequent issue in the late 1990s. Results
from the most recent five year period consider a much more robust underlying market. Notwithstanding,
similar orders of magnitude are seen in the coefficient estimates. The most recent period coefficient
estimate implies that the multiple of Spread Premium to Expected Loss, called multiple, is 2.87. This may
well do as an average estimate but the inspection of the data shows that the multiple varies considerably
both over time, by peril and by level of expected loss. The simple model captures none of these.
Consider, for example, the issuance of the Swiss Re Arbor and Successor Series. In July 2003 Swiss Re
issued Arbor I with a spread of 15.25% and an expected loss of 4.86%. Two years later, in June of 2005,
they re-issued the same series, with the identical risk and peril characteristics, at 12.00%. In June 2006
they issued Successor IIA with a spread of 17.5% and an expected loss of 5.04%. One year after that they
issued the same Successor deal at 14.00%. Just taking the multiple as a simple model it had gone from 3.1
to 2.5 back to 3.5 and down to 2.8 in a five year period. Any model like the simple liner model cannot
capture a major cause of ILS price volatility. The expected loss in those four price observations was
essentially constant.
Table 1. Catastrophe Pricing Model (1)
Since Inception (1997-Q1 2008)
Coefficients Standard Error t Stat P-value
Intercept 0.032049 0.002837 11.29511 4.31356E-24
Expected Loss 2.094608 0.084863 24.68215 2.16711E-68
Multiple R = 0.8445
No intercept
Intercept 0 #N/A #N/A #N/A
Expected Loss 2.694272 0.081477 33.06781 1.66551E-92
Multiple R = 0.9035
Recent Issues (2003-Q1 2008)
Intercept 0.027659 0.003359 8.23534 3.6431E-14
Expected Loss 2.360649 0.093134 25.34691 4.0013E-61
Multiple R = 0.8843
No intercept
Intercept 0 #N/A #N/A #N/A
Expected loss 2.866902 0.081927 34.9934 3.15972E-82
Multiple R = 0.9337
6
Measuring the long term cycle
The multiples or models were shifting due to the then current "hardness" or "softness" of the market. But
that is almost tautological: price can only be explained if we know where we are in the cycle. If this is the
case we need a measure of where we are in the cycle and a pricing model independent of the cycle. A
satisfactory approach might be to try to strip out the cyclic effects and see if the cycle-adjusted price can
be explained by expected loss. It might give a more stable relationship. The proposed approach herein is
thus first to try to separate out cyclic effects and second to explain adjusted or separated spread by
expected loss, then by components of expected loss, i.e., each peril separately.
One measure of cycles in price is demonstrated in Figure 3. It is an index composed of observations of
price from various sources. From 1984 to 2000 it is an index of price movements calculated by the now
defunct company, Paragon. To the best of our knowledge, it is the only long term index available. More
recently Lane Financial LLC has established a new index10 based on observation of secondary market ILS
prices, a pseudo constant expected loss series of original issues from Swiss Re, and observations of ILW
prices. This last series combining several markets, the index is deemed to be more robust in the period
since 2002. That leaves a gap from 2000 to 2002 where experts' judgment has been used to complete the
longer term view. The resulting index is displayed as the thick lines in Figure 3.
This index11 is displayed in the lower panel of Figure 3 against the rate on line12 reported by Guy
Carpenter in 2006 for various zones or perils around the world. It seems to capture the essentials. The
index exceeds 1 when the market is hard and is less than 1 when the market is soft. Over the period 1996-
2008, the index varies from 0.76 (in 2000) to 1.65 (in mid-year 2006). This index is therefore used as the
component to strip out cyclic effect from observed prices.
10 Both the longer term index and the shorter term version were presented in the 2007 Annual Review "That was the
Year that was" published by Lane Financial LLC. See Annex 2 for the list of long term index of reinsurance prices.
11 We have also cross-checked the analysis herein described using purely Paragon and secondary prices as an
alternative cyclic indicator, i.e., without the 2000-2002 "guesstimates", and using secondary prices in that gap. The
main results do not change.
12 The "rate on line" of reinsurance is analogous to premium for limit of coverage, i.e., just like the spread premium
of an ILS. And while Guy Carpenter, and others in the traditional market, regularly report rates on line they do not
report corresponding expected losses.
