Indicator Function and Hattendorff Theorem

INDICATOR FUNCTION

AND

HATTENDORFF THEOREM

Hans U. Gerber,* Bartholomew P.K. Leung,† and Elias S.W. Shiu‡

ABSTRACT This paper presents an integration-by-parts proof of the Hattendorff theorem in the general fully continuous insurance model. The proof motivates a derivation of the theorem in the general fully discrete insurance model. Increments of a martingale over disjoint time intervals are uncorrelated random variables; the paper explains that the Hattendorff theorem can be viewed as an application of this result. A notable feature of the paper is the extensive use of the indicator function.

1. INTRODUCTION

section in Wolthuis (1987) gives an interesting history of the theorem and lists the early researchers.

This pedagogical paper is dedicated to the memory of Professor Cecil J. Nesbitt (1912–2001), our mentor and friend. One goal of this paper is to show that the indicator function can be a useful tool in life contingencies. Another goal is to present a derivation of the Hattendorff theorem (Hattendorff 1868) in the general fully discrete insurance model. In the general fully continuous model, there is an elegant proof of the Hattendorff theorem attributable to Hickman (1964). This proof motivates a derivation of the theorem in the general fully discrete model, which we present in Section 7. Increments of a martingale over disjoint time intervals are uncorrelated. The Hattendorff theorem can be viewed as a special case of this result. This is explained in Section 8. The Hattendorff theorem can be generalized in various directions. Some research papers covering it in the past 15 years are Wolthuis (1987), Ramlau-Hansen (1988), Norberg (1992 and 1996), and Milbrodt (1999 and 2000). The introductory

2. IVERSON’S USE FUNCTION

OF THE INDICATOR

In probability and statistics, the indicator function of a set A, denoted as IA( x) or 1A(x), is the function that takes the value 1 if x is a member of the set A, and takes the value 0 otherwise. In mathematical analysis, such a function is usually called a characteristic function and denoted as ␹A( x). (In probability and statistics, the characteristic function is, up to a multiplicative constant, the Fourier transform of a probability density.) The concept can be adapted as follows. Let P be a statement. Then I(P) takes the value 1 if the statement P is true, and takes the value 0 otherwise. Knuth (1992) calls this Iverson’s convention. Many actuaries are familiar with the elegant computer programming language APL, which was invented by Kenneth E. Iverson. The relational statement is discussed in Iverson’s pioneering book on computer science (1962, p. 11), however he does not use the notation I(P). The notation in Knuth (1992) and Graham, Knuth, and Patshnik (1989) is [P]. The idea of an indicator function can be traced back to the work on mathematical logic by George Boole (1815–1864). Edmund C. Berkeley (1909 –1988), an actuary and pioneer in computer science, published a lengthy article (Berkeley 1937) to introduce Boolean algebra and its

* Hans U. Gerber, A.S.A., Ph.D., is Professor of Actuarial Science, Ecole des H.E.C., Universite´ de Lausanne, CH-1015 Lausanne, Switzerland, e-mail: [email protected]. † Bartholomew P.K. Leung, Ph.D., is Assistant Professor, Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, e-mail: [email protected]. ‡ Elias S.W. Shiu, A.S.A., Ph.D., is Principal Financial Group Foundation Professor of Actuarial Science, Department of Statistics and Actuarial Science, University of Iowa, Iowa City, Iowa 52242-1409, and Visiting Chair Professor of Actuarial Science, Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, e-mail: [email protected].

38

INDICATOR FUNCTION

AND

HATTENDORFF THEOREM

39

insurance applications to the actuarial community. Let us now present an elegant application of the indicator function (Knuth 1992). We are to derive the formula to reverse the order of a double summation: For integers m and n, m ⱕ n,

冘 冘 f共 j, k兲 ⫽ 冘 冘 f共 j, k兲. n

j

n

j⫽m k⫽m

while the expectation of the last expression (the right-hand side) in Equation (4) is

冘v ⬁

冘v ⬁

c k E关I共K ⱖ k兲兴 ⫽

k

k⫽0

ck Pr共K ⱖ k兲

k

k⫽0

冘vc ⬁

n

⫽

(1)

kk

p x.

k⫽0

k⫽m j⫽k

Using the indicator function, we can write any sum as an infinite sum without limits. Thus, the left-hand side of Equation (1) is

k

Hence, we have the formula

冘 冘v ⬁

E关Z兴 ⫽

冘 f共 j, k兲 I共m ⱕ j ⱕ n兲 I共m ⱕ k ⱕ j兲

冘vc ⬁

j

k

ck jpx qx⫹j ⫽

j⫽0 k⫽0

k

p,

kk x

k⫽0

(5)

j,k

⫽

冘 f共 j, k兲 I共m ⱕ k ⱕ j ⱕ n兲 j,k

⫽

冘 f共 j, k兲 I共m ⱕ k ⱕ n兲 I共k ⱕ j ⱕ n兲,

(2)

j,k

which is the right-hand side of Equation (1). A key step in establishing Equation (2) is that, for two statements P and Q, I共P and Q兲 ⫽ I共P兲 I共Q兲,

