First, scroll down and read the previous post.
OK, now that we're all on the same page here, I'd like to write a few things about a stock pricing model known as the Dividend Growth Model, or DGM.
Since I know that non Math-Finance-Economics people read this, I'll take this slowly. What is a stock pricing model? Well, stock pricing models attempt to take financial data about a company and come up with a fair value for their stock price. The DGM basically says the following: "The value of a share of stock XYZ is equal to the present value of its future dividends."
Present value (PV) is money adjusted for interest and inflation. For example, if interest rates are 10%, $110 one year from now is worth only $100 today. This is because you could put $100 in a 10% account today and get $110 in one year. Cute, huh?
I think that the DGM is a bunch of bunk. Why? OK, let's look at the reasons here.
The model says that the value is equal to the PV of all future dividends. That means to infinity. How many of you actually buy anything with the idea that it's going to last to infinity? That isn't realistic in the slightest. Most investors are buying for a shorter horizon than that. Realisitically, you're probably buying stocks for retirement, which should happen in less than 50 years, unless I have much younger readers than I thought I would.
A defender of the DGM would probably explain the model like this: Let's say you buy a share of stock and hold it for one year, during which time you recieve 4 quarterly dividends. Even though you sell it at the end of a year, the price you would get for the stock when you sold it would still be equal to the PV of all future dividends.
Again, I claim that is bunk. It's circular logic. You have to assume that the model works in order for the model to work. Otherwise, the price at the end of the year (or time n for that matter) could be a function of interest rates, consumer confidence, Survivor ratings, womens' skirt lengths in Milan, and a bunch of other factors. Iff (if and only if) you assume that the price at time n exactly equals the PV of all future dividends does the DGM work.
What really frustrates me is that this model is frequently used. Although convenient and easy to calculate (the formula boils down to D1/(k-g), where D1 is the dividend at time 1, k is the required rate of return, and g is the growth rate of dividends), the assumptions of the model are flawed to me, and so I can't really justify its use.
So why is it used? It sounds plausable. I mean it might make sense to think that the value of a stock is worth all its future dividends. It also tends to spit out semi-correct answers (aka prices close to the market price) if you guess around with k and g. In fact, there are reams of literature about how to determine k and g such that they come close to the market price. Most work pretty well, actually, if you use them. Still, I can't really justify the assumptions of the model, and so it seems like a stab in the dark as to whether or not it is correct or not. If you stab enough times in enough dark places, you'll eventually hit something (let's hope not someone).
There are other models which appear to have more solid foundations, so don't worry about the stock market crashing when I hit the publish button to my lower left.
I think that the DGM can be fixed, but I am unsure of how to do that. This may turn into a thesis topic for me, since I had to work with this model extensively this semester. So for all of you out there who I told that my thesis idea was too technical to explain, you now see why. Imagine how much you'd now hate me if I had spent an entire dinner explaining this to you, rather than letting you read it at your leisure. You're welcome.
Overall, the DGM is a flawed model in my eyes and I can't really see why it is used. Maybe I just need to spend a little more time in professors' offices arguing this with them and then I'll understand. Until then, I'm just an insolent multi-disciplinary student who pokes holes in age-old models and doesn't take assumptions lightly.
Monday, December 12, 2005
The Dividend Growth Model
Subscribe to:
Post Comments (Atom)
6 comments:
As I jumped up out of my chair, about to thrust my fist emphatically in the air and shout, "I AGREE!!," I suddenly realized that I am not a financial gnome running around muttering dividend yields with a calculator hidden in my pointy hat. To avoid severe embarrassment, I looked wildly around to make sure there were no real gnomes who would recognize my brazen outburst as pure hogswallop, and then I set to work redeeming myself.
Yes, I admit that my emotions overtook me, but they were merely indicators of my theory that all stock pricing models use unfounded assumptions.
First, I'd like to try and restate the axom of the DGM to make sure I understand it. Stock value now is based on a hypothetical return amount x, calculated by interest rate. Hmm. That was a bit more difficult than I had imagined. It does seem to be saying that the present is determined by the future, which, as you say, is circular logic.
This leads me to a new point, however. How do you determine the best price for a share of stock? My research (assisted by Google) indicates that other models start with observing a pattern in a set stock-growing timeframe. The Binomial Model starts with an expiration or selling date in mind and works backwards from that. Another model, the Black-Scholes (or option pricing theory), uses Browninan motion to establish a stock behavior pattern to work from.
But they still need a place to start. Though the Black-Scholes model comes from a lognormal distribution graph (a smiley bell curve), it still has to make certain assumptions: "1) The stock pays no dividends during the option's life, 2) European exercise terms are used, 3) Markets are efficient, 4) No commissions are charged, 5) Interest rates remain constant and known, and 6) Returns are lognormally distributed."
