Some Fundamental Actuarial Building Blocks for the Curious

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Some Fundamental Actuarial Building Blocks for the Curious

Richard Purvey

2015

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Dedicated to my parents, Patricia and David

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ContentsPreface

Chapter One

The Cohort

Chapter Two

The Value Of A Payment Which Is Guaranteed To Be Made And The Actuarial Present Value Of A Payment

Chapter Three

Expectations

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Preface

Written for the curious “pure mathematician”, this book shows quantities which gave rise to the formalized subject known as life contingencies (this formalized subject along with other formalized subjects making up the formal studies known as actuarial studies).

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Chapter One

The Cohort

Imagine a number of individuals born simultaneously and followed until death. Death is assumed to occur by an identical “mechanism” of failure for these individuals. The number of individuals alive, i.e. remaining, at time, i.e. age, T time units is denoted by lT. 0 ≤ T. lβ/lα is the number of individuals alive, i.e. remaining, at time, i.e. age, β expressed as a fraction of the number of individuals alive, i.e. remaining, at time, i.e. age, α and lα - lβ is the number of individuals alive, i.e. remaining, at time, i.e. age, α minus the number of individuals alive, i.e. remaining, at time, i.e. age, β, this being the decrease in the number of individuals alive, i.e. remaining, between time, i.e. age, α and time, i.e. age, β, which is the number of individuals who die between time, i.e. age, α and time, i.e. age, β. 0 ≤ α ≤ β. And (lα - lβ)/lγ is (lα - lβ) expressed as a fraction of lγ which equals lα/lγ – lβ/lγ, i.e. the number of individuals alive, i.e. remaining, at time, i.e. age, α expressed as a fraction of the number of individuals alive, i.e. remaining, at time, i.e. age, γ minus the number of individuals alive, i.e. remaining, at time, i.e. age, β expressed as a fraction of the number of individuals alive, i.e. remaining, at time, i.e. age, γ. 0 ≤ γ ≤ α ≤ β.

Lim h→0 (lT - lT+h)/h is the instantaneous decrease in the number of individuals alive, i.e. remaining, per unit time at time, i.e. age, T and equals - l’T. And lim h→0 (lT - lT+h)/hlα is lim h→0 (lT - lT+h)/h expressed as a fraction of lα and equals - l’T/lα. 0 ≤ α ≤ T. If α = T then - l’T/lα becomes - l’T/lT and equals – (d/dT)lnlT.

If instead of being continuously re-counted, lT is only re-counted when T is a positive integer then where lT is needed to be known and T is not 0 or a positive integer, lT will have to be interpolated, i.e. assumed. And for a case like this, here is a list showing lT, where T is not 0 or a positive integer, assumed under a choice of three different types of assumption. Also shown are - l’T, where T is not 0 or a positive integer, and - l’T/lT, i.e. – (d/dT)lnlT, where T is not 0 or a positive integer, assumed under each of these three different types of assumption.

1.The assumption that lu+v = lu + v(lu+1 – lu) where u is 0 or a positive integer and 0 < v < 1.

Because (∂/∂v)(u + v) = 1 so that - l’u+v is the same as - (∂/∂v)lu+v, under the assumption that lu+v = lu + v(lu+1 – lu) where u is 0 or a positive integer and 0 < v < 1, - l’u+v = lu – lu+1 where u is 0 or a positive integer and 0 < v < 1 and so under the assumption that lu+v = lu + v(lu+1 – lu) where u is 0 or a positive integer and 0 < v < 1, - l’u+v/lu+v = (lu – lu+1)/(lu + v(lu+1 – lu)) where u is 0 or a positive integer and 0 < v < 1.

2.The assumption that lnlu+v = lnlu + v(lnlu+1 – lnlu) where u is 0 or a positive integer and 0 < v < 1, this being the assumption that lu+v = lu(lu+1/lu)^v where u is 0 or a positive integer and 0 < v < 1.

Because (∂/∂v)(u + v) = 1 so that - l’u+v is the same as - (∂/∂v)lu+v, under the assumption that lu+v = lu(lu+1/lu)^v where u is 0 or a positive integer and 0 < v < 1, - l’u+v = luln(lu/lu+1)(lu+1/lu)^v where u is 0 or a positive integer and 0 < v < 1 and so under the assumption that lu+v = lu(lu+1/lu)^v where u is 0 or a positive integer and 0 < v < 1, - l’u+v/lu+v = ln(lu/lu+1) where u is 0 or a positive integer and 0 < v < 1.

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3.The assumption that 1/lu+v = 1/lu + v(1/lu+1 – 1/lu) where u is 0 or a positive integer and 0 < v < 1, this being the assumption that lu+v = 1/(1/lu + v(1/lu+1 – 1/lu)) where u is 0 or a positive integer and 0 < v < 1.

Because (∂/∂v)(u + v) = 1 so that - l’u+v is the same as - (∂/∂v)lu+v, under the assumption that lu+v = 1/(1/lu + v(1/lu+1 – 1/lu)) where u is 0 or a positive integer and 0 < v < 1, - l’u+v = (1/lu+1 – 1/lu)/(1/lu + v(1/lu+1 – 1/lu))^2 where u is 0 or a positive integer and 0 < v < 1 and so under the assumption that lu+v = 1/(1/lu + v(1/lu+1 – 1/lu)) where u is 0 or a positive integer and 0 < v < 1, - l’u+v/lu+v = (1/lu+1 – 1/lu)/(1/lu + v(1/lu+1 – 1/lu)) where u is 0 or a positive integer and 0 < v < 1.

