Investors, take note: One little equation can change the world. Check it out, and then find four that rule the investing world.
After Isaac Newton developed his equation for the universal law of gravitation, astronomers applied the formula to see if it could explain strange movements in the orbits of Jupiter, Saturn, and Uranus.
The key to understanding the anomalies, according to Newton's theory, is that all objects in the universe attract one another. The pull of gravity isn't one-way. Jupiter is pulled by the sun, but it also pulls back, faintly, and is pulled by other nearby planets.
Sure enough, Newton's equation fully explained the movements of Jupiter and Saturn, but it couldn't explain all the movements of Uranus. Physicist Richard Feynman related the story in a famous series of lectures at Caltech, now compiled into a very readable book called Six Easy Pieces. As Feynman wrote, Newton's law was in danger.
Shortly after the examination of Uranus' orbit, Newton's formula led two astronomers, independently, to predict the existence of another planet close by. In their minds this was the only possibility that could explain Uranus' bizarre path. They directed observatories to train telescopes on the location of the unknown planet and on September 23, 1846 astronomers at the Berlin Observatory discovered Neptune, hidden for centuries in the darkness, suddenly sprung to life by the mathematics of Isaac Newton.
All because of one equation.
Fast-forward to the 20th century. In 1995, Harvard professor Michael Guillen wrote Five Equations That Changed the World, a book that takes us on a tour from Newton to Einstein and explains, in very readable language, how formulas such as Bernoulli's law of hydrodynamic pressure -- which explains how airplanes fly -- changed the way we think and live.
Taking our cue from Guillen, here's a list of investing equations that changed the world, or at least could change your thinking about investing. While investors may not have an equation to rival Bernoulli's law of fluids in motion, there are a few basic formulas worth knowing for their simplicity, utility, reach, and beauty.
1. The balance sheet equation: A = L + E
Assets = Liabilities + Shareholder's Equity
The balance sheet is an illustration of what a company owns (assets) and what it owes (liabilities and equity). The equation reminds us that the accounts on both sides of the ledger must balance.
What this means is that a company is the sum of its investing and financing decisions -- Stickney and Weil provide a good discussion of this in their textbook, Financial Accounting. Therefore, to understand how a business is put together we must know how it's funded and how its managers are spending precious resources.
Think of it this way. A company has two sources of funding: creditors, who loan money that must be paid back with interest, and owners, who don't demand repayment but expect compensation for risk and for delaying consumption. This is the L + E side of the equation. Each of these owners has a claim on the company's assets.
The company then takes this money and invests it in land, buildings, equipment, patents, human beings, and so forth, expecting these assets to generate a return satisfactory to both creditors and shareholders. A = L + E, therefore, is the skeleton frame of every business, and it's difficult to express this delicate structure in simpler terms than with use of the balance sheet formula. Once I understood this, it became clear that capital allocation -- how a company spends money -- is the most important decision a manager makes.
2. Compounding and discounting formulas:
a. FV = PV(1 + i)^n (This is compounding)
b. PV = FV/(1 + i)^n (This is discounting)
FV = future value
PV = present value
i = interest rate
n = time period
^ the caret symbol represents an exponent (Or the "power" to which the previous number is raised. Your calculator does this easily for you. It's usually the "yx" function, where y is the number you want to raise to the xth power.)
These formulas look frightening, but they aren't too hard to understand, and you can perform them easily on your calculator. For starters, it's not really two different equations, but one succinct equation in two different forms. You multiply to compound and divide to discount -- discounting is just the reverse of compounding.
If you can get even a basic grasp of how it works, you're on your way to understanding perhaps the two most important concepts in finance: risk, and the time value of money.
Let's start with the time value of money. I would rather have $1,000 in cash today than $1,000 a year from now, in part because inflation reduces the value of a dollar. This statement is quite obvious, but its implications aren't. Say I decide to delay spending $1,000. I could let the money sit in my wallet until I need more food or a new toy. In one sense, this is exactly what money is: A way to store value. I don't need $10,000 worth of food in my house. I need $300 or $400. The rest I'd rather have in cash to spend or save. People know this instinctively.
But there are options besides letting your money sit around getting eaten by inflation. You can invest it. The reward for delaying consumption and, if you choose, taking additional risk, is called interest. If I invest $1,000 at 10% for five years the compounding formula takes that $1,000 and turns it into $1,610.51. I've not only earned interest, but I've earned interest on my interest. From this equation: FV = PV(1+i)^n, we can see that this part of it, (1 + i)^n, is the compounding engine.
The reverse of this process is discounting, which is just a financial way of saying that money in hand a year from now is worth less than money in hand today. This follows logically from the compounding concept, but is less intuitive. Think of it this way: Say I tell you I'm going to give you $1,000 in one year if you give me $1,000 today. What's in it for you? Nothing.
To make it worth your while, you'd have to give me less than $1,000. That way your money would be earning interest and, perhaps, include a kicker for the risk of loaning to a guy like me, who offered you such a lousy deal in the first place. Risk is a big part of the investing equation, and we can account for it by embedding compensation in the interest rate we choose. Let's say you settle on the same 10% interest rate. In that case, $1,000 in five years is worth $620.92 today. You can see that the discounting engine is the same as in the compounding engine, only we activate it by dividing rather than multiplying. Compounding and discounting are the two sides of the "time value of money coin."
I'm not a math whiz, to say the least, but I can appreciate a formula that allows us to account for time and risk so easily, that is symmetrical forward and backward, and that accomplishes all of this with just a few variables.
Next: Two More Equations »