7
Figure 3. Catastrophe Reinsurance Cycles
Figure 4
400 200%
U.S. CATASTROPHE REINSURANCE PRICE INDICES
(Source 1984-2001: PARAGON, a former subsidiary of the Benfield Group
350 Source 2001-2007 Q1 : LFC averages based on Secondary Cat Bond prices and ILW Prices) 175%
300 150%
secirP
250 125%sulp
ophertsataC exdnI ue
bl
ni ni
200 100% s
of esg nge
x
ndeI anhC haC
150 Post 75% d
LFC Andrew Post earY
ot
nda Katrina ndeert-
100 50% earY de
gonaraP Post
9/11 %
50 25%
0 0%
-50 -25%
1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008
Q1
500
Reinsurance Prices Indices for Different Zones
450
Based on Guy Carpenter Rate on Line Indices (2006 Cat Report) and
Lane Financial LLC Index
400
Australia New Zealand Japan Proportional EQ
Japan EQ Price Index US Cat Property RoL Index
350
World RoL (Ex US) Index World RoL index
France RoL LFC Index
300
250
200
150
100
50
0
1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007
Source: Lane Financial LLC (2008).
8
Adjusting for Cycle
There are many ways to adjust prices for cycle. Two simple approaches are considered; add another
coefficient to the regression to represent the cycle level, or divide the actual spread by the index deflator.
In other words, the simpler linear model displayed in Equation (1) is adjusted so that the relationship
varies depending where one is in the cycle. Thus Model (2) is
(Premium Spread) t = a + b * (Expected Loss) + c *(Cycle Level) t (2)
An alternative form is Model (3):
(Premium Spread)t = (Cycle Level)t*[ a + b * (Expected Loss)] (3)
Or, equivalently,
[(Premium Spread)t/(Cycle Level)t] = a + b * (Expected Loss) (3b)
Some simple interpretations are available. In Model (2) when a=0 then the cycle component may be
interpreted as a load factor which is unrelated to expected loss but which varies over time. It may be
viewed as a loose estimate of capital load that depends on cycle: when the market is hard, that is, when
the demand for capacity exceeds the supply, the cost of capital increases. Conversely, when the market is
soft because of excess capacity the cost of capital decreases. Similarly, in Model (3) when a=0 the b may
be interpreted as long term multiple, but the term [(Cycle Level)t *b ] is the multiple expected to prevail
at different points in time.
The results of fitting these models for the ten year and five year periods are displayed in Table 2 and
Table 3. The degree of explanation provided by the models rises for the additive Model (2) but falls
slightly for the adjusted Model (3). Second, over the five year horizon (lower panel), with a equal to zero
the multiple correlation rises from 0.934 in Model (1) to 0.954 in Model (2), but falls to 0.922 in Model
(3). Therefore, a simple additive load looks best.
To see the effect of the model on Spread Premiums consider Table 4. Various levels of expected loss and
various levels of cycle index are estimated. Then using the five year Model (2) with a=0 (see Table 2 with
recent issues) the spread levels are calculated. The Table clearly shows that the model picks up desirable
features of varying Spread Level estimates by both expected loss level and cycle point: the multiple
increases as the market becomes harder and the expected loss decreases.
Model (3) offers an interesting interpretation of the multiple. When a=0, Equation (3b) can be rewritten
(Premium Spread)t/[(Cycle Level)t*(Expected Loss)] = b (3c)
The estimate parameter b can be interpreted as the cycle-adjusted multiple. Table 3 shows that the cycle-
adjusted multiple is estimated at 2.31 over the period 1997 Q1 2008, and at 2.37 over the period 2003 -
Q1 2008.
9
Table 2. Catastrophe Risk Pricing Model (2)
Since Inception (1997-Q1 2008)
Coefficients Standard Error t Stat P-value
Intercept 0.00194012 0.009581203 0.202492 0.839700601
Expected Loss 2.051656829 0.084239213 24.35513 2.88432E-67
