Act Sci 650 Exam 1 EQUATIONS

Z for continuous whole life

Z = v^{Tₓ} = e^{−δTₓ}

Z for discrete whole life

Z = v^{Kₓ+1}

Z for continuous endowment (term n)

Z = v^{min(Tₓ, n)}

Z for increasing discrete whole life

Z = (Kₓ+1)v^{Kₓ+1}

Continuous whole life EPV Ȧₓ

∫₀^∞ e^{−δt} · ₜpₓ · μ_{x+t} dt

Discrete whole life EPV Aₓ

Σ{k=0}^∞ v^{k+1} · ₖpₓ · q{x+k}

Continuous term EPV Ȧ₍ₓ:n₎¹

∫₀^n e^{−δt} · ₜpₓ · μ_{x+t} dt

Discrete term EPV A₍ₓ:n₎

Σ{k=0}^{n−1} v^{k+1} · ₖpₓ · q{x+k}

Pure endowment nEₓ

v^n · ₙpₓ

Discrete endowment Aₓ:n

A₍ₓ:n₎ + v^n · ₙpₓ

Continuous endowment Ȧₓ:n

Ȧ₍ₓ:n₎¹ + v^n · ₙpₓ

Complete expectation e̊ₓ

∫₀^∞ ₜpₓ dt

Curtate expectation eₓ

Σ_{k=1}^∞ ₖpₓ

Term complete expectation e̊₍ₓ:n₎

∫₀^n ₜpₓ dt

Term curtate expectation e₍ₓ:n₎

Σ_{k=1}^n ₖpₓ

E[Tₓ²]

2 ∫₀^∞ t · ₜpₓ dt

E[Kₓ²]

2 Σ_{k=1}^∞ k · ₖpₓ − eₓ

Var(Tₓ)

E[Tₓ²] − (e̊ₓ)²

Var(Kₓ)

E[Kₓ²] − (eₓ)²

Second moment continuous WL ²Ȧₓ

∫₀^∞ e^{−2δt} · ₜpₓ · μ_{x+t} dt

Second moment discrete WL ²Aₓ

Σ{k=0}^∞ v^{2(k+1)} · ₖpₓ · q{x+k}

Var(SZ) WL

S²(²Aₓ − Aₓ²)

Recursion for Aₓ

v qₓ + v pₓ A_{x+1}

Recursion for eₓ

pₓ(1 + e_{x+1})

Recursion for e₍ₓ:n₎

pₓ(1 + e₍ₓ+1:n−1₎)

Deferred whole life ₙAₓ

ₙEₓ A_{x+n}

Deferred continuous term

Ȧ₍ₓ:u+n₎¹ − Ȧ₍ₓ:u₎¹

General survival (ₜpₓ)

exp(−∫₀^t μ_{x+s} ds)

General death probability (ₜqₓ)

∫₀^t ₛpₓ μ_{x+s} ds

Density of Tₓ (fₓ(t))

ₜpₓ μ_{x+t}

Continuous increasing WL

∫₀^∞ t e^{−δt} ₜpₓ μ_{x+t} dt

Increasing + decreasing identity

(ĪȦ)₍ₓ:n₎¹ + (ĎȦ)₍ₓ:n₎¹ = n Ȧ₍ₓ:n₎¹

Discrete term via WL

Aₓ − v^n ₙpₓ A_{x+n}

Curtate pmf (P(Kₓ = k))

ₖpₓ − ₍ₖ+1₎pₓ

μ_{x+t} (Alternate Equation)

fₓ(t) / ₜpₓ

Constant Force force

μ_{x+t} = μ

Constant Force survival (ₜpₓ)

e^{−μ t}

Constant Force death probability (ₜqₓ)

1 − e^{−μ t}

Constant Force Ȧₓ

μ / (μ + δ)

Gompertz force

B c^x

Gompertz survival (ₜpₓ)

exp(−(B c^x / ln c)(c^t − 1))

Makeham force

A + B c^x

Makeham survival (ₜpₓ)

exp(−A t − (B c^x / ln c)(c^t − 1))

de Moivre future lifetime

Tₓ ~ Uniform(0, ω−x)

de Moivre survival (ₜpₓ )

(ω−x−t)/(ω−x)

de Moivre force (x+t)

1/(ω−x−t)

de Moivre complete expectation (e̊ₓ )

(ω−x)/2

Gompertz–Makeham complete expectation (e̊ₓ )

∫₀^∞ exp(−A t − (B c^x / ln c)(c^t − 1)) dt

Gompertz–Makeham term expectation(e̊₍ₓ:n₎)

∫₀^n exp(−A t − (B c^x / ln c)(c^t − 1)) dt

Annual survival (any model) (1pₓ )

exp(−∫₀^1 μ_{x+s} ds)

Annual mortality (any model) (qₓ )

1 − 1pₓ

UDD m‑thly WL scaling

Aₓ^(m) ≈ (i/ i^(m)) Aₓ

UDD m‑thly term scaling

A₍ₓ:n₎^(m) ≈ (i / i^(m)) A₍ₓ:n₎

UDD m‑thly deferred WL

ₙAₓ^(m) ≈ (i / i)^(m) ₙAₓ

UDD m‑thly increasing WL

(IA)ₓ^(m) ≈ (i / i^(m))(IA)ₓ

UDD m‑thly increasing term

(IA)₍ₓ:n₎^(m) ≈ (i/ i^(m))(IA)₍ₓ:n₎

UDD m‑thly decreasing term

(DA)₍ₓ:n₎^(m) ≈ (i/ i^(m))(DA)₍ₓ:n₎

UDD m‑thly endowment insurance

Aₓ:n^(m) ≈ (i / i^(m)) Aₓ:n

UDD generic scaling for any end‑of‑year‑of‑death benefit

Value^(m) ≈ (i / i^(m)) Value

UDD continuous–annual link

Ȧₓ = (i / δ) Aₓ

UDD continuous–m‑thly link

Ȧₓ ≈ (i^(m) / δ) Aₓ^(m)

