Special Case of Multiple Cash Flows: Annuity

There are some special patterns of multiple cash flows, which are very often used in reality. It is not because they are complex, but because they are actually quite simple. Afterall, as Charles Mingus once put it, "Making the simple complicated is commonplace; making the complicated simple, awesomely simple, that’s creativity."

And we will start to introduce some of these simple patterns from this section.

As we know, the rule of thumb when it comes to multiple cash flow problems is to figure out the PV (the value at time = 0 in above chart). According to our general rule, "find each cash flow's PV and then add them up", the PV of an annuity should be like this:

Here C is the value of each cash flow of the annuity, such as $400 in our mortgage example. r is the interest rate for each period. Here r would refer to the monthly interest rate for your mortgage. Let's say the APR of your loan is 4.8%, so your real monthly interest rate, r is 0.4% (Don't get panic about what APR means, I assure you it is not your fault since we never mention it before. We'll explain it later in this chapter. For now, just take r = 0.4% as given). T is the total number of payments of your mortgage, 36.

Then, to figure out the PV of your mortgage, all you need to do is to plug these numbers into the formula above.

But wait, there are 36 terms in that equation. It would be INSANE to do it term by term. Is there a simple way to do the calculation?

Lucky for you! Since the pattern here is very "obvious" to the mathematicians, (duh), they simplify the the above formula to:

Now, with this new equation, the calculation is much easier, (compared with what we would have to end up with, of course):

So, the total present value of your 36 month mortgage is 13386.35. This is most likely to be more than the MSRP of your car, right? And the difference ($13,386.35 - $11,999 = $1,387.35) should be roughly how much the dealer is making out of it, by simply providing you with the loan and net of the interest. Personally, I would rather go to a bank, negotiate for a loan with the same rate, pay the dealer with the cash I borrowed from the bank, and ride out of the store with that awesome bike.