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Chapter Two

The Value Of A Payment Which Is Guaranteed To Be Made And The Actuarial Present Value Of A Payment

At time, Tv, the value of a payment which is guaranteed to be made (this payment of θj units of money is made at time, Tj) = θj(1 - d(m)/m)^(m(Tj – Tv)) = θj(1 + i(m)/m)^(m(Tv – Tj)). Tj will either be 0 or a positive integer multiplied by 1/m and Tv will either be 0 or a positive integer multiplied by 1/m. If Tv = Tj then no compounding will take place in this formula and it will simply result in θj. If Tv < Tj then compound discount will take place in this formula with Tj – Tv being a positive integer multiplied by 1/m and (Tj – Tv) divided by 1/m, i.e. m(Tj – Tv), being the number of 1/m time units which Tj – Tv is, this being the number of compounds which will take place in this formula. If Tv > Tj then compound interest will take place in this formula with Tv – Tj being a positive integer multiplied by 1/m and (Tv – Tj) divided by 1/m, i.e. m(Tv – Tj), being the number of 1/m time units which Tv – Tj is, this being the number of compounds which will take place in this formula.

m will either be a positive integer (if m→∞ then θj(1 - d(m)/m)^(m(Tj – Tv)) = θj(1 + i(m)/m)^(m(Tv – Tj)) becomes θje^(-d(m)(Tj – Tv)) = θje^(i(m)(Tv – Tj)), meaning that lim m→∞ d(m) = lim m→∞ i(m), because lim m→∞ (1 - d(m)/m)^m, i.e. lim m→∞ (1 + - d(m)/m)^m, is mathematically identical to lim m→∞ e^(-d(m)) and because lim m→∞ (1 + i(m)/m)^m is mathematically identical to lim m→∞ e^i(m). lim m→∞ d(m) = lim m→∞ i(m) is often just denoted by δ) or 1/a positive integer.

Because θj(1 - d(m)/m)^(m(Tj – Tv)) = θj(1 + i(m)/m)^(m(Tv – Tj)), 1 - d(m)/m = (1 + i(m)/m)^(-1) and (1 - d(m)/m)^(-1) = 1 + i(m)/m. 0 < d(m)/m < 1 and d(m) is known as the nominal discount rate per unit time deductible every 1/m time unit and 0 < i(m)/m and i(m) is known as the nominal interest rate per unit time payable every 1/m time unit.

Note: At time, Tv, the sum of the values of payments of θj = lim m→∞ λ/m units of money in each case made every lim m→∞ 1/m time unit (each of these payments is guaranteed to be made) with the first payment made at time x and the last payment made at time x + t equals

x+t

∫x λe^(-δ(T - Tv))dT =

x+t

∫x λe^(δ(Tv - T))dT which, after integration, becomes

λe^(-δ(x - Tv))(1 - e^(-δt))/δ =

λe^(δ(Tv - x))(1 - e^(-δt))/δ. 0 ≤ x < x + t and 0 ≤ Tv.

If, however, a payment is only to be made if an individual is alive to receive it then the actuarial present value of this payment will come into play which equals what would be the present value of this payment if it were guaranteed to be made multiplied by the number of individuals alive, i.e. remaining, at the time this payment is to be made expressed as a fraction of the number of individuals alive, i.e. remaining, at the present. Put more formally, for an individual presently aged y, the actuarial present value of a payment of θj units of money to be made at age a, note that this payment will only be made if the individual presently aged y reaches age a, equals θj(1 - d(m)/m)^(m(a – y))la/ly = θj(1 + i(m)/m)^(m(y – a))la/ly. Tv, i.e. y, will either be 0 or a positive integer multiplied by 1/m,

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Tj, i.e. a, will either be 0 or a positive integer multiplied by 1/m and a will be ≥ y. And so if a = y then no compounding will take place in this formula and it will simply result in θj and if a > y then compound discount will take place in this formula with a – y being a positive integer multiplied by 1/m and (a – y) divided by 1/m, i.e. m(a – y), being the number of 1/m time units which a – y is, this being the number of compounds which will take place in this formula.

m will either be a positive integer (If m ≥ 2, including if m→∞, and if instead of being continuously re-counted, lT is only re-counted when T is a positive integer and if y is not 0 or a positive integer then ly will have to be interpolated, i.e. assumed, and If m ≥ 2, including if m→∞, and if instead of being continuously re-counted, lT is only re-counted when T is a positive integer and if a is not 0 or a positive integer then la will have to be interpolated, i.e. assumed. If m→∞ then θj(1 - d(m)/m)^(m(a – y))la/ly = θj(1 + i(m)/m)^(m(y – a))la/ly becomes θje^(-δ(a - y))la/ly = θje^(δ(y - a))la/ly) or 1/a positive integer.

Note: For an individual presently aged y, the sum of the actuarial present values of payments of θj = lim m→∞ ф/m units of money in each case to be made every lim m→∞ 1/m time unit with the first payment to be made at age x and the last payment to be made at age x + t, note that a particular payment will only be made if the individual presently aged y reaches the age at which this payment is to be made, equals

x+t

∫x (фe^(-δ(T - y))lT/ly)dT =

x+t

∫x (фe^(δ(y - T))lT/ly)dT. 0 ≤ y ≤ x < x + t.

A final note for this chapter;

At time, Tv, the sum of the values of payments of θj = λ/m units of money in each case made every 1/m time unit (each of these payments is guaranteed to be made) with the first payment made at time x and the last payment made at time x + t equals

mt

∑c=0 λ(1 - d(m)/m)^(m(x + c/m - Tv))/m =

mt

∑c=0 λ(1 + i(m)/m)^(m(Tv - x - c/m))/m which, after this geometric series is summed, becomes

λ(1 - d(m)/m)^(m(x - Tv))(1 - (1 - d(m)/m)^(mt + 1))/d(m) =

λ(1 + i(m)/m)^(m(Tv - x) + 1)(1 - (1 + i(m)/m)^(-mt - 1))/i(m).

Tj = x + c/m in each case and x + t will be > x.

If m→∞ then

mt

∑c=0 λ(1 - d(m)/m)^(m(x + c/m - Tv))/m =

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mt

∑c=0 λ(1 + i(m)/m)^(m(Tv - x - c/m))/m, which equals

λ(1 - d(m)/m)^(m(x - Tv))(1 - (1 - d(m)/m)^(mt + 1))/d(m) =

λ(1 + i(m)/m)^(m(Tv - x) + 1)(1 - (1 + i(m)/m)^(-mt - 1))/i(m), becomes

x+t


x+t

∫x λe^(δ(Tv - T))dT which equals

λe^(-δ(x - Tv))(1 - e^(-δt))/δ =

λe^(δ(Tv - x))(1 - e^(-δt))/δ.

