Of Mines, Forests, and Impatience

Capital value is income capitalized, and nothing else.¾ Irving Fisher

Several years ago, my brother-in-law tossed in my direction a well-thumbed tome entitled The Theory of Interest. Grinning like a man relieved of no small burden, he said, "Hey Bill, you’re gonna love this one." Written by Irving Fisher in 1930, the text has been the bane of generations of B-school grads: it’s formidable but quite well written, repetitious enough that you’re not going to miss his basic points.

The book answers the question, "What is a thing worth?" Be it a hot dog, house, or human being. The answer, simply: the pleasure and income it provides. For an immediately-consumed item, the answer is evident—you do not purchase a dinner for $5, $25, or $125 unless it provides you with that measure of sustenance and enjoyment.

Next, suppose someone offers to sell you a meal listed on the menu at $25 for consumption ten years hence? How much would you be willing to pay for it now, ten years in advance? Certainly much less than $25. Say I’ve decided that $5 sounds about right. Punch the numbers into my trusty TI BA-35 and out pops an interest rate of 17.46%. That’s my own personal "restaurant-meal interest rate."

The key is that a pleasure enjoyed today is almost always worth more than one enjoyed in the future. Fisher elegantly calls this the "impatience" for the item in question; it’s synonymous with the rate of interest we calculated. (There are rare cases when the opposite is true, where a negative item-specific rate of interest occurs. Maybe you suspect that in ten years we are likely to fall prey to war or famine. In that case, you might very well be willing to pay a premium today for a meal ten years hence. However, money rates of interest are never negative.)

Of course, you can have different impatience (or interest rates) for different items. Further, the rate for a given item depends on personal circumstances: A starving person’s impatience for food is much higher than that of someone who is well-fed. And looking at it yet another way, a prosperous person is willing to spend much more today for a meal ten years hence than a pauper. There are other intrinsic personal characteristics that determine impatience (or rate of interest)¾ wastrels are by nature profligate and have high impatience; thrifty people have low impatience.

We are talking here about what subsequently became known as discounting. A future pleasure or service is worth considerably less in present value than an immediate one. Thus, its value must be discounted to the present by the appropriate rate of interest.

What of a durable item, such as a house? It provides you with shelter and pleasure, not just now but over a very long period of time. What you must do then is estimate how much benefit the house will give you in each future year. For each year, you must discount the value by the appropriate rate of interest. Finally, you add up all of the discounted-to-the-present values for all years in the future; this infinite sum is the true value of the house to you. (This is quite distinct from the market value of the house. If the market value is higher than the value calculated from the disount method, you are likely better off renting. But that's another article.) For example, let’s suppose that the rate of "house interest" is 9%. The value in each year must be divided by (1.09)ⁿ, where n is the number of years at that point. Thus, in year 10, divide by 2.37 or (1.09)¹⁰; in year 20, divide by 5.6 or (1.09)²⁰.

Investments fit neatly into this scheme. Fisher employs a paradigm that is quaint yet not displeasing to the modern audience: the purchase of a mine, farm, or forest. A mine produces ore for a relatively brief period of time, with the highest returns in the first years, gradually falling to zero as it is played out. A farm, on the other hand, produces a relatively stable output indefinitely into the future. Finally, a newly-planted forest provides no output in its initial years, then gradually increases over time. Let’s represent this with the below table.

Year	Mine	Farm	Forest
1	$2,000	$450	$0
2	$1,800	$450	$0
3	$1,600	$450	$300
4	$1,400	$450	$400
5	$1,200	$450	$500
6	$1,000	$450	$500
7	$800	$450	$500
8	$600	$450	$500
9	$400	$450	$500
10	$200	$450	$500
11	$0	$450	$500
Etc.

Which is the better investment? It all depends on the discount rate. The farm turns out to be the simplest case. It can be considered a perpetuity yielding $450 of income annually. Discounted at 5%, the present value of all its future income is simply $450 divided by 0.05, or $9,000. The present value of the mine or farm must be calculated with a spreadsheet. Below, I’ve plotted the present value of the mine, farm, and forest versus the discount rate (DR) for varying discount rates.

First, note how an increase in the DR decreases the present value of any investment, since in all cases this decreases the present value of future income. Next, note how the mine does best in a high-interest-rate environment. This is because its output is "front-loaded," largely escaping the increasingly corrosive effect of the DR in later years. Last, note how a low-interest-rate environment favors the "back-loaded" long-term return of the forest. This has important implications for the relative returns of value and growth investing, which will be the subject of an article in the next quarter’s journal.

But for now, we’re going to examine just what the model tells us about the New Investment Paradigm. I’m indebted to Cliff Asness of AQR Capital Management for providing me with a framework for understanding what the Fisher model (now known as the Discounted Dividend Model or DDM) tells us about the valuation of technology stocks. Quite obviously, the New Economy is the ultimate "forest," to use Fisher’s nomenclature. Cisco shareholders, for example, have patiently held onto their stock for the past ten years with nary a dividend check in their mailboxes, clearly expecting compensation sometime in the hereafter. Of course, long-time Cisco shareholders have been rewarded with stunning increases in capital value, thanks to the generosity of ever greater fools. But the unhappy truth is, the life of corporations is nasty, brutish, and quite often short: given enough time, all eventually disappear. If a company produces no dividends before it dies, its net return to investors is zero. That the capital gains of the first investors must be exactly counterbalanced by the capital losses of later investors is an accounting identity, a fact that investors in most dot-coms are just beginning to realize. Quite obviously, the rational buyer of Cisco expects a large heap of future dividends, which he then discounts to the present to obtain the current fair market value.

