Best bounds for expected financial payoffs II: Applications

a* (3) d ~< b* (4) d ~b* (6) b* ~
{xp, x*} {xp, x*} {xp, x*} {d*, d} {d, d*} {a, a*} {b*, b} {a, d, b}

W. Hiirlimann /Journal of Computational and Applied Mathematics 82 (1997) 213-227

218

C a s e 2: {d*, d}, d ~> a*. A QP-majorant q ( x ) satisfies the necessary conditions q ( d * ) = O, q'(d*) = O, q(d) = a ( d - p). The unique solution is q(x) =

a ( d - p ) ( x - d*) 2 (d - d*) 2

(4.6)

To be a Q P - m a j o r a n t q ( x ) must lie above the line l(x) = a ( x - p) on [d, b] hence q'(d) >~ a, which implies the restriction p ~< co(d). C a s e 3: {d, d*}, d ~< b*. The degenerate quadratic polynomial q ( x ) = l(x) = a ( x - p) goes through (d, d*). U n d e r the restriction p ~< a one has further q ( x ) >~ 0 on [a, d], hence q ( x ) >~ R z ( x ) on [a, b]. C a s e 4: {a, a*}, d ~< a*. Set u = a, v = a*. A Q P - m a j o r a n t q ( x ) through (u, v) satisfies q(u) = O, q(v) = a ( v - p), q'(v) = a, and the second zero z of q ( x ) lies in the interval ( - ~ , a]. The unique solution is q ( x ) = c ( x -- v) 2 + a ( x - v ) + a ( v - p ) ,

c-

a( (vp- u )- u) z"

(4.7)

The additional condition q(z) = 0 yields the relation: UZ

p = p(z)

--

V2

u + z-

(4.8)

2v"

This increasing function lies for z ~ ( - oo, a] between the two bounds a ~< p ~< a. C a s e 5: {b*, b}, d/> b*. Setting u = b*, v = b a Q P - m a j o r a n t q ( x ) through (u, v) satisfies the conditions q ( u ) = O, q'(u) = O, q(v) = a ( v - p), and the second point of intersection z ofq(x) with the line l(x) = a ( x - p) lies in the interval [-b, ~). The unique solution is (4.9)

a(v( v- - u)p) " c - ------------~

q ( x ) = c ( x - u) 2,

Solving q(z) = a ( z - p) implies the m o n o t o n e relation VZ

--

IA 2

p = p ( z ) -- v + z -- 2u'

(4.10)

from which one obtains the restriction fl ~< p. C a s e 6: {a, d, b}, b* ~< d ~< a*. A Q P - m a j o r a n t q ( x ) through ( a , d , b ) satisfies the conditions q ( a ) = O, q ( d ) = a ( d - p ) , q(b) = a ( b - p), and the second zero z of q ( x ) lies in the interval [b, oo). One finds q ( x ) = c ( x -- a ) ( x -- z),

z=p+

c =

a ( b - p)

a ( d - p)

(b - a)(b - z)

(d - a ) ( d - z ) '

(d - p ) ( b - p)

(a - p)

U n d e r the constraint d t> a one has z/> b if and only if p ~< a.

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219

The above step by step construction of QP-majorants is summarized in Table 8. The only missing case occurs for d ~< b*, p ~> co(d). But in this situation d* > d, hence p ~> co(d) > d. But in the original scale D = o-(d - p) < 0, and Rx(X) does not define a feasible reinsurance payment because the usual constraint 0 ~< Rx(X) <~ X is not satisfied. Finally Table 9 summarizes the special case p = a = -/~/o- discussed by Heijnen and Goovaerts [9], and the Example 4.1 (continued) justifies the practical usefulness of the extended analysis.

Example 4.1 (continued). In the standardized risk scale one has a---,

1 -/~ o-

b=

3 -# o"

,

d-

2 -/~ o"

But a risk is feasible only if a <~ O, b >i 0, ab ~< - 1, hence the restrictions

1 If/~ -- 2, the m a x i m u m is obtained in case (2a) with a maximizing triatomic distribution

Table 8 Maximum expected franchise deductible payoff Conditions

Maximum

Atoms

(1) d ~> a*: (la) p ~< co(d)

(lb) co(d) ~ p ~ fl (lc) p ~> fl

d-p - ½axp b-p

{xa, x* }

(2) b * ~ d ~ a * : (2a) p ~< a (2b) a ~ p ~ (2c) c~i fl

(3) d ~ b*: (3a) p ~< a (3b) a ~ p ~ (3c) c~ ~ p ~< co(d) (3d) p >i co(d)

(- a)'a'\l - ½,~x,,

+a2j

{a, a*} {x,,, x*}

b-p

~(# - p)

(, +pa~

( - a). ,,- \T-g-G~._.:

-- ½o-x,, Payoff function is not feasible

{d, d*} {a, a*}

{~,,, x*}

W. Hiirlimann /Journal of Computational and Applied Mathematics 82 (1997) 213-227

220

Table 9 Special case p = a = - #/a in the standardized a n d original risk scale Conditions

Maximum

Atoms

Standard

Original

Standard

Original

Standard

Original

d • a*

D > 0* = / 2 + --//

1 + d2

D a 2 + (D - / 0 2

(d*, d}

{D*, D} =

d <~ a*

P ~< 0*

/~

/l

{a, a*}

{0, 0"}

/~ - D -

5. The disappearing deductible contract For the special risk support [0, B], the reinsurance contract with disappearing deductible has also been considered in [9]. If X is a risk with arbitrary support [A, B], the financial payoff of such a contract is defined by R x ( x ) = r ( x --

di)+ + (1 - r ) ( x

-

d2)+,

r

d2 -- -

-

d2 -

>/

1, 0 ~< dl

<

d2.

(5.1)

dI

In the standardized risk scale, the mean scale invariant payoff takes the form

Rz(z) = o-r(z - L)+ + o-(1 - r)(z - M ) + =

O,

z <<,L,

a r ( z - L),

L <~ z <<, M ,

o-(z - p),

z~M,

f

(5.2)

where one sets dl -/t

L - - - , o-

M

d2 -

- - , o-

Fz

p = rL + (1-- r)M -

fl (7

<0.

(5.3)

U p to the factor a, this financial payoff looks formally like a 'two-layers stop-loss structure' with however r >~ 1 fixed (cf. Section 6). This fact implies the following two statements: (i) In the interval [-M, b] the line segment a r ( x - L ) lies a b o v e the segment a(x - p). (ii) In the interval [a, M] the line segment o-r(x - L ) lies u n d e r the segment a(x - p). Using these geometric properties a look at the QP-majorants of the 'franchise deductible' and 'stop-loss' contracts show that the m a x i m u m expected payoff is attained as follows: (i) If M >~ a* the best upper bounds are taken from the 'stop-loss' Table 1 by changing b into M, d into L, and multiplying the stop-loss maxima with o-r. (ii) If M ~< a* take the values of the 'franchise deductible' in Table 8 by changing d into M. The result, summarized in Table 10, generalizes that of Heijnen and Goovaerts [-9] obtained for the special case p = a = - p/o-, that is for a support [0, B].

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221

Table 10 Maximum expected disappearing deductible payoff Conditions

Maximum

Atoms

[ 1 + aL", (-a)'(1-----~72)'ar \, - - /

{

(1) M >/a*: (la) L <~ (lb) ~ <~L <~o~(M) (lc) co(M) <<.L < M

-- ½XL'ar M--L (]~-~5)'ar

o,a*}

{M*, M}

(2) b*~
~

'

.(l÷oa

b-~a

M--a

J

{a,M, b}

{a,a*}

(2b) a <<.p <~c~ (2c) ~ <~p <~fl

( - a)" a \1 + aZ J - -tzaxp

{xp, x~}

(2d) p/> fl

a.

{b*,b}

(3) M ~
a(# - p)

{M, M*}

(3b) a ~

(l +pa) ( - a ) . a . \ l + a2J

{a,a*}

(3c) o~<~p <. ~o(m) (3d) p >~co(M)

Payoff function is not feasible

- ½axo

6. The two-layers stop-loss contract A two-layers stop-loss contract is defined by the piecewise linear convex payoff f(x)=r(x-L)+

+(1-r)(x-M)+,

O
a
(6.1)

It is a special case of the n-layers stop-loss contract defined by f ( x ) = ~ r,(x - di)+,

~ r , <<. 1,

ri >>-O,

a = do < dl < "" < dn < b.

(6.2)

A two-layers stop-loss contract has been shown optimal (for any stop-loss order preserving criterion) under the restricted set of reinsurance contracts generated by (6.2) with fixed expected reinsurance net costs E [ f ( X ) ] < E [ X ] (e.g., [24, p. 121; 19, Example VIII.3.1, p. 86-87]). It appears also as optimal reinsurance structure in the theory developed by Hesselager [10]. It belongs to the class of perfectly hedged all-finance derivative contracts introduced by the author [12, 15]. As most 'stop-loss like' treaty it may serve as a valuable substitute in situations a stop-loss contract is not available, undesirable or does not make sense (for this last point see [11, Section 4, Remarque]). To the knowledge of the author, the corresponding optimization probleins have not yet been studied.

222

W. Hiirlimann /Journal o f Computational and Applied Mathematics 82 (1997) 213-227

It suffices to w o r k in the s t a n d a r d i z e d risk scale. The interval I = [a, b] is partitioned into three pieces Io = [a, L], 11 = [L, M ] , I 2 = [M, b]. The piecewise linear segments are described by f i ( X ) = ~i x -3V O~i, ~0 = O, ~1 = r, f12 = 1, ~o = O, ~1 = - r L , ~2 = - - d, where d = r L + (1 - r ) m is the m a x i m u m deductible of the contract. 6.1. D e t e r m i n a t i o n

o f the m i n i m u m

According to C o r o l l a r y 1.5.1 it suffices to construct L - m i n o r a n t s for some X E D f,f~, 3 i = O, 1, 2, a n d Q P - m i n o r a n t s for triatomic distributions of type (T4). L - m i n o r a n t s : The diatomic risk X = {L*, L} belongs to D},yo provided L > a*. By P r o p o s i t i o n 1.5.1, in this d o m a i n of definition, the m i n i m u m is necessarily f , = E f t ( X ) ] - f o ( 0 ) = 0. Similarly, X = {L, L*}, {M*, M} belong to D~,s ' i f M > L* > 0 a n d one has f , = E [ f ( X ) ] =f~(0) = - r L . Finally in D},s, one considers X = {M, M*}, which is feasible provided M < b* a n d leads to the m i n i m u m value f , = E [ f ( X ) ] = f z(0) = - d. These results are reported in Table 13. Q P - m i n o r a n t s : It remains to determine the m i n i m u m in the following regions: (1) 0 < L ~< a* (2) 0 ~< M ~< L* (3) b* ~< M < 0 O n e has to construct Q P - m i n o r a n t s for triatomic distributions of the type (T4), which are listed in Table 11. Their feasible d o m a i n s suggest to subdivide regions (1), (3) into two subregions a n d region (2) into four subregions. This subdivision with the c o r r e s p o n d i n g feasible triatomic distributions is f o u n d in Table 12. Table 11 Triatomic distributions of type (T4) Atoms

Feasible domain

X1 X2 X3 X,~

b* <~ L <. a* b* <~ M <~a* L <~ b * <~ M <~L * L <~a* <~ M <~L *

= = = =

{a, L, b} {a, M, b} {L, M, b} {a, L, M}

Table 12 Triatomic distributions in the subregions Subregion

Type (T4)

(1.1) 0 < L ~a* (2.1) L < b*, M > a*, M ~ a* (2.3) L < b*, 0 ~
Xl ~X2

X1, Xa, X1, X2, X1,

X4 X4 X4 X3 X2

X1, X2 X2, X3

W. Hiirlimann/Journal of Computational and Applied Mathematics 82 0997) 213-227

223

Applying the QP-method it is required to construct quadratic polynomials q~(x) <~f(x), i = 1, 2, 3, 4, such that the zeros of Q~i) = qi(x) - f ( x ) are the atoms of X~. Drawing for help pictures of the situation for i = 1, 2, 3, 4, which is left to the reader, one gets through elementary calculations the following formulas: q,(x)=

(x - a ) ( x - L ) ( b - d ) (b - a)(b - L ) '

q2(x)=c(x-a)(x-z),

(b-d) (M-d) C=(b_a)(b_z)=(M_a)(M_z),

qa(x)=e(x-L)(x-y),

e=

q4(x) =

(b - d) (b - L ) ( b - y)

( M -- d) (M-- L)(M-

z= g~

y)'

d(b-a)-M(b-d) d-a

'

d(b - L) - M ( b - d) d-L

(x - a)(x - L ) ( M - d) ( M -- a ) ( M -- L)

