July 7, 2008

The Gordon Growth Speed Limit

Gordon Growth Model (Value = Payment/(expected rate of return – growth rate)

A student asked why when they assumed a 15% growth rate for a project, the Gordon Growth model gave them a nonsensical negative value for the project. The 15% growth rate exceeded the expected rate of return his learning team was using. This raised a question. Is there a reason why the growth rate must be less than the rate of return? As you can see from the chart, above, as the growth rate approaches the rate or return and the denominator of the Gordon Growth Model goes to zero, the predicted value explodes to infinity.

The Arbitrage Argument

Nobody knows the future. So when we use the Gordon Growth Model we are estimating what we believe will happen in the future. However, there is risk that our predictions will be inaccurate. There are actually two places when making a financial forecast that we can reflect our estimate of risk. The first we have discussed numerous times, that being in our determination of the expected rate of return. This is what we attempt to do with the Capital Asset Pricing Model. However, we also reflect our estimate of risk in our estimates of future cash flows. If we project large and growing cash flows, we are saying that we believe that the risk associated with the project is low.

In fact, what the Gordon Growth Model (PMT/r-g) says is that the value of a stream of growing cash flows stretching out forever and ever is the equivalent of a stream of infinite cash flows of a fixed amount discounted at a lower expected rate of return. This is why the expected growth rate of the cash flows is subtracted from the expected rate of return in the denominator of the model. In other words, the expectation that the cash flows will grow is equivalent to a lower expectation of risk associated with the project.

Assume that you owned this type of asset. We will call it “Asset A.” Assume that the payments are $100 per year and the rate of return is 7%, but the market expects the cash flows will grow as 5%. Thus, the value of the as Asset A would be $100/0.02=$5000. However, assume that the government issues a similar asset paying a $100 per year coupon with an interest rate of 4%, we will call it “Asset B.” The value of this asset would be $100/0.04=$2,500. Since this asset is being issued by the United States government and it is “risk free” a smart investor will sell Asset A for $5,000 and by two units of Asset B. After all as a risk free asset, the investor can purchase an asset with lower risk for a lower price. This is called an arbitrage opportunity. Smart investors will continue to either sell Asset A or short Asset A until its price falls below the price of Asset B. This arbitrage will eventually result in a price for Asset A which reflects its relative risk in the market. In this way, arbitrage will ensure that r-g cannot be less than the risk free rate being paid by the United States government.

The Time Value of Money Argument

A basic assumption in finance is that a dollar today is worth more than a dollar in the future. There are three reasons for this assumption: 1) opportunity cost, 2) inflation, and 3) risk. We will consider each of these and find that even though we might be able to eliminate or reduce two of these reasons, we cannot circumvent opportunity cost as a reason why a dollar today is worth more than a dollar in the future.

Theoretically, we can eliminate risk by holding a risk free asset, such as US Treasury bonds or even US dollars. By holding cash or treasuries we can calculate with certainty the nominal amount which we can claim at some point in the future. However, on a practical level, even these assets involve risk. In finance they are called risk free because the volatility of their expected returns will be zero. However, for the person holding the asset, this does not mean that there is no risk. When I was a growing up, my father had a coffee mug in the bathroom, where he put his toothbrush, which said, “Eat drink and be merry, for tomorrow we die.” Thus, in the hands of a human being, there is never zero risk over time, because all of us are subject to our own mortality and thus, even though we speak of risk free assets, ultimately our own mortality will always lead to some time value of money.

Over the last few decades we have become used to at least a low level of inflation. This was not always true. When money was connected to gold, there were long periods when of deflation as well and inflation. You would not necessarily see inflation as leading to a preference from holding a dollar today over a dollar in the future. However, today an expectation of inflation has been built into the system. Nominal interest rates are thought of as being comprised two parts, the real interest rate and expected inflation. If an interest rate is 10% and inflation is 4%, then the real interest rate would be 6%. Theoretically, it is possible to eliminate inflation expectation from the equation. The Federal Reserve could be extremely tight with its monetary policy and wring all inflationary expectations out of the system. After the bubble burst in Japan in the early 1990s, we saw many years of deflation in Japan. In that case, deflationary expectations could turn the time value of money on its head. If you believed that because of deflation you will be able to buy significantly more with a dollar in five years than you can today, then you may be willing to forego a dollar today with no nominal return and receive a dollar in the future. The lesson to be learned from this is that when we are considering the limits of the Gordon Growth Model we should really be thinking about real returns and not simply the effects of changes in the price level. If we were to use nominal numbers in a deflationary situation, the model could provide us with nonsensical answers.

This leaves opportunity cost at the heart of our belief that a dollar today is worth more than a dollar in the future. By investing today, we are giving up all alternative uses for that money. We are giving up a meal at our favorite restaurant, that new computer you have had your eye on, or putting your money in another investment which would produce a return. In facts as mentioned above, you could always put the money in Treasury bonds and earn the risk free rate. Thus, in real terms opportunity cost will always mean that we will value a dollar today more than a dollar in the future. As mentioned above, because predicting growth in cash flows is simply a way of saying that there is less risk involved with the project, saying that the growth rate exceeds the expected rate of return is saying that either there is no risk (or nonsensically that there is negative risk) or that there is no time value to money.

The Linear Relationship between Risk and Reward

Finance theory tells us that there is a linear relationship between risk and reward. This line starts at the risk free rate and that as risk increases so does the expected rate of return. The basic assumption is that if investors believe that they can receive a higher rate of return while being exposed to lower risk, they will quickly move to purchasing that investment. This increased demand for that investment will drive up the price of the investment and thus drive down the expected return. In this way, market forces will push the risk reward curve into a straight line anytime it deviates.

As mentioned above, saying that future cash flows will grow is simply another way of saying that there is less risk associated with a particular investment. If the risk is really less than would be expected based on the return, the price of the investment will be bid up in the market place and the expected return will be pushed back to the straight risk-reward line.

First consider the value of a risk free investment of infinite duration was a 4% risk free rate of return and a payment of $100 per year. Value = $100/0.04 = $2500. Thus, we know that any investment paying $100 per year must be worth less than $2500, or we would have to assume that the investment has a negative risk. However, as we have discussed above, the world is a risky place. Businesses do not do as well as they expect or they go out of business. Disasters happen. Managers make mistakes. People die. Laws change. Wars are fought.

Looking at the Gordon Growth Model in this way, we have to conclude that our estimated growth amount must be less than our estimates rate of return. In addition, the estimated rate of return must exceed the estimated growth rate by at least the risk free rate. Otherwise we are making the mistaken assumption of negative risk.