Astin Bulletin i 1 (198o) 52-6o
AN E C O N O M I C P R E M I U M
PRINCIPLE
HANS BUHLMANN
*
Eidgen6ssische Technische Hochschule, Ziirich
1. PREMIUM CALCULATION PRINCIPLES VERSUS ECONOMIC PREMIUM PRINCIPLES
(a) T h e notion of p r e m i u m calculation principle has b e c o m e fairly generally
a c c e p t e d in the risk t h e o r y literature. F o r c o m p l e t e n e s s we r e p e a t its definition :
A p r e m i u m calculation principle is a functional ~ assigning to a r a n d o m
v a r i a b l e X (or its distribution function Fx(x)) a real n u m b e r P. I n s y m b o l s
:
premium
calculation
principle
X
--~
P
random
variable
or
Fx(x)
real n u m b e r
~
P
distribution
function
real n u m b e r
T h e i n t e r p r e t a t i o n is r a t h e r obvious. T h e r a n d o m v a r i a b l e X s t a n d s for the
possible claims of a risk whereas P is the p r e m i u m charged for assuming this
risk.
This is of course formalizing the w a y actuaries t h i n k a b o u t p r e m i u m s . I n
a c t u a r i a l terms, the p r e m i u m is a p r o p e r t y of the risk (and nothing else), e.g.
=
EEl,
etc.
(b) Of course, in economics p r e m i u m s are not on]y depending on the risk b u t
also on market conditions. L e t us a s s u m e for a m o m e n t t h a t we can describe
the risk b y a r a n d o m v a r i a b l e X (as u n d e r a)), describe the m a r k e t conditions
b y a r a n d o m v a r i a b l e Z.
T h e n we w a n t to show h o w an economic premium principle
(x, z)
P
pail of
random
variables
real n u m b e r
* This paper is greatly influenced by an exchange of ideas with Flavio Pressaco. I am
also indebted to Hans Gerber for stimulating discussions on this subject.
AN ECONOMIC PREMIUM PRINCIPLE
53
can be constructed. During the development of the paper we will also give a
clear meaning to the random variable Z.
2.
A MODEL
FOR
THE
MARKET:
RISK
EXCHANGE
In the market we are considering agents i = ~, 2, . . . , n. T h e y constitute buyers
of insurance, insurance companies, reinsurance companies.
Each agent i is characterized by his
utility function u~(x) [as usual: u~(x)> o, u~'(x)~< o]
initial wealth w,.
In this section, the risk aspect is modelled by a finite (for simplicity) probability space with states s = 1, 2 . . . . . S and probabilities r:s of state s happening.
(~ = , = ,).
m-l
Terminology
Each agent i in the market has an original risk function X,(s) : p a y m e n t caused
to i if s is happening.
He is buying an exchange ffl~nction Yds) : p a y m e n t received by i if s is happening.
The notion of price for this purchase is given by a vector (p~, Pz, • •., Ps) = P
and the understanding t h a t
s
Price [Y,] = E p~Y, (s).
Hence Ps is tile price for one unit of conditional money (conditional upon
the happening of state s).
Definition
Y = (Y1, Y2 . . . . .
S =
1,2,...,
Y,*) is a risk exchange (REX) if ~ r e ( s ) = o for all
S.
Remark
This condition represents the clearing condition of the market, which must be
satisfied in a n y closed system.
3. EQUILIBRIUM AND EQUILIBRIUM PRICE
Definition
(p, Y) is called an equilibrium of the market if
54
HANS BUHLMANN
8
a) for all i, Z r~8 u,[w,-
X,(s) + Yt (s) - ZpsYt(s)] = m a x for all possible
m,,t
choices of the exchange function Yt.
n
b) X Y,(s) = o for all s = 1, 2 . . . . .
S.
Terminology
If conditions a) and b) are satisfied
p
Y
is called an equilibrium price,
is called an equilibrium R E X .
The following observation, m a d e in the form of a theorem is useful.
Theorem A
Condition a) in the definition of an equilibrium is satisfied if and only if
C) ~ t
p~ u,t [ w , - x,(t) + Y , ( I ) - 2
PJY*(J)] =
= 2 rcsu~[w, - X,(s) + Y,(s) - ZpjY,(j)]
a--I
holds for all t.
The proof of this theorem is obtained by partial differentiation for the
" o n l y if" part and follows from the concavity of u,(x) for the " i f " part.
As a Corollary we see from this theorem t h a t for a n y equilibrium price p we
m u s t have
x p~ = ~.
(~)
e-I
Finally we must remember t h a t an equilibrium as defined in this section
ahvays exists (see e.g. DEBREU [1] or [2]).
4'
THE CASE OF AN A R B I T R A R Y P R O B A B I L I T Y SPACE U N D E R L Y I N G
T H E R I S K ASPECT
The condition t h a t the risk functions Xt(s) and the exchange functions Y~(s)
are r a n d o m variables on a finite probability space is rather unnatural. We
prefer to u n d e r s t a n d t h e m as r a n d o m variables X~(oo), Yl(~o) on an arbitrary
probability space (f~, 2f, lII).
