# Longevity 11 Lyon 7-9 September 2015

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1 Longeviy 11 Lyon 7-9 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Homepage: hp://isfa.univ-lyon1.fr/ norberg/

2 LIFE INSURANCE WITH NO ECONOMIC-DEMOGRAPHIC RISK Life insurance policy issued a ime 0 and erminaing a ime T Sae of policy a ime is Z {0,..., n}, Z 0 = 0, F Z = (F Z) [0,T ] Paymens (benefis less premiums) in [0, ] oal B : db = I b d + b k dn k I = 1 [Z =], sae indicaors N k = {s; s, Z s =, Z s = k}, k, couning processes b rae of annuiy payable in sae a ime, deerminisic b k sum assured payable upon ransiion k a ime, deerminisic Transiion probabiliies p k (s, ) = P[Z() = k Z() = ] (Z is Markov) Raes of ransiion: µ k = lim h 0 p k (, + h)/h, k, deerminisic Rae of ineres a ime : r, deerminisic k 2

3 Principle of equivalence: or T τ 0 e 0 r E [ T 0 e τ 0 r db τ ] p 0 (0, τ) b τ + k;k = 0 (1) b k τ µ k τ Raionale: In a large porfolio of N idenical policies 1 N T [ τ N 0 e 0 r db τ (i) T ] τ E 0 e 0 r db τ i=1 dτ = 0 (2) Equivalence is a benchmark: as N, infinie surplus if premiums are on he safe side and infinie loss if premiums are insufficien. (3) 3

4 Prospecive reserve in sae a ime : V = = E n [ n e τ e τ r g r db τ Z = p g (, τ) b g τ + ] b gh τ µ gh τ h;h g dτ (4) Reserve (insurer s deb o insured) should be non-negaive for any sensible insurance produc. Equivalence V 0 0 = 0. Thiele s differenial equaions d d V = r V b k; k R k µ k (5) R k = b k + V k V (sum a risk) (6) 4

5 LIFE INSURANCE WITH ECONOMIC-DEMOGRAPHIC RISK Y = (r, µ k ), [0, T ] unknown a ime 0, hence sochasic, F Y = (F Y ) [0,T ] Z, [0, T ], follows model above condiional on FT Y Insead of (3) we have 1 N N i=1 T 0 e τ 0 r db (i) τ Equivalence mus now mean E [ T 0 e E [ T τ 0 r db τ F Y T 0 e τ 0 r db τ F Y T which is (2) now wih r, µ k (hence p k ) sochasic. ] ] = 0 (7) Thus, B mus be adaped no only o F Z, bu o F Y F Z. 5

6 WITH PROFIT Paymens b and b k guaraneed a ime 0. Designed by equivalence principle using pruden echnical basis wih elemens r and µ k. Technical surplus by ime is rerospecive reserve based on realized elemens less prospecive reserve based on echnical elemens: S = e 0 r τ 0 e 0 r p 0 (0, τ) b τ + k;k Differeniae using Thiele (5) for he V he p 0 (0, ), d d p0 (0, ) = g;g b k τ µ k τ dτ p 0 (0, ) V and Kolmogorov forward for p 0g (0, ) µ g p 0 (0, ) g;g µ g 6

7 ds = S r d + p 0 (0, ) c d (8) where c ime is he rae a which echnical surplus emerges in sae a c = V (r r ) + k; k R k ( µ k Technical ineres o he safe side if r r. Technical rae of ransiion k o he safe side if sign( µ k µ k ) = sign R k. Inegrae (8), using side condiion S 0 = 0, o recas S = e 0 r τ 0 e 0 r µ k ) (9) p 0 (0, τ) c τ dτ (10) 7

8 Technical surpluses are o be paid back as bonuses, assume here as annuiy dividends a rae δ in sae a ime and assurance dividends δ k upon ransiion k a ime. The ne surplus a ime is W = S e 0 r τ 0 e 0 r p 0 (0, τ) δτ + = e 0 r τ 0 e 0 r p 0 (0, τ) c τ δτ k;k k;k δ k τ δ k τ µ k τ µ k τ dτ dτ This is he balance of he accoun afer accumulaed dividends have been deduced from he echnical surplus (10). 8

9 Dividends are no sipulaed in he conrac: hey are conrolled by he pension fund/insurance company. They need o be non-negaive and o saisfy and W T raes: W 0, = 0, which means equivalence reesablished wih facual T τ 0 e 0 r p 0 (0, τ) b τ + d τ + k;k ( b k τ Remark 1: No model assumpions for r and µ k + δτ k ) µ k τ dτ = 0 Remark 2: Solvency for sure: No longeviy risk (if implemened wih sufficien prudence) 9

10 AUTOMATIC BALANCING MECHANISMS Policy issued a ime 0 specifies baseline raes r, µ k, baseline paymens b, b k, and conracual paymens b and bk adaped o he realized indices r, µ k : b = e 0 r p 0 (0, ) e 0 r p 0 (0, ) T τ 0 e 0 r T τ = 0 e 0 r b b k = e p 0 (0, τ) b τ + p 0 (0, τ) b τ + 0 r p 0 (0, ) µ k e 0 r p 0 (0, ) µ k k;k b k τ k;k µ k τ b k τ dτ µ k τ b k dτ = 0 by choice of baseline elemens a ime 0 10

