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Review

Basic Concepts

– Integrals:

\(\int_{0}^{\infty }{{{t}^{n}}{{e}^{-ct}}dt}=\dfrac{n!}{{{c}^{n+1}}}\)

\(\int_{0}^{u}{{{t}^{n}}{{e}^{-ct}}dt}=\dfrac{1-(1+cu){{e}^{-cu}}}{{{c}^{2}}}\)

\({{(\bar{I}\bar{a})}_{u}}=\dfrac{{{{\bar{a}}}_{\overline{u}}}-u{{v}^{u}}}{\delta }\)

– Geometric Series:

\(\sum\nolimits_{k=0}^{n-1}{a{{r}^{k}}}=a\dfrac{1-{{r}^{n}}}{1-r}\)

\({{i}^{(m)}}=m({{(1+i)}^{{}^{1}/{}_{m}}}-1)\)

\({{d}^{(m)}}=m(1-{{(1+i)}^{-{}^{1}/{}_{m}}})\)

– Survival Function:

\({{S}_{x}}(0)=1\)

\(\underset{t\to \infty }{\mathop{\lim }}\,{{S}_{x}}(t)=0\)

\({{S}_{x}}(t)\) must be a non-increasing function of t

Review from MFE

– Rate of Discount: \(d=\dfrac{i}{1+i}\)

– Discounting Rate: \(v=\dfrac{1}{1+i}=1-d\)

– Continuously Compounded Interest Rate: \(\delta =\ln (1+i)\) 

– Simple Interest Rate i: \({i}_{t}=1+it\) , \({{v}_{t}}=\dfrac{1}{1+it}\)

– PV of an n-Year Immediate Certain Annuity: \({{a}_{\left. {\overline {n}} \right| }}=\dfrac{1-{{v}^{n}}}{i}\)

– PV of an n-Year Immediate Certain Annuity-Due: \({{\ddot{a}}_{\left. \overline{n} \right|}}=\dfrac{1-{{v}^{n}}}{d}\)

 – PV of an n-Year Continuous Certain Annuity-Due: \({{\bar{a}}_{\left. \overline{n} \right|}}=\dfrac{1-{{v}^{n}}}{\delta}\)

– Cumulative Value of an Annuity, the Value at the End of n Years\({{S}_{\left. {\overline {n}}\! \right| }}={{(1+i)}^{n}}{{a}_{\left. {\overline {n}}\! \right| }}\)

Survival Function

\(\Pr ({{T}_{x}}>t+u)=\Pr ({{T}_{x}}>t)\Pr ({{T}_{x+t}}>u)\)

\({{S}_{x}}(t+u)={{S}_{x}}(t){{S}_{x+t}}(u)\)

Actuarial Notation

\({}_{t+u}{{p}_{x}}={}_{t}{{p}_{x}}{}_{u}{{p}_{x+t}}\)
\({}_{\text{t}|u}{{q}_{x}}={}_{t}{{\text{p}}_{x}}-{}_{t+u}{{p}_{x}}={}_{t+u}{{q}_{x}}-{}_{t}{{q}_{x}}\)

Life Table

\({{d}_{x}}={{l}_{x}}-{{l}_{x+1}}\)
\({}_{t}{{p}_{x}}=\dfrac{{{l}_{x+t}}}{{{l}_{x}}}\), \({{q}_{x}}=\dfrac{{{d}_{x}}}{{{l}_{x}}}\)
\({}_{\text{t}|u}{{q}_{x}}=\dfrac{{{l}_{x+t}}-{{l}_{x+t+u}}}{{{l}_{x}}}\)

Survival Function: Moments

Complete Lifetime

Complete Life Expectancy

\({{\overset{\scriptscriptstyle\smile}{e}}_{x}}=\int_{0}^{\infty }{_{t}{{p}_{x}}dt}\), \(E[{{T}_{x}}^{2}]=2\int_{0}^{\infty }{{{t}_{t}}{{p}_{x}}dt}\)

n-Yr. Temporary Life Expectancy

\({{\overset{\scriptscriptstyle\smile}{e}}_{x:\left. {\overline {n}}\! \right| }}=E[\min ({{T}_{x}},n)]=\int_{0}^{n}{_{t}{{p}_{x}}dt}\)

\(E[{{(\min ({{T}_{x}},n))}^{2}}]=2\int_{0}^{n}{t{}_{t}{{p}_{x}}dt}\)

Special Mortality Laws

Constant Force of Mortality

\({{\overset{\scriptscriptstyle\smile}{e}}_{x}}={{\mu }^{-1}}\), \(Var({{T}_{x}})={{\mu }^{-2}}\)
\({{\overset{\scriptscriptstyle\smile}{e}}_{x:\left. {\overline {n}}\! \right| }}=\dfrac{1-{{e}^{-\mu n}}}{\mu }\)

Uniform & Beta (Derive)

For any case in which mortality is uniformly distributed throughout the temporary period, \({{\overset{\smile }{\mathop{e}}\,}_{x:\left. {\overline {n}}\! \right| }}={{n}_{n}}{{p}_{x}}+{{(n/2)}_{n}}{{q}_{x}}\)

\({{\overset{\smile }{\mathop{e}}\,}_{x:\left. {\overline {n}}\! \right| }}={{n}_{n}}{{p}_{x}}+{{(n/2)}_{n}}{{q}_{x}}=n(\dfrac{w-x-n}{w-x})+\dfrac{n}{2}(\dfrac{n}{w-x})\)

Curtate Lifetime

\({{K}_{x}}=\left\lfloor {{T}_{x}} \right\rfloor \),where \(\left\lfloor {{T}_{x}} \right\rfloor \) is the greatest integer less than or equal to x

Curtate Life Expectancy

\({{e}_{x}}=\sum\nolimits_{k=0}^{\infty }{{{k}_{k|}}{{q}_{x}}}=\sum\nolimits_{k=1}^{\infty }{_{k}{{p}_{x}}}\)

n-Yr. Temporary Curtate Life Expectancy

\({{e}_{x:\left. {\overline {\,
n \,}}\! \right| }}=\sum\nolimits_{k=0}^{n-1}{{{k}_{k|}}{{q}_{x}}}+{{n}_{n}}{{p}_{x}}=\sum\nolimits_{k=1}^{n}{_{k}{{p}_{x}}}\)

Second Moment of Curtate Life

\(E[K_{x}^{2}]=\sum\nolimits_{k=0}^{\infty }{{{k}^{2}}_{k|}{{q}_{x}}}=\sum\nolimits_{k=1}^{\infty }{{{(2k-1)}_{k}}{{p}_{x}}}\)

Second Moment of n-Yr. Temporary Curtate Life

\(E[{{(\min ({{K}_{x}},n))}^{2}}]=\sum\nolimits_{k=0}^{n-1}{{{k}^{2}}_{k|}{{q}_{x}}}+{{n}^{2}}_{n}{{p}_{x}}=\sum\nolimits_{k=1}^{n}{{{(2k-1)}_{k}}{{p}_{x}}}\)

Complete Life Expectancy V.S. Curtate Life Expectancy

If Uniform Mortality:
\({{\overset{\scriptscriptstyle\smile}{e}}_{x}}={{e}_{x}}+0.5\), if \(\omega -x\) is a non-negative integer
\({{\overset{\scriptscriptstyle\smile}{e}}_{x:\left. {\overline {n}}\! \right| }}={{e}_{x:\left. {\overline {n}}\! \right| }}+{{0.5}_{n}}{{q}_{x}}\), if n is an integer and \(x+n\le \omega \)

Survival Function: Percentiles & Recursions

Percentiles

\(_{t}{{q}_{x}}=\pi \)

Recursive Formulas for Life Expectancy

Complete Life

\({{e}_{x}}={{e}_{x:\left. {\overline {n}}\! \right| }}{{+}_{n}}{{p}_{x}}{{e}_{x+n}}={{e}_{x:\left. {\overline {\,
n-1 \,}}\! \right| }}{{+}_{n}}{{p}_{x}}(1+{{e}_{x+n}})\)
For n = 1, \({{e}_{x}}={{p}_{x}}+{{p}_{x}}{{e}_{x+1}}={{p}_{x}}(1+{{e}_{x+1}})\)

n-Yr. Temporary Life

For m < n,

\({{e}_{x:\left. {\overline {n}}\! \right| }}={{e}_{x:\left. {\overline {m}}\! \right| }}{{+}_{m}}{{p}_{x}}{{e}_{x+m:\left. {\overline {n-m}}\! \right| }}={{e}_{x:\left. {\overline {m-1}}\! \right| }}{{+}_{m}}{{p}_{x}}(1+{{e}_{x+m:\left. {\overline {n-m}}\! \right| }})\)

For n = 1,

\({{e}_{x}}={{p}_{x}}+{{p}_{x}}{{e}_{x+1:\left. {\overline {n-1}}\! \right| }}={{p}_{x}}(1+{{e}_{x+1:\left. {\overline {n-1}}\! \right| }})\)

and when \({{T}_{x+t}}\) is Uniformly Distributed for \(t\in [0,1)\), then

\({{e}_{x:\left. {\overline {1}}\! \right| }}={{p}_{x}}+0.5{{q}_{x}}\)

Survival Function: Fractional Ages

Uniform Distribution of Deaths (UDD)

\({{l}_{x+s}}=(1-s){{l}_{x}}+s{{l}_{x+1}}={{l}_{x}}-s{{d}_{x}}\),\({}_{s}{{q}_{x}}=s{{q}_{x}}\)

Constant Force of Mortatlity (CFM)

For  \(0\le s\le 1\) and \(0\le t\le 1-s\),\({{\mu }_{x+s}}=\mu\), then \(\mu =-\ln {{p}_{x}}\) and \(_{s}{{p}_{x}}={{e}^{-\mu s}}={{({{p}_{x}})}^{s}}\)

UDD V.S. CFM

Function	UDD	CFM
\({{l}_{x+s}}\)	\({{l}_{x}}-s{{d}_{x}}\)	\({{l}_{x}}{{p}_{x}}^{s}\)
\(_{s}{{q}_{x}}\)	\(s{{q}_{x}}\)	\(1-{{p}_{x}}^{s}\)
\(_{s}{{p}_{x}}\)	\(1-s{{q}_{x}}\)	\({{p}_{x}}^{s}\)
\(_{s}{{q}_{x+t}}\)	\(\dfrac{s{{q}_{x}}}{1-t{{q}_{x}}}\)	\(1-{{p}_{x}}^{s}\)
\({{\mu }_{x+s}}\)	\(\dfrac{{{q}_{x}}}{1-s{{q}_{x}}}\)	\(\mu =-\ln {{p}_{x}}\)
\(_{s}{{p}_{x}}{{\mu }_{x+s}}\)	\({{q}_{x}}\)	\(-{{p}_{x}}^{s}\ln {{p}_{x}}\)
\({{\overset{\scriptscriptstyle\smile}{e}}_{x}}\)	\({{e}_{x}}+0.5\)
\({{\overset{\scriptscriptstyle\smile}{e}}_{x:\left. {\overline { n}}\! \right| }}\)	\({{e}_{x:\left. {\overline {n}}\! \right| }}+{{0.5}_{n}}{{q}_{x}}\)
\({{\overset{\scriptscriptstyle\smile}{e}}_{x:\left. {\overline { 1}}\! \right| }}\)	\({{p}_{x}}+0.5{{q}_{x}}\)

Insurance: Annual and 1/mthly – Moments

Moments of Annual Insurances

1^st Moment

\(E[Z]=\sum\nolimits_{k=0}^{\infty }{{{b}_{k}}{{v}^{k+1}}_{k|}{{q}_{x}}}=\sum\nolimits_{k=0}^{\infty }{{{b}_{k}}{{v}^{k+1}}_{k}{{p}_{x}}{{q}_{x+k}}}\)

2^nd Moment

\(E[{{Z}^{2}}]=\sum\nolimits_{k=0}^{\infty }{{{b}_{k}}^{2}{{v}^{2(k+1)}}_{k|}{{q}_{x}}}=\sum\nolimits_{k=0}^{\infty }{{{b}_{k}}^{2}{{v}^{2(k+1)}}_{k}{{p}_{x}}{{q}_{x+k}}}\)

Standard Insurances and Notation

Whole Life Insurance

A whole life insurance of 1 on (x) pays 1 whenever death occurs.

Term Life Insurance

A term life insurance of 1 on (x) for n years pays 1 if death occurs within n years, 0 otherwise.

Deferred Whole Life Insurance

An n-year deferred life insurance on (x) pays 0 if death occurs within n years, 1 otherwise.

Pure endowment

An n-year pure endowment of 1 on (x) pays 1 at the end of n years if (x) survives to that point, 0 otherwise.

Endowment Insurance

An n-year endowment insurance of 1 on (x) pays 1 at the earlier of the death of (x) or n years.

Insurance	Z	EPV
Whole Life Insurance	\({{v}^{{{K}_{x}}+1}}\)	\({{A}_{x}}\)
n-Yr. Term Life Insurance	\(\left\{ \begin{align} & {{v}^{{{K}_{x}}+1}},{{K}_{x}}<n \\ & 0,{{K}_{x}}\ge n \\ \end{align} \right.\)	\(A_{x:\left. {\overline {\, n \,}}\! \right| }^{1}\)
n-Yr. Deferred Life Insurance	\(\left\{ \begin{align} & 0,{{K}_{x}}<n \\ & {{v}^{{{K}_{x}}+1}},{{K}_{x}}\ge n \\ \end{align} \right.\)	\(_{n|}{{A}_{x}}\)
n-Yr. Deferred Term Insurance	\(\left\{ \begin{align} & 0,{{K}_{x}}<n \\ & {{v}^{{{K}_{x}}+1}},n\le {{K}_{x}}\ge n+m \\ & 0,{{K}_{x}}\ge n+m \\ \end{align} \right.\)	\(_{n|}A_{x:\left. {\overline {\, n \,}}\! \right| }^{1}\)
Pure Endowment	\(\left\{ \begin{align} & 0,{{K}_{x}}<n \\ & {{v}^{n}},{{K}_{x}}\ge n \\ \end{align} \right.\)	\(_{n}{{E}_{x}}\)
Endowment Insurance	\(\left\{ \begin{align} & {{v}^{{{K}_{x}}+1}},{{K}_{x}}<n \\ & {{v}^{n}},{{K}_{x}}\ge n \\ \end{align} \right.\)	\({{A}_{x:\left. {\overline {\, n \,}}\! \right| }}\)

Variance of Z

\(Var(Z)={}^{2}{{A}_{x:\left. {\overline {n}}\! \right| }}-{{({{A}_{x:\left. {\overline {n}}\! \right| }})}^{2}}\)

Calculate Insurance Moments Using Illustrative Life Table

\(_{n|}{{A}_{x:\left. {\overline {n}}\! \right| }}={}_{n}{{E}_{x}}{{A}_{x+n}}\)
\(A_{x:\left. {\overline {n}}\! \right| }^{1}={{A}_{x}}-{}_{n|}{{A}_{x}}={{A}_{x}}-{}_{n}{{E}_{x}}{{A}_{x+n}}\)
\({{A}_{x:\left. {\overline {n}}\! \right| }}=A_{x:\left. {\overline {n}}\! \right| }^{1}+{}_{n}{{E}_{x}}={{A}_{x}}-{}_{n}{{E}_{x}}{{A}_{x+n}}+{}_{n}{{E}_{x}}\)

