# Annuities

An annuity is a **fixed income over a period of time**.

### Example: You get $200 a week for 10 years.

How do you get such an income? **You buy it!**

So:

**you pay them**one large amount, then**they pay you**back a series of small payments over time

### Example: You buy an annuity

It costs you **$20,000**

And in return you get $400 a month for 5 years

*Is that a good deal?*

### Example (continued):

$400 a month for 5 years = $400 × 12 × 5 = **$24,000**

Seems like a good deal ... you get back more than you put in.

Why do you get **more** income *($24,000)* than the annuity originally cost *($20,000)*?

Because **money now** is more valuable than money later.

The people who got your $20,000 can invest it and earn interest, or do other clever things to make more money.

So how much *should* an annuity cost?

## Value of an Annuity

First: let's see the effect of an **interest rate of 10%** (imagine a bank account that earns 10% interest):

### Example: 10% interest on $1,000

$1,000 now could earn $1,000 x 10% = **$100** in a year.

**$1,000 now** becomes **$1,100 in a year's time**.

So $1,100 next year is the **same** as $1,000 now (at 10% interest).

The **Present Value** of $1,100 next year is **$1,000**

So, at 10% interest:

- to go from
**now**to**next year**: multiply by 1.10 - to go from
**next year**to**now**: divide by 1.10

Now let's imagine an annuity of **4 yearly payments of $500**.

Your first payment of $500 is next year ... how much is that worth **now**?

Your second payment is 2 years from now. How do we calculate that? Bring it back one year, then bring it back another year:

The third and 4th payment can also be brought back to today's values:

Finally we add up the 4 payments (in today's value):

We have done our first annuity calculation!

4 annual payments of $500 at 10% interest is worth **$1,584.94 now**

How about another example:

### Example: An annuity of $400 a month for 5 years.

### Use a Monthly interest rate of 1%.

12 months a year, 5 years, that is **60 payments** ... and a LOT of calculations.

We need an easier method. Luckily there is a neat formula:

Present Value of Annuity: PV = P × \frac{1 − (1+r)^{−n}}{r}

**P**is the value of each payment**r**is the interest rate per period, as a decimal, so 10% is 0.10**n**is the number of periods

### First, let's try it on our $500 for 4 years example.

The interest rate per year is 10%, so **r = 0.10**

There are 4 payments, so **n=4**, and each payment is $500, so **P = $500**

**It matches our answer above** (and is 1 cent more accurate)

### Now let's try it on our $400 for 60 months example:

The interest rate is 1% per month, so **r = 0.01**

There are 60 monthly payments, so **n=60**, and each payment is $400, so **P = $400**

Certainly easier than 60 separate calculations.

*Note: use the interest rate per period: for monhtly payments use the monthly interest rate, etc.*

## Going the Other Way

What if you know the annuity value and want to work out the payments?

Say you have $10,000 and want to get a monthly income for 6 years, how much do you get each month (assume a monthly interest rate of 0.5%)

We need to change the subject of the formula above

**r**:P × (1 − (1+r)

^{−n}) = PV × r

**1 − (1+r)**:P

^{−n}*= PV × \frac{r}{1 − (1+r)^{−n}}*

And we get this:

P
* *
= PV × \frac{r}{1 − (1+r)^{−n}}

**P**is the value of each payment**PV**is the Present Value of Annuity**r**is the interest rate per period as a decimal, so 10% is 0.10**n**is the number of periods

### Say you have $10,000 and want to get a monthly income for 6 years out of it, how much could you get each month (assume a monthly interest rate of 0.5%)

The monthly interest rate is 0.5%, so **r = 0.005**

There are 6x12=72 monthly payments, so **n=72**, and **PV = $10,000**

*= PV × \frac{r}{1 − (1+r)^{−n}}*

*= $10,000 × \frac{0.005}{1 − (1.005)^{−72}}*

What do you prefer? **$10,000 now** or 6 years of **$165.73 a month**

### Footnote:

You don't need to remember this, but you may be curious how the formula comes about:

With **n** payments of **P**, and an interest rate of **r** we add up like this:

We can use exponents to help. \frac{1}{1+r} is actually (1+r)^{−1} and \frac{1}{(1+r)×(1+r)} is
(1+r)^{−2} etc:

^{−1}+ P × (1+r)

^{−2}+ P × (1+r)

^{−3}+ ... (n terms)

And we can bring the "P" to the front of all terms:

^{−1}+ (1+r)

^{−2}+ (1+r)

^{−3}+ ... (n terms) ]

*To simplify that further is a little harder! We need some clever work using Geometric Sequences and Sums b**ut trust me, it can be done ... and we get this:*