ACTS 4301 FORMULA SUMMARY Lesson 1. Probability Review (1

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

Bayes’ Theorem P r(B|A)P r(A) P r(A|B) = P r(B)

fY (y|x)fX (x) fX (x|y) = fY (y) Law of total probability

If Bi is a set of exhaustive (in other words, P r(∪iBi) = 1) and mutually exclusive (in other words P r(Bi ∩ Bj) = 0 for i 6= j) events, then for any event A, X X P r(A) = P r(A ∩ Bi) = P r(Bi)P r(A|Bi) i i For continuous distributions, Z P r(A) = P r(A|x)f(x)dx

Conditional Expectation Formulae

Double expectation EX [X] = EY [EX [X|Y ]]

Conditional variance V arX [X] = EY [V arX [X|Y ]] + V arY (EX [X|Y ]) 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 on [a, b] b−a , x ∈ [a, b] θ , x ∈ [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 Bernoulli Shortcut If a random variable can only assume two values a and b with probabilities q and 1 − q respectively, then its variance is V ar(X) = (b − a)2q(1 − q) 2

Name f(x) E[X] Var(X)

Poisson e−λλx/x! λ λ Bernoulli qx(1 − q)1−x q mq(1 − q) m x m−x Binomial x q (1 − q) mq mq(1 − q)

Lesson 2. Introduction to Long Term Insurance

Modern Insurance Contracts 1. Universal Life 2. Unitized-with-proﬁt 3. Equity-linked insurance (Variable Annuity or Equity-indexed Annuity) Underwriting Classes 1. Preferred 2. Normal 3. Rated 4. Uninsurable Types of Annuities 1. Single Premium Deferred Annuity (SPDA) 2. Single Premium Immediate Annuity (SPIA) 3. Regular Premium Deferred Annuity (RPDA) 4. Joint Life Annuity 5. Last Survivor Annuity 6. Reversionary Annuity Other Types of Long Term Insurance 1. Income protection insurance 2. Critical illness insurance 3. Long term care insurance Forms of Ownership of Insurance Companies 1. Mutual 2. Proprietary 3

Lesson 3. Survival Distributions: Probability Functions

1. Actuarial Probability Functions tpx - probability that (x) survives t years tqx - probability that (x) dies within t years t|uqx - probability that (x) survives t years and then dies in the next u years.

t+upx = tpx ·u px+t

t|uqx = tpx ·u qx+t = = tpx −t+u px =

= t+uqx −t qx 2. Life Table Functions lx is the number of lives at exact age x. dx is the number of deaths at age x which is also the number of deaths between exact age x and exact age x + 1. ndx is the number of deaths between exact age x and exact age x + n.

dx = lx − lx+1

ndx = lx − lx+n dx qx = 1 − px = lx lx+t tpx = lx lx+t − lx+t+u t|uqx = lx 3. Mathematical Probability Functions

tpx = ST (x)(t) = P r (T (x) > t)

tqx = FT (x)(t) = P r (T (x) ≤ t)

t|uqx = 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 + u|X > 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) 4

Lesson 4. Survival Distributions: Force of Mortality

f0(x) d µx = = − ln S0(x) S0(x) dx fx(t) d dtpx/dt d lnt px dtqx/dt µx(t) = µx+t = = − ln Sx(t) = − = − = Sx(t) dt tpx dt tpx 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 fT (x)(t) = fx(t) =t px · µx(t) =t px · µx+t Z t+u t|uqx = spx · µx(s) ds t Z t tqx = spx · µx(s)ds 0 ∗ ∗ −kt If µx(s) = µx(s) + k for 0 ≤ s ≤ t, then tpx = tpxe If µx(s) =µ ˆx(s) +µ ˜x(s) for 0 ≤ s ≤ t, then tpx = tpˆxtp˜x ∗ ∗ k If µx(s) = kµx(s) for 0 ≤ s ≤ t, then tpx = (tpx) 5