Moments for Insurance Under Mortality Assumptions

Constant Force of Mortality (CFM)

Assumption

\({{q}_{x+k}}=q\) => \({}_{k|}{{q}_{x}}={}_{k}{{p}_{x}}{{q}_{x+k}}={{(1-q)}^{k}}q\)

EPV

\({{A}_{x}}=\sum\nolimits_{k=0}^{\infty }{{}_{k|}{{q}_{x}}{{v}^{k+1}}}=\sum\nolimits_{k=0}^{\infty }{{{(1-q)}^{k}}q{{v}^{k+1}}}=\sum\nolimits_{k=0}^{\infty }{qv{{((1-q)v)}^{k}}}=\dfrac{qv}{1-(1-q)v}=\dfrac{q}{q+i}\)

Variance

\(E[{{Z}^{2}}]=\dfrac{q}{q+2i+{{i}^{2}}}\)

\(Var(Z)={}^{2}{{A}_{x}}-A_{x}^{2}=\dfrac{q}{q+2i+{{i}^{2}}}-{{(\dfrac{q}{q+i})}^{2}}\)

Uniform Mortality (UDD)

Assumption

\({}_{k|}{{q}_{x}}={}_{k}{{p}_{x}}{{q}_{x+k}}=c\)

EPV

\(E[Z]=\sum\nolimits_{k=m}^{n-1}{{{v}^{k+1}}c}=c{{v}^{m}}{{a}_{\left. {\overline {n-m}}\! \right| }}\)

where \(c=\dfrac{1}{w-x}\)

Summary (EPV for UDD and CFM)

EPV	UDD	CFM
\({{A}_{x}}\)	\(\dfrac{{{a}_{\left. {\overline {w-n}}\! \right| }}}{w-x}\)	\(\dfrac{q}{q+i}\)
\(A_{x:\left. {\overline {n}}\! \right| }^{1}\)	\(\dfrac{{{a}_{\left. {\overline {n}}\! \right| }}}{w-x}\)	\(\dfrac{q}{q+i}(1-{{(vp)}^{n}})\)
\(_{n|}{{A}_{x}}\)	\(\dfrac{{{v}^{n}}{{a}_{\left. {\overline {w-(x+n)}}\! \right| }}}{w-x}\)	\(\dfrac{q}{q+i}{{(vp)}^{n}}\)
\(_{n}{{E}_{x}}\)	\(\dfrac{{{v}^{n}}(w-(x+n))}{w-x}\)	\({{(vp)}^{n}}\)

Normal Approximation

Let \({{Z}^{P}}\) be the present value random variable for the entire portfolio of n insurances of c, then

\(E[{{Z}^{P}}]=cn\times E[Z]\)

\(Var[{{Z}^{P}}]={{c}^{2}}n\times Var[Z]\)

1/mly Insurance

\(A_{x}^{(m)}=\sum\nolimits_{k=0}^{\infty }{{{v}^{\dfrac{(k+1)}{m}}}{}_{\dfrac{k}{m}|\dfrac{1}{m}}{{q}_{x}}}\)

\(A_{x:\left. {\overline {n}}\! \right| }^{(m)}=\sum\nolimits_{k=0}^{nm}{{{v}^{\dfrac{(k+1)}{m}}}_{\dfrac{k}{m}|\dfrac{1}{m}}{{q}_{x}}}\)

Insurance: Continuous-Moments

Moments of Continuous Insurances

\(E[{{Z}^{n}}]=\int_{0}^{\infty }{{{e}^{-n\delta t}}{}_{t}{{p}_{x}}{{\mu }_{x+t}}dt}=\int_{0}^{\infty }{b_{t}^{n}v_{t}^{n}{}_{t}{{p}_{x}}{{\mu }_{x+t}}dt}\)

Definition and General Formulas

Insurance	Z	EPV
Whole Life Insurance	\({v}^{{T}_{x}}\)	\({{\bar{A}}_{x}}=\int_{0}^{\infty }{{{v}^{t}}{{f}_{x}}(t)dt}=\int_{0}^{\infty }{{{e}^{-\delta t}}{}_{t}{{p}_{x}}{{\mu }_{x+t}}dt}\)
n-Yr. Term Life Insurance	\(\left\{ \begin{align} & {{v}^{{T}_{x}}},{{K}_{x}}<n \\ & 0,{{K}_{x}}\ge n \\ \end{align} \right.\)	\({\bar{A}}_{x:\left. {\overline {n}}\! \right| }^{1}\)
n-Yr. Deferred Life Insurance	\(\left\{ \begin{align} & 0,{{K}_{x}}<n \\ & {{v}^{{T}_{x}}},{{K}_{x}}\ge n \\ \end{align} \right.\)	\(_{n|}{{\bar{A}}_{x}}\)
n-Yr. Deferred Term Insurance	\(\left\{ \begin{align} & 0,{{K}_{x}}<n \\ & {{v}^{{T}_{x}}},n\le {{K}_{x}}\ge n+m \\ & 0,{{K}_{x}}\ge n+m \\ \end{align} \right.\)	\(_{n|}{\bar{A}}_{x:\left. {\overline {n}}\! \right| }^{1}\)
Endowment Insurance	\(\left\{ \begin{align} & {{v}^{{T}_{x}}},{{K}_{x}}<n \\ & {{v}^{n}},{{K}_{x}}\ge n \\ \end{align} \right.\)	\({{\bar{A}}_{x:\left. {\overline {n}}\! \right| }}\)

Constant Force of Mortality

Assumption

\({}_{t}{{p}_{x}}={{e}^{-\mu t}}\)

EPV

EPV	Expression
\({{\bar{A}}_{x}}\)	\(=\dfrac{\mu }{\mu \text{+}\delta }\)
\(_{n}{{E}_{x}}\)	\(={{e}^{-(\mu +\delta )n}}\)
\(_{n|}{{\bar{A}}_{x}}\)	\(={}_{n}{{E}_{x}}{{\bar{A}}_{x+n}}={{e}^{-(\mu +\delta )n}}\dfrac{\mu }{\mu +\delta }\)
\(\bar{A}_{x:\left. {\overline { n}}\! \right| }^{1}\)	\(={{\bar{A}}_{x}}-{}_{n}{{E}_{x}}{{\bar{A}}_{x+n}}={{\bar{A}}_{x}}(1-{}_{n}{{E}_{x}})=\dfrac{\mu }{\mu +\delta }(1-{{e}^{-(\mu +\delta )n}})\)
\(_{n|}\bar{A}_{x:\left. {\overline { m}}\! \right| }^{1}\)	\(={{\bar{A}}_{x}}({}_{n}{{E}_{x}}-{}_{m+n}{{E}_{x}})=\dfrac{\mu {{e}^{-(\mu +\delta )n}}}{\mu +\delta }(1-{{e}^{-(\mu +\delta )m}})\)
\({{\bar{A}}_{x:\left. {\overline { n}}\! \right| }}\)	\(=\dfrac{\mu }{\mu +\delta }(1-{{e}^{-(\mu +\delta )n}})+{{e}^{-(\mu +\delta )n}}\)

Uniform Survival Function

Assumption

\(_{t}{{p}_{x}}=\dfrac{1}{w-x}\)

EPV

EPV	Expression
\({{\bar{A}}_{x}}\)	\(\dfrac{{{{\bar{a}}}_{\left. {\overline {\, w-x \,}}\! \right| }}}{w-x}\)
\(_{n}{{E}_{x}}\)	\(\dfrac{{{e}^{-\delta n}}(w-(x+n))}{w-x}\)
\(_{n|}{{\bar{A}}_{x}}\)	\(\dfrac{{{e}^{-\delta n}}{{{\bar{a}}}_{\left. {\overline {\, w-(x+n) \,}}\! \right| }}}{w-x}\)
\(\bar{A}_{x:\left. {\overline { n}}\! \right| }^{1}\)	\(\dfrac{{{{\bar{a}}}_{\left. \overline{n} \right|}}}{w-x}\)
\(_{n|}\bar{A}_{x:\left. {\overline { m}}\! \right| }^{1}\)
\({{\bar{A}}_{x:\left. {\overline { n}}\! \right| }}\)

Variance of Endowment Insurance

Let Z₁ be the random variable for the term insurance, Z₂ for the pure endowment, and Z₃ for the endowment insurance, then

Z₃ = Z₁ + Z₂

Var(Z₃) = Var(Z₁) + Var(Z₂) + 2Cov(Z₁, Z₂)

Cov(Z₁, Z₂) = E[Z₁Z₂] + E[Z₁]E[Z₂] = -E[Z₁]E[Z₂]

Normal Approximation

\(E[Z]=\sum\nolimits_{i=1}^{m}{{{n}_{i}}E[{{Z}_{i}}]}\)

\(Var[Z]=\sum\nolimits_{i=1}^{n}{{{n}_{i}}Var[{{Z}_{i}}]}\)

Insurance: Probabilities and Percentiles

Probabilities for Continuous Insurance Variables

Steps

1. Calculate the times of death T_x that correspond to the desired range of Z. This calculation involves the interest rate only; it does not involve the mortality function in any way. It involves finding T such that some function of v^T is in the desired range.

2. Calculate the probability of death in the range of T_x determined in the first step. This calculation involves the mortality function only; it does not involve the interest rate in any way.

Percentiles

Steps

1. First calculate the time t at which the value of the present value random variable is at the desired percentile. This calculation uses the mortality function only, not the interest rate.

2. Now calculate the value of the present value random variable if death occurs at the time determined in the first step. This calculation uses the interest rate only, not the mortality function. The present value is some function of v^t.

Insurance: Recursive Formulas, Varying Insurance

Recursive Formulas

EPV	Expression
\(A_{x}\)	\(=v{{q}_{x}}+v{{p}_{x}}{{A}_{x+1}}\), for x < w-1 \(=v(1)+v(0){{A}_{w}}=v\), for x = w-1
\({{A}_{x:\left. {n}\! \right| }} \)	\(=v{{q}_{x}}+v{{p}_{x}}{{A}_{x+1:\left. {n-1}\! \right| }}\), for x < n – 1 \(=v{{q}_{x+n-1}}+v{{p}_{x+n-1}}=v\), for x = n – 1
\(A_{x:n}^{1}\)	\(=v{{q}_{x}}+v{{p}_{x}}A_{x+1:\left. {n-1}\! \right| }^{1}\), for x < n – 1 \(=v{{q}_{x+n-1}}\), for x = n – 1
\({}_{n|}{{A}_{x}}\)	\(=v{{p}_{x}}\centerdot {}_{n-1|}{{A}_{x+1}}\), for x < n \(=A_{x+n}\), for x = n

Varying Insurance

Definition

\({{(\bar{I}\bar{A})}_{x}}=\dfrac{\mu }{{{(\mu +\delta )}^{2}}}\)

\(E[{{Z}^{2}}]=\dfrac{2\mu }{{{(\mu +2\delta )}^{3}}}\)

Relationships

\((\bar{I}\bar{A})_{x:\left. {n}\! \right| }^{1}+(\bar{D}\bar{A})_{x:\left. {n}\! \right| }^{1}=n\bar{A}_{x:\left. {n}\! \right| }^{1}\)

\((I\bar{A})_{x:\left. {\bar{n}} \right|}^{1}+(D\bar{A})_{x:\left. {\bar{n}} \right|}^{1}=(n+1)\bar{A}_{x:\left. {\bar{n}} \right|}^{1}\)

\((IA)_{x:\left. {\bar{n}} \right|}^{1}+(DA)_{x:\left. {\bar{n}} \right|}^{1}=(n+1)A_{x:\left. {\bar{n}} \right|}^{1}\)

\((IA)_{x:\left. {\bar{n}} \right| }^{1}=A_{x:\left. {\bar{n}} \right| }^{1}+v{{p}_{x}}(IA)_{x+1:\left. {\bar{n-1}} \right| }^{1}\)

Insurance: Relationships Between \(A_{x}\), \(A_{x}^{(m)}\), \({{\bar{A}}_{x}}\)

UDD

\({{\bar{A}}_{x}}=\dfrac{i}{\delta }{{A}_{x}}\)

\(\bar{A}_{x\left. {n}\! \right| }^{1}=\dfrac{i}{\delta }A_{x\left. {n}\! \right| }^{1}\)

\({}_{n|}{{\bar{A}}_{x}}=\dfrac{i}{\delta }{}_{n|}A\)

\({{\bar{A}}_{x:\left. {n}\! \right| }}=\frac{i}{\delta }A_{x:\left. {n}\! \right| }^{1}+A_{x:\left. {n}\! \right| }^{…1}\)

\(A_{x}^{(m)}=\dfrac{i}{{{i}^{(m)}}}{{A}_{x}}\)

\({}^{2}{{\bar{A}}_{x}}=\frac{2i+{{i}^{2}}}{2\delta }{}^{2}{{A}_{x}}\)

Claims Acceleration Approach

\({{\bar{A}}_{x}}={{(1+i)}^{0.5}}{{A}_{x}}\)

\(\bar{A}_{x\left. {n}\! \right| }^{1}={{(1+i)}^{0.5}}A_{x\left. {n}\! \right| }^{1}\)

\({}_{n|}{{\bar{A}}_{x}}={{(1+i)}^{0.5}}{}_{n|}A\)

\({{\bar{A}}_{x:\left. {n}\! \right| }}={{(1+i)}^{0.5}}A_{x:\left. {n}\! \right| }^{1}+A_{x:\left. {n}\! \right| }^{…1}\)

\(A_{x}^{(m)}={{(1+i)}^{{}^{(m-1)}/{}_{2m}}}{{A}_{x}}\)

\({}^{2}{{\bar{A}}_{x}}=(1+i){}^{2}{{A}_{x}}\)

