Discussion Paper No.

576

WHEN SHOULD A CAT INDEX FUTURES
BE CREATED?

Kazuhiko Ōhashi

March 2003

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

When should a CAT index futures
be created?
Kazuhiko Ohashi
Graduate School ofInternational Corporate Strategy,
Hitotsubashi University∗
First draft: August 2002
This draft: March 2003

Abstract
Traditionally, insurance risks are borne in reinsurance markets. In
1990s, however, after the sequence of huge natural disasters and huge insurance payments, the reinsurance markets reduced its capability to bearrisks, especially those related to catastrophic natural disasters. CatastropheLinked Securities (CLS) were invented in order to fill the need for additional reinsurance capacity by transferring insurance risks to the capital
markets. The CAT (catastrophe) index futures is one of the several typesof CLS’s. This paper investigates conditions under which the index derivatives, such as the CAT index futures, of the insurance risks can be tradedby the non-insurance investors and is beneficial from the insurers’ andthe exchange’s viewpoints.

∗
I thank Yoshiyasu Ono, Fumio Ohtake, Shinsuke Ikeda, Masaki Aoyagi, Atsushi Tsuneki,
Atsushi Kajii, Daiji Kawaguchi and all seminar participants at the ISER, Osaka University.
This paper is still preliminary. Comments are very welcome. (e-mail: kohashi@ics.hit-u.ac.jp)

1. INTRODUCTION
Insurance risks are traditionally borne in reinsurance markets. That is, after
originating insurance contracts and bearing risks of customers, the insurers
spread and reallocate such insurance risks among themselves and reinsurers.
Lloyd’s is a notable example. Munich Re. and Swiss Re. are some of therepresentative reinsurers.
In 1990’s, however, the capability of the reinsurance markets to bear risks
was at stake. A sequence of huge natural disasters, including Hurricane Andrew(1992) and Northridge Earthquake (1994), produced huge amount, more than $30 billion just for Hurricane Andrew and Northridge Earthquake, in industrywide insured losses. There was the need for additional reinsurance capacity.
Capital markets, or the non-insurance investors in them, became the natural
target. (See Ganapati et. al. (1997).)
Then, innovation occurred. Catastrophe-linked securities (CLS’s) were created, which enabled the investors outside the insurance and reinsurance industry
to trade the catastrophe insurance risks, so that the capital markets could provide additional capacity for bearing the insurance risks. A CAT (catastrophe)
index futures contract is among the several types of CLS’s.1
The CAT index futures was introduced by Chicago Board of Trade (CBOT)in 1992, based on an index provided by the Insurance Services Oﬃce (ISO). Since
1995, the index has been changed to that provided by Property Claims Services(PCS), which is now regarded as the industry accepted standards. Other indexes
are used in other markets. In the Bermuda Commodity Exchange (BCOE),
CAT index derivative instruments are based on the Guy Carpenter Catastrophe
Index (GCCI). In over-the-counter (OTC) markets, Risk Management Solutions(RMS) Index and Sigma Index by Swiss Re. are utilized. (See Kist and Meyers(1999).)
Roughly speaking, these indexes are designed to reflect the industry-wide
insurance losses, not the individual insurer’s losses. From the purpose of riskreallocation, such design of the indexes limits the hedging eﬀectiveness of the
CAT index derivatives for the individual insurers. The risk traded by the index
is inevitably diﬀerent from the risks that the individual insurers wish to hedge.
There is basis risk. Thus, if the insurers and the investors outside the insurance
industry have the same information about the insurance risks, there is no benefit
for creating an index derivatives for both sides. Trading each insurer’s risk
separately avoids the basis risk in trading the index and provides better risksharing opportunities.
1

Securitization of catastrophe insurance risks is another example.