At time, Tv, the sum of the values of payments of θj = λ/m units of money in each case made every 1/m time unit (each of these payments is guaranteed to be made) with the first payment made at time x + 1/m and the last payment made at time x + t equals

mt

∑c=1 λ(1 - d(m)/m)^(m(x + c/m - Tv))/m =

mt

∑c=1 λ(1 + i(m)/m)^(m(Tv - x - c/m))/m which equals

λ(1 - d(m)/m)^(m(x - Tv))(1 - (1 - d(m)/m)^(mt + 1))/d(m) - λ(1 - d(m)/m)^(m(x + 0/m - Tv))/m =

λ(1 + i(m)/m)^(m(Tv - x) + 1)(1 - (1 + i(m)/m)^(-mt - 1))/i(m) - λ(1 + i(m)/m)^(m(Tv - x - 0/m))/m which equals

λ(1 - d(m)/m)^(m(x - Tv))(1 - (1 - d(m)/m)^(mt + 1))/d(m) - λ(1 - d(m)/m)^(m(x - Tv))/m =

λ(1 + i(m)/m)^(m(Tv - x) + 1)(1 - (1 + i(m)/m)^(-mt - 1))/i(m) - λ(1 + i(m)/m)^(m(Tv - x))/m which equals

λ(1 - d(m)/m)^(m(x - Tv) + 1)(1 - (1 - d(m)/m)^(mt))/d(m) =

λ(1 + i(m)/m)^(m(Tv - x))(1 - (1 + i(m)/m)^(-mt))/i(m).


If m→∞ then

mt

∑c=1 λ(1 - d(m)/m)^(m(x + c/m - Tv))/m =

mt


λ(1 - d(m)/m)^(m(x - Tv) + 1)(1 - (1 - d(m)/m)^(mt))/d(m) =

λ(1 + i(m)/m)^(m(Tv - x))(1 - (1 + i(m)/m)^(-mt))/i(m), becomes

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x+t


x+t


λe^(-δ(x - Tv))(1 - e^(-δt))/δ =

λe^(δ(Tv - x))(1 - e^(-δt))/δ.

At time, Tv, the sum of the values of payments of θj = λ/m units of money in each case made every 1/m time unit (each of these payments is guaranteed to be made) with the first payment made at time x and the last payment made at time x + t – 1/m equals

mt-1

∑c=0 λ(1 - d(m)/m)^(m(x + c/m - Tv))/m =

mt-1

∑c=0 λ(1 + i(m)/m)^(m(Tv - x - c/m))/m which equals

λ(1 - d(m)/m)^(m(x - Tv))(1 - (1 - d(m)/m)^(mt + 1))/d(m) - λ(1 - d(m)/m)^(m(x + mt/m - Tv))/m =

λ(1 + i(m)/m)^(m(Tv - x) + 1)(1 - (1 + i(m)/m)^(-mt - 1))/i(m) - λ(1 + i(m)/m)^(m(Tv - x - mt/m))/m which equals

λ(1 - d(m)/m)^(m(x - Tv))(1 - (1 - d(m)/m)^(mt + 1))/d(m) - λ(1 - d(m)/m)^(m(x + t - Tv))/m =

λ(1 + i(m)/m)^(m(Tv - x) + 1)(1 - (1 + i(m)/m)^(-mt - 1))/i(m) - λ(1 + i(m)/m)^(m(Tv - x - t))/m which equals

λ(1 - d(m)/m)^(m(x - Tv))(1 - (1 - d(m)/m)^(mt))/d(m) =

λ(1 + i(m)/m)^(m(Tv - x) + 1)(1 - (1 + i(m)/m)^(-mt))/i(m).


If m→∞ then

mt-1

∑c=0 λ(1 - d(m)/m)^(m(x + c/m - Tv))/m =

mt-1


λ(1 - d(m)/m)^(m(x - Tv))(1 - (1 - d(m)/m)^(mt))/d(m) =

λ(1 + i(m)/m)^(m(Tv - x) + 1)(1 - (1 + i(m)/m)^(-mt))/i(m), becomes

x+t


x+t


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λe^(-δ(x - Tv))(1 - e^(-δt))/δ =

λe^(δ(Tv - x))(1 - e^(-δt))/δ.

For an individual presently aged y, the sum of the actuarial present values of payments of θj = ф/m units of money in each case to be made every 1/m time unit with the first payment to be made at age x and the last payment to be made at age x + t, note that a particular payment will only be made if the individual presently aged y reaches the age at which this payment is to be made, equals

mt

∑c=0 ф(1 - d(m)/m)^(m(x + c/m - y))lx + c/m/mly =

mt

∑c=0 ф(1 + i(m)/m)^(m(y - x - c/m))lx + c/m/mly.

a = x + c/m in each case and x + t will be > x.

If m→∞ then

mt


mt

∑c=0 ф(1 + i(m)/m)^(m(y - x - c/m))lx + c/m/mly becomes

x+t


x+t

∫x (фe^(δ(y - T))lT/ly)dT.

For an individual presently aged y, the sum of the actuarial present values of payments of θj = ф/m units of money in each case to be made every 1/m time unit with the first payment to be made at age x + 1/m and the last payment to be made at age x + t, note that a particular payment will only be made if the individual presently aged y reaches the age at which this payment is to be made, equals

mt


mt



If m→∞ then

mt


mt


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x+t


x+t


For an individual presently aged y, the sum of the actuarial present values of payments of θj = ф/m units of money in each case to be made every 1/m time unit with the first payment to be made at age x and the last payment to be made at age x + t – 1/m, note that a particular payment will only be made if the individual presently aged y reaches the age at which this payment is to be made, equals

mt-1


mt-1



If m→∞ then

mt-1


mt-1


x+t


x+t


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Chapter Three

Expectations

At the present, i.e. at time, x, the time remaining until the beginning of the 1/r time unit in which time, T, is equals int(r(T – x))/r. x will either be 0 or a positive integer multiplied by 1/r and T will be ≥ x. (T – x) divided by 1/r, i.e. r(T – x), is the number of 1/r time units which T – x is and int(r(T – x)) is this number of 1/r time units rounded down to the lower integer and int(r(T – x)) multiplied by 1/r, i.e. int(r(T – x))/r, is T – x rounded down to the lower 1/r time unit.

And for an individual presently aged x, the expectation of int(r(T – x))/r between the ages x and x + t is the expected time remaining until the beginning of the 1/r time unit in which death occurs between the ages x and x + t. x will either be 0 or a positive integer multiplied by 1/r and x + t will be a positive integer multiplied by 1/r and x + t will be > x.