Annuities: Discrete, Expectation

Annuities-Due

Annuity-Due	Annual Payment at time k	PV	EPV
\({\ddot{a}}_{x}\)	\(1,0\le k\le {{K}_{x}}\)	\({{\ddot{a}}_{\overline{{{K}_{x}}+1}}}\)	\(\sum\nolimits_{k=0}^{\infty}{{{v}^{k}}{}_{k}{{p}_{x}}}\)
\({{\ddot{a}}_{x:\overline{n}}}\)	\(1,0\le k\le \min ({{K}_{x}},n-1) \) \(0,k>\min ({{K}_{x}},n-1) \)	\({{\ddot{a}}_{\overline{{{K}_{x}}+1}}},{{K}_{x}}<n\) \({{\ddot{a}}_{\overline{n}}},{{K}_{x}}\ge n\)	\(\sum\nolimits_{k=0}^{n-1}{{{v}^{k}}{}_{k}{{p}_{x}}}\)
\({}_{n|}{{\ddot{a}}_{x}}\)	\(0,0\le k<n,or,k>{{K}_{x}}\) \(1,n\le k\le {{K}_{x}}\)	\(0,{{K}_{x}}<n\) \({{\ddot{a}}_{\overline{{{K}_{x}}+1}}}-{{\ddot{a}}_{\overline{n}}},{{K}_{x}}\ge n\)	\(\sum\nolimits_{k=n}^{\infty }{{{v}^{k}}_{k}{{p}_{x}}}\)
\({}_{n|}{{\ddot{a}}_{x:\overline{m}}}\)	\(0,0\le k<n\) \(1,n\le k<\min (n+m,{{K}_{x}}+1) \) \(0,k\ge \min (n+m,{{K}_{x}}+1) \)	\(0,{{K}_{x}}<n\) \({{\ddot{a}}_{\overline{{{K}_{x}}+1}}}-{{\ddot{a}}_{\overline{n}}},n\le {{K}_{x}}<n+m\) \({{\ddot{a}}_{\overline{n+m}}}-{{\ddot{a}}_{\overline{n}}},{{K}_{x}}\ge n+m\)	\(\sum\nolimits_{k=n}^{m-1 }{{{v}^{k}}_{k}{{p}_{x}}}\)
\({{\bar{a}}_{\overline{x:\overline{m}}}}\)	\(1,0\le k\le \max ({{K}_{x}}+1,n) \) \(0,k\ge \max ({{K}_{x}}+1,n) \)	\({{\ddot{a}}_{\overline{n}}},{{K}_{x}}<n\) \({{\ddot{a}}_{\overline{{{K}_{x}}+1}}},{{K}_{x}}\ge n\)

Relationships

\({{\ddot{a}}_{\overline{x:\overline{n}}}}={{\ddot{a}}_{\overline{n}}}+{}_{n|}{{\ddot{a}}_{x}}\)

\(_{n|}{{\ddot{a}}_{x}}={}_{n}{{E}_{x}}{{\ddot{a}}_{x+n}}\)

\({{\ddot{a}}_{x}}={{\ddot{a}}_{x:\overline{n}}}+{}_{n|}{{\ddot{a}}_{x}}\)

\({{\ddot{a}}_{x:\overline{n}}}={{\ddot{a}}_{x}}-{}_{n}{{E}_{x}}{{\ddot{a}}_{x+n}}\)

EPV

\({{\ddot{a}}_{x:\overline{n}}}=\sum\nolimits_{k=1}^{n}{{{{\ddot{a}}}_{\overline{k}}}{}_{k-1}{{p}_{x}}{{q}_{x+k-1}}}+{{\ddot{a}}_{\overline{n}}}{}_{n}{{p}_{x}}\)

OR

\({{\ddot{a}}_{x:\overline{n}}}=\sum\nolimits_{k=1}^{n-1}{{{{\ddot{a}}}_{\overline{k}}}{}_{k-1}{{p}_{x}}{{q}_{x+k-1}}}+{{\ddot{a}}_{\overline{n}}}{}_{n-1}{{p}_{x}}\)

Annuities-Immediate

\(A_{x:\overline{n}}^{1}=v{{\ddot{a}}_{x:\overline{n}}}-{{a}_{x:\overline{n}}}\)

Relationships Between Annuities-Due and Annuities-Immediate

\({{\ddot{a}}_{x}}={{a}_{x}}+1\)

\({{\ddot{a}}_{x:\overline{n}}}={{a}_{x:\overline{n-1}}}+1={{a}_{x:\overline{n}}}+1-{}_{n}{{E}_{x}}\)

\({}_{n|}{{\ddot{a}}_{x}}={}_{n|}{{a}_{x}}+{}_{n}{{E}_{x}}\)

1/mthly Annuities

\(\ddot{a}_{x}^{(m)}=\sum\nolimits_{k=1}^{\infty }{\dfrac{1}{m}{{v}^{k/m}}{}_{k/m}{{p}_{x}}}\)

\(\ddot{a}_{x}^{(m)}=\dfrac{1-A_{x}^{(m)}}{{{d}^{(m)}}}\)

Annuities: Continuous, Expectation

Annuity	Payment Per Annum at Time t	PV	EPV
\({{\bar{a}}_{x}}\)	\(1,t\le T\)	\({{\bar{a}}_{\overline{T}}}\)	\({{\bar{a}}_{x}}=\dfrac{1-{{{\bar{A}}}_{x}}}{\delta }\)
\({{\bar{a}}_{x:\overline{n}}}\)	\(1,t\le \min (T,n)\) \(0,t>\min (T,n)\)	\({{\bar{a}}_{\overline{T}}},T\le n\) \({{\bar{a}}_{\overline{n}}},T>n\)	\({{\bar{a}}_{x:\overline{n}}}=\dfrac{1-{{A}_{x:\overline{n}}}}{\delta }\)
\({}_{n|}{{\bar{a}}_{x}}\)	\(0,t\le n,or,t>T\) \(1,n<t\le T\)	\(0,T\le n\) \({{\bar{a}}_{\overline{T}}}-{{\bar{a}}_{\overline{n}}},T>n\)	\({}_{n|}{{\bar{a}}_{x:\overline{m}}}={{\bar{a}}_{x:\overline{n+m}}}-{{\bar{a}}_{x:\overline{m}}}\) \(=\dfrac{1-{{A}_{x}}}{\delta }-\dfrac{1-{{A}_{x:\bar{n}}}}{\delta }=\dfrac{{{A}_{x:\bar{n}}}-{{A}_{x}}}{\delta }\)
\({}_{n|}{{\bar{a}}_{x:\overline{n}}}\)	\(0,t\le n,or,t>T\) \(1,n<t\le n+m,and,t\le T\) \(0,T>n+m\)	\(0,T\le n\) \({{\bar{a}}_{\overline{T}}}-{{\bar{a}}_{\overline{n}}},n<T\le n+m\) \({{\bar{a}}_{\overline{n+m}}}-{{\bar{a}}_{\overline{n}}},T>n+m\)
\({{\bar{a}}_{\overline{x:\overline{n}}}}\)	\(1,t\le \max (T,n)\) \(0,t>\max (T,n)\)	\({{\bar{a}}_{\overline{n}}},T\le n\) \({{\bar{a}}_{\overline{T}}},T>n\)

EPV

\({{\bar{a}}_{x}}=\int_{0}^{\infty }{{{{\bar{a}}}_{\overline{t}}}{}_{t}{{p}_{x}}{{\mu }_{x+t}}dt}=\int_{0}^{\infty }{{{v}^{t}}{}_{t}{{p}_{x}}dt}\), since \({{\bar{a}}_{\overline{T}}}=\dfrac{1-{{v}^{T}}}{\delta }\)

\({{\bar{a}}_{x:\overline{n}}}=\int_{0}^{n}{{{v}^{t}}{}_{t}{{p}_{x}}dt}\)

\({}_{n|}{{\bar{a}}_{x}}=\int_{n}^{\infty }{{{v}^{t}}{}_{t}{{p}_{x}}dt}\)

\({}_{n|}{{\bar{a}}_{x:\overline{m}}}=\int_{n}^{m}{{{v}^{t}}{}_{t}{{p}_{x}}dt}\)

CFM

\({{\bar{a}}_{x}}=\dfrac{1}{\mu +\delta }\)

\(_{n|}{{\bar{a}}_{x}}={}_{n}{{E}_{x}}{{\bar{a}}_{x}}=\dfrac{{{e}^{-(\mu +\delta )n}}}{\mu +\delta }\)

\({{\bar{a}}_{x:\overline{n}}}={{\bar{a}}_{x}}(1-{}_{n}{{E}_{x}})=\dfrac{1-{{e}^{-(\mu +\delta )n}}}{\mu +\delta }\)

Annuities: Variance

	\(Var[\gamma ]\) with respect to Insurance	\(Var[\gamma ]\) with respect to annuity
\({{\bar{a}}_{x}}\)	\(=\dfrac{{}^{2}{{{\bar{A}}}_{x}}-{{{\bar{A}}}_{x}}^{2}}{{{\delta }^{2}}}\)	\(=\dfrac{2({{{\bar{a}}}_{x}}-{}^{2}{{{\bar{a}}}_{x}})}{\delta }-{{({{\bar{a}}_{x}})}^{2}}\), since \({}^{2}{{\bar{A}}_{x}}=1-(2\delta ){}^{2}{{\bar{a}}_{x}}\)
\({{\bar{a}}_{x:\overline{n|}}}\)	\(=\dfrac{{}^{2}{{{\bar{A}}}_{x:\overline{n|}}}-{{({{{\bar{A}}}_{x:\overline{n|}}})}^{2}}}{{{\delta }^{2}}}\)	\(=\dfrac{2({{{\bar{a}}}_{x:\overline{n|}}}-{}^{2}{{{\bar{a}}}_{x:\overline{n|}}})}{\delta }-{{({{\bar{a}}_{x:\overline{n|}}})}^{2}}\), since \({}^{2}{{\bar{A}}_{x:\overline{n|}}}=1-(2\delta ){}^{2}{{\bar{a}}_{x:\overline{n}|}}\)
\({{\ddot{a}}_{x}}\)	\(=\dfrac{{}^{2}{{A}_{x}}-{{({{A}_{x}})}^{2}}}{{{d}^{2}}}\)	\(=\dfrac{2({{{\ddot{a}}}_{x}}-{}^{2}{{{\ddot{a}}}_{x}})}{d}-{}^{2}{{\ddot{a}}_{x}}-{{({{\ddot{a}}_{x}})}^{2}}\), since \({}^{2}{{A}_{x}}=1-{}^{2}d{}^{2}{{\bar{a}}_{x}}=1-(2d-{{d}^{2}}){}^{2}{{\bar{a}}_{x}}\)
\({{\ddot{a}}_{x:\overline{n|}}}\)	\(=\dfrac{{}^{2}{{A}_{x:\overline{n|}}}-{{({{A}_{x:\overline{n|}}})}^{2}}}{{{d}^{2}}}\)	\(=\dfrac{2({{{\ddot{a}}}_{x:\overline{n|}}}{{-}^{2}}{{{\ddot{a}}}_{x:\overline{n|}}})}{d}{{-}^{2}}{{\ddot{a}}_{x:\overline{n|}}}-{{({{\ddot{a}}_{x:\overline{n|}}})}^{2}}\), since \(^{2}{{A}_{x:\overline{n|}}}=1{{-}^{2}}{{d}^{2}}{{\bar{a}}_{x:\overline{n|}}}=1-{{(2d-{{d}^{2}})}^{2}}{{\bar{a}}_{x:\overline{n|}}}\)

Definition of the Second Moment of a Continuous Whole Life Annuity

\(=\int_{0}^{\infty }{\bar{a}_{\overline{t|}}^{2}{}_{t}{{p}_{x}}{{\mu }_{x+t}}dt}\)
\(=\int_{0}^{\infty }{{{(\dfrac{1-{{v}^{t}}}{\delta })}^{2}}{}_{t}{{p}_{x}}{{\mu }_{x+t}}dt}\)

The variance of an n-year temporary annuity-immediate

= The variance of an n + 1 year temporary life annuity-due

Since they differ only by a time-0 certain payment of 1.

Discrete Annuities from First Principles

Variance of a Discrete Whole Life Annuity

\(E[\ddot{\gamma }_{x}^{2}]=\sum\nolimits_{k=1}^{\infty }{\ddot{a}_{\overline{k|}}^{2}{}_{k-1|}{{q}_{x}}}\)

\(E[\ddot{\gamma }_{x:\overline{n|}}^{2}]=\sum\nolimits_{k=1}^{n}{\ddot{a}_{\overline{k|}}^{2}{}_{k-1|}{{q}_{x}}}+{}_{n}{{p}_{x}}\ddot{a}_{\overline{n|}}^{2}=\sum\nolimits_{k=1}^{n-1}{\ddot{a}_{\overline{k|}}^{2}{}_{k-1|}{{q}_{x}}}+{}_{n-1}{{p}_{x}}\ddot{a}_{\overline{n|}}^{2}\)

Normal Approximation

Combinations of Annuities and Insurances with No Variance

Continuous

If \(\gamma\) is a whole life continuous annuity,

\(Z\) is a whole life insurance payable at the moment of death, then

\(Z={{v}^{T}}\) and \(\gamma =\dfrac{1-{{v}^{T}}}{\delta }\), so \(\delta \gamma +{{v}^{T}}=1\). This implies that \(Var(\delta \gamma +{{v}^{T}})=0\).

The same conclusion applies if \(\gamma\) is an n-year temporary continuous annuity and \(Z\) an n-year endowment insurance payable at the moment of death.

Discrete

If \(\gamma\) is a whole life annuity-due,

\(Z\) a whole life insurance payable at the end of the year of death, then

\(Z={{v}^{K+1}}\) and \(\gamma =\dfrac{1-{{v}^{K+1}}}{d}\), so \(d\gamma +Z=1\). This implies that \(Var(d\gamma +Z)=0\).

The same conclusion applies if \(\gamma\) is an n-year temporary life annuity-due and \(Z\) an n-year endowment insurance payable at the end of the year of death. ,

For annuities-immediate, the corresponding statement is that \(Var(i\gamma +(1+i)Z)=0\).

Annuities: Probabilities and Percentiles

Probabilities for Continuous Annuities

Unlike insurances, annuity random variables always increase as a function of survival time. They are always monotonically increasing. The longer one lives, the more annuity payments one receives.

Distribution Functions of Annuity Present Values

Probabilities for Discrete Annuities

Percentiles

Since annuities monotonically increase in value as survival time increases, calculating an annuity percentile reduces to calculating the percentile of T_x and then calculating \({{\ddot{a}}_{\overline{{{T}_{x}}|}}}\) for that value.

Probabilities and Percentiles

1. To calculate a probability for an annuity, calculate the \(t\) for which \({{\bar{a}}_{\overline{t|}}}\) has the desired property. Then calculate the probability \(t\) is in that range.

2. To calculate a percentile of an annuity, calculate the percentile of \({{T}_{x}}\), then calculate \({{\bar{a}}_{\overline{{{T}_{x}}|}}}\).

3. Some adjustments may be needed for discrete annuities or non-whole-life annuities, as discussed in the lesson.

4. 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 expected present value is \(\Pr ({{\bar{a}}_{\overline{{{T}_{x}}|}}}>{{\bar{a}}_{x}})={{(\frac{\mu }{\mu +\delta })}^{\mu /\delta }}\)

Annuities: Varying Annuities, Recursive Formulas

Increasing and Decreasing Annuities

Geometrically increasing annuities

Let g be the rate of geometrically increasing, then

\({i}’=\dfrac{1+i}{1+g}\), and use it to calculate the life annuity \(a\)

Arithmetically increasing annuities

\({{(\bar{I}\bar{a})}_{x}}\)	The EPV of a continuous life annuity with a continuously increasing rate of payment, paying t per year at time t. For a temporary increasing life annuity, \({{(\bar{I}\bar{a})}_{x:\overline{n|}}}\) is used.
\({{(I\bar{a})}_{x}}\)	The EPV of a continuous life annuity which pays at a rate of t per year in year t.
\({{(Ia)}_{x}}\)	The EPV of a life annuity that pays k at the end of year k.
\({{(I\ddot{a})}_{x}}\)	The EPV of a life annuity that pays k at the beginning of year k.
\({{(\bar{D}\bar{a})}_{x:\overline{n|}}}\)	The EPV of a continuously decreasing continuous life annuity that pays at a rate of n – t at time t until time n.
\({{(D\bar{a})}_{x:\overline{n|}}}\)	The EPV of a continuous life annuity that pays at a rate of n – t + 1 in year t until time n.
\({{(Da)}_{x:\overline{n|}}}\)	The EPV of a life annuity that pays n – k + 1 at the end of year k until time n.
\({{(D\ddot{a})}_{x:\overline{n|}}}\)	The EPV of a life annuity that pays n – k + 1·at the beginning of year k until time n.