(3)

which is a fundamental formula in Boolean algebra. Next, we use the indicator function to show the equivalence between the aggregate payment form and the current payment form of the actuarial present value of a life annuity. In most of this paper, we shall use the notation in Bowers et al. (2000). The present value of a life annuity to ( x) with annual payment of ck on survival to age x ⫹ k, k ⫽ 0, 1, 2, . . . , is

冘v

k⫽0

冘v

k

ck ⫽

k

c k I共K ⱖ k兲,

冘冘v

k⫽0

j⫽0 k⫽0

冘 冘v ⬁

j

k

v t c共t兲 dt ⫽

0

c k Pr共K ⫽ j兲 ⫽

冕

⬁

I共T ⬎ t兲v t c共t兲 dt.

(6)

0

Taking expectation of each side of Equation (6), we have E

冋冕

T

册冕

⬁

v c共t兲 dt ⫽ t

0

E关I共T ⬎ t兲兴vt c共t兲 dt.

(7)

0

Because E关I共T ⬎ t兲兴 ⫽ Pr共T ⬎ t兲 ⫽ tpx,

(4)

where K ⫽ K( x) is the curtate future lifetime random variable. The expectation of the middle expression in Equation (4) is the double sum ⬁

冕

T

⬁

K

Z⫽

which is usually derived by summation by parts or by applying Equation (1). Note that Bowers et al. (2000, Exercise 5.39.a) has an expression equivalent to Equation (4). For the continuous analogue of the above, let c(t), t ⱖ 0, be the payment rate function of a life annuity payable to (x). That is, the amount c(t) dt is paid between time t and time t ⫹ dt, if ( x) survives to time t. With T ⫽ T(x) denoting the future lifetime random variable, the present value of the life annuity is

(8)

Equation (7) becomes

冕 冋冕 ⬁

0

s

0

册

v c共t兲 dt spx ␮x共s兲 ds ⫽ t

冕

⬁

p vt c共t兲 dt,

t x

0

(9)

j

j⫽0 k⫽0

k

ck jpx qx⫹j,

which is usually derived by integration by parts or by reversing the order of integration.

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NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 7, NUMBER 1

3. HEAVISIDE FUNCTION DELTA FUNCTION

With Y ⫽ T, we have

AND

E关I共t ⬍ T ⱕ t ⫹ dt兲兴 ⫽ tpx ␮x共t兲 dt,

The function

再

1 t⬎0 H共t兲 ⫽ 0 t ⬍ 0

(10)

is called a Heaviside function or Heaviside unit function in honor of the British scientist Oliver Heaviside (1850 –1925). Its derivative, H⬘(t), is called a delta function or Dirac delta function in honor of physicist Paul A. M. Dirac (1902–1984), who won a Nobel prize in 1933. Because H⬘(0) is infinite, H⬘(t) is not a function but a generalized function. An important property of the delta function is that, for each continuous function g,

冕

a formula that will be used in Sections 5 and 6. We end this section with a solution to Exercise 5.32 in Bowers et al. (2000). We are to differen៮ a៮ )x:n with respect to n. The first derivatiate (D tive of the positive part function, t⫹ ⫽

(11)

៮ a៮ 兲 x:n ⫽ 共D

⫽ I共t ⬍ y ⱕ t ⫹ dt兲,

(13) ⫽

applying which to Equation (11) yields

冕

(19)

(20)

冕

⬁

共n ⫺ s兲⫹ sEx ds

(21)

that ⳵ ៮ a៮ 兲 x:n ⫽ 共D ⳵n

⫽ I共t ⫹ dt ⱖ y兲 ⫺ I共t ⱖ y兲

,

0

Now, H⬘共t ⫺ y兲 dt ⫽ H共t ⫺ y ⫹ dt兲 ⫺ H共t ⫺ y兲

0 t⬍0

It follows from

The value of H(0) can be defined arbitrarily. By setting H(0) ⫽ 1, we have (12)

tⱖ0

t

d t ⫽ H共t兲, t ⫽ 0. dt ⫹

⫺⬁

H共t兲 ⫽ I共t ⱖ 0兲.

再

is the Heaviside function:

⬁

g共t兲 H⬘共t ⫺ y兲 dt ⫽ g共 y兲.

(18)

冕 冕 冕

⬁

0

⳵ 共n ⫺ s兲⫹ sEx ds ⳵n

⬁

H共n ⫺ s兲 s Ex ds

0

⬁

g共t兲 I共t ⬍ y ⱕ t ⫹ dt兲 ⫽ g共 y兲.

(14) ⫽

⫺⬁

g共Y兲 ⫽

冕

g共t兲 I共t ⬍ Y ⱕ t ⫹ dt兲,

(15)

⫺⬁

E关I共t ⬍ Y ⱕ t ⫹ dt兲兴 ⫽ Pr共t ⬍ Y ⱕ t ⫹ dt兲 ⫽ dFY共t兲, (16) where FY(t) is the cumulative distribution function of Y. If the probability density function of Y, fY(t), exists, then E关I共t ⬍ Y ⱕ t ⫹ dt兲兴 ⫽ fY共t兲 dt.