In this situation the notion of price is given by a function q~: Y/--7 IR, and
the u n d e r s t a n d i n g t h a t Price [Y~] = f Y~(¢o) q~(¢o) d[I(¢o).
f~
AN ECONOMIC PREMIUM P R I N C I P L E
55
We then define an equilibrium as the " p a i r " (qo, Y) such t h a t
a') for all i
f ul[w~ - X~(co) + Y~(co) - J" Y~(co') q~(~o') dl-[(oo')] dII(co) = m a x for all
possible choices of the exchange function Y,(o)).
n
b') E Y,(o) = o
~-,
for all co.
Remark
T h e existence of such an equilibrium can not be p r o v e d in the same way as in
the case of a finite probability space. Actually it is not clear w h e t h e r an equilibrium always exists. In this paper, however, we take the p r a g m a t i c point of
view of assuming existence. At least in the special case t r e a t e d from section
5 on, this assumption turns out to be correct since we can explicitely calculate
qo and Y.
T h e analogous t h e o r e m to A in section 3 then is the following:
Theorem A'
Condition a') in the definition of an equilibrium is satisfied if and only if
c'~ '4 [w* - x,(~) + Y,(~) - f Y,(o') ~(~o') drl(,~')]
= ~,(~) I u; [~, - x~(~) + Y,(oo) - I Y~(oo') ~(o;) dn(~')] drl(,o) = C, ,O(~)
for almost all co.
Remark 1
Observe t h a t c') is Borch's [3] condition characterizing a P a r e t o optimal risk
exchange, t-tence we have as a consequence of t h e o r e m A' the
Corollary
A price equilibrium is a P a r e t o optimal risk exchange.
Remark 2
The converse of the corollary only holds if one allows for additional deterministic transfer p a y m e n t s .
Proof of Theorem A'
(i) if p a r t : First observe t h a t u n d e r condition c') we have J" qo(co) dF[(o~) = 1
.< ,,,
[w,-
x~(~o) + Y~(~o)- I Y,(oo')q,(oo')dn(~')]
+ u; [w, - X{(co) + Y,(oo) - ]"Y,(oo') qo(oo') dri(¢o')]
x [Y{(o~) - Y{(o~) - f [Yi(o/) - Y{(oo')] ¢~(o~') dl](~o')]
56
HANS BUHLMANN
The last summand can be rewritten using c') as
C,q~(o) [Y,(¢0) - Y,(oo) -
f [Y,(o') - Y,(o')] ~(c0') dH(o')]
I n t e g r a t i n g this t e r m with dfI(o) and using the fact t h a t f ~(o) d H ( o ) = 1
we obtain zero which completes the proof.
(if) o n l y if p a r t • Choose an a r b i t r a r y measurable set A and a real n u m b e r
¢, and replace Y,(o) b y Y~(o) + ¢[A(o)
g(~) = I u, [~,- x,(~) + Y,(~) + da(~)
-
- f {Y,(o') + ¢IA(o')} q~(00')dlfI(o')]dl-I(o)
is a differentiable real function with m a x i m u m at ~ = 0.
Hence we must have
'
t
o = g'(o) = f u~ [w, - X , ( o ) + Y,(o~) - .f Y,(o ) ~ ( o ' ) dlH(o')]
z(o)
× [zA(o~) - f IA(o') ,p(o') dn(~')] un(,~)
Hence
I z(~0) a n ( o ) = ; ~,(o) a n ( o )
A
• ; z ( o ) tin(k0)
A
Since this relation is true for a n y measurable set A , w e must have
Z(o) = $(o~) f Z(c0) dl-I (o~)
for almost all o,
which is condition c') in a b b r e v i a t e d notation.
5" THE CASE OF E X P O N E N T I A L U T I L I T Y FUNCTIONS
We are now applying T h e o r e m A' in the case of exponential utilities
1
u,(x) =
-~, [ 1 - e-~'~q
where
u~(x) = e -~'~.
0~, stands for the risk aversion,
1
- - stands for the risk tolerance unit.
Condition c') then becomes
(2)
exp [ - ~, [w, - X,(o0) + Y,(~) - J" Y~(o') q~(o0') d[I(o')]] = C,q~(o) a.s.
As a n y optimal Y,(o0) is only d e t e r m i n e d up to an additive constant, there is
no loss of generality if we assume t h a t
d') ; Y,(o') ~o(o') a n ( ~ ' ) = o.
AN ECONOMIC PREMIUM P R I N C I P L E
Absorbing
e -~'w'
57
into the c o n s t a n t C~, which then becomes C,, we have
exp [0c,X,(co) - a,Yt(o)] = C,qo(o)
a.s.
or
1
-
(3)
=
-
[in c , + zn ~o(o~)]
a.S.
