Wednesday, June 29, 2016

The Brexit Effect: The Signals amidst the Noise


There are few events that catch markets by complete surprise but the decision by British voters to leave the EU comes close. As markets struggle to adjust to the aftermath, analysts and experts are looking backward, likening the event to past crises and modeling their responses accordingly. There are some who see the seeds of a market meltdown, and believe that it is time to cash out of the market. There are others who argue that not only will markets bounce back but that it is a buying opportunity. Not finding much clarity in these arguments and suspicious of bias on both sides, I decided to open up my crisis survival kit, last in use in August 2015, in the midst of another market meltdown.

The Pricing Effect
I am sure that you have been bombarded with news stories about how the market has reacted to the Brexit vote and I won't bore you with the gory details. Suffice to say that, for the most part, it has followed the crisis rule book: Government bond rates in developed market currencies (the US, Germany, Japan and even the UK) have dropped, gold prices have risen, the price of risk has increased and equity markets have declined. The picture below captures the fallout of the vote:


While most of the reactions are not surprising, there are some interesting aspects worth emphasizing. 
  1. Currency Wars: If this is a battle, the British Pound is on the front lines and taking heavy fire, down close to 10% over the last week against the US dollar and approaching three-decade lows, with the Euro seeing collateral damage against the US dollar and the Japanese Yen.
  2. Old EU, New EU and the Rest of the World : The damage is greatest in the EU, but even within the EU, it is the old EU countries (primarily West European, that joined the EU prior to 2000) that have borne the biggest pain, with sovereign CDS spreads rising and stock prices falling the most. The new EU countries (mostly East European) have been hurt less than Britain's other trading partners (US, Australia and Canada) and the damage has been muted in emerging markets. At least for the moment, this is more a European crisis first than a global one.
  3. Banking Problems? Though I have seen news stories suggesting that financial service companies are being hurt more than the rest of the market by Brexit and that smaller companies are feeling the pain more than larger ones, the evidence is not there for either proposition at the global level. At more localized levels, it is entirely possible that it does exist, especially in the UK, where the big banks (RBS, Barclays) have dropped by 30% or more and mid-cap stocks have done far worse than their  large-cap counterparts.
While I did stop the assessment as of Friday (6/24), the first two days of this trading week have continued to be volatile, with a big down day on Monday (6/27) followed by an up day on Tuesday (6/28), with more surprises to come over the next few days.

The Value Effect
As markets make their moves, the advice that is being offered is contradictory. At one end of the spectrum, some are suggesting that Brexit could trigger a financial crisis similar to 2008, pulling markets further down and the global economy into a recession, and that investors should therefore reduce or eliminate their equity exposures and batten down the hatches. At the other end are those who feel that this is much ado about nothing, that Brexit will not happen or that the UK will renegotiate new terms to live with the EU and that investors should view the market drops as buying opportunities. Given how badly expert advice served us during the run-up to Brexit, I am loath to trust either side and decided to go back to basics to understand how the value of stocks could be affected by the event and perhaps pass judgment on whether the pricing effect is under or overstated. The value of stocks collectively can be written as a function of three key inputs: the cash flows from existing investment, the expected growth in earnings and cash flows and the required return on stocks (composed of a risk free rate and a price for risk). The following figure looks at the possible ways in which Brexit can affect value:

Embedded in this picture are the most extreme arguments.  Those who believe that Brexit is Lehman-like are arguing that it will lead to systemic shocks that will lower global growth (not just growth in the UK and the EU) and increase the price of risk. In this story, these shocks will come from banking problems spilling over into the rest of the economy or an unraveling of the EU.  Those who believe that Brexit’s effects are more benign are making a case that while it may reduce UK or even EU growth in the short term, the effects of global growth are likely to be small and/or not persistent and that the risk effect will dissipate once investors feel more reassured. 

I see the truth as falling somewhere in the middle.  I think that doomsayers who see this as another Lehman have to provide more tangible evidence of systemic risks that come from Brexit. At least at the moment, while UK banks are being hard hit, there is little evidence of the capital crises and market breakdowns that characterized 2008. It is true that Brexit may open the door to the unraveling of the EU, a bad sign given the size of that market but buffered by the fact that growth has been non-existent in the EU for much of the last six years. If the European experiment hits a wall, it accelerate the shift towards Asia that is already occurring in the global economy. I also think that those who believe that is just another tempest in a teapot are too sanguine. The UK may be only the fifth largest economy in the world but it has a punch that exceeds its weight because London is one of the world's financial centers. I think that this crisis has potential to slow an already anemic global economy further. If that slowdown happens, the central banks of the world, which already have pushed interest rates to zero and below in many currencies will run out of ammunition. Consequently, I see an extended period of political and economic confusion that will affect global growth and some banks, primarily in the UK and the US, will find their capital stretched by the crisis and their stock prices will react accordingly. 

