Lessons in FM: Part I - Present Value

I'm going to start a series of educational posts about the basic concepts of Financial Mathematics, my major. These are not intended to be overly complex and I will do my best to keep the calculus out of it. There will be a mix of mathematics and economics, and the symbols used will likely be actuarial in nature. This convention won't bother those who are new to the topic, but might if you have some familiarity with it already from another discipline, like Finance.

Present value tells you the value of some amount at one point in time at another point in time. Usually, you're trying to bring payments at some future time back and find out what they're worth today. To find this, you need the future value(s) of payments (FV), the interest rate (i), and the times at which these payments will be made (t or t_k). We'll work with just one payment for now, and we'll also be using compound interest.

The equation we start with is pretty basic:

Present Value * Interest = Future Value

in symbols,

PV * (1+i)^t = FV

To solve for PV, we divide both sides by (i+i)^t. However, remember that actuarial notation I mentioned? Actuaries use this method (dividing by the interest factor) so often that they have their own symbol for it, v ("vee").

Definition: v = 1 / (1 + i)

Using this notation, we solve for PV and get

PV = FV * v^t

And that's it! Just plug your numbers in after that! So long as you have three of the variables, you can solve for the fourth.

So that's great and all, Guardian, but what if I have more than one payment? Well, fortunately, you can just add them up. Say you have two payments - one at time t = 1 and the other at time t = 3. To find what they're worth today, just add them. You get

PV₁ = FV₁*v¹
PV₃ = FV₃*v³

PV₁ + PV₃ = PV_total = FV₁*v¹ + FV₃*v³

The tricky thing is finding the interest rate which makes that equation balance. This is a problem that often has no direct solution and must be done iteratively using numerical methods. That's part of that scary math I promised I wouldn't get into. (As an aside, if people really want to see that mathematics, comment that you do and I'll whip something up.)

So what good does this do in the Second Life capital markets? Well, theoretically (and in Second Life, I mean very theoretically), that is what all the stock prices are based on. Investors value securities according to what they think the future value of it is, discounted with interest (this interest rate likely has a risk premium built into it, however). Whether that future value is based off of reselling the security, dividend payments, or both is something that each investor must decide for themselves. However, if you know your own preferences and you can make guesses as to the dividend amounts and/or stock prices, you can use this method to value what "your" price of the securities is and whether or not it's a good buy for you.

So what if there are a lot of payments, but they occur in regular intervals? I'll tackle that one in Part II.