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精算·寿险
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精算·寿险
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生命表与生存曲线
Life Table & Survivorship Curve
l0基数 · lx存活/qx死亡率/dx=lx·qx · 生存曲线 · Σdx=l0
Recurse lx from a radix l₀; read qx, px, dx
死亡力 μ(x)
Force of Mortality μ(x)
μ(x)=−dlnlx/dx · μ≈qx · U形:婴儿高→少年谷→衰老升
The U/J-shaped instantaneous hazard and the μ↔qx link
冈珀茨-梅克汉死亡律
Gompertz–Makeham Law
μ=A+B·c^x · Makeham floor+Gompertz衰老 · 倍增ln2/lnc≈7年
μ(x) = A + B·c^x: accident floor A and geometric senescence B·c^x
简约与完全平均余命 e_x、ė_x
Curtate & Complete Life Expectancy e_x, ė_x
e_x=Σkpx · 完全≈e_x+½ · 随年龄降但x+e_x升
From a life table: curtate e_x = Σ kpx, complete ė_x ≈ e_x + ½
生存 tpx、死亡 tqx 与延期死亡 u|tqx
Survival tpx, Mortality tqx & Deferred u|tqx
kpx=l(x+k)/lx · tqx=1−tpx · 递延u|tqx=upx−(u+t)px
From a life table: tpx = l_{x+t}/l_x, tqx = 1 − tpx, deferred u|tqx = upx − (u+t)px
终身与定期期初生命年金现值 ä_x、ä_{x:n}
Present Value of a Whole-Life / Temporary Annuity-Due ä_x, ä_{x:n}
ä_x=Σv^k·kpx · v=1/(1+i) · 定期n年 · i↑则ä↓
Pays 1 at the start of each year while alive; ä_x = Σ v^k·kpx, v = 1/(1+i)
寿险趸缴净保费 A_x
Whole-life net single premium A_x
A_x=Σv^(k+1)·kpx·q · 恒等A_x=1−d·ä_x · 年末给付
Whole life: benefit 1 paid at end of year of death; A_x is the sum of discounted death terms
均衡净保费 P_x = A_x / ä_x
Net level premium P_x = A_x / ä_x
均衡原理 · P=A_x/ä_x · P·ä_x=A_x · 随投保年龄升
Level annual premium set by the equivalence principle
未来法责任准备金 tV = A_{x+t} − P·ä_{x+t}
Prospective policy reserve tV = A_{x+t} − P·ä_{x+t}
tV=A(x+t)−P·ä(x+t) · 0V≈0升向面额1 · 未来法
Issue age 45, whole life, i = 5%: reserve rises from 0 toward the face amount 1
换算函数 Dx · Nx · Cx · Mx
Commutation Functions Dx · Nx · Cx · Mx
Dx=v^x·lx/Nx=ΣDx/Cx=v^(x+1)dx/Mx=ΣCx · ä=N/D A=M/D
Commutation columns compress the life table and interest into single columns; annuities and insurances become simple ratios ä_x=Nx/Dx, A_x=Mx/Dx
限额期望值、免赔额与损失消除比
Limited Expected Value, Deductible & LER
E[X∧u]限期望 · 免赔(X−d)+ · LER=E[X∧d]/E[X] · 指数θ(1−e^(−u/θ))
Modify a loss X by a deductible d or a policy limit u: E[X∧u], E[(X−d)₊] and LER(d)
聚合索赔 S = X₁+…+X_N 的复合泊松矩
Aggregate Claims S = X₁+…+X_N (Compound Poisson)
S=ΣXi N~Poi(λ) · E[S]=λE[X]=50000 · Var=λE[X²] SD=7071
Claim count N ~ Poisson(λ), severities Xᵢ iid; E[S]=λ·E[X], Var[S]=λ·E[X²]