Leverage. Leverage is debt used for investment purposes, and is extremely important. Why?

• Debt, when invested, multiplies return (profits and losses)

Leverage is a multiplier, a super-power. Super-strength is great when times are good, and horrific when you accidentally “bite your tongue” (it’s super-strength, not invulnerability). Concepts like leverage are casually mentioned, but let’s see why the dominoes fall.

Get Rich Quick

I’ve got a great investment plan for you. Ready?

• Step 1: Withdraw all your money
• Step 2: Go to Las Vegas
• Step 3: Bet it all on red in roulette (Get it right and double your money — get it wrong and lose it all)

It’s perfect! We’ll double our money in one step.

Sure, there’s a “chance” that things go wrong. But even then it’s no so bad — we’ll be at zero, like the day we were born. Presumably naked and crying as well.

Double My Money? But I Want More!

The plan sounds interesting, but there’s a problem — what if I only have $100? Doubling to $200 is nice, but not life-changing.

A few wild thoughts later, and we’re onto a better idea: let’s borrow money! The plan becomes exciting:

• Take our $100 and borrow $1,000,000 from friends, families, banks, and unsavory characters. (How? Well, show different people our $100 and ask to borrow another $100, with our original cash as collateral).
• Go to Vegas
• Bet the $1M dollars on black! I mean, red!

What happens now?

If we win: we get $2M, pay back our $1M loan, and are sitting pretty with our profit of $1M.

And if we lose? Uh oh. Now we’re worse than naked: we’ve lost our shirts and everyone else’s too! Because we took debt, our worst case scenario is no longer going broke — it’s going negative.

Notice how the loan changed the outcomes — neither wild riches nor debtor’s prison were possible without the loan.

The Risk Multiplier

What just happened? Debt multiplies our risk and reward. The good times get great, and the bad times become awful. In our example, we went from winning or losing $100 to winning or losing $1M — a 10,000x difference in profit and loss!

This effect from investing debt is called “leverage”. Why?

I suppose it’s because a lever lets you move one end a tiny bit (one inch) and have the other side move a large amount (1 foot). It’s also called a leverage or gearing ratio — move the big gear one cycle and move the small gear many cycles.

My inner geek cringes, since the sides of a lever move in opposite directions (one side up, one side down) and same with the gears (one side clockwise, the other counter-clockwise). Remember that with financial leverage, both sides move the same way.
I imagine leverage as a game of follow-the-leader: I push my money in one direction (making a bet), and the huge pile of money I borrowed does the same.

Use whatever analogy works for you — the key is if your money wiggles up or down, the borrowed money does the same.

The Risk and Benefit of Leverage

Why does leverage work? At its heart, you are borrowing someone’s assets and reaping the benefits. It’s like borrowing a cow and selling the milk! What a great idea!

It’s great until the cow runs off. Now you’re stuck — you owe a cow and don’t have one to return. The risk of leverage is investing that debt and losing what you borrowed, which can wipe out any profits.

Let’s try a more realistic example then roulette: investing in a house. Suppose you have 10k and borrow 90k, to purchase a $100k house. You have a leverage ratio of 10:1 — for every 10 dollars of the asset, you’ve put in 1 dollar of equity (your own money).

If house prices rise by 10%, how much did you make? At first blush we’d say 10%, which is true — but you made 10% on the entire 100k! The house is now worth 110k, and after paying your 90k debt you’re left with 20k. That 10% growth became 100% profit on your initial investment!

• leverage ratio = asset / equity
• return = leverage ratio * percent change

Again, with 10x leverage, 10% growth becomes 100% return on our initial equity. From our analogy, we were in “control” of 10 dollars for every 1 we put in. So, we gained 10x the profit.

Now what about the reverse — when the house falls 10% to 90k?

Well, we can sell the house for 90k, pay off our loan (90k) and are left with… zero! Similarly, a 10% dip in prices becomes a 100% loss of equity — we’re wiped out! We get 10x the loss when prices go south.

And if the house price falls 20% (impossible! improbable! unlikely!), we suffer a 200% loss — we’ve lost our initial 10k and owe 10k beyond that (sell the house for 80k, but the loan is still 90k).

Hopefully the magnifying effect of borrowed money is becoming clear. You lose your equity when the investment drops 1/leverage ratio — in this case, 1/10 or 10%. With a 25x leverage ratio, the investment only needs to drop 4% in order to be wiped out. One way to think about it: you’re paying for losses out of your own pocket, not the borrowed money (you always have to pay it back). Your pocket is only 1 dollar of the 10, so once you lose it (1 dollar out of 10, or 10%) you are wiped out. Any more, and you’re in debt.