1However, this is not the case for insurance risks, including catastrophe risk.
Usually, the outside non-insurance investors know little about the risks that theinsurance contracts deal with, while the insurers and the reinsurers know themwell. Asymmetric information is prevalent between the insiders and the outsiderof the insurance markets. Trading an index of insurance risks is expected tomitigate such asymmetric information and to facilitate the participation of the
non-insurance investors in trading. (See Kist and Meyers (1999), and Major(1999).)
Despite such device, the CAT index derivative markets have not been successful. For all contracts, trading has been thin. The exchanges have shut down
most of these markets. Although this is partly because of the recovery of capacity of the reinsurance markets due to the lack (and luck) in occurrence ofhuge natural disasters, from the theoretical viewpoint, the developments of the
CAT index derivatives should be meaningful steps in achieving incremental risk
transfer to the capital markets.
What was wrong? Under what condition, can the index derivatives, such as
the CAT index futures, of the insurance risks be traded and preferred to thereinsurance contracts? These are the questions that we address in this paper.
In what follows, we focus on the situation where each insurance risk of each
insurer cannot be traded separately with the non-insurance outside investors
because of the adverse selection. As practical way to mitigate such adverse
selection problem, we consider an index futures where the index is defined tobe the average of the payoﬀs of the insurance risks that the individual insurers
bear. We first investigate the condition under which bundling these risks into
an index futures mitigates the adverse selection enough for the index futures tobe traded between the insurers and the outside investor.
We then compare the attractiveness of the index futures with the reinsurance
markets from the insurers’ viewpoints. Participants in the reinsurance markets
are presumably professionals in insurance. They know equally well about theinsurance risks that they trade. Adverse selection problem about the insurance
risks is small in the reinsurance markets. Thus, we model the reinsurance
markets as one representative reinsurer who has the same information as theinsurers, but is less risk tolerant than the representative non-insurance outside
investor. We consider the idealized reinsurance markets where the insurers andthe reinsurer have the symmetric information, trade their risks competitively,
and attain the optimal risk allocation. We then investigate the condition underwhich each insurer, ex-ante, wish to trade the index futures rather than thereinsurance contracts.
We also investigate the attractiveness of creating the index futures from the2exchange’s viewpoint..
In section 2, we formulate the model. In section 3, we investigate the condition under which the index futures can be traded between the insurers andthe outside non-insurance investors. In section 4, we investigate the idealized
reinsurance markets. In section 5, we compare the ex-ante attractiveness of theindex futures with the reinsurance contracts to the insurers. In section 6, we investigate the attractiveness of the index futures from the exchange’ss viewpoint.
In section 7, we conclude with some comments on further investigation.
2. THE MODEL
All random variables are defined on a probability space (Ω, F, P ) . Throughout we denote by V ar [χ], the unconditional variance of random variable χ, andby V ar [χ|η], the conditional variance of χ given η. Similarly Cov [χ, ξ] denotes
the unconditional covariance between χ and ξ, and Cov [χ, ξ|η] the conditional
covariance.
There are N + 2 agents, with von Neumann-Morgenstern utility functions
displaying constant absolute risk aversion. We refer to the first N agents insurers
and assume that each insurer n (n = 1, ..., N) has an asset, as an insurance
contract, which yields a stochastic payoﬀ zn at some terminal date. The N +1st agent is an aggregated reinsurer who provides reinsurance for the insurers.
The N + 2nd agent is an aggregated outside investor who represents the (noninsurance) capital market as a whole.
There are three dates in the economy. At the ex-ante stage, date 0, theinsurers decide which contract and with whom they trade at date 1, an index
futures with the outside investors or reinsurance contracts with the reinsurer.
At the interim stage, date 1, the insurers and the reinsurer receive some private
signals about the payoﬀs of the insurance risks, while the outside investors
receives no signal. Right after the signal is received, the insurers trade either an
index futures with the uninformed outside investor, or a reinsurance contract
with the reinsurer, as they planned at date 0. At the final stage, date 2, payoﬀs
are realized and all signals become public.
More precisely the insurer n (n = 1, ..., N) has the utility function E[− exp(− r1I W )]over the consumption W at date 2, is endowed with an asset that has a payoﬀ z
at date 2, and receives a vector S of information signals at date 1. The reinsurer
has the utility function E[− exp(− r1R W )] over the consumption W at date 2,
receives the same vector S of information signals that the insurers do at date 1,
but has no asset. The outside investor has the utility function E[− exp(− r1u W )]3over the consumption W at date 2, but has neither endowment of an asset nor
an information signal.
For tractability we assume that zn is given by a product of two randomvariables xn and en , namely zn = en xn , where xn represents the per unit payoﬀ
of the n’s insurance risk and en represents the size, in terms of units, of the n’s
insurance risk. We also assume thatxn = xc + εn .
For each n, denote by sn a signal on εn . Define e ≡ [e1 , ..., eN ]| , ε ≡[ε1 , ..., εN ]| , x ≡ [x1 , ..., xN ]| , and s ≡ [s1 , ..., sN ]| .The vector of information
signals S that the insurers and the reinsurer receives at date 1 is given byS ≡ (s| , e| )| .
All underlying random variables are normally distributed with 0 means, andare independent except that for each n,C[εn , sn ] = ρ (ρ 6= 0).
We assume that V ar[en ] is the same for all n, that V ar[εn ] is the same for alln, and that V ar[sn ] is the same for all n.
There are two kinds of possible contracts that the insurers
PN choose to trade.
1
One is an index futures contract with a payoﬀ F = N n=1 xn . The otheris reinsurance contracts with payoﬀs x = (x1 , ..., xN )| . The insurers decidecollectively whether they trade the index futures or the reinsurance contracts.
In the former case, the insurers share their endowment risks with the uninformed
non-insurance investor in the capital markets through trading the index futures.
In the latter case, the insurers share their endowment risks with the equally
informed reinsurer through the reinsurance market.
Note that although the investor in the capital markets is less informed than
the insurers and the reinsurer, typically the capital markets is much larger
than the reinsurance market in its size. Thus, with symmetric information, theaggregate investor of the larger capital market would be more risk tolerant than
the reinsurer in the smaller reinsurance market.
3. THE INDEX FUTURES
We first investigate the case where the insurers trade the index futures withthe outside investor in the capital market. Here, we assume that the reinsurer