(17)

Ex ds ⫽ a៮ x:n .

(22)

We now calculate the second derivative by applying Equation (11): ⳵2 ៮ a៮ 兲 x:n ⫽ 共D ⳵n 2

a formula that will be needed in Section 5. Also, note that

s

0

Replacing y by a random variable Y, we have ⬁

n

⫽

冕 冕 冕

⬁

0

⳵2 共n ⫺ s兲⫹ sEx ds ⳵n2

⬁

H⬘共n ⫺ s兲 s Ex ds

0

⫽

⬁

0

H⬘共s ⫺ n兲 s Ex ds ⫽ nEx.

(23)

INDICATOR FUNCTION

HATTENDORFF THEOREM

AND

41

4. MODEL OF A GENERAL FULLY DISCRETE INSURANCE Here we consider the model, presented in Bowers et al. (2000, Sect. 8.2) and also in Gerber (1997, Sect. 5.5), of a general fully discrete insurance on (x). For j ⫽ 1, 2, 3, . . . , the death benefit in the j-th policy year is bj, payable at time j, which is the end of the policy year of death; the benefit premium payment in the j-th policy year is ␲j⫺1, payable at time j ⫺ 1, which is the beginning of the policy year. The random variable of insurer’s loss is

冘v ␲. K

L⫽v

K⫹1

b K⫹1 ⫺

k

(24)

k

k⫽0

Note that the time-0 reserve, 0V, is not assumed to be zero in this paper. A main purpose of this paper is to derive the Hattendorff theorem, which gives an elegant formula for Var(L) and, similarly, for Var(hL兩K ⱖ h). Using Equation (26) to determine the variance of L results in a complicated formula, because {Ck} are correlated random variables, as shown in Bowers et al. (2000, Exercise 8.4). However, in the general fully continuous model, there is a rather simple way to derive the Hattendorff theorem, and the proof motivates an approach in the general fully discrete model. We end this section by deriving a recursion formula for benefit reserves, which will be needed in Section 6. Since

Now,

冘 ⬁

冘v

v hhL ⫽ v h C h ⫹

⬁

v K⫹1 b K⫹1 ⫽

k⫹1

k⫽h⫹1

b k⫹1 I共K ⫽ k兲

we have a recursion formula for prospective loss random variables,

k⫽0

and

冘v␲ ⫽冘v␲

h

⬁

K

k

k

k

k⫽0

k

I共K ⱖ k兲.

Hence, with the definition C k ⫽ vb k⫹1 I共K ⫽ k兲 ⫺ ␲ k I共K ⱖ k兲,

(25)

h⫹1

冘v C. ⬁

h

(26)

k

L ⫽ 共1 ⫹ i兲 h

vb h⫹1 E关I共K ⫽ h兲兩K ⱖ h兴 ⫺ ␲h E关I共K ⱖ h兲兩K ⱖ h兴

冘vC⫽冘v

while the second expectation, by conditioning on whether the life will survive another year, is E关h⫹1L兩K ⫽ h兴 Pr关K ⫽ h兩K ⱖ h兴 ⫹ E关h⫹1L兩K ⱖ h ⫹ 1兴 Pr关K ⱖ h ⫹ 1兩K ⱖ h兴 ⫽ 0 qx⫹h ⫹ h⫹1V px⫹h.

⬁

k

k

k⫽h

k⫺h

Ck

(27)

With these two formulas, Equation (31) becomes

k⫽h

h

and h

V ⫽ E关hL兩K ⱖ h兴, V ⫽ E关L兴.

V ⫽ vb h⫹1 q x⫹h ⫺ ␲ h ⫹ v

h⫹1

V p x⫹h ,

(32)

(28)

which is equation (8.3.9) in Bowers et al. (2000). For the application in Section 6, we rewrite Equation (32) as

(29)

␲ h ⫽ 共b h⫹1 ⫺ h⫹1 V兲vq x⫹h ⫹ 共v

respectively. Hence, 0L ⫽ L, and 0

(31)

⫽ vbh⫹1 qx⫹h ⫺ ␲h,

The random variable Ck is the time-k value of the net cash flow in policy year k ⫹ 1, and Equation (25) is equivalent to equation (8.3.1) in Bowers et al. (2000). For a nonnegative integer h, the time-h prospective loss random variable and benefit reserve are defined by

h

(30)

V ⫽ E关Ch兩K ⱖ h兴 ⫹ vE关h⫹1L兩K ⱖ h兴.

k⫽0

⬁

L,

The first expectation on the right-hand side of Equation (31) is

Equation (24) becomes k

L ⫽ Ch ⫹ v

which is equation (8.3.7) in Bowers et al. (2000). Substituting Equation (30) in Equation (28) yields

k⫽0

L⫽

v k C k ⫽ v h C h ⫹ v h⫹1 h⫹1L,

h⫹1

V ⫺ h V兲,

(33)

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NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 7, NUMBER 1

which is equation (8.3.14) in Bowers et al. (2000). This formula shows that the benefit premium has two components. The first component is the oneyear term insurance benefit premium for the net amount at risk, and the second is a savings component that adjusts the benefit reserve.

