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On the Schuette-Nesbitt paradox - IA|BE · On the Schuette-Nesbitt Paradox Nariankadu D. Shyamalkumar1 Abstract. In the fully continuous model, addition of with- ... ,nand t∈ (0,1)

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Page 1: On the Schuette-Nesbitt paradox - IA|BE · On the Schuette-Nesbitt Paradox Nariankadu D. Shyamalkumar1 Abstract. In the fully continuous model, addition of with- ... ,nand t∈ (0,1)

On the Schuette-Nesbitt ParadoxNariankadu D. Shyamalkumar1

Abstract. In the fully continuous model, addition of with-drawal as a cause of decrement with withdrawal benefit equalto the previous reserve leaves both the reserve and the bene-fit premium unchanged - a result found in actuarial texts likeBowers et al (1997). The concept of equity about the treat-ment of lapsing and continuing policyholders arising from thisunchanged price-benefit structure for continuing policyhold-ers is the guiding principle behind the regulatory definition ofnonforfeiture benefits in the United States. But the regulationin the United States uses the discrete model where, as shownby Schuette and Nesbitt (1988), the reserve as the withdrawalbenefit fails to be equitable. This makes the identification andstudy of such an equitable withdrawal benefit, in the discretecase, interesting - the topic of this paper.

Keywords: Withdrawal Benefit, Non-forfeiture Benefit

1 Introduction

In the fully continuous model, addition of withdrawal as acause of decrement with withdrawal benefit equal to the pre-vious reserve leaves both the reserve and the benefit premiumunchanged - a result found in actuarial texts like Bowers etal (1997). The concept of equity about the treatment of thelapsing and continuing policyholders arising from this un-changed price-benefit structure for continuing policyholdersis the guiding principle behind the regulatory definition ofnonforfeiture benefits in the United States. The regulation inthe United States uses the discrete model where, as shown bySchuette and Nesbitt (1988), the reserve as the withdrawalbenefit fails to be equitable. For example, in Bowers et al(1997) this quirk of the discrete case is attributed to the factthat, in the discrete model, the probability of withdrawal de-pends on the force of mortality. Along similar lines, it hasbeen pointed out, for example by Greville, T. N. E. in Nesbitt(1964) and in Milbrodt and Stracke (1997), that under restric-tive conditions on the probability of withdrawal the reserveas the withdrawal benefit continues to be equitable in the dis-crete case. All of the above study the neutrality of the reserveand while concluding its non-neutrality in the general discretecase fail to identify and study such an equitable withdrawalbenefit. And as the attainment of equity could be the focus ofinterest, the study of such an equitable withdrawal benefit isinteresting.

The next section summarizes the known results in the con-tinuous case. The third and main section concerns the discretecase. It begins with the numerical example of Schuette and

1 Department of Statistics and Actuarial Science, The University of Iowa

Nesbitt (1987) as it is the basis for all of the figures. Then itdefines the equitable withdrawal benefit in the discrete caseand studies its properties. We end this section with a brief de-scription of the notation.

In the following we will look at either solely death orboth death and withdrawal as possible cause(s) of decrement.Death will be termed as cause 1 and withdrawal as cause 2.The random variables T and J denote the time and cause ofdecrement, respectively. Quantities which refer to the doubledecrement model will carry a superscript of 2, for example thereserve at time t in the double decrement continuous modelwill be denoted by tV

2. In all of the following models the

premium and benefit functions are defined on [0, n] with bndenoting the amount of the pure endowment. Also, we willassume a constant force of interest for ease in exposition; thetheory holds without such an assumption. The models we usein this paper are those found in Bowers et al (1997) and Gerber(1997).

2 Continuous CaseHere we show that the reserve as the withdrawal benefit in thecontinuous case is equitable. First, we define the single anddouble decrement models in the continuous case.

Single Decrement Model - Premiums are paid at a contin-uous rate of π(·) until death and the benefit, paid at the timeof death, say t, is given by b(1)(t). The force of decrement isgiven by µ(1)

x (·).Double Decrement Model - Premiums are paid at a con-

tinuous rate of π2(·) until decrement and the benefit, paid atthe time of decrement, say t, for cause j (∈ {1, 2}) is given byb(j)(t). The force of decrement is given by µ(1)

x (·) and µ(2)x (·)

for death and withdrawal, respectively.Note that we have assumed the presence of the withdrawal

leaves the force of mortality unchanged. This will be a cen-tral assumption in both the continuous and the discrete case.Whether it is realistic depends on the insurance product; forexample it will be unrealistic in the case of an insurance prod-uct with significant levels of selective lapsation.