Real-world Examples

Leverage appears all over finance, but sometimes in disguise. Let’s take a look:

Getting a mortgage: As we saw, borrowing money to buy a house is a form of leverage. With 5% down (a 20x gearing ratio), your house only needs to drop by 5% to lose money. With 0% down, your house has to drop… wait for it… any amount for you to lose! And after your house is worth less than your mortgage, there’s little incentive to pay it off (better to go bankrupt, depending on the debt).

Lending money for a mortgage: Even banks are levered. When they offer money, where do you think they get it? From deposits! They borrow your deposits to loan it out to other people. If they have 10k of deposits they might loan out 100k (there’s some magic that happens here, how banks loan more money than they have; that’s for another time). If they loaned money for a house, and the house drops 10% in value and the debtor doesn’t pay, the bank has lost all if its deposits.

Offering mortgage insurance: Insurance companies might have 10k worth of cash, and offer 100k worth of insurance coverage to banks (C’mon, did you really think the insurance company had enough to pay off everyone’s claims?). Of course they don’t expect everyone to file a claim, but if even 10% of people do, they are wiped out. There isn’t an explicit loan, but the insurance company has created an obligation to pay (called the insurance leverage ratio).

See the set up? When prices are rising:

• Owners are making a lot of money (they leveraged the house)
• Banks are making a lot of money (they leveraged their loans, and earn a lot of interest on the borrowed money)
• Insurance companies are making a lot of money (they’re offer more coverage, which means more premiums)

If the music stops and house prices fall, problems arise:

• Borrowers lose equity — a 5% drop when 20x levered means the borrower is wiped out. Any more and the loan is worth more than the house.
• Banks lose loans — if 5% of loans go bad, the banks have to pay for the lost value themselves.
• Insurance companies lose money — if 5% of claims are called in, when the insurance company is 20x levered, it means the company has lost all of its assets!

For 20x leverage, a 5% drop would wipe you out to zero equity. Any more and you’re going negative — you’re at zero equity and still owe money!

The Lessons

I don’t understand the crisis, but I’m getting a grasp on leverage: it’s the gravity that pulls down the dominoes. In fact, it can multiply the dominoes as they fall! Here are the key points:

• Leverage multiplies profits and losses: You can make a “regular” investment swing as wildly as you like by borrowing money.
• Return = leverage ratio * percent change: A meager 10x gearing ratio can double your money with a 10% gain, or wipe you out with a 10% loss. By the end of a crisis, some banks increased their leverage ratio to 30:1 — if prices fell even 3% they would be wiped out!
• Leverage appears everywhere: Companies have debt/equity ratios (how levered they are) and stock portfolios have beta (riskiness beyond the market average, which is increased by debt). Whenever you see debt or investment, look to see if it’s leveraged in some way.

Leverage make the boom times better and the busts harsher. The financial crisis has many other effects in play (such as the liquidity crisis, which makes it difficult to sell the assets you have to pay off your debts), but let’s take one idea at a time. A good friend found some podcasts on the crisis — if you’ve found a resource that helps you get the crisis, feel free to share it below.

Understanding these concepts will help understand finance (mortgages & savings rates), along with the omnipresent e and natural logarithm. Here’s our cheatsheet:

Term	Return	Description & Usage
Simple		Fixed, non-growing return (bond coupons)
Compound (Annual)		Changes each year (stock market, inflation)
Compound (n times per year)		Changes each month/week/day (savings account)
Continuous Growth		Changes each instant (radioactive decay, temperature)
APR	Annual Percentage Rate (compounding not included)
APY	Annual Percentage Yield (all compounding effects included)

And a quick calculator to convert APR to APY:

Why the fuss?

Interest rates are complex. Like Roman numerals and hieroglyphics, our first system “worked” but wasn’t quite ideal.

In the beginning, you might have had 100 gold coins and were paid 12% per year (percent = per cent = per hundred — those Roman numerals still show up!). It’s simple enough: we get 12 coins a year. But is it really 12?

If we break it down, it seems we earn 1 gold a month: 6 for January-June, and 6 for July-December. But wait a minute — after our June payout we’d have 106 gold in July, and yet earn only 6 during the rest of the year? Are you saying 100 and 106 earn the same amount in 6 months? By that logic, do 100 and 200 earn the same amount, too? Uh oh.