5. HATTENDORFF THEOREM FOR GENERAL FULLY CONTINUOUS INSURANCE In this section, we consider the model presented in Bowers et al. (2000, Sect. 8.2) of a general fully continuous insurance on ( x). The death benefit payable at the moment of death, t, is bt, and benefit premiums are payable continuously with the amount ␲(t) dt payable between time t and time t ⫹ dt. The random variable of insurer’s loss is L៮ ⫽ v T b T ⫺

冕

T

v t ␲共t兲 dt,

(34)

and t

v bT ⫽ T

冕

⬁

v b t I共t ⬍ T ⱕ t ⫹ dt兲. t

(35)

0

冕

T

v t ␲共t兲 dt ⫽

0

冕

v t I共T ⬎ t兲␲共t兲 dt.

(36)

0

Thus, if we define the net cash flow rate, C(t), by C共t兲 dt ⫽ b t I共t ⬍ T ⱕ t ⫹ dt兲 ⫺ I共T ⬎ t兲␲共t兲 dt, t ⱖ 0,

(37)

then Equation (34) becomes L៮ ⫽

冕

⫽

冕

⬁

v2t共bt ⫺ tV៮ 兲2tpx ␮x共t兲 dt,

a special case of which is Exercise 8.24 in Bowers et al. (2000). To prove Equation (42), we need, instead of Equation (34), another expression for L៮ . It follows from Equations (40), (39), (37), (18), and (8) that v t t V៮ ⫽

冕

⬁

v s E关C共s兲 ds兩T ⬎ t兴

t

⫽

1 Pr共T ⬎ t兲 1 tpx

冕

冕

⬁

vs E关C共s兲 ds兴

t

⬁

vs关bs spx ␮x共s兲 ⫺ spx ␲共s兲兴 ds.

t

Differentiating Equation (43) with respect to t yields the following form of the Thiele differential equation (Bowers et al. 2000, Exercise 8.22.b): d 共v t t V៮ 兲 ⫽ ␮ x 共t兲v t t V៮ ⫺ v t 关bt ␮x共t兲 ⫺ ␲共t兲兴. dt

(38)

L៮ ⫽ v T b T ⫺

冕冋 T

which corresponds to Equation (26). For t ⱖ 0, the time-t prospective loss random variable and benefit reserve are defined by

冕

册

t

Integrating and rearranging, we have L៮ ⫺ 0 V៮ ⫽ v T 共b T ⫺ T V៮ 兲 ⫺

⬁

v s C共s兲 ds

(44)

d 共v t t V៮ 兲 ⫹ v t 共b t ⫺ t V៮ 兲␮ x 共t兲 dt. dt

0

L៮ ⫽ 共1 ⫹ i兲 t

(43)

We can apply Equation (44) to replace the integrand, vt␲(t), in Equation (34):

0

t

(42)

0

⬁

v t C共t兲 dt,

(41)

Var共L៮ 兲 ⫽ E共关vT共bT ⫺ TV៮ 兲兴2兲

⫽

⬁

V៮ ⫽ E关L៮ 兴.

The Hattendorff theorem for the general fully continuous model is:

0

As in Equation (6), we have

(40)

respectively. Hence, 0L៮ ⫽ L៮ , and

0

where T ⫽ T(x) is the future lifetime random variable. It follows from Equation (15) that

V៮ ⫽ E关tL៮ 兩T ⬎ t兴,

(39)

冕

T

v t 共b t ⫺ t V៮ 兲␮ x 共t兲 dt.

0

(45)

INDICATOR FUNCTION

HATTENDORFF THEOREM

AND

43

By Equation (41), the left-hand side of Equation (45) is L៮ ⫺ E关L៮ 兴. Consequently, Var共L៮ 兲

冉冋

冕

⫽ E vT共bT ⫺ TV៮ 兲⫺

T

册冊

Var共sL៮ 兩T ⬎ s兲,

2

vt共bt ⫺ tV៮ 兲␮x共t兲 dt

0

.

(46)

Now Equation (42) will follow, once we establish

冉冋冕

T

E

0

册冊 2

vt共bt ⫺ tV៮ 兲␮x共t兲 dt

冉冋

冕

⫽ 2E vT共bT ⫺ TV៮ 兲

T

册冊

(47)

s ⱖ 0.

(48)

0

冕

s

v t 共b t ⫺ t V៮ 兲␮ x 共t兲 dt,

Var共sL៮ 兩T ⬎ s兲

冕

⬁

vt共bt ⫺ tV៮ 兲␮x共t兲 dt .

s ⱖ 0,

by considering ␲(s ⫹ t) and bs⫹t in place of ␲(t) and bt, respectively. The result is:

⫽

v2t共bs⫹t ⫺ s⫹tV៮ 兲2tpx⫹s ␮x共s ⫹ t兲 dt, s ⱖ 0.

