Eksamen i Forsikringsmatematik, Forsikring og Jura. ved Københavns Universitet, Blok , , 2 timer

Transcription

1 ksaen i Forsikringsateatik, Forsikring og Jura ved Københavns Universitet, Blok , , 2 tier MS, All written aids and writing with pencil is allowed Alle skriftlige hjælpeidler og besvarelse ed blyant er tilladt) Proble In this proble we study a life annuity for a arried couple in the following sense: The life annuity starts at tie k and pays at the end of every period an annuity benefit of 2 units as long as both spouses are alive så længe begge ægtefæller er i live) and an annuity benefit of as long as one and only one spouse is alive så længe een og kun een ægtefælle er i live) The annuity stops at tie k + n Let T x and T y denote the reaining life ties of the ale x and the feale y aged x and y, respectively, at tie 0 where the insurance contract is issued) Let the insurance copany assue unisex ortality ie the sae ortality assuption for both spouses a) Argue that the strea of benefits can be foralized by b t) = k<t k+n) 2 Tx>t,T y>t) + Tx>t,T y t) + Tx t,t y>t)), and argue why a couple would be interested in such a benefit strea b) Assue that T x and T y are independent uafhængige) Show that the expected k+n ] present value of benefits, t=0 e t b t), is given by 2 k+n t=k+ e t tp x t p y + k+n t=k+ e t tp x t q y + k+n t=k+ e t tq x t p y, and use this to derive the following ore coprised expression for the sae expected present value brug dette til at udlede følgende ere koprierede udtryk for den sae forventede nutidsværdi), k+n t=k+ e t tp x + k+n t=k+ e t tp y

2 Conclude that the benefits are priced siply as a su of two separate deferred teporary life annuities on each spouse in the couple so en su af to separate opsatte ophørende livrenter på hver ægtefælle i parret) Hint: Use that, for independent variables T x and T y, Tx>t,T y>t)] = Tx>t)] Ty>t)], Tx>t,T y t)] = Tx>t)] Ty t)], and Tx t,t y>t)] = Tx t)] Ty>t)] ; use also that tp y + t q y = ) c) The calculations in b) were based on the assuption of independence antagelsen o uafhængighed) The question is whether the interpretation fortolkningen) of the product as two separate life annuities also holds in the case where the two reaining life ties are dependent Show that, even for dependent reaining life ties, b t) = k<t n) Tx>t) + Ty>t)) Use this to prove that, no atter the dependence between T x and T y, Hint: Tx t) + Tx>t) = ) k+n t=0 b t) e t ] = k a xn + k a yn d) Assue that the benefit strea is paid for by a level preiu π løbende præie π) paid at the beginning of every period as long as x is alive until tie k Derive the equivalence preiu π in ters of k a xn, k a yn, and ä xk Proble 2 In this proble we consider the pricing of a portfolio of non-life insurance contracts We assue that the clais Y, Y 2, are exponentially distributed eksponentialfordelt) which eans that the density tætheden) of each clai is given by, f y) = e y a) Argue that the exponential distribution is a special case of a Gaa distribution with a particular choice of paraeters) Use the general forula for the Gaa distribution, Y k e ay ] = Γ γ + k) Γ γ) γ + a) γ+k, to specify a forula for the kth oent of the exponential distribution and conclude that Y ] = and V ar Y ] = Is the exponential distribution heavy-tailed? 2 Hint: You can use that Γ n) = n )! for n =, 2, and that Γ ) = ) 2

3 b) Assue that the portfolio consists of insurance contracts and each contract causes giver anledning til) a clai with probability p Then the total clai nuber is binoially distributed, p) Show that X] = p and V ar X] = p 2 p) and write down the total preiu π std X) topspcript std abbreviates standard deviation) based on the standard deviation principle with risk loading 2 paraeter a standardafvigelsesprincippet ed risikotillægsparaeter a) c) We assue that the portfolio consists of 0 6 policies and that each contract causes a clai with the probability 0 3 Furtherore we assue that the expected value of each clai equals 0 6 Show that the equivalence preiu per contract ækvivalenspræien pr contract), ie the preiu that each contract holder has to pay such that the equivalence principle ækvivalensprincippet) is fulfilled, equals πe X) = 000 topscript e abbreviates equivalence) Show that the risk loading per contract, in case the insurance copany uses the standard deviation principle with loading paraeter a on the portfolio of policies, equals a 999Hint: Use your result fro b)) d) What is the risk loading per contract if the insurance copany instead of assuing a binoially distributed total clai nuber,, p) = 0 6, 0 3 ), assues a Poisson distributed total clai nuber with N] = 000, ie λ = 0 3? Will the risk loading be larger or saller than this the answer you just found in the first part of question d)) if instead a negative binoial distribution with N] = 000 is used? TH ND 3

4 SKTCH OF SOLUTIONS a) The indicator function k<t k+n) specifies that the payents are paid at the end of every period fro period k to period k + n The ter 2 Tx>t,T y>t) specifies the double annuity as long as both are alive and the ters Tx>t,T y t) + Tx t,t y>t) specify the single annuity as long as one and only one of the spouses is alive The product reflects that a couple has a need for twice as uch oney copared to a single It is also based on the assuption that each spouse has the sae need for oney b) k+n ] t=0 e t b t) k+n = ) ] t=0 e t k<t k+n) 2 Tx>t,T y>t) + Tx>t,T y t) + Tx t,t y>t) t=k+ e t 2 Tx>t,T y>t) + Tx>t,T y t) + Tx t,t y>t) = 2 k+n t=k+ e t tp x t p y + k+n t=k+ e t tp x t q y + k+n t=k+ e t tq x t p y t=k+ e t tp x t p y + k+n t=k+ e t tp x t p y + k+n t=k+ e t tp x t q y + k+n t=k+ e t tq x t p y t=k+ e t tp x + k+n t=k+ e t tp y, corresponding to the price of two separate life annuities c) b t) = k<t k+n) 2 Tx>t,T y>t) + Tx>t,T y t) + Tx t,t y>t) = k<t k+n) Tx>t) Ty>t) + Tx>t) Ty>t) + Tx>t,T y t) + Tx t,t y>t) = k<t k+n) Tx>t) Ty>t) + Tx>t) Ty>t) + Tx>t) Ty t) + Tx t) Ty t) d) = k<t k+n) Tx>t) + Ty>t)) k+n t=0 b t) e t ] t=k+ e t tp x + k+n t=k+ e t tp y = k a xn + k a yn πä xk = k a xn + k a yn π = k a xn + k a yn ä xk 2 a) The Gaa density reduces to the exponential density for γ =, noting that Γ ) = But then, Y k] = k k! 4

5 such that Y ] =, V ar Y ] = = 2 b) c) d) X = NX = p, V arx = N] V ar Y ] + 2 Y ] V ar N] = p p p) = p 2 2 p), X + a V arx = p p + a 2 2 p) X + a V arx X + a V arx = p + p a 2 2 p) = a ) = a = a = a 999 = risk loading = λ + a λ Y 2 ] = a = a = a = a 2000 = risk loading Since the variance for the negbin distribution is larger than the expected value for the binoial distribution the variance is saller than the expected value and for the Poisson distribution they are the sae) we get a larger variance for fixed expectation and therefore the preiu becoes larger This could of course also be spelled out in atheatics) 5