Cycle Index 0.033209231 0.010112302 3.284043 0.001173341
Multiple R = 0.8517
No intercept
Intercept 0 #N/A #N/A #N/A
Expected Loss 2.052184059 0.084034015 24.42087 1.38768E-67
Cycle Index 0.035168649 0.0029309 11.99927 2.11937E-26
Multiple R = 0.9404
Recent Issues (2003-Q1 2008)
Intercept -0.00140586 0.010650567 -0.13199 0.895134073
Expected Loss 2.333766308 0.091786229 25.42611 3.76097E-61
Cycle Index 0.030405597 0.010596176 2.869488 0.004609275
Multiple R = 0.8898
No intercept
Intercept 0 #N/A #N/A #N/A
Expected Loss 2.332482952 0.091019005 25.62633 8.57089E-62
Cycle Index 0.02907543 0.003266839 8.900171 6.08881E-16
Multiple R = 0.9035
Table 3. Catastrophe Risk Pricing Model (3)
Since Inception (1997-Q1 2008)
Coefficients Standard Error t Stat P-value
Intercept 0.034679 0.002444 14.18696 9.43757E-34
Expected Loss 1.663705 0.073108 22.75669 2.3071E-62
Multiple R = 0.8239
No intercept
Intercept 0 #N/A #N/A #N/A
Expected Loss 2.31257 0.07682 30.10386 1.87423E-84
Multiple R = 0.8869
Recent Issues (2003-Q1 2008)
Intercept 0.027595 0.002911 9.478194 1.58295E-17
Expected Loss 1.86544 0.080734 23.10603 1.32376E-55
Multiple R = 0.8654
No intercept
Intercept 0 #N/A #N/A #N/A
Expected Loss 2.370522 0.074119 31.98266 3.65308E-76
Multiple R = 0.9222
10
Table 4. Indicated Levels of Premium Spreads from Model (2)
Expected Loss coefficient Cycle Index Coefficient Model Indicated Premium Multiples
2.33 2.91% Spread Levels
Mid Market Cycle Levels
0.50% 1.00 4.08% 8.15
1.00% 1.00 5.24% 5.24
2.00% 1.00 7.57% 3.79
5.00% 1.00 14.56% 2.91
Hard Market Cycle Levels
0.50% 1.60 5.82% 11.64
1.00% 1.60 6.99% 6.99
2.00% 1.60 9.32% 4.66
5.00% 1.60 16.31% 3.26
Soft Market Cycle Levels
0.50% 0.75 3.35% 6.70
1.00% 0.75 4.51% 4.51
2.00% 0.75 6.84% 3.42
5.00% 0.75 13.83% 2.77
Component Perils
Since 1997 almost 40% by value of all ILS issued have involved more than one peril, that is, are multi-
peril bonds. The rest are single peril. When single peril bonds are issued the risk analysis will be
provided for that specific peril. When multi-peril bonds are issued, the prospectus typically also provides
the fraction of overall expected loss that emanates from the component perils of the bond. The historical
result of this by expected loss and by potential limit was shown in Figure 1.
The combination of this expected loss by peril data means that Model (1) can be amended to allow for
peril exposure. Thus Model (1') is:
Premium Spread = a + p [bp* (Expected Loss)p] (4)
Where parameter bp is the coefficient associated with peril p and (Expected Loss)p is the expected loss
from peril p. In the empirical analysis that follows, the set of perils considered is the same as the one
displayed in Figure 1: US Wind, US Quake, Euro Wind, Euro Quake, Japan Wind, Japan Quake and
"Other". Other includes catastrophes in other parts of the world rather than the six already listed:
examples include CAT-Mex covering Mexico earthquake and GlobeCat LAQ covering Guatemala
earthquake. Clearly, ILS issues to date have been dominated by the four major perils; less than 10% of all
limits (and an even smaller share of expected loss) have been Euro Quake, Euro Wind or Other. Model
(2) is also expanded to include component perils, and it denoted Model (2').
Tables 5 and 6 show the result of refining the expected loss data into its component perils using Model
(1') and Model (2'), respectively, over the same ten and five year periods.13. The explanatory power of
these models is high, with a Multiple R higher than 90% (See Tables 4 and 5). However, some
parameters are not statistically significant. "Other Perils" is not statistically significant in Model (1') and
Model (2').
13 It can be shown that the degree of explanation under Model (2'), with Multiple R = 0.9672, is higher than the
degree of explanation under Model (3'), that is Model (3) with decomposition by peril, with Multiple R = 0.9278.