(51)

0

6. MOVING FROM CONTINUOUS TO DISCRETE

To prove Equation (47), we write ␾共s兲 ⫽

properties that are special to the valuation date of time 0, other than that the insured is alive at time 0. In particular, it was not assumed that 0V៮ ⫽ 0. Hence, the above can be readily extended to the determination of the conditional variance,

In his discussion on Hickman’s paper, Nesbitt wrote:

0

“The author has neatly obtained Hattendorf’s theorem for the continuous case discussed in the paper. For such case, there are a number of calculus devices which permit one to prove the theorem but leave one somewhat in the dark as to why it holds” (Hickman 1964, p. 149).

Then, the left-hand side of Equation (47) is E共关␾共T兲兴2兲 ⫽

冕

⬁

0

⫽⫺

关␾共s兲兴2 dFT共s兲

冕

⬁

关␾共s兲兴2 d关1 ⫺ FT共s兲兴.

(49)

0

Integrating by parts, we have E共关␾共T兲兴2兲 ⫽ 2

冕

⬁

␾共s兲␾⬘共s兲关1 ⫺ FT共s兲兴 ds.

(50)

0

Since ␾⬘共s兲 ⫽ v s 共b s ⫺ s V៮ 兲␮ x 共s兲 and ␮ x 共s兲关1 ⫺ F T 共s兲兴 ⫽ f T 共s兲, we see that Equation (50) is Equation (47). This elegant proof of the Hattendorff theorem in the general fully continuous model can be found at the end of Section I in Hickman (1964). In deriving Equation (42), we did not use any

We hope that the remainder of this paper can give some clarification. In fact, various ideas in the rest of this paper can be found in Nesbitt’s discussion and in the two editions of Bowers et al. (1986, 2000). In the derivation of Equation (42), there are two key steps. The first is Equation (45), expressing the loss random variable, L៮ , in the “pure risk” or “net amount at risk” form. The second is Equation (50), the integration-by-parts calculation. The first step has a ready counterpart in the general fully discrete model. Applying Equation (33) to Equation (24), we have

冘 关共v K

L⫽v

K⫹1

b K⫹1 ⫺

k⫹1

k⫹1

V ⫺ v k k V兲

k⫽0

⫹ vk⫹1共bk⫹1 ⫺ k⫹1V兲qx⫹k兴,

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NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 7, NUMBER 1

or

It follows from Equation (45) and Equation (57) that

L ⫺ 0 V ⫽ v K⫹1 共b K⫹1 ⫺ K⫹1 V兲

冘v

L៮ ⫺ 0 V៮ ⫽

K

⫺

k⫹1

共b h⫹1 ⫺ h⫹1 V兲q x⫹h .

(52)

As for the second step, one may attempt the technique of summation by parts. However, there is a hurdle: The summation-by-parts formula is more intricate than the integration-by-parts formula. The method of integration by parts is a consequence of the product rule, (53)

⌬关 g共k兲h共k兲兴 ⫽ g共k兲⌬h共k兲 ⫹ h共k兲⌬g共k兲 (54)

Here ⌬ denotes the forward difference operator, ⌬g共k兲 ⫽ g共k ⫹ 1兲 ⫺ g共k兲.

(58)

By Equations (57), (18), and (8), E关⌳共t兲 dt兴 ⫽ 共bt ⫺ tV៮ 兲关tpx ␮x共t兲 dt ⫺ tpx ␮x共t兲 dt兴 ⫽ 0. (59) Hence, Var关⌳共t兲 dt兴 ⫽ 共bt ⫺ tV៮ 兲2 E共关I共t ⬍ T ⱕ t ⫹ dt兲 ⫺ I共T ⬎ t兲␮x共t兲 dt2兲.

while summation by parts follows from a more awkward formula,

⫹ 关⌬g共k兲兴关⌬h共k兲兴.

v t ⌳共t兲 dt.

0

k⫽0

d关 g共t兲h共t兲兴 ⫽ g共t兲 dh共t兲 ⫹ h共t兲 dg共t兲,

冕

⬁

(60) Because the square of an indicator function is itself, we have 关I共t ⬍ T ⱕ t ⫹ dt兲 ⫺ I共T ⬎ t兲␮ x 共t兲 dt兴 2 ⫽ I共t ⬍ T ⱕ t ⫹ dt兲

(55)

It turns out that the Hattendorff theorem in the general fully discrete model is:

⫹ terms of order higher than dt. Thus, Equation (60) is Var关⌳共t兲 dt兴 ⫽ 共bt ⫺ tV៮ 兲2tpx ␮x共t兲 dt

Var共L兲 ⫽ E共关vK⫹1共bK⫹1 ⫺ K⫹1V兲兴2 px⫹K兲. (56) With the extra survival probability factor, Equation (56) is more complicated than Equation (42). To find a derivation for the Hattendorff theorem in the general fully discrete model, we compare Equation (34) with Equation (45). Taking note of Equation (37), we define ⌳(t) by

(61)

by applying Equation (18) and ignoring those terms of order higher than dt. Comparing Equation (61) with Equation (42), we see that Var共L៮ 兲 ⫽

冕

⬁

v2t Var关⌳共t兲 dt兴.