The Bigger Lessons
It is easy to get caught up in the crisis of the moment but there are general lessons that I draw from Brexit that I hope to use in molding my investment strategies.
  1. Markets are not just counting machines: One of the oft-touted statements about markets is that they are counting machines, prone to mistakes but not to bias. If nothing else, the way markets behaved in the lead-up to Brexit is evidence that markets collectively can suffer from many of the biases that individual investors are exposed to. For most of the last few months, the British Pound operated as a quasi bet on Brexit, rising as optimism that Remain would prevail rose and falling as the Leave campaign looked like it was succeeding. There was a more direct bet that you would make on Brexit in a gamblers' market, where odds were constantly updated and probabilities could be computed from these odds. Since Brexit was also one of the most highly polled referendums in history, you would expect the gambling to be closely tied to the polling numbers, right? The graph below illustrates the divide.
    While the odds in the Betfair did move with the polls, the odds of the Leave camp winning never exceeded 40% in the betting market, even as the Leave camp acquired a small lead in the weeks leading up to the vote. In fact, the betting odds were so sticky that they did not shift to the Leave side until almost a third of the votes had been counted. So, why were markets so consistently wrong on this vote? One reason, as this story notes,  is that the big bets in these markets were being made by London-based investors tilting the odds in favor of Remain. It is possible that these investors so wanted the Remain vote to win and so separated from this with a different point of view that they were guilty of confirmation bias (looking for pieces of data or opinion that backed their view). In short, Brexit reminds us that markets are weighted, biased counting machines, where if big investors with biases can cause prices to deviate from fair value for extended periods, a lesson perhaps that we learned from value investors piling into Valeant Pharmaceuticals.
  2. No one listens to the experts (and deservedly so): I have never seen an event where the experts were all so collectively wrong in their predictions and so completely ignored by the public. Economists, policy experts and central banks all inveighed against exiting the EU, arguing that is would be catastrophic, and their warnings fell on deaf years as voters tuned them out. As someone who cringes when called a valuation expert, and finds some of them to be insufferably pompous,  I can see why experts have lost their cache. First, in almost every field including economics and finance, expertise has become narrower and more specialized than ever before, leading to prognosticators who are incapable of seeing the big picture. Second, while economic experts have always had a mixed track record on forecasting, their mistakes now are not only more visible but also more public than ever before. Third, the mistakes experts make have become bigger and more common as we have globalized, partly because the interconnections between economies means there are far more uncontrollable variables than in the past. Drawing a parallel to the investment world, even as experts get more forums to be public, their prognostications, predictions and recommendations are getting far less respect than they used to, and deservedly so.
  3. Narrative beats numbers: One of the themes for this blog for the last few years has been the importance of stories in a world where numbers have become more plentiful. In the Brexit debate, it seemed to me that the Leave side had the more compelling narrative (of a return to an an old Britain that some voters found appealing) and while the Remain side argued that this narrative was not plausible in today's world, its counter consisted mostly of numbers (the costs that Britain would face from Brexit). Looking ahead to similar referendums in other EU countries,  I am afraid that the same dynamic is going to play out, since few politicians in any EU country seem to want to make a full-throated defense of being Europeans first. 
  4. Democracy can disappoint (you): The parallels between political and corporate governance are plentiful and Brexit has brought to the surface the age-old debate about the merits of direct democracy. While some (mostly on the winning side) celebrate the power of free will, those who have never trusted people to make  reasoned judgments on their futures view the vote as vindication of their fears. In corporate governance, this tussle has been playing out for a while, with those who believe that shareholders, as the owners of public corporations, should control outcomes, at one end, and those who argue that incumbent managers and/or insiders are more knowledgeable about businesses and should therefore be allowed to operate unencumbered, at the other. I am sure that there are many in the corporate world who will look at the Brexit results and cheer for the Facebook/Google model of corporate governance, where shares with different voting rights give insiders control in perpetuity. As someone who has argued strongly for corporate democracy and against entrenching incumbent managers, it would be inconsistent of me to find fault with the British public for voting for Brexit.  In a democracy, you will get outcomes you do not like and throwing a tantrum (as some in the Remain camp are doing right now) or threatening to move (to Canada or Switzerland) are not grown-up responses.  You may not like the outcome, but as an American political consultant said after his candidate lost an election, "the people have spoken... the bastards".
The End Game
I have not bought or sold anything since the Brexit results for the simple reason that almost anything I do in the midst of a panic is more likely to be counter productive than helpful. To those who would argue that I should move my money away from Europe, the markets have already done that for me (by marking down my European stocks) and I see little to be gained by overdoing it. To those who assert that this is the time to buy, I am not a fan of blind contrarianism but I will be looking at UK-based companies that have significant non-European operating exposure in the hope that markets have knocked down their prices too much. Finally, to those who posit that this is a financial meltdown, I will keep a wary eye on the numbers, looking for early signs that the worst case scenario is playing out. In my view, bank stocks will be the canaries in the coal mine, and especially so if the damage spreads to non-UK banks, and I will continue to estimate equity risk premiums for the S&P 500 and perhaps add the UK and Germany to the list to get a measure of how equity markets are repricing risk. 