11
Table 5. Catastrophe Risk Pricing Model (1) with Component Perils
Since Inception (1997-Q1 2008)
No intercept
Coefficients Standard Error t Stat P-value
US Wind 3.295162 0.123135 26.76055 5.46E-74
US Quake 3.043272 0.276142 11.02068 4.01E-23
Euro Wind 1.944646 0.290497 6.694208 1.52E-10
Euro Quake 1.412992 1.030007 1.371827 0.171398
Japan Wind 3.970125 1.117267 3.553424 0.000458
Japan Quake 2.55942 0.453563 5.642922 4.7E-08
Other Perils -0.51689 0.444247 -1.16352 0.245774
Multiple R = 0.9317
Recent Issues (2003-Q1 2008)
No intercept
US Wind 3.317867 0.129194 25.68123 4.36E-61
US Quake 2.832238 0.299626 9.452575 2.25E-17
Euro Wind 1.793549 0.323083 5.551354 1.04E-07
Euro Quake 1.774677 1.082363 1.639632 0.102888
Japan Wind 2.975919 1.281416 2.322368 0.021371
Japan Quake 2.523373 0.492235 5.12636 7.81E-07
Other Perils 1.421159 0.842704 1.686428 0.093506
Multiple R = 0.9420
Table 6. Catastrophe Risk Pricing Model (2) with Component Perils
Since Inception (1997-Q1 2008)
No intercept
Coefficients Standard Error t Stat P-value
US Wind 2.723048 0.095658 28.46636 9.71E-79
US Quake 1.835132 0.212636 8.630401 8.7E-16
Euro Wind 1.383443 0.210754 6.564255 3.22E-10
Euro Quake 0.119274 0.740626 0.161045 0.872194
Japan Wind 0.995709 0.821662 1.211824 0.226777
Japan Quake 2.034328 0.325811 6.2439 1.93E-09
Other Perils -0.31434 0.317601 -0.98974 0.3233
Cycle Index 0.035704 0.002347 15.21153 5.3E-37
Multiple R = 0.9659
Recent Issues (2003-Q1 2008)
No intercept
US Wind 2.787485 0.108703 25.64306 7.9E-61
US Quake 1.752864 0.246679 7.105855 3E-11
Euro Wind 1.314467 0.249004 5.278894 3.86E-07
Euro Quake 0.282996 0.832554 0.339913 0.734334
Japan Wind 0.830354 0.991505 0.837468 0.403485
Japan Quake 2.090056 0.37582 5.561329 1E-07
Other Perils 0.342672 0.647094 0.529556 0.597099
Cycle Index 0.033549 0.002958 11.34069 1.21E-22
Multiple R = 0.9672
12
The parameters associated with Euro Quake, Japan Wind are all less than, or borderline statistically
significant. Thus we cannot rely on the parameter estimates for those perils in the same way we can for
US Wind, for example. The reason for the statistical caution may be due to the fact that there is not
enough data on these issues to be confident about the statistical robustness of the estimates.
Nevertheless, qualitative if not exact quantitative observations can be made. Compare the most recent
period before and after disaggregation into component perils. The multiple for expected loss as a whole is
2.332, with a cyclic load coefficient of 2.91% (see Table 2). However, when the expected loss is broken
down to its component perils, the multiple for each peril is different. The US Wind peril demands the
highest multiple of 2.787 whereas Euro Quake has a multiple of 1.314 (see Table 6). In other words, when
pricing a bond with pure US Wind with an expected loss of, say, 2% the premium would be 5.575% plus
the cyclic load component -- 3.355% in a neutral market (index =1) -- for a price of 8.930 (a multiple of
4.5). By the same reasoning using Model (2') for a Euro Wind risk with an expected loss of 2% would
give a price of 5.984%. Table 5 also shows that Japan Wind has a multiple of less than one. A 2% Japan
Wind risk, in the same neutral environment (index=1) would price at 5.05% - a multiple of 2.5.Therefore,
a significant component of price is over and above expected loss, but it varies considerably by peril.
Coefficients of expected loss less than one could mean that credit is given for diversification. And in the
very rare perils Euro Quake and Other the diversification credit is high. This is subject to the
statistical significance of these estimates, which may not be always highly reliable. Furthermore, even
when the coefficient is less than one (e.g., Japan Wind), it does not mean that the premium will be less
than expected loss: there is an additional factor to be considered, the cyclic load, which is also part of the
capital requirements.
The usefulness of these models for relative pricing of individual securities is not the subject of this paper;
however, it can be illuminating to view these models from that perspective, as illustrated in Figure 4
below, where spreads are estimated using Model (2').
Figure 4. Actual vs Predicted Spreads for CAT ILS Post 2002.
13
Cross Section Analysis of Secondary Market Prices
After an ILS is issued in security form it is eligible to be traded in the secondary market. Investors who
bought a particular transaction may later find that they see new opportunities and wish to readjust their
portfolio by selling the ILS. Others may reassess the risk and want to sell. Still others may want to acquire
more after failing to get a big enough initial allocation. All these motivations to trade are typical of
security markets. Such after-issue trading is not, however, typical of the underlying reinsurance market.
Reinsurers write a risk and hold it. Thus the ILS market brings to reinsurance an element not available in
the traditional market, that is, secondary trading. With it comes the price of such secondary transactions
and the information about current market conditions. Given a robust secondary market, transaction
pricing is invaluable contemporary information to issuers and investors alike.
The ILS secondary market cannot be considered as robust, but it does exist and is growing. Dealers
regularly provide prices at which they might transact, if asked to do so, or that they have recently seen
transacted. This is the information contained in price indication sheets. And, while far from perfect, it is
nevertheless valuable. The average of such price indication sheets from several dealers is available for
two periods: end Q2 2006 and end Q1 2008. These two points respectively represent the hardest point in
the recent post-Katrina hard market and the most recent quarter. Some 41 separately priced ILS tranches
were outstanding in Q2 2006 and 83 were outstanding14 in Q1 2008, which illustrates how much the
market has grown in the last two years.