And for an individual presently aged x, the expectation of int(r(T – x))/r between the ages x and x + t equals

x+t

∫x (-(int(r(T – x))/r)l’T/lx)dT which, expressed as

rt-1 x+c/r+1/r

∑c=0 ∫x+c/r (-(int(r(T – x))/r)l’T/lx)dT, becomes

rt-1 x+c/r+1/r

∑c=0 ∫x+c/r (-(c/r)l’T/lx)dT because int(r(T – x)) equals c.

And this equals

rt-1 x+c/r+1/r

∑c=0 c ∫x+c/r (-l’T/rlx)dT which equals

rt-1

∑c=0 c(lx + c/r – lx + c/r + 1/r)/rlx which equals

rt-1

∑c=0 clx + c/r/rlx –

rt-1

∑c=0 clx + c/r + 1/r/rlx which equals

-(rt – 1)lx+t/rlx +

rt-1

∑c=1 lx + c/r/rlx which equals

lx+t/rlx – rtlx+t/rlx +

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rt-1


-rtlx+t/rlx +

rt


-tlx+t/lx +

rt

∑c=1 lx + c/r/rlx.

Adding

tlx+t/lx to this answer results in

rt


r will either be a positive integer (If r ≥ 2, including if r→∞, and if instead of being continuously re-counted, lT is only re-counted when T is a positive integer and if x is not 0 or a positive integer then lx will have to be interpolated, i.e. assumed, and If r ≥ 2, including if r→∞, and if instead of being continuously re-counted, lT is only re-counted when T is a positive integer and if x + c/r is not a positive integer then lx+c/r will have to be interpolated, i.e. assumed. If r→∞ then int(r(T – x))/r becomes r(T – x)/r and equals T - x) or 1/a positive integer.

Because at the present, i.e. at time, x, the time remaining until the beginning of the 1/r time unit in which time, T, is equals int(r(T – x))/r, at the present, i.e. at time, x, the time remaining until the beginning of the 1/r time unit in which time, T + 1/r, is, this being the end of the 1/r time unit in which time, T, is, equals int(r(T + 1/r – x))/r which equals int(r(T – x) + 1)/r which equals (int(r(T – x)) + 1)/r. x will either be 0 or a positive integer multiplied by 1/r and T will be ≥ x. (T – x) divided by 1/r, i.e. r(T – x), is the number of 1/r time units which T – x is and int(r(T – x)) is this number of 1/r time units rounded down to the lower integer and int(r(T – x)) + 1 is it rounded up to the higher integer and (int(r(T – x)) + 1) multiplied by 1/r, i.e. (int(r(T – x)) + 1)/r, is T – x rounded up to the higher 1/r time unit.

And for an individual presently aged x, the expectation of (int(r(T – x)) + 1)/r between the ages x and x + t is the expected time remaining until the end of the 1/r time unit in which death occurs between the ages x and x + t. x will either be 0 or a positive integer multiplied by 1/r and x + t will be a positive integer multiplied by 1/r and x + t will be > x.

And for an individual presently aged x, the expectation of (int(r(T – x)) + 1)/r between the ages x and x + t equals

x+t

∫x (-((int(r(T – x)) + 1)/r)l’T/lx)dT which, expressed as

rt-1 x+c/r+1/r

∑c=0 ∫x+c/r (-((int(r(T – x)) + 1)/r)l’T/lx)dT, becomes

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rt-1 x+c/r+1/r

∑c=0 ∫x+c/r (-((c + 1)/r)l’T/lx)dT because int(r(T – x)) equals c.

And this equals

rt-1 x+c/r+1/r

∑c=0 (c + 1) ∫x+c/r (-l’T/rlx)dT which equals

rt-1

∑c=0 (c + 1)(lx + c/r – lx + c/r + 1/r)/rlx which equals

rt-1

∑c=0 (c + 1)lx + c/r/rlx –

rt-1

∑c=0 (c + 1)lx + c/r + 1/r/rlx which equals

–rtlx+t/rlx +

rt-1


-tlx+t/lx +

rt-1


Adding


rt-1


r will either be a positive integer (If r ≥ 2, including if r→∞, and if instead of being continuously re-counted, lT is only re-counted when T is a positive integer and if x is not 0 or a positive integer then lx will have to be interpolated, i.e. assumed, and If r ≥ 2, including if r→∞, and if instead of being continuously re-counted, lT is only re-counted when T is a positive integer and if x + c/r is not 0 or a positive integer then lx+c/r will have to be interpolated, i.e. assumed. If r→∞ then (int(r(T – x)) + 1)/r, i.e. int(r(T + 1/r – x))/r, becomes r(T – x)/r and equals T - x) or 1/a positive integer.

At the present, i.e. at time, x, the value of a payment of θj = Ψ units of money which is guaranteed to be made at the beginning of the 1/m time unit in which time, T, is equals Ψ(1 - d(m)/m)^int(m(T – x)). x will either be 0 or a positive integer multiplied by 1/m and T will be ≥ x. If T < x + 1/m then no compounding will take place in this formula and it will simply result in Ψ otherwise compound discount will take place in this formula with (T – x) divided by 1/m, i.e. m(T – x), being the number of 1/m time units which T – x is and int(m(T – x)) being this number of 1/m time units rounded down to the lower integer, this being the number of compounds which will take place in this formula.

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And for an individual presently aged x, the expectation of Ψ(1 - d(m)/m)înt(m(T – x)) between the ages x and x + t is the expected present value of a payment of θj = Ψ units of money to be theoretically made at the beginning of the 1/m time unit in which death occurs between the ages x and x + t, this being the cost of insuring an individual presently aged x for θj = Ψ units of money to be theoretically paid at the beginning of the 1/m time unit in which death occurs between the ages x and x + t. x will either be 0 or a positive integer multiplied by 1/m and x + t will be a positive integer multiplied by 1/m and x + t will be > x.

And for an individual presently aged x, the expectation of Ψ(1 - d(m)/m)înt(m(T – x)) between the ages x and x + t equals

x+t

∫x (-Ψ(1 - d(m)/m)înt(m(T – x))l’T/lx)dT which, expressed as

mt-1 x+c/m+1/m

∑c=0 ∫x+c/m (-Ψ(1 - d(m)/m)înt(m(T – x))l’T/lx)dT, becomes

mt-1 x+c/m+1/m

∑c=0 ∫x+c/m (-Ψ(1 - d(m)/m)^cl’T/lx)dT because int(m(T – x)) equals c.