\({{(\bar{I}\bar{a})}_{x}}\) VS \({{(I\bar{a})}_{x}}\)

 \(\int_{k-1}^{k}{tdt}=\dfrac{{{t}^{2}}}{2}|_{k-1}^{k}=\dfrac{2k-1}{2}<k\) , so \({{(\bar{I}\bar{a})}_{x}}<{{(I\bar{a})}_{x}}\)

Similarly, \({{(\bar{D}\bar{a})}_{x:\overline{n|}}}<{{(D\bar{a})}_{x:\overline{n|}}}\)

Recursive Formulas

\({{\ddot{a}}_{x}}\)	\(=v{{p}_{x}}{{\ddot{a}}_{x+1}}+1\)
\(\ddot{a}_{x}^{(m)}\)	\(={{v}^{{}^{1}/{}_{m}}}{}_{{}^{1}/{}_{m}}{{p}_{x}}\ddot{a}_{x+{}^{1}/{}_{m}}^{(m)}+\dfrac{1}{m}\)
\({{a}_{x}}\)	\(=v{{p}_{x}}{{a}_{x+1}}+v{{p}_{x}}\)
\({{\bar{a}}_{x}}\)	\(=v{{p}_{x}}{{\bar{a}}_{x+1}}+{{\bar{a}}_{x:\overline{1|}}}\)
\({{\ddot{a}}_{x:\overline{n|}}}\)	\(=v{{p}_{x}}{{\ddot{a}}_{x+1:\overline{n-1|}}}+1\)
\({{a}_{x:\overline{n|}}}\)	\(=v{{p}_{x}}{{a}_{x+1:\overline{n-1|}}}+v{{p}_{x}}\)
\({{\bar{a}}_{x:\overline{n|}}}\)	\(=v{{p}_{x}}{{\bar{a}}_{x+1:\overline{n-1|}}}+{{\bar{a}}_{x:\overline{1|}}}\)
\({}_{n|}{{\ddot{a}}_{x}}\)	\(=v{{p}_{x}}{}_{n-1|}{{\ddot{a}}_{x+1}}\)
\({}_{n|}{{a}_{x}}\)	\(=v{{p}_{x}}{}_{n-1|}{{a}_{x+1}}\)
\({}_{n|}{{\bar{a}}_{x}}\)	\(=v{{p}_{x}}{}_{n-1|}{{\bar{a}}_{x+1}}\)
\({{\ddot{a}}_{\overline{x:\overline{n|}}}}\)	\(=1+v{{q}_{x}}{{\ddot{a}}_{\overline{n-1|}}}+v{{p}_{x}}{{\ddot{a}}_{\overline{x+1:\overline{n-1|}}}}\)
\({{a}_{\overline{x:\overline{n|}}}}\)	\(=v+v{{q}_{x}}{{a}_{\overline{n-1|}}}+v{{p}_{x}}{{a}_{\overline{x+1:\overline{n-1|}}}}\)
\({{\bar{a}}_{\overline{x:\overline{n|}}}}\)	\(={{\bar{a}}_{\overline{1|}}}+v{{q}_{x}}{{\bar{a}}_{\overline{n-1|}}}+v{{p}_{x}}{{\bar{a}}_{\overline{x+1:\overline{n-1|}}}}\)

Annuities: 1/m-thly Payments

For Low interest rates,

\(\ddot{a}_{x}^{(m)}\approx {{\ddot{a}}_{x}}-\dfrac{m-1}{2m}\)

\(a_{x}^{(m)}\approx {{a}_{x}}+\dfrac{m-1}{2m}\)

Uniform Distribution of Deaths Assumption

\(\alpha (m)=\dfrac{id}{{{i}^{(m)}}{{d}^{(m)}}}\), \(\beta (m)=\dfrac{i-{{i}^{(m)}}}{{{i}^{(m)}}{{d}^{(m)}}}\)

\(\ddot{a}_{x}^{(m)}\approx {{\ddot{a}}_{x}}-\dfrac{m-1}{2m}\)	\(=\alpha (m){{\ddot{a}}_{x}}-\beta (m)\) ♥
\(\ddot{a}_{x:\overline{n|}}^{(m)}\)	\(=\alpha (m){{\ddot{a}}_{x:\overline{n|}}}-\beta (m)(1-{}_{n}{{E}_{x}})\)
\({}_{n|}\ddot{a}_{x}^{(m)}\)	\(=\alpha (m){}_{n|}{{\ddot{a}}_{x}}-\beta (m){}_{n}{{E}_{x}}\)
\(a_{x}^{(m)}\)	\(={{\ddot{a}}_{x}}-\dfrac{1}{m}=\alpha (m){\ddot{a}_{x}}-\beta (m)-\dfrac{1}{m}\)

Woolhouse’s Formula

Using the simple approximation of \(\ddot{a}_{x}^{(m)}={{\ddot{a}}_{x}}-\dfrac{m-1}{2m}\), ♥

By adding correction terms: \(\dfrac{{{m}^{2}}-1}{12{{m}^{2}}}({{\mu }_{x}}+\delta )\), where: ♥

\({{\mu }_{x}}\approx -\dfrac{1}{2}(\ln {{p}_{x-1}}+\ln {{p}_{x}})\), since \({{\mu }_{x}}=-\dfrac{\partial }{\partial x}\ln {{p}_{x}}\) if an exact value of \({{\mu }_{x}}\) is not available.

\(\ddot{a}_{x}^{(m)}\)	\(\approx {{\ddot{a}}_{x}}-\dfrac{m-1}{2m}-\dfrac{{{m}^{2}}-1}{12{{m}^{2}}}({{\mu }_{x}}+\delta )\)
\({{\bar{a}}_{x}}\)	\(\approx {{\ddot{a}}_{x}}-\dfrac{1}{2}-\dfrac{1}{12}({{\mu }_{x}}+\delta )\)
\(\ddot{a}_{x:\overline{n|}}^{(m)}\)	\(\approx {{\ddot{a}}_{x:\overline{n|}}}-\dfrac{m-1}{2m}(1-{}_{n}{{E}_{x}})-\dfrac{{{m}^{2}}-1}{12{{m}^{2}}}({{\mu }_{x}}+\delta -{}_{n}{{E}_{x}}({{\mu }_{x+n}}+\delta ))\)
\({}_{n|}\ddot{a}_{x}^{(m)}\)	\(\approx {}_{n|}{{\ddot{a}}_{x}}-\dfrac{m-1}{2m}{}_{n}{{E}_{x}}-\dfrac{{{m}^{2}}-1}{12{{m}^{2}}}{}_{n}{{E}_{x}}({{\mu }_{x+n}}+\delta )\)
\({{\overset{\scriptscriptstyle\smile}{e}}_{x}}\)	\(\approx {{e}_{x}}+\dfrac{1}{2}-\dfrac{1}{12}{{\mu }_{x}}\)

Premiums: Net Premiums for Discrete Insurances

Future Loss

Future loss is the present value of the benefits, and possibly expenses, paid under the contract minus the present value of the premiums collected.

\(Pa=A\)

Premium Formulas

Discrete Whole Life

\({{P}_{x}}=\dfrac{{{A}_{x}}}{{{{\ddot{a}}}_{x}}}=\dfrac{1-d{{{\ddot{a}}}_{x}}}{{{{\ddot{a}}}_{x}}}=\dfrac{1}{{{{\ddot{a}}}_{x}}}-d\)

\({{P}_{x}}=\dfrac{{{A}_{x}}}{{{{\ddot{a}}}_{x}}}=\dfrac{{{A}_{x}}}{(1-{{A}_{x}})/d}=\dfrac{d{{A}_{x}}}{1-{{A}_{x}}}\)

Discrete Term Life

\({{P}_{x}}=\dfrac{{{A}_{x:\overline{n|}}}}{{{{\ddot{a}}}_{x:\overline{n|}}}}=\dfrac{1-d{{{\ddot{a}}}_{x:\overline{n|}}}}{{{{\ddot{a}}}_{x:\overline{n|}}}}=\dfrac{1}{{{{\ddot{a}}}_{x:\overline{n|}}}}-d\)

\({{P}_{x:\overline{n|}}}=\dfrac{{{A}_{x:\overline{n|}}}}{{{{\ddot{a}}}_{x:\overline{n|}}}}=\dfrac{{{A}_{x:\overline{n|}}}}{(1-{{A}_{x:\overline{n|}}})/d}=\dfrac{d{{A}_{x:\overline{n|}}}}{1-{{A}_{x:\overline{n|}}}}\)

Expected Value of Future Loss

 \({}_{0}L=b{{v}^{{{K}_{x}}+1}}-\pi {{\ddot{a}}_{\overline{{{K}_{x}}+1|}}}=b{{v}^{{{K}_{x}}+1}}-\pi (\dfrac{1-{{v}^{{{K}_{x}}+1}}}{d})={{v}^{{{K}_{x}}+1}}(b+\dfrac{\pi }{d})-\dfrac{\pi }{d}\)

\(E{{[}_{0}}L]=b{{A}_{x}}-\pi {{\ddot{a}}_{x}}={{A}_{x}}(b+\dfrac{\pi }{d})-\dfrac{\pi }{d}\)

International Actuarial Premium Notation

\({{P}_{x}}\) is the premium for a fully discrete whole life insurance, or \({{P}_{x}}=\dfrac{{{A}_{x}}}{{{{\ddot{a}}}_{x}}}\)

\(P_{x:\overline{n|}}^{1}\) is the premium for a fully discrete n-year term insurance, or \(P_{x:\overline{n|}}^{1}=\dfrac{A_{x:\overline{n|}}^{1}}{{{{\ddot{a}}}_{x:\overline{n|}}}}\)

\(P_{x:\overline{n|}}^{..1}\) is the annual premium for an n-year pure endowment, or \(P_{x:\overline{n|}}^{..1}=\dfrac{A_{x:\overline{n|}}^{…1}}{{{{\ddot{a}}}_{x:\overline{n|}}}}\)

\({{P}_{x:\overline{n|}}}\) is the premium for a fully discrete n-year endowment insurance, or \({{P}_{x:\overline{n|}}}=\dfrac{{{A}_{x:\overline{n|}}}}{{{{\ddot{a}}}_{x:\overline{n|}}}}\)

Premiums: Net Premiums for Fully Continuous Insurances

Whole Life Insurance

\(P=\dfrac{1-\delta {{{\bar{a}}}_{x}}}{{{{\bar{a}}}_{x}}}=\dfrac{1}{{{{\bar{a}}}_{x}}}-\delta \)

\(P=\dfrac{{{{\bar{A}}}_{x}}}{(1-{{{\bar{A}}}_{x}})/\delta }=\dfrac{\delta {{{\bar{A}}}_{x}}}{1-{{{\bar{A}}}_{x}}}\)

Term Life Insurance

\(P=\dfrac{1-\delta {{{\bar{a}}}_{x:\overline{n|}}}}{{{{\bar{a}}}_{x:\overline{n|}}}}=\dfrac{1}{{{{\bar{a}}}_{x:\overline{n|}}}}-\delta \)

\(P=\dfrac{{{{\bar{A}}}_{x:\overline{n|}}}}{(1-{{{\bar{A}}}_{x:\overline{n|}}})/\delta }=\dfrac{\delta {{{\bar{A}}}_{x:\overline{n|}}}}{1-{{{\bar{A}}}_{x:\overline{n|}}}}\)

Future Loss

\({}_{0}L={{v}^{{{T}_{x}}}}(b+\dfrac{\pi }{\delta })-\dfrac{\pi }{\delta }\)

\(E[{}_{0}L]={{\bar{A}}_{x}}(b+\dfrac{\pi }{\delta })-\dfrac{\pi }{\delta }\)

Premiums: Variance of Future Loss, Discrete

Variance of Net Future Loss by Formula

Whole Life Insurance

\({}_{0}L={{v}^{{{K}_{x}}+1}}(b+\dfrac{\pi }{d})-\dfrac{\pi }{d}\)

\(Var({}_{0}L)=({}^{2}{{A}_{x}}-{{A}_{x}}^{2}){{(b+\dfrac{\pi }{d})}^{2}}\)

\(Var({}_{0}L)={{b}^{2}}(\dfrac{{}^{2}{{A}_{x}}-{{A}_{x}}^{2}}{{{(1-{{A}_{x}})}^{2}}})\), if \(\pi\) is net premium

Term Life Insurance

\(Var({}_{0}L)=({}^{2}{{A}_{x:\overline{n|}}}-{{A}_{x:\overline{n|}}}^{2}){{(b+\dfrac{\pi }{d})}^{2}}\)

\(Var({}_{0}L)={{b}^{2}}(\dfrac{{}^{2}{{A}_{x:\overline{n|}}}-{{A}_{x:\overline{n|}}}^{2}}{{{(1-{{A}_{x:\overline{n|}}})}^{2}}})\), if \(\pi\) is net premium

Variance of Net Future Loss from First Principles

\(Var({}_{t}L|{{K}_{x}}\ge k)=E[{}_{t}{{L}^{2}}|{{K}_{x}}\ge k]-E{{[{}_{t}L|{{K}_{x}}\ge k]}^{2}}=\sum\nolimits_{0}^{T}{{{l}_{t}}^{2}{}_{t|}{{q}_{x+k}}}-({{\sum\nolimits_{0}^{T}{{{l}_{t}}{}_{t|}{{q}_{x+k}})}}^{2}}\)

\(t\)	\({}_{t|}{{q}_{x+k}}\)	\(PV{{B}_{{{K}_{x}}+1}}\)	\(PV{{P}_{t}}\)	\({l}_{t}=PV{{B}_{{{K}_{x}}+1}}-PV{{P}_{t}}\)	\({l}_{t}{}_{t|}{{q}_{x+k}}\)	\({l}_{t}^{2}{}_{t|}{{q}_{x+k}}\)
0
1
…
> T

Variance of Gross Future Loss

\({}_{0}L=(b+E){{v}^{{{K}_{x}}+1}}+({{e}_{f}}-e)-(G-e){{\ddot{a}}_{\overline{{{K}_{x}}+1|}}}\)

\(Var({}_{0}L)=({}^{2}{{A}_{x}}-{{A}_{x}}^{2}){{(b+E+\dfrac{G-e}{d})}^{2}}\)

Premiums: Variance of Future Loss, Continuous

Variance of Net Future Loss

Whole Life Insurance

\(Var{{(}_{0}}L)={{(}^{2}}{{\bar{A}}_{x}}-{{\bar{A}}_{x}}^{2}){{(b+\dfrac{\pi }{\delta})}^{2}}\)

\(Var{{(}_{0}}L)={{b}^{2}}(\dfrac{^{2}{{{\bar{A}}}_{x}}-{{{\bar{A}}}_{x}}^{2}}{{{(1-{{{\bar{A}}}_{x}})}^{2}}})\), if \(\pi \) is net premium

Term Life Insurance

\(Var{{(}_{0}}L)={{(}^{2}}{{\bar{A}}_{x:\overline{n|}}}-{{\bar{A}}_{x:\overline{n|}}}^{2}){{(b+\dfrac{\pi }{\delta })}^{2}}\)

\(Var{{(}_{0}}L)={{b}^{2}}(\dfrac{^{2}{{{\bar{A}}}_{x:\overline{n|}}}-{{{\bar{A}}}_{x:\overline{n|}}}^{2}}{{{(1-{{{\bar{A}}}_{x:\overline{n|}}})}^{2}}})\), if \(\pi\) is net premium

Variance of Gross Future Loss

\(Var{{(}_{0}}L)={{(}^{2}}{{\bar{A}}_{x}}-{{\bar{A}}_{x}}^{2}){{(b+E+\dfrac{G-e}{\delta })}^{2}}\)

Premiums: Probabilities and Percentiles of Future Loss

Probabilities

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 probability 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 than 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 than the deferral period minus the probability that survival time is less than the deferral period.

