# Stocks, Options, and Dividends

I've been thinking lately about stock prices and the value of employee stock options. This is based on my own brainstorming, so some of it may well be obvious or completely wrong - if so, I rely on y'all to correct me.

The basic question I set out to answer was how to approximately value a stock option, which of course depends on the future price of the underlying stock. I wasn't looking for an exact solution or fancy math, but merely a general, intuitive idea of things like "What is the chance that this option will be in the money in a year? 2 years?" Which of course boils down to "What is the chance that the stock will be higher? Lower?".

For some reason I had taken the efficient markets hypothesis to mean that, since stock prices are a random walk, the chance of a stock being higher in the future should be exactly 50%. But this turns out to be completely wrong when there is growth or retained earnings, both of which are common. Lets see why.

First, we'll consider a company that has no growth prospects and distributes all earnings as dividends, hence pays steady predictable dividends forever. Essentially it is a riskless annuity. As Brandon Berg pointed out to me a few months ago, the price of such a security fluctuates, rising as you get closer to the dividend date, then dropping when the dividend is issued. (A stock that will give 1$ every January 1st is worth just about a buck less on January 1st than on Dec. 31st). Since its long-term future never changes, an option would be worth nothing.

Now suppose we add some uncertainty to this basic model. Earnings have no expected growth, but they have some random fluctuation, and earnings are announced before dividends are distributed. Now options have some value because they are a freeroll - if earnings are higher than usual, stock price will jump and the option will have value. If they are the same or lower, the option is worth nothing. This is where fancy mathematical options pricing comes in, and that's exactly what I want to ignore, so let's forget about that component of option value. The component I am curious about is that which comes from an expected increase in stock price.

Expected increase in stock price? But isn't that contrary to the random walk theory? Couldn't you make money trading on that stock? No and No are the answers.

Earnings growth is the first factor that can lead to an expected increase in stock price. Suppose that a company's dividends are steadily increasing - the market expects $0.10/share this year, $0.20 next year, $0.40 the year after that, and $0.80 the year after that. Just as the price of the annuity goes up as you get closer to the payout, the price of this growing company should go up as you get closer to the big payouts. The expected value of the future stream of dividends is not constant because $0.20/share in a year is worth more than $0.20/share in 2 years. In fact, the price difference is very convenient, since its just the interest rate. So the stock price should go up at the interest rate, with jumps down when each dividend is issued. And of course, there is no money to be made knowing that a security will rise at the interest rate!

The other situation is a company which retains earnings. Remember the yearly fluctuations of that dividend producing stock. As you get closer to the payoff date, the stock goes up in value because you're gonna gaid paid sooner. Another, more value-based way of looking at it is that the company is banking its earnings, hence is worth more. When it distributes them, its book value has dropped, and so does the stock price. What happens if it never distributes the dividend? Its book value should just keep going up, and hence its share price.

How fast should this share price go up? Here we have a subtle disconnect between book value and share price, and the EMH finally gets to have a say. First, suppose we approximated share price for a company which never distributed dividends as book value. That is, the stock is still worth something even without dividends because the company might someday get sold, and book value should be a reasonable approximation for sale price. But if growth and retained earnings make book value grow faster than the market rate of return, we run afoul of the EMH, since knowing this actually would let a trader make money. However, the EMH does give us a nice simple answer to the question of how fast such a growth company's stock should increase - at exactly the market rate of return. No one would bid it up faster than that, or they'd lose money, and if they bid it up slower than that, they would miss out on profits. The actual value of the stock, the constant that is growing at the interest rate, depends on the exact future stream of book values or dividends. But given that starting point, the stock price should grow at the interest rate.

So, back in an uncertain world, we can see that the value of a stock option is not due simply to its "heads I win, tails I break even" nature. There is another component, which is that for a company with growth or retained earnings, we actually expect the stock price to go up, and hence the option to have value. As the time horizon gets longer, and the chance of being out-of-the-money approaches zero, the main determining factor shifts from the first to the second.

In other words, an employee stock option recipient should not look at it as a pure gamble, something that will be worth nothing half the time. Rather, they should expect the stock price to increase in value at approximately the interest rate for its level of risk (with lots of fluctuations along the way), and hence the option value accordingly. You can look at it as earning the market rate of return on the "base value". So if you are given 1,000 options at $100, they should on average earn the market rate of return on $100,000 (plus risk).

