On the evaluation of saving-consumption plans

Transcription

1 On the evauation of saving-consumption pans Steven Vanduffe Jan Dhaene Marc Goovaerts Juy 13, 2004 Abstract Knowedge of the distribution function of the stochasticay compounded vaue of a series of future (positive and/or negative) payments is needed for soving severa probems in an insurance or finance environment, see e.g. Dhaene et a. (2002 a,b). In Kaas et a. (2000), convex ower bound approximations for such a sum have been proposed. In case of changing signs of the payments however, the distribution function or the quanties of the ower bound are not easy to determine, as the approximation for the random compounded vaue of the payments wi in genera not be a comonotonic sum. In this paper, we present a method for determining accurate and easy computabe approximations for risk measures of such a sum, in case one first has positive payments (savings), foowed by negative ones (consumptions). This particuar cashfow pattern is observed in saving - consumption pans. In such a pan, a person saves money on a reguar basis for a certain number of years. The amount avaiabe at the end of this period is then used to generate a yeary pension for a fixed number of years. Using the resuts of this paper one can find accurate and easy to compute answers to questions such as: "What is the minima required yeary savings effort α during a fixed number of years, such that one wi be abe to meet, with a probabiity of at east (1 ε), a given consumption pattern during the withdrawa period?" 1

2 1 Introduction In a finance or insurance context, one is often interested in the distribution function (d.f.) of a random variabe (r.v.) S given by S = n α i e Z i. (1) Here the α i are rea numbers and (Z 0,Z 1,..., Z n ) is a mutivariate norma random vector. Theaccumuatedvaueattimen of a series of future deterministic saving amounts α i at times i, i =0,...,n 1, can be written in the form (1), where Z i denotes the random accumuation factor over the period [i, n]. The present vaue of a series of future deterministic payments α i at times i, i =1,...,n, can be written in the form (1), where now Z i denotes the random discount factor over the period [0,i]. Determining the price of Asian or basket options in a Back & Schoes setting bois down to computing stop-oss premiums of a r.v. of the type described in (1), see for instance Abrecher, Dhaene, Goovaerts & Schoutens (2003) or Vanmaee, Dhaene, Deestra, Liinev & Goovaerts (2004). As mentioned in Dhaene, Vanduffe, Goovaerts, Kaas & Vyncke (2004) or Vanduffe, Dhaene, Goovaerts & Kaas (2003), setting provisions and capita requirements in an insurance context comes down to determining risk measures reated to a r.v. of the type (1). As the r.v. S defined in (1) is a sum of non-independent ognorma r.v. s, its d.f. cannot be determined anayticay. Therefore a variety of approximation techniques for determining d.f. s of this type have been proposed in the iterature. Practitioners often use a (first two moments) matching ognorma approximation for the d.f. of S. Mievsky & Posner (1998) and Mievsky & Robinson (2000) propose a moment matching reciproca Gamma approximation for the d.f. of S. Kaas, Dhaene and Goovaerts (2000) and Dhaene, Denuit, Goovaerts, Kaas & Vyncke (2002a,b) derive a ower bound approximation for the d.f. of S. The ower bound is defined by E[S Λ] for an appropriate choice of the conditioning r.v. Λ. It is caed a ower bound approximation since (the d.f.) of E[S Λ] is smaer in the convex order sense than (the d.f. of) S. This means that the expectations of both r.v. s are equa, whereas the stop-oss premiums of the ower bound are smaer than the corresponding stop-oss 2

