The Price Earnings Ratio is perhaps the most commonly used valuation tool. We all know and love the Price Earnings Ratio (P/E Ratio). But do we know what it means? And do we know what in our opinion it should be? We all know how to calculate it: It is Price Divided by Earnings. But that gives you what it is. It does not tell you what it means. To understand what it means, you must understand the Math of the Multiple. And that will allow you to both understand what it means, and calculate what, in your opinion, it should be.

Simply put, the P/E Ratio reflects the present value of future cash flows that a person can expect from each $ of earnings. Mathematically it is expressed as:

$1 * (1 + G%) * (1 - Rr)/(VLT ERP + Rf - G%), where G% is the earnings growth rate, Rr is the reinvestment rate, VLT ERP is the equity risk premium measured as β * (Rm - Rf), Rm is the expected investor return from the market and Rf is the risk free rate.

This formula calculates the present value of future cash flows an investor can expect to receive from future distributions.

To calculate a market multiple most appropriate for you as an investor, means having to have a point of view on your assessment of growth expectations, reinvestment rates, equity risk premiums, return expectations and risk free rates. I have set out below my expectations. You can use your own.

**Growth**

For the market, over the very long-term earnings growth in U.S. has been in-line with nominal GDP growth of around 4.5%. In recent years earnings growth has accelerated to a higher level. This has occurred for two reasons.

The first is that in recent years, companies have started returning capital via share buybacks. When a company buys back shares, earnings per share will rise even if there is no growth in the company's earnings. While earnings per share growth will be higher as a result of buybacks, what we are looking for is the percentage growth in total market earnings. It is a growth in total market earnings, not growth in earnings per share that raises the absolute value of the market. The total value of the company will rise with growth. With buybacks, a smaller number of shareholders may cause the value of each remaining share to rise, even if the worth of the company as a whole does not rise. As a result, the impact of buyback activity on growth is best ignored as far as assessing long-term growth rates is concerned. Buybacks are a mechanism to return value to shareholders, and its impact on the multiple is best assessment as part of market return expectations [Rm].

The second is globalization. With globalization, several companies listed in U.S., including those on the S&P 500, see their earnings growth rates influenced by global growth rates, which are higher than U.S. growth rates.

Global nominal GDP growth potential is about 8% (4.2% real and 3.8% inflation), while U.S. nominal GDP expectations are 4.5% (2.5% real and 2% inflation). In my view the growth potential for the S&P 500 has increased to 6.25% (about 50% from 8% Global Nominal GDP growth and 50% from 4.5% U.S. Nominal GDP growth).

Keep in mind that my nominal Global GDP growth estimates are optimistic by many standards. My estimate of 4.2% real Global GDP growth potential is significantly over the Conference Board estimate of 3.1% for 2014-2019. IMF on the other hand estimates 2014 global GDP growth at 3.7%, with it rising to 3.9% in 2015 - this is more in line with my expectation of a rising trend to a 4.2% real GDP growth potential.

**Re-investment Rate**

The re-investment rate is that percentage of earnings that a company must re-invest to generate the target growth for future years. If you subtract the re-investment rate from 1, you are left with that percentage of earnings that are available to shareholders. I refer to it as the notional payout ratio which is the amount a company could return to shareholders through dividends, buybacks or re-investment to generate growth higher than market growth rates.

One way to estimate the re-investment rate is by using growth rates and the return on equity we expect companies to earn on incremental equity invested. Simply take the growth rate and divide it by the return on incremental equity invested. If a company can be expected to grow at a 6.25% rate, and it earns a return on incremental equity investments of 13.25%, then it must re-invest 47% of its earnings to generate that growth.

I estimate the return on incremental equity investments at 13.25% based on a historic analysis of as reported earnings divided by the book value since 1999, which is included at the end of this section.

The data on operating earnings, as reported earnings, and book values for the S&P 500 included on the below analysis can be downloaded from S&P Dow Jones Indices website here.

With growth expectations of 6.25% and a 13.25% return on incremental equity, the re-investment rate is expected to be just over 47%, and that leaves a notional payout of 53%.

Over the very long-term, the re-investment rate in U.S. has tended towards 40%. The rise in the re-investment rate to 47% has occurred partly as a result of a rise in the long-term earnings growth expectations from 4.5% to 6.25%, and partly as a result in return on incremental equity rising from near 11%, to an expected 13.25%. The growth acceleration comes about as a result of globalization and access to global markets, while the increase in return on equity comes about as a result of shifts in the composition of the economy from one driven by manufacture to one driven by services.