For a deferred annuity with a single premium, the loss at issue is a negative constant during the deferral period, then increases. If regular premiums are payable during the deferral period, the loss at issue decreases until the end of the deferral period and increases thereafter.

Fully Continuous Insurances

Annuities

Percentiles

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 probability 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 than 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 than the deferral period minus the probability that survival time is less than the deferral period.

For a deferred annuity with a single premium, the loss at issue is a negative constant during the deferral period, then increases. If regular premiums are payable during the deferral period, the loss at issue decreases until the end of the deferral period and increases thereafter.

Premiums: Special Topics

The Portfolio Percentile Premium Principle

1. For the case n = 1, namely one policy, we know how to calculate exact percentiles of future loss. However, the portfolio percentile principle may lead to silly results when n = 1. For example, the 95th percentile of the present value of benefits on a short-term term policy will be 0 if the probability of death is less than 5% in the term period. Then the portfolio percentile principle at the 95th percentile leads to a premium of 0.

2. For n > 1 policies, we cannot calculate the exact percentiles. Instead, we use a normal approximation. We know how to calculate the mean and variance of future loss.

3. The premium per policy based on the normal approximation is of the form E[_oL] + Z_p(Var(_oL)/n)^1/2 since the variance of average future loss for a policy in a portfolio is the variance of the future loss for an individual policy divided by n. The premium depends on n, and is always higher than the equivalence principle premium, but converges to the equivalence principle premium as n ~ ∞.

4. Both E[_oL] and Var(_oL) are functions of the premium.

Reserves: Prospective Net Premium Reserve

The net premium reserve, is E[_tL | T_x > t], where the net premium is calculated based on the equivalence principle and using the same mortality and interest assumptions as are used in calculating the expected value of future loss. 

It is denoted by _tV, or sometimes _tVⁿ if it is necessary to distinguish this reserve from a gross premium reserve.

\({{(}_{k}}V+{{\pi }_{k}})(1+i)={{b}_{k+1}}{{q}_{x+k}}+{{p}_{x+k}}_{k+1}V\)

Reserves: Gross Premium Reserve and Expense Reserve

Gross Premium Reserve

The gross future loss _tL is the present value at time t of future benefits and expenses minus future gross premiums assuming the policy is in force at that time. Unlike the net future loss, expenses are taken into account. The gross premiums are not necessarily calculated using the equivalence principle.

The gross premium reserve at time t, _tV^g, is E [_tL | T_x ≥ t]. It is calculated using a set of mortality and interest assumptions called the reserve basis.

Expense Reserve

When the gross premium is calculated using the equivalence principle and the premium basis is the same as the reserve basis, then _oV^g = 0. At issue, the present value of future benefits and expenses equals the present value of future premiums.

It is then possible to split the gross premium into the net premium pⁿ and the expense loading or expense premium p^e, so that p^g = pⁿ + p^e, where p^g is the gross premium and pⁿ is the net premium. We can then calculate an expense reserve v^e, which is the expected present value of future expenses minus the expected present value of future expense loadings.

The gross premium reserve equals the net premium reserve plus the expense reserve, i.e.  V^g = Vⁿ + V^e

Expense Loadings = – V^e

Reserves: Retrospective Formula

Reserve = The Accumulated Value of Premiums – The Accumulated Cost of Insurance

Retrospective Reserve Formula

Fully Discrete Whole Life Insurance

\(_{t}V=\dfrac{P{{{\ddot{a}}}_{x:\overline{t|}}}-A_{x:\overline{t|}}^{1}}{_{t}{{E}_{x}}}\)

Relationships Between Premiums (Omitted)

Premium Difference and Paid Up Insurance Formulas (Omitted)

Reserves: Special Formulas for Whole Life and Endowment Insurance

Annuity-Ratio Formula

Whole Life Insurance

\(_{k}V=1-\dfrac{{{{\ddot{a}}}_{x+k}}}{{{{\ddot{a}}}_{x}}}\)

n-Year Endowment Insurance

\(_{k}V=1-\dfrac{{{{\ddot{a}}}_{x+k:\overline{n-k|}}}}{{{{\ddot{a}}}_{x:\overline{n|}}}}\)

Chaining Reserve

\(1{{-}_{m+n}}{{V}_{x}}=(1{{-}_{m}}{{V}_{x}})(1{{-}_{n}}{{V}_{x+m}})\)

Insurance-Ratio formula

\(_{k}V=\dfrac{{{A}_{x+k:\overline{n-k|}}}-{{A}_{x:\overline{n|}}}}{1-{{A}_{x:\overline{n|}}}}\)

Reserves: Variance of Loss Reading: Actuarial

Continuous Insurance

\(Var{{(}_{t}}L|{{T}_{x}}\ge t)=Var(Z){{(b+\dfrac{P}{\delta })}^{2}}\)

Continuous Premium Continuous Whole Life Insurance

\(Var{{(}_{t}}L|{{T}_{x}}\ge t)=\dfrac{{}^{2}{{{\bar{A}}}_{x+t}}-{{{\bar{A}}}^{2}}_{x+t}}{{{(1-{{{\bar{A}}}_{x}})}^{2}}}\)

Continuous Premium Continuous Endowment Insurance

\(Var{{(}_{t}}L|{{T}_{x}}\ge t)=\dfrac{{}^{2}{{{\bar{A}}}_{x+t:\overline{n-t|}}}-{{{\bar{A}}}^{2}}_{x+t:\overline{n-t|}}}{{{(1-{{{\bar{A}}}_{x:\overline{n|}}})}^{2}}}\)

Fully Discrete Insurance

\(Var{{(}_{k}}L|{{K}_{x}}\ge k)=Var(Z){{(b+\dfrac{P}{\delta })}^{2}}\)

\(Var{{(}_{k}}{{L}^{g}}|{{K}_{x}}\ge k)=Var(Z){{(b+E+\dfrac{G-e}{d})}^{2}}\)

Annual Premium Annual Whole Life Insurance

\(Var{{(}_{t}}L|{{T}_{x}}\ge t)=\dfrac{{}^{2}{{A}_{x+k}}-{{A}^{2}}_{x+k}}{{{(1-{{A}_{x}})}^{2}}}\)

Annual Premium Annual Endowment Insurance

\(Var{{(}_{t}}L|{{T}_{x}}\ge t)=\dfrac{{}^{2}{{A}_{x+t:\overline{n-k|}}}-{{A}^{2}}_{x+t:\overline{n-k|}}}{{{(1-{{A}_{x:\overline{n|}}})}^{2}}}\)

Reserves: Recursive Formulas

Net Premium Reserve

\({{(}_{k-1}}V+{{\pi }_{k-1}})(1+i)={{b}_{k}}\centerdot {{q}_{x+k-1}}{{+}_{k}}V\centerdot {{p}_{x+k-1}}\)

\(_{k}V=\dfrac{{{(}_{k-1}}V+{{\pi }_{k-1}})(1+i)-{{b}_{k}}\centerdot {{q}_{x+k-1}}}{{{p}_{x+k-1}}}\)

Net Amount at Risk (NAR)

\({{(}_{k-1}}V+{{\pi }_{k-1}})(1+i)=({{b}_{k}}{{-}_{k}}V)\centerdot {{q}_{x+k-1}}{{+}_{k}}V\), where \({{b}_{k}}{{-}_{k}}V\) is the NAR

Terminology

To start the recursion off, you can start at time 0, since _oV = 0. You can also start at the end, at which point the net premium reserve is 0. The following rules help:

1. For paid up insurance, the net premium reserve is the net single premium.

2. For term insurance, the net premium reserve at expiry is 0.

3. For endowment insurance, the net premium reserve right before maturity is the endowment benefit.

4. Deferred annuities and insurances have no benefits during the deferral period, so omit the b_k•q_x+k-l term for recursions during the deferral period.

Insurances and Annuities with Payment of Reserve upon Death

\({{(}_{k-1}}V+{{\pi }_{k-1}})(1+i)=({{b}_{k}}+{}_{k}V)\centerdot {{q}_{x+k-1}}+{}_{k}V\centerdot {{p}_{x+k-1}}\)

\({}_{k}V={{(}_{k-1}}V+{{\pi }_{k-1}})(1+i)-{{b}_{k}}\centerdot {{q}_{x+k-1}}\), which means the reserve and \(\pi\) at the time of k-1 is accumulated at the guaranteed interest rate i.

Level Premium π and Level Face Amount b

\(_{k}V=\pi {{\ddot{s}}_{\overline{k|}}}-b\sum\nolimits_{j=1}^{k}{{{q}_{x+j-1}}{{(1+i)}^{k-j}}}\), which is the accumulated premiums – accumulated costs of insurance

Develop Formulas

Constant Mortality

\(_{k}V=(\pi -bvq){{\ddot{s}}_{\overline{k|}}}\)

Gross Premium Reserve

\(_{k}V=\dfrac{{{(}_{k-1}}V+{{G}_{k-1}}-{{e}_{k-1}})(1+i)-{{q}_{x+k-1}}({{b}_{k}}+{{E}_{k}})}{1-{{q}_{x+k-1}}}\)

If the equivalence principle is used for calculating the gross premium and the reserve basis is the same as the premium basis, then ₀V =0. Otherwise, if you want to develop a table of all reserves using recursion, the recursion has to start at the end (expiry or maturity) and work backwards.

Reserves: Modified Reserves

When we discussed expense reserves, they’re usually negative, since expenses are front loaded. As a result, the gross premium reserve is less than the net premium reserve.

A lower modified net premium is used in the calculation in the first year, and higher modified net premiums are used in the later years. Such a reserve is called a modified
reserve.

Full Preliminary Term (FPT)

The policy is treated as if it is a one-year term insurance followed by an insurance issued to a life one year older.

After the first year, treat the policy as if it were issued one year later. In the first year, treat it as if it is a one-year term insurance.

EPV: \(\alpha +\beta {{a}_{x:\overline{n-1|}}}\)

\(\alpha \)

In the full preliminary term method, the modified net premium per unit, which we’ll call the valuation premium, is \(P_{x:\overline{1|}}^{1}\) in the first year. This modified net premium is also called \(\alpha \)

\(\beta \)

The renewal valuation premium (which for whole life would be \({{P}_{x+1}}\) ) is called \(\beta \)

Summary of Modified Reserves Concepts

For the full preliminary term method, ₀V=0. Thereafter, the reserve is the net premium reserve for an otherwise similar policy (with the same maturity date as the original policy) issued one year later.

For any modified reserve method, the expected present value of the valuation premiums must equal the expected present value of the benefits.

Reserves: Other Topics

Reserves on Semi-Continuous Insurance

Whole Life Insurance

The semi-continuous reserve is i/δ times the fully discrete reserve.

Endowment Insurance

Endowment Insurance can be decomposed into a term reserve and a pure endowment reserve; the former is multiplied by i/δ.

Reserves Between Premium Dates

\({}_{k+s}V=\dfrac{({}_{k}V+{{\pi }_{k+1}}){{(1+i)}^{s}}-{{b}_{k+1}}\centerdot {}_{s}{{q}_{x+k}}{{v}^{1-s}}}{{}_{s}{{p}_{x+k}}}\)

UDD

\(_{k+s}V=\dfrac{{{(}_{k}}V+{{\pi }_{k+1}}){{(1+i)}^{s}}-s{{b}_{k+1}}\centerdot {{q}_{x+k}}{{v}^{1-s}}}{1-{}_{s}{{q}_{x+k}}}\)

Thiele’s Differential Equation

\(\dfrac{d}{dt}{}_{t}V={{\delta }_{t}}{}_{t}V+{{G}_{t}}-{{e}_{t}}-({{b}_{t}}+{{E}_{t}}-{}_{t}V){{\mu }_{[x]+t}}\)

\(_{t+h}V{{-}_{t}}V\approx h({{\delta }_{t}}_{t}V+{{G}_{t}}-{{e}_{t}}-({{b}_{t}}+{{E}_{t}}{{-}_{t}}V){{\mu }_{[x]+t}})\)

\(_{t}V=\frac{_{t+h}V-h({{G}_{t}}-{{e}_{t}}-({{b}_{t}}+{{E}_{t}}){{\mu }_{[x]+t}})}{1+h({{\mu }_{[x]+t}}+{{\delta }_{t}})}\)

Policy Alterations

1. The policyholder may cancel the policy. In many cases, the company pays all or part of the reserve to the policyholder. The cash surrender value at time t, _tCV is often based on the net premium reserve or on a regulatory formula.

_tCV + EPV of future net premiums, altered contract = EPV(future benefits, altered contract)

2. The policyholder may wish to continue the policy but not pay any more premiums. In this case, the benefit amount is reduced.

3. As a variation of the previous change, for a whole life insurance, premiums could be discontinued and the benefit not reduced. Instead, the coverage period is reduced. The policy becomes a paid up term policy. For an endowment insurance, sometimes the pure endowment benefit is reduced instead.

Surrender: Continue as a Paid-Up Term Policy (Extended Term)

\({}_{t}{{W}_{x}}=\dfrac{{}_{t}C{{V}_{x}}}{{{A}_{x+t}}}\)

\({}_{t}C{{V}_{x}}=A_{x+t:\overline{n|}}^{1}\)

\({}_{t}C{{V}_{x}}=A_{x+t:\overline{n-t|}}^{1}+P{{E}_{n-t}}{{E}_{x+t}}\), where \(PE\) is the pure endowment amount

Surrender: Continue as a Paid-Up Policy for a Reduced Face Amount (Reduced Paid Up)

Markov Chains: Discrete-Probabilities

Introduction

Definition of Markov chains

In a homogeneous Markov chain, the probability of leaving a state is independent of x as well. In other words, it’s a constant. Otherwise, the chain is non-homogeneous.

