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A
PRIMER ON EQUITY VALUATION
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Most market observers use both theoretical concepts and empirical data to reach meaningful investment conclusions. The former attempts to identify the way stocks should be valued, whereas the latter concentrates on how stocks have been valued. Thanks to a number of economists and financial analysts, we now have a sound base of theoretical concepts upon which to draw from. One of the most important contributors was Irving Fisher, an economist and the father of modern capital theory. When analyzing bonds, Fisher cited several empirical relationships, the most important of which was that a "marked correlation exists between interest rates and a weighted average of past price-level changes, reflecting effects that are distributed over time"
(1). Fisher attributed this relationship to imperfect foresight about future inflation rates and the resulting inclination to extrapolate past price-level changes into the future in order to adjust interest rates for expected changes. In other words, the behavioral mechanism (uncertainty) used to value bonds is dependent upon recent history and expectations about the future. This has come to be known as the "Fisher effect". The same effect can be observed in the way today's investors value not only bonds, but stocks as well. In the case of equities, it is the forecast of sustainable growth rate that is the product of recent history and analyst expectations, i.e., the Fisher effect.
In 1930 Robert F. Wiese stated that "the proper price of any security, whether a stock or a bond, is the sum of all the future income payments discounted at the current rate of interest in order to arrive at the present
value" (2). We believe this was the first statement of the present value theory applied to common stocks; however, present value had been used for many years prior to 1930 as the basis for the construction of bond tables.
In 1938, John Burr Williams published his book, The Theory of Investment
Value (3). He used essentially the same definition of investment value as had Wiese. He stated:
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"Let us define investment value of a stock as the present worth of all dividends to be paid upon it . . . To appraise the investment value then it is necessary to estimate the future payments. The annuity of payments, adjusted for changes in the value of money itself, may be discounted at the pure interest rate demanded by the investor.
A stock is worth the present value of its future dividends, with future dividends dependent on future earnings. Value thus depends on the distribution rate for earnings, which rate is itself determined by the reinvestment needs of the
business." |
The original formula used by Williams was mathematically presented as:
V = D0
+ D1
+ D2
+ ........ Dn
1+k (1+k)^2
(1+k)^n
Where: V is the present value
D0 is the dividend initially
Dn is the dividend in the nth year
k is the discount rate, or the desired
rate of return |
If the future growth rate (g) of dividends can be projected, then:
| V
+ D0 (1+g / 1+k) + (1+g / 1+k)^2
+ ........ + (1+g / 1+k)^n |
There are two important concepts that Williams captured:
1. Future dividends are dependent on future earnings.
2. The distribution of dividends is determined by the reinvestment needs (opportunities) of the business.
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Here is the rub: Companies with high incremental returns on capital will reinvest their profits back into the company rather than increase the dividend payout ratio. This fact has made the computation of value much more difficult using the Williams model. It then becomes much more difficult to know when corporate return on capital will become so low that directors decide to payout more in the form of dividends. Recall that a stock hypothetically has a perpetual life, and therefore meaningful dividend payouts might happen decades into the future. (Beyond forty years the present value of the next dividend becomes negligible.) Assumptions concerning the
magnitude of payouts and when they might occur can have a huge impact on a model's current theoretical valuation.
To circumvent this problem, theoreticians began to calculate a terminal price for the holding period, much like the maturity price and date of a bond. Myron Gordon
(4) introduced such a model, traditionally known as the Gordon growth model, that very compactly illustrated the connection between a stock's price, the current level of its dividend, the expected growth rate of dividends, and the discount rate. Gordon also formulated a special case wherein one could convert from classical valuation models to a special case model for a price/earnings ratio. |
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PRICE-EARNING
RATIOS (PE)
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If we assume that a company has a
constant earnings retention rate (the inverse of dividend payout ratio), a
constant growth in earnings per share and a constant discount rate, then we can convert the Williams model to one that defines a special case price/earnings ratio. This is known as the Gordon-Shapiro valuation equation:
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P / E =
1-b
k-g
Where: b = earnings retention
k = discount rate
g = growth rate
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As an example, let's assume that the constant earnings retention rate is 64% (a dividend payout ratio of 36%), and that sustainable growth is 8.8%. To determine a suitable discount rate we need to consider what return one might be able to achieve with much greater certainty (i.e., less risk) such as that available from long term Treasury bonds. To this rate (the yield to maturity) we add a premium to compensate for the increased risk of owning a stock. This is known as the "equity risk premium". In this example we'll assume a risk premium of 4% over Treasury bonds that were yielding 6%.