Now we can see part of why tech companies don't distribute dividends, and the odd difference in incentives between options holders and share holders. First, as rapidly growing companies they believe they can invest the money internally at a higher than market rate of return, hence shareholders win from retaining earnings. But second, every penny they distribute to shareholders reduces the stock price, which costs the shareholders nothing, but transfers money directly from options issuers to options holders (if the options are in the money).

So there is a serious conflict of interest between options holders and share holders about whether to distribute dividends. In terms of power, on the one hand the share holders have all the power, since they can vote. But on the other hand, if options are used heavily for compensation, morale at all levels, including senior management, is likely to be greatly hurt by this. It looks like sort of an implied contract - which makes me wonder why it isn't made into an explicit contract, perhaps a bylaw of the corporation that they will not distribute dividends for X years.

Another, much simpler factor is that inflation obviously benefits the options holder, since the strike price is in nominal dollars, but the share price is in real dollars. Just as inflation benefits debtors at the expense of creditors, it benefits options holders at the expense of options issuers. For an in the money option, the strike price is essentially a type of nominal-valued debt owed on a real-valued security, so naturally inflation acts in its favor.

## Jason: The value of the firm

Jason:

The value of the firm is essentially Net Present Value minus Investments. Dividends don’t factor in at all, thus whether or not a firm pays out a dividend is irrelevant.If you buy the stock of a company that's paid dividends in the past, you're buying future earnings. If you buy the stock of a company that's retained all past earnings, you're buying future earnings and past earnings. The latter is obviously worth more.

I don’t think that inflation would benefit the option holder, because the seller of the option can proceed to reinvest her income at a higher rate then the option holder, whose cash is locked up.The problem is that the value of options increases much more quickly than the value of the underlying stock. If you have an option with a $10 strike price on an $11 stock, a 9.1% increase in the stock's value will double the value of the option (ignoring time value, which doesn't matter when you can't sell the option).

At any rate (no pun intended) if there is any discrepancy, then you would expect it to be rapidly arbitraged away.Unlike options purchased on the market, employee stock options usually have have a term of several years (mine---which, incidentally, are now worthless due to my company's recent decision to issue a huge dividend---last either seven or ten years). If inflation increases unexpectedly during this time, the company can't renegotiate the terms of the options to adjust for it.

Brian:

If a company were guaranteed to retain earnings forever and never liquidate its assets, its stock would be worthless. What retained earnings do is (in theory) allow a growing company to invest its earnings internally and allow shareholders to earn a better rate of return than they would be able to earn elsewhere if the earnings were distributed. Eventually, the company will no longer be able to do this, at which point it should begin to distribute earnings as dividends. Because the company has grown so much, these dividends should be large enough to make up for the fact that it never distributed dividends in the past.

In other words, the NPV of the expected dividends of a company which does not currently distribute earnings should be higher than the NPV of the expected dividends of a company which does currently distribute dividends. If this isn't the case, then the company can't use its cash efficiently and has no business retaining earnings.

Of course, taxes complicate the issue. When a company distributes earnings, shareholders have to pay taxes on the dividends. When it retains earnings, they don't, which increases the rate of compounding.

Also, who threw Stephan's comment down the memory hole?

## Options typically have

Options typically have little to no value at time of grant, so I don't see how that would cause a tax problem. The nightmare I've heard has to do w/ a combination of AMT and doing exercise-and-hold. w/ AMT, tax is assessed at time of exercise, so if you have giant profits at exercise time, and then the stock goes down, you end up w/ lots of tax and no capital gains. I assume this is the same situation you are talking about.

## Patri, Not to address your

Patri,

Not to address your theoretical issues, all practical issues with equity (and option) compensation must take both corporate and individual taxes into account.

You may or may not be aware that thousands? of employee recipients of stock and option grants have been assessed income taxes based on values at the time of grant, and have subsequently ended up with nothing of value except a enormous tax bill. This has often been true even if the employees had no chance to ever realize the temporary value of their taxed 'holdings'. I'm not familiar with the details, but thought a warning might be appropriate.