4

does not participate in trading the futures. We would like to know the condition under which bundling the insurers’ endowment risks into an index futures
facilitates risk sharing between the insurers and the outside investor.
Note that if the insurers and the outside investor have the same information
about the risks that they trade, there is no benefit for creating an index futures
for both sides. Trading each insurer’s endowment risk separately provides better
opportunities for risk sharing.
However, this is not the case for typical insurance markets, including CAT
insurance. Usually, the outside non-insurance investor knows little about therisks that the insurance contracts deal with, while the insurers know them well.
Asymmetric information is prevalent between the insiders and the outsider ofthe insurance markets.
Thus, we focus on the situation where each insurer’s endowment risk cannot
be traded separately with the outside investor because of adverse selection,
but bundling these risks into an index futures mitigates the adverse selection
problem so that the index futures can be traded between the insurers and theoutside investor.
For this purpose, let us start with the hypothetical case where each insurern creates an insurance futures contract with the payoﬀ xn aiming to trade it
separately with the outside investor. We assume that each insurer does nottrade the other insurers’ contracts.
Denote by θn the position that the insurer n takes. Each insurer n tries totrade the futures xn 2 with the outside investor strategically i.e., taking accountof the price impact of his position θn . Let θ = (θ1 , ..., θN )| . We assume thatthe price is given by a linear functionX
Pn (θ) = hn + kθn + l
θm .
m6=n

The insurer n ’s date 2 wealth after trading this futures is given by
Wn = en xn + θn (xn − Pn (θ)) .
At date 1, the insurer n solves the following problem to obtain the optimal
position θn .
Maxθn [− exp(

−1Wn ) | S].
γI

2Equivalently, we can assume that the contract is not a futures, that its price is paid at
date 1, and that there exists a riskless asset between date 1 and date 2 whose rate of returnis normalized to one.

5

Its first order condition isE[xn |S] − hn − 2kθn − l

X

m6=n

θm −

1
{V ar[xn |S]en + V ar[xn |S]θn } = 0
γIand its second order condition is2k +1
V ar[xn |S] > 0.
γI

Note that the optimal θn ’s are simultaneously determined. Define qn = E[xn |S]−1
V ar[xn |S] , Q = (q1 , ..., qN )| , and H = (h1 , ..., hN )| . Denote by IN the N ×N
γI
identity matrix and by JN the N × N matrix whose elements are all 1. Then, if
the second order conditions are satisfied for all n, then the vector of the insurers’
optimal positions θ = (θ1 , ..., θN )| is given by
θ=

l

1

{IN −
JN }(Q − H).
2k + γ1 V ar[xn |S] − l
2k + γ1 V ar[xn |S] + (N − 1)lI
I

Denote by θun the position that the outside investor u takes in trading thenth futures. Let θu = (θu1 , ..., θuN )| and x = (x1 , ..., xN )| . The outside investor
is assumed to represent a large number of identical investors and hence behavescompetitively. He has rational expectations and uses the observed prices to
update beliefs about the payoﬀ of the traded securities. Then, the outside
investor ’s date 2 wealth after trading the futures is given by
Wu = θ|u (x − P (θ))
where p(θ) = (p1 (θ), ..., pN (θ))| . At date 1, the outside investor solves thefollowing problem to obtain the optimal position θu .
Maxθu [− exp(

−1Wu ) | θ].
γuIts first order condition isE[x|θ] − P (θ) −1V ar[x|θ]θu = 0γuand its second order condition is1
V ar[x|θ] > 0.
γu6This second order condition is satisfied by assumption. Hence, the vector of theoutside investor’s optimal position is given byθu = γ u V ar[x|θ]−1 {E[x|θ] − P (θ)}.
A linear rational expectation equilibrium is a set (P (·), θ, θu ) such that (a)P (·) is given by (3), (b) agents maximize, and (c) markets clear, that is θ +θu = 0. From the insurers’ second order condition (4), we obtain the following
condition for the individual insurance risks to be traded separately in the capital
market:Lemma 1:
An equilibrium exists if and only ifJ=

Cov 2 [εn , sn ]
1
2
> 0.
V
ar
[x
|s
]V
ar[e
]
−
n
n
n
γ 2I
V ar[sn ]Lemma 1 shows that if J 5 0, this economy fails to have an equilibrium. The
first term of J is related to the hedging demand of the informed insurers as
indicated by its dependence on the degree of risk aversion. The second termis related to the informational motive for trading. An equilibrium exists if theformer dominates the latter. If this is not the case, the adverse selection problem
is so severe that no equilibrium exists.
In the following, we assume the following:
Assumption 1:

J 5 0.