3 premiums of S. The ower bound approximation performs very accurate when an optima choice is made for the conditioning r.v. Λ, see for instance Vanduffe, Hoedemakers & Dhaene (2004). We point out that the ognorma and the reciproca Gamma approximation have been proposed in the context of series of positive cash fows, whereas the ower bound approximation can aso be appied in case of changing signs of the α i. In this case however, and this in contrast to the situation that a α i have equa signs, it is not possibe to find a conditioning r.v. Λ that eads to an accurate approximation for the d.f. of S such that E[S Λ] = P n α i E e Z i Λ is a sum of non-decreasing functions of Λ. This impies that distortion risk measures reated to E[S Λ] cannot be obtained by simpy summing the corresponding risk measures of the individua terms in the sum, as is the case when a α i are positive, see Dhaene, Vanduffe, Tang, Goovaerts, Kaas & Vyncke (2004). Goovaerts, De Schepper, Hua, Darkiewicz & Vyncke (2003) propose an approach that uses convex bounds for the positive and negative sum separatey. They connect the two r.v. s invoved by a copua of a particuar famiy. In this paper, we foow another path to determine accurate and easy computabe approximations for a sum S as defined in (1), in the specia case that one first has positive payments α i (savings) foowed by negative ones (withdrawas). An important situation where one encounters this particuar cash fow pattern, and hence where our resuts can be appied, is the saving - consumption probem. Take as an exampe a 20/65/95 pension pan in a defined contribution pension scheme. A person of age 20 intends to save money for 45 consecutive years (unti retirement). After his retirement, he wants to withdraw money from his pension account on a reguar basis and this for a period of 30 years. Assume that his yeary savings are constant and equa to α, whie his yeary consumption(pension) is constant and equa to 1. A reevant question to answer is: "What is the minima required yeary savings effort α such that this person wi be abe to meet his consumption pattern during the 30 year withdrawa period, with a probabiity of at east (1 ε)? In this paper, we wi present a methodoogy for answering these types of questions. The paper is organized as foows. In Section 2, we describe the savingconsumptionprobem. InSections3and4,wepresentaccurateandeasy computabe approximations for the quanties of fina weath random variabes in the saving-consumption probem. The specia case of constant savings and 3

4 constant withdrawas is considered in Section 5. Finay, in Section 6 the theoretica resuts presented in this paper are iustrated by some numerica exampes. 2 Probem description Consider a set of deterministic amounts α 0,α 1,,α with n 1,m 0. Thefirst n amounts α 0,α 2,,α n 1 are nonnegative and correspond to saving amounts that are put on an investment account at respective times 0, 1,,n 1. Theastm +1amounts α n,α n+1,,α are nonpositive and correspond to withdrawas from the account at times n, n+1,,, respectivey. We wi assume that α 0 > 0 and α n < 0. We wi ca a pan as described above a saving - consumption pan. We assume that the return on the account is generated by a geometric Brownian motion process. An amount of 1 avaiabe on the account at time i 1, is assumed to grow to the random amount e Y i at time i. Hence,the r.v. Y i is the random return over the year [i 1,i]. The r.v. s Y i are i.i.d. and norma distributed, with parameters µ σ2 and 2 σ2. Let V j denote the surpus at time j. By convention, the surpus at time j has to be understood as the surpus just after saving or withdrawa. Starting from the initia vaue V 0 = α 0,thesurpusV j avaiabe at time j is given by the foowing recursive reation: V j = V j 1 e Y j + α j, j =1,,n+ m. (2) For the moment, we aow the surpus to become negative, which means that shortseing of units of the investment account is aowed. Note that for the first saving - ater consuming cash fow pattern as described in (2), we have that once the surpus becomes negative, it wi stay negative over the whoe remaining time period and no recovery is possibe anymore. This ruin scenario can ony occur from time n (retirement) on. Soving the recursion (2), we find that the fina surpus V can be written as V = α i e Z i, (3) 4

5 where the r.v. s Z i are given by Z i = j=i+1 Y j, i =0,,n+ m, (4) and where, by convention, P s j=r a j =0if r>s.ther.v. Z i is the accumuation factor over the period [i, n]. Its mean and variance are given by E [Z i ]=(n + m i) µµ σ2 (5) 2 and σ 2 Z i =(n + m i)σ 2. (6) As it is impossibe to determine the d.f. of V anayticay, we propose to approximate it by the d.f. of V = E [V Λ] (7) for some convenient choice of the conditioning r.v. Λ. Wehavethat(the d.f. of) V is a ower bound in the sense of convex order for (the d.f. of) V, see for instance Kaas, Dhaene & Goovaerts (2000). In particuar, we have that E [V ]=E V = α i e ( i)µ. (8) In the seque, we wi consider a conditioning r.v. Λ which is a inear combination of the compounded returns Z i : Λ = j=0 γ j Z j, (9) for suitabe choices of the parameters γ j. This r.v. can aso be written in terms of the yeary returns: Λ = j=1 where the reation between the β j and the γ j is given by β j Y j, (10) j 1 β j = γ k. (11) k=0 5