Regards, Don

## Jason- If I'm reading Patri

Jason-

If I'm reading Patri correctly, there is at least the potential for a difference between internal ROI and market ROI, so the decision of the company to issue dividends or not would have a material effect on NPV, no? If the value of the stock is as you say (given the theorems, value is NPV net of Investments) then if they issued dividends the expected income flow would be part of the NPV and thus, I suppose paradoxically, *increase* the stock value, while if they retained the earnings the stock value would go down.

Since that seems to be opposite of what Patri's saying, I'm pretty sure I'm mistaken about something here, but it certainly seems to be paradoxical on first glance...

## Patri, This is really,

Patri,

This is really, really back of the envelope, but look at the company's earnings estimates, multiply them by the current P/E ratio and subtract your strike price. Full of holes, yes, but easy.

GE has a mean '07 estimate of 2.291 and a P/E of 19.318. It is currently trading at about 34 and has averaged about the same for '05, so let's assume that your strike is also 34. Based on this simple method, it should trade at 44 in '07 meaning that your option today could be worth 10.

Ford has a mean '07 estimate of 1.275 and a P/E of 5.279. It is currently trading at about 8.70 and has averaged about 10.50 the same for '05, so let's assume that your strike is also 10.50. Based on this simple method, it should trade at 6.75 in '07 meaning that your option today could be worth -3.75.

If the company does not have earnings estimates or is not public, find one the is comparable and use it.

## Patri, Just a couple of

Patri,

Just a couple of quick thoughts.

1) While we're operating in the world of Perfect Capital Markets & EMH, Modigliani Miller Theorem states that how a company chooses to distributes its income, either via retaining it or issuing dividends, makes no difference to the company's valuation. The value of the firm is essentially Net Present Value minus Investments. Dividends don't factor in at all, thus whether or not a firm pays out a dividend is irrelevant. Of course, numerous assumptions abound. Technically, shareholders should be indifferent.

2) The problem with most option pricing models is that they are horribly non-intuitive, to me at any rate. The thing about the Black Scholes model is that it explicitly does away with "expected increase in stock price". We just need to solve for "implied volatility" - we are given the strike price, expiration date, current market price, and the historical volatility of the stock.

3) I don't think that inflation would benefit the option holder, because the seller of the option can proceed to reinvest her income at a higher rate then the option holder, whose cash is locked up. At any rate (no pun intended) if there is any discrepancy, then you would expect it to be rapidly arbitraged away.

## Tanger, My company did not

Tanger,

My company did not have any policy about LEAPs per se, but most companies have some sort of ownership disclosure agreement with employees. They don't want you to influence/be influenced by companies in which you are a significant owner.

Technically, you may not qualify as an investor for regular tradable options even though you have employee stock options because the employee stock options are different than regular stock options. Employee stock options are a reward for good behavior that are non-transferable and only saleable or exercisable through a designated custodian. Regular stock options are only available to investors with a fair amount of liquidity and sophistication.

If your company has not yet gone public, you could negotiate your own long term put contract with a financial intermediary, but it would be expensive.

## Brandon, Apologies, I

Brandon,

Apologies, I didn't make this clear in my original post: everything there was in reference to the theoretical world of perfect markets and everything else. In the real world, yes, but I was talking in reference to the world of Modigliani Miller, i.e. no taxes, infinitely divisible shares, no transactions costs, etc. In that universe, investors are dividend indifferent because they can create their own synthetic dividends, selling stocks that pay too low a dividend, and buying stocks that give too high a dividend. Obviously, that isn't quite so useful in the real world, but I think it's important to figure out where the more bizarre implications of option pricing come from.

If you're worried about interest rates, you don't exactly care about the underlying price movement, you focus on rho (the % increase of the option relative to change in the risk free interest rate). But all the greeks try to measure how changes in the market affect implied volatility, the only real unknown in the option. And it is endlessly furstrating that the greeks always change.

I'm not familiar with employee stock options, unfortunately - though if you have a publicly traded company, and you are worried about your company paying a huge dividend, wouldn't you be able to offset the option you have by selling the opposite of that option (there are LEAPS out there that should match the maturity date of your option)? Or is there a company policy against that?