Monday, June 6, 2016

Icahn exits, Buffett enters, Whither Apple? Value and Price Effects of Big Name Investing

In my last post, I looked at Apple, arguing, with a Monte Carlo simulation, that the stock was a good investment at the prevailing market price ($93 at the time of the analysis). I appreciate the many comments that I got on the analysis, some taking issue with the distributions that I used for profit margins and revenue growth and some taking me to task for ignoring the fact that big name investors were either entering and exiting the stock. Those who felt that my valuation was optimistic pointed out that Carl Icahn, a long time and very vocal investor in Apple, had decided to sell his stake in the company on April 28. Some who concurred with my value judgment on Apple pointed out that Berkshire Hathaway (and by extension, Warren Buffett or his proxies) had invested in the company on May 16.  Should Carl Icahn’s decision to sell Apple or Berkshire Hathaway’s choice to buy it change my assessments of value or views on its price? More generally, should the decisions by "big name" investors to buy or sell a specific company affect your investment judgments about that company? 

Price versus Value: The Set Up
To set up the discussion of whether, and if so how, the actions of other investors, especially those with big names and reputations to match, affect your investing choices, I will fall back on a device that I have used before, where I contrast the value and pricing processes.


Put simply, the value process is driven by a company's fundamentals (cash flows, growth and risk) or at least your perception of those fundamentals, whereas  the pricing process is driven by demand and supply, with mood, momentum and liquidity all playing big roles in determining price. In an earlier post,  I argued that it these processes that separate investors from traders, with investors focused on the drivers of value and traders on the pricing process, and that the skills and tools that you need to be a successful trader are different from those that you need to be a successful investor. To understand how and why the entry of a big name investor may alter your assessments of value and price, I would suggest categorizing that investor into one of four types.
  1. An Insider, who is either part of management or has privileged access to management.
  2. An Activist, who plans to change the way the firm is run or financed.
  3. A Trader, whose skill lies in playing the pricing game, with the power to either reinforce or reverse price momentum
  4. A Value Investor, who has valued the company and is willing to take a position based upon that value, on the expectation that the pricing gap will close.
Each type of big name investor has the potential to change how you view the dynamics of price and value, though the place where the change occurs will depend on the investor type.

The entry (or exit) of a big name insider or big name activist can alter your estimate of value for a company, by either changing your perceptions of cash flows, growth and risk or by having the potential to change the company's operating and financing characteristics. As a trader, the entry or exit of a big name trader may cause you to move from one side of the pricing game to the other, i.e., shift you from being a buyer to a seller. Finally, as a long term value investor who believes that a stock is mis-priced but has little or no power to cause the pricing gap to close, the entry of a big name value investor can provide a catalyst for the correction.

The Value Effect 
If you asked a value purist whether the actions of other investors affect his or her value, the answer will almost always be "of course not". After all, the essence of intrinsic value is that it is determined not by what others think about the company but the company's capacity to generate cash flows  over time. That said, there are two ways that the investment action (to buy or sell) of a big name investor can change your assessment of value. 

  1. The first is if the big name investor has private information or is perceived as knowing more about the firm than you do. While that may walk awfully close to the insider trading line in the United States, it is entirely possible that the investor's information is diffuse enough to not be in violation of the law. In this case, it is entirely rational for you, as an investor, to reassess your cash flows and risk, based upon the insiders' actions. That is perhaps why we are so fascinated by insider trading, where the perception is that insider buying is value increasing and insider selling is value selling. In some emerging markets, where possessing proprietary information is neither illegal nor unusual, and the decision by an investor who is perceived as having this information (an insider, manager or family member) to buy (or sell) is an indicator that your value should be increased (decreased). 
  2. The other scenario is where the big name investor is an activist who plans to push for changes in the way the company operates, how it is financed or how much and how it returns cash to investors. The potential effects of these changes can be most easily seen using a financial balance sheet:

To the extent that you believe that the company will have to respond to activist pressure, your assessment of value will change. An asset restructuring can alter he cash flows and risk characteristics of a business, changing your estimate of value, though the direction of the value change and its magnitude will depend on how you see these operating changes playing out in cash flows and growth.  Adding debt to your financing mix can add value to a firm (because of the tilt in the tax code towards debt) or destroy value (because it exposes companies to bankruptcy risk). If you are valuing a company, the entry of a big name activist investor in the ranks with a history of pushing for more debt could lead you to reassess your value estimate as well. Returning more cash to stockholders in special dividends or buybacks can change value either upwards (if the market is discounting the cash on the presumption that the company would waste the cash on bad investments/acquisitions) or downwards (if returning the cash will expose the firm to default risk or substantial financing costs in the future).