We now use the clearing condition b')
n
for all oo
Z Y~(co) = o
n
and the abbreviation G Xt(co) = Z(oo).
Summing equation (3) over
Z(o~)
(4)
=
i, w e
1
- l n qo(o~) + k ,
O~
obtain
where-
1
~
= Z--
O~
I-I
1
0~
k a constant i n d e p e n d e n t of co and i
n
(k
=
11-1.
-- In
0if
C,).
The last equation yields
and the condition J" qo(oo) dII(~) = 1 finally leads to
e~Z (~)
(5)
q'(~)
-
E[e~
I n t r o d u c i n g the solution for ~(o~) into (3), we also obtain
1
X,(oo) - Y¢(oo) =
e-~ Z + k,
1
o~
1
(6)
= y , Z + kt,
where 'y, = - 1
Gt
and E k, = o.
58
IIANS BOHLMANN
Remark
From an easy exercise using BORCH'S theorem [3] one finds that the "allocation after the exchange" which is described by equation (6) is Pareto
optimal. As we know, this result, verified by explicit calculation in the special
case of exponential utility functions, is generally true. (See corollary after
Theorem A').
Finally the constants k~ appearing in (6) can be determined from
f Y,(~) q~(oo) dII(oo) = o. Hence we have
Ix&o)
~,(~) an(oo) = "r, I z(~o) q,(oo)dI](oo) + k~
OF
E [ X , e~,Z~
E[Ze ~'z]
- + ki
E[e~Z] - yi E[eaz]
(7)
for all i.
Hence
(8)
Observe that our explicitely calctflated (qo, Y) is an equilibrium which a
posteriori justifies our original assumption of existence of such equilibrium.
6.
AN ECONOMIC PREMIUM PRINCIPLE
Equation (5) defines an economic premium principle for all random variables
X: f~ --~ IR. We have
G E X , z~ -
E[Xe*,z~,
where Z = 2 X, is the sum of original risk
E[e~,Z~
functions in the market.
It is most interesting to observe that in the case where X and Z - X are
independent, one obtains
E[Xe~.~ E[e~,(z-x)]
E [ X e ~x]
~ [ X , Z] = E[e~,X~E[e~ (z- x)] - E[e~.r] •
Hence in this case our economic principle reduces to a "standard" premium
calculation principle
~[X~ -
E[Xe~,X~
E[e:,X] , which one might call tile Esscher principle (because of its formal connection with the Esscher
transform).
59
AN ECONOMIC P R E M I U M P R I N C I P L E
1
The parameter -,
as we h a v e seen, has the intuitive m e a n i n g of tim risk
tolerance unit of the whole m a r k e t .
L e t us c o m p a r e this (new) Esscher principle with the e x p o n e n t i a l principle
for a g e n t i
with risk tolerance unit
1
54',) [,x3 = ~, In EEe~'X].
-
Tim two principles carl also be written b y m e a n s of the m o m e n t g e n e r a t i n g
function M x ( t ) = E[e tx] a n d its d e r i v a t i v e 2l'/~y(t) = E[XdX].
Hence
t
M x (~)
d
~ : [ " q - Mx(~) - dt [ln Mx(t)] I t_,
1
-
In M x ( a d .
O b s e r v e t h a t gp~[.X~ f as o~ f and hence
g
# ~ ) EX] = g f i)(t)
i x ( t ) dt
o
M)(~)
-
-
Mx(~)
= &E.X3.
Hence
(9)
and approximately
(io)
5~ ~' E-\3 ~ ,Y~[~3
F o r m u l a e (9) a n d (lo) give rise to an interesting i n t e r p r e t a t i o n .
A n e w c o m e r ( n m n b e r e d as ,n + l) to the m a r k e t , who only offers cover b u t
does not bring a n y additional risks (e.g. a professional reinsurer) can easily
check individual r a t i o n a l i t y b y testing w h e t h e r
n+l
;#)
Dx3 .~ @£~3,
where
1
1
- -- ~ '
-
According to (lO), the b o r d e r c a s e for individual r a t i o n a l i t y is a b o u t characterized b y
O~n+z =
2~
hence
~n+l
-~-
+ ... +
{
+ ..~ - -
,
~n+l
60
HANS BUHLMANN
1
or
1
otn+l
I
+ ...
~1
+ -Otn
which leads to the "rule of t h u m b " t h a t the m a r k e t should only be joined b y
such a n e w c o m e r if he offers at least as m a n y new risk tolerance units as
a l r e a d y present in the m a r k e t before him.
REFERENCES
[1] DEBREU, O. (1974). F o u r Aspects of t h e M a t h e m a t i c a l T h e o r y of E c o n o m i c Equilibrium, Proceedings of the International Congress of Mathematwmns, Vancouver.
[2] DEBREU, G. (1976). Werttheorie (see Anhang), Springer Verlag.
[3] BORCH, K. (196o). The Safety loading of Reinsurance Premiums, Skandinavisk
A ktuarietidskrift.