Probably the hardest decision for most stock investors is knowing when to buy or sell a stock. Some investors trade at a fast rate, buying and selling very actively. Others buy stocks on a regular schedule, but don't know when, if ever, to sell. Some investors believe the answer is to "never" sell, buying and holding essentially forever. There is a whole spectrum of philosophies and approaches.

What makes the most sense? Is there even a single right answer?

In order to get started thinking logically about this all-important issue, let's create a simple model of how stock prices change. The model is idealized and represents no real stock, but it is a powerful tool for thinking about the questions of when to buy and when to sell.

Here's the model:

Picture a simple sine wave, with a horizontal line straight through the center of it. The straight line represents time, while the sine wave represents the changing price of your stock over time. The price starts at the left end of the timeline, or "time = 0," which could be right now. The sine wave starts at the centerline, rises for a while, levels off at a peak, declines for a while, passes down through the centerline (so that the price goes below where it started), levels off again forming a trough, rises smoothly back up through the centerline, goes on to another peak, and so on. Each full rise, fall, and re-rise to the centerline is a cycle.

Anybody familiar with stock price movements knows that prices are volatile. They go up, they come down. None of them, of course, traces a perfect sine wave shape, but the sine wave picture is a simplifying assumption: It is a smoothed-out version of what stock prices actually do.

For our idealized model, let's say that each peak in the cycle is 20% above the centerline, and that each trough is 20% below the centerline. So there is a 40% difference between the peak price and the lowest price of each cycle. That happens to be the difference in the real world between many stocks' high and low prices for a year. So in our model, let's make each cycle one year long.

Finally, tilt the whole thing upwards slightly, so that the centerline, rather than being horizontal, is pointed upward at 10% per year. This represents the average return of the stock market over the past century or so.

That's our idealized model. Let's call the company that it represents Sine, Inc. Sine's stock has behaved like this since the company went public 100 years ago, and it will behave like this infinitely into the future.

What can we learn from this simple model? Plenty!

Question: What would be the ideal times to buy and sell Sine? There are at least four good answers:

(1) Since we know that the model is tilted upwards at 10% per year, just buy the stock at time = 0 (when the sine wave is at the center line) and hold it as long as possible. Or if you are buying Sine in chunks over an extended period as money becomes available, you can make your purchases at any time. You don't care where Sine is in its cycle, because you know that, over time, you'll make 10% per year on your average chunk. You know that because the centerline is tilted upwards at a 10% grade. There's a name for this approach: Dollar cost averaging. You buy, say, $100 of Sine every month, so you're buying it at every point along its cycle, sometimes getting a good price, sometimes not. Your blended return from all those purchases, however, will match the 10% upward tilt of the chart itself. This is a widely recommended approach.

(2) But you can do better. Wait a few months and purchase the stock at the exact bottom of its price cycle. There's a name for this approach too: Buy on the dip. That will increase your returns by a surprising amount, because you will get more shares for your money. For example, if Sine's price is $100 at time = 0, and you wait nine months until the cycle hits its low point at $80, then $1000 will get you 12½ shares instead of 10. That's 25% more shares for the same amount of money. You'll benefit from those extra shares forever. By the way, this is exactly what value investors aim to do. This is also a widely recommended approach, although in the real world it is impossible to know exactly when the exact bottom of the cycle has been hit.

(3) Next, let's surmise that you have perfect knowledge about Sine's price behavior and know that it is going to keep repeating its steady performance year after year, cycle after cycle. Then you can improve on #2 above. Buy at the bottom of a cycle, hold until the top of the cycle, sell right there, bide your time for six months until the next bottom, re-buy, sell at the next top, and so on. Your returns would be astronomical. Let's just follow this through two years of the cycle (and make it simple by ignoring the tilt). Your first purchase would get you 12.5 shares at $80 each, same as in #2 above. At the top of the cycle (six months later), you would sell those shares for $112 each (40% more than you paid), or $1400 total. Wait six months for the next trough, and that money will buy you 17.5 shares at the bottom of the cycle. Wait six more months, and the sale of the 17.5 will bring in $1960 at the top. Wait six more months and the $1960 will buy you 24.5 shares. After 6 more months, the cycle will reach another peak, and your shares will have gone up 40% again to $2744. And so on. In the first 18 months from time = 0, you make 174% ($2744 divided by your original $1000), and every year after that it gets better and better as everything compounds. And that's ignoring the 10% upward tilt, which brings in even more money. Your $1000 will turn into $1,000,000 in just a few years, even though on average the stock's price is rising just 10% per year. The name for this is timing or trend following.

(4) Actually, trend following has an additional component that increases returns even more. Rather than biding your time during the downward portion of the cycle, many trend followers simply reverse their position from long (owning the stock) to short (betting against the stock). That way, they make money when the stock's price is declining. This adds more orders of magnitude to the theoretical returns of our idealized model.

OK, that is the model. Now let's inject reality into the model and see how it impacts our real-life decisions.

First and most obviously, no one knows the future. No stock's price in the real world is guaranteed to have a clear trend line pointing upwards at 10% a year, or any other percentage. No one can foresee with confidence that such a trend will last into perpetuity. We only can look at what has happened in the past and try to discern probabilities of what is likely to happen in the future.

Second, stock prices in the real world do not follow smooth sine waves. Their movement is jagged, subject to sudden reversals, unclear as to both long-term and short-term trends, and certainly not as predictably cyclical as is our idealized model.

Nevertheless, the model points to a systematic method of looking for advantageous buy and sell points. The model epitomizes "buy low and sell high."

For the Sensible Stock Investor, the model tells us that value investors have it figured out exactly right on the "buy" side of the equation. Wait for a low valuation, a low price, the price at the very trough of its cycle. That's the best time to buy a stock.

How do you know when a stock has hit its low? You don't, exactly. But by insisting on favorable valuations-when the stock's price appears to be low compared to the long-term value of the company-you can come pretty close to buying at the trough of the sine wave. Some investors wait for a turn upwards from an actual recent low price to confirm that the stock just hit its low. They are willing to forego a little bit of the upturn in return for a little more certainty that the stock isn't going to keep going down after they buy it.

How do you know when a stock has hit its high? You don't know that exactly, either. But when valuations become high, when they suggest that the price is too much for the underlying worth of the company, it becomes more likely that the stock is approaching a peak, and that the market will soon "correct" the price of the stock. What the Sensible Stock Investor can do is use a trailing sell-stop to protect himself or herself on the downside. Set the stop at, say, 15% below the stock's current price. Reset it once per week, and keep moving it up as long as the price keeps going up. If the stock starts to reverse, you can depend on your trailing stop to get you out before too much damage has been done. In practice, you will sometimes find (especially for stocks that are not especially volatile) that your trailing sell-stop never triggers a sale, and you become a long-term holder of an excellent stock. In other cases, your trailing stop will trigger and preserve most of your gains if the stock's price goes into a protracted decline. You will have caught and sold the stock near the peak of its cycle.