The Pricing Effect
In some cases, the big name investing in the stock is a trader, doing so on the expectation that momentum will either continue, sustaining the pricing trend, or that momentum will reverse, causing the trend to reverse as well. Since this trade is not motivated by either new information or the desire to change how the company is run, there is no value effect, but there can be a price effect for two reasons. 
  1. The volume effect: If the big name trader has enough money to back his or her trade, there will be a liquidity effect, where a buy will push the price up higher and a sell will push it lower. 
  2. The bandwagon effect: To the extent that there are some in the market who perceive the big name trader as better at perceiving momentum swings than the rest of us, they will follow the investor in buying or selling the stock. 
In contrast to a value effect, which is long term and sustained, the pricing effect will have a shorter half life. To the extent that the big name trader's time horizon may be even shorter, he or she can still make money from the bandwagon effect. To get a measure of the pricing effect of a big name trade, you have to look at both the resources commanded by the trader as well as the liquidity/trading volume in the stock. A trader with billions under his control investing in a lightly traded and lightly followed stock will have a much bigger pricing effect than in a very liquid, large market capitalization company. 

The Catalyst Effect
It is an undeniable and frustrating truth about value investing that for most of us, it is not just enough to be right in your assessment of value but you have to get the market to correct its mistakes to make money on your investments. If you are a small investor, there is little that you can do to close the pricing gap because you have neither the money or the megaphone to close the gap. A big name value investor, though, may be more successful for two reasons: he or she can take a larger position in the stock and as with the big name trader, create a bandwagon effect where other value investors will follow into the stock.  Again, the magnitude of the catalyst effect will vary across both investors and companies. The extent of the impact on the pricing gap will depend in large part on the history of success that the big name investor brings into the investment, with sustained success in the past going with a larger impact. 

Apple, Icahn and Buffett
It has taken me a while to get to the point of this post, which was ostensibly about Apple and how Icahn’s exit and Buffett’s entry into the stock affect my thinking. At first sight, this graph shows how the market reacted to their actions:

While it does look like Icahn's sale had a negative effect (albeit mild) and Berkshire's buy had a positive effect (almost as mild), I plan to use the framework of the last section to assess each of these investors and gauge how it should affect my thinking about the stock.

Icahn, the Activist Trader
Through much of his tenure, Carl Icahn has been labeled an activist investor but I will take issue with at least a portion of that label. It is true that Icahn is an activist, though he is much more active on the financing/dividend dimension (pushing companies to borrow money and return cash) than on the operating dimension. I do think that Icahn is more of a trader than an activist, more focused on momentum and pricing than on value and this is illustrated by the tools that brings to the assessment. When Icahn was asked why he invested in Lyft in 2015, his response was that it looked cheap relative to Uber, a classic pricing argument. With Apple, in his bullish days, Icahn argued that it was cheap, but consider how he justified his contention in May 2015, that Apple, then trading at $100, should really be trading at $240. In effect, he forecast out earnings per share in 2016 to be $12, applied a PE ratio of 18 and added the cash balance of $24.44/share. Not only is this definitely not an intrinsic valuation, it is at best "casual pricing", i.e., the type of pricing you would do on the back of an envelope after you have had a little too much to drink.

Before you point out to me that Icahn is worth billions and I am not, let me hasten to add that there is nothing ignoble about trading and that Icahn has been an incredibly successful trader over the last few decades, testimonial to his targeting and trading skills. It does color how I viewed Icahn’s investment in Apple in January 2014, his push at Apple for more dividends and more debt during his days as a Apple investor and his decision to sell his holdings on April 2016. I was already an investor in Apple in January 2014, when Icahn bought his shares, and while I did not view his decision to buy the shares as vindication of my valuation, I welcomed him to the shareholder ranks both because Apple was badly in need of a momentum shift and Icahn was playing both an activist and a catalyst role. I am glad that he put pressure on Apple to get over its unwillingness to borrow money and to return more cash in dividends and buybacks. His decision to depart does tells me two things. First, Icahn has recognized the limitations of financing and dividend policy changes in driving Apple’s value and is moving on to companies where the payoff is greater from financial reengineering. Second, it is possible that Icahn’s momentum detector is telling him that while Apple’s stock price may not be going lower, it has little room to go higher either, at least in the short term, and given his trading track record, I would take that signal seriously,

Buffett Buys In?
The decision by Berkshire Hathaway to invest in Apple about three weeks after Icahn’s departure mollified some worried Apple investors, since there is no more desirable endorsement in all of value investment than Warren Buffett’s buy order. I am not privy to the inner workings in Omaha, but I have a feeling that this decision was made more by Todd Combs and Ted Wechsler, the co-heads that Buffett hired as his successors, than by Buffett, but let’s assume that they are following the Buffett playbook. What does that tell you about Apple stock? The good news is that the greatest value investor of this generation now considers Apple to be a value stock. The bad news is that this investor's biggest investment in a technology company has been in IBM, a company that delivers solid dividends and cash flows but has been liquidating itself gradually over the last ten years. If my value judgment on Apple had required substantial growth for value to be delivered, Buffett’s investment could very well have adversely affected my view on the company. In this case, though, I agree with his assessment that Apple is a mature company, with enough cash flows to cover dividends for a generation. 

