Annuities
An annuity is a regular payment over a period of time.
Example: You get $400 a month for 5 years.
That’s an annuity: a regular payment (in this case $400 each month) for a fixed time (5 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 on.
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 $1,000 now is the same as $1,100 next year (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
Example: 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 $1584.94 now
As a table the calculations are:
| Payment Time | Payment Amount | Present Value Calculation | Present Value Now |
|---|---|---|---|
| 1 year | $500 | $500 ÷ 1.10 | $454.55 |
| 2 years | $500 | $500 ÷ 1.102 | $413.22 |
| 3 years | $500 | $500 ÷ 1.103 | $375.66 |
| 4 years | $500 | $500 ÷ 1.104 | $341.51 |
| $1584.94 |
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 per period, as a decimal, so 10% is 0.10
- n is the number of periods
First, let's try it on our 4 yearly payments of $500 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 monthly payments use the monthly interest rate, and so on.
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 can "change the subject" of the formula above:
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
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. 1 1+r is actually (1+r)-1 and 1 (1+r)×(1+r) is (1+r)-2 and so on:
And we can bring the "P" to the front of all 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:
Summary
- An annuity is regular payments over time
- Present value lets us compare money in the future to money now
- The annuity formula PV = P × 1 − (1+r)-n r can save time