Lessons in FM: Part II - Annuities

Note: This is a continuation of the series Lessons in Financial Mathematics. Please read Part I - Present Value if you have not already done so. It will aid greatly in your understanding of subsequent parts of the series.

Now that you understand the basics of present value, you might see how this could possible get a little cumbersome. For example, if you had monthly payments due for 30 years on a mortgage, you could do the sum of all the present values...all 30*12=360 of them...but it would take awhile. Fortunately, there is a better way: geometric series to the rescue!

From your early algebra classes (and I know all of you treasured those greatly), you might recall that if you have a sum where the terms increase or decrease by a common ratio (r) then you can easily condense the sum into a compact formula. The proof is very compact, but I will summarize the findings here:

a + a*r + a*r² + a*r³ + ... + a*rⁿ = a*(1-rⁿ⁺¹)/(1-r)

There's also a version for an infinite sum, provided that |r|<1:>2 + a*r³ + ... = a* (1/(1-r))

If we had a series of payments occurring at regular intervals, they would likely look something like this, where P is the payment, PV is present value, v is (as defined in Part I) 1/(1+i), i is the interest rate, and there are n payments:

P + P*v + P*v² + P*v³ + ... + P*v^n-1 = PV

(we use n-1 because we started at time t=0, so going from that to n-1 gives n payments) using the geometric series above, this simplifies to:

PV = P*(1-vⁿ)/(1-v)

and 1-v can be simplified to i/(1+i) (I leave that proof to the reader). In FM, we define the discount rate d = i/(1+i). Therefore, the entire formula reduces to:

PV = P*(1-vⁿ)/d

Cute, huh? Note that this formula is applicable if payments are made at the beginning of the period. If they are made at the end of the period, then you replace the d with an i (for the interest rate) and the formula still works (I'll leave the proof of that to the reader). Actuaries have a special symbol for that, which will be rather difficult to put in Blogger. It's named "a-double-dot angle n", and is a lowercase "a" with two dots over it. I can find an image of it from the Wikipedia article on actuarial notation:




The lowercase a represents the fact that it's an annuity (as opposed to being a capital A, which would represent an insurance). The angle-n represents that the payment is guaranteed for n periods. The double-dot represents that the payment occurs at the beginning of the period (if it is missing, then it occurs at the end of the period). The i after the angle is often omitted, but is simply a reminder of what interest rate is being used.

For the remainder of this post, I will assume end of period payments (thus eliminating the need for double dots), and spell out "a-angle-n" whenever I need to use it.

Let's do an example. Suppose you want to take out a loan for L$10,000 and the banker wants to charge you 7% interest. You agree to pay the loan back over 30 periods. The question then becomes, how much is the payment.

To solve this, you use the formula above. You know the present value (10,000), the interest rate (0.07), and the number of periods (n), so you just solve for the payment.

PV = P * "a-angle-30"
10,000 = P * (1 + (1/1.07)^30)/(0.07)
10,000 = P * (12.40904)
P = 805.86

Your payment would be L$805.86 at the end of each period.

You might be saying "there has to be an easier way" at this point. Fortunately, there is. I give you, the financial calculator! To show you how to use this, we'll go over the previous example again with the calculator.

To start, note that the general way of inputting numbers to the right-hand side of that calculator is to enter the number first, and then press the value that you want it to go to. Also note that we enter percentages, so 7% is entered at 7, not 0.07. To do the previous example, follow these steps:

1. Enter 10000
2. Press PV
3. Enter 7
4. Press i%
5. Enter 0
6. Press FV (the loan is worth zero at the end)
7. Enter 30
8. Press N
9. Press Calculate
10. Press PMT

Now, observe that the number it spit out was the same as I calculated above (rounded): 805.86. Funny how that works.

I haven't actually defined that funny word in the title of the article yet: annuities. An annuity is a set of payments, guaranteed or conditional, that occur at regular intervals. There is a lot more detail that can be set up with annuities, such as varying the frequency of payment (say, to quarterly), the amount of payment (increasing/decreasing), the time frame (infinite annuity = perpetuity, by the way), etc. These are annuities, and they are the basis of loans, dividends, millions of people's retirements, insurance, etc. Annuities are quite flexible and applicable in a variety of financial settings.

I'll leave you to play with that calculator. Note that it only appears to do end-of-period calculations. You can enter any four of the five variables, and it'll compute the last one (provided it is solvable). The next segment will have less math (I think...) and deal with a more economic topic: Risk.