By application of Theorem 1.4.1 it is possible to determine when the qi(x) are QP-admissible. However, in this relatively simple situation, the graphs of the qi(x)'s show that the following equivalent criteria hold: qx(x) < . f ( x ) ¢~ q , ( M ) <~f(M) = M -- d ~ d <~ 4, q2(x) <~f (x) ¢~ q2(L) <~f (L) = 0 . ~ d >~ 4, q3(x) <~f(x) ¢~ q3(a) <~f(a) = 0 ~ d <~ 4,

q4(x) <~f(x) ¢~ q4(b) <~f(b) = b - d ¢~ d >~ 4, where ~ = ( b M - aL)/[b + M - (a + L)] has been setted. Use these criteria for each subregion in Table 12 to get the remaining minimum values as displayed in Table 13. 6.2. Algorithmic evaluation o f the maximum

Through application of Theorem 1.3.1 one determines first the possible types of triatomic distributions for which the maximum may be attained.

Proposition 6.1.

Triatomic distributions necessarily of the following types:

X EDaf,q, for which there may exist a QP-majorant, are

{XL, X*}, (XL, X*)e; [a, L] × [L, M];

(D1) {Xd, X~'}, (Xa, X~')e [a, L] x [M, b]; {xM, x*}, (xM, x*) ~ [L, M] x [M, b]; (D2) {a, a*}, {b*, b};

(T1) {u,v,w} such t h a t u : = L - r ( M - - L ) > ~ a , v : = L + r ( M - L ) , w := M + (1 - r ) ( M - L) <~ b, and u <~ w* < 0, w* ~< v ~< u*;

224

W. Hiirlimann / J o u r n a l o f Computational a n d Applied Mathematics 82 (1997) 2 1 3 - 2 2 7

Table 13 Minimum expected two-layers stop-loss payoff Conditions

Minimum

Atoms

L>a* M>L* M
0

{L*, L} {L, L*}, {M*, M} {M, M*}

d<. O~L

>0

-rL -d

bM - aL b + M - (a + L)

1 +aL

<~a*

b* <~ L <~ 0, b* <~ M L ~ b * <~ M <~ L * d>~

(b - d)

(b - L)(b - a) 1 +Ly

( M - d) ( M - L ) ( M - y) d(b - L) - M ( b - d) Y= d-L

{a, L, b}

{L, M, b)

bM - aL b + M--(a

+ L)

O < L <~ a*, M <<.a* L <~ O, O <~ M <. a*

1 +az • (b -

d)

(b - a)(b - z) d(b - a) - m ( b - d) Z= d-a

{a, M, b}

b * < ~ M <.O O < L <~ a*, M >~ a*

1 +aL ( M -- a ) ( M - L)

(M - d)

a* <~ M <~ L*

(T2) {a, va, w,} such that

w.:=a-(l-~r)'(-(d-a)+~/r(L-a)(d-a))e[M,b],

and a ~< w* < O, w* ~< v. ~< a* {Ub, Vb, b} such that

Ub:= b - 2 " ( b - d - x / ( 1 - r)(b - M)(b - d)) Vb:=b+!'((1-r)(b-M)-x/(1-r)(b-M)(b-d))~[g,M],

and

Ub <<. b* < O, b* <~ Vb <<. U~

{a, L, M}

W. Hiirlimann /Journal of Computational and Applied Mathematics 82 (1997) 213-227

(T3)

{a, V,,b, b} such t h a t V,,b:=

a x / ( 1 -- r)(b - M ) + b %/(1

r)(b - M) + ~

--

rx/~ - a)

225

[L, M ] ,

-- a)

a n d b* <<. V,,b <<. a*. P r o o f . L e a d i n g to a m i n i m u m the type (T4) can be eliminated. T y p e (D1) is i m m e d i a t e l y settled o b s e r v i n g t h a t doa = L, do2 = d, d12 = M. F o r the type (D2) the cases where L, M are r a n d points are i n c l u d e d as limiting cases of the type (D1) a n d this omitted. T y p e (T1) is clear. F o r (T2) three cases m u s t be distinguished. If w e Io t h e n necessarily w = a because w ¢ L, d. T h e case w e 11 is impossible because o n e s h o u l d have w ~ L, M. If w e I 2 t h e n w = b because w ~ d, M. Similarly type (T3) m a y be possible in three ways. If(v, w) e Io x 11 then v = a, w = L because v ~ d, w ~ M. If (v, w) ~ Io x 12 t h e n v = a, w = b because v :~ L, w ¢ M. If (v, w) ~ 11 x I 2 t h e n v = M, w = b because v ¢ L, w ¢ d. H o w e v e r the two types {u, M, b}, {a, L, u} can be eliminated. I n d e e d d r a w i n g Table 14 Maximizing QP-admissible triatomic distributions Atoms