Below is a theorem, adapted from Hoem (1988) andSchuette & Nesbitt (1988), which summarizes the neutral-ity of the reserve as the withdrawal benefit in the continuousmodel. The first part in a sense defines lapse supported insur-ance products and concludes that reserves in the single decre-ment model for such products will be higher than those in thedouble decrement model.

c© BELGIAN ACTUARIAL BULLETIN, Vol. 6, No. 1, 2006

Page 2: On the Schuette-Nesbitt paradox - IA|BE · On the Schuette-Nesbitt Paradox Nariankadu D. Shyamalkumar1 Abstract. In the fully continuous model, addition of with- ... ,nand t∈ (0,1)

Theorem 2.1. For the above described models, the followinghold.

i. If π2(·) = π(·) then for each t in [0, n] for which

b(2)(s) ≤ sV, ∀s ∈ [t, n] (1)

we havesV

2 ≤ sV, ∀s ∈ [t, n] (2)

Moreover, equality in (1) implies equality in (2).ii. If π2(·) = C ∗ π(·), for some C where both are assumed to

be benefit premium rates for their respective models, then

b(2)(t) = tV, ∀t ∈ (0, n) (3)

implies

C = 1 and tV2

= tV, ∀t ∈ [0, n] (4)

Proof. The Thiele differential equations for the single decre-ment model is given by

ddt tV = π(t) + δ tV − µ(1)

x (t)[b(1)(t) − tV

](5)

and for the double decrement model by

ddt tV

2= π(t)2 + δ tV

2 − µ(1)x (t)

(b(1)(t) − tV

2)

− µ(2)x (t)

(b(2)(t) − tV

2)

(6)

We have from (5) and (6)

ddt

[νt

tp(τ)x

(tV − tV

2)]

=

νttp

(τ)x

(π(t) − π2(t) + µ(2)

x (t)[b(2)(t) − tV

])(7)

Note that nV is equal to nV2

as the endowment amount isthe same under both the models. Hence, integrating the abovefrom t to n, for t in [0, n), we have

νttp

(τ)x

(tV − tV

2)

=∫ n

t

νssp

(τ)x

(π2(s) − π(s) + µ(2)

x (s)[

sV − b(2)(s)])

ds

(8)

The first part easily follows from (8). For the second part, theabove relation with t equal to zero along with the assumptionof π(·) and π2(·) being benefit premiums - implying 0V isequal to 0V

2- gives the result.

3 Discrete ModelIn the discrete case the single decrement reserve as the with-drawal benefit, unlike the continuous case, is no longer equi-table. This was shown in Schuette and Nesbitt (1988) fromwhich the following numerical example is taken.

Example 3.1. Consider a fully discrete annual term insuranceon a life aged 40 for an unit death benefit and with the with-drawal benefit being the reserve from the single decrementmodel. The interest rate is taken to be 6%, the mortality ratesare taken from the illustrative life table of Bowers et al (1997)and the withdrawal rates are as shown below. We shall use theuniform distribution assumption for each year of age in theassociated single decrements, see Bowers et al (1997), whichis equivalent to

tq′(1)x = t q

′(1)x and tq

′(2)x = t q

′(2)x ,∀t ∈ (0, 1) (9)

Table 1 shows the results for both the single decrement anddouble decrement models.

s orq

′(1)s q

′(2)s q

(1)s q

(2)s tV

401

:10tV

2

401

:10t+3940 2.7812 100 2.6421 99.8609 1.032 1.024941 2.9818 95 2.8402 94.8584 1.9282 1.915642 3.2017 90 3.0576 89.8559 2.6611 2.644643 3.4427 85 3.2964 84.8537 3.1994 3.180844 3.707 80 3.5587 79.8517 3.5077 3.488945 3.9966 75 3.8467 74.8501 3.546 3.528946 4.3141 70 4.1631 69.849 3.2691 3.255247 4.6621 65 4.5106 64.8485 2.6258 2.616248 5.0436 60 4.8923 59.8487 1.5579 1.553249 5.4617 55 5.3115 54.8498 0 0

Table 1. Schuette-Nesbitt Example (figures are per 1000)

For the purpose of our main theorem, below we define thesingle and double decrement models in the discrete annualcase. Extension to any other discrete case is easy.