That is, we focus on the situation where each insurer n cannot trade his own endowed insurance risk separately with the non-insurance outside investor becauseof the strong adverse selection between them.
One practical way to mitigate this adverse selection problem is to create anindex of the average of the payoﬀs of the insurers’ endowment risks and to create
a futures contract whose payoﬀ is determined by this index. (See Ganapati et.
al. (1997), Kist and Meyers (1999), and Major (1999).) In this average index, it
is expected that by the law of large number, each insurers’ specific risks would
be diversified enough, if the number of insurers is large enough. The index
7

would depend largely on the market-wide common risk. Since informational
asymmetry is much less for the market-wide common risk than for the insurers’
specific risks, the adverse selection for the average index futures is much lessthan that for the
P individual insurance contracts.
Let F = N1 N
n=1 xn be the payoﬀ of the average index futures. Denote byθF n the position that the insurer n takes in trading the index futures. Let θF =(θF 1 , ..., θF N )| . Each insurer n tries to trade the index futures F strategicallywith the outside investor i.e., taking account of the price impact of his position
θF n . We assume that the price of the index futures is given by a linear function
PF (θ) = hF + kF

N
X

θn .

n=1

The insurer n ’s date 2 wealth after trading the index futures is given by
WF n = en xn + θF n (F − PF (θ)) .
At date 1, the insurer n solves the following problem to obtain the optimal
position θF n .
MaxθF n [− exp(

−1
WF n ) | S].
γI

Its first order condition isE[xn |S] − hF − kF {2θF n +

X

m6=n

θm } −

1{Cov[F, xn |S]en + V ar[F |S]θF n } = 0
γIand its second order condition is
2kF +1
V ar[F |S] > 0.
γI

Define qF n = E[F |S] − γ1 Cov[F, xn |S]en , QF = (qF 1 , ..., qF N )| , and HF =I
(hF , ..., hF )| . Note that IN is the N ×N identity matrix and that JN is the N ×N
matrix whose elements are all 1. If the second order conditions are satisfied for
all n, then the vector of the insurers’ optimal positions θF = (θF 1 , ..., θF N )| isgiven by
θF =

1
kF
{IN −
JN }(QF − HF ).
1
kF + γ V ar[F |S](N + 1)kF + γ1 V ar[F |S]
I
I
8

P
Define θF I ≡ N
n=1 θ F n . Then, θ F I is the aggregate position in the index futuresby the insurers, andN
X
1
1{NE[F |S] − Cov[F, xn |S]θF I =en − NhF ).
γI(N + 1)kF + γ1 V ar[F |S]
n=1
I

Denote by θF u the position that the outside investor u takes in trading theindex futures. The outside investor behaves competitively. He has rational
expectations and uses the observed price to update beliefs about the payoﬀ of
the index futures. Then, the outside investor ’s date 2 wealth after trading theindex futures is given by
WF u = θF (F − PF (θF ))At date 1, the outside investor solves the following problem to obtain the optimal
position θF u .
MaxθF u [− exp(

−1
WF u ) | θF ].
γu

Its first order condition isE[F |θF ] − PF (θ) −1V ar[F |θF ]θF u = 0γuand its second order condition is1
V ar[F |θF ] > 0.
γuThis second order condition is satisfied by assumption. Hence, the vector of theoutside investor’s optimal position is given by
θF u = γ u V ar[F |θF ]−1 {E[F |θF ] − PF (θ)}.
A linear rational expectation equilibrium is a set (PF (·), θF , θF u ) such that(a) PF (·) is given by (9), (b) agents maximize, and (c) markets clear, that isθF + θF u = 0. From the insurers’ second order condition (10), we obtain thefollowing condition for the index futures to be traded:9

Lemma 2:
An equilibrium exists for the index futures market if and only if
JF ≡1
Cov 2 [εn , sn ]
2
{(N
−
1)V
ar[x]
+
V
ar[x
|s
]}
V
ar[e
]
−
> 0.
n
n
n
γ 2I
V ar[sn ]Similarly to lemma 1, the first term of JF is related to the hedging demand of
the informed insurers for the index futures. The second term is related to theinformational motive for trading. An equilibrium exists if the former dominates
the latter. Comparing this lemma with lemma 1, we now obtain the following
result:Proposition 1:
Between the insurers and the outside investor, the average index
futures is traded, while the individual insurance risks cannot be
traded separately, if and only if
1{(N − 1)V ar[xc ] + V ar[xn |sn ]}2 V ar[en ]γ 2I
Cov 2 [εn , sn ]>V ar[sn ]1> 2 V ar2 [xn |sn ]V ar[en ]γIHence, there is a case where creating an index futures whose payoﬀ is the averageof the insurers endowment risks mitigates the adverse selection enough for theindex futures to be traded, although each insurer’s individual insurance risk
cannot be traded with the uninformed outside investor.
Observe that if N = 1, the condition JF > 0 is equivalent to J > 0. Observealso that the larger N is, the weaker the restriction JF > 0 is. Thus, the futuresof each insurer’s individual risk is a special case of the average index futures,
and as the number of the insurers in the average index increases, the index
futures is more likely to be traded.
10Proposition 1 also shows that for the average-indexing to work, it is necessarythat V ar[xc ] > 0. That is, if the insurers’ endowment payoﬀs are independent,
averaging is useless to mitigate the adverse selection problem. The larger the
portion of the (non-informational) common part xc of the payoﬀs xn ’s is (interms of their variances), the more eﬀective the averaging is for mitigating theadverse selection.