(62)

0

In contrast,

⌳共t兲 dt ⫽ 共bt ⫺ tV៮ 兲关I共t ⬍ T ⱕ t ⫹ dt兲

Var共L៮ 兲 ⫽ Cov共L៮ , L៮ 兲

⫺ I共T ⬎ t兲 Pr共T ⱕ t ⫹ dt兩T ⬎ t兲兴

⫽ Cov

⫽ 共bt ⫺ tV៮ 兲关I共t ⬍ T ⱕ t ⫹ dt兲 ⫺ I共T ⬎ t兲␮x共t兲 dt兴.

(57)

If ( x) is alive at time t, ⌳(t) dt is the loss for a term insurance from time t to time t ⫹ dt, with death benefit being the net amount at risk bt ⫺ tV៮ .

冕冕 ⬁

⫽

冋冕

0

⬁

0

v ⌳共s兲 ds, s

冕

⬁

0

册

vt⌳共t兲 dt

⬁

vs⫹t Cov关⌳共s兲 ds, ⌳共t兲 dt兴.

(63)

0

In view of Equations (62) and (63), a key to the Hattendorff theorem is that ⌳(s) ds and ⌳(t) dt are uncorrelated for s ⫽ t.

INDICATOR FUNCTION

AND

HATTENDORFF THEOREM

45

7. HATTENDORFF THEOREM FOR GENERAL FULLY DISCRETE INSURANCE

related random variables. To show this, we note that ⌳ h ⫽ ⌳ h I共K ⱖ h兲

Motivated by the above, we now define

and, for 0 ⱕ j ⬍ h,

⌳h ⫽ v共bh⫹1 ⫺ h⫹1V兲关I共K ⫽ h兲

⌳ j I共K ⱖ h兲 ⫽ v共b j⫹1 ⫺ j⫹1 V兲共⫺qx⫹j兲I共K ⱖ h兲.

⫺ I共K ⱖ h兲 Pr共K ⫽ h兩K ⱖ h兲兴

Hence, applying Equation (66), we obtain, for 0 ⱕ j ⬍ h,

⫽ v共bh⫹1 ⫺ h⫹1V兲关I共K ⫽ h兲 ⫺ I共K ⱖ h兲qx⫹h兴, h ⫽ 0, 1, 2, . . .

Cov共⌳j, ⌳h兲 ⫽ E关⌳j ⌳h兴

(64) If (x) is alive at time h, ⌳h is the loss for a one-year term insurance in policy year h ⫹ 1, with death benefit being the net amount at risk bh⫹1 ⫺ h⫹1V. See also Bowers et al. (2000, Exercise 8.31.a) and Gerber (1997, eq. 6.7.2). It follows from Equations (52) and (64) that

⫽ ⫺v共bj⫹1 ⫺ j⫹1V兲qx⫹j E关⌳h兴 ⫽ 0.

(70)

This proves Equation (68). Substituting Equation (67) in Equation (68) yields

冘v ⬁

Var共L兲 ⫽

共bh⫹1 ⫺ h⫹1V兲2hpx px⫹h qx⫹h.

2共h⫹1兲

(71)

h⫽0

冘v⌳, ⬁

L ⫺ 0V ⫽

h

h

(65)

h⫽0

which is analogous to Equation (58). We now prove the following three formulas, which are analogous to Equations (59), (61), and (62), respectively: E关⌳h兴 ⫽ 0, Var共⌳h兲 ⫽ v2共bh⫹1 ⫺ h⫹1V兲2hpx px⫹h qx⫹h,

Var共kL兩K ⱖ k兲

(67)

冘v ⬁

⫽

共bh⫹1 ⫺ h⫹1V兲2h⫺kpx⫹k px⫹h qx⫹h,

2共h⫺k⫹1兲

(72)

h⫽k

(66)

which is Theorem 8.5.1.b in Bowers et al. (2000).

冘v

REMARKS

⬁

Var共L兲 ⫽

Using an argument similar to that at the end of Section 5, we can extend the derivation of Equation (71) to obtain

2h

Var共⌳h兲.

(68)

Formula (65) can be generalized as

Formula (66) is an immediate consequence of Equation (64). It follows from Equation (66) that Var共⌳h兲 ⫽ E关⌳h2兴 ⫽ v 2 共b h⫹1 ⫺ h⫹1 V兲 2 ⫻ E共关I共K ⫽ h兲 ⫺ I共K ⱖ h兲qx⫹h兴2兲,

冘v ⬁

h⫽0

(69)

which is analogous to Equation (60). Because the square of an indicator function is itself and I共K ⫽ h兲 I共K ⱖ h兲 ⫽ I共K ⫽ h兲, we obtain Equation (67) from Equation (69). Formula (68) will follow from Equation (65) once we show that {⌳h} is a sequence of mutually uncor-

k L ⫺ k V I共K ⱖ k兲 ⫽

h⫺k

⌳ h,

(73)

h⫽k

which should be compared with Equation (27). This formula, which is the same as equation (8.5.9) in Bowers et al. (2000), implies Equation (72). A consequence of Equation (73) is the recursion formula k

L ⫺ k V I共K ⱖ k兲 ⫽ v j 关 k⫹j L ⫺ k⫹j V I共K ⱖ k ⫹ j兲兴

冘

k⫹j⫺1

⫹

v h⫺k ⌳ h ,

(74)

h⫽k

which should be compared with

冘

k⫹j⫺1 j k L ⫽ v k⫹j L ⫹

h⫽k

v h⫺k C h .

