#### Introduction to annuities

Annuity is a summary term for any investment that consists of a series of payments at equal intervals. These payments can either be made by you or to you, dependent on the type of annuity we are talking about. Some examples include

- Saving plans, where you pay a constant amount of money into a savings pot at regular intervals
- Retirement plans, where you pay a lump sum up front in order to receive regular payments over a given period of time
- repayment loans and mortgages, where you receive a lump sum loan amount and pay it back using regular payments over a given period of time.

###### Compound interest

Obviously it is not just about the payments at equal intervals but rather the interest that you receive/pay on the amount that you saved or borrowed. In all of the above cases interest is paid at certain intervals (often monthly) which is shorter than the overall length of the annuity. The result

is that over time you do not only receive interest on the original amount but also on the interest accumulated during previous periods. This form of interest is referred to as **compound interest** and the process is called **compounding.**

This means that the total return for a single payment can be calculated as follows based on the number of periods it will incur interest:

```
\small
\begin{aligned}
&\text{One Period} \\
&&return_{(\textcolor{red}{1})} &&&= \space \space pmt * (1+interest)^\textcolor{red}{1} &&&&&\space \space\{1\} \\
\\
&\text{Two Periods} \\
&&return_{(2)} &&&= \space \space pmt * (1+interest) (1+interest) &&&&&\{2.a\} \\
&&return_{(\textcolor{red}{2})} &&&= \space \space pmt * (1+interest)^\textcolor{red}{2}&&&&&\{2.b\} \\
&&&&&\vdots \\
&\text{N Periods} \\
&&return_{(\textcolor{red}{n})} &&&= \space \space pmt * (1+interest)^\textcolor{red}{n}&&&&&\{n\} \\
\end{aligned}
```

One further note should be made about compounding interest. Often interest is compounded on a monthly bases while the interest rate is given as an annual percentage. In these instances the interest rate that needs to be used is the annual interest rate divided by 12. The same logic needs to be applied in cases where compounding happens quarterly (divided by 4) or biannually (divided by 2) etc.

##### Savings plan – Ordinary annuity vs annuity due

Armed with the understanding of compound interest lets look at the first example of an annuity – a savings plan. There are two types of annuities we can distinguish – **ordinary annuities** and **annuities due.**

In the case of an ordinary annuity the first payment is only made at the end of the first period and the last payment while the last payment is made right at the end of the annuity. Therefore the first payment only accrues interest for `(n-1)`

periods while the last payment accrues no interest at all.

With an annuity due payments are made at the beginning of each period. This means there are a full `n`

interest payments on the first payment and the last payment receives a single interest payment. Therefore an annuity due provides a higher return than an ordinary annuity.

The two different types of annuities are exemplified in the following graph for a 4 year annuity with annual payments and interest returns.

###### Future Value

Looking at these examples the total amount received at the end of the annuity also referred to as the **future value** can be calculated as follows:

```
\text{Ordinary Annuity} \newline
\space \newline
FV = pmt * (1+i)^{\color{red}(n-1)} + pmt * (1+i)^{(n-2)} + \dots + pmt * (1+i)^{\color{red}0} \quad (1) \newline
\space \newline
\space \newline
\text{Annuity due} \newline
\space \newline
FV = pmt * (1+i)^{\color{red}n} \quad+ pmt * (1+i)^{(n-1)} + \dots + pmt * (1+i)^{\color{red}1} \quad (2) \newline
\space \newline
\space \newline
\kern{-18em}
\begin{alignedat}{4}
&\footnotesize FV &&= \space \footnotesize\text{total return} \newline \\
&\footnotesize i &&= \space \footnotesize\text{compounding interest} \newline \\
&\footnotesize pmt &&= \space \footnotesize\text{regular payments} \newline \\
&\footnotesize n &&= \space \footnotesize\text{total number of periods} \newline \\
\end{alignedat}
```

While these functions allow us to calculate the value of an annuity by hand, they don’t provide a simple way of calculating annuities that run over many years. But if we look closer at above formulas it becomes clear that they are very similar to geometric series and therefore can be solved in the same manner using the following approach.