4. THE REINSURANCE MARKETS
Traditionally, insurance risks are traded in reinsurance markets. That is,after originating insurance contracts and bearing risks of customers, the insurers
reallocate such risks among themselves and reinsurers. Lloyd’s is a notable
example. Munich Re. and Swiss Re. are some of the representative reinsurers.
Participants in the reinsurance markets are presumably professionals in insurance. They know equally well about insurance risks that they trade. At
least, asymmetric information is much less among insurers and reinsurers thanbetween insurers and non-insurance outside investors in the capital markets.
Adverse selection problem about insurance risks is small in the reinsurance
markets.
However, the size of the reinsurance markets is much smaller than that of thecapital markets. This limits the capacity of the reinsurance markets to bear the
insurance risks, especially those associated with large natural disasters. In fact,catastrophic losses by earthquakes and typhoons in 1990’s, in conjunction withthe slump in business by Lloyd’s, caused the shortage of capacity of reinsurance,
which resulted in the sharp rise of the reinsurance premiums. Naturally, theinsurers looked for the way that enabled them to trade their insurance risks
with the non-insurance outside investors in the capital markets. The CAT
index futures is one of the inventions.
Thus, we model the reinsurance markets as one representative reinsurer who
has the same information as the insurers, but is less risk tolerant than the representative non-insurance outside investor. We consider the idealized reinsurance
markets where the insurers and the reinsurer have the symmetric information,
trade their risks competitively, and attain the optimal risk allocation.
In this idealized reinsurance markets, the insurance risks x = (x1 , ..., xN )| of
the insurers are traded. Denote by θRn = (θRn1 , ..., θRnN )| the position that theinsurer n takes in the reinsurance markets. Each insurer n trades the futures
xn competitively. Let PR = (PR1 , ..., PRN )| be the price of the reinsurance11

futures contracts. Then, the date 2 wealth of the insurer n after trading in thereinsurance markets is given by
WRn = en xn + θ|Rn (x − PR ) .
It is more convenient to use the gross position φRn that is defined by φRn ≡(θR1 , ..., θRn + zn , ..., θRN )| for all n. In terms of the gross futures position, at
date 1, the insurer n solves the following problem to obtain the optimal position
φRn .
MaxφRn [− exp(

−1WRn ) | S].
γI

Its first order condition isE[x|S] − PR −1
V ar[x|S]φn = 0
γI

and its second order condition is
V ar[x|S] > 0.
Since the second order condition is satisfied by assumption, the insurer n’s
optimal gross position φRn = (φR1 , ..., φRN )| is given by
φRn = γ I V ar[x|S]−1 {E[x|S] − PR }.
Denote by θRE = (θRE1 , ..., θREN )| the position that the reinsurer takes. The
reinsurer has the same information as the insurers, and trade the reinsurance
contracts competitively. Then, the reinsurer’s date 2 wealth after trading thereinsurance contracts is given by
WRE = θ|RE (x − PR )At date 1, the reinsurer solves the following problem to obtain the optimal
position θRE .
MaxθRE [− exp(

−1WRE ) | S].
γ RE

Its first order condition isE[x|S] − PR −

1
γ RE
12

V ar[x|S]θRE = 0

and its second order condition is
V ar[x|S] > 0.
Since the second order condition is satisfied by assumption, the reinsurer’s optimal position is given by
θRE = γ RE V ar[x|S]−1 {E[x|S] − PR }.
Note that since the reinsurer has no endowment, this position θRE is equal to
the gross position φRE.
P
PN
Let e = N
n=1 en . In an equilibrium,
n=1 φRn +φRE = e. From this marketclearing condition, we obtain the equilibrium price PR as follows:
PR = E[x|S] −1
V ar[x|S]eNγ I + γ RE

Hence, the investor n’s equilibrium gross position is given by
φRn =

γI
e,
Nγ i + γ RE

and the reinsurer’s equilibrium gross position is given by
φRn =

γ RE
e.
Nγ i + γ RE

That is, in an equilibrium in the idealized reinsurance markets, the aggregatedinsurance risks e is allocated to the market participants according to their risk
tolerance..
5. CREATION OF THE INDEX FUTURES BY THE INSURERS
Traditionally, the insurance risks that the individual insurers bear for theircustomers are reallocated among the insurers and the reinsurers through thereinsurance markets. In 1990’s, however, several huge natural disasters hit theinsurance industry with huge amount of insurance payments for them, whichshrank the capacity of the reinsurance markets to bear suck risks. The reinsurance premiums soared, and the insurers were forced to invent the means toshare their insurance risks outside the reinsurance markets. The CAT index
futures is among them.
13