Single Decrement Model - Premiums are paid annually atthe start of each policy year until death - at the start of the kth

policy year, it will be of the amount πk−1. The benefit is paidat the end of the year of death - if the death occurs in the kth

policy year the benefit will be of the amount b(1)k . The force

of mortality is given by µ(1)x (·).

Double Decrement Model - Premiums are paid annually atthe start of each policy year until decrement - at the start of thekth policy year, it will be of the amount π2

k−1. The benefit ispaid is paid at the end of the year of decrement - if the decre-ment happens in the kth policy year for cause j (∈ {1, 2}),then it is of the amount b(j)

k . The force of decrement is givenby µ(1)

x (·) and µ(2)x (·) for death and withdrawal, respectively.

The main result of this article (Theorem 3.2) is the specifi-cation of such an equitable withdrawal benefit in the discretecase. This benefit is given by the sequence {ψk}1≤k≤n , de-fined as

ψk = βx+k−1b(1)k + (1 − βx+k−1) kV, k = 1, 2, . . . , n

where

βx+k−1 =

(q

′(1)x+k−1 − q(1)

x+k−1

q(2)x+k−1

), k = 1, 2, . . . , n

50

Page 3: On the Schuette-Nesbitt paradox - IA|BE · On the Schuette-Nesbitt Paradox Nariankadu D. Shyamalkumar1 Abstract. In the fully continuous model, addition of with- ... ,nand t∈ (0,1)

To understand the above benefit sequence {ψk}1≤k≤n , notethat if the withdrawal benefit is paid at the moment of decre-ment then the equitable amount is the single decrement re-serve, T V. This follows from the Cantelli theorem, see Mil-brodt and Stracke (1997). But since the benefit is actuallypaid at the end of the year, one can adjust this amount bythe factor (1 + i)(k−T ) and by doing this preserve financialparity. The catch is that at the end of the year all the with-drawals during the year will receive the same withdrawal ben-efit. Such a benefit payable at the end of the year, which pre-serves the actuarial present value, can be defined as the condi-tional expected value on withdrawal sometime during the yearof (1+i)(k−T )

T V. The above leads to an alternative definitionof the benefit sequence {ψk}1≤k≤n as,

ψk = E((1 + i)(k−T )

T V|T ∈ (k − 1, k];J = 2),

k = 1, 2, . . . , n (10)

Note that the single decrement reserve for non-integral valuesused above is defined for k = 1, 2, . . . , n and t ∈ (0, 1) by,

k−1+tV = 1−tq′(1)x+k−1+tν

(1−t)b(1)k

+ 1−tp′(1)x+k−1+tν

(1−t)kV. (11)

Conditional expectation of a similar nature have been usedbefore in the literature, for example see the definition of theretrospective reserve of Norberg (1991). Combining (10) and(11) we get

ψk = E(

k−T q′(1)x+T |T ∈ (k − 1, k];J = 2

)(b(1)

k − kV)

+ kV, k = 1, 2, . . . , n (12)

It is easy to check, see Exercise 10.32 (in the double decre-ment case with j = 1) of Bowers et al (1997), that

E(

k−T q′(1)x+T |T ∈ (k − 1, k]; J = 2

)= βx+k−1 (13)

This implies that the above two definitions of {ψk}1≤k≤n co-incide. While it is true that the above heuristics can be con-verted to a proof, we imitate the above proof of the continu-ous case. It is important to note that paying a constant amountfor all withdrawals during a policy year implies that we willhave to assume that the withdrawal pattern does not changeas a result of policyholders exercising their right to withdrawat the optimal time (end of the year). In fact, if they do onlywithdraw at the end of the year then the reserve continues tobe equitable.