This issue didn’t seem to bother the ancient Egyptians, but did raise questions in the 1600s and led to Bernoulli’s discovery of e (sorry math fans, e wasn’t discovered via some hunch that a strange limit would have useful properties). There’s much to say about this riddle — just keep this in mind as we dissect interest rates:

• Interest rates and terminology were invented before the idea of compounding. Heck, loans were around in 1500 BC, before exponents, 0, or even the decimal point! So it’s no wonder our discussions can get confusing.
• Nature doesn’t wait for a human year before changing. Interest earnings are a type of “growth”, but natural phenomena like temperature and radioactive decay change constantly, every second and faster. This is one reason why physics equations model change with “e” and not “(1+r)^n”: Nature rudely ignores our calendar when making adjustments.

Learn the Lingo

As a result of these complications, we need a few terms to discuss interest rates:

• APR (annual percentage rate): The rate someone tells you (“12% per year!”). You’ll see this as “r” in the formula.
• APY (annual percentage yield): The rate you actually get after a year, after all compounding is taken into account. You can consider this “total return” in the formula. The APY is greater than or equal to the APR.

APR is what the bank tells you, the APY is what you pay (the price after taxes, shipping and handling, if you get my drift). And of course, banks advertise the rate that looks better.

Getting a credit card or car loan? They’ll show the “low APR” you’re paying, to hide the higher APY. But opening a savings account? Well, of course they’d tout the “high APY” they’re paying to look generous.

The APY (actual yield) is what you care about, and the way to compare competing offers.

Simple Interest

Let’s start on the ground floor: Simple interest pays a fixed amount over time. A few examples:

• Aesop’s fable of the golden goose: every day it laid a single golden egg. It couldn’t lay faster, and the eggs didn’t grow into golden geese of their own.
• Corporate bonds: A bond with a face value of $1000 and 5% interest rate (coupon) pays you $50 per year until it expires. You can’t increase the face value, so $50/year what you will get from the bond. (In reality, the bond would pay $25 every 6 months).

Simple interest is the most basic type of return. Depositing $100 into an account with 50% simple (annual) interest looks like this:

You start with a principal (aka investment) of $100 and earn $50 each year. I imagine the blue principal “shoveling” green money upwards every year.

However, this new, green money is stagnant — it can’t grow! With simple interest, the $50 just sits there. Only the original $100 can do “work” to generate money.

Simple interest has a simple formula: Every period you earn P * r (principal * interest rate). After n periods you have:

This formula works as long as “r” and “n” refer to the same time period. It could be years, months, or days — though in most cases, we’re considering annual interest. There’s no trickery because there’s no compounding — interest can’t grow.

Simple interest is useful when:

• Your interest earnings create something that cannot grow more. It’s like the golden goose creating eggs, or a corporate bond paying money that cannot be reinvested.
• You want simple, predictable, non-exponential results. Suppose you’re encouraging your kids to save. You could explain that you’ll put aside $1/month in “fun money” for every $20 in their piggybank. Most kids would be thrilled and buy comic books each month. If your last name is Greenspan, your kid might ask to reinvest the dividend.

In practice, simple interest is fairly rare because most types of earnings can be reinvested. There really isn’t an APR vs APY distinction, since your earnings can’t change: you always earn the same amount per year.

Really Understanding Growth

Most interest explanations stop there: here’s the formula, now get on your merry way. Not here: let’s see what’s really happening.

First, what does an interest rate mean? I think of it as a type of “speed”:

• 50 mph means you’ll travel 50 miles in the course of an hour
• r = 50% per year means you’ll earn 50% of your principal in the course of a year. If P = $100, you’ll earn $50/year (your “speed of money growth”).

But both types of speed have a subtlety: we don’t have to wait the full time period!

Does driving 50 mph mean you must go a full hour? No way! You can drive “only” 30 minutes and go 25 miles (50 mph * .5 hours). You could drive 15 minutes and go 12.5 miles (50 mph * .25 hours). You get the idea.

Interest rates are similar. An interest rate gives you a “trajectory” or “pace” to follow. If you have $100 at a 50% simple interest rate, your pace is $50/year. But you don’t need to follow that pace for a full year! If you grew for 6 months, you should be entitled to $25. Take a look at this:

We start with $100, in blue. Each year that blue contributes $50 (in green) to our total amount. Of course, with simple interest our earnings are based on our original amount, not the “new total”. Connecting the dots gives us a trendline: we’re following a path of $50/year. Our payouts look like a staircase because we’re only paid at the end of the year, but the trajectory still works.

Simple interest keeps the same trajectory: we earn “P*r” each year, no matter what ($50/year in this case). That straight line perfectly predicts where we’ll end up.

The idea of “following a trajectory” may seem strange, but stick with it — it will really help when understanding the nature of e.