In this section, we consider such situation where the insurers create the index
futures to share their insurance risks with the outside investors i.e., the capital
markets, instead of the reinsurance markets. We would like to know under what
condition the insurers wish to have such index futures rather than reinsurance
contracts.
For this purpose, we assume that the insurers trade either the index futuresor the reinsurance contracts, and then compare the insurer’s ex-ante utility
levels between when the index futures is traded and when reinsurance contracts
are traded.
We first obtain the ex-ante utility level for the index futures.
Lemma 3:
When the index futures is traded, the ex-ante utility of the insurer
n is given by
E[uI (WF n )]222
= −[(1 − 2 V ar[xn ]V ar[en ]) + V ar[θF n ]α3 (1 − 2 V ar[xn ]V ar[en ])
γI
γI
γI
−1
1
1
2
+ V ar[en ]α3 {2Cov[xn , sn ]α0 α2 + Cov 2 [xn , sn ]α20 + V ar[en ]V ar[xn ]α22 }] 2
γI
γI
γI
where
α0 ≡
α1

Cov[xn , sn ] 1
1
1(N + 1)kF + γ V ar[F |S] V ar[sn ] N
I

kF
1
1
≡
Cov[F, xn |S]
1
1
kF + γ V ar[F |S] (N + 1)kF + γ V ar[F |S] γ I
I

α2 ≡

I

NkF + γ1 V ar[F |S]
1
1
I
Cov[F, xn |S]
1
1
kF + γ V ar[F |S] (N + 1)kF + γ V ar[F |S] γ I
II

1V ar[F |S]2γ I
V ar[θF n ] = N α20 V ar[sn ] + (N − 1)α21 V ar[en ] + α22 V ar[en ]α3 ≡ kF +We next obtain the ex-ante utility level for the reinsurance contracts.

14

Lemma 4:
When the reinsurance contracts are traded, the ex-ante utility of theinsurer n is given by
E[uI (WRn )]1(2N − 1)γ I + 2γ R
1}V ar[εn |sn ]var[en ]= −[1 − 2 V ar[xn ]V ar[en ] + { 2 −γIγIγ I (Nγ I + γ R )2
(2N − 1)γ I + 2γ R
1−+{ 2 −γIγ I (Nγ I + γ R )2
1+

V ar[en ](N − 1)V ar[en ](Nγ I +γ R )2}
V ar[en ]
{V
ar[ε
|s
]
+
(N
−
1)V
ar[e
]}
n
n
n
2
(Nγ I +γ R )−1{(N − 1)γ I + γ R }2
2
×
V
ar[x
]V
ar[e
]
]
c
n
γ 2I (Nγ I + γ R )2−
V ar[en ]V ar[en ]×[{1 +(V ar[εn |sn ] + (N − 1)V ar[en ])}{1 +V ar[εn |sn ] }N−2 ] 2
22(Nγ I + γ R )(Nγ I + γ R )By comparing both utility levels, we can find when the insurers want the index
futures.
Proposition 2:
The insurers wish to create an index futures, if and only if
E[uI (WF n )] > E[uI (WRn )]The following numerical example describes the situation shown in this proposition. We take V ar[xc ] = V ar[εn ] = V ar[en ] = 1, Cov[εn , sn ] = 0.6, γ I = γ u =5. The number of insurers N varies from 1 to 12. The coeﬃcient of risk toleranceγ R of the reinsurer varies from 0.5 to 1.
In this situation, by Proposition 1, we know that the individual insurer’s own
risk cannot be traded with the uninformed outside investor due to the strong
adverse selection. Indeed, N, the number of the insurers, or the number of theinsurers’ risks included in the index futures, should be larger than or equal to
3 for the index futures to be traded. This is why the graph of figure 1 starts
from N = 3.
In general, the larger N is, the less is the adverse selection in trading theindex futures due to the diversification of the private risks. This makes it easierfor the insurers to trade the index futures with the uninformed outside investors.
However, this also increases the basis risks that the insurers face in trading the15

futures. The benefit of trading the index futures is determined by the trade-oﬀ
between the decrease of the adverse selection and the increase of the basis risks.
In this example, the former is more eﬀective than the latter so that the utilityof the insurer increases as N increases.3[Insert Fig. 1]
On the other hand, the benefit of using the reinsurance market is determinedby the reinsurer’s risk tolerance γ R . The larger γ R is, the more risk hedge theinsurers have from the reinsurance market. The following graph in figure 2
shows this situation.4[Insert Fig. 2]
For the insurers, whether to use the CAT index futures or the reinsurance
markets depends on which gives them the better hedging opportunities. Figure
3 shows the diﬀerence between the utility levels obtained through trading theindex futures and that through the reinsurance market.
Take, for example, the case where the reinsurer’s coeﬃcient γ R of risk tolerance is equal to 0.5. In this case, since the risk tolerance of the uninformed
outside investor is much larger, despite the adverse selection, the index futures
provides better hedging opportunity to the insurers when the number N of
the insurers is small. As N increases, since the insurers share the insurance
risks through reinsurance, the reinsurance market provides the better hedging
opportunity than the index futures.
Take also the case where the number N of the insurers is 3. As the reinsurer’s
coeﬃcient γ R of risk tolerance increases, though γ R is much smaller than γ u ,the risk tolerances of the uninformed outside investor, the reinsurance market
becomes more attractive than the index futures to the insurers. This is because
the index futures suﬀers from the adverse selection, while the reinsurance market
does not.
3

Note that in this example, as N increases, the number of the insureres increases, and so
the number of the traders in the futures market also increases. Since the calculated utility
levels are aﬀected by the change of the number of market participants, we need to be careful
to interpret the eﬀect of increasing N . Also, note that the reinsurer’s risk tolerance does not
aﬀect the utility het because the reinsurer is assumed not to trade thr index futures.
4
Note that in this model. we assume that the insurers bear the risks of the other insurers
through the reinsurance market. This assumption tends to make the risk allocation throughthe reinsurance market more eﬃcient than that throught the index futures, and hence theutility levels from the reinsurance tend to be larger than those from trading the index futuresin this example.

16

Recall that in this example, the index futures cannot be traded when N = 1
and 2. Thus, the insurers use the reinsurance market when N = 1 and 2, theCAT index futures when N = 3 to 5 if γ R is suﬃciently small, and again thereinsurance market when N is large enough.
[Insert Fig. 3]

6. CREATION OF THE INDEX FUTURES BY THE EXCHANGE
Although the CAT index futures is created mainly for the benefit of theinsurers, it is the exchange who indeed creates the futures and who may have adiﬀerent objective from the insurers. In this section, we look into this possibility.
We assume that the exchange decides, ex-ante, whether to create the index
futures depending on its expected trading volume.
Lemma 5:The expected volume of the index futures is given byr1
2 2[α0 NV ar[sn ] + α21 (N − 1)V ar[en ] + α22 V ar[en ]] 2 .
EVF =π
Also, the normalized expected volume of the index futures is given byr
1
NEVF = EVF × V ar[xc ] + V ar[εn ].
N
We consider the normalized expected volume in order to keep the variance ofthe payoﬀ of the normalized index futures equal to unity, which enables us tocompare more accurately the expected volumes among the index futures with
diﬀerent N’s.
The following numerical example shows that for some parameter values,
increase in the number N of the insurers in the index leads to decrease of boththe expected and the normalized expected trading volume. We take V ar[xc ] =V ar[εn ] = V ar[en ] = 1, Cov[εn , sn ] = 0.6, γ I = γ u = 5. The number of insurers
N varies from 1 to 5. The green line shows how the expected trading volume
changes as N increases. The blue line shows the normalized expected trading
volume, and the red line shows the insurer’s utility level + 1.
17

[Insert Fig. 4]
Combined with the discussion in section 3, this observation has an important
implication, which we states as the following proposition:
Proposition 3:
In some situation, while it is necessary to have the number N of theinsurers in the index large enough to mitigate the adverse selectionand encourage the outside investor to trade the index futures, it at
the same time may reduce the (normalized) expected trading volume
and discourage the exchange to create such index futures.
Thus, creating an index futures, though it is a useful device for mitigating
adverse selection, may not be a useful device for the exchange to obtain profitable opportunities from creating the futures. It is possible that this is a partof the reason why the CAT index futures have had such thin markets and havenever been successful, though their creation is a meaningful step to open a pathof additional risk transfer from the reinsurance markets to the capital markets.
7. CONCLUSIONS
We investigated the conditions under which an average index futures ofseveral insurance risks can be traded in the presence of asymmetric information.
We found that even when each risks cannot be traded because of strong adverse
selection, averaging these risks may mitigate the adverse selection enough sothat the index futures can be traded, if the number of the risks included inaveraging is large enough. However, we also found that averaging may reduce
the attractiveness of the futures for both the insurers and the exchange. Thismay be a part of the reason why the CAT index futures are so unsuccessful,despite its clear theoretical importance as a new tool to transfer insurance risks
to the capital markets.
For the ease of the analysis, we have investigated only the cases where theinsurers trade either the index futures with the outside non-insurance investor or
the reinsurance contracts with the reinsurer and where the reinsurance markets
are idealized in that the insurers and the reinsurer can trade all reinsurance
contracts with no frictions. Examining the case where the insurers can trade
only the limited kinds of reinsurance contracts is a possible extension of theanalysis.
18

Finally, though we have just compared the use of the capital markets andthat of the reinsurance markets from the insurers’ viewpoint, they indeed interact with each other. Allowing the reinsurer to trade both the reinsurance
contracts and the index futures at the same time may clarify the role of thereinsurer as an intermediary of the insurance risks among several markets. Thisis an immediate task in further research.

REFERENCES
1. Allen, F. and D. Gale (1994), Financial Innovation and Risk Sharing,
MIT Press, Cambridge MA, USA.
2. Bhattacharya, U., P. J. Reny, and M. Spiegel (1995), Destructive Interference in an Imperfectly Competitive Multi-Security Market, Journal of
Economic Theory, 65, 136-170.
3. Bhattacharya, U. and M. Spiegel (1991), Insiders, Outsiders, and Market
Breakdowns, Review of Financial Studies, 4, 255-282.
4. DeMarzo, P. and D. Duﬃe (1999), A Liquidity-Based Model of Security
Design, Econometrica, (1999) 67, 65-90.
5. Duﬃe, D. and R. Rahi (1995), Financial Innovation and Security Design,
Journal of Economic Theory, 65, 1-42.
6. Ganapati, S., M. Retik, P. Puleo, and B. Starr (1997), Catastrophe-Linked
Securities, Chapter 20 in Handbook of Structured Financial Products,”
ed. by F. Fabozzi (1998) Frank J. Fabozzi Associates, New Hope, Pennsylvania, USA.
7. Kist, F. O. and G. Meyers (1999), Evaluating the eﬀectiveness of indexbased insurance derivatives in hedging property/casualty insurance transaciton, American Academy of Actuaries Index Securitization Task Force.
8. Major, J. A. (1999), Index Hedge Performance: Insurer Market Penetration and Basis Risk, Chapter 10 in The Financing of Catastrophe Risk,”ed. by K. A. Froot, The University of Chicage Press, Chicago, Illinois,
USA.