Lesson 5. Survival Distributions: Mortality Laws

Exponential distribution or Constant Force of Mortality

µx(t) = µ −µt tpx = e ∗ −µ1t ∗ µ1, 0 < t ≤ t e , 0 < t ≤ t µx+t = ∗ ⇒ tpx = −µ t∗ −µ (t−t∗) ∗ µ2, t > t e 1 · e 2 , t > t Uniform distribution or De Moivre’s law 1 µ (t) = , 0 ≤ t ≤ ω − x x ω − x − t ω − x − t p = , 0 ≤ t ≤ ω − x t x ω − x t q = , 0 ≤ t ≤ ω − x t x ω − x u q = , 0 ≤ t + u ≤ ω − x t|u x ω − x Beta distribution or Generalized De Moivre’s law α µ (t) = x ω − x − t ω − x − tα p = , 0 ≤ t ≤ ω − x t x ω − 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 6

Lesson 6. Survival Distributions: Moments

Complete Future Lifetime

Z ∞ Z ∞ ˚ex = ttpxµx+t dt = tpx dt 0 0 1 ˚e = for constant force of mortality x µ ω − x ˚e = for uniform (de Moivre) distribution x 2 ω − x ˚e = for generalized uniform (de Moivre) distribution x α + 1 Z ∞ h 2i E (T (x)) = 2 ttpx dt 0 1 V ar (T (x)) = for constant force of mortality µ2 (ω − x)2 V ar (T (x)) = for uniform (de Moivre) distribution 12 α(ω − 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 Z n ˚ex:n = t · tpxµx+t dt + nnpx = tpx dt 0 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 ∞ Z n h 2i 2 E (T (x) ∧ n) = t ·t pxµx+t dt = 2 ttpx dt 0 0

Curtate Future Lifetime

∞ ∞ X X ex = kk|qx = kpx k=1 k=1 ex =˚ex − 0.5 for uniform (de Moivre) distribution ∞ ∞ h 2i X 2 X E (K(x)) = k k|qx = (2k − 1)kpx k=1 k=1 1 V ar (K(x)) = V ar (T (x)) − for uniform (de Moivre) distribution 12 7 n-year Temporary Curtate Future Lifetime n−1 n X X ex:n = kk|qx + n ·n px = kpx k=1 k=1 ex:n =˚ex:n − 0.5nqx for uniform (de Moivre) distribution n−1 n h 2i X 2 2 X E (K(x) ∧ n) = k k|qx + n ·n px = (2k − 1)kpx k=1 k=1

Sum of Integers

n X n(n + 1) k = 2 k=1 n X n(n + 1)(2n + 1) k2 = 6 k=1 8

Lesson 7. Survival Distributions: Percentiles and Recursions

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 ∞ 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 ∞ 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 9

Lesson 8. 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 10

Lesson 9. 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. 11

Lesson 10. Survival Distributions: Models for Mortality Improvement Projected mortality for a single factor scale q(x, t) = q(x, 0)(1 − φ(x))t Central death rate R 1 q spxµx+s ds m = x = 0 x R 1 R 1 0 spx ds 0 spx ds Constant force of mortality between integral ages:

mx = µ −µ −mx qx = 1 − e = 1 − e

mx = − ln(1 − qx)

Uniform distribution between integral ages: qx qx mx = 1 = R 1 − 0.5qx 0 (1 − sqx) ds mx qx = 1 + 0.5mx µx+0.5 = mx Lognormal moments 2 E[X] = eµ+σ /2 2 2 V ar(X) = e2µ+σ eσ − 1 Lee Carter model:

lm(x, t) = αx + βxKt + (x, t) Usually we assume Kt is a random walk:

Kt+1 = Kt + c + σkZt Constraints: x t Xω Xn βx = 1 Kt = 0 x=x0 t=t0 Consequence of constraints: Ptn lm(x, t) t=t0 αx = tn − t0 + 1 Central death rate improvement factor: m(x, t) φ(m)(x, t) = 1 − m(x, t − 1) (m) For a Lee Carter model, 1 − φ (x, t) is lognormal with µ = cβx and σ = βxσk. CBD model Logit function: q(x, t) lq(x, t) = ln 1 − q(x, t) elq(x,t) q(x, t) = 1 + elq(x,t) Original form of CBD model: (1) (2) lq(x, t) = Kt + Kt (x − x¯), 12 wherex ¯ is the mean of ages in the data set. (i) We assume Kt are random walks with drift: (i) (i) (i) (i) Kt+1 = Kt + ci + σk Zt , (i) (1) (2) (i) (i) where Zt are standard normal, Zt and Zt have ﬁxed correlation ρ, and Zt and Zu are inde- pendent for t 6= u. CBD M7 model: (1) (2) (3) 2 2 lq(x, t) = Kt + Kt (x − x¯) + Kt (x − x¯) − sx + Gt−x, where Pxmax (x − x¯)2 2 x=xmin sx = xmax − xmin + 1 (3) and Kt and Gt−x are time series.