6 For more detais about these kind of approximations, its reation with the concept of comonotonicity and its appications in insurance and finance, see Dhaene, Denuit, Goovaerts, Kaas & Vyncke (2002 a,b). Let U be a uniform(0,1) r.v. and et Φ denote the standard norma d.f. After some straightforward derivations we find that the r.v. V defined in (7) with Λ given by (9) is distributed as V α i e E[Z i]+ 1 2 (1 r2 i )σ2 Z +r i σ Zi Φ 1 (U) i, (12) d = where = d stands for equaity in distribution and where the coefficients r i are given by r i = cov (Z P i, Λ) j=i+1 = β j σ Zi σ q P, i =0,,n+ m 1 (13) Λ n + m i j=1 β2 j and r =0. (14) Notice that the ast term in (12), and aso in (3), reduces to the constant number α. For any rea-vaued r.v., we wi denote its distribution function Pr [ x] by F (x). The ower quanties Q[] and the upper quanties Q + p [] of are defined by Q p [] = inf{x R F (x) p}, (15) Q + p [] = sup{x R F (x) p}, p (0, 1), where by convention, inf φ =+ and sup φ =. When F is stricty increasing, we have that Q p [] =Q + p []. Let and g() be rea-vaued r.v. s. If g is non-decreasing and continuous, then Q p [g()] = g (Q p []), (16) Q + p [g()] = g Q + p [], p (0, 1). Furthermore, for any rea number x number, we have p F (x) Q p [] x, (17) Pr [ <x] p x Q + p [], p (0, 1). 6

7 If a the terms α i e E[Z i]+ 1 2 (1 r2 i )σ2 Z +r i σ Zi Φ 1 (U) i in (12) are non-decreasing functions of U (or a are non-increasing functions of U), then we say that V is a comonotonic sum. In case a terms are non-decreasing, then aso the continuous function f defined by f(p) = a i e E[Z i]+ 1 2 (1 r2 i )σ2 Z i +r i σ Zi Φ 1 (p), p (0, 1), (18) is non-decreasing. From (16) we find that in this case, the p-quantie of V is given by Q + p [V] =f(p), p (0, 1). (19) Moreover, in case V is a comonotonic sum, any distortion risk measure (such as VaR p and TVaR p ) reated to V equas the sum of the risk measures reated to the margina terms in (12), see for instance Dhaene, Vanduffe,Tang,Goovaerts,Kaas&Vyncke(2004). Aconditioning r.v. Λ that makes f non-decreasing (and continuous) can aways be found. Indeed, if we take a β j 0 for j =1, 2,...n, whereas β j =0for j = n +1,n+2,..., n + m, then one has that f is non-decreasing, so that (19) hods in this case. Of course, we cannot expect that such a choice of the parameters β j wi in genera ead to an accurate approximation for the d.f. of V. It is cear that for appropriate choices of Λ, ther.v. V wi not necessariy be a comonotonic sum of ognorma r.v. s. Hence, the function f(p) wi not be non-decreasing on the whoe interva (0, 1). This impies that the quanties of V cannot be determined easiy in this case because (16) can not be appied. As noticed above, the cashfow pattern that we consider (first saving - ater consuming) impies that once the vaue V j become negative, no recovery is possibe in the sense that a future vaues V k,k j, wi be negative too. From now on, we wi assume that shortseing is not aowed. Hence, once the surpus reaches eve 0, no further withdrawas from the pension account are aowed. One can easiy verify that under this assumption, the weath W k avaiabe on the account at time k can be expressed as foows in terms of the surpus V k,defined in (2): W k =max[v k, 0], k =0,,n+ m. (20) Quantities heping to decide whether or not to underwrite the saving - consumption pan are the quanties and the distribution function of fina weath 7