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P/E =
(1 - .64) / (.10 - .088) = .36 / .012 = 30 |
The above is a P/E model of the S&P Industrials for the year 1999. It says that at a price-earnings ratio of 30, at this growth rate and this earnings retention rate, the expected rate of return is 10.0%. Once the growth rate (g) exceeds the discount rate (k), however, the denominator turns negative and the formula becomes unusable in this form.
One of the most notable contributors to the store of knowledge about price/earning ratios was Nicholas
Molodovsky (5). Molodovsky reasoned that to make the classical valuation model operational, as well as theoretically sound, a relationship between dividends and earnings had to be found to make the latter become the basic input for the model. He wrote:
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Low dividend payouts are a result of high investment return, which in turn causes high earnings growth. High dividend payouts are a result of low investment return, which in turn causes low earnings growth. Our Tables are computed by using the hypothesis that the payout ratio is a function of earnings growth.
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| Molodovsky's tables were a stroke of genius. To compute the relevant price/earnings ratio he used two growth periods, one for constant growth and one for diminishing growth. For the first time the concepts of Williams (see above) were translated so that a price/earnings ratio might be easily computed. In fact, Molodovsky's work showed that high growth companies can have very high price-earnings multiples under the right circumstances. |
While Molodovsky's work took into account a dynamic growth rate, the Federal Reserve's model does not. The Fed's model of the stock market relates the yield on the ten year Treasury bond to the earnings yield (earnings/price) of the S&P 500, the inverse of the price/earnings ratio (P/E). Through most of the past two decades (except 1980-1982), this model has worked rather well. Mr. Greenspan made the Fed's model public in his Humphrey-Hawkins testimony on July 22,
1997 (6). By January of 2000, the Fed model appeared to be 60% overvalued! (See
Exhibit 1 & 2). Unfortunately, the Fed's model is a very simple one that works when long-term growth expectations are rather static. When expectations change significantly, it muddies up the model results.
Another way to think about valuation is expected return, similar to the yield-to-maturity of a bond. In an equilibrium state, expected return is the same as the discount rate, or (k). If price is known and other inputs remain the same, one can solve for (k). This is known as the Dividend Discount Model. This implied expected return can then be contrasted with different levels of risk, liquidity, or the competitive returns expected from other asset types such as bonds.
You will recall that in the price-earnings model above, we added a premium to the bond yield to determine an appropriate discount rate. This is known as the equity risk premium (ERP). Work done by academicians in the 1970's indicated that the historical difference between the discount rate (k) for the S&P 500 and Treasury bonds (the same as yield-to-maturity) was about 2 ½ percentage points. Later studies suggested that the difference was closer to
7.4% (7). Exhibit 3 plots the difference between the expected return (k) and long bond yields. In 1999 the difference was approximately 400 basis points, or 4%. (As one can see, the ERP as defined this is quite volatile.) During severe market drops, the ERP can be driven upwards because of declining stock prices in the face of declining bond yields. Such was the case in 1982, 1995, and 1998. Contrary examples occurred in 1983 and 1987. The average spread between the S&P 500 expected return and long-term interest rates in this example was 420 basis points. |
Many analysts have observed the expected rate of return (k) vs. other factors such as risk and liquidity. Much of this work can be attributed to Bill
Sharpe (8) and the financial analysts at Wells Fargo Investment Advisors (to which Sharpe was a consultant). They and others developed the Capital Market Theory which posited that the risk premiums investors implicitly charge is scaled in a linear fashion to the systemic risk (beta) of each common stock. Analysts computed capital market lines that depicted reward per unit of risk to explain how the market pricing mechanism worked.
In a study of how risk and liquidity impact discount rates, Fouse (9) found that all other things being equal, required return (the discount rate) is
less for the more liquid stocks than for the less liquid. (In this case, liquidity sectors were defined by the dollar value of trading needed to elicit a one percent price variation). When Fouse plotted expected rates of return (the same as the spot discount rate) vs. beta it described what was known as the Market Line. He presented graphs that depicted the change in the market line during different market periods.