And this equals

mt-1 x+c/m+1/m

∑c=0 Ψ(1 - d(m)/m)^c ∫x+c/m (-l’T/lx)dT which equals

mt-1

∑c=0 Ψ(1 - d(m)/m)^c(lx + c/m – lx + c/m + 1/m)/lx which equals

mt-1

∑c=0 Ψ(1 - d(m)/m)^clx + c/m/lx –

mt-1

∑c=0 Ψ(1 - d(m)/m)^clx + c/m + 1/m/lx which, expressed as

mt-1

∑c=0 (1 - d(m)/m)Ψ(1 - d(m)/m)^(c – 1)lx + c/m/lx –

mt-1

∑c=0 Ψ(1 - d(m)/m)^clx + c/m + 1/m/lx, becomes

Ψ(1 - (1 - d(m)/m)^(mt – 1)lx+t/lx –

mt-1

∑c=1 d(m)(1 - d(m)/m)^(c – 1)lx + c/m/mlx) which, expressed as

Ψ(1 – (d(m)/m + 1 - d(m)/m)(1 - d(m)/m)^(mt – 1)lx+t/lx –

mt-1

∑c=1 d(m)(1 - d(m)/m)^(c – 1)lx + c/m/mlx), becomes

Ψ(1 – d(m)(1 - d(m)/m)^(mt – 1)lx+t/mlx – (1 - d(m)/m)(1 - d(m)/m)^(mt – 1)lx+t/lx –

17

mt-1

∑c=1 d(m)(1 - d(m)/m)^(c – 1)lx + c/m/mlx) which equals

Ψ(1 – (1 - d(m)/m)^(mt)lx+t/lx –

mt

∑c=1 d(m)(1 - d(m)/m)^(c – 1)lx + c/m/mlx) which, because 1 - d(m)/m = (1 + i(m)/m)^(-1), can be expressed as

Ψ(1 – (1 + i(m)/m)^(-mt)lx+t/lx –

mt

∑c=1 i(m)(1 + i(m)/m)^(-c)lx + c/m/mlx).

m will either be a positive integer (If m ≥ 2, including if m→∞, and if instead of being continuously re-counted, lT is only re-counted when T is a positive integer and if x is not 0 or a positive integer then lx will have to be interpolated, i.e. assumed, and If m ≥ 2, including if m→∞, and if instead of being continuously re-counted, lT is only re-counted when T is a positive integer and if x + c/m is not a positive integer then lx+c/m will have to be interpolated, i.e. assumed, and If m ≥ 2, including if m→∞, and if instead of being continuously re-counted, lT is only re-counted when T is a positive integer and if x + t is not a positive integer then lx+t will have to be interpolated, i.e. assumed. If m→∞ then Ψ(1 - d(m)/m)înt(m(T – x)) becomes Ψ(1 - d(m)/m)^(m(T – x)) and, because m→∞, equals Ψe^(-δ(T - x))) or 1/a positive integer.

Because at the present, i.e. at time, x, the value of a payment of θj = Ψ units of money which is guaranteed to be made at the beginning of the 1/m time unit in which time, T, is equals Ψ(1 - d(m)/m)înt(m(T – x)), at the present, i.e. at time, x, the value of a payment of θj = Ψ units of money which is guaranteed to be made at the beginning of the 1/m time unit in which time, T + 1/m, is, this being the end of the 1/m time unit in which time, T, is, equals Ψ(1 - d(m)/m)înt(m(T + 1/m – x)) which equals Ψ(1 - d(m)/m)înt(m(T – x) + 1) which equals Ψ(1 - d(m)/m)^(int(m(T – x)) + 1). x will either be 0 or a positive integer multiplied by 1/m and T will be ≥ x. Compound discount will take place in this formula with (T – x) divided by 1/m, i.e. m(T – x), being the number of 1/m time units which T – x is and int(m(T – x)) being this number of 1/m time units rounded down to the lower integer and int(m(T – x)) + 1 being it rounded up to the higher integer, this being the number of compounds which will take place in this formula.

And for an individual presently aged x, the expectation of Ψ(1 - d(m)/m)^(int(m(T – x)) + 1) between the ages x and x + t is the expected present value of a payment of θj = Ψ units of money to be made at the end of the 1/m time unit in which death occurs between the ages x and x + t, this being the cost of insuring an individual presently aged x for θj = Ψ units of money to be paid at the end of the 1/m time unit in which death occurs between the ages x and x + t. x will either be 0 or a positive integer multiplied by 1/m and x + t will be a positive integer multiplied by 1/m and x + t will be > x.

And for an individual presently aged x, the expectation of Ψ(1 - d(m)/m)^(int(m(T – x)) + 1) between the ages x and x + t equals

x+t

∫x (-Ψ(1 - d(m)/m)^(int(m(T – x)) + 1)l’T/lx)dT which, expressed as

18

mt-1 x+c/m+1/m

∑c=0 ∫x+c/m (-Ψ(1 - d(m)/m)^(int(m(T – x)) + 1)l’T/lx)dT, becomes

mt-1 x+c/m+1/m

∑c=0 ∫x+c/m (-Ψ(1 - d(m)/m)^(c + 1)l’T/lx)dT because int(m(T – x)) equals c.