An ILS issued with a premium spread of 8% over LIBOR might trade in the secondary market at 6% one
year later. This is said to be its current yield. It is the premium spread at which an identical new bond
might be issued. A fall of premium spread from 8% to 6% would indicate a considerable softening in the
market. If expected losses were the same then the load above expected loss would have dropped
significantly. This cyclical impact on pricing was identified and quantified in the previous section. The
cross section analysis provides another light on the effect of market cycles by comparing the ILS priced
on the secondary market at two different periods.
Before proceeding, however, a note of caution is needed. After initial issue, prices may change in the
secondary market, but so may estimates of expected loss. While investors are given an initial estimate of
expected loss, no one provides them with updates as time and new information becomes available to
change an expected loss estimate. It is the responsibility of each ILS holder to make those estimates or to
engage someone to do it for them. The upshot is that there is no information about secondary estimates of
expected loss. In what follows, secondary market yields are compared with original issue estimates of
expected loss.
Model (1) is adapted to capture the current yield and no intercept is allowed. Model (1b) is:
Current Yield = b * (Original Expected Loss) (5)
The basic results of the cross-section analysis are displayed Table 7. The two periods are displayed
alongside each other for comparison purposes. The comparison is quite dramatic: the multiple falls from
3.3374 to 1.6814 a fifty percent reduction. The cross section analysis confirms what the cycle adjusted
14 The deals considered that the cross-section analyses contained certain Wind transactions that were eliminated
from the statistical analysis because they were too close to maturity (3 months or less). At 3 months to maturity most
seasonal risk has disappeared and pricing tends to be erratic.
14
time series showed, namely if the amount over expected loss is considered to be load then it varies
considerably over time.
Table 7. Cross Section analysis Model (1b)
Coefficients Standard Error t Stat P-value
Second Quarter 2006 (41 observations)
Original
Expected Loss 3.337374 0.191179 17.45681 2.6E-20
Multiple R = 0.9402
First Quarter 2008 (83 observations)
Original
Expected Loss 1.681447 0.142756 11.77848 2.57E-19
Multiple R = 0.7928
Component Perils
The expected loss can also be broken into component perils, as it was done in the previous section.
Model (1'), where the intercept is set at zero, becomes Model (1b')
Current Yield = p [bp* (Original Expected Loss)p] (6)
However, note that in the case of Q2 2006 only 41 deals are available, so that the degrees of information
for separating out the perils are even less than in the time series analysis. The results are shown in Table
8. The presence of so many "not significant" variables commends caution15.
Table 8. Cross Section Analysis Model (1b) with Component Perils
Coefficients Standard Error t Stat P-value
Second Quarter 2006 (41 observations)
US Wind 3.44806 0.323957 10.64359 1.62E-12
US Quake 4.465869 1.40003 3.189838 0.002999
Euro Wind 3.365641 0.760508 4.425514 8.97E-05
Euro Quake 0 0 N/A N/A
Japan Wind 6.921184 6.968441 0.993218 0.327421
Japan Quake 1.984895 0.900147 2.20508 0.034112
Other Perils 1.777366 1.848708 0.96141 0.342942
Multiple R = 0.9459
First Quarter 2008 (83 observations)
US Wind 2.396642 0.252253 9.500951 1.47E-14
US Quake 2.187185 0.39567 5.527794 4.38E-07
Euro Wind -1.0669 0.618171 -1.72591 0.088428
Euro Quake 1.702749 1.169533 1.455922 0.149533
Japan Wind 1.726069 1.521742 1.134272 0.260246
Japan Quake 1.114602 0.785889 1.418269 0.1602
Other Perils 2.199775 1.031283 2.133046 0.036149
Multiple R = 0.8355
15 Regressions have been re-run eliminating the "not significant" variables. The coefficients of the remaining
variables change but not markedly so we have deliberately left the poorer fit to show the larger picture.
15
The component perils coefficients display the same drop in value as in the Model (1'). US Wind drops
from 3.4481 to 2.3966. US Quake drops from 4.4659 to 2.1872. The Quake coverage was in much
demand in Q2 2006, and there is anecdotal evidence for this. After Hurricane Katrina in August 2005, it
became apparent that while physical property damage generate the main demand for catastrophe
coverage, catastrophes can also affect workers compensation claims. Accordingly, reinsurers began to
increase their coverage in US quake zones to allow for additional non physical claims from an
earthquake.
Cross Section Analysis including ILWs and Re-Modeled Risk Statistics
Beyond the ILS market there is a related market for insurance risk that trespasses outside the traditional
realm; it is the Industry Loss Warranty Market, or ILW market. Since much of the ILW trading takes
place in the capital market. However, whereas the ILS quantities are a well known, the ILW sizes are not.