(Exhibit 4) However, when he created Market Lines for each of five liquidity sectors it revealed a large spread between expected returns for those of the highest liquidity sector and those of the lowest.
Exhibit 5 shows the "liquidity fan" of these Market Lines as of year-end 1974. This implies that all other things being equal, P/Es for small capitalization (less liquid) stocks will be lower than those of larger capitalization (more liquid) stocks. Moreover, as the market moves from a period of high anxiety to complacency, the relationship between the liquidity sectors changes in
unpredictable ways.
Price-earning ratios and other valuation metrics are difficult to predict. Crossectional analysis of many stocks with different growth rates, payout rates, risk levels, and liquidity tells us only about relationships at a point in time. Systemic "tilts" due to sector or stock type favoritism are not easily detected using this approach. Additionally, other factors such as the quality and depth of management, anticipated use of free cash flow, the independence of the Board of Directors, and even such mundane factors as the willingness of the specialist to maintain a liquid market in the stock will affect valuation. On the other hand, by analyzing a company from a time-series perspective (i.e., over time), the impact of many of these intangible factors can be minimized. A more difficult task is to decipher the tangible and intangible changes to index valuations as company weightings within the index change. |
We prefer the use of time-series data to value securities because typically there is a behavioral consistency by investors toward any given company. That is to say we prefer to employ an
empirically based valuation solution that has good theoretical underpinnings. Below are the basic components of our models:
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1. Enterprise Value (EV). Both debt and equity owners have claims on a
firm, and the earnings of the firm support
both. We combine net debt per
share with price per share to calculate the
firm's enterprise value (EV).
This then adjusts for differences between the
leverage employed by firms
and how they are valued. (This applies only
to non-financial companies).
EV then becomes the numerator of the
valuation multiplier.
2. EBIDA. These initials stand for Earnings Before
Interest, Depreciation,
and Amortization. Since we included debt in
the numerator, we must add
the interest on that debt back to the
denominator. Adding back deprecia-
tion puts companies with different
depreciation cycles on a level playing
field. As for amortization of intangibles,
one needs to make adjustments
to that for goodwill, and possibly depletion
allowances.
3. The valuation multiple we prefer is EV/EBIDA, which is
often used by
Street analysts. Since the stream of
operating profits represents receipts
to which both debt and equity holders have a
claim, we believe the EV
rather than price is the relevant
numerator in the valuation ratio.
4. Expected Total Return is the combination of
sustainable growth plus
current yield. This is the classical formula
for total return but makes
intuitive sense as well. This is a proxy for
DDM's expected rate of
return. |
As seen above, classical equity valuation and P/E models are dependent on current dividend rates, the growth rate of dividends and earnings, payout ratios and the sum of their present values. Sustainable growth (SGR) and the base of earnings from which it is measured are very important concepts.
Expected growth itself is probably the most important element in determining how a stock is valued. It is a function of a company's incremental return on equity (ROE) and its earnings retention rate (b). From an historical perspective, the fundamental underpinnings for SGR are derived from changes in income and balance sheet
item (10). The basic formula is:
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As a practical matter, we do not always use historical ROE as the basis for our growth estimates. In most cases, we use the consensus growth rate (CGR) as an input. The CGR is an estimate of the analysts following a stock regarding its projected five-year growth rate. Both the CGR and SGR are highly correlated. With more cyclical companies we will make significant changes to current year estimates in order to "normalize" them so that the growth rate applies to the correct base level.
One could use reported return on equity, but often it does not incorporate what's in the future for a company and miscalculates what has occurred historically because of accounting changes and the impact of inflation on asset values. We prefer to use the components of ROE that are "current" values such as profit margins, sales/capital spending, and interest coverage ratios to approximate true ROE. These margin and asset turnover ratios are the key components of classical ROE.
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ROE = (Sales/Assets) * (Profits/Sales)
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Implied growth is that which has been incorporated into historical valuations. In most cases that we have observed, implied growth is a function of historical growth rates, CGR and SGR.
