arithmetic_variation_insurances: Life insurance with arithmetic-variation benefit (increasing/decreasing, fractional claims)

Description

This help page groups two term life insurances with **arithmetic variation** of the benefit, both with optional deferment m and k fractional claim times (claims at end of subperiods):

IAxn: increasing arithmetic term insurance (benefit grows linearly with time);
DAxn: decreasing arithmetic term insurance (benefit declines linearly with time).

Usage

IAxn(
  actuarialtable,
  x,
  n,
  i = actuarialtable@interest,
  m = 0,
  k = 1,
  type = "EV",
  power = 1
)
DAxn(
  actuarialtable,
  x,
  n,
  i = actuarialtable@interest,
  m = 0,
  k = 1,
  type = "EV",
  power = 1
)

Value

A numeric value: the APV (or one simulated realization if type="ST").

Arguments

actuarialtable: A lifetable or actuarialtable object.
x: Attained age at inception.
n: Coverage length in years. If missing, it is set to getOmega(actuarialtable) - x - m.
i: Annual effective interest rate. Defaults to actuarialtable@interest.
m: Deferment (years). Default 0.
k: Fractional periods per year ($k \ge 1$). Default 1.
type: Output type: "EV" (expected value, default) or "ST" (one stochastic realization via rLifeContingencies).
power: Power applied to discounted cash flows before expectation (default 1).

<code>IAxn</code> — Increasing arithmetic term

Computes the APV of an n-year **increasing** term insurance on a life aged x, with k fractional claim times and optional deferment m. The benefit at the $j$-th subperiod equals $j/k$.

Fractional timing conventions

For **insurance** benefits in this package, fractional claims are assumed to occur at the **end** of each subperiod (i.e., at $t_j + 1/k$). This matches the implementation that multiplies survival to $t_j$ and a fractional death probability over the next subperiod: $$ v^{t_j + 1/k}\; {}_{t_j}p_x\; q_{x+t_j}^{(1/k)}. $$ By contrast, **annuities** use a payment-timing flag ("immediate" vs "due") which changes the evaluation times; insurance here has a fixed claim timing at end-subperiod.

<code>DAxn</code> — Decreasing arithmetic term

Computes the APV of an n-year **decreasing** term insurance on a life aged x, with k fractional claim times and optional deferment m. The benefit at the $j$-th subperiod equals $n - (j-1)/k$.

Details

Let $t_j = m + (j-1)/k$, $j=1,\dots,nk$. With **fractional claims at end of subperiods**, the EV implementations follow the pattern already used in Axn:

IAxn (increasing): $$ \mathrm{IA}_{\overline{n}|}^{(k)} = \sum_{j=1}^{nk} \Big(\frac{j}{k}\Big)\, v^{t_j + 1/k}\; {}_{t_j}p_x\; q_{x+t_j}^{(1/k)}, $$ where $v=(1+i)^{-1}$, computed via pxt(...) and qxt(..., t=1/k).

DAxn (decreasing) is analogous with benefit $(n - (j-1)/k)$; see its subsection below.

DAxn (decreasing): $$ \mathrm{DA}_{\overline{n}|}^{(k)} = \sum_{j=1}^{nk} \Big(n - \frac{j-1}{k}\Big)\, v^{t_j + 1/k}\; {}_{t_j}p_x\; q_{x+t_j}^{(1/k)}, \qquad t_j = m + \frac{j-1}{k}. $$ See “Fractional timing conventions” above for claim timing assumptions.

References

Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones, D. A., Nesbitt, C. J. (1997). Actuarial Mathematics, 2nd ed., SOA.

Examples

Run this code

## Setup (legacy examples)
data(soaLt)
soa08Act <- with(soaLt, new("actuarialtable", interest=0.06, x=x, lx=Ix, name="SOA2008"))

## IAxn: increasing arithmetic term, 10 years, age 25 (legacy)
IAxn(actuarialtable = soa08Act, x = 25, n = 10)

## More examples (k>1 and deferment)
IAxn(actuarialtable = soa08Act, x = 40, n = 20, k = 12)     # monthly claims
IAxn(actuarialtable = soa08Act, x = 40, n = 15, m = 5, k = 4) # deferred 5y, quarterly

## DAxn: decreasing arithmetic term, 10 years, age 25 (legacy)
DAxn(actuarialtable = soa08Act, x = 25, n = 10)
## More examples (k>1 and deferment)
DAxn(actuarialtable = soa08Act, x = 45, n = 10, k = 2)       # semiannual
DAxn(actuarialtable = soa08Act, x = 45, n = 12, m = 3, k = 12) # deferred 3y, monthly

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