Bill Totten's Weblog

Thursday, May 20, 2010

The Discounted Cashflow Paradox

aka The Saint Petersburg Paradox (May 21 2007)

The "Discounted Cashflow" model is the generally considered to be the most conservative valuation method for pricing stocks and investments. It's the model that Warren Buffett uses. Warren Buffett even mentioned this paradox at one of his annual shareholder meetings. The idea is that money in the future is worth less than money in the present. The value of a business is the value of all its future profits, discounted to the present. The paradox is that, under very reasonable assumptions, the Discounted Cashflow pricing model gives an infinite stock price.

I tried using the Discounted Cashflow model myself to price things, and came up with ridiculous results. Only recently, I figured out what was going on.

This is not to be confused with what I call the Saint Petersburg Math Paradox. This is the following game: You flip a fair coin until a tail ("T") shows up, at which point the game ends. For each head ("H"), you receive a payment of $1*2^n ($1 times two to the nth power), where n is the number of consecutive heads already seen. For an outcome of T (probability "p" = 1/2), you receive $0. For an outcome of HT (p = 1/4), you receive $1. For HHT (p = 1/8), you receive $1 + $2 = $3. For HHHT (p = 1/16), $1 + $2 + $4 = $7. For HHHHT (p = 1/32), $1 + $2 + $4 + $8 = $15. If you take the weighted infinite sum, you get that this game has an infinite value. Obviously, no game has an infinite value. If you were playing this game against a real opponent, at some point your opponent would be unable to pay off a sufficiently large string of heads. The value of this game, if someone ever actually offered it to you, would be finite, with the value based on how much money your opponent had.

The Saint Petersburg Math Paradox is completely unrelated to the Saint Petersburg Financial Paradox, which I describe and explain below. The Saint Petersburg Financial Paradox has a perfectly rational explanation.

The idea of the Discounted Cashflow pricing model is that all future income is discounted to the present. The rate used for discounting is typically the long-term bond rate or the risk-free interest rate. That's the return you could get without any effort or risk by buying government bonds. I will use a fixed value of five percent here, to keep things simple. (Technically, the interest rate should vary based on the timeframe. However, the yield curve is pretty flat right now, so a flat five percent interest rate is a reasonable assumption.)

Historically, stocks have improved their dividends by more than the bond yield rate, in the long-term. This has been true in every historic period of substantial size, so it's a very reasonable assumption. There are short-term fluctuations, but for now let's assume we're a long-term investor.

Let's price a stock that pays a dividend of $1 immediately and we expect will grow its dividend by six percent per year. This is a very conservative assumption, because most indices have had their dividend go up at a much higher rate historically.

I also will assume that the stock we invest in will never go bankrupt, that it will keep growing its dividend forever. If you like, consider this to be an investment in an index rather than a single stock. (For example, if every company in the S&P 500 goes bankrupt, you probably have bigger problems than the value of your investments.)

After one year, we receive a dividend of $1.06. To get $1.06 guaranteed a year from now, we would have needed to invest $1.06 / 1.05 = $1.0095 at the risk-free rate a year ago. The present value of the $1.06 dividend is only $1.0095.

So, running our model for one year, we get a price of $1 + $1.0095 or $2.0095.

Similarly, after two years, we get a dividend of $1.124, which discounts to $1.019. Our stock price is now $3.03.

Typically, these discounted cashflow models are run for something like ten or twenty years. Running the model for ten and twenty years gives a stock price of $11.53 and $23.13, respectively. A price of $23.13 sounds reasonable for an established company that pays a dividend of $1 now - that's a yield of just over four percent, comparable to companies like Verizon, AT&T, Bank of America, and Citigroup.

However, we're missing something. We're assigning a value of $0 to all payments later than twenty years from now. Surely an established, profitable company will still have some value then. We should be able to sell our shares for something, if we needed to. The person we sell our shares to twenty years from now is going to do his own discounted cashflow pricing, starting from the most recent dividend of $3.21.

A forty year discounted cashflow calculation gives a price of $49.87. A 100 year simulation gives a price of $168.50.

Do you see the problem? We're summing an increasing geometric series. Under the assumption that the company never goes bankrupt and keeps increasing its dividend faster than our discount rate, we get an infinite stock price.

An economist would say that I'm doing it wrong. I really should be discounting at a rate that's higher than my expected dividend yield growth, such as seven percent. Under those assumptions, it's a decreasing geometric series and a finite price.

But why should I discount at seven percent instead of five percent? I can't find an investment that will give me a guaranteed seven percent. I can only get a guaranteed five percent. A stock investment is "riskier" because its price may fluctuate, but when you take into account the increased expected return, the stock investment is actually less risky. Historically, stocks really have outperformed bonds in every significant time period. As long as the corporation never goes bankrupt, and doesn't get bought out for less than its value, your stock investment should eventually outperform the risk-free investment, no matter what price you paid.

Using the discounted cashflow model, I validly calculated that the correct price for any stock is infinite. If you know the company never goes bankrupt or gets bought out for too little, it will be a good investment provided you hold onto it for long enough, no matter what price you pay now.

Where's the mistake? I was bothered by this result for a long time before I figured out the answer.

The mistake is that a dollar is intrinsically worthless. The government will give me a five percent return if I invest in bonds, but the government prints new money at a rate of six to ten percent per year! (Or more!) That guarantees that if I keep my savings in bonds forever, it will be eventually be worthless, losing its value to inflation.

Even in the past, when the dollar was on a gold standard, the government from time to time devalued the dollar, decreasing the amount of gold each dollar represented. This happened often enough that anybody who owned bonds got cheated.

A stock is backed by something concrete, the productive value of a corporation. Even when the government prints new money, the stock will naturally rise in price to offset inflation. The corporation will see the prices of its supplies rise, but it will in turn increase the price it charges customers by the same amount. Even though corporate executives usually dilute your ownership by granting themselves shares and options, they don't do it at the same rate that the government prints new money. Plus, they usually try to repurchase enough shares to offset any grants they give themselves. Executive stock grants and share repurchases are a drag on a corporation's earnings, but this drag is already factored into the stock price when you buy. Unlike when the government prints new money, new share grants are clearly stated in the annual report. The actual amount of dollars in circulation is not published. (The Federal Reserve recently stopped publishing M3, because the values published would have been too embarrassing.)

That's the answer to the Discounted Cashflow Paradox. A dollar is inherently worthless. In the long run, you can't trust the government to not dilute its value via inflation. The bond rate is insufficient compensation for inflation.

A stock doesn't have infinite value. A dollar has zero value and there's a division by zero error when you run the Discounted Cashflow model for a sufficiently long investment horizon. A dollar has temporary value, because there are only a finite number of dollars in circulation right now, and people haven't caught onto the scam yet. The government has the right to print as many dollars as it desires, and it exercises this right as much as it can. The moral is that there's no point holding dollars, or bonds that pay dollars in the future. You'd better convert your dollars to a tangible asset as soon as possible. Only keep as much cash as you need to cover your short-term cashflow needs.

The Discounted Cashflow model can still be used to price stocks. Provided you use the same time horizon, and assume the same long-term trend growth rate, you get a valid comparison. The Discounted Cashflow model will tell you which investments are most efficient for you to purchase with your soon-to-be-worthless dollars.

So there's your answer. If you ever thought economics was just so much nonsense, now you know why. Economics is nonsense because the fundamental unit of value, the dollar, has a value of zero. All of economics is just one big division by zero error.

Bill Totten


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