(75)

46

NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 7, NUMBER 1

respect to {Ik}. For each pair of nonnegative integers h and k,

It follows from Equation (74) that Var共kL兩K ⱖ k兲

E关Yk⫹h兩Ik兴 ⫽ E关E关L兩Ik⫹h兴兩Ik兴 ⫽ E关L兩Ik兴 ⫽ Yk;

⫽ v2j Var共k⫹jL兩K ⱖ k兲

冘

the first and third equality follow from Equation (77), and the second equality is obtained by applying the law of iterated expectations. It follows from Equation (80) that, for h ⱖ 0 and k ⱖ 0,

k⫹j⫺1

⫹

v2共h⫺k兲 Var共⌳h兩K ⱖ k兲

h⫽k

⫽ v2jjpx⫹k Var共k⫹jL兩K ⱖ k ⫹ j兲

冘

E关⌬Yk⫹h兩Ik兴 ⫽ E关Yk⫹h⫹1兩Ik兴 ⫺ E关Yk⫹h兩Ik兴

k⫹j⫺1

⫹

(80)

v

共bh⫹1 ⫺ h⫹1V兲

2共h⫺k⫹1兲

2

p

h⫺k x⫹k

px⫹h qx⫹h,

h⫽k

⫽ Y k ⫺ Y k ⫽ 0.

(81)

(76) which is Theorem 8.5.1.c in Bowers et al. (2000).

Because of Equation (77), the difference ⌬Yk is a function of the elements of Ik⫹1. Hence, for h ⱖ 1, k ⱖ 0,

8. MARTINGALE APPROACH HATTENDORFF THEOREM

Cov共⌬Yk, ⌬Yk⫹h兲 ⫽ E关共⌬Yk兲共⌬Yk⫹h兲兴

TO

The Hattendorff theorem has been a rather mysterious result for many actuaries, since the ⌳’s are obviously not independent random variables. A purpose of this section is to point out that the framework of martingales provides a way to understand the result better. Such a framework can be found in publications such as Bu¨ hlmann (1976), Gerber (1976 and 1979, Chap. 3), and Patatriandafylou and Waters (1984). The key idea is that increments of a martingale over disjoint time intervals are uncorrelated random variables. Rewriting Equation (25) as

⫽ E关E关共⌬Yk兲共⌬Yk⫹h兲兩Ik⫹1兴兴 ⫽ E关⌬Yk E关⌬Yk⫹h兩Ik⫹1兴兴 ⫽ 0

by Equation (81). This shows that the martingale differences, {⌬Yk}, are uncorrelated. Thus,

冉 冘 冊 冉冘 冊 k⫺1

Var共Yk兲 ⫽ Var Y0 ⫹

Y k ⫽ E关L兩Ik兴,

k ⫽ 0, 1, 2, . . . ,

(77)

where Ik denotes the set of k ⫹ 1 indicator random variables 兵I共K ⱖ 0兲, I共K ⱖ 1兲, I共K ⱖ 2兲, . . . , I共K ⱖ k兲其. (78) Since we assume that (x) is alive at time 0, we have I(K ⱖ 0) ⬅ 1 and Y 0 ⫽ E关L兴 ⫽ 0V.

(79)

The stochastic process {Yk} is a martingale with

⌬Yh

h⫽0

k⫺1

⫽ Var

⌬Yh

h⫽0

Cj ⫽ vbj⫹1关I共K ⱖ j兲 ⫺ I共K ⱖ j ⫹ 1兲兴 ⫺ ␲j I共K ⱖ j兲, we see from Equation (26) that the loss random variable, L, is a function of the random variables I(K ⱖ 0), I(K ⱖ 1), I(K ⱖ 2), . . . . Now, consider the following sequence of random variables,

(82)

冘 Var共⌬Y 兲.

k⫺1

⫽

h

(83)

h⫽0

In the next paragraph, we shall show that ⌬Y h ⫽ v h ⌳ h .

(84)

Consequently, Equation (83) can be rewritten as

冘v

k⫺1

Var共E关L兩Ik兴兲 ⫽

2h

Var共⌳h兲.

