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UDD annuity approximations

annuity_approximations_udd.Rd

UDD-based approximations for Chapter 8 annuity functions.

Computes $$\ddot{a}_x^{(m)} \approx \alpha(m)\ddot{a}_x - \beta(m).$$

Computes $$\ddot{a}_{x:\overline{n}|}^{(m)} \approx \alpha(m)\ddot{a}_{x:\overline{n}|} - \beta(m)(1-{}_nE_x).$$

Computes $${}_{n\mid}\ddot{a}_x^{(m)} \approx \alpha(m)\,{}_{n\mid}\ddot{a}_x - \beta(m)\,{}_nE_x.$$

Computes $$a_x^{(m)} \approx \alpha(m)a_x + \gamma(m).$$

Computes $$a_{x:\overline{n}|}^{(m)} \approx \alpha(m)a_{x:\overline{n}|} + \gamma(m)(1-{}_nE_x).$$

Computes $${}_{n\mid}a_x^{(m)} \approx \alpha(m)\,{}_{n\mid}a_x + \gamma(m)\,{}_nE_x.$$

Computes $$\ddot{s}_{x:\overline{n}|}^{(m)} \approx \alpha(m)\ddot{s}_{x:\overline{n}|} - \beta(m)\left(\frac{1}{{}_nE_x}-1\right).$$

Computes $$s_{x:\overline{n}|}^{(m)} \approx \alpha(m)s_{x:\overline{n}|} + \gamma(m)\left(\frac{1}{{}_nE_x}-1\right).$$

Computes $$\bar{a}_x \approx \frac{id}{\delta^2}\ddot{a}_x - \frac{i-\delta}{\delta^2}.$$

Uses the identity $$\bar{a}_{x:\overline{n}|} \approx \frac{1-\bar{A}_{x:\overline{n}|}}{\delta}$$ together with the package's existing Chapter 7 UDD insurance approximation for \(\bar{A}_{x:\overline{n}|}\).

Computes $${}_{n\mid}\bar{a}_x \approx {}_nE_x \, \bar{a}_{x+n}.$$

Usage

adotx_m_udd(x, m, i, model, ...)

adotxn_m_udd(x, n, m, i, model, ...)

nadotx_m_udd(x, n, m, i, model, ...)

ax_m_udd(x, m, i, model, ...)

axn_m_udd(x, n, m, i, model, ...)

nax_m_udd(x, n, m, i, model, ...)

sdotxn_m_udd(x, n, m, i, model, ...)

sxn_m_udd(x, n, m, i, model, ...)

abarx_udd(x, i, model, ...)

abarxn_udd(x, n, i, model, ...)

nabarx_udd(x, n, i, model, ...)

Arguments

x

Age.

m

Number of payments per year.

i

Effective annual interest rate.

model

Survival model.

...

Additional model parameters.

n

Term.

Value

Numeric vector.

Details

These functions implement the standard Uniform Distribution of Deaths approximations linking annual, m-thly, and continuous annuity values.

The exported functions documented on this page are:

  • ax_m_udd()

  • axn_m_udd()

  • nax_m_udd()

  • adotx_m_udd()

  • adotxn_m_udd()

  • nadotx_m_udd()

  • sxn_m_udd()

  • sdotxn_m_udd()

  • abarx_udd()

  • abarxn_udd()

  • nabarx_udd()

Note that this function relies on the already-existing Abarxn_udd() implementation in the package, so extra survival-model arguments are not used.