**Source: My Analysis of Data Available for Download on S&P Dow Jones Indices website****here.**

**Risk Free Rate**

The risk free rate is the theoretical return on an investment with zero risk over a specified period of time. I tend to use the ten-year US government securities as a measure of the risk free rate. At present the rate is 2.71%. However, over the coming several years, I expect it to rise towards its ten-year median levels of 3.69%, and so I use 3.5% as an estimated risk free rate for the coming five year investment horizon. However, if the long-term growth rates do rise to 6.25% as a result of globalization, as I expect they will, the long-term risk free rate will also rise.

In my view, an appropriate risk free rate should offer a 1.5%-1.6% real return over a composite inflation rate of 2.9% (Global Inflation 3.8% * 50% + U.S. Inflation 2% *50% = 2.9%). That implies a forward risk free rate of 4.5%.

**Source: My Analysis of Data Provided By Dr. Robert Shiller Available for Download****here****.**

**Expected Market Return**

People buy and sell stocks. Stocks make up the market. And the market exists in perpetuity. The owners of the market, be they you or another, will receive dividends, buybacks & real growth of dividend & buyback distributions in perpetuity. That is it, nothing else: your only return expectation is from future corporate distributions. Thus what shareholders can expect to receive by way of returns is the dividend yield, the buyback yield, plus real future growth.

Over the very long-term, the market has delivered returns of 9%. This has been made up of a 4.5% yield from a combination of dividends and buybacks, and another 4.5% through growth. Today with long-term growth expectation creeping up to 6.25% for the reasons expressed in the growth section of this post, I suspect the returns investors expect from the market will be higher.

In recent times, between 12/31/2007 and 9/30/2013, the dividend plus the buyback yield averaged 5.33%. Excluding the 6/30/2009 to 9/30/2009 crisis period, the dividend plus the buyback yield averaged 4.78%.

Over the long-term, with the notional payout ratio declining from 60% to 53%, I would expect the dividend plus the buyback yield to decline from 4.5% to 4%. To that we add 6.25% from growth to arrive at a total investor return expectation of 10.25%. This is the investor return an investor should expect over the long-term if the market is bought at the computed multiple. It is not the return expectation based on where the market is trading at the present point in time.

To run a quick check on expected returns from markets as currently priced you can multiply the earnings by the 53% notional payout, divide the result by market price, and then add 6.25% to the result. For example, with the market at 1,800, and operating earnings at $108, over the long term we can expect 3.2% ($108 * 53%/1800) via the dividend and the buyback yield, and 6.25% from future growth. The result is a total return expectation of 9.45%. This is below target market return expectations of 10.25%, indicating that the market is somewhat expensive.

The information on dividend and buyback yield is based on the data below, which is based on an analysis of the most recent data available on the S&P Dow Jones Indices website, which you can download here.

**Source: My Analysis of Data Available for Download on S&P Dow Jones Indices website, which you can download****here****.**

**Equity Risk Premium (ERP)**

The equity risk premium is the excess return demanded by investors in equity to compensate them for the higher risk associated with equity. It is calculated as the difference between the market return and the risk free rate multiplied by β. Since the market has a β of 1, the equity risk premium for the S&P500 is estimated as the difference between expected market return and the risk free rate. In my view, the very long-term equity risk premium ought to be 5.75% (Rm 10.25% minus Rf 4.5%).

Over the very long-term equity risk premiums have tended to 4.5%, or 50% of the market return expectations, and 100% of growth expectations. My projections imply that forward ERP is 56% of forward market return expectation and 92% of forward growth expectations.

**Back to the Multiple: This is Value**

Thus the multiple most appropriate for me as an investor is 14. That is $1 * (1 + G%) * (1 - Rr)/(VLT ERP + Rf - G%) = $1 * 106.25% * (1-47%)/5.75% + 4.5% - 6.25%).

We have the target multiple. But what do we multiply it with? Which of the E's is applicable? Is it trailing 12 month earnings, is it current year earnings, is it forward four quarter estimates, is it forward year earnings, is it cyclically adjusted earnings (say an average or median of ten or six year earnings), is it as reported earnings, is it operating earnings?

