**Summary**

I'm a big supporter of the works of Professor Shiller and the duo Andrew Smithers and Stephen Wright which popularized the concepts of CAPE (Cyclical Adjusted Price to Earnings ratio) and q (or Tobin's q).

In this paper the market is identified with the S&P 500 index (SPY, VOO, IVV, SSO).

These measures give a framework to calculate the fair value of the market.

CAPE and q give a similar picture of the current state of the market: both highlight a strong overvaluation between 70% (CAPE) and 50% ((q)).

But current environment is very different from what we had experienced in the past; I'm referring here to the unprecedented stimulus orchestrated by the central banks of the whole world with real rates deep negative and heavy doses of quantitative easing.

Only the improperly called Fed model for the valuation of the market tells that the market is currently undervalued. The Fed model simply compares the Earning Yield (the inverse of the current P/E, 4.23% at the 3rd of April 2017) with the 10 year yield of the US Treasury note (2.33%) and therefore concludes that, at present conditions, the market should be considered still undervalued.

The picture depicted by CAPE / q and, on the other side, by the Fed model is therefore very different.

My goal in this paper is to modify the CAPE calculation in order to take into account current level of interest rates to better check if the market is as overvalued as it seems by original CAPE.

It will result that at current levels the market is almost exactly fairly valued.

My conclusion is that probably the market is not as overvalued as it seems according to the original CAPE and q, in any case I fear that the market could be very sensitive to a rise in interest rates level and to the end of quantitative easing, which makes present conditions very risky.

**Introduction to CAPE**

Cyclically adjusted Price to Earning (CAPE) is simply the ratio between price of the index (S&P 500) and the average of ten years inflation adjusted earnings (geometric mean).

Nobel laureate professor Shiller popularized this concept, but the first idea came from Graham and Dodd who in their classic Security Analysis text argued for smoothing a firm's earnings over the past five to ten years, because one year earnings were too volatile to offer a good idea of a firm's true earning power through the economic cycle.

CAPE has some predictive power to future equity returns over timescales of 10 to 20 years, with higher than average CAPE values implying lower than average long-term annual average returns and vice versa.

**Introduction to q**

Tobin's q is the ratio between physical asset's market value and its replacement value. q can be used to value the whole market dividing the valuation of the stock market by the aggregate corporate net worth. USA publishes regularly this type of statistics.

The average historical value of the q ratio is supposed to be the fair value of the market (similar to the average of the CAPE which is the historical fair multiple); however the average value of q is not perfectly coincident with one, because it is difficult to exactly value corporate net worth and in particular intangible assets. q has the good property to specify an economic reason why it should be mean reverting trough fair value. In particular, high q values encourage companies to invest more in capital because they are "worth" more than the price they paid for them and possibly issue more stocks to pay for the physical expansion. This is because market value is greater than the value of the company's recorded assets. When q is low, companies can for example repurchase shares to close the valuation gap or liquidate some assets. So this form of arbitrage should keep q not too far from the equilibrium value.

**Historical CAPE and q**

You can see below the historical values of CAPE and q since 1900 on the left scale of the graph and also the logarithm of the inflation adjusted S&P 500 on the right scale. In general, low values of CAPE / q are associates with subsequent rising markets (1920, 1933, 1982 and 2009) and high values of CAPE / q with declining S&P 500 (for example 1929, 2000, 1969, 1937).

(Source: Created by author)

You can see that at the moment CAPE and q are very high with respect to their historical average values; they are only below the peaks of 1929 and 2000. S&P 500 is currently overvalued by 90% according to CAPE and almost 70% to q.

CAPE is higher than q and this should currently depend on the real estate sector which in the last years increased its tangible value more than it earning power.

You can see that CAPE and q were very similar in the past, in fact they provided almost the same information and, for ease of calculation, I will limit the following analysis to CAPE, which can be calculated quite easily for a lot of different markets.

**CAPE predictions**

CAPE and q give us a quick answer to the amount of over / under valuation of the market, but it is also interesting to better understand the historical speed of mean reversion of these indicators to their historical means. To this end, I considered a variation of the CAPE indicator and in particular two series, log E_{10} (the natural logarithm of the average of ten years inflation adjusted earnings) and log S&P 500. Using then the average CAPE of 17.25, I can construct the theoretical fair value of the S&P 500 multiplying the average earnings by the average P/E.

In the graph below I plotted the difference between the real value of the S&P 500 and its theoretical value in order to better show the peaks and valleys of the market valuation since 1900. The graph is quite similar to the previous one, as it should be.

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The next step is to check if there is a cointegration relationship between the two variables log S&P 500 and log(E_{10} * PE_{avg}). Cointegration means a statistical relationship between variables, a long term link between their values. In this case, a cointegration relationship between stock prices and earnings means that these two variables are mean reverting, i.e. when for example stock prices go sufficiently above fair value, there is a force that brings the relationship to its equilibrium value.

Standard statistical techniques affirm that with high probability there is a cointegration relationship between these two variables and thereafter a model can be constructed. With the model in place, a forecast can be done with a standard Monte Carlo approach.

Below there are the results, first for the log variables and then for the real variables, with the average forecast path and their one-standard deviations bands.