The Apple End Game
In early May, when I analyzed Apple, I knew that Carl Icahn had already closed out his position and it had no impact on my value estimate or investment judgment. Icahn’s decision to sell was an indication to me that the price might not recover quickly and that momentum could work against me in the near term, but I was okay with that, since my time horizon was not constrained. Buffett’s decision (if it was his) to jump in, a couple of weeks later, may be an indication that the best days of Apple are behind it, but I had already made the same judgment in my valuation. If there is a silver lining, it is that Buffett's followers, with their large numbers and unquestioning, will imitate him and perhaps get the price gap to close. 

The Dark Side of Big Name Investing
While I am open to the possibility that the entry of a big name into a company has the potential to change the way I think about the company and perhaps my investment decisions, there are dangers embedded in doing so.
  1. Confirmation bias: It is a well-established fact that investors look for evidence that confirms decisions that they have already made and ignore evidence that contradicts it and big name investors feed into this bias. Thus, if you have bought a stock, you are far more likely to focus in on those big name investors who agree with you (and are either bullish on the stock or buy it) and screen out big name investors who do no.
  2. Mixed Motives: It is entirely possible that you (as an investor) may be misreading or misunderstanding the motives that caused the big name trade in the first place. In particular, the insider, who you assumed was trading because he or she had private information, may be selling the stock for tax or liquidity reasons. The activist, who you assumed was pushing for real changes in the company, may be more interested in collecting a payoff from the company to leave it alone. The trader, who you assumed had skills playing the momentum game, may himself be following the crowd rather than assessing momentum shifts. Finally, the value investor, who you assumed had valued the company and was pushing for the price/value gap to close, may be more trader than investor, quick to give up, if the stock moves in the wrong direction.
There are some investors, including many institutional investors, whose entire investment strategy seems to be built around watching what big name investors do and imitating them. While imitation may be the best form of flattery, it is inauthentic and a poor basis for an investment philosophy, no matter who the big name investor that you are imitating is and how successful he or she has been. We are too quick to attribute investment success to skill and wisdom and that much of what passes for "smart" money is really "lucky" money. My advice is that if you have an investment thesis that leads you to buy the stock, do so and stop worrying about what the talking heads on CNBC or Bloomberg tell you about it. If you have so little faith in your reasoning that you doubt it and are ready to abandon it the moment it is contested by a big name, you should consider investing in index funds instead.

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Monday, May 23, 2016

DCF Myth 3.2: If you don't look, its not there!

In this, the last of my three posts on uncertainty, I complete the cycle I started with a look at the responses (healthy and unhealthy) to uncertainty and followed up with an examination of the Margin of Safety, by taking a more extended look at one approach that I have found helpful in dealing with uncertainty, which is to run simulations. Before you read this post, I should warn you that I am not an expert on simulations and that the knowledge I bring to this process is minimalist and my interests are pragmatic. So, if you are an expert in statistics or a master simulator, you may find my ramblings to be amateurish and I apologize in advance. 

Setting the Stage
The tools that we use in finance were developed in simpler times, when data was often difficult (or expensive) to access and sophisticated statistical tools required machine power that was beyond the reach of most in the finance community. It should come as no surprise then that in discounted cash flow valuation, we have historically used point estimates ( single numbers that reflect best judgments at the time of the valuation) for variables that have probability distributions attached to them. To illustrate, in my valuation of Apple in February 2016, I used a revenue growth rate of 2.2% and a target operating margin of 25%, to arrive at my estimate of value per share of $129.80.

It goes without saying (but I will say it anyway) that I will be wrong on both these numbers, at least in hindsight, but there is a more creative way of looking at this estimation concern. Rather than enter a single number for each variable, what if I were able to enter a probability distribution? Thus, my estimate for revenue growth would still have an expected value of 2.2% (since that was my best estimate) but would also include a probability distribution that reflected my uncertainty about that value. That distribution would capture not only the magnitude of my uncertainty (in a variance or a standard deviation) but also which direction I expect to be wrong more often (whether the growth is more likely to be lower than my expected value or higher).  Similarly, the expected value for the operating margin can stay at 25% but I can build in a range that reflects my uncertainty about this number.

Once you input the variables as distributions, you have laid the foundations for a probabilistic valuation or more specifically, for a simulation, where in each run, you pick one outcome out of each distribution (which can be higher or lower than your expected values) and estimate a value for the company based on the drawn outcomes. Once you have run enough simulations, your output will be a distribution of values across simulations. If the distributions of your variables are built around expected values that match up to the numbers that you used in your point estimate valuation, the expected value across the simulations will be close to your point estimate value. That may seem to make the simulation process pointless, but there are side benefits that you get from simulations that enrich your decision process. In addition to the expected value, you will get a measure of how much variability there is in this value (and thus the risk you face), the likelihood that you could be wrong in your judgment of whether the stock is under and over valued and the potential payoffs to be right and wrong. 

