What is Apple Worth? Part 2: Paradox Resolved?

In part 1 of this series I talked about Daniel Bernoulli's brilliant idea of using a concave utility function to solve the St. Petersburg paradox, a problem originally posed 300 years ago (and a couple of decades before Daniel's solution) by his cousin Nicolas. I explicitly suggested a strong parallel with the way Apple is being undervalued as it gets bigger and bigger, and ended with a discussion of how the overused modern phrase "law of large numbers" (aka lol numbers) as misused by current mainstream financial media talking heads (when referring to the "limits of growth" concept) ironically is in contradiction with the actual fundamental probabilistic concept called the Law of Large Numbers (LLN), first proved by Daniel's and Nicolas' uncle Jacob Bernoulli: If Jacob's LLN applies, the likelihood that Apple will continue to grow would seem higher, given the historical empirical record, not lower (as the lol numbers trite cliche implies). On the other hand, if in fact there are limits to growth being reached, then the actual LLN is violated (independence of trials). In short, the semantic confussion and logical contradiction of invoking a misnamed "law" to justify persistent multiple compression despite continued strong growth in Apple's earnings might get resolved by reinterpreting it not necessarily as limits to growth but instead as an expression of the concept of diminishing marginal utility (although these two concepts might be related after all).

For this part 2 my goal was to explain why and how such a simple solution of a concave utility function works in assessing the pragmatic value placed by most people on such big yet unlikely potential returns as well as why and how the specific logarithmic utility proposed by Bernoulli might not work in every case, and how the theory eventually got refined over the years, decades and centuries. This is no simple task: to wade through 300 years of academic literature on economic theory, and most recently financial theory, from the incipient probability research to modern portfolio theory and CAPM. For an example of the nature of the literature, check out this 850 page book (many pages omitted in that Google Books preview) focused only on a singular very specific thread (the Kelly Criterion) within the broader subject. Even if I could digest and condense all of it into a blog post (which I definitely can't), I wouldn't want to submit my readers to such a dry academic dissertation. Besides, it's mostly way over my head. Hence, after being set back on my promise to post this within a couple of weeks of part 1, I've finally decided to go with a very simplified intuitive exposition, but leaning heavily on the multiple referenced links provided (with the caveat to only rely on Wikipedia for a superficial glance and to conduct your own research through more reliable sources if a deeper understanding is needed). Apologies if this ends up with significant theoretical holes.

Fiscal 4Q 2012 Final Estimates [Updated 10/24]

[10/24 UPDATE: In light of information revealed during the October 23 event and, to a lesser extent, widely reported PC industry weakness, I'm revising iPad and Mac estimates for Q4. I now estimate 4m fewer iPads and 0.5m fewer Macs (mainly inventory drawdowns) results in about $3b less revenue and $0.80 lower EPS. Gross margin percent goes up about 70 bps due to mix. I'll be posting a detailed revision later today (here).]

As of Tuesday's closing price of $661.31, AAPL is trading at a 10.1 multiple on my forward EPS estimate (8.1x ex-cash). Valuation remains attractive despite the 9.1% increase in share price and downward revenue/EPS estimate revisions for FY13 of 8.7/10.4% over the last 3 months.

Detailed estimates below:

3mo ending Sep-2012   Rev($M)   EPS($)
-------------------   -------   ------
Apple guidance         34,000     7.65
Analysts consensus     36,270     8.88
Deagol estimates       39,136    10.48

What is Apple Worth? Part 1: Regarding Large Numbers

A Big Lottery

“Peter tosses a coin and continues to do so until it should land "heads" when it comes to the ground. He agrees to give Paul one ducat if he gets "heads" on the very first throw, two ducats if he gets it on the second, four if on the third, eight if on the fourth, and so on, so that with each additional throw the number of ducats he must pay is doubled. Suppose we seek to determine the value of Paul's expectation.”

This is how Daniel Bernoulli rephrased a problem nagging a tight circle of mathematicians almost 300 years ago. The issue is that, if you go by the mathematical expected value of the winnings you end up with an infinite value for the lottery: you have 1/2 probability of winning one ducat on the first throw, 1/4 probability of winning two ducats in 2 throws (which adds another 1/2 to the expected value), 1/8 probability of winning 4 ducats in 3 throws (adding another 1/2), and so on adding 1/2 a ducat in value an infinite number of times.

The expected value suggests that a player should be willing to pay any amount asked for a chance to play this lottery. Yet, in real life no one in their right sense would pay more than a few ducats to play this, perhaps somewhere around 5 ducats. Why is it that the mathematical expected value doesn’t reflect the real value placed by real players on this lottery?

Fiscal 3Q 2012 Final Estimates

As of Friday's closing price of $605.88, AAPL is trading at a 8.9 multiple on my forward EPS estimate (7.0x ex-cash). For guidance and forward-looking valuation I'm assuming an iPhone launch in September (which paradoxically is the conservative view for the long term). This implies about 6m units shifted into this fiscal year instead of next, and affects the replacement cycle timing well into 2014. Still, I'm including estimates for the October launch scenario below so you can pick your own.

Detailed estimates below:

3mo ending Jun-2012   Rev($M)   EPS($)
-------------------   -------   ------
Apple guidance         34,000     8.68
Analysts consensus     37,370    10.34
Deagol estimates       40,641    12.16

Fiscal 2Q 2012 Final Estimates

At today's closing price of $628.44, AAPL is trading at a 10.4 multiple on my forward EPS (8.5x ex-cash).

The valuation metrics provided below are intended to track the market price of the shares at the given timeframes, and thus exclude any dividend paid up to that point in time because the dividend is subtracted from the cash balance, as it should. However, because my underlying theory is that shares should be (roughly) valued on a forward-one-year basis, to compensate for this one needs to add back into the value the next 12-months worth of dividends.

For example, the current fair value is based on 10 times the EPS estimate of $60.39 for the next 12 months which gives $604, plus the estimated cash balance at the end of the next 12 months of $165/sh (which already excludes any dividends paid over that period), and adding back the $8 worth of dividends paid over the next 12 months which a buyer today would in fact be entitled to, resulting in a $777 fair value. The one-year target is computed similarly given the annual EPS, dividends and cash balance for the 12 months ending 2 years from now.

I consider the "trailing" metric relevant for the past 3 to next 3 months, the "fair value" metric relevant for roughly the next 3 to 9 months, and the "1yr target" metric relevant for roughly the next 12-18 months. Of course this is all my own subjective/empirical sense.