# 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?

\$500 ÷ 1.10 = \$454.55 now (to nearest cent)

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

\$500 ÷ 1.10 ÷ 1.10 = \$413.22 now

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

\$500 ÷ 1.10 ÷ 1.10 ÷ 1.10 = \$375.66 now
\$500 ÷ 1.10 ÷ 1.10 ÷ 1.10 ÷ 1.10 = \$341.51 now

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

Annuity Value = \$454.55 + \$413.22 + \$375.66 + \$341.51
Annuity Value = \$1,584.94

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:

### 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 × 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

PV = \$500 × 1 − (1.10)−4 0.10
PV = \$500 × 1 − 0.68301... 0.10
PV = \$500 × 3.169865...
PV = \$1584.93

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

PV = \$400 × 1 − (1.01)−60 0.01
PV = \$400 × 1 − 0.55045... 0.01
PV = \$400 × 44.95504...
PV = \$17,982.02

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

Start with:PV = P × 1 − (1+r)−n r
Swap sides:P × 1 − (1+r)−n r = PV
Multiply both sides by r:P × (1 − (1+r)−n) = PV × r
Divide both sides by 1 − (1+r)−n :P = PV × r 1 − (1+r)−n

And we get this:

P = PV × 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

P = PV × r 1 − (1+r)−n
P = \$10,000 × 0.005 1 − (1.005)−72
P = \$10,000 × 0.016572888...
P = \$165.73

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:

P × 1 1+r + P × 1 (1+r)×(1+r) + P × 1 (1+r)×(1+r)×(1+r) + ... (n terms)

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

P × (1+r)−1 + P × (1+r)−2 + P × (1+r)−3 + ... (n terms)

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

P × [ (1+r)−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 but trust me, it can be done ... and we get this:

PV = P × 1 − (1+r)−n r