Random variable under Constant Force

Tₓ ~ Exponential(μ), support t ≥ 0

Random variable support for de Moivre

Tₓ ~ Uniform(0, ω−x), support 0 ≤ t ≤ ω−x

Random variable density (general model)

f{Tₓ}(t) = ₜpₓ μ{x+t}, support t > 0

Random variable for curtate lifetime

Kₓ = ⌊Tₓ⌋, support {0,1,2,…}

Continuous–discrete link for second moment (WL)

²Ȧₓ = ((1+i)² − 1)/(2δ) · ²Aₓ

Continuous–discrete link for second moment (term)

²Ȧ₍ₓ:n₎¹ = ((1+i)² − 1)/(2δ) · ²A₍ₓ:n₎

Continuous–discrete link for second moment (deferred WL)

²(Ȧₙ|x) = ((1+i)² − 1)/(2δ) · ²(ₙAₓ)

Temporary complete expectation e̊ₓ:n

e̊ₓ:n° = ∫₀ⁿ t · ₜpₓ · μₓ₊ₜ dt + n · ⁿpₓ

UDD conversion eₓ → ėₓ

ėₓ = eₓ + ½

Complete Expected Future Lifetime Under UDD Recursion

e̊ₓ = qₓ (1/2) + pₓ (1 + e̊ₓ₊₁)

Whole Life Annuity BoY First Principles

äₓ = ∑ₖ₌₀^∞ vᵏ ₖpₓ

Temporary Annuity BoY First Principles

äₓ:ₙ = ∑ₖ₌₀^ₙ₋₁ vᵏ ₖpₓ

Expected Value of Whole Life Annuity BoY

äₓ = (1-Aₓ)/d

Expected Value of Temporary Annuity BoY

äₓ:ₙ = (1-Aₓ:ₙ)/d

Random Variable for Discrete Life Annuity

äₖₓ+1=(1-vᵏˣ+1)/d

Random Variable for Whole Life Continuous Annuity

äₜₓ+1=(1-vᵀˣ+1)/δ

Whole Life Continuous Annuity First Principles

āₓ=∫₀^∞e⁻ᵟᵗ ₜpₓ dt

Whole Life Annuity BoY Equation

äₓ = äₓ:ₙ + ₙ| äₓ

Deferred Whole Life Annuity BoY equation

ₙ| äₓ = ₙEₓ · äₓ+ₙ

Aₓ in Terms of äₓ

Aₓ = 1 - d·äₓ

Random Variable for BoY Discrete Temporary Annuity

ä[min(kx+1,n)] = (1-vmin(kx+1,n))/d

Order of Greatest Value to Least Value Annuities

äₓ, äₓ(m), āₓ, aₓ (m), aₓ

Equation for Certain and Life Annuity

ä⁅ₓ:ₙ⁆ = äₙ⌉ + ₙEₓ · äₓ+ₙ
m-thly is ä⁅ₓ:ₙ⁆ = äₙ⌉^(m) + ₙEₓ · äₓ+ₙ^(m)

Random Variable for Continuous Temporary Annuity

ä[min(Tx+1,n)] = (1-vmin(Tx+1,n)) / δ

Variance for a BoY Discrete Annuity

Var(Y) = (²Aₓ - Aₓ²) / d²

Variance for a Continuous Annuity

Var(Y) = (²Āₓ - Āₓ²) / δ²

BoY Discrete Annuity if i = 0%

äₓ = eₓ +1

EoY Discrete Annuity if i = 0%

aₓ = eₓ

Continuous Annuity if i = 0%

āₓ = ėₓ

How to Develop äₓ(m) from äₓ Under UDD

äₓ(m) = [(i · d) · äₓ / (i(m) · d(m))] - [(i - i(m)) / (i(m) · d(m))]

Equation for α(m)

α(m) = (i · d) / (i(m) · d(m))

Equation for β(m)

β(m) = (i - i(m)) / (i(m) · d(m))

How to Develop äₓ(m) from äₓ Under UDD Using α(m) and β(m)

äₓ(m) = [α(m) · äₓ] - β(m)

Equation for a BoY Discrete Term Annuity Using α(m) and β(m)

äₓ:ₙ(m) = [α(m) · äₓ:ₙ] - [β(m) · (1 - ₙEₓ)]

Woolhouse Equation (Given on SULT)

äₓ(m) ≈ [äₓ ] - (m-1) / 2m ] - (m² - 1) · (δ + μₓ) / (12m²) ]

Summation Formula for an Arithmetically Increasing BoY Annuity

(Iä)ₓ = ∑ₖ₌₀^∞ (k+1) vᵏ ₖpₓ

Summation Formula for an Arithmetically Increasing Continuous Annuity

(Iā)ₓ = ∑ₖ₌₀^∞ (k+1) ₖ| āₓ:₁

Actuarial Equivalence Principle

E[PV(Benefit)] = E[PV(Prem)]

Net Loss at Issuance Conceptual Equation

Lⁿ₀ = PV(Insurance) - PV(Net Premium Income)