I won't go into the formula, which is quite intimidating, but one of the variables in it is volatility. Volatility is by nature unpredictable. Peter Hoadley's options analysis website even says: "Whilst implied volatility -- the volatility of the option implied by current market prices -- is commonly used, the argument that this is the "best" estimate is somewhat circular." In other words, it's very complicated to get a good estimate. And how do you use current market price to determine current market price?
How about the Binomial Model? This model starts with the current market price and calculates possible future prices by steps up to the expiration date. Then, it reworks the possibilities going back down the "tree," adding in adjustments, and ends up with one stock option price again. I'm not sure how this determines a new value. Plus, each step on the tree is calculated again using volatility.
Please understand that although I did a little research into this, I am still far from being a stock gnome. I could very well have misinterpreted or misunderstood more than one concept. The only stock I really know is my stock-ing. However, I did feel the urge to put on a tall pointy hat.
~your Evenstar
I think that you fail to understand the dividend growth model fully.
While it is true that the dividend growth model is incomplete due to the fact that it focuses purely on a single aspect of a stock and ignores a variety of factors such as capital appreciation, book value, cash flows, etc. And of course, it obviously doesn't work on stocks that have no dividend. However, that does not mean that the theory behind it is flawed.
If we assume for a moment that the dividend model is adequate to summarize a stock's value, we can see that it actually does work and is completely logical. First of all, please notice the assumptions inherent in this model.
The first one is that we assume the business will operate into perpetuity (though realistically, it only needs to last maybe 20 years to approach its ultimate value from payouts, after that time the future cash flows start to become worth very little). However flawed this may be, it is a fundamental assumption made in all aspects of a business. In accounting it is referred to as the "going concern" principle. Anyway, since a business will likely continue to operate in some form or another, it is probably best to just assume it will last forever due to the inherent uncertainty that exists in business. It truly is the only tennable position.
The second assumption is that it is possible to know the growth rate of the dividend. Obviously this is impossible, however, that has never stopped anyone before. In creating a model to predict the future, you have to make assumptions. As you are constantly reminded in finance, past performance is not indicative of future results. Therefore, it is purely logical to assume a growth rate. Also, in many dividend paying industries, you will find that the growth rates are actually fairly stable, making this a valuable model.
So, now that we've establised our assumptions, let's move on to the fundamental objection here. You claim that essentially the whole model is based upon the model working. You could make this same argument about currency as well. Currency is only as valuable as what the next person will give you in return for it. Therefore, currency is also a self-fulfilling thing and is quite circular. However, I think that the DGM is fundamentally different than this example and stands on a very firm foundation.
First of all, I hope we are in agreement that the time value of money is a fact and is something that we can base our valuations on. I'd say that it's a pretty fundamental fact that money now is worth more than the same amount of money in the future (assuming we are not in a deflationary environment). So, let's continue shall we. If someone is going to pay you payments yearly as long as the company exists, it should be obvious that this stream of cash is worth the sum of the present values. In fact, this fundamental principle is pretty much what all valuation methods are based on (excluding the frequently debunked technical analysis models based upon the inherent risk of a security. If you want to talk about circular, those are circular). So, since we have established that a discounted cash flow method makes sense for these kinds of situations (cash payments going into the future), it should not be a big leap from there to understand how the DGM works. You are correct in assuming that no one holds a security forever, however, this model is not predicated on this fact. It is merely predicated on the fact that someone, somewhere, will hold this security. You objection would only be accurate if you only paid the price that you paid for the cash flows you receive. However, you are ignoring the residual value of the future cash flows. Where you hold it for 1 year or 100 years, assuming that the stock still pays a dividend, it should still be worth the PV of all future dividends when you are down with it and therefore, anyone in the future will be willing to pay that amount.
You are correct that if you are not going to hold it forever, this requires you to assume that other people will use the model. But, the fact is, they will due to the fact that the model holds up.
This can all be summed up as a string of future cash flows is worth the present value of those cash flows. Therefore, all other purchasers will use this model as well and everything will work fine.
If you want a more concrete example, look at fixed income securities. The market regularly proves that this fundamentally works and is always true.
The DGM may have its flaws, but in a situation where no capital gains can be expected and book value (what you, the stock holder, actually own) is ignored, the DGM holds up. It better, because DCF models are essentially what all valuations are based on.
I will quote the third paragraph of your argument:
"If we assume for a moment that the dividend model is adequate to summarize a stock's value, we can see that it actually does work and is completely logical."
You had to make the assumption that the model works in order to even start your argument. Later on, you mention:
"However, you are ignoring the residual value of the future cash flows. Where you hold it for 1 year or 100 years, assuming that the stock still pays a dividend, it should still be worth the PV of all future dividends when you are down with it and therefore, anyone in the future will be willing to pay that amount."