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To create an example of the dynamics of valuation in a simplistic fashion, we will present a case study of the Standard & Poor's Industrial Average. (Because S&P did not compute book values prior to 1977, we have always used the Industrials to measure return on equity and valuation). The industrials account for approximately 85% of the larger index. Below are three periods of time over the last seventeen years. Each had different dynamics that caused an upward movement in valuations.
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S&P INDUSTRIALS
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1983
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1989
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% CHG
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Valuation |
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Price/Earnings |
12.20 |
12.40 |
1.6% |
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EV/EBIDA
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5.71
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7.49
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31.1% |
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Variables
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Long Treasury Bond
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10.84%
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8.59%
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20.8%(a) |
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SGR
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7.08%
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8.93%
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SGR + Yield
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11.12%
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11.96%
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7.6% |
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Consensus Growth
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9.00%
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11.10%
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CGR + Yield
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13.04%
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14.13%
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8.4% |
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Related Variables
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Dividend Yield
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4.04%
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3.03%
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Operating Margins
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13.64%
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14.69%
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LT Debt/Capital
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27.30%
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38.60%
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(a) A decline in the bond yield variable is counted as a positive for valuation. |
(b) Return on equity not comparable due to accounting change for financial subsidiary debt in 1987.
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The period above represents an excellent example of an increase in enterprise valuation (31.1%) caused primarily by the decline of interest rates (20.8%) and to a lesser extent by an increase in expected total return (8.4%). The increase in the EV/EBIDA multiple compared to the small change in the P/E multiple is due to the higher debt levels relative to total capital and slower cash-flow per share growth than earnings per share growth.
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S&P INDUSTRIALS
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1989 |
1993 |
% CHG
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Valuation |
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Price/Earnings |
12.40
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16.80
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35.5%
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EV/EBIDA
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7.49
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9.52
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27.1%
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Variables
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Long Treasury Bond
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8.59%
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6.46%
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24.8%(a) |
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Consensus Growth
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11.10%
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11.50%
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CGR + Yield
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14.13%
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13.90%
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-1.6% |
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Related Variables
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Dividend Yield
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3.03%
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2.40%
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Operating Margins
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14.70%
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13.60%
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LT Debt/Capital
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38.60%
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41.00%
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(a) A decline in the bond yield variable is counted as a positive for valuation. |
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The period in Table II was one of a modest decline in growth expectations and another significant decline in bond yields. Although consensus growth increased modestly, current yields declined, making expected total return essentially unchanged. The increase in price/earnings ratio vs. the EV/EBIDA ratio was due to changes to the tax law that allowed for depreciation charges to accelerate thereby allowing earnings per share to grow less rapidly than EBIDA.
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S&P INDUSTRIALS
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1993 |
1999 |
% CHG
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Valuation |
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Price/Earnings |
16.8
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27.7
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64.9%
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EV/EBIDA
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9.52
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9.52
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67.6% |
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Variables
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Long Treasury Bond
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6.46% |
6.00% |
7.1%(a) |
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SGR (4 year)
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8.10% |
15.80% |
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SGR + Yield
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10.5% |
16.89%
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60.8% |
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Consensus Growth
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11.5%
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16.60% |
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CGR + Yield
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13.9%
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17.69%
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27.2% |
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Related Variables
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Dividend Yield
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2.40%
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1.09%
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Operating Margins
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13.61%
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16.40%
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LT Debt/Capital
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41.0%
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37.3%
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(a) A decline in the bond yield variable is counted as a positive for valuation.
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The period covered in Table III exhibits only a modest decline in interest rates but a significant increase is projected growth rates. Both the SGR and CGR grew significantly. This is due in part to a more consistent U.S. economy and improved productivity, however, most of the change is due to modifications to the composition of index.
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One can get a sense of the dynamics of valuation from the above segments of time. In order to understand how the various factors interrelate over the entire time-frame, we need to create a model of the historical relationships. To achieve this end we need to determine how EV/EBIDA is correlated with estimated growth rate, dividend yield, and bond yield. Using a basic multiple regression model, we can identify the coefficients for each
variable (11) and produce a valuation model. The summary results can be observed in
Table IV.