One point: the trajectory is “how fast” a bank account is growing at a certain moment. With simple interest, we’re stuck in a car going the same speed: $50/year, or 50 mph. In other cases, our rate may change, like a skydiver: they start off slow, but each second fall faster and faster. But at any instant, there’s a single speed, a single trajectory.

(The math gurus will call this trajectory a “derivative” or “gradient”. No need to hit a mosquito with the calculus sledgehammer just yet.)

Basic Compound Interest

Simple interest should make you squirm. Why can’t our interest earn money? We should use the bond payouts ($50/year) to buy more bonds. Heck, we should use the golden eggs to fund research into cloning golden geese!

Compound growth means your interest earns interest. Einstein called it “one of the most powerful forces in nature”, and it’s true. When you have a growing thing, which creates more growing things, which creates more growing things… your return adds up fast.

The most basic type is period-over-period return, which usually means “year over year”. Reinvesting our interest annually looks like this:

We earn $50 from year 0 – 1, just like with simple interest. But in year 1-2, now that our total is $150, we can earn $75 this year (50% * 150) giving us $225. In year 2-3 we have $225, so we earn 50% of that, or $112.50.

In general, we have (1 + r) times more “stuff” each year. After n years, this becomes:

Exponential growth outpaces simple, linear interest, which only had $250 in year 3 (100 + 3*50). Compound growth is useful when:

• Interest can be reinvested, which is the case for most savings accounts.
• You want to predict a future value based on a growth trend. Most trends, like inflation, GDP growth, etc. are assumed to be “compoundable”. Yearly GDP growth of 3% over 10 years is really (1.03)^10 = 1.344, or a 34.4% increase over that decade.

Interest as a Factory

The typical interpretation sees money as a “blob” that grows over time. This view works, but sometimes I like to see interest earnings as a “factory” that generates more money:

Here’s what’s happening:

• Year 0: We start with $100.
• Year 1: Our $100 creates a $50 “bond”.
• Year 2: The $100 generates another $50 bond. The $50 generates a $25 bond. The total is 50 + 25 = 75, which matches up.
• Year 3: Things get a bit crazy. The $100 creates a third $50 bond. The two existing $50 bonds make $25 each. And the $25 makes a 12.50.
• Years 4 to infinity: Left as an exercise for the reader. (Don’t you love that textbook cop out?)

This is an interesting viewpoint. The $100 just mindlessly cranks out $50 “factories”, which start earning money independently (notice the 3 blue arrows from the blue principal to the green $50s). These $50 factories create $25 factories, and so on.

The pattern seems complex, but it’s simpler in a way as well. The $100 has no idea what those zany $50s are up to: as far as the $100 knows, we’re only making $50/year.

So why’s this viewpoint useful?

• You can separate the impact of the parent ($100) from the children. For example, at Year 3 we have $328 total. The parent has earned $150 (“3 * 50% * $100 = $150″, using the simple interest formula!). This means the “children” have contributed $328 – $150 – $100 = $128, or about 1/3 the total value.
• Breaking earnings into components helps understand e. Knowing more about e is a good thing because it shows up everywhere.

And besides, seeing old ideas in a new light is always fun. For one of us, at least.

Understanding the Trajectory

Oh, we’re not done yet. One more insight — take a look at our trajectory:

With simple interest, we kept the same pace forever ($50/year — pretty boring). With annually compounded interest, we get a new trajectory each year.

We deposit our money, go to sleep, and wake up at the end of the year:

• Year 1: “Hey, waittaminute. I’ve got $150 bucks! I should be making $75/year, not $50!”. You yell at your banker, crank up the dial to $75/year, and go to sleep again.
• Year 2: “Hey! I’ve got $225, and should be making $112.50 per year!”. You scream at your bank and get the rate adjusted.

This process repeats forever — we seem to never learn.

Compound Interest Revisited

Why are we waiting so long? Sure, waiting a year at a time is better than waiting “forever” (like simple interest), but I think we can do better. Let’s zoom in on a year:

Look at what’s happening. The green line represents our starting pace ($50/year), and the solid area shows the cash in our account. After 6 months, we’ve earned $25 but don’t see a dime! More importantly, after 6 months we have the same trajectory as when we started. The interest gap shows where we’ve earned interest, but stay on our original trajectory (based on the original principal). We’re losing out on what we should be making.

Imagine I took your money and returned it after 6 months. “Well, ya see, I didn’t use it for a full year, so I don’t really owe you any interest. After all, interest is measured per year. Per yeeeeeaaaaar. Not per 6 months.” You’d smile and send Bubba to break my legs.