The residual value of these cash flows is valued according to the model you have assumed is correct. Once again, the model works because you have assumed it works. If we were to come up with some completely different model to value stock and place that at a time n when you sold the security, the DGM would not be accurate to this new stock price prediction.
As to the argument that money, too, is circular, your method of valuing it, "Currency is only as valuable as what the next person will give you in return for it," holds not only for currency but every other type of financial instrument. I agree with you that this is the case. However, you are interpreting the real-life value of these financial instruments, which of course is determined by thousands, if not millions, of buyers and sellers in the marketplace. What I'm writing about here is the theoretical value of the financial instruments. If there is a theoretical way of valuing cash, I'm pretty sure most people do not use it. However, people do use theoretical models for buying and selling stock, which is why the DGM is so different from currency.
Fixed-income securities are relatively easy to value. I contend that, except for preferred stocks, this is because they are finite. The DGM takes the model out into infinity, where no investor could possibly see. yes, if we assume the DGM works, then it is like a series of lighthouses each of which can see the next one, thus effectively seeing into infinity. I contend, however, that assuming something works is not a good assumption to base your model off of. If the DGM's success were not assumed in the model, then it would be like a single lighthouse, which can only see so far into the darkness of time.
Who are you anyway, and how did you find my blog?
Let us assume momentarily that we are referring to preferred stock whose value is based entirely upon dividend payments.
So, let's examine the key differences between preferred stock (which I will now just refer to as stock) and fixed income securities. As you already stated, the key difference is the fact that fixed income securities are finite (though some are impossibly long term, such as the new hybrid equity-bond, used for tax benefits that lasts up to 80 years). So, if you can admit that fixed income securities can be fairly valued using a DCF model, then it should not be difficult to take this one step further and use it on something that is infinite.
When you sell a 30 year bond that you've held for 10 years, it has some inherent value based upon the future payments. Now, when you bought the bond, the price you paid at the time was a affected by required return, number of payments remaining, and the amount of those payments. You had to pay for all of the payments, not just the payments that you would receive during the relatively short time that you held it. You did this because obviously, the next person would value the bond based upon the remaining payments. Therefore, it has residual value when you are done with it.
So how does this relate to investors in stocks? Well, it's basically the same concept. The fixed-income investor who buys your bond halfway through is paying you based upon the future payments that remain. Stocks are the same way, however, since it is assumed that dividend payments are infinite, there are still an infinite number remaining. Therefore, an investor can be just as assured that the underlying security will have residual value as a fixed-income holder. As a result, it makes perfect sense for it to be an infinite calculation because that is what everyone else will be using to value his holdings since it is in fact true.
As a practical matter, none of this matters much because after a while, you start to approach a point where it doesn't matter anyway. For instance, using 10% interest rate, the PV of a $100 dollar payment 50 years from now is only $.85. Obviously not very much and it will only be making incremental changes.
Another way to look at it would be with limits. In fact, I encourage you to view it that way. The way you would value the cash flows from a fixed income security is Payment*((1/r)-(1/r(1+r)^n)). I'm writing it this way to make it easier. Now, we all agree that this valuation makes sense. So, what happens to this as n, the number of payments, reaches infiniti? The second term of this becomes 0. Therefore, we are left with Payment*(1/r). Now, this obviously is ignoring growth, but I think this is close enough. So, isn't that pretty much like the DGM with 0% growth? Therefore, this model is simply derived from a proven model that is completely accepted.
I ask again, who are you, and how did you find my blog? I'm not going to come after you with a calculus-pointed shovel - quite the opposite - I'm thrilled that my blog is actually inspiring intelligent conversation.
I don't know if you know actuarial terms or not, but your derivation of 1/r is the same as taking Lim N -> Infinity of "A angle N," for any of you actuaries tuning in out there.
That formula works just fine, so long as we assume that there is no redemption value or selling price. Many investors, however, buy a stock for capital gains, not just dividends. I would also venture to guess that many investors sell their investment long before the time value of money has essentially worn it away to zero. 50 years at 10% is significant, but 5 years at 4% is not so striking (.855). The question to answer is which is more like the typical investor?
For awhile, I toyed with the idea that the price of a stock would grow at the same rate as the dividends, and that brought up some very interesting results, which were not equivilant to the DGM, and that is what origianlly sparked this idea in my mind. (If you would provide me with your email address, I will gladly send you a scanned copy of these calculations.) The values that this newly developed model gave were nearly the same as the DGM for large holding periods, but dropped off drastically for short periods. I have been meaning to work with that model more, but I have not yet gotten the chance.
The point is that I derived another formula which limits to the same as the DGM, but which now includes time as a variable. Granted, this was based on a shaky assumption of my own, that price grows at the same rate as dividends, but nevertheless, it shed some light in a dark corner for me.
I congratulate, what words..., a brilliant idea
Post a Comment