Annuities (Finance)
Annuity Immediate
An annuity immediate is a regular series of payments at the end of every period. The accumulated future value, at time $t=n$, of one unit of capital paying at the end of every period for $n$ periods is denoted by $S_{n|}$. The present value, at time $t=0$, of one unit of capital paying at the end of every period for $n$ periods is denoted by $a_{n|}$.
The accumulated value for an annuity immediate, $S_{n|}$, paying one unit of capital is given by: \[S_{n|}=\frac{(1+i)^n-1}{i}\] The present value for an annuity immediate, $a_{n|}$, paying one unit of capital is given by: \[a_{n|}=\frac{1-v^n}{i}\] The present value can also be written as: \[a_{n|}=v^n S_{n|}\] where $v=\frac{1}{1+i}$ (the discount factor)
Worked Example 1
Worked Example
What is the present and future value of an annuity immediate that pays $£200$ per year for five years at a rate of $6.5\%$?
Solution
\begin{align} &S_{5\vert}=250\times52 \frac{(1+0.05)^5-1}{0.04881306056}\approx£73,579.90\\ &a_{5\vert}=250\times52 \frac{~1-(1+0.05)^{-5}~}{0.04881306056}\approx£57,651.78. \end{align}
Video Example
Dr Graham Murphy solving an annuity immediate problem.
Annuity Due
An annuity due is a regular series of payments at the beginning of every period. The accumulated future value for one unit of capital is denoted by $\ddot S _{n|}$ whilst the present value is denoted by $\ddot a _{n|}$.
The accumulated value for an annuity due, $\ddot S _{n|}$, paying one unit of capital is given by: \[\ddot S _{n|}=\frac{(1+i)^n-1}{d}\] The present value for an annuity due, $\ddot a _{n|}$, paying one unit of capital is given by: \[\ddot a _{n|}=\frac{1-v^n}{d}\]
Worked Example 2
Worked Example
What is the present and future value of an annuity due that pays $£200$ per year for five years at a rate of $6.5\%$?
Solution
\begin{align} &d=iv=\frac{0.065}{1.065}=0.06103286385\\ &\ddot S _{n\vert}=200\frac{(1+0.065)^5-1}{0.06103286385}\approx£1,212.75\\ &\ddot a _{n\vert}=200 \frac{~1-(1+0.065)^{-5}~}{0.06103286385}\approx£885.16. \end{align}
Video Example
Dr Graham Murphy solving an annuity due problem.
Annuities Payable $p$-thly
Annuity Immediate
An annuity immediate payable $p$-thly is a regular series of payments at the end of every period. The present value of a standard annuity paying $p$ times a year for $n$ years with payments of $\frac{1}{p}$ at the end of every period is denoted by $a_{n|}^{(p)}$ whilst the future value is denoted by $S_{n|}^{(p)}$.
The accumulated value for an annuity immediate payable $p$-thly is given by: \[S_{n|}^{(p)}=p \frac{(1+i)^n-1}{i^{(p)}}\] The present value of an annuity immediate payable $p$-thly is given by: \[a_{n|}^{(p)}=p \frac{1-v^n}{i^{(p)}}\] where:
- $i^{(p)} = $ the nominal interest rate.
Worked Example 3
Worked Example
What is the future and present value of an annuity immediate that pays $£250$ per week for five years at an interest rate of $5\%$?
Solution
\begin{align} &1+0.05=\left(~1+\frac{i^{(52)}~}{52}\right)^{52}\\ &\Rightarrow i^{(52)}=0.04881306056\\ &S_{5\vert}^{(52)}=250\times52 \frac{(1+0.05)^5-1}{0.04881306056}\approx£73,579.90\\ &a_{5\vert}^{(52)}=250\times52 \frac{~(1-(1+0.05)^{-5}~}{0.04881306056}\approx£57,651.78. \end{align}
Annuity Due
An annuity due payable $p$-thly is a regular series of payments at the beginning of every period. The present value of a standard annuity paying p times a year for n years with payments of $\frac{1}{p}$ at the start of every period is denoted by $\ddot a_{n|}^{(p)}$ whilst the future value is denoted by $\ddot S_{n|}^{(p)}$.
The accumulated value for an annuity due payable $p$-thly is given by: \[\ddot S_{n|}^{(p)}=p \frac{(1+i)^n-1}{d^{(p)}}\] The present value of an annuity due payable $p$-thly is given by: \[\ddot a_{n|}^{(p)}=\frac{1-v^n}{d^{(p)}}\] where:
- $d^{(p)} = $the nominal discount rate.
Worked Example 4
Worked Example
What is the future and present value of an annuity due that pays $£250$ per week for five years at an interest rate of $5\%$?
Solution
\begin{align} &d=iv=\frac{0.05}{1.05}=0.04761904762\\ &1-0.04761904762=\left(1-\frac{~d^{52}~}{52}\right)^{52}\\ &\Rightarrow d^{(52)}=0.04876728209\\ &\ddot S_{5\vert}^{(52)}=250\times52 \frac{(1+0.05)^5-1}{0.04876728209}\approx £73,648.97\\ &\ddot a_{5\vert}^{(52)}=250 \times 52 \frac{~1-(1+0.05)^{-5}~}{0.04876728209}\approx £57,705.90 \end{align}