Neither is there any declared expected loss associated with a particular ILW transaction. They are very
much private and confidential transactions. There are, however, indicated prices and modeling agencies
that will provide third party expected loss estimates.
We use in this analysis remodeled ILW and ILS estimates provided by Air Worldwide, which is one of
the three largest catastrophe risk modeling firms with RMS and EQECAT. As one of three independent
modelers they provide roughly one third of the original expected loss estimates and risk analyses for all
issued ILS. An important question is whether all modeling firms model on the same basis. Is a 2%
expected loss from AIR directly comparable to a 2% expected loss from RMS? Generally the answer is
positive but there are differences that market participants are well aware of. One company tends to be
conservative with a particular peril than another while another is more liberal elsewhere. Over time
investors have been sensitized to such nuances and have come to demand for themselves analyses of risk
on the same, or at least on a consistent, basis. The model companies have responded by re-modeling
nearly all deals that are outstanding whether originally analyzed by themselves or their competitors.
One consequence of having a remodeled set of estimates is that they may differ from those originally
provided with an ILS. We thus distinguish the original expected loss, oEL, and the remodeled expected
loss, rEL. By way of double checking the series, Model (1b) is re-run with both series for the Q1 2008
ILS set with secondary yields. The results are shown in Table 9. The parameter estimates look very
similar. Apparently, the market looks mostly at the original data, and not yet at remodeled statistics.
Table 9. Comparing the Use of Original vs Re-Modeled Expected Loss
Coefficients Standard Error t Stat P-value
Observed data: First Quarter 2008 (83 observations)
Intercept 0.041456 0.00435 9.529676 7.14E-15
Expected Loss 0.883247 0.12939 6.826238 1.46E-09
Multiple R = 0.6043
Remodeled data: First Quarter 2008 (83 observations)
Intercept 0.040351 0.005137 7.854396 1.45E-11
Expected loss 0.872826 0.159259 5.48055 4.67E-07
Multiple R = 0.5201
Another virtue of having data modeled on the same basis is that we can make estimates of other statistics
beyond expected loss. Theoretically this is possible with original risk analysis as well but prospectus
16
information is often given in such a way as to make the exercise difficult, or at least cumbersome. Two
risk measures: Standard Deviation and Tail Value at Risk.
Standard Deviation is a well known measure of risk; it is essential to much capital market pricing where
returns tend to be symmetric. Even in the reinsurance world there are those who believe prices should be
set using standard deviation. Kreps (2005) has proposed the following pricing model:
Premium Spread = Expected Loss + alpha*[Standard Deviation] (7)
where alpha represents a fraction of the standard deviation that should be loaded on to the expected loss.
It can be rewritten as:
{Premium Spread Expected Loss} = alpha *[Standard Deviation] (7')
Or, equivalently, since {Premium Spread Expected Loss} can be viewed as the Expected Excess Return
(EER)
EER = alpha *[Standard Deviation] (7")
Then alpha*[Standard Deviation] can also be viewed as the expected return on capital required by the
market on ILS of that riskiness.
An alternative risk measure is Tail Value at Risk. In this, a probability level is set, such as x%, and the tail
value at risk is defined as the expected level of loss experienced in the next worst (1-x)% of outcomes.
This is denoted as TVaR(1-x). For example, TVaR99 is the expected level of loss to be experienced in
the worst 1% of outcomes. The level of loss experienced exactly at the chosen probability level is itself a
risk measure known as Value at Risk or VaR. It has been popular for some time, but is has some less than
satisfactory properties and generally the market is moving toward the use of TVaR.. Adapting the simple
model for TVaR gives a linear function:
Premium Spread = Expected Loss + alpha * TVaR (8)
Reasoning similar to the standard deviation case shows that the term alpha*TVaR can be interpreted as
the expected return on capital for an ILS with that level of TVaR riskiness. In what follows the
regressions are conducted at several levels of TVaR.
Table 10 shows the result of the cross section analysis for Q2 2006 and Q1 2008 using secondary ILS
prices, ILW price indications and expected losses. All measures are on a re-modeled consistent basis and
in each case the regressions are conducted with
rEER = Current Yield rEL = alpha*[risk measure],
where rEER is the remodeled Expected Excess Return and the risk measure is either the standard
deviation or TVaR.