\({}_{t}{{p}_{x}}^{ij}\)

The probability that someone in state i at time x is in state j (where j may equal i) at time x +t

\(_{t}{{p}_{x}}^{\overline{ii}}\)

The probability that someone in state i at time x stays in state i continuously until time x + t

\(_{t}{{p}_{x}}^{\overline{ii}}\le{}_{t}{{p}_{x}}^{ii}\), since the latter includes the probability of leaving state i and then returning later while the former doesn’t.

Discrete Markov Chains

Transition Matrix: \({{P}^{(t)}}\)

 In discrete Markov chains, we only keep track of the state one is in at integral times. While transitions can occur at any time, we only care about whether a transition occurred between time k and time k + 1, where k is an integer, and do not care about the exact transition time.

Chapman-Kolmogorov equations

\({}_{k}{{p}_{x}}^{ij}=\sum\nolimits_{m=1}^{n}{{}_{t}{{p}_{x}}^{im}{}_{k-1}{{p}_{x+1}}^{mj}}\)

It says that the probability of transitioning from state i to state j ink steps equals the sum of the probabilities of transitioning from state i to some intermediate state m in l steps followed by transitioning from state m to state j in the remaining k – l steps, summed up over all possible intermediate states m.

To calculate the probability that the state is s₂ at time k₂ given that it is s₁ at time k₁,

Use the transition matrix at time k to calculate the probabilities of each of the states at time k₁ + 1. We’ll call the vector of these probabilities “the state vector”.

Multiply this vector by the transition matrix at time k₁ + 1. The result is the vector of probabilities of the states at time k₁ + 2.

Repeat this step until you reach time k₂. In other words, multiply the state vector at time m by the transition matrix at time m to obtain the state vector at time m + 1. Repeat this until m + 1 = k₂

Markov Chains: Continuous-Probabilities

A continuous Markov chain is similar to a discrete Markov chain in that it satisfies the Markov property: there is no memory.

The probabilities of transitions do not depend on the amount of time in a state. Unlike discrete Markov chains, in which transitions only occur at integral times, transitions can occur at any time. In order to describe the model, we need to specify the probabilities of transitions within h units of time for any h. But usually, rather than specifying probabilities of transitions, we specify forces of transition.

Assumptions

The probability of 2 transitions in a small amount of time is small

Pr(2 transitions in time h) = o(h), where o(h) is a function g(h) having the property \(\underset{h\to \infty }{\mathop{\lim }}\,\dfrac{g(h)}{h}=0\).

\({}_{t}{{p}_{x}}^{ij}\) is differentiable for all i and j.

\({{\mu }_{x}}^{ij}=\underset{h\to 0}{\mathop{\lim }}\,\dfrac{{}_{h}{{p}_{x}}^{ij}}{h}\)

\(_{h}{{p}_{x}}^{ij}=h{{\mu }_{x}}^{ij}+o(h)\)

\(_{h}{{p}_{x}}^{ij}={}_{h}{{p}_{x}}^{\overline{ii}}+o(h)\)

\({}_{h}{{p}_{x}}^{\overline{ii}}=1-h\sum\limits_{j\ne i}{{{\mu }_{x}}^{ij}}+o(h)\)

Probabilities-Direct Calculation

\(_{h}{{p}_{x}}^{\overline{ii}}=\exp (-\int_{0}^{t}{\sum\limits_{j\ne i}{\mu _{x+s}^{ij}}ds})\)

Examples of Transitioning From One State to Another

Suppose (x) is in state 1 at time 0. The probability of transitioning directly (without first returning to state 0) to state 2 by time t is:

\(\int_{0}^{t}{{}_{s}p_{x}^{\overline{11}}\mu _{x+s}^{12}ds}\)

Suppose (x) is in state 0 at time 0. The probability of transitioning to state 1 at least once by time t is:

\(\int_{0}^{t}{_{s}p_{x}^{\overline{00}}\mu _{x+s}^{01}ds}\)

Suppose (x) is in state 0 at time 0. The probability of transitioning to state 1 exactly once and then staying in state 1 until time t is:

\(\int_{0}^{t}{{}_{s}p_{x}^{\overline{00}}\mu _{x+s}^{01}{}_{t-s}p_{x+s}^{\overline{11}}ds}\)

General Form

\({}_{t}p_{x}^{ij}={{\mu }^{ij}}(\dfrac{{{e}^{-{{\mu }^{i\centerdot }}t}}}{{{\mu }^{j\centerdot }}-{{\mu }^{i\centerdot }}}+\dfrac{{{e}^{-{{\mu }^{j\centerdot }}t}}}{{{\mu }^{i\centerdot }}-{{\mu }^{j\centerdot }}})\)

Kolmogorov’s Forward Equations

\(\dfrac{d}{dt}{}_{t}p_{x}^{ij}=\sum\limits_{k=0,k\ne j}^{n}{{{(}_{t}}p_{x}^{ik}\mu _{x+t}^{kj}{{-}_{t}}p_{x}^{ij}\mu _{x+t}^{jk})}\)

i may equal j in these equations.

They’re saying that the change in the probability of going from state i to state j is the rate at which those outside state j are entering state j from any of the other states, minus the rate at which those in state j are leaving state j to one of the other states.

\(\sum\limits_{k=0,k\ne j}^{n}{_{t}p_{x}^{ik}\mu _{x+t}^{kj}}\)

The rate at which state j is entered from state i at time t. To get to state j at time t from state i, you must have been in some state other than j right before time t, and we sum up the probability of being in state k right at time t times the rate at which one goes from state k to state j at time t for all possible k.

\(\sum\limits_{k=0,k\ne j}^{n}{_{t}p_{x}^{ij}\mu _{x+t}^{jk}}\)

The rate at which state j is exited. To exit state j, you must go to some other state, and we sum up the probability of being in state j at time t times the rate at which one goes to state k for all possible k at time t.

\({}_{t+h}p_{x}^{ij}\approx {}_{t}p_{x}^{ij}+h\sum\limits_{k=0,k\ne j}^{n}{{{(}_{t}}p_{x}^{ik}\mu _{x+t}^{kj}{{-}_{t}}p_{x}^{ij}\mu _{x+t}^{jk})}\)

Markov Chains: Premiums and Reserves

Premiums

Life Insurance

\(\bar{A}_{x}^{ij}=\int_{0}^{\infty }{\sum\limits_{k\ne j}{{{e}^{-\delta t}}{}_{t}p_{x}^{ik}\mu _{x+t}^{kj}}dt}\)

Annuity

The EPV of an annuity on someone age x currently in state i which pays 1 per year continuously in state j, where j may equal i:

\(\bar{a}_{x}^{ij}=\int_{0}^{\infty }{{{e}^{-\delta t}}{}_{t}p_{x}^{ij}dt}\)

An annuity-due pays 1 at the beginning of the year while in state j, and its EPV is:

\(\ddot{a}_{x}^{ij}=\sum\nolimits_{k=0}^{\infty }{{{v}^{k}}_{k}p_{x}^{ij}}\)

Reserves

Thiele’s Differential Equation

\(\dfrac{d}{dt}{}_{t}{{V}^{(i)}}={{\delta }_{t}}{}_{t}{{V}^{(i)}}-{{B}_{t}}^{(i)}-\sum\limits_{j=0,j\ne i}^{n}{\mu _{x+t}^{ij}({{b}_{t}}^{(ij)}+{}_{t}{{V}^{(j)}}-{}_{t}{{V}^{(i)}})}\)

\({}_{t-h}{{V}^{(i)}}{{=}_{t}}{{V}^{(i)}}(1-{{\delta }_{t}}h)+{{B}_{t}}^{(i)}h+h\sum\limits_{j=0,j\ne i}^{n}{\mu _{x+t}^{ij}({{b}_{t}}^{(ij)}{{+}_{t}}{{V}^{(j)}}{{-}_{t}}{{V}^{(i)}})}\)

Multiple Decrement Models: Probabilities

Probabilities

A multiple decrement model is a Markov chain with the following properties:

1. It has m + 1 states.

2. State 0 is the starting state, and the other m states are exit states.

3. The only transitions possible are from state 0 to one of the other states.

\({}_{t}q_{x}^{(j)}\)

The probability of transition from state 0 to state j, also called the probability of decrement (j), within t years.

\(_{t}p_{x}^{(\tau )}\)

The probability of no transition from state 0, or no decrement, occurring within t years.

\(_{t}p_{x}^{(\tau )}=1-\sum\limits_{j=1}^{m}{_{t}q_{x}^{(j)}}\)

\(_{t}q_{x}^{(\tau )}\)

The probability of no decrement occurring is the complement of the probability that one of the decrements occurs:

\(_{t}q_{x}^{(\tau )}=1-_{t}p_{x}^{(\tau )}\)

The probability that one of the decrements occurs is the sum of the probabilities of the individual decrements:

\(_{t}q_{x}^{(\tau )}=\sum\limits_{j=1}^{m}{_{t}q_{x}^{(j)}}\)

Sum up over all the years the probability of survival from all decrements to the start of each year times the probability of the decrement occurring in that year.

\(_{t}q_{x}^{(j)}=\sum\limits_{k=0}^{t-1}{_{k}p_{x}^{(\tau )}q_{x+k}^{(j)}}\)

\(_{t}q_{x}^{(\tau )}\)

The probability that decrement j occurs in year k + 1 is the probability of survival from all decrements for k years followed by succumbing to decrement j in the following year.

\(_{k|}q_{x}^{(j)}=_{k}p_{x}^{(\tau )}q_{x+k}^{(j)}\)

\(_{t}q_{x}^{(\tau )}\)

The probability of succumbing to decrement j after t years but not after t + u years:

\(_{t|u}q_{x}^{(j)}{{=}_{t}}p{{_{x}^{(\tau )}}_{u}}p_{x+t}^{(j)}=\sum\limits_{k=t}^{t+u-1}{_{k}p_{x}^{(\tau )}q_{x+k}^{(j)}}\)

Life tables

\(l_{x}^{(\tau)}\)

The total number of lives alive at exact age x.

\(l_{x+k+1}^{(\tau )}=l_{x+k}^{(\tau )}-d_{x+k}^{(\tau )}\)

\(d_{x}^{(j)}\)

The number of lives failing due to decrement (j) in the age interval (x, x + 1].

\(d_{x}^{(\tau)}\)

The total number of lives failing in the age interval (x, x + 1]:

\(d_{x}^{(\tau )}=\sum\limits_{j=1}^{m}{d_{x}^{(j)}}\)

\(l_{x}^{(\tau )}\)

The total number of lives alive at exact age x.

Discrete Insurances

\(A=\sum\limits_{k=1}^{\infty }{{{v}^{k}}{}_{k-1}p_{x}^{(\tau )}\sum\limits_{j}{q_{x+k-1}^{(j)}b_{k}^{(j)}}}\)

Multiple Decrement Models: Forces of Decrement

\(\mu _{x}^{(j)}\)

\(_{t}q_{x}^{(j)}=\int_{0}^{t}{_{s}p_{x}^{(\tau )}\mu _{x+s}^{(j)}ds}\)

\(\mu _{x+t}^{(j)}=\dfrac{d{}_{t}q_{x}^{(j)}/dt}{{}_{t}p_{x}^{(\tau )}}\)

\({}_{t}p_{x}^{(\tau )}=\exp (-\int_{0}^{t}{\mu _{x+s}^{(\tau )}ds})\), where \(\mu _{x+t}^{(\tau )}=\sum\limits_{j=1}^{n}{\mu _{x+t}^{(j)}}\)

Probability Framework for Multiple Decrement Models (Skipped)

Fractional ages

Assume uniform distribution of the decrement in the multiple decrement table:

\({}_{s}q_{x}^{(j)}=sq_{x}^{(j)}\), for \(0\le s\le 1\)

Assume constant forces of decrement:

\({}_{s}q_{x}^{(j)}=\dfrac{q_{x}^{(j)}}{q_{x}^{(\tau )}}(1-{{(p_{x}^{(\tau )})}^{s}})\), and \({}_{s}q_{x}^{(j)}={{({}_{s}p_{x}^{(\tau )})}^{s}}\)

Multiple Decrement Models: Associated Single Decrement Tables

\(_{t}p_{x}^{‘(j)}=\exp (-\int_{0}^{t}{\mu _{x+s}^{(j)}}ds)\)

\(_{t}q_{x}^{‘(j)}=1{{-}_{t}}p_{x}^{‘(j)}\)

\(_{t}q_{x}^{‘(j)}=\exp (-\int_{0}^{t}{_{t}p_{x}^{‘(j)}\mu _{x+s}^{(j)}}ds)\)

\(\prod\limits_{j=1}^{n}{{}_{t}p_{x}^{‘(j)}}={}_{t}p_{x}^{(\tau )}\) for associated single decrement rates

\(\sum\limits_{j=1}^{n}{{}_{t}q_{x}^{(j)}}={}_{t}q_{x}^{(\tau )}\) for multiple decrement rates

Multiple Decrement Models: Relations Between Multiple Decrement Rates and Associated Single Decrement Rates

Constant Force of Decrement

\({}_{s}p_{x}^{‘(j)}={{({}_{s}p_{x}^{(\tau )})}^{q_{x}^{(j)}/q_{x}^{(\tau )}}}\), where \(0\le s\le 1\) ♥

for \(s>1\), calculate separately for x, x+1, etc.