Mr. Asness suggests an easy way to examine the market’s assumptions concerning growth stocks. Here’s how it works: Imagine a stock selling at a PE of 100. Further assume it has $1 of earnings per share and thus a share price of $100. In order to command such a multiple, it must be growing its earnings rapidly—say at 40% per year. It is, of course, yielding no dividends yet. Generous suppositions would be as follows:

1. That this stock maintains its high growth rate for five years without paying a dividend.
2. After these first five years, it begins paying out 50% of its earnings as dividends.
3. In years six to ten, its earnings growth gradually falls to that of the average company.
4. Finally, that the market’s overall earnings growth rate is 6%.

We now have defined the company’s dividend stream, which we can discount at a given rate. This discount rate is, of course, the stock’s expected return. Thus, we have two parameters to play with: the stock’s initial growth rate and its discount rate (or expected return). We fiddle with these two until the sum of all the discounted dividends equals the market price. This is a bit complex, so I’ll show you what the top of the spreadsheet table looks like, where I’ve matched the market price with the sum of all the discounted dividends. (The sum, an infinite sum of all values in the last column, is not visible in the table. As you can see, after peaking at $3.22 in year 10, the discounted dividend then falls off slowly; the infinite sum is a finite value, in this case $101.43.)

Year	Growth	Earnings	Dividend	Discounted Dividend
1	42.0%	$1.42
2	42.0%	$2.02
3	42.0%	$2.86
4	42.0%	$4.07
5	42.0%	$5.77
6	36.0%	$7.85	$3.93	$2.22
7	30.0%	$10.21	$5.10	$2.62
8	24.0%	$12.66	$6.33	$2.95
9	18.0%	$14.94	$7.47	$3.17
10	12.0%	$16.73	$8.36	$3.22
11	6.0%	$17.73	$8.87	$3.11
12	6.0%	$18.80	$9.40	$2.99
13	6.0%	$19.92	$9.96	$2.89
14	6.0%	$21.12	$10.56	$2.78
15	6.0%	$22.39	$11.19	$2.68
16	6.0%	$23.73	$11.86	$2.58
17	6.0%	$25.15	$12.58	$2.49
18	6.0%	$26.66	$13.33	$2.40
19	6.0%	$28.26	$14.13	$2.31
20	6.0%	$29.96	$14.98	$2.23
Etc.

In order to understand the above table, the stock is assumed to have earnings per share of $1.00 in year zero, growing initially at 42% per year. Its dividend stream, beginning in year six, is then discounted at 10% each successive year, yielding the value of these dividends discounted to the present in the last column. These discounted dividends are then infinitely summed, obtaining the present value of the stock. What is done in practice is vary both the DR and the initial growth rate to produce the desired present value, say $100, which corresponds to a current PE of 100. In the above example, the initial growth rate of 42% combined with a DR of 10% yields a stock price (PE) of $101.43. Below, I’ve plotted some "Asness Curves" which demonstrate how PE, initial growth rate, and DR (expected return) relate to each other.

As you can see, it’s not a pretty picture. Focus on the blue (PE = 100) curve. Most investors would expect such a stock to return at least 15% per year. The plot shows that a DR/return of 15% and a PE of 100 implies an initial earnings growth rate of 63% for the first five years, followed by a gradual fall to 6% over the next five years, as postulated in the model. Lower rates of growth produce lower rates of return. Start out at a growth rate of only 28% and you wind up with 8% returns. And, alas, the investor who expects a 20% return needs a 78% initial growth rate. While it is possible with 20/20 hindsight to pick out the odd company that has turned this trick, the probability of your doing so prospectively with a given stock is nil.

Next, consider that the 100 largest stocks on the Nasdaq sell at a PE of 100. We’re talking now about approximately one-quarter of the capitalization of the U.S. stock market. The probability that this huge chunk of the economy will grow en bloc at 40% to 60% for the next five years is about the same as that of the Empire State Building spontaneously levitating to Beardstown by breakfast tomorrow.

A much more reasonable supposition is that the Nasdaq 100 sits at the far left end of the blue curve with earnings growth in the teens, yielding long-term bond-like returns accompanied by Ivana Trump-like volatility. There is nothing to prevent these shares from rising another 50% in the next year. But in the long run, the grim picture painted above is not idle conjecture or opinion. It is mathematical fact.

Finally, I want to be clear about one thing: the Asness model is wildly optimistic (as he himself admits). The real world decay of the most glamorous companies' earnings growth is breathtaking. In their landmark study of earnings growth persistence, Fuller, Huberts, and Levinson (Journal of Portfolio Management, Winter 1993) looked at stocks sorted by PE. They found that the top quintile¾ the most popular growth stocks¾ increased their earnings about 10% faster than the market in year one, 3% faster in year two, 2% faster in years three and four, and about 1% faster in years five and six. After that, their growth was the same as the market's. In other words, you can count on a growth stock increasing its earnings, on average, about 20% cumulatively more than the market over six years. After that, nothing.

Perhaps times have changed since the 1973-1990 period analyzed in the above-cited study. Let's be generous and assume that in the New Era the top quintile can manage a 50% cumulative growth advantage over the market. As I'm writing this, the top quintile of the 1,000 largest stocks with positive earnings sells at an average multiple of 78. A one-time 50% earnings growth advantage does not do much to justify such a valuation relative to the rest of the market (which sells at a PE of about 20).

It would seem, then, that a prerequisite for investing in the New Era is an inability or unwillingness to run the numbers. At some point in the not too distant future, we shall shake our heads and wonder how so many folks confused the forest with a gold mine.

Copyright © 2001, William J. Bernstein. All rights reserved.

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