Statistical Distributions: A Short Preview
It is a sad truth that most of us who go through statistics classes quickly consign them to the “I am never going to use this stuff” heap and move on, but there is no discipline that is more important in today’s world of big data and decision making under uncertainty. If you are one of those fortunate souls who not only remembers your statistics class fondly but also the probability distributions that you encountered during the class, you can skip this section. If, like me, the only memory you have of your statistics class is of a bell curve and a normal distribution, you need to expand your statistical reach beyond a normal distribution, because much of what happens in the real world (which is what you use probability distributions to capture) is not normally distributed. At the risk of over simplifying the choices, here are some basic classifications of uncertainties/ risks::
  1. Discrete versus Continuous Distributions: Assume that you are valuing an oil company in Venezuela and that you are concerned that the firm may be nationalized, a risk that either occurs or does not, i.e., a discrete risk. In contrast, the oil company's earnings will move with oil prices but take on a continuum of values, making it a continuous risk. With currency risk, the risk of devaluation in a fixed exchange rate currency is discrete risk but the risk in a floating rate currency is continuous.
  2. Symmetric versus Asymmetric Distributions (Symmetric, Positive skewed, Negative skewed): While we don't tend to think of upside risk, risk can deliver outcomes that are better than expected or worse than expected. If the magnitude and likelihood of positive outcomes and negative outcomes is similar, you have a symmetric distribution. Thus, if the expected operating margin for Apple is 25% and can vary with equal probability from 20% to 30%, it is symmetrically distributed. In contrast, if the expected revenue growth for Apple is 2%, the worse possible outcome is that it could drop to -5%, but there remains a chance (albeit a small one) that revenue growth could jump back to 25% (if Apple introduces a disruptive new product in a big market), you have an a positively skewed distribution. In contrast, if the expected tax rate for a company is 35%, with the maximum value equal to the statutory tax rate of 40% (in the US) but with values as low as 0%, 5% or 10% possible (though not likely), you are looking at a negatively skewed distribution.
  3. Extreme outcome likelihood (Thin versus Fat Tails): There is one final contrast that can be drawn between different risks. With some variables, the values will be clustered around the expected value and extreme outcomes, while possible, don't occur very often; these are thin tail distributions. In contrast, there are other variables, where the expected value is just the center of the distribution and actual outcome that are different from the expected value occur frequently, resulting in fat tail distributions.
I know that this is a very cursory breakdown, but if you are interested, I do have a short paper on the basics of statistical distributions (link below), written specifically with simulations in mind. 

Simulation Tools
I was taught simulation in my statistics class, the old fashioned way. My professor came in with three glass jars filled with little pieces of paper, with numbers written on them, representing the different possible outcomes on each variable in the problem (and I don't even remember what the problem was). He then proceeded to draw one piece of paper (one outcome) out of each jar and worked out the solution, with those numbers and wrote it on the board. I remember him meticulously returning those pieces of paper back into the jar (sampling with replacement) and at the end of the class, he proceeded to compute the distribution of his solutions.

While the glass jar simulation is still feasible for simulating simple processes with one or two variables that take on only a few outcomes, it is not a comprehensive way of simulating more complex processes or continues distributions. In fact, the biggest impediment to using simulation until recently would have been the cost of running one, requiring the use of a mainframe computer. Those days are now behind us, with the evolution of technology both in the form of hardware (more powerful personal computers) and software. Much as it is subject to abuse, Microsoft Excel has become the lingua franca of valuation, allowing us to work with numbers with ease. There are some who are conversant enough with Excel's bells and whistles to build simulation capabilities into their spreadsheets, but I am afraid that I am not one of those. Coming to my aid, though, are offerings that are add-ons to Excel that allow for the conversion of any Excel spreadsheet almost magically into a simulation.

I normally don't make plugs for products and services, even if I like them, on my posts, because I am sure that you get inundated with commercial offerings that show up insidiously in Facebook and blog posts. I am going to make an exception and praise Crystal Ball, the Excel add-on that I use for simulations. It is an Oracle product and you can get a trial version by going here. (Just to be clear, I pay for my version of Crystal Ball and have no official connections to Oracle.) I like it simply because it is unobtrusive, adding a menu item to my Excel toolbar, and has an extremely easy learning curve.

My only critique of it, as a Mac user, is that it is offered only as a PC version and I have to run my Mac in MS Windows, a process that I find painful. I have also heard good things about @Risk, another excel add-on, but have not used it.

Simulation in Valuation
There are two aspects of the valuation process that make it particularly well suited to Monte Carlo simulations. The first is that uncertainty is the name of the game in valuation, as I noted in my first post in the series. The second is that valuation inputs are often estimated from data, and that data can be plentiful at least on some variables, making it easier to estimate the probability distributions that lie at the heart of simulations. The sequence is described in the picture below:



Step 1: Start with a base case valuation
The first place to start a simulation is with a base case valuation. In a base case valuation, you do a valuation with your best estimates for the inputs into value from revenue growth to margins to risk measures. Much as you will be tempted to use conservative estimates, you should avoid the temptation and make your judgments on expected values. In the case of Apple, the numbers that I use in my base case valuation are very close to those that I used just a couple of months ago, when I valued the company after its previous earnings report and are captured in the picture below:
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In my base case, at least, it looks like Apple is significantly under valued, priced at $93/share, with my value coming in at $126.47, just a little bit lower my valuation a few months ago. I did lower my revenue growth rate to 1.50%, reflecting the bad news about revenues in the most recent 10Q.