(85)

h⫽0

Letting k tend to ⬁ in Equation (85) yields Equation (68), proving the Hattendorff theorem. It remains to prove Equation (84). By writing I(K ⫽ h) in Equation (64) as I共K ⱖ h兲 ⫺ I共K ⱖ h ⫹ 1兲,

INDICATOR FUNCTION

AND

HATTENDORFF THEOREM

47

we see that ⌳h is a function of the elements in Ih⫹1. Thus, E关⌳h兩Ik兴 ⫽ ⌳h,

k ⱖ h ⫹ 1.

(86)

Equation (66) can readily be generalized as E关⌳h兩Ik兴 ⫽ 0, k ⱕ h.

(87)

It follows from Equations (77), (65), (86), and (87) that

冘 v E关⌳ 兩I 兴 ⫽ V ⫹ 冘 v ⌳ . ⬁

Y k ⫽ 0V ⫹

k⫺1

h

h k

h

0

h⫽0

h

(88)

h⫽0

Thus, Y k⫹1 ⫺ Y k ⫽ v k ⌳ k , which is Equation (84). REMARK

Applying Equation (26) to Equation (77) yields Yk ⫽

冋 冏册

冘 v C ⫹E 冘v C I ⬁

k⫺1

j

j

j⫽0

j

j

k

.

(89)

j⫽k

Now, if K ⬍ k, then the conditional expectation, ⬁ E[¥j⫽k vjCj兩Ik], takes the value 0; however, if K ⱖ k, then it takes the value

冋冘 冏 册 ⬁

E

vj Cj K ⱖ k ⫽ vkkV.

j⫽k

Thus, Equation (89) is

冘 v C ⫹v

k⫺1

Yk ⫽

j

j

k

k

V I共K ⱖ k兲,

(90)

j⫽0

which should be compared with Equation (88).

ACKNOWLEDGMENTS We thank the anonymous referees for their valuable comments and suggestions. Elias Shiu gratefully acknowledges the generous support from the Principal Financial Group Foundation and Robert J. Myers, FCA, FCAS, FSA. REFERENCES BERKELEY, E. C. 1937. “Boolean Algebra (the Technique for Manipulating ‘And,’ ‘Or,’ ‘Not,’ and Conditions) and Applica-

tions to Insurance,” Record of the American Institute of Actuaries 26: 373– 414; Discussion in 27: 167–76. BOWERS, N. L., JR., H. U. GERBER, J. C. HICKMAN, D. A. JONES, AND C. J. NESBITT. 1986. Actuarial Mathematics. Itasca, IL: Society of Actuaries. ———. 2000. Actuarial Mathematics. 2nd ed. Reprint with corrections. Schaumburg, IL: Society of Actuaries. BU¨ HLMANN, H. 1976. “A Probabilistic Approach to Long-Term Insurance (Typically Life Insurance),” Transactions of the 20th International Congress of Actuaries in Tokyo (4): 267–76. GERBER, H. U. 1976. “A Probabilistic Model for (Life) Contingencies and a Delta-Free Approach to Contingency Reserves,” Transactions of the Society of Actuaries 28: 127– 41; Discussions 143– 8. ———. 1979. An Introduction to Mathematical Risk Theory. Huebner Foundation Monograph 8, distributed by Richard D. Irwin, Homewood, IL. ———. 1997. Life Insurance Mathematics, 3rd ed. Berlin: Springer. GRAHAM, R. L., D. E. KNUTH, AND O. PATSHNIK. 1989. Concrete Mathematics: A Foundation for Computer Science. Reading, MA: Addison-Wesley. HATTENDORFF, K. 1868. “Das Risiko bei der Lebensversicherung,” Maisius’ Rundschau der Versicherungen 18: 169 – 83. HICKMAN, J. C. 1964. “A Statistical Approach to Premiums and Reserves in Multiple Decrement Theory,” Transactions of the Society of Actuaries 16: 1–16; Discussion 149 –54. IVERSON, K. E. 1962. A Programming Language. New York: Wiley. KNUTH, D. E. 1992. “Two Notes on Notation,” American Mathematical Monthly 99: 403–22. MILBRODT, H. 1999. “Hattendorff’s Theorem for Non-Smooth Continuous-Time Markov Models I: Theory,” Insurance: Mathematics & Economics 25: 181–95. ———. 2000. “Hattendorff’s Theorem for Non-Smooth Continuous-Time Markov Models II: Applications,” Insurance: Mathematics & Economics 26: 1–14. NORBERG, R. 1992. “Hattendorff’s Theorem and Thiele’s Differential Equation Generalized,” Scandinavian Actuarial Journal 2–14. ———. 1996. “Addendum to ‘Hattendorff’s Theorem and Thiele’s Differential Equation Generalized,’ ” Scandinavian Actuarial Journal 50 –3. PATATRIANDAFYLOU, A., AND H. R. WATERS. 1984. “Martingales in Life Insurance,” Scandinavian Actuarial Journal 210 –30. RAMLAU-HANSEN, H. 1988. “Hattendorff’s Theorem: A Markov Chain and Counting Process Approach,” Scandinavian Actuarial Journal 143–56. WOLTHUIS, H. 1987. “Hattendorff’s Theorem for a ContinuousTime Markov Model,” Scandinavian Actuarial Journal 157–75.

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