Annual payouts are man-made artifacts, used to keep things simple. But in reality, money should be earned all the time. We can pay interest after 6 months to reduce the gap:

Here’s what happened:

• We start with $100 and a trajectory of $50/year, like normal
• After 6 months we get $25, giving us $125
• We head out using the new trajectory: 50% * $125 = $62.5/year
• After 6 months we collect 62.5/year times .5 year = 31.25. We have 125 + 31.25 = 156.25.

The key point is that our trajectory improved halfway through, and we earned 156.25, instead of the “expected” 150. Also, early payout gave us a smaller gap area (in white), since our $25 of interest was doing work for the second half (it contributed the extra 6.25, or $25 * 50% * .5 years).

For 1 year, the impact of rate r compounded n times is:

In our case, we had (1 + 50%/2)^2. Repeating this for t years (multiplying t times) gives:

Compound interest reduces the “dead space” where our interest isn’t earning interest. The more frequently we compound, the smaller the gap between earning interest and updating the trajectory.

Continuous Growth

Clearly we want money to “come online” as fast as possible. Continuous growth is compound interest on steroids: you shrink the gap into oblivion, by dividing the year into more and more time periods:

The net effect is to make use of interest as soon as it’s created. We wait a millisecond, find our new sum, and go off in the new trajectory. Except it’s not every millisecond: it’s every nanosecond, picosecond, femtosecond, and intervals I don’t know the name for. Continuous growth keeps the trajectory perfectly in sync with your current amount.

Read the article on e for more details (e is a special number, like pi, and is roughly 2.718). If we have rate r and time t (in years), the result is:

If you have a 50% APR, it would be an APY of e^(.50) = 64.9% if compounded continuously. That’s a pretty big difference! Notice that e takes care of the icky parts, like dividing by an infinite number of periods.

Why’s this useful?

• Most natural phenomena grow continuously. As mentioned earlier, physical phenomena grows on its own schedule: radioactive material doesn’t wait for the Earth to go around the Sun before deciding to decay. Any physical equation that models change is going to use e^rt.
• e^rt is the adjustable, one-size-fits-all exponential. It sounds strange, but e can even model the jumpy, staircase-like growth we’ve seen with compound interest. We’ll get into this in a later article.

Most interest discussions leave e out, as continuous interest is not often used in financial calculations. (Daily compounding, (1 + r/365)³⁶⁵, is generous enough for your bank account, thank you very much. But seriously, daily compounding is a pretty good approximation of continuous growth.)

The exponential e is the bridge from our jumpy “delayed” growth to the smooth changes of the natural world.

A Few Examples

Let’s try a few examples to make sure it’s sunk in. Remember: the APR is the rate they give you, the APY is what you actually earn (your true return).

• Is a 4.5 APY better than a 4.4 APR, compounded quarterly? You need to compare APY to APY. 4.4% compounded quarterly is (1 + 4.4%/4)^4 = 4.47%, so the 4.5% APY is still better.
• Should I pay my mortgage at the end of the month, or the beginning? The beginning, for sure. This way you knock out a chunk of debt early, preventing that “debt factory” from earning interest for 30 days. Suppose your loan APY is 6% and your monthly payment is $2000. By paying at the start of the month, you’d save $2000 * 6% = $120/year, or $3600 throughout a 30-year mortgage. And a few grand is nothing to sneeze at.
• Should I use several small payments, or one large payment?. You want to pay debt off as early as possible. $500/week for 4 weeks is better than $2000 at the end of the month. Each payment stops a few weeks’ worth of interest. The math is a bit tricker, but think of it as 4 $500 investments, each getting different return. In a month, the first payment saves 3 week’s worth of interest: 500 * (1 + daily rate)²¹. The next saves 2 weeks: 500 * (1 + daily rate)¹⁴. The third saves a week 500 * (1 + daily rate)⁷ and the last payment doesn’t save any interest. Regardless of the details, prepayment will save you money.

The general principle: When investing, get interest paid early, so it can compound. When borrowing, pay debt early to prevent that interest from compounding.

Onward and Upward

This is a lot for one sitting, but I hope you’ve seen the big picture:

• The interest rate (APR) is the “speed” at which money grows.
• Compounding lets you adjust your “speed” as you earn more interest. The APR is the initial speed; the APY is the actual change during the year.
• Man-made growth uses (1+r)^n, or some variant. We like our loans to line up with years.
• Nature uses e^{rt}. The universe doesn’t particularly care for our solar calendar.
• Interest rates are tricky. When in doubt, ask for the APY and pay debt early.

Treating interest in this funky way (trajectories and factories) will help us understand some of e’s cooler properties, which come in handy for calculus. Also, try the Rule of 72 for a quick way to compute the effect of interest rates mentally (that investment with 6% APY will double in 12 years). Happy math.