19

9. Marin, J. M. and R. Rahi (1999), Speculative Securities, Economic Theory, 14, 653-668.
10. Ohashi, K. (1995), Endogenous Determination of the Degree of MarketIncompleteness in Futures Innovation, Journal of Economic Theory, 65,
198-217.
11. Rahi, R. (1996), Adverse Selection and Security Design, Review of Economic Studies, 63, 287-300.
12. Subrahmanyam, A. (1991), A Theory of Trading in Stock Index Futures,
Review of Financial Studies 4, 17-51.

20

APPENDIX : Key mathematical results.
Math. Lemma 1:
Let IN be the N × N identity matrix, JN be the N × N matrix withall elements equal to unity, and a and b be scalars where a 6= 0.
Then,
1
b
(aIN + bJN )−1 = (IN −JN ).
aa + bN
Proof of Math. Lemma 1:
Direct computation.||Math. Lemma 2:
Let A be a symmetric N × N matrix, B be an N × 1 vector, and C
be a scalar. Suppose that e ∼ N(0, Σ). Then,−1
1
E[exp(e| Ae+B | e+C)] = |IN − 2ΣA| 2 exp( B | (IN −2ΣA)−1 ΣB+C)
2

if and only if IN − 2ΣA is positive definite.
Proof of Math.Lemma 2:

=
=
=
=

E[exp(e| Ae + B | e + C)]
Z
−1
−N
−1
exp(e| Ae + B | e + C)(2π) 2 |Σ| 2 exp( e| Σ−1 e)de
2
N
ZR
−1
−N
−1
1
(2π) 2 |Σ| 2 exp[ (e − H)| (Σ−1 − 2A)(e − H) + B | (Σ−1 − 2A)−1 B + C]de
2
2
RN
−1
¯
¯
−1
1
|Σ| 2 ¯(Σ−1 − 2A)−1 ¯ 2 exp[ B | (Σ−1 − 2A)−1 B + C]
2
−1
1
|IN − 2ΣA| 2 exp( B | (IN − 2ΣA)−1 ΣB + C)2

where H = (Σ−1 − 2A)−1 B. ||21

Proof of Lemma 1:
Since θu + θ = 0 in equilibrium, we have H = 0,
k =
l =

1
n ,qn ]
1 − 2 Cov[xV ar[qn ]1{
1

n ,qn ] γ
1 − Cov[xuV ar[qn ]Cov[xn , qn ] 1
1
V ar[xn |sn ] + V ar[xn |qn ]}, andV ar[qn ] γ IγuV ar[xn ].

Substituting them, the second order condition of the insurers willbe1
1
2
{ V ar[xn |sn ] + V ar[xn |qn ] > 0,
Cov[xn ,qn ] γγu
1 − 2 V ar[qn ]Iwhich is satisfied if and only if
1−2

Cov[xn , qn ]> 0.
V ar[qn ]Calculation shows that this is equivalent to1
Cov 2 [xn , qn ]
2
V
ar
[x
|s
]V
ar[e
]
−
> 0.
n n
n
γ 2I
V ar[qn ]||Proof of Lemma 2:
Similar to the proof of Lemma 1. ||Proof of Proposition 1:
Clear from Lemma 1 and 2. ||Proof of Lemma 3:
Direct calculation shows that the insurer’s utility E[uI (WF n )] is
given by−1
E[uI (WF n )] = E[− exp( ξ > AF ξ)]
γI
22

wheresn
ξ =  en  ,θn
1 Cov[xn ,sn ]00
2 V ar[sn ]n ,sn ]
−1
AF =  12 Cov[xV ar[xn |sn ] 0  , andV ar[sn ]
2γ I
0
0
α3


θn = α0NX

sn + α1

m=1

N
X

m6=n

en − α2 en .

Applying Math. Lemma 2, we obtain the desired result. ||Proof of Lemma 4:
Similar to the proof of Lemma 3. ||Proof of Lemma 5:
Note thatθn = α0NX

sn + α1

m=1

N
X

m6=n

en − α2 en .

thus, θn is normally distributed with mean equal to zero. The expected value of the absolute value of a normally distributed randomvariable with mean zero is well known. Apply this formula to θn we
obtain the desired result. ||Proof of Proposition 3:
Clear from the numerical example. ||23

Figure 1:

24

Figure 2:

25

Figure 3:

26

Figure 4:27