1) Multiply both sides of the formula with $$(1-i)$$

2) Subtract the original formula from this new formula

3) Solve the resulting formula for $$FV$$

```
\text{\textbf{Ordinary Annuity}} \newline[1em]
\small
\begin{alignedat}{5}
&FV * (1+r) &&= \hphantom{{\color{blue}{pmt * (1+i)^{n}}} + {}} pmt * (1+i)^{n-1} + pmt * (1+i)^{n-2} +pmt * (1+i)^{n-3} +\dots + pmt * (1+i)^{1} + {\color{blue} pmt * (1+i)^{0}} &&&\quad(1)\\[1em]
&FV * (1+r) &&= {\color{blue}{pmt * (1+i)^{n}}} + pmt * (1+i)^{n-1} + pmt * (1+i)^{n-2} +pmt * (1+i)^{n-3} +\dots + pmt * (1+i)^{1} &&&\quad(2)\\[1em]
\hline \\
&FV(1+i)-FV &&= pmt * (1+i)^n - pmt * (1+i)^0 &&&(3)\\[1em]
&FV*(1+i-1) &&= pmt * ((1+i)^n -1) &&&(3.a)\\[1em]
&FV &&= pmt * \frac {(1+i)^n-1} i &&&(3.b) \\
\end{alignedat} \newline
\space\newline
\large
\colorbox{lightgreen}{$FV = pmt * \frac {(1+i)^n-1} i$} \newline[2em]
\normalsize
\text{\textbf{Annuity due}} \newline[1em]
\small
\begin{alignedat}{4}
&FV * (1+r) &&= \hphantom{{\color{blue}{pmt * (1+i)^{n+1}}} + {}} pmt * (1+i)^{n} + pmt * (1+i)^{n-1} +pmt * (1+i)^{n-2} +\dots + pmt * (1+i)^{2} + {\color{blue} pmt * (1+i)^{1}} &&&\quad(1)\\[1em]
&FV * (1+r) &&= {\color{blue}{pmt * (1+i)^{n+1}}} + pmt * (1+i)^{n} + pmt * (1+i)^{n-1} +pmt * (1+i)^{n-2} +\dots + pmt * (1+i)^{2} &&&\quad(2)\\[1em]
\hline\\
&FV * (1+r) - FV &&= \color{blue}{pmt * (1+i)^{n+1}} - {\color{blue}pmt * (1+i)^{1}} &&&(3)\\[1em]
&FV * \big((1+r)-1\big) &&= pmt * \big((1+i)^{n+1} - (1+i)\big) &&&(3a)\\[1em]
&FV * r &&=pmt * \big((1+i)^{n+1} - (1+i)\big) &&&(3b)\\[1em]
&FV &&=pmt*\frac {(1+i)^{n+1} - (1+r)} i \\
\end{alignedat}\newline
\newline[1em]
\large
\colorbox{lightgreen}{$FV=pmt * \frac{(1+i)^{n+1} - (1+i)} i$}
\newline[1em]
```

Now we have a handy formula to calculate the future value for any savings plan involving regular payments.

##### Retirement Annuity

Retirement Annuities are very common. The difference between a retirement annuity and a savings plan is that with a retirement annuity we pay a lump sum at the beginning which will provide us with regular income over a certain amount of time.

###### Present Value

In order to understand this better we need to talk about the time value of money and the concept of **present value**. Going back to our savings plan we know exactly how much money we will have at the end of it – in ten, fifteen or twenty years. But how much would this future return be worth today. If you wanted to "buy" this future return today how much would you have to pay. This is referred to as the **present value** of an annuity. Another way of looking at this is to ask yourself how much money would I have to put in the bank today to receive the same amount as the annuity in the end.

Assuming the same interest rate and schedule as for the annuity we can easily calculate this.