It is up to you to decide. It should represent the level of earnings from which you expect sustainable growth at the estimated levels to be feasible over the long term. I tend to focus on trailing twelve months and a six year median earnings as action levels.

At present, with 2013 expected to close out with operating earnings of $108 and inflation adjusted six year median operating earnings at $106, I expect 1,485 to 1,515 to be a good buy range to return to allocation. At 1732, a multiple of 16 times, I would be comfortable in the context of historically consistent animal spirits in multiples.

**Animal Spirits and the Multiple: This is Price**

Over the past several decades, markets delivered returns of 9%, half through a dividend plus a buyback yield, and half through growth. The dividend and the buyback yield has been accomplished with a payout of about 60%. Using these numbers to compute a multiple which would have correctly calculated the present value of future cash flows from each $ of earnings we get 13.98. Yet on average markets have traded at multiple of close to 16. Why?

We are animals. And there are animal spirits at work. Animal spirits tend to squeeze the equity risk premiums as investors accept lower market returns.

Like price, value changes, but the value walks at a more leisurely pace: it changes slowly with the passage of time and performance. You may find that the price cause the markets to be over-valued (or undervalued) for an extended duration. Thus while a market trading at or below the value is a buy signal, the market being higher than value level is not necessarily a sell signal.

Over the years when growth ran at long-term rates of 4.5%, and long-term market returns were delivered at 9%, the market traded at an average multiple of 16. This implies that much of the time investors were satisfied with a return of 7.9%. This implies that the equity risk premium of 4.5% declined to 3.4% as a result of animal spirits.

Since we are creatures of habit, and bond yields are low, we could find investors embracing risk for return. For instance, if today investors were willing to accept a 7.9% return, it would imply a long-term equity risk premium of 3.4% (7.9% minus long-term risk free rate 4.5%). This is in-line with historic equity risk premiums adjusted by the work of animal spirits. Indeed, with current risk free rates at 2.71%, the apparent equity risk premium of 5.2% (7.9% minus current risk free rates 2.7%) is wide compared to long-term equity risk premiums of 4.5%, and to the 3.4% equity risk premium as adjusted for animal spirits. This suggests that while markets are expensive in the context of value, they are not expensive in the context of price.

Today, with the markets trading near 1,800, a multiple of 16.63 trailing 12 month operating earnings, assuming my expectations of 6.25% long-term earnings growth are not misplaced, my market return expectation is 9.44%: a 53% notional payout multiplied by $108 in earnings, divided by 1,800 plus 6.25%. If my growth estimates are incorrect, and growth comes in at 5.65% (Conference Board U.S. Growth 2.4% + 2% Inflation & Conference Board Global Growth 3.1% + 3.8% Inflation implies S&P500 earnings growth of 5.65% {[4.4%+6.9%]/2}), the market return expectation with S&P500 at 1,800 is 8.83%: a 53% notional payout multiplied by $108 in earnings, divided by 1,800 plus 5.65%.

The 9.44% to 8.83% range of return expectations is higher than the 7.9% that has been acceptable to investors in the past. The implied risk free rate expectation using my long-term equity risk premium target of 5.75% is 3.69% (9.44% - 5.75%) to 3.08% (8.83% - 5.75%).

Given that the Conference Board estimates are likely more credible (and certainly more influential) than mine, the current implied risk free rate expectation of 3.08% suggests that the market is vulnerable to a decline if the risk free rate rises from its present level of near 2.7% to 3.08% and beyond. However, unless growth expectations start revising downwards, compared with the recent strengthening of growth expectations, I doubt that the market is priced for a dramatic decline.

This is comforting: at present there is no compelling reason to buy or to sell.

**Confronting the Confounding Growth Multiple Conundrum**

Look at the mathematical expression: $1 * (1 + G%) * (1 - Rr)/(VLT ERP + Rf - G%). We know that VLT ERP + Rf is the market return expectation. What would we do if the market return expectation was lower than or equal to growth? We get either an incoherent or an insolvable value!

Approaching growth multiples is different, and I'll get into it in a post another day. We have all seen situations where buying a high multiple paid out manifold, as well as situations where buying a high multiple ended in tears. What occurs is not a result of the multiple: the multiple simply is what it is. The outcome is influenced by the accuracy of the various estimates included in the components of the multiple.

Working through the math of the multiple gives you a better chance of seeing whether today's high multiple, will become a lower multiple and a profitable investment tomorrow.

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