Theorem 3.2. In the discrete case, the following hold.

i. If π2(·) = π(·) then for each m in {1, 2, . . . , n} for which

b(2)k ≤ ψk, k = m,m+ 1 . . . , n (14)

we have

kV2 ≤ kV, k = m− 1,m, . . . , n (15)

Moreover, equality in (15) implies equality in (16).

ii. If π2· = C ∗ π·, for some C where both are assumed to be

benefit premium rates for their respective models, then

b(2)k = ψk, k = 1, 2, . . . , n (16)

implies

C = 1 and kV2 = kV, k = 0, 1, . . . , n (17)

Proof. The proof is similar to that in the continuous case. Therecurrence formula in the single decrement model is given by

k−1V + πk−1 = νq′(1)x+k−1b

(1)k + νp

′(1)x+k−1 kV,

k = 1, 2, . . . , n (18)

and in the double decrement model by

k−1V2 + π2k−1 = νq(1)

x+k−1b(1)k + νq(2)

x+k−1b(2)k

+ νp(τ)x+k−1 kV2, k = 1, 2, . . . , n (19)

Using (18) and (19), we get for k in {1, 2, . . . , n}

∆[νk−1

k−1p(τ)x

(k−1V − k−1V2

)]=

νk−1k−1p

(τ)x

(πk−1 − π

2k−1 + νq(2)

x+k−1

[b(2)

k − ψk

])(20)

Note that nV is equal to nV2 as the endowment amount is thesame under both the models. Hence summing the above fromm to n, for m in {1, 2, . . . , n}, we have

νm−1m−1p

(τ)x

(m−1V − m−1V2

)=

n∑k=m

νk−1k−1p

(τ)x

2k−1 − πk−1

+ νq(2)x+k−1

[ψk − b(2)

k

])(21)

The first part easily follows from (21). For the second part, theabove relation with m equal to one along with the assumptionthat the premiums are the benefit premiums - implying 0V isequal to 0V2 - gives the result.

It is interesting to compare ψk, the equitable withdrawalbenefit, with the reserve. Here we will assume that the deathbenefit is constant across policy years and, without loss ofgenerality, equal to one. First, the following simple inequal-ity,

kV ≤ ψk ≤ q′(1)x+k−1 +

(1 − q

′(1)x+k−1

)kV, k = 1, 2, . . . , n

(22)The lower limit is attained if q

′(1)x+k−1 is equal to q(1)

x+k−1 andthis is the case where the reserve as the withdrawal benefitis equitable, see Milbrodt and Stracke (1997). One particularcase where this is true is when the withdrawals are assumedto happen at the end of the year while the time of death is

51

Page 4: On the Schuette-Nesbitt paradox - IA|BE · On the Schuette-Nesbitt Paradox Nariankadu D. Shyamalkumar1 Abstract. In the fully continuous model, addition of with- ... ,nand t∈ (0,1)

Policy Year (k)

β x+k

−1q x

+k−1

'(1)

UniformConstant Force

1 2 3 4 5 6 7 8 9 10

0.50

10.

503

0.50

50.

507

0.50

9

Figure 1. Ratio of β· and q′(1)·

continuously distributed, see Atkison and Dallas (2000). Theupper limit, though attainable, is not of any practical interest.

Second, we note that the value of β· under the uniform dis-tribution assumption for each year of age in the associatedsingle decrements is given by

βx+k−1 =

(q

′(1)x+k−1

2 − q′(1)x+k−1

), k = 1, 2, . . . , n (23)

and under the constant force assumption

βx+k−1 = 1 −

q′(2)x+k−1 log

(p(τ)

x+k−1

)q(τ)

x+k−1 log(p

′(2)x+k−1

) p

′(1)x+k−1,

k = 1, 2, . . . , n (24)

Under both of the above assumptions it can be easily seenthat βx+k−1 is approximately equal to 0.5q

′(1)x+k−1, for small

values of q′(1)x+k−1 and q

′(2)x+k−1, implying

ψk ≈ 0.5q′(1)x+k−1 + (1 − 0.5q

′(1)x+k−1)kV (25)

Interestingly, β· does not depend on the independent rate ofwithdrawal (q

′(2)· ) under the uniform assumption. Hence in

this case the definition of ψ· becomes free of the withdrawalrate. More generally, this happens to be the case when thewithdrawal decrement assumption (on the associated singledecrement) is free of the independent withdrawal rate. This

Policy Year (k)ψ

kR

eser

ve

UniformConstant Force

1 2 3 4 5 6 7 8 9

1.6

1.8

2.0

2.2

2.4

2.6

Figure 2. Ratio of ψ· and ·V

class is similar to that described in Wilmott (1997) whichhas the fractional independence property. The families offractional age assumptions described in Jones and Mereu(2000,2002), while possessing other desirable properties, re-sult in β· being a function of the independent rate of with-drawal.

Observe that it is not necessary that the withdrawal benefitis actually paid at the end of the policy year. The above onlyrequires that

b(2)(t) = b(2)(dte)νdte−t (26)

In other words, the withdrawal benefit paid at two differenttime points during the kth policy year must be equal in presentvalue terms.

4 Conclusion

We have studied the neutral withdrawal benefit in the discretecase which is not always equal to the single decrement re-serve. It is important to point out that the withdrawal bene-fit, ψ·, only guarantees that the double decrement reserve isequal to the single decrement reserve at the end of all pol-icy years. The behavior of the difference between the doubledecrement and single decrement reserves at non-integral timepoints is plotted in Figure 3. To understand its behavior, con-tinuing with the above heuristics, note that the actual break-even amount due to those withdrawing at time k − 1 + t (tin (0, 1)) is given by k−1+tV(1 + i)(1−t). This satisfies the

52

Page 5: On the Schuette-Nesbitt paradox - IA|BE · On the Schuette-Nesbitt Paradox Nariankadu D. Shyamalkumar1 Abstract. In the fully continuous model, addition of with- ... ,nand t∈ (0,1)

Policy Year (k)

105 (V

2−

V)

0 1 2 3 4 5 6 7 8 9 10

01

23

4

Figure 3. The plot of tV2 − tV

relation

k−1+tV(1 + i)(1−t) = 1−tq′(1)x+k−1+t b(1)

k

+ 1−tp′(1)x+k−1+t kV (27)

which implies that, under constant death benefit, it is decreas-ing in t. Hence, ψk being its average will initially be lowerthan the actual and then higher, justifying the pattern in Figure3. The assumption stated before Theorem 3.2 was that the pol-icyholders do not exploit this intra-year inequity. Moreover,under it, we have shown that ψ· leads to the equitable treat-ment of the two classes of policyholders - the ones lapsingand the ones continuing.

Finally, we note that the content of this paper has a closeconnection with Loewy premiums and reserves, for examplesee Loewy (1917) and Bicknell and Nesbitt (1956). The aboveresults can be derived using this theory but it will be ratherdifficult to beat the clarity of the above heuristics.

ACKNOWLEDGEMENTS

I thank Prof. Elias Shiu for motivating me to bring the ideato its current form as well as many fruitful discussions on thetopic.

REFERENCES[1] D. B. Atkinson and J. W. Dallas. Life Insurance - Products and Finance.

Society of Actuaries, Schaumburg, Ill., 2000.

[2] W. S. Bicknell and C. J. Nesbitt. Premiums and reserves in multipledecrement theory. Transactions of the Society of Actuaries, 8:344–377,1956.

[3] N. Bowers, H. Gerber, J. Hickman, D. Jones, and C. J. Nesbitt. Ac-tuarial Mathematics. Society of Actuaries, Schaumburg, Ill., Secondedition, 1997.

[4] H. Gerber. Life Insurance Mathematics. Springer, Berlin, Third edition,1997.

[5] J. M. Hoem. The versatility of the markov chain as a tool in the mathe-matics of life insurance. Transactions of the 23rd Congress of Actuar-ies, R:171–202, 1988.

[6] B. L. Jones and J. A. Mereu. A family of fractional age assumptions.Insurance: Mathematics and Economics, 27:261–276, 2000.

[7] B. L. Jones and J. A. Mereu. A critique of fractional age assumptions.Insurance: Mathematics and Economics, 30:363–370, 2002.

[8] A. Loewy. Zur theorie anwendung der intensitaten in der versicherungs-mathematik. Sitz. Ber. Heidleberg Akd. Wiss., 8 A., 1917.

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[10] C. J. Nesbitt. Discussion of ”a statistical approach to premiums andreserves in multiple decrement theory”. Transactions of the Society ofActuaries, 15:149–153, 1964.

[11] Ragnar Norberg. Reserves in life and pension insurance. ScandinavianActuarial Journal, pages 3–24, 1991.

[12] D. R. Schuette and C. J. Nesbitt. Withdrawal benefit equal to re-serve: Non-neutrality in the discrete case. Actuarial Research ClearingHouse, (2):179–190, 1988.

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