Uniform in the Multiple-Decrement Tables

\({}_{s}p_{x}^{‘(j)}={{({}_{s}p_{x}^{(\tau )})}^{q_{x}^{(j)}/q_{x}^{(\tau )}}}\), where \(0\le s\le 1\) ♥

\(\dfrac{{{\ln }_{s}}p_{x}^{‘(j)}}{{{\ln }_{s}}p_{x}^{(\tau )}}=\frac{q_{x}^{(j)}}{q_{x}^{(\tau )}}\)

\(q_{x}^{(j)}=q_{x}^{(\tau )}(\dfrac{\ln p_{x}^{‘(\tau )}}{\ln p_{x}^{(\tau )}})\) ♥

Uniform in the Associated Single-Decrement Tables

Associated Single Decrement Rates also has the uniform property: \({{\mu }_{x+s}}=\dfrac{q_{x}^{‘(j)}}{1-sq_{x}^{‘(j)}}\)

Two Decrements

\({}_{t}q_{x}^{(1)}=q_{x}^{‘(1)}(1-\dfrac{q_{x}^{‘(2)}}{2})\), when t=1

Three Decrements

\(_{t}q_{x}^{(1)}=q_{x}^{‘(1)}(1-\dfrac{q_{x}^{‘(2)}+q_{x}^{‘(3)}}{2}+\dfrac{q_{x}^{‘(2)}q_{x}^{‘(3)}}{3})\), when t=1

Multiple Decrement Models: Continuous Insurances

\(\bar A=\int_{0}^{\infty }{{{v}^{t}}_{t}p_{x}^{(\tau )}\sum\limits_{j=1}^{n}{\mu _{x+t}^{(j)}b_{x}^{(j)}dt}}\)

\(E[{{Z}^{2}}]=\int_{0}^{\infty }{{{v}^{2t}}_{t}p_{x}^{(\tau )}\sum\limits_{j=1}^{n}{\mu _{x+t}^{(j)}{{(b_{x}^{(j)})}^{2}}dt}}\)

The EPV of the additional benefit paid for this decrement (beyond what is paid regardless of which decrement applies) is \({{b}^{*}}{{\mu }^{(2)}}{{\bar{a}}_{x:\overline{n|}}}\)

Multiple Lives: Joint Life Probabilities

Markov Chain Model

Actuarial Notation	Markov Chain Notation
\(_{t}{{p}_{xy}}\)	\(_{t}p_{xy}^{00}\)
\(_{t}{{q}_{xy}}\)	\(_{t}p_{xy}^{0\bullet }\) or \(_{t}p_{xy}^{01}{{+}_{t}}p_{xy}^{02}{{+}_{t}}p_{xy}^{03}\)

\(_{t}{{p}_{xy}}{{+}_{t}}{{q}_{xy}}=1\)

\(_{t+u}{{p}_{xy}}{{=}_{t}}{{p}_{xy}}{{\centerdot }_{u}}{{q}_{x+t:y+t}}\)

\(_{t|u}{{q}_{xy}}{{=}_{t}}{{p}_{xy}}{{\centerdot }_{u}}{{q}_{x+t:y+t}}{{=}_{t}}{{p}_{xy}}{{-}_{t+u}}{{p}_{xy}}\)

\({}_{t}{{p}_{xy}}={}_{t}p_{xy}^{00}=\exp (-\int_{0}^{t}{\mu _{x+s:y+s}^{0\centerdot }ds})\)

Independent Lives

\(1{{-}_{t}}{{q}_{xy}}=(1{{-}_{t}}{{q}_{x}})(1{{-}_{t}}{{q}_{y}})\) ⇒ \(_{t}{{q}_{xy}}{{=}_{t}}{{q}_{x}}{{+}_{t}}{{q}_{y}}{{-}_{t}}{{q}_{x}}_{t}{{q}_{y}}\)

\(_{t}{{p}_{xy}}{{=}_{t}}{{p}_{x}}{{\cdot }_{t}}{{p}_{y}}\)

\({{\mu }_{x+t:y+t}}={{\mu }_{x+t}}+{{\mu }_{y+t}}\)

\(_{t}{{p}_{xy}}=\exp (-\int_{0}^{t}{({{\mu }_{x+s}}+{{\mu }_{y+s}})ds})\)

Joint Distribution Function Model

\({{F}_{xy}}(t)=\Pr ({{T}_{xy}}\le t)\)

\({{S}_{xy}}(t)=\Pr ({{T}_{xy}}>t)\)

\(f({{t}_{1}},{{t}_{2}})=\frac{\partial }{\partial {{t}_{2}}}\frac{\partial }{\partial {{t}_{1}}}F({{t}_{1}},{{t}_{2}})\)

\(F({{t}_{1}},{{t}_{2}})=\int_{0}^{{{t}_{1}}}{\int_{0}^{{{t}_{2}}}{f({{s}_{1}},{{s}_{2}})d{{s}_{2}}d{{s}_{1}}}}\)

\(S({{t}_{1}},{{t}_{2}})=\int_{{{t}_{1}}}^{\infty }{\int_{{{t}_{2}}}^{\infty }{f({{s}_{1}},{{s}_{2}})d{{s}_{2}}d{{s}_{1}}}}\)

\({{F}_{{{T}_{xy}}}}(t)={{F}_{{{T}_{x}},{{T}_{y}}}}(t,\infty )+{{F}_{{{T}_{x}},{{T}_{y}}}}(\infty ,t)-{{F}_{{{T}_{x}},{{T}_{y}}}}(t,t)\)

Multiple Lives: Last Survivor Probabilities

Properties

The time when the last life of a set of lives fails (the last survivor status as a status that fails only when every member of the status fails): \({T}_{\overline{xy}}\)

\({{T}_{x}}+{{T}_{y}}={{T}_{xy}}+{{T}_{\overline{xy}}}\)

\({}_{t}{{p}_{xy}}+{}_{t}{{p}_{\overline{xy}}}={}_{t}{{p}_{x}}+{}_{t}{{p}_{y}}\) ⇒ \(_{t}{{p}_{\overline{xy}}}{{=}_{t}}{{p}_{x}}{{+}_{t}}{{p}_{y}}{{-}_{t}}{{p}_{xy}}\)  ♥

Under independence, \(_{t}{{p}_{\overline{xy}}}{{=}_{t}}{{p}_{x}}{{+}_{t}}{{p}_{y}}{{-}_{t}}{{p}_{x}}{{\cdot }_{t}}{{p}_{y}}\)

The probability that exactly one life of two is alive:

\({}_{t}{{p}_{\overline{xy}}}-2{}_{t}{{p}_{xy}}={_{t}}{{p}_{x}}{{+}_{t}}{{p}_{y}}{{-}2_{t}}{{p}_{xy}}\)

Under independence, the probability that the last status fails:

\(_{t}{{p}_{\overline{xy}}}{{+}_{t}}{{q}_{\overline{xy}}}=1\)

\(_{t|u}{{q}_{\overline{xy}}}{{=}_{t}}{{p}_{\overline{xy}}}{{-}_{t+u}}{{p}_{\overline{xy}}}\) ♥

False Statement

\(_{t+u}{{p}_{\overline{xy}}}{{\ne }_{t}}{{p}_{\overline{xy}}}{{\cdot }_{u}}{{p}_{\overline{x+t:y+t}}}\)

\(_{t|u}{{q}_{\overline{xy}}}{{\ne }_{t}}{{p}_{\overline{xy}}}{{\cdot }_{u}}{{q}_{\overline{x+t:y+t}}}\)

Force of Mortality

The force of mortality is complicated even when the two lives are independent. We will use the notation \({{\mu }_{\overline{xy}}}(t)\) instead of  \({{\mu }_{\overline{x+t:y+t}}}\) because the force of mortality for the last survivor status is not independent of the starting ages x y.

For constant force of mortality and lives are independent:

\({{\mu }_{\overline{xy}}}(t)=\dfrac{{}_{t}{{q}_{x}}{}_{t}{{p}_{y}}{{\mu }_{y+t}}+{}_{t}{{q}_{y}}{}_{t}{{p}_{x}}{{\mu }_{x+t}}}{{}_{t}{{p}_{\overline{xy}}}}\)

Distribution Function of The Last Survivor Status

This distribution function is related to the joint distribution function of T_x and T_y by:

\({{F}_{{{T}_{\overline{xy}}}}}(t)={{F}_{{{T}_{x}},{{T}_{y}}}}(t,t)=\int_{0}^{t}{\int_{0}^{t}{{{f}_{S,T}}(s,t)ds}dt}\)

Multiple Lives: Contingent Probabilities

Sequence

The Probability That (x) Dies First by Time t:

\(_{t}q_{xy}^{1}\) indicates that at time t, (x) is dead, and died before (y). This chain could be in state 2 at time t, but could also be in state 3 if the route was 0 ⇒ 2 ⇒ 3

\(_{t}q_{xy}^{1}=\int_{0}^{t}{_{s}p_{xy}^{00}\mu _{x+s:y+s}^{02}ds}\)

The Probability That (x) Dies Second by Time t:

\(_{t}q_{xy}^{2}\) indicates the chain is in state 3 at time t, but also that the route was 0 ⇒ 1 ⇒ 3

\(_{t}q_{xy}^{2}=\int_{0}^{t}{{}_{s}p_{xy}^{00}\mu _{x+s:y+s}^{01}\int_{0}^{t-s}{{}_{u}p_{x+s}^{11}\mu _{x+s+u}^{13}du}ds}\)

Relationship

If (x) dies first then (y) must die second:

\(_{\infty }q_{xy}^{1}{{=}_{\infty }}{{q}_{x}}_{y}^{2}\)

Either (x) dies first or (y) dies first:

\(_{\infty }q_{xy}^{1}{{+}_{\infty }}{{q}_{x}}_{y}^{1}=1\)

One of them must die second:

\(_{\infty }q_{xy}^{2}{{+}_{\infty }}{{q}_{x}}_{y}^{2}=1\)

The probability that (x) dies first within n years:

(y) dies second within n years

(x) dies first within n years and (y) survives n years

\(_{n}q_{xy}^{1}{{=}_{n}}{{q}_{x}}_{y}^{2}{{+}_{n}}{{q}_{x}}{{\cdot }_{n}}{{p}_{y}}\)

The joint-life status fails within n years:

Either (x) dies first within n years

Or (y) dies first within n years

\(_{n}{{q}_{xy}}{{=}_{n}}q_{xy}^{1}{{+}_{n}}{{q}_{x}}_{y}^{1}\)

The last-survivor status fails within n years:

One of the lives must have died second

\(_{n}{{q}_{\overline{xy}}}{{=}_{n}}q_{xy}^{2}{{+}_{n}}{{q}_{x}}_{y}^{2}\)

(x) dies in a given period:

(x) must die either first or second in that period

\(_{n}{{q}_{x}}{{=}_{n}}q_{xy}^{1}{{+}_{n}}q_{xy}^{2}\)

Multiple Lives: Common Shock

The common shock model allows for both lives dying at the same time, for example through a car accident.

Multiple Lives: Insurances

\({{V}^{{{T}_{x}}}}+{{V}^{{{T}_{y}}}}={{V}^{{{T}_{xy}}}}+{{V}^{{{T}_{\overline{xy}}}}}\)

\({{\bar{a}}_{x}}+{{\bar{a}}_{y}}={{\bar{a}}_{xy}}+{{\bar{a}}_{\overline{xy}}}\)

\({{\bar{A}}_{x}}+{{\bar{A}}_{y}}={{\bar{A}}_{xy}}+{{\bar{A}}_{\overline{xy}}}\)

Similar equalities are true for any insurance or annuity, including discrete insurances, term insurances, endowment insurances, deferred insurances, temporary annuities, deferred annuities.

However, no such statement holds for annual premiums, which are quotients of insurances and annuities.

Multiple Lives: Insurances

\({{A}_{x}}_{y}^{2}\)

\({{A}_{x}}_{y}^{2}=\int_{0}^{\infty }{{{v}^{t}}{}_{t}{{q}_{x}}{}_{t}{{p}_{y}}{{\mu }_{y+t}}dt}\), integrate over y

\({A}_{xy}\)

\({{A}_{xy}}=\int_{0}^{\infty }{{{v}^{t}}_{t}{{p}_{x}}_{t}{{p}_{y}}({{\mu }_{x+t}}+{{\mu }_{y+t}})dt}\), integrate over x:y

\(A_{xy}^{1}\)

\(A_{xy}^{1}=\int_{0}^{\infty }{{{v}^{t}}_{t}{{p}_{y}}_{t}{{p}_{x}}{{\mu }_{x+t}}dt}\), integrate over x

Non-Independent Lives

If \({{\mu }^{01}}\ne {{\mu }^{23}}\) or \({{\mu }^{02}}\ne {{\mu }^{13}}\), the lives are not independent

Insurance on the Joint-Life Status

\(\int{{{v}^{t}}{}_{t}p_{xy}^{00}(\mu _{x+t:y+t}^{01}+\mu _{x+t:y+t}^{02})dt}\)

Annuity on the Joint-Life Status

\(\int{{{v}^{t}}{}_{t}p_{xy}^{00}dt}\)

Insurance on the Last Survivor Status

\(\int{{{v}^{t}}({}_{t}p_{xy}^{01}\mu _{x+t:y+t}^{13}+{}_{t}p_{xy}^{02}\mu _{x+t:y+t}^{23})dt}\)

Annuity on the Last Survivor Status

\(\int{{{v}^{t}}({}_{t}p_{xy}^{00}+{}_{t}p_{xy}^{01}+{}_{t}p_{xy}^{02})dt}\)

Contingent Insurances

An insurance on (x) if (x) dies first plus an insurance on (y) if (y) dies first is the same as an insurance on the joint status:

\(\bar{A}_{xy}^{1}+{{\bar{A}}_{x}}_{y}^{1}={{\bar{A}}_{xy}}\)

The same goes for insurances for second deaths and the last survivor status:

\(\bar{A}_{xy}^{2}+{{\bar{A}}_{x}}_{y}^{2}={{\bar{A}}_{xy}}\)

An insurance on (x) if (x) dies first plus an insurance on (y) if (y) dies first is the same as an insurance on the joint status

\({}_{n}\bar{A}_{xy}^{1}+{}_{n}{{\bar{A}}_{x}}_{y}^{1}={}_{n}{{\bar{A}}_{xy}}\)

If (x) and (y) both survive n years and then (x) dies first, \(\bar{A}_{xy:\overline{n|}}^{2}\) would pay a benefit since the failure of (x) was second (the certain status m was the first failure), while \({}_{n}\bar{A}_{xy}^{2}\) would 

Common Shock Insurances (Skipped)

Multiple Lives: Annuities

Three techniques for calculating expected present values of annuities

Two-annuity technique: Calculate two single annuities, then adjust the joint annuity

Three-annuity technique: Calculate two single-life annuities and one joint-life annuity

Reversionary annuity technique: Calculate a full annuity and cancel out the part that is not paid

A reversionary annuity is an annuity that makes regular payments to one status after another status has failed. An example would be an annuity that makes an annual payment to (y) only after (x) has died.

\({{a}_{x|y}}={{a}_{y}}-{{a}_{xy}}\)

Pension Mathematics

Categories of Pensions

Defined Benefit (DB)

These pay a monthly amount to retirees determined by formula. (For simplicity, we will often assume an annual amount, rather than a monthly amount, is paid.) Typicallythey pay a percentage of final average salary, but they may also pay a percentage of career average salary. The percentage is determined based on length of service. Final average salary is the average salary over the last few years-often the last 3 years. Career average salary is the average salary over an employee’s entire career. The employer bears the investment risk for investments made during the employee’s career to fund the pension.

Defined Contribution (DC)

These specify the amount contributed to the pension fund each year of service. The fund grows with interest and is distributed to the employee at retirement. The amount that is contributed to the fund is a percentage of salary. The amount in the fund at retirement is not known in advance, and depends on investment performance. The employee bears the investment risk.

Calculating the Contribution for a Defined Contribution Plan

Final Salary

Final salary is defined as the salary earned in the 12 months preceding the retirement date.

Replacement Ratio (R)

The replacement ratio R is the ratio of the pension paid in the first year of retirement to the final salary.

Rate of Salary Function

A rate of salary function is an instantaneous concept, whereas a salary scale is an annual concept. To go from a rate of salary function to the corresponding salary scale, you’d integrate the rate of salary function over one year:

\({{s}_{x}}=\int_{0}^{1}{{{{\bar{s}}}_{x+t}}dt}\)

\({{\bar{s}}_{x}}={{s}_{x-0.5}}\)

Service Table

The states of employees in a pension plan may be modeled as a multiple decrement model. State 0 is the active state, and there are four exits from this state: death(d), disability(i), withdrawal(w), and retirement(r).