For xmax − xmin even, n(n + 1) s2 = , x 3 where n = (xmax − xmin)/2. 13

Lesson 12. Insurance: Annual and 1/m-thly: Moments

∞ ∞ X k+1 X k+1 Ax = k|qxv = 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 ω−x Whole life q+i ω−x

q n an n-year term q+i (1 − (vp) ) ω−x

vna q n ω−(x+n) n-year deferred life q+i (vp) ω−x

n vn(ω−(x+n)) n-year pure endowment (vp) ω−x 14

Lesson 13. Insurance: Continuous - Moments - Part 1

Z ∞ −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 whole life insurance |A¯ vT T > n n x

0 T ≤ n T ¯1 ¯ Deferred term insurance v n < T ≤ n + m n|Ax:m =n |mAx 0 T > n

0 T ≤ n Pure endowment insurance A 1 or E vn T > n x:n n x

vT T ≤ n Endowment insurance A¯ vn T > n x:n 15

Lesson 14. Insurance: Continuous - 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 ω−x Whole life µ+δ ω−x

µ −n(µ+δ) a¯n n-year term µ+δ 1 − e ω−x

e−δna¯ µ −n(µ+δ) ω−(x+n) n-year deferred life µ+δ e ω−x

−n(µ+δ) e−δn(ω−(x+n)) n-year pure endowment e ω−x

j-th moment Multiply δ by j in each of the above formulae

Gamma Integrands Z ∞ n −δt n! t e dt = n+1 0 δ Z u u −δt 1 h −δui 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] 16

Lesson 15. 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 . Identify parts of the graph that are below the horizontal line Z = z, and the corresponding t’s. Then calculate the probability of T being in the range of those t’s. Note that ∗ ∗ ∗ ∗ t ∗ ln z ∗ ln z ln z P r(Z < z ) = P r(v < z ) = P r t > − = ∗ p , where t = − = − δ t x δ ln(1 + i) The following table shows the relationship between the type of insurance coverage and corresponding probability:

Type of insurance P r(Z < z∗)

Whole life t∗ px

∗ n npx z ≤ v n-year term ∗ n t∗ px z > v

∗ n q + ∗ p z < v n-year deferred life n x t x 1 z∗ ≥ vn

0 z∗ < vn n-year endowment ∗ n t∗ px z ≥ v

 ∗ m+n nqx +n+m px z < v  m+n ∗ n n-year deferred m-year term nqx +t∗ px v ≤ z < v  1 z∗ ≥ vn

• In the case of constant force of mortality µ and interest δ, P r(Z ≤ z) = zµ/δ • For discrete Z, identify T and then identify K + 1 corresponding to those T . In other words, ∗ ∗ ∗ ∗ P r(Z > z ) = P r (T < t ) =k qx, where k = bt c - the greatest integer smaller than t ∗ ∗ ∗ ∗ P r(Z ≤ z ) = P r (T ≥ t ) =k px, where k = bt c - the greatest integer smaller than t Be careful with the case when t∗ = K + 1 for some integer K. • To calculate percentiles of continuous Z, draw a graph of Z as a function of T . Identify where the lower parts of the graph are, and how they vary as a function of T. For example, for whole life, higher T lead to lower Z. For n-year deferred whole life, both T < n and higher T lead to lower Z. Write an equation for the probability Z less than z in terms of mortality probabilities expressed in terms of t. Set it equal to the desired percentile, and solve for t or for ekt for any k. Then solve for z (which is often vt) 17

Lesson 16. Insurance: Recursive Formulae, Varying Insurances Recursive Formulae

Ax = vqx + vpxAx+1 2 2 Ax = vqx + v pxqx+1 + v2pxAx+2

Ax:n = vqx + vpxAx+1:n−1 1 1 Ax:n = vqx + vpxAx+1:n−1

n|Ax = vpxn−1|Ax+1 1 1 Ax:n = vpxAx+1:n−1 Increasing and Decreasing Insurance 1 1 1 (IA)x:n + (DA)x:n = (n + 1)Ax:n ¯1 ¯1 ¯1 IA x:n + DA x:n = (n + 1)Ax:n ¯ ¯1 ¯ ¯1 ¯1 IA x:n + DA x:n = nAx:n The Constant Force Increasing Insurance Z ∞ n −δt n! t e dt = n+1 0 δ Z u u −δt 1 −δu a¯n δ − uv te dt = 2 1 − (1 + δu) e = 0 δ δ µ I¯A¯ = for constant force x (µ + δ)2 2µ E[Z2] = for Z a continuously increasing continuous insurance, constant force. (µ + 2δ)3

Recursive Formulas for Increasing and Decreasing Insurance 1 1 1 (IA)x:n = Ax:n + vpx (IA)x+1:n−1 1 1 1 (IA)x:n = Ax:1 + vpx (IAA)x+1:n−1 1 1 1 (DA)x:n = nAx:1 + vpx (DA)x+1:n−1 1 1 1 (DA)x:n = Ax:n + (DA)x:n−1 18

Lesson 17. Relationships between Insurance Payable at Moment of Death and Payable at End of Year Summary of formulas relating insurances payable at moment of death to insurances payable at the end of the year of death assuming uniform distribution of deaths

i A¯ = A x δ x i A¯1 = A1 x:n δ x:n i |A¯ = |A n x δ n x i IA¯1 = (IA)1 x:n δ x:n i ID¯1 = (ID)1 x:n δ x:n i A¯ = A1 + A 1 x:n δ x:n x:n i A(m) = A x i(m) x 2i + i2 2A¯ = 2A x 2δ x 1 1 I¯A¯1 = IA¯1 − A¯1 − x:n x:n x:n d δ Summary of formulas relating insurances payable at moment of death to insurances payable at the end of the year of death using claims acceleration approach

0.5 A¯x = (1 + i) Ax ¯1 0.5 1 Ax:n = (1 + i) Ax:n 0.5 n|A¯x = (1 + i) n|Ax ¯ 0.5 1 1 Ax:n = (1 + i) Ax:n + Ax:n (m) (m−1)/2m Ax = (1 + i) Ax 2 2 A¯x = (1 + i) Ax 19

Lesson 19. Annuities: Discrete, Expectation Relationships between insurances and annuities 1 − A a¨ = x x d Ax = 1 − da¨x 1 − A a¨ = x:n x:n d Ax:n = 1 − da¨x:n 1 1 1 Ax:n = va¨x:n − ax:n (1 + i)Ax + iax = 1 Relationships between annuities

a¨x:n =a ¨n +n |a¨x a¨x:n =a ¨x −n Exa¨x+n

n|a¨x =n Exa¨x+n

a¨x =a ¨x:n +n |a¨x

ax =a ¨x − 1

a¨x:n = ax:n + 1 −n Ex = ax:n−1 + 1 Other annuity equations ∞ ∞ X k X k a¨x = v kpx, ax = v kpx k=0 k=1 n−1 X a¨x:n = a¨k k−1pxqx+k−1 +a ¨nn−1px k=1 n−1 n X k X k a¨x:n = v kpx, ax:n = v kpx k=0 k=1 1 + i a¨ = , if q is constant x q + i x Accumulated value a¨x:n s¨x:n = nEx sx:n =s ¨x+1:n−1 + 1 1 sx:n = sx+1:n−1 + n−1Ex+1 1 sx:n =s ¨x:n + 1 − nEx mthly annuities ∞ k (m) X 1 m a¨x = v k px m m k=0 20

Lesson 20. Annuities: Continuous, Expectation Actuarial notation for standard types of annuity

Name Payment per Present value Symbol for actuarial annum at time t random variable present value

Whole life 1 t ≤ T a¯T a¯x annuity

1 t ≤ min(T, n) a¯T T ≤ n Temporary life a¯x:n 0 t > min(T, n) a¯n T > n annuity

0 t ≤ n or t > T 0 T ≤ n Deferred life annuity n|a¯x 1 n < t ≤ T a¯T − a¯n T > n

0 t ≤ n or t > T 0 T ≤ n Deferred temporary 1 n < t ≤ n + m and t ≤ T a¯T − a¯n n < T ≤ n + m n|a¯x:m 0 T > n + m a¯n+m − a¯n T > n + m life annuity