`PV * (1+i)^n = FV`

Using the formulas for the future value of an ordinary annuity or an annuity due, we end up with the following formulas for the present value of an annuity

```
\normalsize
\text{\textbf{Present Value Of An Ordinary Annuity}} \newline[2em]
\small
\begin{alignedat}{4}
&PV * (1+r)^n &&= pmt * \frac {(1+i)^n -1} i \qquad \qquad \qquad \big | * \frac 1 {(1+r)^n} &&&\quad (1)\\[1.5em]
&PV * \frac {(1+r)^n} {(1+r)^n} &&= pmt * \frac {(1+i)^n -1} i * \frac 1 {(1+r)^n} &&&\quad (2)\\[1.5em]
&PV * 1 &&= pmt * \frac {(1+i)^n -1} i * (1+r)^{-n} &&&\quad (3)\\[1.5em]
&PV &&= pmt * \frac {1 - (1+i)^{-n}} i &&&\quad (4) \\
\end{alignedat}\newline[2em]
\large
\colorbox{lightgreen} {$PV = pmt * \frac {1 - (1+i)^{-n}} i$} \newline[3em]
\normalsize
\text{\textbf{Present Value Of An Annuity Due}} \newline[2em]
\small
\begin{alignedat}{4}
&PV * (1+r)^n &&= pmt * \frac {(1+i)^{n+1} - (1+i)} i \qquad \qquad \qquad \big | * \frac 1 {(1+r)^n} &&&\quad (1)\\[1.5em]
&PV * \frac {(1+r)^n} {(1+r)^n} &&= pmt * \frac {(1+i)^{n+1} - (1+i)} i * \frac 1 {(1+r)^n} &&&\quad (2)\\[1.5em]
&PV * 1 &&= pmt * \frac {(1+i)^{n+1} - (1+i)} i * (1+r)^{-n} &&&\quad (3)\\[1.5em]
&PV &&= pmt * \frac {(1+i) - (1+i)^{-n}} i &&&\quad (4) \\
\end{alignedat}\newline[2em]
\large
\colorbox{lightgreen} {$PV = pmt * \frac {(1+i) - (1+i)^{-n}} i$} \newline[3em]
```

In the context of a savings plan these formulas allow us to calculate todays value of these savings plans.

In the context of a retirement annuity they provide us with the lump sum we have to spend today in order to receive the respective payments for the specified amount of time. In effect we are calculating how much these future payments to us are worth today and that’s the amount we have to spend today.

##### Repayment mortgages

In the case of a repayment mortgage we borrow money from a bank or building society. But instead of paying them back at the end of the mortgage period giving them one big lump sum of money including the interest we owe, we pay it back in monthly fixed instalments – looks like another annuity to me.

###### Mortgage payment

Lets actually assume for a minute we are going to pay the bank back at the end. In this scenario we would owe the bank/building society `\big[\space loan \space amount * (1+i)^n \space \big]`

at the end of the mortgage period taking into account all the compounded interest we would have to pay. As a prudent person we would set up a monthly savings plan, ideally with the same interest rate as the mortgage and with the same running period. When we look at it this way, the money we borrow is equivalent to the present value of the annuity. The future value we need to achieve with our "savings plan" is determined by the present value (loan value), the mortgage period, the payment schedule (how often interest is compounded) and the effective interest rate. Armed with that knowledge we can now calculate the monthly payments.

Based on the modalities of a repayment mortgage we can calculate:

```
\text{\textbf{Mortgage Payment}}\newline[2em]
\small
\begin{alignedat}{4}
PV &= pmt * \frac {1 - (1+i)^{-n}} i \qquad \big | * \frac i {1 - (1+i)^{-n}} &&&\quad (1) \\
pmt &= PV * \frac i {1 - (1+i)^n} &&& (2)
\end{alignedat}\newline[2em]
\large
\colorbox{lightgreen} {$pmt = PV * \Large\frac i {1 - (1+i)^{-n}}$}
```

###### Remaining loan amount

Now that we know the amount we have to pay every month, it also is important to understand how much we still owe after we have made some of the payments. It would be tempting to just subtract the monthly payment from the amount we borrowed but its not quite as simple.