So What’s Accounting About, Anyway?

To be blunt, accounting is about tracking stuff (yes, there’s more to it, but hang with me). What kind of stuff can we track?

• Assets: “Stuff” inside the company
• Liabilities: “Stuff” that belongs to others
• Owner’s Equity (aka Capital): “Stuff” that belongs to the owners

Simple enough. Now how are these related?

Assets = Liabilities + Owner’s Equity

In layman’s terms, everything the company has belongs to the owners or someone else. Think of the equation like this:

• assets = liabilities + owner’s equity
• stuff the company has = other people’s stuff + owner’s stuff

This formula (also called ALOE) might seem strange at first. Why do we add liabilities? Because we’re looking from the point of view of the company, not the shareholders. If the company has something, it could be owed to someone else.

From the owner’s point of view, owner’s equity = assets – liabilities. This equation looks more natural, but often we aren’t interested in the owner’s point of view. We want to know about the company.

What’s a balance sheet?

A balance sheet is a document that tracks a company’s assets, liabilities and owner’s equity at a specific point in time. As you know, if the company’s has something, it belongs to someone. The sides must balance. So let’s do an example.

Suppose we start a company with $100 cash:

Assets:
  Cash: 100
Liabilities:
  None
Owner's Equity:
  Stock: 100

The company has $100 in short-term investments, and the owners have $100 worth of stock (how ownership is represented in a company).

Now suppose we take a bank loan for $150. The balance sheet becomes this:

Assets:
  Cash: 250
Liabilities:
  Loans: 150
Owner's Equity
  Stock: 100

Now our company has $250, but $150 belongs to the bank and $100 belongs to the owners. Sorry guys — you can’t take out a loan and make your share of the company more valuable.

Next, let’s buy a building for $200:

Assets:
  Cash: 50
  Building: 200
Liabilities:
  Loans: 150
Owner's Equity
  Stock: 100

Buying a building doesn’t make our company more valuable: we re-arranged our assets. Instead of $250 in cash, we have $50 in cash and $200 in “building”. Our share of the company ($100) didn’t change a lick. And we still owe the bank $150.

That’s not how it really works, is it?

It is. Well, real accountants use fancier terms (“accounts receivable” vs “deadbeats who owe me”), and have a bigger, badder balance sheet. But the core idea is the same: show what the company’s worth, and who owns what.

Take a look at the balance sheet for a small internet company:

Assets are broken into short-and long-term categories; the company is worth about $18 billion on the books (as of Dec 2006). This is up from $10B in 2005.

There’s many, many reasons why assets may be over or under-valued on the books. How do you measure momentum? Employee morale? A brand? Customer loyalty?

Accountants try to quantify items like this with intangible terms like “Goodwill”, but it’s not easy. In reality, most companies are worth several times their reported assets; Google’s market cap is over 10x the book value (but read more about stocks to see why market cap is not quite right).

Now examine the other side of the equation, liabilities and owner’s equity:

Wow — Google doesn’t have many liabilities! Only $1.4B (of the total $18B) and there’s no long-term debt. What it does owe are ”accounts payable” — the equivalent of a credit-card bill (usually paid within a short timeframe).

Now you can examine a company and see what it’s worth (on paper) and where the value lies. Google has no “inventory” (ever bought an off-the-shelf product from them?) but has a lot of cash, investments, and equipment. There’s very little debt and other liabilities, so it seems like a very stable company on paper; they won’t be going bankrupt anytime soon (there’s other documents that show how profitable the company is).

Blockbuster, for example, has 2.5B in assets but 1.9B is owed to others (saved balance sheet here). Shareholders aren’t left with much. In fact, it has 700M in “intangible assets”, so it actually has a negative amount of real, tangible assets. Not a good sign — if you liquidated the company today, it couldn’t pay off its debt.

The Rules of the Game

Accounting has many rules, but a basic one is this: use double-entry bookkeeping.

This fancy term means that all changes happen in pairs:

• If assets go down, liabilities or owner’s equity should decrease also
• If assets go up, liabilities or owner’s equity must increase as well

Every change to assets must have a corresponding change to keep the equation in balance. There’s a formal system of “debits and credits” that describes these changes, but the concept is simple: if you make a change to one side, you must make one on the other as well.

There’s More to Learn

There’s much more to accounting, but you’ve got an idea of the basics:

• If a company has something, someone had better own it
• A balance sheet lists assets, liabilities and owner’s equity at a point in time; everything must add up
• Changes must be made in pairs: if assets, liabilities or owner’s equity changes, something else much change as well

Any system can be interesting (even “fun”) if you look at the reasons it was created and the problem it’s trying to solve. Could you have made a simpler way to report what a company is worth and who is owed what?