Step 2: Identify your driver variables
While there are multiple inputs into valuation models that determine value, it remains true that a few of these inputs drive value and that the rest go along for the ride. But how do you find these value drivers? There are two indicators that you can use. The first requires trial and error, where you change each input variable to see which ones have the greatest effect on value. It is one reason that I like parsimonious models, where you use fewer inputs and aggregate numbers as much as you can. The second is more intuitive, where you focus on the variable that investors in the company seem to be most in disagreement about. My Apple valuation is built around four inputs: revenue growth (growth), operating margin (profitability), the sales to capital ratio (investment efficiency) and cost of capital (risk). The graph below captures how much value changes as a function of these inputs:

As you can see the sales to capital ratio has little effect on value per share, largely because the base case growth rate that I use for Apple is so low. Revenue growth and operating margin both affect value significantly and cost of capital to a much lesser degree. Note that the value per share is higher than the current price though every single what-if analysis, but that reflects the fact that only variable at a time in being changed in this analysis. It is entirely possible that if both revenue growth and operating margins drop at the same time, the value per share will be lower than $93 (the stock price at the time of this analysis) and one of the advantages of a Monte Carlo simulation is that you can build in interconnections between variables. Looking at the variables through the lens that investors have been using to drive the stock price down, it seems like the front runner for value driver has to be revenue growth, as Apple reported its first year on year negative revenue growth in the last quarter and concerns grow about whether the iPhone franchise is peaking. Following next on the value driver list is the operating margin, as the competition in the smart phone business heats up.

Step 3: The Data Assessment
Once you have the value drivers identified, the next step is collecting data on these variables, as a precursor for developing probability distributions. In developing the distributions, you can draw on the following:
  1. Past data: If the value driver is a macroeconomic variable, say interest rates or oil prices, you can draw on historical data going back in time. My favored site for all things macroeconomic is FRED, the Federal Reserve data site in St. Louis, a site that combines great data with an easy interface and is free. I have included data on interest rate, inflation, GDP growth and the weighted dollar for those of you interested in US data in the attached link. For data on other countries, currencies and markets, you can try the World Bank data base, not as friendly as FRED, but rich in its own way.
  2. Company history: For companies that have been in existence for a long time, you can mine the historical data to get a measure of how key company-specific variables (revenues, operating margin, tax rate) vary over time. 
  3. Sector data: You can also look at cross sectional differences in key variables across companies in a sector. Thus, to estimate the operating margin for Amazon, you could look at the distribution of margins across retail companies.: If the value driver is a macroeconomic variable, say interest rates or oil prices, you can draw on historical data going back in time. My favored site for all things macroeconomic is FRED, the Federal Reserve data site in St. Louis, a site that combines great data with an easy interface and is free. I have included data on interest rate, inflation, GDP growth and the weighted dollar for those of you interested in US data in the attached link. For data on other countries, currencies and markets, you can try the World Bank data base, not as friendly as FRED, but rich in its own way.
In the case of Apple, I isolated my data assessment to three variables: revenue growth, operating margin and the cost of capital.  To get some perspective on the range and variability in revenue growth rates and operating margins, I started by looking at the values for these numbers annually from 1990 to 2015:


This extended time period does distract from the profound changes wrought at Apple over the last decade by the iPhone. To takes a closer look at its effects, I looked at growth and margins at Apple for every quarter from 2005 to the first quarter of 2016 :
Superimposed on this graph of gyrating revenue growth, I have traced the introduction of the different iPhone models that have been largely responsible for Apple's explosive growth over the last decade. There are a few interesting patterns in this graph. The first is that revenue growth is clearly driven by the iPhone cycle, peaking soon after each new model is introduced and fading in the quarters after. The second is that the effect of a new iPhone on revenue growth has declined with each new model, not surprising given the scaling up of revenues as a result of prior models. The third is that the operating margins have been steady through the iPhone cycles, with only a midl dip in the last cycle. There is good news and bad news in this graph for Apple optimists. The good news is that the iPhone 7 will deliver an accelerator to the growth but the bad news is that it will be milder that the prior versions; if the trend lines hold up, you are likely to see only a 10-15% revenue growth in the quarters right after its introduction. 

To get some perspective on what the revenue growth would look like for Apple, if it's iPhone franchise fades, I looked at the compounded annual revenue growth for US technology firms older than 25 years that were still listed and publicly traded in 2016:

Of the 343 firms in the sample, 26.2% saw their revenues decline over the last 10 years. There is a sampling bias inherent in this analysis, since the technology firms with the worst revenue growth declines over the period may not have survived until 2016. At the same time, there were a healthy subset of aging technology firms that were able to generate revenue growth in the double digits over a ten-year period. 

