Universidad Carlos III de Madrid. Licenciatura en Ciencias Actuariales y Financieras Survival Models and Basic Life Contingencies

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1 Universidad Carlos III de Madrid Licenciatura en Ciencias Actuariales y Financieras Survival Models and Basic Life Contingencies PART II Lecture 3: Commutation Functions In this lesson, we will introduce the Commutation Functions (or Symbols) There is currently an interesting debate about the use of these functions in the international actuarial community We will define all the common commutation function, but the practical use of these functions is mainly related with increasing forms of benefits: Increasing Annuities and Increasing Insurance 1 Commutation Functions First, we define the basic functions using V x+1 and either l x+1 or d x, life or death functions respectively Later, the advanced functions are the infinite sums (series) of the basic ones In the following presentation, the formulae for the continuous case are also introduced, the proofs are a very good theoretical exercise not included in this short review 11 Life Commutation Functions (Annuities) The basic commutation function is D y V y l y 1

2 In practice, this basic function is only used for integer y The first use of the comutation is to compute the expected value of pure endowement ne x V n np x V n+x ln+x D n+x Now, the second generation of Life Commutation Functions is N x x integer N x +j +z dz See the equivalence for the expected value of life perpetuities-due ä x V j jp x V j l x+j l x a x N x, and n-year temporary annuity-due V x+j l x+j +j ä x:ne ä x +n ä x+n N x +n N x+n +n N x N x+n N x a x:ne N x N x+n The third generation of commutation functions are most interesting for us because they are specially deviced for increasing benefits Observe the following relationships S x N x+j (j +1)+j (j +1)V x+j l x+j S x N x+z dz z +z dz, using the Current Payments Technique, the expected value for an increasing perpetuity-due is 2

3 (Iä) x (j +1)V j jp x (j +1)V j l x+j (j +1) V x+j l x+j l x P (j +1)V x+j l x+j S x (Ia) x 1 E x (Iä) x+1 +1 S x+1 +1 S x+1 (Ia) x S x 12 Death Commutation Functions (Insurance) In the case of Death commutation function, the basic function is defined C x V x+1 d x C y V y l y µ(y) Then, the second generation of Death Commutation Functions is obtained as M x M x C x+j C x+z dz See the equivalence for the expected value of a whole of life insurance A x V j+1 jp x q x+j V j+1 l x+j d x+j V x+j+1 d x+j C x+j l x l x+j M x A x M x and an n-year temporary term insurance A1 x:ne A x +n A x+n M x +n M x+n +n M x M x+n A1 x:ne M x M x+n 3

4 The third generation of commutation functions is useful for increasing insurance Again we derive some relationships first R x M x+j (j +1)C x+j (j +1)V x+j+1 d x+j R x M x+z dz z C x+z dz The expected value of a whole of life increasing insurance is obtained using the third generation of Death commutation functions (IA) x (j +1)V j+1 jp x q x+j (j +1)V j+1 l x+j l x d x+j l x+j (j +1) V x+j+1 d x+j P (j +1)V x+j+1 d x+j R x (IA) x R x 2 Temporary Increasing Benefits In the previous section we presented a method of obtaining the expected values of the increasing whole life benefits (insurance and annuity) Now we will study the expected value of the present value of temporary annuities and insurance The incresing benefits: annuities or term insurance can be studied using the following picture (triangle): (IA) x+n n A x+n (IA) x 4

5 The formula for the expected value of an increasing temporary term insurance is (IA) x n n A x n (IA) x R x n +n M x+n +n R x+n +n +n R x R x+n n M x+n If we consider the case of n-year endowment (IA) x:ne (IA) x n n A x n (IA) x + n n E x R x R x+n n M x+n + n +n In the continuous case (IA) x n n A x n (IA) x R x R x+n n M x+n and remember that assuming UDD i δ For the increasing annuities-due we observe (Iä) x+n n ä x+n (Iä) x:ne (Iä) x and deduce the formua for its expected value (Iä) x:ne (Iä) x n n ä x n (Iä) x S x n +n N x+n +n S x+n +n +n S x S x+n n N x+n 5