Enjoy.

If you’re like most of us, probably not. Here’s why stock markets rock:

• They match buyers and sellers efficiently
• All prices are completely transparent and you see what other people have paid/sold for
• You pick your own price and will get that amount if there’s a willing partner

Most explanations jump into the minor details — not here. Today we’ll see why the stock market works as it does.

iPods Ahoy!

I’m told iPods are popular with the 18-35 demographic. A market research firm asked me to find a good selling price, so I’ll pass the question onto you:

Me: You, the coveted 18-35 year old demographic, want an iPod. What’s it worth?
You: Dude, just get the price. Duh.

Ok hotshot, riddle me this: what is the price, exactly?

• What you can buy it for? (Your best bid)
• What you can sell it for? (What you’d ask for it)

So which price is the “real one”? Both.

You see, buyers and sellers each have prices in mind. When prices match, whablamo, there’s a transaction (no match, no whablamo).

The idea of two prices for every item is key to understanding any market, not just stocks. Everything has a bid and an ask, and each shopping model has a different way of handling them. This leads to different advantages for buyers and sellers.

Shopping Time

Suppose we want to buy an iPod from Amazon. You see the selling price of $200 (Amazon’s ask), and personally decide if it’s “worth it” (i.e. less than or equal to your bid):

In the store model, Amazon shows a public asking price ($200). Each buyer has a secret bidding price, some more than others. Buyers willing to bid $200 or more purchase the iPod; the rest hold off ($199 and below).

Amazon picks a price that attracts the most bidders yet still keeps a profit. In the store model:

• Buyer pro: Buyers know the price and can pay less than their internal value
• Buyer con: Buyers have to visit multiple stores to find the best price
• Seller con: Sellers don’t know what each buyer is willing to pay; it’s difficult to set the pricing. Do low sales mean a bad price or a bad product?

Even though buyers are “in control”, they may have to search around to find a store that meets their bid (if any). That’s inefficient.

Onto eBay

Now suppose we want to sell our new, unopened gadget (you, the 18-35 demographic, are fickle like that; the survey said so). Sure, we could try to sell it on Amazon — now we’re our own store and need a price we think people will pay. We’re in the same boat as Amazon, and could set the price too low. That’s no fun.

Instead, we auction off the new iPod on eBay to maximize profits:

In the eBay model, buyers have public bids and compete for the product. The seller keeps their minimum price secret and hopes to make a profit by having someone “overpay”. In the auction model:

• Seller pro: Sellers have a secret ask (reserve or minimum price) and can get paid above this.
• Seller pro: Buyers’ demand is transparent. They can easily see if they are pricing too high.
• Buyer con: Difficult to buy a product.

eBay is great for sellers — you have the chance of making extra profit. For buyers, it’s not so great: you can lose auctions by $1 (paying 201 when 202 was the highest bid), even though the seller would have been happy with 201. You could enter multiple auctions with $201 but risk getting two iPods.

Want Ads and Hagglers

There’s other trading approaches also:

• Want ad: Publicly announce your desire for an iPod and let sellers fight it out.
• Haggle: Find someone with an iPod, and without knowing a selling price, make an offer. You both haggle back and forth, trying to eke the other person out of a few bucks. If you’ve gone car shopping you know how fun this is.

In want ads, the asks are transparent while the bids (your value) are hidden. When haggling, both prices are hidden which can lead to a stressful situation.

It’s About Supply and Demand

Each model has similar concepts, namely:

• Supply: sellers provide asks
• Demand: buyers provide bids

The phrase liquidity refers to how effectively you can trade; how easily cash can flow. When buyers and sellers have to argue or haggle, trading freezes up. In particular, there’s a common problem in the market above:

• There’s secret prices and a lack of transparency
• There’s multiple vendors and a lack of consolidation

When buyers and sellers need to search to find each other, and haggle when they get there, trading slows down.

Enter the Market

But hope is not lost! Surprisingly, the very symbol of capitalism is an “open source” model:

• All prices are transparent
• Buyers write public bids (buying price)
• Sellers write public asks (selling price)
• There’s one location to get a particular stock; there’s no searching
• Dealers/specialists help match buyers and sellers

And here’s what it looks like:

Every iPod seller lists their asking price (210, 205, 201, 200). Every iPod buyer lists their buying price (190, 195, 199, 200). When prices match, a transaction happens: the buyer who wants to pay 200 gets matched with the seller who wants 200. They’re happy.