Going back to our earlier analogy. The amount we borrow is equivalent to the present value of an annuity. As we have said, the money we owe at the end of the mortgage is equal to `PV * (1+i)^n`

if we had to repay the loan as a single lump sum. The repayment is equivalent to an ordinary annuity savings plan and at the end of the annuity the amount of money in the "savings plan" is equivalent to the money we owe. At any given time point before that the money we owe is equivalent to `PV * (1+i)^{a_t}`

, and the amount we have paid back equivalent to `\big[\space pmt * \frac {(1+i)^{a_t}-1} i \space \big]`

and the amount we owe is bigger than the amount we have paid back to that date.

```
\text{\textbf{Reamining loan amount}}\newline[2em]
\small
\begin {alignedat}{2}
&\begin{alignedat}{4}
&\hphantom{{PV * (1+i)^{a_t}}} &&\hphantom{\small{= PV * (1+i)^{a_t} - PV * \frac i {1 - (1+i)^{-n}} * \frac {(1+i)^{a_t}-1} i}}\\
&PV * (1+i)^{a_t} &&= pmt * \frac {(1+i)^{a_t}-1} i + R &&& (1)\\
&&R &= PV * (1+i)^{a_t} - pmt * \frac {(1+i)^{a_t}-1} i &&& \quad(2)\\[2em]
\end{alignedat} \\
& \text{subsitute } pmt \text{ with } \footnotesize \quad PV * \scriptsize \frac i {1 - (1+i)^{-n}} \\[1em]
&\small\begin{alignedat} {4}
% the hphontom is needed to align with previous equation group properly
&\hphantom{PV * (1+i)^{a_t}} \\
&&R &= PV * (1+i)^{a_t} - PV * \frac i {1 - (1+i)^{-n}} * \frac {(1+i)^{a_t}-1} i &&&\quad (3)\\
&&R &= PV * \Big ((1+i)^{a_t} - \frac {(1+i)^{a_t}-1} {1 - (1+i)^{-n}} \Big) &&& (4)\\[1.5em]
&&R &= PV * \Big (\frac {(1+i)^{a_t} * (1 - (1+i)^{-n})}{1 - (1+i)^{-n}} - \frac {(1+i)^{a_t}-1} {1 - (1+i)^{-n}} \Big) &&& (5)\\[1.5em]
&&R &= PV * \Big (\frac {(1+i)^{a_t} - (1+i)^{a_t-n}}{1 - (1+i)^{-n}} - \frac {(1+i)^{a_t}-1} {1 - (1+i)^{-n}} \Big) &&& (6)\\[1.5em]
&&R &= PV * \Big (\frac {(1+i)^{a_t} - (1+i)^{a_t-n} - ((1+i)^{a_t}-1)} {1 - (1+i)^{-n}} \Big) &&& (7)\\[1.5em]
&&R &= PV * \Big (\frac {(1+i)^{a_t} - (1+i)^{a_t-n} - (1+i)^{a_t}+1} {1 - (1+i)^{-n}} \Big) &&& (8)\\[1.5em]
&&R &= PV * \Big (\frac {- (1+i)^{a_t-n} +1} {1 - (1+i)^{-n}} \Big) &&& (9)\\[1.5em]
&&R &= PV * \Big (\frac { 1- (1+i)^{a_t-n}} {1 - (1+i)^{-n}} \Big) &&& (10)\\[3em]
\end{alignedat}\\
&\hphantom{PV * (1+i)^{a_t}} \colorbox{lightgreen} {$ \Large R = PV * \Large\frac { 1- (1+i)^{a_t-n}} {1 - (1+i)^{-n}}$}\\[2.5em]
&\footnotesize\begin{alignedat}{3}
&R &&= \text{ remaining amount left to pay}\\
&PV &&= \text{ loan amount} \\
&a_t &&= \text{ periods already elapsed = current period}\\
&n &&= \text{ total number of periods}\\
&i &&= \text{effective interest rate}
\end{alignedat}
\end{alignedat}
```

###### Interest and Principal of payment

Last but not least, when it comes to a mortgage it might be interesting to know how much of the monthly payment covers interest and how much actual goes to reducing the loan (i.e principal). Again this is not quite as simple as it looks, as the proportion of interest to principle changes over the length of the mortgage. At the beginning most of the monthly payment covers interest, whereas towards the end it mostly covers principle. The reason is quite simple. The interest amount that is due at the end of each period is based on the amount we owe at the beginning of the period (i.e the amount we owe at the end of the previous period.)

Consider this – at the end of the first period we owe interest on the full loan amount, but we also pay some of the actual loan back. Therefore at the end of the second period we only have to pay interest on the remaining loan amount after the first month. Makes sense?!