The forecast is done both for the earnings and for the market, first for a 10 years period and then for 50 years, because the mean reversion is quite slow to converge. Earnings are predicted to grow as they have done in the past and this of course should push stock price higher, but in the current situation, the unbalance between market price and fair value is too big and so the cointegration, mean reverting force should prevail and bring down the S&P 500 toward its fair value.

Maybe the best way to represent those forecasting results is to compute the probability of realizing capital gains in real terms in the next 10 and 50 years which is respectively 27.8% and 20.6%. By the way, current dividend yield of S&P 500 is around 2% quite similar to recent inflation and to the Fed target, so we have that real, inflation adjusted, returns should coincide with nominal returns. You can see that there is a small probability of obtaining positive real/nominal returns in the next years, suggesting a prudent allocation to the equity market.

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**Introduction to the Fed Model**

The "Fed model" is a very simple theory of equity valuation that compares the stock market's earnings yield (E/P) to the yield on long-term government bonds. When the two yields are very similar the markets are in equilibrium as there is no convenience to the switch between bonds and equities.

The model is very basic and could be improved simply by using after tax average corporate investment grade yield instead of government bond yield, in case it is a quick and easy test of relative convenience between stocks and bonds.

Equity valuation should be the present value of future cash flows, so it's natural to link stock valuation to interest rates.

Since 1980 correlation between earning yields and interest rates was quite high, even if for longer past periods this relationship was much lower. Moreover this model lacks a strong predictive power.

At the 3rd of April 2017 current earning yield is at 4.23% and 10 US government bond yield is at 2.33%, so for the Fed model the market is right now undervalued.

**Benjamin Graham formula**

Benjamin Graham in its book The Intelligent Investor (for me the best financial text ever written) proposed a formula to quickly determine the fair value of stocks. The original 1962 formula was

P = EPS x (8.5 + 2g)

Where g is the expected grow rate in the following 10 years and 8.5 represents the base P/E for a no growth company.

In 1974 Graham revised the formula to take into account interest rates level as

P = EPS x (8.5 + 2g) x 4.4 / Y

Where Y is the current yield on 20yeras AAA corporate bond and 4.4 is the average yield of high grade corporate bonds in 1962 in order to recover the previous formula for that year.

**My version of CAPE, ICAPE**

Creatively combining Fed model's and Graham formula's suggestions into the CAPE framework, I devised an adjusted and dynamic P/E to use instead of the fixed average CAPE of 17.25.

The resulting graph of the difference between the logarithm of S&P 500 and of the fair value of the market is quite similar to the previous one, except for the last period from 2008, when world's central banks started unprecedented monetary policies of negative real and nominal interest rates and quantitative easing. The effect of these policies is that according to ICAPE the market seems at fair value right now, and was a buy since 2009, whereas the standard CAPE considered the market almost always overvalued in the same period. So the operative indications of ICAPE were much better than those of CAPE in the last 8 years of low interest rates.

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**Conclusions**

The classical CAPE or q approaches to stock valuations scream sell the market, whereas the CAPE adjusted for the interest rate level portrays a neutral conclusion.

A note of caution is deserved in my opinion, fair value depends heavily on the 10 years US Treasury yield, which is at historic lows. Thereafter, probably the best days for the equity returns were in the past years / decades when interest rates were higher and investors benefited later of the secular bond market rally with the resulting P/E multiple expansion.

Classical CAPE and q tells us that long term equity futures returns will be very low with respect to the historical standard, as it is quite intuitive as earning yields are at historical low and corporate profit margins are at historical high. Classical CAPE is indeed a good indication of future returns.

ICAPE gives us an explanation of current market multiples which were rich since many years and also suggest to keep an eye on the world's central banks policies. Negative real interest rates, unprecedented quantitative easing and the Fed put (the practice of the Fed to support the equity market at soon at it falls a little bit) created the conditions for another global asset bubble and the underestimation of the risks (volatility is one of the few thinks that are cheap today). The transition to a more normal monetary policy will be very difficult and this could be the factor that could blow up the bubble.

At the 3rd of April 2017 conditions (10 years Treasury yield of 2.33%) the market is, as said, at fair value (less than 3% overvalued), but if the rate would be for example at 3% (current Fed Funds terminal rate according to the plot of FOMC) the same market would be over 30% overvalued. The 90% overvaluation of CAPE corresponds to a "normal" interest rate of 4.3% with ICAPE, not far from the 4.4% equilibrium level in the Graham's formula.

Buying the S&P 500 at these levels implies an earning yield around 3% p.a. according to CAPE and I think that this is the best average outcome in terms of real returns one should expect holding the index for a 10 years period. 3% p.a. is not a big bargain even if real bond yield are at zero. Prudence should be the mantra of the conservative investor. Highlighting the fact that this paper represent my current good-faith view and is not meant as investment advice, for my personal portfolio I keep my investments in low multiple, low rated and out of favor stocks, like for example Surgutneftegas preference (OTCPK:SGTPY), waiting for Mister Market to realize their fair value. At higher values of the S&P 500 index when ICAPE would move to a clear overvalued area, it would be possible to hedge some long stocks positions with a short on the S&P 500 (SDS, SPXU-OLD) to reduce portfolio's risk.

**Disclosure:** I am/we are long SGTPY.

I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.

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