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 you give the $20,000 to could 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.

present value $1000 vs future value $1100
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:

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 × 1 − (1+r)−n r

  • P is the value of each payment
  • r is the interest rate 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 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 monthly interest rate is 1%, 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.

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 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