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ACTS 4301 FORMULA SUMMARY Lesson 1: Probability Review 1. Var ACTS 4301 FORMULA SUMMARY Lesson 1: Probability Review 1. Var(X)= E[X2]- E[X]2 2. V ar(aX + bY ) = a2V ar(X) + 2abCov(X; Y ) + b2V ar(Y ) ¯ V ar(X) 3. V ar(X) = n 4. EX [X] = EY [EX [XjY ]] Double expectation 5. V arX [X] = EY [V arX [XjY ]] + V arY (EX [XjY ]) Conditional variance Distribution, Density and Moments for Common Random Variables Continuous Distributions Name f(x) F (x) E[X] Var(X) − x − x 2 Exponential e θ /θ 1 − e θ θ θ 1 x a+b (b−a)2 Uniform b−a ; x 2 [a; b] θ ; x 2 [a; b] 2 12 x x α−1 − θ α R 2 Gamma x e =Γ(α)θ 0 f(t)dt αθ αθ Discrete Distributions Name f(x) E[X] Var(X) Poisson e−λλx=x! λ λ m x m−x Binomial x p (1 − p) mp mp(1 − p) 2 Lesson 2: Survival Distributions: Probability Functions, Life Tables 1. Actuarial Probability Functions tpx - probability that (x) survives t years tqx - probability that (x) dies within t years tjuqx - probability that (x) survives t years and then dies in the next u years. t+upx = tpx ·u px+t tjuqx = tpx ·u qx+t = = tpx −t+u px = = t+uqx −t qx 2. Life Table Functions dx = lx − lx+1 ndx = lx − lx+n dx qx = 1 − px = lx lx+t tpx = lx lx+t − lx+t+u tjuqx = lx 3. Mathematical Probability Functions tpx = ST (x)(t) tqx = FT (x)(t) tjuqx = P r (t < T (x) ≤ t + u) = FT (x)(t + u) − FT (x)(t) = F (x + t + u) − F (x + t) = P r (x + t < X ≤ x + t + ujX > x) = X X sX (x) Sx(t + u) Sx+u(t) = Sx(u) S0(x + t) Sx(t) = S0(x) F0(x + t) − F0(x) Fx(t) = 1 − F0(x) 3 Lesson 3: Survival Distributions: Force of Mortality f0(x) d µx = = − ln S0(x) S0(x) dx fx(t) d µx+t = = − ln Sx(t) Sx(t) dx Z t Z t Z x+t Sx(t) =t px = exp − µx+s ds = exp − µx(s) ds = exp − µs ds 0 0 x dtpx=dt d lnt px µx(t) = − = − tpx dt fT (x)(t) = fx(t) =t px · µx(t) Z t+u tjuqx = spx · µx(s) ds t Z t tqx = spx · µx(s)ds 0 ∗ ∗ −kt If µx(t) = µx(t) + k for all t; then spx = spxe If µx(t) =µ ^x(t) +µ ~x(t) for all t; then spx = sp^xsp~x ∗ ∗ k If µx(t) = kµx(t) for all t; then spx = (spx) 4 Lesson 4: Survival Distributions: Mortality Laws Exponential distribution or Constant Force of Mortality µx(t) = µ −µt tpx = e Uniform distribution or De Moivre's law 1 µ (t) = ; 0 ≤ t ≤ ! − x x ! − x − t ! − x − t p = ; 0 ≤ t ≤ ! − x t x ! − x t q = ; 0 ≤ t ≤ ! − x t x ! − x u q = ; 0 ≤ t + u ≤ ! − x tju x ! − x Beta distribution or Generalized De Moivre's law α µ (t) = x ! − x − t ! − x − tα p = ; 0 ≤ t ≤ ! − x t x ! − x Gompertz's law: x µx = Bc ; c > 1 Bcx(ct − 1) p = exp − t x ln c Makeham's law: x µx = A + Bc ; c > 1 Bcx(ct − 1) p = exp −At − t x ln c Weibull Distribution n µx = kx −kxn+1=(n+1) S0(x) = e 5 Lesson 5. Survival Distributions: Moments Complete Future Lifetime Z 1 ˚ex = ttpxµx+t dt 0 Z 1 ˚ex = tpx dt 0 1 ˚e = for constant force of mortality x µ ! − x ˚e = for uniform (de Moivre) distribution x 2 ! − x ˚e = for generalized uniform (de Moivre) distribution x α + 1 Z 1 h 2i E (T (x)) = 2 ttpx dt 0 1 V ar (T (x)) = for constant force of mortality µ2 (! − x)2 V ar (T (x)) = for uniform (de Moivre) distribution 12 (! − x)2 V ar (T (x)) = for generalized uniform (de Moivre) distribution (α + 1)2(α + 2) n-year Temporary Complete Future Lifetime Z n ˚ex:n = ttpxµx+t dt + nnpx 0 Z n ˚ex:n = tpx dt 0 ˚ex:n =n px(n) +n qx(n=2) for uniform (de Moivre) distribution ˚ex:1 = px + 0:5qx for uniform (de Moivre) distribution 1 − e−µn ˚e = for constant force of mortality x:n µ Z n h 2i E (T (x) ^ n) = 2 ttpx dt 0 Curtate Future Lifetime 1 X ex = kkjqx k=1 1 X ex = kpx k=1 ex =˚ex − 0:5 for uniform (de Moivre) distribution 1 h 2i X E (K(x)) = (2k − 1)kpx k=1 1 V ar (K(x)) = V ar (T (x)) − for uniform (de Moivre) distribution 12 6 n-year Temporary Curtate Future Lifetime n−1 X ex:n = kkjqx + nnpx k=1 n X ex:n = kpx k=1 