1 t ≤ max(T, n) a¯n T ≤ n Certain-and-life a¯x:n 0 t > max(T, n) a¯T T > n

Relationships between insurances and annuities 1 − A¯ a¯ = x x δ A¯x = 1 − δa¯x 1 − A¯ a¯ = x:n x:n δ A¯x:n = 1 − δa¯x:n A¯ − A¯ |a¯ = x:n x n x δ General formulas for expected value

Z ∞ a¯x = antpxµx+t dt 0 Z ∞ t a¯x = v tpx dt 0 Z n t a¯x:n = v tpx dt 0 Z ∞ t n|a¯x = v tpx dt n

Formulas under constant force of mortality 21

1 a¯ = x µ + δ 1 − e−(µ+δ)n a¯ = x:n µ + δ e−(µ+δ)n |a¯ = n x µ + δ

Relationships between annuities

a¯x =a ¯x:n +n Exa¯x+n

a¯x:n =a ¯n +n |a¯x 22

Lesson 21. Annuities: Variance General formulas for second moments Z ∞ ¯ 2 2 E[Yx ] = a¯t tpxµx+t dt 0 ∞ ¨ 2 X 2 E[Yx ] = a¨k k−1|qx k=1 n n−1 ¨ 2 X 2 2 X 2 2 E[Yx:n] = a¨k k−1|qx + npxa¨n = a¨k k−1|qx + n−1pxa¨n k=1 k=1 Special formulas for variance of whole life annuities and temporary life annuities 2A¯ − (A¯ )2 2(¯a −2 a¯ ) V ar(Y¯ ) = x x = x x − (¯a )2 x δ2 δ x 2A¯ − (A¯ )2 2(¯a −2 a¯ ) V ar(Y¯ ) = x:n x:n = x:n x:n − (¯a )2 x:n δ2 δ x:n 2A − (A )2 2(¨a −2 a¨ ) V ar(Y¨ ) = V ar(Y ) = x x = x x +2 a¨ − (¨a )2 x x d2 d2 x x 2A − (A )2 V ar(Y¨ ) = V ar(Y ) = x:n x:n x:n x:n−1 d2 23

Lesson 22. Annuities: Probabilities and Percentiles

• To calculate a probability for an annuity, calculate the t for whicha ¯t has the desired property. Then calculate the probability t is in that range. • To calculate a percentile of an annuity, calculate the percentile of T , then calculatea ¯T . • Some adjustments may be needed for discrete annuities or non-whole-life annuities • If forces of mortality and interest are constant, then the probability that the present value of payments on a continuous whole life annuity will be greater than its actuarial present value is µ µ/δ P r(¯a > a¯ ) = T (x) x µ + δ 24

Lesson 23. Annuities: Varying Annuities, Recursive Formulas Increasing/Decreasing Annuities 1 I¯a¯ = , if µ is constant x (µ + δ)2 ¯ ¯ Ia¯ x:n + Da¯ x:n = na¯x:n

(Ia)x:n + (Da)x:n = (n + 1)ax:n Recursive Formulas Whole life annuities:

a¨x = vpxa¨x+1 + 1 1 a¨(m) = v1/m · p · a¨(m) + x 1/m x x+1/m m ax = vpxax+1 + vpx

a¯x = vpxa¯x+1 +a ¯x:1 Temporary annuities:

a¨x:n = vpxa¨x+1:n−1 + 1

ax:n = vpxax+1:n−1 + vpx

a¯x:n = vpxa¯x+1:n−1 +a ¯x:1 Deferred life annuities:

n|a¨x = vpx · n−1|a¨x+1

n|ax = vpx · n−1|ax+1

n|a¯x = vpx · n−1|a¯x+1 Certain-and-life annuities: a¨ = 1 + vq a¨ + vp a¨ x:n x n−1 x x+1:n−1 a = v + vq a + vp a x:n x n−1 x x+1:n−1 a¯ =a ¯ + vq a¯ + vp a¯ x:n 1 x n−1 x x+1:n−1 25

Lesson 24. Annuities: 1/m-thly Payments In general:

1 a(m) =a ¨(m) − x x m !m !−m i(m) d(m) 1 + i = 1 + = 1 − m m

(m) 1 i = m (1 + i) m − 1

(m) − 1 d = m 1 − (1 + i) m

For small interest rates:

m − 1 a¨(m) ≈ a¨ − x x 2m m − 1 a(m) ≈ a + x x 2m

Under the uniform distribution of death (UDD) assumption:

m − 1 a¨(m) =a ¨ − x x 2m m − 1 a(m) = a + x x 2m (m) a¨x = α(m)äx − β(m) (m) a¨x:n = α(m)äx:n − β(m)(1 −n Ex) (m) n|a¨x = α(m)n|a¨x − β(m)nEx a¯x = α(∞)äx − β(∞) 1 1 a(m) =a ¨(m) − + E x:n x:n m m n x 1 − A(m) a¨(m) = x x d(m)

Similar conversion formulae for converting the modal insurances to annuities hold for other types of insurances and annuities (only whole life version is shown).

id α(m) = i(m)d(m) i − i(m) β(m) = i(m)d(m) i(∞) = d(∞) = ln(1 + i) = δ 26

(m) Woolhouse’s formula for approximatinga ¨x : m − 1 m2 − 1 a¨(m) ≈ a¨ − − (µ + δ) x x 2m 12m2 x 1 1 a¯ ≈ a¨ − − (µ + δ) x x 2 12 x m − 1 m2 − 1 a¨(m) ≈ a¨ − (1 − E ) − (µ + δ − E (µ + δ)) x:n x:n 2m n x 12m2 x n x x+n m − 1 m2 − 1 |a¨(m) ≈ |a¨ − E − E (µ + δ) n x n x 2m n x 12m2 n x x+n 1 1 ˚e = e + − µ x x 2 12 x When the exact value of µx is not available, use the following approximation: 1 µ ≈ − (ln p + ln p ) x 2 x−1 x 27

Lesson 26. Premiums: Net Premiums for Discrete Insurances - Part 1 Assume the following notation

(1) Px is the premium for a fully discrete whole life insurance, or Ax/a¨x 1 1 (2) Px:n is the premium for a fully discrete n-year term insurance, or Ax:n/a¨x:n 1 1 (3) Px:n is the premium for a fully discrete n-year pure endowment, or Ax:n/a¨x:n (4) Px:n is the premium for a fully discrete n-year endowment insurance, or Ax:n/a¨x:n 28

Lesson 27. Premiums: Net Premiums for Discrete Insurances - Part 2 Whole life and endowment insurance beneﬁt premiums 1 Px = − d a¨x dAx Px = 1 − Ax 1 Px:n = − d a¨x:n dAx:n Px:n = 1 − Ax:n

For fully discrete whole life and term insurances: 1 If qx is constant, then Px = vqx and Px:n = vqx. Future loss at issue formulas for fully discrete whole life with face amount b: P P L = vKx+1 b + x − x 0 d d P P E[L ] = A¯ b + x − x 0 x d d Similar formulas are available for endowment insurances. Refund of premium with interest To calculate the beneﬁt premium when premiums are refunded with interest during the deferral period, equate the premiums and the beneﬁts at the end of the deferral period. Past premiums are accumulated at interest only. Three premium principle formulae 1 1 nPx − Px:n = Px:n Ax+n 1 Px:n −n Px = Px:n (1 − Ax+n) 29

Lesson 28. Premiums: Net Premiums Paid on an 1/m-thly Basis If premiums are payable mthly, then calculating the annual beneﬁt premium requires dividing by an mthly annuity. If you are working with a life table having annual information only, mthly annuities can be estimated either by assuming UDD between integral ages or by using Woolhouse’s formula (Lesson 20). The mthly premium is then a multiple of the annual premium. For example, for h-pay whole life payable at the end of the year of death,

(m) Ax hPxax:h hPx = = a(m) a(m) x:h x:h 30

Lesson 29. Premiums for Fully Continuous Expectation The equivalence principle: The actuarial present value of the beneﬁt premiums is equal to the actuarial present value of the beneﬁts. For instance:

whole life insurance A¯x = P¯ A¯x a¯x n − year endowment insurance A¯x:n = P¯ A¯x:n a¯x:n ¯ 1 ¯ ¯ 1 n − year term insurance Ax:n = P Ax:n a¯x:n ¯ ¯ ¯ n − year deferred insurance n|Ax =n P n|Ax a¯x:n n − pay whole life insurance A¯x =n P¯ A¯x a¯x:n ¯ n − year deferred annuity n|a¯x = P (n|a¯x)a ¯x:n