17
Table 10. Secondary Market Remodeled Expected Excess Return vs Risk Measures
ILS and ILW 06Q2 (148 observation)
Coefficients Standard Error t Stat P-value
Std Dev 0.448869 0.025233 17.78891 1.76E-38
Multiple R = 0.8263
TVaR90 0.17059 0.010945 15.58562 6.03E-33
Multiple R = 0.7893
TVaR99 0.099455 0.005626 17.67798 3.3E-38
Multiple R = 0.8247
ILS and ILW 08Q1 (213 observation)
Coefficients Standard Error t Stat P-value
Std Dev 0.247743 0.025202 9.830178 4.98E-19
Multiple R = 0.5596
TVaR90 0.091548 0.010599 8.637379 1.39E-15
Multiple R = 0.5102
TVaR99 0.054198 0.005134 10.55756 3.28E-21
Multiple R = 0.5870
First, consider the standard deviation model. In Q2 2006 individual ILS , or for that matter individual
ILWs, appear to have been priced on the basis of expected loss plus 44.88% of the standard deviation
embedded in that particular ILS (see Table 10). For example, an ILS with an expected loss of 2% and a
standard deviation of 5% would be priced at 4.2445%. That is with a load of 2.2445%, or equivalently an
expected profit of 2.2445%. By Q1 2008, however, the estimated load for the standard deviation had
dropped to 24.77% of the standard deviation. So the same ILS re-priced in 2008 would have a premium of
3.2385%. The near fifty percent drop of the estimated load has led in this example to a premium drop of
some 25%.
Similar stories can be derived from the TVaR regressions. The TVaR90 coefficient drops from 0.1706 to
0.0915. Similarly, the TVaR99 coefficient drops from 0.0995 to 0.0542. In both cases they are cut in half.
To give some intuition to the TVaR measures consider an ILS with an expected loss of 2% and a TVaR90
of 40%. In other words, in the worst 10% of cases the investor would expect to lose 40% of principal. The
models would suggest premium shift for each of the periods from 8.824% to 5.366%, a drop of some
40%. The TVaR and the standard deviation models demonstrate dramatically the effect of a shift from a
hard market to a soft market.
Concluding Remarks
Theory suggests that an important driver of reinsurance pricing is the amount of capital that is necessary
to protect against extreme risk (see, for example, Froot (2007)). In practice, catastrophic risk pricing
fluctuates quite dramatically following extreme events. However, identifying or quantifying an exact
relationship is more elusive. Some of the principal practitioners of reinsurance are private companies
located offshore where disclosure requirements are not as stringent as for onshore primary insurers. Even
if required to report to regulators, however, it is unlikely they would reveal either their pricing practice or
their capital allocation rules. Moreover, many of the companies that write catastrophe reinsurance do it as
part of a multi-line business. There are only a few catastrophe reinsurance specialists. The industry has
traditionally been quite opaque.
18
A window has now been opened to reinsurance pricing by the introduction during the last decade of
Insurance Linked Securities (ILS), such as catastrophe bonds. These bonds, distributed broadly to
investors in the capital markets, contain catastrophe risk and have explicit prices associated with them.
They also associate premiums with a risk analysis. To the extent the embedded risks are typical of the
traditional market the components of price can now be forensically examined. Since inception in 1996, in
excess of 250 separately priced ILS tranches have been issued and priced. These bonds contain a wide
variety of catastrophe perils. The principal peril classifications are windstorm and earthquake risk in the
US, Europe and Japan. These have been complimented by others including earthquake risk in Mexico and
Guatemala, and windstorm in the Caribbean. This set of perils is quite representative of the traditional
reinsurance market as a whole.
This paper has found that the long term average over the whole set of bonds (1997 Q1 200) is 2.69
times the expected loss. The most recent issues of the last five years show an average of 2.87. Whether
this is too high is not a subject of discussion here; clearly these are freely derived market prices, so they
are correct if we assume that these markets are competitive.
There is suggestion that the multiple will decline over time as the ILS market becomes more and more
competitive. However, this is not a conclusion that can be easily drawn from the data. Simple multiples
differ according to the point in the cycle; the level of risk attachment; the peril and zone; the risk profile
of the coverage; and how it diversifies or concentrates the existing portfolio. Stripping out the volatile
elements of the hard/soft market reinsurance cycles indicates that the cycle-adjusted multiple is estimated
at 2.33.
Differentiating by peril shows that the multiple for US Wind peril is a relatively high 2.78 plus a cycle
adjustment. On the other hand, Japanese wind shows a multiple of 0.88 plus the cycle adjustment. This is
strong evidence that diversification pays. Japanese Wind deals appear to get a diversification credit from
the market, i.e., they are priced at less than expected loss (when cyclic components are excluded). Similar
remarks appear to apply to other non-peak perils.