Value of A

Values of ¢, r/

Conditions

{xL, xZ}

Ao12 (xL, x*)

None

A~<0

Ao21(Xa, x*) A120(XM,X~t)

//2 None None

/32 - 8o - , / 5 {Xd, X*} {XM:, X~I}

2Col(x*)

A~<0 A~<0 - /31+.,/5

¢o

{a, a*}

2c12(x*) (d - a * ) 2

A2ol (a*, a)

t/o = d

Alo2 (a*, a)

None

Ao21(b*, b)

q2 = d -{

A120(b*, b)

None ~o-

{u, v, w}

None

A >0and~o~a M <~a*, A <~O, tlo ~ a

d-a

L~a*
/32 - / 3 1 - , / 5 ~12 - -

{b*, b}

A > 0 and q2 ~ b

2Clo(a) (d - - b * ) 2 b-d

L <~a* < M, A >0, ¢/2 ~>b b* <~L, A <~O, tlz >~b L
/30 - / 3 1 + , / 5

L < b* <~M , A >0,¢o~
2c12(b)

None

{a, va, wa}

None

r/o = d

{uh, Vb, b}

None

//2 =

{a, Va,b, b}

None

qo = L

None (d - w~) 2

tlo <~a

d-a (m -

vb)2

m + b ~ (Va,b - - L ) 2

L--a (M -- V,,b)2 rl2 = M - l b-M

rl 2 >/ b

~/o ~/b

226

w. Hiirlimann/Journal of Computational and Applied Mathematics 82 (1997) 213-227

a g r a p h in these s i t u a t i o n s s h o w s t h a t n o Q P - m a j o r a n t c a n be c o n s t r u c t e d . T h e a d d i t i o n a l c o n s t r a i n t s o n the a t o m s follow f r o m L e m m a 1.2.1. T h e precise c o n d i t i o n s u n d e r which the t r i a t o m i c d i s t r i b u t i o n s of P r o p o s i t i o n 6.1 allow the c o n s t r u c t i o n o f a Q P - m a j o r a n t follow f r o m T h e o r e m 1.4.1 a n d are d i s p l a y e d in T a b l e 14. It suffices to c h o o s e a p p r o p r i a t e l y (u, v, w ) e Ii x 1i x Ik such t h a t T h e o r e m 1.4.1 applies. Details are left to the reader. I n p a r t i c u l a r d r a w i n g pictures s h o w t h a t the d o u b l e - s i d e d interval c o n s t r a i n t s are in fact o n e - s i d e d c o n s t r a i n t s . A numerical algorithm to e v a l u a t e the m a x i m i z i n g d i s t r i b u t i o n c o n t a i n s the following steps: Step 1: G i v e in the s t a n d a r d i z e d risk scale the values a < L < M < b, r e ( 0 , 1), a < 0 < b,

ab <<, - 1 . Step 2: F i n d the finite set of t r i a t o m i c distributions, which satisfy the conditions of P r o p o s i t i o n 6.1. Step 3: F o r the finite set o f t r i a t o m i c d i s t r i b u t i o n s f o u n d in step 2, c h e c k the Q P - a d m i s s i b l e c o n d i t i o n s of T a b l e 14. I f a Q P - a d m i s s i b l e c o n d i t i o n is fulfilled, the c o r r e s p o n d i n g t r i a t o m i c d i s t r i b u t i o n is a m a x i m i z i n g distribution. Step 4: T r a n s f o r m the result b a c k to the original risk scale.

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[20] P.M. Kahn, Some remarks on a recent paper by Borch, ASTIN Bull. 1 (1961) 265-272. [21] J. Ohlin, On a class of measures of dispersion with application to optimal reinsurance, ASTIN Bull. 5 (1969) 249-266. [22] M. Pesonen, Optimal reinsurances, Scand. Actuarial J. (1984) 65-90. [23] D. Stoyan, Bounds for the extrema of the expected value of a convex function of independent random variables, Studia Scientiarum Mathematicarum Hungarica 8 (1973) 153-159. [24] A.E. van Heerwaarden, Ordering of risks: theory and actuarial applications, Ph.D. Thesis, Tinbergen Research Series no. 20. Amsterdam, 1991. [25] S. Wang, Insurance pricing and increased limits ratemaking by proportional hazards transforms, Insurance: Math. Econom. 17 (1995) 43-54.