And this equals

mt-1 x+c/m+1/m

∑c=0 Ψ(1 - d(m)/m)^(c + 1) ∫x+c/m (-l’T/lx)dT which equals

mt-1

∑c=0 Ψ(1 - d(m)/m)^(c + 1)(lx + c/m – lx + c/m + 1/m)/lx which equals

mt-1

∑c=0 Ψ(1 - d(m)/m)^(c + 1)lx + c/m/lx –

mt-1

∑c=0 Ψ(1 - d(m)/m)^(c + 1)lx + c/m + 1/m/lx which, expressed as

mt-1

∑c=0 (1 - d(m)/m)Ψ(1 - d(m)/m)^clx + c/m/lx –

mt-1

∑c=0 Ψ(1 - d(m)/m)^(c + 1)lx + c/m + 1/m/lx, becomes

Ψ(1 – (1 - d(m)/m)^(mt)lx+t/lx –

mt-1

∑c=0 d(m)(1 - d(m)/m)^clx + c/m/mlx) which, because 1 - d(m)/m = (1 + i(m)/m)^(-1), can be expressed as

Ψ(1 – (1 + i(m)/m)^(-mt)lx+t/lx –

mt-1

∑c=0 i(m)(1 + i(m)/m)^(-c - 1)lx + c/m/mlx).

m will either be a positive integer (If m ≥ 2, including if m→∞, and if instead of being continuously re-counted, lT is only re-counted when T is a positive integer and if x is not 0 or a positive integer then lx will have to be interpolated, i.e. assumed, and If m ≥ 2, including if m→∞, and if instead of being continuously re-counted, lT is only re-counted when T is a positive integer and if x + c/m is not 0 or a positive integer then lx+c/m will have to be interpolated, i.e. assumed, and If m ≥ 2, including if m→∞, and if instead of being continuously re-counted, lT is only re-counted when T is a positive integer and if x + t is not a positive integer then lx+t will have to be interpolated, i.e. assumed. If m→∞ then Ψ(1 - d(m)/m)^(int(m(T – x)) + 1), i.e. Ψ(1 - d(m)/m)^int(m(T + 1/m – x)), becomes Ψ(1 - d(m)/m)^(m(T – x)) and, because m→∞, equals Ψe^(-δ(T - x))) or 1/a positive integer.

At the present, i.e. at time, x, the time remaining until time, T, equals T – x. 0 ≤ x ≤ T.

And for an individual presently aged x, the expectation of T – x between the ages x and x + t is the expected time remaining until death between the ages x and x + t. 0 ≤ x < x + t.

19

And for an individual presently aged x, the expectation of T – x between the ages x and x + t equals

x+t

∫x (-(T - x)l’T/lx)dT which equals

x+t

∫x ((x - T)l’T/lx)dT which, integrating by parts with u(T) = x – T, v’(T) = l’T/lx, u’(T) = – 1 and v(T) = lT/lx, becomes

x+t

[(x - T)lT/lx] x +

x+t

∫x (lT/lx)dT which equals

(x – (x + t))lx+t/lx – (x – x)lx/lx +

x+t


-tlx+t/lx – 0 +

x+t


-tlx+t/lx +

x+t

∫x (lT/lx)dT.

Adding


x+t

∫x (lT/lx)dT.

At the present, i.e. at time, x, the value of a payment of θj = Ψ units of money which is guaranteed to be made at time, T, equals Ψe^(-δ(T - x)). 0 ≤ x ≤ T.

And for an individual presently aged x, the expectation of Ψe^(-δ(T - x)) between the ages x and x + t is the expected present value of a payment of θj = Ψ units of money to be made at death between the ages x and x + t, this being the cost of insuring an individual presently aged x for θj = Ψ units of money to be paid at death between the ages x and x + t. 0 ≤ x < x + t.

And for an individual presently aged x, the expectation of Ψe^(-δ(T - x)) between the ages x and x + t equals

x+t

∫x (-Ψe^(-δ(T - x))l’T/lx)dT which, integrating by parts with u(T) = - Ψe^(-δ(T - x)), v’(T) = l’T/lx, u’(T) = Ψδe^(-δ(T - x)) and v(T) = lT/lx, becomes

x+t

[-Ψe^(-δ(T - x))lT/lx] x -

20

x+t

∫x (Ψδe^(-δ(T - x))lT/lx)dT which equals

- Ψe^(-δ(x + t - x))lx+t/lx – (-Ψe^(-δ(x - x))lx/lx) -

x+t


Ψ - Ψe^(-δt)lx+t/lx -

x+t


Ψ(1 - e^(-δt)lx+t/lx -

x+t

∫x (δe^(-δ(T - x))lT/lx)dT) which can be expressed as


x+t

∫x (δe^(δ(x - T))lT/lx)dT).

For an individual presently aged y, the actuarial present value of the cost of insuring an individual aged x for θj = Ψ units of money to be theoretically paid at the beginning of the 1/m time unit in which death occurs between the ages x and x + t, note that this cost will only be met if the individual presently aged y reaches age x, equals

this cost multiplied by (1 - d(m)/m)^(m(x – y))lx/ly =

this cost multiplied by (1 + i(m)/m)^(m(y – x))lx/ly and so equals

Ψ(1 – (1 - d(m)/m)^(mt)lx+t/lx –

mt

∑c=1 d(m)(1 - d(m)/m)^(c – 1)lx + c/m/mlx) multiplied by (1 - d(m)/m)^(m(x – y))lx/ly =

Ψ(1 – (1 + i(m)/m)^(-mt)lx+t/lx –

mt

∑c=1 i(m)(1 + i(m)/m)^(-c)lx + c/m/mlx) multiplied by (1 + i(m)/m)^(m(y – x))lx/ly which equals

Ψ((1 - d(m)/m)^(m(x – y))lx/ly – (1 - d(m)/m)^(m(x + t – y))lx+t/ly –

mt

∑c=1 d(m)(1 - d(m)/m)^(m(x + (c – 1)/m – y))lx + c/m/mly) =

Ψ((1 + i(m)/m)^(m(y – x))lx/ly – (1 + i(m)/m)^(m(y - x – t))lx+t/ly –

mt

∑c=1 i(m)(1 + i(m)/m)^(m(y – x – c/m))lx + c/m/mly).

21

For an individual presently aged y, the actuarial present value of the cost of insuring an individual aged x for θj = Ψ units of money to be paid at the end of the 1/m time unit in which death occurs between the ages x and x + t, note that this cost will only be met if the individual presently aged y reaches age x, equals

this cost multiplied by (1 - d(m)/m)^(m(x – y))lx/ly =

this cost multiplied by (1 + i(m)/m)^(m(y – x))lx/ly and so equals

Ψ(1 – (1 - d(m)/m)^(mt)lx+t/lx –

mt-1

∑c=0 d(m)(1 - d(m)/m)^clx + c/m/mlx) multiplied by (1 - d(m)/m)^(m(x – y))lx/ly =

Ψ(1 – (1 + i(m)/m)^(-mt)lx+t/lx –

mt-1

∑c=0 i(m)(1 + i(m)/m)^(-c - 1)lx + c/m/mlx) multiplied by (1 + i(m)/m)^(m(y – x))lx/ly which equals

Ψ((1 - d(m)/m)^(m(x – y))lx/ly – (1 - d(m)/m)^(m(x + t – y))lx+t/ly –

mt-1

∑c=0 d(m)(1 - d(m)/m)^(m(x + c/m – y))lx + c/m/mly) =

Ψ((1 + i(m)/m)^(m(y – x))lx/ly – (1 + i(m)/m)^(m(y - x – t))lx+t/ly –

mt-1

∑c=0 i(m)(1 + i(m)/m)^(m(y – x – (c + 1)/m))lx + c/m/mly).