11 PENSIONS A pension scheme is inroduced a ime 0. A any ime > 0, every person aged x pays he amoun a (x) per ime uni: a (x) = { c (x), if x < z (conribuion) b (x), if x z (benefi) z is reiremen age. l (x) dx members a age (x, x + dx) a ime. The oal fund a ime is U wih dynamics du = U r d + saring from U 0 0. ( z x=0 c (x) l (x) dx T x=z b (x) l (x) dx ) d 11

12 Some obvious facs can be read ou of he formula. For insance, wih ime-independen conribuions and benefis funcions c(x), b(x), and z, we see ha a sufficienly big drop in he number of enrans will in due course lead o negaive ne paymen ino he scheme, and he same goes for a sufficienly big drop in he moraliy raes a ages x > z. 12

13 Life annuiies in privae insurance: Pension producs offered on an individual and volunary basis by he life insurance offices mus be based on he principle of (individual) equivalence. This means ha, for given moraliy and ineres raes, expeced discouned conribuions mus cover expeced discouned benefis so ha balance is obained in a large porfolio. The reason for his is ha enrollmen in he scheme is volunary and canno be anicipaed; if he company would seek o cover a defici by charging premiums in excess of he equivalence rae, hen poenial new cusomers would be deerred and exising cusomers would cancel heir policies. The problem wih such schemes, wih conribuions and benefis se ou in he conrac a he ouse, is ha he solvency of he scheme depends on nondiversifiable indices. This problem is ackled in wih-profi schemes. 13

14 There is no iniial fund, so 0 r τ 0 e 0 r τ U = e a τ(x) l τ (x) dx dτ 0 Change variables s = τ x, u = τ o see paymens per generaion: U = e 0 r τ 0 r a u (u s) l u (u s) du ds s=0 u=s e Equivalence on an individual basis (or for each generaion) means s e τ 0 r a u (u s) l u (u s) du = 0 Since a u (u s) is firs posiive for u < s + z (conribuions) and hen negaive for u s + z (benefis), τ 0 r a u (u s) l u (u s) du > 0, and so U > 0. s e 14

15 Occupaional pension schemes. Enries ino such schemes are due o employmen (occupaional schemes) We lis wo commonly used occupaional schemes: Defined benefis: The funcions c (x), b (x), and z are fixed for a cerain ime period. Conribuions mus be se sufficienly high o ensure U 0 for all likely developmens of r, µ, and ϕ over he period. Defined conribuions: c (x) is fixed for a cerain ime period. Benefis b (x) may be regulaed currenly o ensure U 0 over he period. 15

16 Usually conribuions and benefis are relaed o income on an individual basis, and in principle hey are se such ha he presen value of conribuions less benefis is 0 on he average. This means ha here is a posiive reserve a any ime, ha is, U > 0. Sae pensions do no necessarily guaranee a cerain level of benefis. Benefis are conrolled such ha he U says posiive, and solvency is no an issue. This is all differen in occupaional pension schemes and for annuiy producs offered by privae life insurers. 16

17 Sae pensions. Surpluses and losses may be ransferred across groups of paricipans and also across generaions, and i is up o poliical and governmenal bodies o decide when and how in view of experience from he pas and predicions abou he fuure. The main characerisics of a given scheme are he funcions c (x), b (x), and z and, in paricular, he exen o which hey can be conrolled and adaped o he developmen of he unconrollable processes r, µ (x). 17

18 Pay-as-you-go: The funcions c (x), b (x), and z are currenly chosen such ha conribuions mach benefis a any ime. Thus, for all, da = 0 or z c (x)l (x) dx = b (x)l (x) dx x=0 z Thus, if U 0 = 0, hen U = 0 for all so here is no savings elemen in he scheme. There is no need o predic r and µ (or even o know wha hey are oday). 18

19 INTERGENERATIONAL RISK SHARING IN PENSIONS Y = (r, µ (x), x (0, T ), 0 ( calendar ime, x age) h s ds new enrans in (s, s + ds), all aged 0 (say) The number of members a age (x, x + dx) a ime is l (x) dx, where l (x) = h x e x µ u(u ( x)) du Ne surplus by ime is W : s µ u(u s) du dw = W r d + h s ds e [ s=0 (r r + µ ( s) µ( s)) V s δ ( s) ] d HOW TO POSE THE PROBLEM? 19

20 REFERENCES: Bohn, H. (2010): Privae versus public risk sharing: Should governmens provide reinsurance? Universiy of California Sana Barbara. Cui, J., De Jong, F., Ponds, E. (2011). Inergeneraional risk sharing wihin funded pension schemes. Journal of Pension Economics and Finance 10, 1-29 doi: / S Gollier, C. (2008): Inergeneraional risk-sharing and risk-aking of a pension fund. Journal of Public Economics 92, Norberg,R (1999): A heory of bonus in life insurance. Finance and Sochasics 3, Norberg (2006): The pension crisis: is causes, possible remedies, and he role of he regulaor. In Erfaringer og ufordringer, 20 years Jubilee Volume of Krediilsyne, he Financial Supervisory Auhoriy of Norway. Preliminary version a hp://isfaserveur.univ-lyon1.fr/ norberg/links/publicaions.hml 20

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