1 − δa¯x 1 P¯ A¯x = = − δ a¯x a¯x ¯ ¯ Ax δAx P¯ A¯x = = (1 − A¯x)/δ 1 − A¯x 1 P¯ A¯x:n = − δ a¯x:n ¯ δAx:n P¯ A¯x:n = 1 − A¯x:n For constant force of mortality, P¯ A¯x and P¯ A¯x:n are equal to µ. Future loss formulas for whole life with face amount b and premium amount π: 1 − vT π π L = bvT − πa¯ = bvT − π = vT b + − 0 T δ δ δ π π E[ L] = bA¯ − πa¯ = A¯ b + − 0 x x x δ δ Similar formulas are available for endowment insurance. 31

Lesson 30. Premiums: Gross Premiums g The gross future loss at issue L0 is the random variable equal to the present value at issue of benefits plus expenses minus the present value at issue of gross premiums: g g L0 = PV (Ben) + PV (Exp) − PV (P ) To calculate the gross premium P g by the equivalence principle, equate P g times the annuity-due for the premium payment period with the sum of 1. An insurance for the face amount plus settlement expenses 2. P g times an annuity-due for the premium payment period of renewal percent of premium expense, plus the excess of the first year percentage over the renewal percentage 3. An annuity-due for the coverage period of the renewal per-policy and per-100 expenses, plus the excess of first year over renewal expenses 32

Lesson 31. Premiums: Variance of Future Loss, Discrete

The following equations are for whole life and endowment insurance of 1. For whole life, drop n. P 2 V ar( L) = 2A − (A )2 1 + 0 x:n x:n d 2 2 Ax:n − (Ax:n) If equivalence principle premium is used: V ar(0L) = 2 (1 − Ax:n) q(1 − q) For whole life with equivalence principle and constant force of mortality only: V ar( L) = 0 q +2 i If the beneﬁt is b instead of 1, and the premium P is stated per unit, multiply the variances by b2. For two whole life or endowment insurances, one with b0 units and total premium P 0 and the other with b units and total premium P , the relative variance of loss at issue of the ﬁrst to second is ((b0d + P 0) / (bd + P ))2. 33

Lesson 32. Premiums: Variance of Future Loss, Continuous

The following equations are for whole life and endowment insurance of 1. For whole life, drop n. P 2 V ar(L ) = 2A¯ − A¯ 2 1 + 0 x:n x:n δ 2 2 A¯x:n − A¯x:n V ar(L ) = , If equivalence principle premium is used 0 2 1 − A¯x:n µ V ar(L ) = , For whole life with equivalence principle and constant force of mortality only 0 µ + 2δ For whole life and endowment insurance with face amount b: P 2 V ar(L ) = 2A¯ − A¯ 2 b + 0 x x δ P 2 V ar(L ) = 2A¯ − A¯ 2 b + 0 x:n x:n δ If the beneﬁt is b instead of 1, and the premium P is stated per unit, multiply the variances by b2. For two whole life or endowment insurances, one with b0 units and total premium P 0 and the other with b units and total premium P , the relative variance of loss at issue of the ﬁrst to second is ((b0δ + P 0) / (bδ + P ))2. 34

Lesson 33. Premiums: Probabilities and Percentiles of Future Loss • For level benefit or decreasing benefit insurance, the loss at issue decreases with time for whole life, endowment, and term insurances. To calculate the probability that the loss at issue is less than something, calculate the probabiitiy that survival time is greater than something. • For level benefit or decreasing benefit deferred insurance, the loss at issue decreases during the deferral period, then jumps at the end of the deferral period and declines thereafter. – To calculate the probability that the loss at issue is greater then a positive number, calculate the probability that survival time is less than something minus the probability that survival time is less than the deferral period. – To calculate the probability that the loss at issue is greater than a negative number, calculate the probability that survival time is less than something that is less than the deferral period, and add that to the probability that survival time is less than something that is greater then the deferral period minus the probability that survival time is less than the deferral period. • For a deferred annuity with premiums payable during the deferral period, the loss at issue decreases until the end of the deferral period and increases thereafter. • The 100pth percentile premium is the premium for which the loss at issue is positive with probability p. For fully continuous whole life, this is the loss that occurs if death occurs at the 100pth percentile of survival time. 35

Lesson 34. Premiums: Special Topics