8 W.Aswehavethat Q + p [W ]=sup{x Pr [W >x] 1 p}, (21) the quantie Q + p [W ] can be interpreted as the argest amount of money that wi be eft at time n + m, with a probabiity of at east (1 p). A possibe requirement for a saving - consumption pan to be considered as feasibe coud be Q [W ] > 0. Forapanfufiing this condition, there is a probabiity of (at east) 95% that one wi be abe to meet the desired consumption pattern, hence that there wi be no consumption shortfa. For a given pan, one coud be interested in the probabiity that no consumption shortfa wi occur. This probabiity is given by Pr [W > 0]. In genera, the probabiities Pr [W >x] and the quanties Q + p [W ] cannot be determined anayticay. Therefore, we propose to approximate (the d.f. of ) W by (the d.f. of) W =max[v, 0]. From (12), we find that d =max[f(u), 0], (22) W where the function f is defined by (18). We propose to approximate the tai probabiities Pr [W >x] and the quanties Q + p [W ] by Pr W >x and Q + p [W ], respectivey. In the next section we wi prove that under rather weak conditions the function max[f(p), 0] is non-decreasing and continuous. From (16) we find a straightforward way to compute the quanties of W in this case. 3 Approximations for the quanties of W The foowing emma gives sufficient conditions for the function max[f(p), 0] to be non-decreasing. Lemma 1 If a β j > 0, for j =1, 2,...n+ m, then for a p in the unit interva (0, 1), one has that f(p) 0 impies f 0 (p) > 0. Proof. From (13), we find that r i σ Zi = σ P j=i+1 β j q P j=1 β2 j, i =0,,n+ m 1. 8

9 Hence, a correations r i > 0 for i =0, 1,...n + m 1 and the sequence {r i σ Zi } 0 r is stricty decreasing and stricty positive. By appication of the chain rue, we find for p (0, 1) that f 0 (p) = 1 1 a Φ 0 [Φ 1 i e E[Z i]+ 1 2 (1 r2 i )σ2 Z +r i σ Zi Φ 1 (p) i r i σ Zi. (p)] Now assume that f(p) 0 for some vaue of p in the unit interva (0, 1). When n + m =1, it is straightforward to prove that f 0 (p) > 0. 1 Let us now assume that n + m>1. Since > 0 and α Φ 0 [Φ 1 (p)] 0, we find that f 0 (p) > r 1 n 1σ Zn 1 a Φ 0 [Φ 1 i e E[Z i]+ 1 2 (1 r2 i )σ2 Z +r i σ Zi Φ 1 (p) i (p)] which ends the proof. Notice that = r n 1σ Zn 1 Φ 0 [Φ 1 (p)] (f(p) α ) 0, σ 2 β j = Cov (Y j, Λ), j =1,,n+ m. (23) Hence, the condition that a β j > 0 in Lemma 1 means that any yeary return Y j is stricty positive correated with the conditioning random variabe Λ. One can easiy prove that when a β j > 0, one has that im f(p) =a 0 (24) p 0 and im f(p) =+. (25) p 1 When a β j are assumed to be stricty positive, we find from Lemma 1 that the function max [f(p), 0] is non-decreasing (and continuous) on the interva (0, 1). From (22) and (16) we see that the quanties of W can easiy be determined anayticay in this case: Q p [W ] =Q + p [W ] =max[f(p), 0], p (0, 1). (26) 9

10 Under the conditions of Lemma 1, we find that the d.f. of W can be determined from f(f W (x)) = x, x 0, (27) Indeed, we have that for any x 0, n o F W (x) = sup p (0, 1) p F W (x) = sup p (0, 1) Q p [W] x ª = sup{p (0, 1) max[f(p), 0] x} = sup{p (0, 1) f(p) x}, (28) so that the reation (27) foows from (24), (25), Lemma 1 and the fact that f is continuous on (0, 1). 4 On the choice of the conditioning r.v. Λ In order to determine the optima vaues for the coefficients β j, we foow the procedure as expained in Vanduffe, Hoedemakers & Dhaene (2004). They consider the case were a cash fow payments α i are positive. But the procedure can easiy be extended to the case of genera α i. We have that Var[V ] and Var V are given by and Var [V ]= Var V = j=0 j=0 α i α j e E[Z i]+e[z j ]+ 1 2 (σ2 Z i +σ 2 Zj ) (e Cov(Z i,z j ) 1) (29) α i α j e E[Z i]+e[z j ]+ 1 2 (σ2 Z i +σ 2 Zj ) (e r ir j σ Zi σ Zj 1), (30) 10