Eventually the matches cease and we come to a standstill.

Drop and Spread ‘em.

Trades don’t last forever: there’s a standoff and an awkward pause. The lowest sellers want $201, and the highest bidder wants $199; this $2 gap is called the spread. The last price of a transaction was $200.

Now what happens? Buyers and sellers can do:

• Limit order: put their bid/ask in the queue.
• Market order: buy or sell immediately.

When you place a limit order (“Buy an iPod for 195″), your order gets added to the bid queue (similar for asks).

If you need to trade right now (“buy it now!” or “sell it now!”), then you use a market order. You’ll get the best price available:

• Market order to sell: You can unload your iPod for $199 (the highest bid). The “last” price is now 199.
• Market order to buy: You can buy for $201 (the lowest price). The “last” price is now 201.

Now this is interesting. Notice how market orders take items off the queue and change the last price. When people place market orders, the stock price fluctuates. Yes, it’s “just” supply and demand, but it’s pretty cool to know it’s happening real-time in the stock market.

If there’s a lot of buyers, they’ll “use up” the ask queue and the price will rise. If there’s a lot of sellers, they’ll “use up” the bid queue and the price will fall.

This explains why it’s hard to buy and sell for the same price. If you buy for 201, and no new bids come in, you’ll only be able to sell for 199.

So Who Runs This Popsicle Stand?

The NYSE and NASDAQ are the two major American exchanges. There are differences, but at the core they provide:

• A single market to trade. All stocks for Microsoft (MSFT), are traded on the NASDAQ exchange. All stocks for Ford (F) are on the NYSE.
• A market maker or “specialist” (not the kind that kills people). These people make the market liquid: they help collect and match bids and asks. The NYSE has one specialst per stock; NASDAQ has several market makers (dealers) who compete on price.

How Do They Make Money?

Well, often they don’t. In the NYSE, 88% of the trades happen between the public without needing the specialist (remember those guys waving papers and screaming at each other? I wouldn’t want to get involved with them either).

But sometimes they are needed. The market makers literally “create a market” by providing liquidity: you can buy and sell stocks to them at the bid and ask prices. Popular stocks have a small spread due to the demand and volume.

But how do market makers make money?

Well, it’s a bit like a currency exchange at a bank, where’s there’s a different rate for buying and selling. Let’s say Sue has an iPod to sell, and Bob wants to buy an iPod. It might go like this:

• Hey Sue, I’ll take your iPod. Here’s 199.
• Hi Bob, I’ll sell you an iPod. That’ll be 201.

See what happened? The market maker bought an iPod for 199 and sold it for 201: it pocketed the spread of $2. Dealers constantly change their prices based on the bids and asks; they can even lose money depending on the trades coming in. But usually it’s a pretty good gig.

You, the investor, can avoid paying “the spread” by placing limit orders to sell or buy at a certain price. But then you aren’t guaranteed to make a trade.

It’s All About Timing

Bill Gates has a lot of shares of Microsoft. People naively put this wealth as “shares times price”, but you know that doesn’t really work. If he tried to sell all his shares, he’d use up the bids.

Each block of shares would be sold for a lower and lower value — and potential buyers would panic and reduce their bids, thinking something was amiss. Sellers would fear the worst and lower their asks to compete. Pandemonium would ensue. So the actual liquidation value of his shares is really some fraction of the reported amount. But it’s still nothing to sneeze at.

Similarly, large institutions must spread their stock trades over time so they don’t disrupt the market (and evaporate their profits).

The market has built-in shock absorbers: as you sell more, the price you get is smaller and smaller, so you sell less. As you buy more, the price you pay gets higher and higher, so you buy less. So it makes sense to take things slow. Nifty.

There’s Much to Learn

I’ve simplified a lot of things and only scratched the glossed-over surface. Each market has its own rules to create a trading-friendly environment. Read more here:

• Invest-faq on the NASDAQ and NYSE. The NYSE is an “auction market” where bids and asks are public (this is different from eBay auctions, where only bidders compete in a given auction). The NASDAQ is a “dealer market” where you buy/sell from a dealer’s personal inventory.
• Investopedia on the difference between a market maker and specialist
• See the current bid/ask for Microsoft or Google (and # of shares at that price)

But, my goal wasn’t to fill your head with details. I want to share insight:

• Markets exist to match supply and demand
• The stock market is fast, transparent, and efficient
• Every stock has a bid and ask
• Buying or selling changes the trading price in a direct, measurable way

Want a stock tip? Don’t listen to stock tips. (Stolen from a Charles Schwab ad). This article is about looking at a system as one way to solve a larger problem. Happy investing.