For an individual presently aged y, the actuarial present value of the cost of insuring an individual aged x for θj = Ψ units of money to be paid at death between the ages x and x + t, note that this cost will only be met if the individual presently aged y reaches age x, equals

this cost multiplied by e^(-δ(x - y))lx/ly =

this cost multiplied by e^(δ(y - x))lx/ly and so equals


x+t

∫x (δe^(-δ(T - x))lT/lx)dT) multiplied by e^(-δ(x - y))lx/ly =


x+t

∫x (δe^(δ(x - T))lT/lx)dT) multiplied by e^(δ(y - x))lx/ly which equals

Ψ(e^(-δ(x - y))lx/ly - e^(-δ(x + t – y))lx+t/ly -

x+t

∫x (δe^(-δ(T - y))lT/ly)dT) =

Ψ(e^(δ(y - x))lx/ly - e^(δ(y – x - t))lx+t/ly -

22

x+t

∫x (δe^(δ(y - T))lT/ly)dT).

To finish this chapter, and this book, here is the following;

For an individual presently aged x, the expectation of (ф(1 - d(m)/m)/d(m))(1 - (1 - d(m)/m)înt(m(T - x))) between the ages x and x + t (x will either be 0 or a positive integer multiplied by 1/m and x + t will be a positive integer multiplied by 1/m and x + t will be > x) equals

x+t

∫x (-(ф(1 - d(m)/m)/d(m))(1 - (1 - d(m)/m)înt(m(T - x)))l’T/lx)dT which equals

(ф(1 - d(m)/m)/d(m)) multiplied by

x+t

∫x (-(1 - (1 - d(m)/m)înt(m(T - x)))l’T/lx)dT which equals


x+t

(∫x (-l’T/lx)dT –

x+t

∫x (-(1 - d(m)/m)înt(m(T - x))l’T/lx)dT) which equals


x+t

([-lT/lx] x –

x+t



(- lx+t/lx + lx/lx –

x+t



(1 - lx+t/lx –

x+t

∫x (-(1 - d(m)/m)înt(m(T - x))l’T/lx)dT). It can be seen from earlier that the integral,

x+t

∫x (-(1 - d(m)/m)înt(m(T - x))l’T/lx)dT, equals

1 – (1 - d(m)/m)^(mt)lx+t/lx –

mt

∑c=1 d(m)(1 - d(m)/m)^(c – 1)lx + c/m/mlx meaning that

23


(1 - lx+t/lx –

x+t

∫x (-(1 - d(m)/m)^int(m(T - x))l’T/lx)dT) equals


(1 - lx+t/lx – (1 – (1 - d(m)/m)^(mt)lx+t/lx –

mt

∑c=1 d(m)(1 - d(m)/m)^(c – 1)lx + c/m/mlx)) which equals


(1 - lx+t/lx – 1 + (1 - d(m)/m)^(mt)lx+t/lx +

mt



( - lx+t/lx + (1 - d(m)/m)^(mt)lx+t/lx +

mt



(( - 1 + (1 - d(m)/m)^(mt))lx+t/lx +

mt



(-(1 - (1 - d(m)/m)^(mt))lx+t/lx +

mt


-ф(1 - d(m)/m)(1 - (1 - d(m)/m)^(mt))lx+t/d(m)lx +

mt

∑c=1 ф(1 - d(m)/m)^clx + c/m/mlx which, because 1 - d(m)/m = (1 + i(m)/m)^(-1), can be expressed as

-ф(1 - (1 + i(m)/m)^(-mt))lx+t/i(m)lx +

24

mt

∑c=1 ф(1 + i(m)/m)^(-c)lx + c/m/mlx.

Adding

ф(1 - d(m)/m)(1 - (1 - d(m)/m)^(mt))lx+t/d(m)lx =

ф(1 - (1 + i(m)/m)^(-mt))lx+t/i(m)lx to this answer results in

mt

∑c=1 ф(1 - d(m)/m)^clx + c/m/mlx =

mt

∑c=1 ф(1 + i(m)/m)^(-c)lx + c/m/mlx, this being, for an individual presently aged x, the sum of the actuarial present values of payments of θj = ф/m units of money in each case to be made every 1/m time unit with the first payment to be made at age x + 1/m and the last payment to be made at age x + t, note that a particular payment will only be made if the individual presently aged x reaches the age at which this payment is to be made.

m will either be a positive integer (If m ≥ 2, including if m→∞, and if instead of being continuously re-counted, lT is only re-counted when T is a positive integer and if x is not 0 or a positive integer then lx will have to be interpolated, i.e. assumed, and If m ≥ 2, including if m→∞, and if instead of being continuously re-counted, lT is only re-counted when T is a positive integer and if x + c/m is not a positive integer then lx+c/m will have to be interpolated, i.e. assumed. If m→∞ then (ф(1 - d(m)/m)/d(m))(1 - (1 - d(m)/m)^int(m(T – x))) becomes (ф(1 - d(m)/m)/d(m))(1 - (1 - d(m)/m)^(m(T – x))) and, because m→∞, equals (ф/δ)(1 - e^(-δ(T - x)))) or 1/a positive integer.