ex:n =˚ex:n − 0:5nqx for uniform (de Moivre) distribution n h 2i X E (K(x) ^ n) = (2k − 1)kpx k=1 7 Lesson 6: Survival Distributions: Percentiles, Recursions, and Life Table Concepts Recursive formulas ˚ex =˚ex:n +n px˚ex+n ˚ex:n =˚ex:m +m px˚ex+m:n−m; m < n ex = ex:n +n pxex+n = ex:n−1 +n px(1 + ex+n) ex = px + pxex+1 = px(1 + ex+1) ex:n = ex:m +m pxex+m:n−m = = ex:m−1 +m px(1 + ex+m:n−m) m < n ex:n = px + pxex+1:n−1 = px 1 + ex+1:n−1 Life Table Concepts Z 1 Tx = lx+tdt; total future lifetime of a cohort of lx individuals 0 Z n nLx = lx+tdt; total future lifetime of a cohort of lx individuals over the next n years 0 Z 1 Yx = Tx+tdt 0 Tx ˚ex = lx nLx ˚ex:n = lx Central death rate and related concepts ndx nmx = nLx qx mx = for uniform (de Moivre) distribution 1 − 0:5qx nmx = µx for constant force of mortality L − l a(x) = x x+1 the fraction of the year lived by those dying during the year dx 1 a(x) = for uniform (de Moivre) distribution 2 8 Lesson 7: Survival Distributions: Fractional Ages Function Uniform Distribution of Deaths Constant Force of Mortality Hyperbolic Assumption s lx+s lx − sdx lxpx lx+1=(px + sqx) s sqx sqx 1 − px sqx=(1 − (1 − s)qx) s spx 1 − sqx px px=(1 − (1 − s)qx) s sqx+t sqx=(1 − tqx); 0 ≤ s + t ≤ 1 1 − px sqx=(1 − (1 − s − t)qx) µx+s qx=(1 − sqx) − ln px qx=(1 − (1 − s)qx) s 2 spxµx+s qx −px(ln px) pxqx=(1 − (1 − s)qx) 2 mx qx=(1 − 0:5qx) − ln px qx=(px ln px) Lx lx − 0:5dx −dx= ln px −lx+1 ln px=qx ˚ex ex + 0:5 ˚ex:n ex:n + 0:5nqx ˚ex:1 px + 0:5qx 9 Lesson 8: Survival Distributions: Select Mortality When mortality depends on the initial age as well as duration, it is known as select mortality, since the person is selected at that age. Suppose qx is a non-select or aggregate mortality and q[x]+t; t = 0; ··· ; n − 1 is select mortality with selection period n. Then for all t ≥ n; q[x]+t = qx+t: 10 Lesson 9: Insurance: Payable at Moment of Death - Moments - Part 1 Z 1 −δt A¯x = e tpxµx(t) dt 0 Actuarial notation for standard types of insurance Name Present value random variable Symbol for actuarial present value T Whole life insurance v A¯x vT T ≤ n Term life insurance A¯1 0 T > n x:n 0 T ≤ n Deferred life insurance jA¯ vT T > n n x 0 T ≤ n T ¯1 ¯ Deferred term insurance v n < T ≤ n + m njAx:m =n jmAx 0 T > n 0 T ≤ n Pure endowment A 1 or E vn T > n x:n n x vT T ≤ n Endowment insurance A¯ vn T > n x:n 11 Lesson 10: Insurance: Payable at Moment of Death - Moments - Part 2 Actuarial present value under constant force and uniform (de Moivre) mortality for insurances payable at the moment of death Type of insurance APV under constant force APV under uniform (de Moivre) a¯ µ !−x Whole life µ+δ !−x µ −n(µ+δ) a¯n n-year term µ+δ 1 − e !−x e−δna¯ µ −n(µ+δ) !−(x+n) n-year deferred life µ+δ e !−x −n(µ+δ) e−δn(!−(x+n)) n-year pure endowment e !−x j-th moment Multiply δ by j in each of the above formulae Gamma Integrands Z 1 n −δt n! t e dt = n+1 0 δ Z u u −δt 1 −δu a¯u − uv te dt = 2 1 − (1 + δu)e = 0 δ δ Variance If Z3 = Z1 + Z2 and Z1;Z2 are mutually exclusive, then V ar(Z3) = V ar(Z1) + V ar(Z2) − 2E[Z1]E[Z2] 12 Lesson 11: Insurance: Annual and m-thly: Moments 1 1 X k+1 X k+1 Ax = kjqxv = kpxqx+kv 0 0 Actuarial present value under constant force and uniform (de Moivre) mortality for insurances payable at the end of the year of death Type of insurance APV under constant force APV under uniform (de Moivre) a q !−x Whole life q+i !−x q n an n-year term q+i (1 − (vp) ) !−x vna q n !−(x+n) n-year deferred life q+i (vp) !−x n vn(!−(x+n)) n-year pure endowment (vp) !−x 13 Lesson 12: Insurance: Probabilities and Percentiles Summary of Probability and Percentile Concepts • To calculate P r(Z ≤ z) for continuous Z, draw a graph of Z as a function of T .
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