Step 4: Distributional Assumptions
There is no magic formula for converting the data that you have collected into probability distributions, and as with much else in valuation, you have to make your best judgments on three dimensions.
  1. Distribution Type: In the section above, I broadly categorized the uncertainties you face into discrete vs continuous, symmetric vs skewed and fat tail vs thin tail. At the risk of being tarred and feathered for bending statistical rules, I have summarized the distribution choices based on upon these categorizations. The picture is not comprehensive but it can provide a road map though the choices:
  2. Distribution Parameters: Once you have picked a distribution, you will have to input the parameters of the distribution. Thus, if you had the good luck to have a variable be normally distributed, you will only be asked for an expected value and a standard deviation. As you go to more complicated distributions, one way to assess your parameter choices to look at the full distribution, based upon your parameter choices, and pass it through the common sense test.
In the case of Apple, I will use the historical data from the company, the cross sectional distribution of revenue growth across older technology companies as well as a healthy dose of subjective reasoning to pick a lognormal distribution, with parameters picked to yield values ranging from -4% on the downside to +10% on the upside. On the target operating margin, I will build my distribution around the 25% that I assumed in my base case and assume more symmetry in the outcomes; I will use a triangular distribution to prevent even the outside chance of infinite margins in either direction.
Note the correlation between the two, which I will talk about in the next section.

Step 5: Build in constraints and correlations
There are two additional benefits that come with simulations. The first is that you can build in constraints that will affect the company's operations, and its value, that are either internally or externally imposed. For an example of an external constraint, consider a company with a large debt load. That does not apply to Apple but it would to Valeant. If the company's value drops below the debt due, you could set the equity value to zero, on the assumption that the company will be in default. As another example, assume that you are valuing a bank and that you model regulatory capital requirements as part of your valuation. If the regulatory capital drops below the minimum required, you can require the company to issue more shares (thus reducing the value of your equity).  The second advantage of a simulation is that you can build in correlations across variables, making it more real life. Thus, if  you believe that bad outcomes on margins (lower margins than expected) are more likely to go with bad outcomes on revenue growth (revenue growth lower than anticipated), you can build in a positive correlation between the variables. With Apple, I see few binding constraints that will affect the valuation. The company has little chance of default and is not covered by regulatory constraints on capital. I do see revenues and operating margins moving together and I build in this expectation by assuming a correlation of 0.50 (lower than the historical correlation of 0.61 between revenues and operating margin from 1989 to 2015 at Apple).

Step 6: Run the simulations
Using my base case valuation of Apple (which yielded the value per share of $126.47) as my starting point and inputting the distributional assumptions for revenue growth and operating margin, as well as the correlation between the two, I used Crystal Ball to run the simulations (leaving the number at the default of 100,000) and generated the following distribution for value:

The percentiles of value and other key statistics are listed on the side. Could Apple be worth less than $93/share. Yes, but the probability is less than 10%, at least based on my assumptions. Having bought and sold Apple three times in the last six years (selling my shares last summer), this is undoubtedly getting old, but I am an Apple shareholder again. I am not a diehard believer in the margin of safety, but if I were, I could use this value distribution to create a more flexible version of it, increasing it for companies with volatile value distributions and reducing it for firms with more stable ones.

The most serious concern that I have, as an investor, is that I am valuing cash , which at $232 billion is almost a third of my estimated value for Apple, as a neutral asset (with an expected tax liability of $28 billion). Some of you, who have visions of Apple disrupting new businesses with the iCar or the iPlane may feel that this is too pessimistic and that there should be a premium attached for these future disruptions. My concern is the opposite, i.e., that Apple will try to do too much with its cash, not too little. In my post on aging technology companies, I argued that, like aging movie stars in search of youth, some older tech companies throw money at bad growth possibilities. With the amount of money that Apple has to throw around, that could be deadly to its stockholders and I have to hope and pray that the company remains restrained, as it has been for much of the last decade.

Conclusion
Uncertainty is a fact of life in valuation and nothing is gained by denying its existence. Simulations offer you an opportunity to look uncertainty in the face, make your best judgments and examine the outcomes. Ironically, being more open about how wrong you can be in your value judgments  will make you feel more comfortable about dealing with uncertainty, not less. If staring into the abyss is what scares you, take a peek and you may be surprised at how much less scared you feel.

YouTube video


Attachments
  1. Paper on probability distributions
  2. Apple valuation - May 2016
  3. Link to Oracle Crystal Ball trial offer
  1. If you have a D(discount rate) and a CF (cash flow), you have a DCF.  
  2. A DCF is an exercise in modeling & number crunching. 
  3. You cannot do a DCF when there is too much uncertainty.
  4. The most critical input in a DCF is the discount rate and if you don’t believe in modern portfolio theory (or beta), you cannot use a DCF.
  5. If most of your value in a DCF comes from the terminal value, there is something wrong with your DCF.
  6. A DCF requires too many assumptions and can be manipulated to yield any value you want.
  7. A DCF cannot value brand name or other intangibles. 
  8. A DCF yields a conservative estimate of value. 
  9. If your DCF value changes significantly over time, there is something wrong with your valuation.
  10. A DCF is an academic exercise.