For an individual presently aged x, the expectation of (ф/d(m))(1 - (1 - d(m)/m)^(int(m(T - x)) + 1)) between the ages x and x + t (x will either be 0 or a positive integer multiplied by 1/m and x + t will be a positive integer multiplied by 1/m and x + t will be > x) equals

x+t

∫x (-(ф/d(m))(1 - (1 - d(m)/m)^(int(m(T - x)) + 1))l’T/lx)dT which equals

(ф/d(m)) multiplied by

x+t

∫x (-(1 - (1 - d(m)/m)^(int(m(T - x)) + 1))l’T/lx)dT which equals


x+t


x+t

∫x (-(1 - d(m)/m)^(int(m(T - x)) + 1)l’T/lx)dT) which equals


x+t

([-lT/lx] x –

25

x+t




x+t



(1 - lx+t/lx –

x+t

∫x (-(1 - d(m)/m)^(int(m(T - x)) + 1)l’T/lx)dT). It can be seen from earlier that the integral,

x+t

∫x (-(1 - d(m)/m)^(int(m(T - x)) + 1)l’T/lx)dT, equals

1 – (1 - d(m)/m)^(mt)lx+t/lx –

mt-1

∑c=0 d(m)(1 - d(m)/m)^clx + c/m/mlx meaning that


(1 - lx+t/lx –

x+t

∫x (-(1 - d(m)/m)^(int(m(T - x)) + 1)l’T/lx)dT) equals


(1 - lx+t/lx – (1 – (1 - d(m)/m)^(mt)lx+t/lx –

mt-1

∑c=0 d(m)(1 - d(m)/m)^clx + c/m/mlx)) which equals


(1 - lx+t/lx – 1 + (1 - d(m)/m)^(mt)lx+t/lx +

mt-1

∑c=0 d(m)(1 - d(m)/m)^clx + c/m/mlx) which equals


( - lx+t/lx + (1 - d(m)/m)^(mt)lx+t/lx +

26

mt-1



(( - 1 + (1 - d(m)/m)^(mt))lx+t/lx +

mt-1



(-(1 - (1 - d(m)/m)^(mt))lx+t/lx +

mt-1


-ф(1 - (1 - d(m)/m)^(mt))lx+t/d(m)lx +

mt-1

∑c=0 ф(1 - d(m)/m)^clx + c/m/mlx which, because 1 - d(m)/m = (1 + i(m)/m)^(-1), can be expressed as

-ф(1 + i(m)/m)(1 - (1 + i(m)/m)^(-mt))lx+t/i(m)lx +

mt-1

∑c=0 ф(1 + i(m)/m)^(-c)lx + c/m/mlx.

Adding

ф(1 - (1 - d(m)/m)^(mt))lx+t/d(m)lx =

ф(1 + i(m)/m)(1 - (1 + i(m)/m)^(-mt))lx+t/i(m)lx to this answer results in

mt-1

∑c=0 ф(1 - d(m)/m)^clx + c/m/mlx =

mt-1

∑c=0 ф(1 + i(m)/m)^(-c)lx + c/m/mlx, this being, for an individual presently aged x, the sum of the actuarial present values of payments of θj = ф/m units of money in each case to be made every 1/m time unit with the first payment to be made at age x and the last payment to be made at age x + t – 1/m, note that a particular payment will only be made if the individual presently aged x reaches the age at which this payment is to be made.

m will either be a positive integer (If m ≥ 2, including if m→∞, and if instead of being continuously re-counted, lT is only re-counted when T is a positive integer and if x is not 0 or a positive integer then lx will have to be interpolated, i.e. assumed, and If m ≥ 2, including if m→∞, and if instead of being continuously re-counted, lT is only re-counted when T is a positive integer and if x + c/m is not 0 or a positive integer then lx+c/m will have to be interpolated, i.e. assumed. If m→∞ then (ф/d(m))(1 -

27

(1 - d(m)/m)^(int(m(T - x)) + 1)), i.e. (ф/d(m))(1 - (1 - d(m)/m)^int(m(T + 1/m – x))), becomes (ф/d(m))(1 - (1 - d(m)/m)^(m(T - x))) and, because m→∞, equals (ф/δ)(1 - e^(-δ(T - x)))) or 1/a positive integer.

For an individual presently aged x, the expectation of (ф/δ)(1 - e^(-δ(T - x))) between the ages x and x + t (0 ≤ x < x + t) equals

x+t

∫x (-(ф/δ)(1 - e^(-δ(T - x)))l’T/lx)dT which equals

(ф/δ) multiplied by

x+t

∫x (-(1 - e^(-δ(T - x)))l’T/lx)dT which equals


x+t


x+t

∫x (-e^(-δ(T - x))l’T/lx)dT) which equals


x+t

([-lT/lx] x –

x+t




x+t



(1 - lx+t/lx –

x+t

∫x (-e^(-δ(T - x))l’T/lx)dT). It can be seen from earlier that the integral,

x+t

∫x (-e^(-δ(T - x))l’T/lx)dT, equals

1 - e^(-δt)lx+t/lx -

x+t

∫x (δe^(-δ(T - x))lT/lx)dT meaning that


28

(1 - lx+t/lx –

x+t

∫x (-e^(-δ(T - x))l’T/lx)dT) equals


(1 - lx+t/lx – (1 – e^(-δt)lx+t/lx –

x+t

∫x (δe^(-δ(T - x))lT/lx)dT)) which equals


(1 - lx+t/lx – 1 + e^(-δt)lx+t/lx +

x+t

∫x (δe^(-δ(T - x))lT/lx)dT) which equals


( - lx+t/lx + e^(-δt)lx+t/lx +

x+t



(( - 1 + e^(-δt))lx+t/lx +

x+t



(-(1 - e^(-δt))lx+t/lx +

x+t


- ф(1 - e^(-δt))lx+t/δlx +

x+t

∫x (фe^(-δ(T - x))lT/lx)dT which can be expressed as

- ф(1 - e^(-δt))lx+t/δlx +

x+t

∫x (фe^(δ(x - T))lT/lx)dT.

Adding

29

ф(1 - e^(-δt))lx+t/δlx to this answer results in

x+t

∫x (фe^(-δ(T - x))lT/lx)dT =

x+t

∫x (фe^(δ(x - T))lT/lx)dT, this being, for an individual presently aged x, the sum of the actuarial present values of payments of θj = lim m→∞ ф/m units of money in each case to be made every lim m→∞ 1/m time unit with the first payment to be made at age x and the last payment to be made at age x + t, note that a particular payment will only be made if the individual presently aged x reaches the age at which this payment is to be made.

END

Some Fundamental Actuarial Building Blocks for the Curious

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Transcript of Some Fundamental Actuarial Building Blocks for the Curious