The **interest amount** that is due for a particular period can therefore be calculated as

```
\text{\textbf{Interest amount for period}}\newline[2em]
\colorbox{lightgreen} {$I_{a_t} = PV * \Large \frac { 1- (1+i)^{a_{\tiny t-1}-n}} {1 - (1+i)^{-n}} * \normalsize i$} \newline[2em]
\footnotesize
\begin {alignedat}{3}
&I_{a_t} &&= \text{Interest amount due at a particular period}\\
&PV &&= \text{loan amount}\\
&a_{t-i} &&= \text{preceding period $a_{t-1} = a_t -1$}\\
&i &&= \text{interest rate}\\
\end{alignedat}
```

Last but not least we can now calculate the **principal** as this is simply our monthly payment minus the Interest for the period.

```
\text{\textbf{Principal for period}}\newline[2em]
\small\begin{alignedat}{3}
&P_{a_t} = pmt - PV * \frac { 1- (1+i)^{a_{t-1}-n}} {1 - (1+i)^{-n}} * i && \quad(1)\\[1.5em]
\text{subsitute } & pmt \text{ with } \footnotesize \quad PV * \scriptsize \frac i {1 - (1+i)^{-n}} && \quad(2) \\[1.5em]
&P_{a_t} = PV * \Big(\frac i {1 - (1+i)^{-n}} - \frac { 1- (1+i)^{a_{t-1}-n}} {1 - (1+i)^{-n}} * i \Big) && \quad (3)\\[1.5em]
&P_{a_t} = PV * \Big(\frac i {1 - (1+i)^{-n}} - \frac { i- i* (1+i)^{a_{t-1}-n}} {1 - (1+i)^{-n}} \Big) && \quad (4)\\[1.5em]
&P_{a_t} = PV * \Big(\frac {i - i + i* (1+i)^{a_{t-1}-n}} {1 - (1+i)^{-n}} \Big) && \quad (5)\\[1.5em]
&P_{a_t} = PV * \Big(\frac {i* (1+i)^{a_{t-1}-n}} {1 - (1+i)^{-n}} \Big) && \quad (6)\\[1.5em]
&P_{a_t} = PV * i * \Big(\frac {(1+i)^{a_{t-1}-n}} {1 - (1+i)^{-n}} \Big) && \quad (7)\\[3 em]
\end{alignedat} \newline
\colorbox{lightgreen} {$\large P_{a_t} = PV * i * \Large\frac {(1+i)^{a_{\tiny t-1}-n}} {1 - (1+i)^{-n}} $} \newline
\begin{alignedat}{3}
&\hphantom{P_{a_t}} &&\hphantom{= PV * \Big(\frac i {1 - (1+i)^{-n}} - \frac { 1- (1+i)^{a_{t-1}-n}} {1 - (1+i)^{-n}} * i \Big) \quad (3)}\\[1.5em]
&P_{a_t} &&= \text{Principal amount due at a particular period}\\
&PV &&= \text{loan amount}\\
&a_{t-i} &&= \text{preceding period $a_{t-1} = a_t -1$}\\
&i &&= \text{interest rate}\\
\end{alignedat} \newline
```

#### Final thoughts

This was quite a marathon. I hope the explanations on how the different formulas can be derived was helpful. Obviously you can use these formulas to calculate other aspects of an annuity provided you know all the other variables. For instance you can calculate the monthly amount that you need to save if you have a particular savings goal over the next 10 years for instance, or the amount you can expect to receive monthly if you have a known lump sum. I think you get the picture.

###### Cash flow

I just want to make one final comment though – that is cash flow, i.e. the direction money is moving. For instance, in the case of a savings plan, the regular payments leave our account, but at the end the annuity is paid out to us.

When it comes to the retirement plan we are paying a lump sum at the beginning, i.e. money is leaving our account but we receive regular payments from then on.

In the case of the repayment mortgage the situation is reversed.

While the direction of the money movement is irrelevant for calculating the value they matter when we look at these from an accounting perspective. Just keep that in mind when you are using proprietary software that might expect you to provide this information – normally by designating a positive value as money influx and a negative value as money that is leaving our account.

Well that’s all for annuities, future values, present values, compounded interest and mortgage repayments.

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