The market is expensive. If we work with a risk free rate expectation of 4.50%, a long-term equity risk premium of 5.75%, we are setting our long-term return expectations at 10.25%. This we expect from a market where long-term nominal growth can be expected at 6.25%, with shareholder value returned via a mix of dividends and buybacks of 53% of earnings. You can read more about where I get my estimates for long-term market returns, long-term nominal growth, and the equity risk premium here.

With as reported earnings for 2014 estimated at $106, if we use a very long-term growth expectation of 6.99%, the S&P 500 is worth 1,845.00. S&P 500 Value = [1 + Long-term Growth Rate] * Sustainable Earnings * Adjusted Payout Ratio / [Long-term Return Expectation-Long-term Growth Rate] = 106.99% * $106 * 53% / (10.25%-6.99%) = $1,845. At this price, it is likely that an investor with a return expectation of 10.25% will be satisfied. My estimate of long-term nominal earnings growth for the S&P 500 is 6.25%, based on a mix of long-term nominal U.S. growth and long-term nominal global growth. Alpha is the difference between actual returns and the risk adjusted return expectation. And if I am correct, the market is over-valued and pricing seventy-four basis points, of negative alpha. If we use S&P 500 operating earnings estimates of $116 for 2014, we get forty-four basis points of negative alpha.

As of now, negative alpha of forty-four to seventy-four basis points suggests that markets are expensive, but not terrifyingly so. It is time for lower risk appetite or rising risk aversion levels, and for rotation out of expensive pockets of the market, to cheaper areas. Churn, not exit is what is evident. The S&P 500 is attractively valued at 1,492, if you rely on as reported earnings and at 1,633 if you rely on operating earnings. This does not necessarily suggest that a bear market is imminent: the implied over-valuation of 11% to 19% is very much in-line with the customary level of over-valuation during periods outside of recessions.

In my view, selling a market simply because it is expensive is not a good idea. Markets can go from being expensive to very expensive, and from being cheap to very cheap. And no one knows which it will be. But negative alpha is an important market timing tool for asset allocators - it hints that it might be time to return to allocation, by selling some of the expensive asset class, and buying another asset class.

During times when alpha expectations turn negative, it pays to listen closely to the voice of the market, for that is where we can see changing levels of risk aversion. In U.S. we have seen small and midcap stocks out-perform large capitalization stocks for a long while. We have also seen growth stocks outperform value stocks for an extended duration. This signals low levels of risk aversion amongst market participation. But in recent times this has changed. There are clear signs of a shift in bias back to value from growth and from small and midcap stocks to large cap stocks. This suggests that risk aversion levels are rising. And so the more active asset allocator might also use an expectation of negative alpha to adopt a more defensive equity portfolio.

In this post I am looking at Apple (NASDAQ:AAPL), because it is a value opportunity hiding in plain sight. And it is hard to believe that this value will not be recognized.

**How do different market participants view Apple?**

A couple of years ago, I had written some code to facilitate stock selection. It would help if you read about the build-out of that system here, as that will allow you to appreciate the model output later in this post better.

*AOM Statistical Scores*

The AOM statistical scores are a statistical evaluation of thirty-eight key indicators for the company, grouped into value, growth, quality, and momentum categories. It illustrates how the key indicators for the stock, perform in comparison to the market capitalization weighted scores for the market, the stocks sector and the stocks industry of operation.

Apple scores high on value, particularly in comparison to the market as a whole, and in comparison with stocks in its sector of operations. It scores high on quality, across the board. The score for growth is mediocre, while momentum is weak, particularly in comparison with stocks in its industry of operation.

**Source:****Alpha Omega Mathematica**

*AOM Model Recommendation*

This stock appeals to value and growth style stock selectors, regardless of whether they allocate capital at sector, industry, or with no sector/industry bias. The balanced investor who considers value, growth, quality and momentum equally, might appreciate this stock too, though a balanced investor allocating capital at industry level would be inclined to hold. Momentum investors and investors with no stock selection style bias allocating capital at market level would be inclined to buy, while those allocating capital at sector or industry level would hold.

**Source:****Alpha Omega Mathematica**

Overall, after analyzing fifteen stock selection and capital allocation strategy combinations, the system assigns an AOM Score of 68% and an AOM Buy Recommendation for Apple.

The AOM statistical scores for each of the fifteen strategy combinations are unique and not comparable with each other. The AOM Score is very different from AOM Statistical scores: it evaluates and rates the AOM Statistical scores for each of the fifteen strategy combinations, and uses a unique technique to make the statistical scores across the strategy combinations comparable. The output is the AOM Score: a quantitative assessment of the output from the fifteen strategy combinations. The AOM Recommendation is a plain English recommendation based on the quintile the AOM Score falls in.

I'll hasten to add that this is a package aimed at generating ideas, it does not intend to, and nor does it replace the due diligence we must do as investors. It is a tool that uses quantitative techniques to understand the behavior of different market participants, and then brings that data together so that users can hear the voice of the market through the noise. The AOM system can guide you where to look, but make no mistake about it - it cannot look for you.

**The Case for Apple:**

*Why look at Apple now?*

Firstly, Apple is a mega cap stock. This gives it a defensive character, which appeals to me when I perceive the markets are expensive. Secondly, Apple pays a dividend of $12.20 per share, which provides a dividend yield to 2.3%, which is a premium to the broad market dividend yield. And in the coming quarter the dividend is likely to be hiked. This too provides defensive characteristics to the stock. Thirdly, Apple enhances its defensive character by returning very significant capital to owners via buybacks.

**Beta, co-efficient of determination and alpha intercept considerations**

Value Line reports a beta of 0.90 for Apple. The Value Line beta is calculated as a five-year regression of weekly closing prices of the stock, relative to weekly closing prices of the market, adjusted for beta's tendency to converge towards one.

I calculate the raw beta based on the five-year regression of weekly closing prices of the stock, relative to weekly closing prices of the S&P 500 at 1.04, and I adjust it to 1.01 on account of the beta's tendency to converge towards one. This low beta adds defensive characteristics to the stock.

The coefficient of determination for Apple is 31.55%. This suggests that only 31.55% of the price movement in Apple is explained by movements in the market: the residual price movement is based on company specific factors. This low coefficient of determination suggests that the market related risks are low. And because company specific risks can be diversified, Apple is a great pick for most portfolios at the present time.

Apple also has an alpha intercept of 0.32%, which means that if the S&P 500 returns 0%, the stock can be expected to return 0.32%. But because of the low coefficient of determination, the raw beta and alpha are less meaningful.

An interesting observation is that the raw beta based on a three-year regression of weekly closing prices of the stock, relative to weekly closing prices of the S&P 500 is at 0.90, which adjusted for beta's tendency to converge towards one is 0.98. The coefficient of determination falls to 21.03%, while the alpha intercept drops to 0.16%.

**Cyclicality and Apple**

In my view, the U.S. economy is getting ready to shift from mid-cycle to late-cycle conditions. And during late-cycle, stocks in the consumer discretionary sector tend to underperform. If I am right, this is negative for Apple investors. While Apple is viewed as a technology company, its business is greatly influenced by discretionary spending by consumers.

You can have a look at this information from Fidelity here to understand their take on sectors and the business cycle. One note of caution: I find reading the business cycle is getting increasingly difficult with globalization - for instance today I think U.S. is exhibiting classic signals of a move towards late-cycle conditions. However, the global business cycle is quite out of sync with the U.S. business cycle. And since many U.S. companies are influenced by the global business cycle, it is more difficult to figure out how U.S. sectors will behave. For example, discretionary spending is expected to underperform in late cycle conditions. But if Europe, other developed markets, or emerging markets shift into early-cycle conditions, which is a time when the discretionary sector typically outperforms, U.S. companies with exposure to discretionary spending could well outperform. Let's not forget that Apple derives over 61% of its sales from outside the United States.

**Analyst price expectations**

Recently Apple traded at $521.32. From Yahoo Finance we know that forty-five analysts expect an average price target of $592.49 (median $600.00), with a high target of $777 and a low target of $270. This is a wide dispersion in expectations, which suggests risks are high. It is early in the year. So far, neither the bulls looking for $777, nor the bears looking for $270 are in control.

**Valuation**

We might believe that Apple is attractively valued. But thus far, its attractiveness has been viewed relative to other stocks in its sector, industry or the coverage universe in the analysis of the perception of different market participants. We also know that Apple is cheap relative to the broad markets. What we do not know is whether the stock is priced to deliver a long-term return in-line with our long-term expectations on a standalone basis and regardless of broad market valuations.

Mathematically, the worth of Apple is estimated as [1 + Long-term Growth Rate] * Sustainable Earnings * Adjusted Payout Ratio / [Long-term Return Expectation-Long-term Growth Rate].

What is our long-term return expectation for a stock with a beta of 1.04, a long-term risk free rate of 4.50% and an equity risk premium of 5.75%? You can read more about where I get my estimates for long-term market returns and equity risk premium here. It is calculated as Risk Free Rate plus Beta Multiplied by Market Return less Risk Free Rate. Thus for Apple, we should be targeting a long-term return of 10.48%. Is the stock priced to deliver that return?

Earnings tend to be volatile from year-to-year over the course of the economic cycle. When I speak of sustainable earnings, I mean the level of earnings that can be expected to occur over the course of an economic cycle, which can be grown at estimated growth rates over a long period of time. This chart below displays normalized trailing twelve month earnings over the past ten years, together with analyst expectations for the current and coming three years. It also shows Apple historic revenues and sales estimates for the current and coming fiscal years. I am very comfortable with $40.32 representing a bottom in earnings, and expect growth to accelerate with a pick-up in the global economic cycle.

Fifty-two analysts included on Reuters data estimate average earnings of $42.77 (High: $47.04, Low: $39.57) during the year ended September 14, with forty-nine analysts estimating that it will rise to an average of $46.50 (High: $53.45, Low: $40.23) for the year ending September 15. Five analysts assess long-term growth rates at 21.28% on average, with a high estimate of 29.40% and a low estimate of 15%.

I am comfortable with $42.77 as a fair representation of sustainable earnings.

The adjusted payout potential is that part of sustainable earnings that we can expect the company to return to shareholders via dividends and buybacks, net of dilution. I expect Apple will pay out approximately 60% of earnings via dividends and buybacks (approximately 35% to 45% via dividends and another 25% to 15% via buybacks) over the long term. An adjusted payout ratio of 60%, assuming nominal earnings growth of 8%, implies a return on incremental equity of 20%: the 40% of earnings retained, invested at a 20% return on equity, delivers the required 8% (40% * 20%) growth. This return on incremental equity is not unreasonable to expect, considering that the recent return on equity is 28.81%, and it has averaged 36.56% over the past five years.

If we use a very long-term growth expectation of 5.3%, Apple is worth $521.32. Apple Value = [1 + Long-term Growth Rate] * Sustainable Earnings * Adjusted Payout Ratio / [Long-term Return Expectation-Long-term Growth Rate] = 105.3% * $42.77 * 60% / (10.48%-5.3%) = $521.32. At this price, it is likely that an investor with a return expectation of 10.48% will be satisfied.

The growth estimate implied by the current market price of 5.3% is ridiculously low. Alpha is the difference between actual returns and the risk adjusted return expectation. If Apple grows at a long-term rate of 8%, in-line with nominal global growth expectations, we have growth alpha of 2.7%. And an investor buying at present levels can expect a long-term return of 13.18%. If earnings grow at the lowest long-term growth estimate of 15% for the next five years, and we assume they reduce to 8% for the following forty-five years, we get a composite very long-term growth rate of 8.68%, which raises the growth alpha to 3.38%, and the total long-term return expectation to 13.86%. I will go with 2.7% as a conservative estimate of alpha available.

An investor with a shorter time horizon might do quite well, too. A price target of $777 implies confidence in long-term earnings growth rising to 6.95% from 5.3% at present. And this level of expectation is well within the realm of possible outcomes. Apple Value = [1 + Long-term Growth Rate] * Sustainable Earnings * Adjusted Payout Ratio / [Long-term Return Expectation-Long-term Growth Rate] = 106.95% * $42.77 * 60% / (10.48%-6.95%) = $777.

To cut a long story short, Apple is priced at a level that offers between 2.7% of alpha, compared with the S&P 500, which prices between forty-four to seventy-four basis points of negative alpha. This indicates that Apple creates alpha relative to the S&P 500 of between 3.14% and 3.88%. This is huge with the magic of compounding. And it represents 31% to 38% over the total long-term market return expectation of 10.25%. Apple represents value hidden in plain sight. One day it will be recognized.

**Explanatory Note: **The formula above helps to compute a fair value for a stock. It is simple, and you can use it to calculate a value based on your expectations, as opposed to mine. But if you do so, be cautious.

If you alter your risk free rate assumptions, you will need to evaluate how that will impact market return expectations and the equity risk premium. You will also need to think about how the risk free rates will impact long-term growth. If you alter the adjusted pay-out assumptions, think about how that might impact the growth assumption and the capital structure. For instance, higher growth may signal a need for fresh equity or debt. And if the capital structure changes, so will beta. When you alter growth assumptions, think about how it might change the operational mix, and the impact that might have on the stock's beta and stock return expectations.

There is a high degree of inter-connectivity between beta, growth, adjusted pay-out ratios, risk free rates, and market and stock return expectations. And reading the inter-connectivity wrong will result in an answer that ranges from absurd to obscene: this model comes with warts. To avoid falling into this trap, here are some ideas, which I hope will help.

1. Since valuation is about what you are willing to pay for a stock, perhaps the most important consideration is the growth risk premium: that is long-term return expectation minus long-term growth rate. The question to ask yourself is that if you expect a stock to grow at a certain rate, how much over and above that growth would you want by way of a stock return expectation, to compensate you for the risk that the long-term growth rate might not be in line with your expectations. This spread will be low for a fast growing stock, where there is great confidence in forward growth expectations, and higher for stocks where the confidence in growth is low. In the very long-term, the growth risk premium has tended toward 4.5% for the market.

2. When you look at sustainable earnings for a growth stock, you need to look at where you expect earnings to be a few years down the road. And then discount that number to its present value using your stock return expectations to obtain today's sustainable earnings

3. When you look at long-term growth rates, remember it is not the next years' growth or the next five years' growth you are looking for. You are looking for a composite long-term growth rate expected over the life of the company. This will be made up of foreseeable growth rates for some years, reversion to market growth rates, and finally a terminal growth rate. The terminal growth rate used by many is the risk free rate. I tend to use the very long-term market growth rates, since if the terminal growth rate is below market growth rates, I as an investor have the option to exit and enter the broad market. The life-expectancy of a typical Fortune 500 company is 40 to 50 years: you can read more about this here. So we can consider using a five-year forward rate, and then assume growth shall revert to being in-line with market growth expectations for the following 45 years. What this signals is that I'm willing to pay a premium for currently foreseeable growth expectations, but after that period, I expect to share fully in the reward of growth over market growth rates. On excel you can calculate a composite growth rate for a company growing at 15% per year for five years, followed by growth at 8% for the following 45 years as 8.68% [=POWER(1/1*115%^5*108%^45,1/50)-1].

4. Test your adjusted payout expectations. Take your growth rate and divide it by 1 minus the payout ratio. The result will estimate the return on equity implied in the model for the company. Review the return on equity to see if it is broadly consistent with the return on equity for the industry over the long-term. If it is high, review the return on equity in the context of the leverage employed by the company. A levered company will have a higher return on equity, and so a high return on equity for a company may be perfectly justifiable in some circumstances. However, higher than industry leverage implies higher financial risk and this implies a higher beta, and a higher market return expectation. If you see a low beta with higher than industry leverage, you may want to compute a bottom up beta for the company, instead of one generated using regression analysis.

**Disclosure: **I am long AAPL. 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.