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.
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 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
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
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 × \frac{1 − (1+r)^{−n}}{r} | |
Swap sides: | P × \frac{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 × \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 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. \frac{1}{1+r} is actually (1+r)^{−1} and \frac{1}{(1+r)×(1+r)} is (1+r)^{−2} etc:
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: