02-level-annuity.Rmd

# Level Annuity

**Annuity immediate**

$$a_{\overline{n}\mid}=\frac{1-v^n}{i}$$
$$s_{\overline{n}\mid}=(1+i)^na_{\overline{n}\mid}=\frac{(1+i)^n-1}{i}$$

**Annuity due**

$$\ddot{a}_{\overline{n}\mid}=\frac{1-v^n}{d}=(1+i)a_{\overline{n}\mid}=1+a_{\overline{n-1}\mid}$$

$$\ddot{s}_{\overline{n}\mid}=(1+i)^n\ddot{a}_{\overline{n}\mid}$$


**Deffered annuity**

$$_{m|}a_{\overline{n}\mid}=v^ma_{\overline{n}\mid}=a_{\overline{m+n}\mid}-a_{\overline{n}\mid}$$

**Perpetuity**

$$a_{\overline{\infty}\mid}=\frac{1}{i}$$
$$\ddot{a}_{\overline{\infty}\mid}=\frac{1}{d}$$

**$m$-thly payable annuity**

$$a^{(m)}_{\overline{n}\mid}=\frac{1-v^n}{i^{(m)}}=\frac{i}{i^{(m)}}a_{\overline{n}\mid}$$
$$\ddot{a}^{(m)}_{\overline{n}\mid}=\frac{1-v^n}{d^{(m)}}=\frac{d}{d^{(m)}}{\ddot{a}}_{\overline{n}\mid}$$
$$a^{(m)}_{\overline{\infty}\mid}=\frac{1}{i^{(m)}}$$
$$\ddot{a}^{(m)}_{\overline{\infty}\mid}=\frac{1}{d^{(m)}}$$
$$\ddot{a}^{(m)}_{\overline{\infty}\mid}=(1+i)^{1/m}a^{(m)}_{\overline{\infty}\mid}$$

**Continuous payable annuity**

$$\bar{a}_{\overline{n}\mid}=\frac{i}{\delta}{a}_{\overline{n}\mid}=\frac{d}{\delta}\ddot{a}_{\overline{n}\mid}=\frac{1-v^n}{\delta}$$
$$\bar{a}_{\overline{\infty}\mid}=\frac{1}{\delta}$$

**Key relations**

```{r,echo=FALSE,out.width="50%",fig.align = 'center'}
knitr::opts_chunk$set(fig.pos = "!H", out.extra = "")
knitr::include_graphics("./plots/annuity-1.png")
```