Time to pass the old maid?
IN PRINT ARCHIVE CIR Spring 1999
Time To Pass the Old Maid?  
By Laurence Booth  
As the TSE 300 continues to climb, even as commodity prices remain depressed, concern as to whether equity prices are involved in a speculative bubble increases. For a finance professor brought up with the theory of efficient markets it is hard to run counter to the idea that the market distills all expectations, and that at any one point in time half the money will be predicting higher values and half the opposite, so that disagreement as to the level of prices is a natural condition of financial markets. Increasingly, however, finance theory points to the role of noise traders^{1} in determining prices, as well as investors who rationally value shares in the manner described in finance textbooks. The distinction between noise traders and rational investors is important. Rational investors will look at all the relevant information to come up with a value based on "fundamentals," largely independent of other investors. Noise traders, in contrast, are interested in market sentiment and momentum, as such the critical value is where they feel the market is heading, not whether or not that value is sound. This view of the markets has been characterized as the "bigger fool theory," because if you think the markets are overvalued, but you buy because they are going up, you are a fool. However, the "bigger fool" is the person you plan on selling to, so you can get out before the market crashes. The idea is not recent. In discussing the role of professional investors, John Maynard Keynes stated in 1935, "They are concerned, not with what an investment is really worth to a man who buys it "for keeps," but with what the market will value it at, under the influence of mass psychology, three months or a year hence." ^{2} Note, however, that even with the "bigger fool" theory, a forecast of future prices is needed, since an estimate has to be made as to when the bubble has run its course. In another famous analogy, Keynes^{3} compared the stock market with a game of Old Maid in which "He is victor who passes the old maid to his neighbour before the game is over." We need to know how much longer the game has to be played in order to work out when to pass the old maid. 

A FRAMEWORK  
One key ratio looked at by investors is the priceearning ratio (PER). Typical graphs of PERS show skyrocketing values during recessions, not because values are high, but simply because corporate earnings almost disappear. To correct the trailing earnings problem of the standard PER, we can say that "on average" over long periods of time, companies pay out 50% of their earnings as dividends, and that they are very reluctant to cut their dividends. As a result, twice the average dividend yield is an estimate of the underlying "permanent" earnings yield, or the reciprocal of the PER based on permanent earnings. By taking the reciprocal you estimate this "implied PER." Comparing implied PER values to historic values of approximately 15 explains why some feel that the market is overvaluedthe TSE 300 over 30, and the industrial, interest sensitive and consumer sectors over 20. The rejoinder to the information from the PERs is that current earnings may still be biased low. As a result, there is more than average growth potential to corporate earnings, which is in turn reflected in the high PERs. Moreover, the implied PER is biased high for two reasons. First, alternative ways of distributing dividends such as share buy backs, have become increasingly important. Second, fundamental changes are occurring in the economy, in that knowledge and service based firms (which do not pay dividends) are displacing old line sectors. As a result, the implied PER, which is just the reciprocal of twice the average dividend yield, is biased high.^{4} The question, then, is what drives PERs? The constant dividend growth model first derived by Myron Gordon of the University of Toronto^{5} determines that the stock price is the expected dividend divided by the equity discount rate minus the expected long run growth rate. The "Gordon" growth model is derived under some very special restrictions, the implications of which are often not fully appreciated. For example, the growth rate is assumed to be some long run average that goes on forever. As a result, people often get frustrated when they try to value firms that are growing very rapidly: inserting large growth numbers frequently gives nonsensical values. As a result, they lose confidence in the Gordon model, as well as other quantitative models. The problem is that the Gordon model can only value securities that fit the model's assumptions. Firms that are growing rapidly, or that don't pay dividends, do not fit the assumptions and can not be valued using it. In contrast to individual stocks, the market as a whole fits the assumptions of the Gordon model more closely. Based on historic levels, a long run real growth rate of 3% would not be unreasonable.^{6} Adding this to the 13% inflation targets of the Bank of Canada would indicate that the earnings for the market as a whole should grow at 46% over the long term. The second parameter that is easier to value for the market as a whole rather than an individual firm, is the equity discount rate. For individual firms there are data problems in determining the market's risk assessment, which is also frequently changing. For the market as a whole we have data on the riskiness of equities versus bonds going back to 1924^{7} for Canada and 1802 for the U.S.^{8} For Canada, the equity risk premium has been about 3.25% since the TSE 300 indices were created in 1956. If instead we go back to 1924, the risk premium increases to 6% primarily because in the earlier period there was very little interest rate risk. As a result, equities were six times riskier than bonds, whereas since 1956 they have been only twice as risky. With current long Canada bonds at about 5.5% and an equity discount rate range of 8.510.5%, implying an equity risk premium of 3.05.0% would seem to bracket most analysts' estimates. With the above equity discount rate and nominal growth rate ranges and a 50% long run dividend retention rate, we can calculate the PER. With 4% GDP growth and an 8.5% equity discount rate the dividend multiplier is 23, so with 50% of earnings paid out in dividends the PER is 12X. As the equity discount rate increases, the PER falls to 10X and then 8X. Conversely, as the growth rate increases to 6%, the PER increases to 21X with the 8.5% equity discount rate and then to 15 and 12X as the discount rate increases. That it is possible to generate a PER of over 20X by choosing the appropriate values for the equity discount rate and the growth rate is not surprising. The interesting point is that what would normally be regarded as extreme values have to be chosen to get close to current PERs. Real GDP growth at 3% forever would exceed the experience of the last 20 plus years, 3% inflation is at the top of the Bank of Canada's range, while an 8.5% equity discount rate implies a real return of only 5.5%, which is considerably lower than the historic experience.^{9} As soon as more traditional values are used in the Gordon model, the PER drops away. The fact that it is difficult to get PERs of higher than 25X from the Gordon model does not mean the markets are overvalued. It does imply that the stable, "equilibrium" long run PER can not be greater than 25X. However, the Canadian market can have earnings growth that exceeds the above 46% range for a number of years, it is just that it can not exceed that range forever. For example, the market may be pricing robust economic growth and an improvement in profits for the next three years before settling in at the long run level. To take this possibility into account we can use a two stage dividend growth model^{10 }where the current dividend increases at one rate for the next three years, after which it reverts to the long run growth rate. Recognizing that the dividend is just the earnings times one, minus the retention rate converts the two stage dividend growth model to the two stage PER model. Forecasting individual growth model is hazardous, but in aggregate for the market there are more constraints. The first constraint, that of GDP, we have already discussed: it may be that in the longer run the Canadian economy just can not grow much faster than the 23% that the U.S. and the UK economies grow at. However, in the short run, since there is relatively more slack in the Canadian economy, it may have superior shortterm growth prospects. Suppose, contrary to the forecasts, the Canadian economy grows at 5% over the next three years. The second constraint relates to the share of GDP taken by corporate profits or "aggregate" earnings. For example, profits are still rebounding from the effects of the recession. Suppose the market is anticipating a further recovery of profits, perhaps to the 1988 level of 10.5% of GDP, the last good year before the Bank of Canada started slowing the economy down and NAFTA took hold. If we throw in some optimism, a good target would be 11% of GDP and an outside high 12%, which would mean a 2233% profit increase. The combination of a further profit rebound and continuing strong economic growth means that profits can in aggregate increase faster than the longterm range of 46%. If we assume an optimistic scenario of three years of further recovery in Canada, we can calculate the earnings growth rate implied by these assumptions. For example, if $100 of current GDP produces $9 in pretax profits, the current average level, and we take the optimistic real growth rate scenario of 5%, and add 2% inflation, then at the end of three years GDP will be $122.50. If we then take the extreme profit assumption of 12% of GDP, then the $9 in pre tax profits will have grown to $14.7 for a 17.8% compound growth rate. A high shortterm growth rate forecast would then be 17.8% for the next three years before reverting to the long run 46% growth rate. With an optimistic range of shortterm growth rates pegged at 12.317.8% we can insert this "supernormal" growth into the two stage growth model. However, we still need the PER at the end of the three year period, and can use the historic average PER of 15X plus a "high" of 20X.^{11} The result is: 