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Projected |
Annual |
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| 1987 |
11.3 |
7.42 |
8.34 |
1.16 |
| 1988 |
11.0 |
7.27 |
6.93 |
0.94 |
| 1989 |
11.0 |
7.77 |
7.49 |
0.96 |
| 1990 |
11.2 |
8.23 |
7.95 |
0.96 |
| 1991 |
11.8 |
8.61 |
8.59 |
1.00 |
| 1992 |
12.0 |
9.45 |
8.96 |
0.94 |
| 1993 |
11.3 |
10.23 |
9.52 |
0.92 |
| 1994 |
11.6 |
8.96 |
9.21 |
1.03 |
| 1995 |
11.6 |
9.44 |
9.61 |
1.02 |
| 1996 |
12.3 |
10.25 |
10.49 |
1.03 |
| 1997 |
13.5 |
11.5 |
12.41 |
1.09 |
| 1998 |
14.4 |
14.77 |
14.37 |
1.01 |
| 1999 |
16.0 |
15.76 |
15.96 |
0.98 |
In the last few years there has been a pervasive concern about the valuation of the market and price-earnings multiples in general. Using classical valuation techniques to value the market has proven extremely difficult. Simplistic models such as the Federal Reserve's offer little or no value in diagnosing the market. Dividend discount models (DDM) have provided useful insights into the behavior of stocks as they pertain to relative risk and liquidity. Unfortunately, the relationship between interest rates and DDM expected return, the equity risk premium, has proven to be too volatile to be useful.
As more security analysts convert to using enterprise valuations, they find their results becoming more relevant. Using the sustainable growth rate (SGR) and/or the consensus growth rate (CGR) as inputs, one can create a time-series valuation model using multiple regression techniques. Knowing how an index or stock has been valued historically can provide empirical insights into how it might be valued in the future. It is the
inputs regarding incremental ROE (growth), and the cost of capital (interest rates) that then can be explored to project future outcomes.
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| 1. Irving Fisher,
The
Theory of Interest. (New York: Macmillan, 1930), p. 438.
2. Robert F. Wiese,
"Investing for True Value", Barron's, (September 8,
1930),
p. 5.
3. J.B. Williams, The
Theory of Investment Value (1938). (Reprint: Amsterdam, The
Netherlands: North Holland Publishing Company, 1964).
4. Myron J. Gordon,
The Investment, Financing, and Valuation of the Corporation.
(Homewood, Ill.: R.D. Irwin, 1962).
5. Nicholas Molodovsky,
"Common Stock Valuation Principles, Tables and
Application", Financial Analysts Journal (March-April
1965).
6. Alan Greenspan,
Monetary Policy Report to Congress Pursuant to the Full
Employment & Balanced Growth Act of 1998, (Superintendent of
Documents, July 22, 1997).
7. Bradford Cornell,
The Equity Risk Premium (New York, NY: John Wiley & Sons,
Inc. 1999).
8. William F. Sharpe,
Portfolio Theory and Capital Markets (New York: McGraw-Hill,
1970).
9. William L. Fouse,
"Risk and Liquidity: The Keys to Stock Price
Behavior", Financial Analysts Journal, (June 1976).
10. Guilford C.
Babcock, "The Concept of Sustainable Growth",
Financial Analysts Journal, (May-June 1970), p. 108.
11. See Appendix.
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A linear regression is a technique for determining whether and by how much a change in one variable will result in a change in another variable. The coefficient of determination (r-squared) will indicate the percent of variance in one variable that is explained by the second variable. Multiple regression permits the prediction of an unknown variable that depends on not one but on several other known variables. Each of these independent variables is provided a weight depending on its contribution to determining the dependent variable.
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r2=.95
t stat=.43
log EV/EBIDA= .1884 + (1.563*log ETR) + (-1.174*log TBOND)
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OTHER VALUATION MODELS TO CONSIDER
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Russell J. Fuller and Chi-Cheng Hsia, "A Simplified Common Stock
Valuation Model", Financial Analysts Journal, (September/October
1984), pp. 49-55.
Jarrod W. Wilcox, "The P/B-ROE Valuation Model", Financial
Analysts Journal, (January/February 1984), pp. 58-66.
Bartley J. Madden, CFROI Valuation, (Jordan Hill, Oxford: Butterworth-Heinemann,
1999).
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Source: Dr. Edward Yardeni, Deutche Bank Alex. Brown
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Source:
Financial Analysts Journal, May-June 1976
Back
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Source:
Financial Analysts Journal, May-June 1976
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