11 respectivey. Consider the foowing first order approximation for Var V : Var V = j=0 j=0 α i α j e E[Z i]+e[z j ]+ 1 2 (σ2 Z i +σ 2 Zj ) (r i r j σ Zi σ Zj ) (31) µ α i α j e E[Z i]+e[z j ]+ 1 2 (σ2 Z +σ 2 i Zj ) Cov[Zi, Λ] Cov[Z j, Λ] Var(Λ) = (Cov(P α i e E[Z i]+ 1 2 σ2 Z i Z i, Λ)) 2 Var(Λ) =(Corr( α j e E[Z j]+ 1 2 σ2 Z j Zj, Λ)) 2 Var( α j e E[Z j]+ 1 2 σ2 Z j Zj ). j=0 In genera, we have that Var V < Var[V ] hods, uness V = V. Intuitivey, it seems reasonabe to choose Λ such that Var V is maximized and hence as cose as possibe to Var[V ]. In order to find an easy computabe approximation, we propose to choose Λ such that the first order approximation (31) for Var V is maximized: Λ = j=0 j=0 α j e E[Z j]+ 1 2 σ2 Z j Zj. (32) This means that the coefficients β j,j=1,,n+ m, in (10) are given by j 1 j 1 j 1 β j = γ i = α i e E[Z i]+ 1 2 σ2 Z i = α i e ( i)µ. (33) Our main resut is stated in the foowing Theorem. Theorem 2 If the β j,j=1, 2,..., n + m, are defined by (33), and if then the quanties of W are given by E [V ] > 0, (34) Q + p [W ] =max[f(p), 0], 0 <p<1, (35) d 11

12 whereas the d.f. of W foows from with f(p) defined by (18). f(f W (x)) = x, x 0, (36) Proof. It is straightforward to verify that the condition (34) impies that a β j > 0, for j =1, 2,...n + m. The stated resut then foows from (26) and (27). It is cear that any reasonabe first saving- ater consuming pan shoud fufi the condition (34), which states that the average fina surpus E[V ] shoud be non-negative. 5 The case of constant savings and consumptions In the remainder of this paper, we wi consider the specia case that a saving amounts are equa: α 0 = α 1 = = α n 1 = α, and aso that a withdrawas are equa: α n = α n+1 = = α = 1. In the seque, we wi aways assume that the β j coefficients, j=1,,n+ m, needed to define Λ, are given by (33). The condition (34) can be rewritten as α>α = 1 e (m+1)µ. (37) e nµ 1 When the condition (37) is fufied, we find from Theorem 3 and (18) that the approximated quanties Q + p [W ] and the approximated probabiities F W (x) foow from (35) and (36) with f(p) f α (p) given by n 1 f α (p) =α e ( i)µ e 1 2 r2 i σ2 Z +r i σ Zi Φ 1 (p) i e ( i)µ e 1 2 r2 i σ2 Z +r i σ Zi Φ 1 (p)) i. (38) Let W (α) and W (α) denote the (approximated) fina weath for a given saving-consumption pan with a saving α and consumption eve of 1. Simiary, V (α) is the surpus of the α -pan. i=n 12