This table of PERs includes almost all the key drivers that describe current PERs: there is supernormal growth coming from a strengthening economy and a profit rebound, there is a longterm "equilibrium" PER, and there is an appropriate equity discount rate. However, even allowing for robust profit growth, adding 67X to the historic average PER of 15 cannot get to the level of PERs that we are currently seeing for the nonresource sector. This implies that the market can not just be valuing profit recovery, something more fundamental must be going on. Whether or not the equity markets are currently in a bubble depends less on shortterm profit growth, and more on the "equilibrium" PER assumed for three years into the future. If this PER is 20X, then there is enough potential profit growth in an "optimistic" short run scenario to sustain a current PER into the mid to high 20s. However, even though I believe that equity risk has declined significantly relative to bond market risk, it is difficult to generate the values needed to support a long run 20X PER. To my mind it is time to think about passing the old maid. 

ENDNOTES  
This paper is an edited version of a longer paper with the same title prepared for a valuation executive program delivered at the University of Toronto, summer 1998. The complete paper may be found at www.mgmt.utoronto.ca/~Booth. 1. See the presidential address of the late Fisher Black to the American Finance Association, "Noise Trading", Journal of Finance, May 1989. 2. John Maynard Keynes, Chapter 12, The State of Long Run Expectations in The General Theory of Employment, Interest and Money, Macmillan, 1936, page 154. 3. Keynes op cit, page 156. 4. See C. Farrel, "Does the Dividend Yield Matter?," Business Week, May 11 , 1998 for similar arguments. 5. See. M. J. Gordon, The Investment, Financing and Valuation of the Corporation, R. Irwin, Homewood Ill, 1962, for the fullest treatment. 6. Average real growth rates in Canada for the last 10 and 20 years have been 1.9% and 2.5% respectively, but these are probably biased low due to the restructuring that occurred as a result of NAFTA. Consensus Economics (May 1998) forecasts 1999 real growth for Canada at 2.9%, for the U.K. 1.8%, U.S. 2.2%, Germany 2.7% and the world average at 2.6%. 7. Laurence Booth, "On Shaky Ground," Canadian Investment Review, Spring 1995. 8. J. Siegel, "The Equity Premium: Stock and Bond Returns since 1802," Financial Analyst's Journal, (JanFeb 1992). 9. U.S. evidence shows a real equity rate of return averaging 9.12% over the period 18711997. The real equity rate of return in Canada was 8.79% for 19241996, unfortunately data is not available for a longer period. 10.
11. This is the standard way of valuing companies: fix an intermediate period and then use a long run multiple as a terminal value. The advantage is that it captures deviations from normal due to extraordinary events and the business cycle. n 

Laurence Booth is a professor of finance, Joseph L. Rotman School of Management, University of Toronto. 