Further detailed examination of these phenomena is conducted at two points in time using secondary
market prices and Industry Loss Warranty (ILW) prices instead of the time series. This view points to the
volatile nature of the markets. In the last hard market, June 2006, the capital load as represented by the
long-term average multiple was 3.34. The most recent, first quarter 2008, was 1.68. And,
correspondingly, the peril-specific multiples change dramatically between periods. US Wind peril falls
from 3.45 to 2.4. US Quake falls from 4.47 to 2.19. Similar effects are perceived in the more diversifying
perils. In other words, there is even a diversifying credit that is not immune to the larger capital market
cycle.
The capital cost in a transaction is also affected by the profile of the embedded risk. Thus a deal with a
long tail in its risk distribution may require more capital than one with more concentrated outcomes. Risk
profiles are best summarized by risk measures such as Standard Deviation and Tail Value at Risk. This
paper shows how higher capital loads are associated with higher risk measures. This, too, varies by cycle.
Catastrophe reinsurance prices, as represented by the ILS market, are determined freely in the capital
markets. Dissecting the prices reveals that any ILS price is a function of a) expected losses, b) the wider
capital market cycle, c) the risk profile of the transaction, d) the perils contained.
This paper has not proposed, nor discovered, any new models of catastrophe risk pricing behavior. Instead
it has pulled together data from a number of disparate, and sometimes proprietary, sources to examine
19
certain simple intuitive ideas about pricing. The list of data includes: original issue ILS data, secondary
market ILS data, a cyclic price index, ILW price indication sheets and a set of remodeled scenarios
allowing evaluation of individual ILS and ILW and portfolios of the same. The quality of the data is
uneven but to our knowledge has not been previously brought together in this fashion. The exercise is
comforting in that the various relationships tested herein make intuitive sense and, just as important,
highlight areas of insufficient knowledge.
Catastrophe risk pricing has for too long been a subject of actuarial scrutiny of statistical costs (expected
losses) with little attention paid to market driven prices. The analysis herein shows that market prices
follow a cycle, and they vary considerably by peril. Even low premium, low correlation perils appear to
follow the same cycle, even if the cycle is amplified for peak perils. Pricing also depends on the risk
characteristics of individual transactions.
None of these are surprising in themselves, and yet the analysis shows that each component is significant.
It remains, of course, to bring all the elements together in a single model and quantify that relationship.
That is for the future. The present study we believe takes the first step in that direction.
20
Annex 1. Data, Sources and Units
The data used in the regression exercises conducted in this paper comes from three proprietary sources;
the ILS Data Base of Lane Financial LLC, Price Sheets of Access Re and the remodeled scenario set of
AIR Worldwide. The secondary market ILS data is a compilation of Price Indication Sheets produced by
various brokers at regular intervals. The indications are averaged at mid-market quotes. Similarly, ILW
prices from Access Re are "indication only" prices. They represent neither actionable nor transacted
prices. As such, statistical exercise results should be viewed with appropriate caution. Further, while
every attempt has been made to be accurate, any errors that may have occurred are the responsibility of
the author, not the source.
Where possible in the following, data on spreads and expected losses, etc., are quoted in decimal form.
Thus a spread of 0.095 is the equivalent of 9.5% or 950 basis points. Unfortunately, not all sources use
the same convention so attention should be paid to context. Also, while data is in that form, text may use
the conventions interchangeably.
21
Annex 2. Lane Financial Long Term Index of Catastrophe Reinsurance prices
Lane Financial Long Term Index of Catastrophe Reinsurance
Prices
Ten Year Index Ten Year Index
12/31/1996 1.00 3/31/2003 1.05
3/31/1997 0.97 6/30/2003 1.07
6/30/1997 1.01 9/30/2003 1.03
9/30/1997 0.93 12/31/2003 0.99
12/31/1997 0.87 3/31/2004 0.92
3/31/1998 0.85 6/30/2004 0.96
6/30/1998 0.89 9/30/2004 0.92
9/30/1998 0.84 12/31/2004 0.91
12/31/1998 0.79 3/31/2005 0.87
3/31/1999 0.80 6/30/2005 0.90
6/30/1999 0.80 9/30/2005 0.86
9/30/1999 0.79 12/31/2005 1.02
12/31/1999 0.77 3/31/2006 1.08
3/31/2000 0.76 6/30/2006 1.65
6/30/2000 0.77 9/30/2006 1.57
9/30/2000 0.76 12/31/2006 1.46
12/31/2000 0.76 3/31/2007 1.30
3/31/2001 0.76 6/30/2007 1.35
6/30/2001 0.79 9/30/2007 1.19
9/30/2001 0.79 12/31/2007 1.06
12/31/2001 0.83 3/31/2008 1.06
3/31/2002 0.90
6/30/2002 1.01
9/30/2002 1.05
12/31/2002 1.07
Source: Lane Financials LLC (2008).
22
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