Valuing Pension Plan Benefits

A defined benefit pension plan may have benefits upon retirement or death. In addition, a deferred pension may be payable to those who withdraw. To value a defined benefit pension plan, calculate the projected benefits (possibly using a salary scale), then multiply each benefit by the probability of receiving it (possibly using a service table) and discount it.

Funding the Benefits

The accrued liability of a defined benefit pension plan, whether calculated using the traditional unit method or the projected unit method, is called the actuarial liability. It corresponds to a reserve for an insurance policy, and we will use the same notation, _tV, for it.

Normal Contribution

Let C_t be the contribution the employer must make to the plan, which we will call the normal contribution. Then

\({{C}_{t}}=v{}_{1}p_{x}^{00}{}_{t+1}V-{}_{t}V+EPV(Benefits\_Paid\_for\_Mid-Year\_Exits)\)

If the accrued liability is computed using the traditional unit method, then we say that the normal contribution is computed using the traditional unit credit (TUC) method and if the accrued liability is computed using the projected unit method then we say that the normal contribution is computed using the projected unit credit (PUC) method.

Traditional Unit Credit (TUC)

1. Calculate the Final Average Salary at time x+t

2. Annual Annuity Benefit for Retirement Age T₁ = The Proportion c × Years of Service t × Final Average Salary at time x+t

    Annual Annuity Benefit for Retirement Age T₂ = The Proportion c × Years of Service t × Final Average Salary at time x+t × (1 – Early Retirement Deduction)

    …

EPV of Accrued Retirement Benefits = Sum of (Annual Annuity Benefit × Discount Factor × the Probability of Retirement × Annuity Factor)

Projected Unit Credit (PUC)

1. Project the Final Average Salaries for all possible retirement ages

2. Retirement Benefit for Retirement Age T₁ = The Proportion c × Years of Service t × Final Average Salary at time T₁ × Annuity Factor

    Retirement Benefit for Retirement Age T₂ = The Proportion c × Years of Service t × Final Average Salary at time T₂ × (1 – Early Retirement Deduction) × Annuity Factor

    …

EPV of Accrued Retirement Benefits = Sum of (Annual Annuity Benefit × Discount Factor × the Probability of Retirement × Annuity Factor)

3. Future Benefit for Retirement Age T₁ = The Proportion c × ( T₁ – t) × Final Average Salary at time T₁ × Annuity Factor

    Future Benefit for Retirement Age T₂ = The Proportion c × ( T₂ – t) × Final Average Salary at time T₁ × (1 – Early Retirement Deduction × Annuity Factor

EPV of Future Benefits = Sum of (Annual Annuity Benefit × Discount Factor × the Probability of Retirement × Annuity Factor)

Interest Rate Risk: Replicating Cash Flows

Spot Rate

\({{y}_{t}}={{(1/v(t))}^{1/t}}-1\), y_t is the effective interest rate paid by a zero-coupon bond maturing at time t

Forward Rate

\(f(t,t+k)={{(\dfrac{v(t)}{v(t+k)})}^{1/k}}-1\)

\({{(1+f(t,T))}^{T-t}}=\dfrac{v(t)}{v(T)}=\dfrac{{{(1+{{y}_{T}})}^{T}}}{{{(1+{{y}_{t}})}^{t}}}\)

Interest Rate Risk: Diversifiable and Non-Diversifiable Risk

\(Var(\dfrac{{{L}^{p}}}{n})=Var(E[{{L}^{i}}|I])+\dfrac{E[Var({{L}^{1}}|I)]}{n}\)

As n → ∞, the second summation goes to 0 but the first one doesn’t. The first term is the non-diversifiable part, the second term is the diversifiable part.

Profit Tests: Asset Shares

Asset Shares

An asset share measures the accumulation of cash income per surviving policy. Asset shares are computed retrospectively by recursion. Start at policy inception with an asset share
of 0.

\({}_{k}AS=\dfrac{({}_{k-1}AS+{{G}_{k-1}}-{{e}_{k-1}})(1+i)-({{b}_{k}}+E_{k}^{(d)})q_{x+k-1}^{(d)}-({}_{k}CV+E_{k}^{(w)})q_{x+k-1}^{(w)}}{1-q_{x+k-1}^{(d)}-q_{x+k-1}^{(w)}}\), where

\({}_{k}AS\) is the asset share at the end of year k;

\({{G}_{k-1}}\) is the gross premium paid at time k -1; in other words, the gross premium for year k;

\({{e}_{k-1}}\) is the expense paid at time k – 1;

\({{b}_{k}}\) is the face amount for year k;

\(q_{x+k-1}^{(d)}\) and \(q_{x+k-1}^{(w)}\) are death and withdrawal probabilities;

\(E_{k}^{(d)}\) and \(E_{k}^{(w)}\) are settlement expenses paid on death and surrender respectively;

\({}_{k}CV\) is the cash value at duration k, the amount the policyholder gets if he surrenders the policy at time k.

Profit Tests: Profits for Traditional Products

Profits by Policy Year

\({{\Pr }_{k}}=({}_{k-1}V+{{G}_{k-1}}-{{e}_{k-1}})(1+i)-({{b}_{k}}+E_{k}^{(d)})q_{x+k-1}^{(d)}-(C{{V}_{k}}+E_{k}^{(w)})-{}_{k}Vp_{x+k-1}^{(\tau )}\)

\({\Pr }_{k}\) is the net profit at the end of year k per policy in force at the beginning of year k, while \(\prod\nolimits_{k}{{}}\) is the net profit at the end of year t per policy issued. The vector of \(\prod\nolimits_{k}{{}}\) ‘s is called the profit signature.

\(\prod\nolimits_{k}{{}}={}_{k-1}{{p}_{x}}{{\Pr }_{k}}\)

Change in Reserve

\({{\Delta }_{k}}V{{=}_{k-1}}V(1+i){{-}_{k}}Vp_{x+k-1}^{(\tau )}\)

Profit Measures

IRR

The internal rate ofreturn (IRR) is the interest rate j such that the present value of the profit signature is 0:

\(\sum\limits_{k=0}^{n}{\dfrac{\prod\nolimits_{k}{{}}}{{{(1+j)}^{k}}}=0}\)

NPV

The expected present value of future profits (EPVFP), better known as the net present value (NPV), is the present value of the components of the profit signature, discounted at a special interest rate r used for discounting risk. This rate is also called the hurdle interest rate.

\(NPV=\sum\limits_{j=0}^{\infty }{\prod\nolimits_{j}{v_{r}^{j}}}\)

Partial NPV, denoted NPV(k), which is the expected present value of all cash flows up to and including time k:

\(NPV(k)=\sum\limits_{j=0}^{k}{\prod\nolimits_{j}{v_{r}^{j}}}\)

Profit Margin

The profit margin is the ratio of the NPV to the (expected) present value of future gross premiums.

Discounted Payback Period

The smallest k for which NPV(k) ≥ 0.

Zeroization

of reserves means setting the reserves so that the profit is 0 in each year except the first.

Determining the Reserve Using a Profit Test

If negative profits occur only in early years, early reserves may be too high. To avoid this situation, the reserve can be adjusted so that profits in later years are 0. This process is called zeroization of the reserve and the resulting reserve is called a zeroized reserve.

Handling Multiple-State Models

1. Calculate net cash flows per policy in force at beginning of each year for each non-terminal state. For the benefit cash flow, calculate the probability of each transition out of the state times the benefit paid for transition out of the state, and sum it up.

2. Calculate profits per policy in force at beginning of each year, Pr_r⁽ⁱ⁾, for each non-terminal state. Add to the cash flows the increases in reserves, which are the sums of the probabilities of transition to the same or different states times the reserve in each stater minus the starting reserve in the starting state raised at interest.

3. Calculate the profit signature components \(\prod\nolimits_{t}{{}}\) as weighted averages of the Pr_r⁽ⁱ⁾ ; weight the profits with the probability of being in that state. In other words, for t > 0: 

\(\prod\nolimits_{t}{{}}=\sum\limits_{i}{{}_{t-1}p_{x}^{0j}\Pr _{t}^{(i)}}\)

While for t = 0, \(\prod\nolimits_{0}{{}}={{\Pr }_{0}}\) (initial expenses are independent of state). The weights \({}_{t-1}{{p}^{0i}}\) will probably add up to less than 1 due to the possibility of transition to a terminal state.

Profit Tests: Participating Insurance

Reversionary Bonus

Face Amount of Reversionary Bonuses

\({Div}_{k}=c{{\Pr }_{k}}\)

Paid-up Insurance

\({{B}_{k}}=\dfrac{{Div}_{k}}{{{A}_{x+k}}}=\dfrac{c{{\Pr }_{k}}}{{{A}_{x+k}}}\) ♥

Cumulative Reversionary Bonuses

\(R{{B}_{k}}=R{{B}_{k-1}}+{{B}_{k}}\)

Bonus Rate

Bonus Rate = Paid-up Insurance / (Death Benefits + Cumulative Face Amount of Reversionary Bonuses) ♥

Annual dividends used to buy paid-up insurance are called “reversionary bonuses”. These come in three styles:

1. Simple reversionary bonuses are based only on the original face amount of the policy.

The bonus rate is calculated by: dividing the bonus by the original face amount

2. Compound reversionary bonuses are paid on the face amount of the policy plus the face amount of cumulative reversionary bonuses from past years.

The bonus rate is calculated by: dividing the bonus by the original face amount plus the face amount of cumulative reversionary bonuses through the previous duration

3. Super-compound reversionary bonuses are also paid on the face amount of the policy and the face amount of cumulative reversionary bonuses, but different rates are used.

The bonus rate is calculated by: a rule needs to be given as to how to split the bonus between the original face amount and the cumulative reversionary bonuses. Based on this rule, divide the two components of the bonus by the original face amount and the cumulative reversionary bonuses through the previous duration respectively

Profit

\({{\Pr }_{k}}=({}_{k-1}{{V}^{original}}+{}_{k-1}{{V}^{Bonus}}+{{G}_{k}}-{{e}_{k}})(1+{{i}_{k}})-(b_{k}^{(d)}+F){{q}_{x+k-1}}-({}_{k}{{V}^{original}}+{}_{k}{{V}^{Bonus}}){{p}_{x+k-1}}\), where

\(F\)= Cumulative Face Amount of Reversionary Bonuses

\({}_{k}{{V}^{Bonus}}=F\cdot {{A}_{x+k}}\)

Universal Life

Account Value

End of AV = Start of AV + Premium – Expense Charge – Mortality Charge + Interest Credited

 Type A design (FA Paid upon Death)

\(A{{V}_{t}}=\dfrac{(A{{V}_{t-1}}+{{P}_{t}}-{{e}_{t}})(1+i)-FA{{q}_{x+t-1}}}{1-{{q}_{x+t-1}}}\) ♥

\(CO{{I}_{t}}=(FA-A{{V}_{t}}){{v}_{q}}\cdot {{q}_{x+t-1}}\) ♥

\(CO{{I}_{t}}={{v}_{q}}\cdot {{q}_{x+t-1}}(\dfrac{FA-(A{{V}_{t-t}}+{{P}_{t}}-{{e}_{t}})(1+i)}{1-{{v}_{q}}\cdot {{q}_{x+t-1}}(1+i)})\)

 Type B design (FA + AV_t Paid upon Death)

\(A{{V}_{t}}=(A{{V}_{t-1}}+{{P}_{t}}-{{e}_{t}}-FA{{v}_{q}}{{q}_{x+t-1}})(1+i)\) ♥

\(CO{{I}_{t}}=FA{{v}_{q}}{{q}_{x+t-1}}\) ♥

Corridor

The death benefit must be at least as high as the account value times the corridor factor.

First of all, calculate the account value ignoring the corridor factor. If the resulting account value times the corridor factor is less than or equal to the death benefit, you’re done. Otherwise, let \(\gamma \) be the corridor factor and set the death benefit equal to \(\gamma A{{V}_{t}}\).

\(A{{V}_{t}}\cdot {{p}_{x+t-1}}=(A{{V}_{t-1}}+{{P}_{t}}-{{e}_{t}}-\gamma A{{V}_{t}}\cdot {{v}_{q}}\cdot {{q}_{x+t-1}})(1+i)\) ♥

\(A{{V}_{t}}=\dfrac{(A{{V}_{t-t}}+{{P}_{t}}-{{e}_{t}})(1+i)}{1+{{v}_{q}}\cdot {{q}_{x+t-1}}(1+i)(\gamma -1)}\)

\(CO{{I}_{t}}={{v}_{q}}\cdot {{q}_{x+t-1}}(\dfrac{(A{{V}_{t-t}}+{{P}_{t}}-{{e}_{t}})(1+i)(\gamma -1)}{1+{{v}_{q}}\cdot {{q}_{x+t-1}}(1+i)(\gamma -1)})\)

Profit Tests

1. Project account value at the end of the period

2. Calculate the current year’s profit:

+ Account Value at the beginning of time t

– Expected Expenses

+ Expected Interests Earned

– Expected Death Benefits and Associated Processing Costs

– Expected Surrender Benefits and Associated Processing Costs (After Deaths)

– Account Value at the end of time t (After Deaths, Surrenders)

Profit Tests: Gain by Source

Traditional Products: Profit Per Policy in Force at the Beginning of the Year

\({{\Pr }_{k}}=({}_{k-1}V+{{G}_{k-1}}-{{e}_{k-1}})(1+i)-q_{x+k-1}^{(d)}({{b}_{k}}+E_{k}^{(d)})-q_{x+k-1}^{(w)}(C{{V}_{k}}+E_{k}^{(w)})-p_{x+k-1}^{(\tau )}\cdot {}_{k}V\)

Gross Premium Reserve

\({}_{k}V=\dfrac{({}_{k-1}V+{{G}_{k-1}}-{{e}_{k-1}})(1+i)-q_{x+k-1}^{(d)}({{b}_{k}}+E_{k}^{(d)})-q_{x+k-1}^{(w)}(C{{V}_{k}}+E_{k}^{(w)})}{p_{x+k-1}^{(\tau )}}\)

Expected Profit V.S. Actual Profit

Expected profit is profit computed using the interest, mortality, surrender, and expense rates assumed in pricing the product; in other words, the assumptions used in computing the premium.

Actual profit is computed using actual experience for interest, mortality, surrender, and expense.

Total Profit

\(({}_{k-1}V+{{G}_{k}}-{{e}_{k}})(1+i)-{{q}_{x+k-1}}({{b}_{k}}+{{E}_{k}})-{{p}_{x+k-1}}\cdot {}_{k}V\)

Total Gain = Actual Profit – Expected Profit

Interest

\((i’-i*)({}_{k-1}V+{{G}_{k-1}}-{{e}_{k-1}})\)

Expense

\((e_{k}^{*}-e_{k}^{‘})(1+{i}’)+{{q}_{x+k-1}}(E_{k}^{*}-E_{k}^{‘})\)

Mortality

\((q_{x+k-1}^{*}-q_{x+k-1}^{‘})({{b}_{k}}+{{E}_{k}}{{-}_{k}}V)\)

Lapse

\((q_{x+k-1}^{(w)*}-q_{x+k-1}^{(w)’})({}_{k}CV+E_{k}^{(w)}-{}_{k}V)\)