13 For α as defined in (37), one has that F W (α )(0) > 0.5. (39) Indeed, α corresponds to the saving-consumption pan with expected surpus E[V (α )] equa to zero. From (8), we find n 1 E [V (α )] = a e ( i)µ i=n e ( i)µ =0. (40) From this expression we see that a β j,j=1,,n + m, asdefined in (33) are stricty positive. Hence, from (27) we find that the probabiity of consumption shortfa foows from f α (F W (α )(0)) = 0 From (18) and the fact that the sequence {r i σ Zi } 0 r is non-increasing and positive, one finds that n 1 f α (0.5) = a e ( i)µ e 1 2 r2 i σ2 Z i 1 i=n e ( i)µ e 1 2 r2 i σ2 Z i < 0. so that F W (α )(0) > 0.5, as stated in (39). In practica situations, one wi often be interested in the minima savings amount α that is required such that the probabiity of a consumption shortfa F W (α)(0) is at most equa to ε. Hence, et us consider the case that one wants to determine α(ε) whichisdeterminedby α(ε) =inf α F W (α)(0) ε ª, 0 <ε<1. (41) The probabiity ε is typicay smaer than 10%, et s say. In genera, F W (α)(0) is stricty decreasing and continuous in α. Thisimpies that α(ε) foows from F W (α(ε))(0) = ε. (42) We propose to approximate α(ε) by α (ε) which is defined by F W (α (ε))(0) = ε. (43) 13

14 Now, in case ε is such that α (ε) >α we have that E V (α (ε)) > 0. Hence,from(36),wehavethattheapproximatedsavingseffort α (ε) can be found from f α (ε)(ε) =0 if α (ε) >α. (44) Notice that we can expect that for any reasonabe ε<0.5, the approximated savings effort α (ε) can be found from (44). To see this, notice that from (39) we have that F W (α )(0) > 0.5. Furthermore, we have that F W (α)(0) is in genera stricty decreasing and continuous in α. From these two observations we find that it is very ikey that for a typica ε<0.5 we wi have that the condition α (ε) >α in (44) is fufied. 6 Numerica iustration In this section we wi numericay iustrate the resuts of this paper, appied to the specia case of α-pans as considered in the previous section. We wi assume that the yeary returns Y i have expectation and variance given by µ σ2 and 2 σ2,whereµ =0.075 and σ =0.15. First, we consider a saving-consumption pan that consists of 10 constant savings α 0 = α 2 = = α 9 =1, foowed by 10 constant withdrawas α 10 = α 11 = = α 19 = 1. Inthiscasewefind that E[V 19 ]=16.02 > 0, so that Theorem 3 can be used for determining the quanties of W19. In Tabe 1 we compare the approximated quanties Q + p [W19] with the simuated quanties Q + p [W 19 ] This tabe iustrates the accuracy of the ower bound based approximation W19. From this tabe we find for instance that there is a 90% probabiity that the fina weath at time 19 wi exceed 1.75 (simuated vaue). The approximated vaue for this fina weath is From (27) it foows that the (approximated) probabiity of consumption shortfa Pr W19 =0 equas 4.83%. In case the investor finds this probabiitytoohigh,hewihavetoincreasehissavingsefforts of 1 per year during the first ten years. As a second appication, we consider the 20/65/95 saving-consumption pan as mentioned in the introduction. This pan consists of 45 constant savings α 0 = α 2 = = α 44 = α>0, thefirst one at the age of 20 and the ast one at the age of 64. After retirement, yeary withdrawas equa to -1 wi be made unti the age of 95 has been reached, hence α 45 = α 46 = = α 75 = 1. The condition (34), or equivaenty (37) can be expressed as: α>

15 p Q + p [W19] Q + p [W 19 ] s.e Tabe 1: Approximate and simuated vaues for the quanties of W 19. α Pr W75 = % % % % % % Tabe 2: The approximated probabiity of consumption shortfa for different vaues of the savings effort. Tabe 2 contains the approximated probabiities of consumption shortfa Pr W 75 =0,fordifferent saving amounts α. These probabiities foow from (27). From (44) we find that the approximated minima savings amount α (0.05) that guarantees that the probabiity of consumption shortfa is ess than or equa to 5% is given by Acknowedgement 3 The authors acknowedge the financia support by the Onderzoeksfonds K.U.Leuven (GOA/02: Actuariëe, financiëe en statistische aspecten van afhankeijkheden in verzekerings- en financiëe portefeuies). 15

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