Blog #63 Dumb Alphas, Smart Betas (Last Post)

Investors the world over chase alpha but struggle to get it. Nevertheless, their zeal for alphas does have one lasting effect: it nourishes an active fund management industry that is worth trillions. In the US alone, mutual funds oversee some $16 trillion of assets, six times more than that managed by ETFs (2017 Investment Company Fact Book). Some lessons, it seem, are hard to learn.

Having made my point that alphas are expensive but betas (broad factor exposures) are cheap, I don’t want to flog a dead horse. Instead, I want to move on and say a bit more about betas. What are the interesting factors driving stock prices? Why should we care? How rewarding are these factor exposures? How to get them? What are the pitfalls?

There is a fundamental difference between alphas and betas that confuses many investors. Alpha is “excess returns over a benchmark”. If I invest in a well diversified Singapore-focused equity mutual fund, a natural benchmark is the FTSE ST All Share Index. If my mutual fund returned 10% after fees last year while the FTSE ST All Share Index made 7%, my fund manager has handed me an alpha of 3%. This is kind of result that every mutual fund investor hopes for, but seldom get on a consistent basis.

What about betas? When you buy an ETF that mirrors the FTSE ST All Share Index, your beta is one because that’s the market’s beta. Unbeknownst to many investors, the market itself is a “factor”, because a factor is defined as a persistent force that drives the returns of assets like stocks or bonds.

There are two main types of factors that have driven stock returns. The first are macroeconomic factors – things like GDP growth, inflation, interest rates, energy prices and the like. As you might imagine, quantifying how stock returns move with these factors is complicated, the job of an economist (and he better be a well-trained one!). Fortunately, there is a short-cut – every day, the movements of a stock market index already aggregate the views of millions of investors, each of whom are trying to make sense of vast amounts of macroeconomic data. That is the beauty of market indices – they already do the job of many economists!  If you recall my previous discussions, the near random walk property of stock prices imply that for predicting tomorrow’s or next month’s stock price, its hard to do better than use the price you observed today.  This is just another way of saying that stock prices are pretty random because they buffeted by many, many forces.

The other type of factors are style factors. Style factors help to explain risk and returns within asset classes. A good example is the returns of small versus big firms. Another example is the returns of past winners (stocks that performed well over the last 12 months) versus the returns of past losers. And so on.

Do these style factors have anything to do with fundamentals or risk?  Some do, some apparently don’t. There is an ongoing debate among academics on which style factor is a reward for bearing risk, and which is “wonky” (unrelated to risk, but more like capturing investor sentiment or psychology).

Style investing is what I will focus on in the rest of this blog. This is partly because interesting styles show up quite persistently in the data and also because now you can buy exposures to some of these style factors through ETFs. Importantly, since betas are what you get when you buy styles, style-ETFs give you exposure beta-driven returns you at the characteristically low cost of ETFs. Compared to mutual funds, investing in a style-ETF is pretty much a passive affair. Unlike alpha hunting, you don’t need an active fund manager to monitor style portfolios 24/7 because with style-ETFs: (1) it is broad characteristics of stocks, not the details of each stock’s fundamentals you want, and (2) if those characteristics can be counted on to give nice, stable returns, it basically runs on autopilot and the less a fund manager meddles with the portfolio, the better!

The style approach described above is now branded by the industry as the Smart Betas approach. The tagline of this approach is that “beta is the new alpha”.

As alluded above, Smart Beta essentially tries to capture high returns by forming portfolios of stocks that have exposures to certain rewarding “factors”. Since this is my last post, I’ll let you on a secret: I employ a core-satellite strategy for my equity investments. In plain English, about 60% of my equity investments (the core) is in ETFs that track broadly market indices like the FTSE ST All Share for Singapore and the S&P 500 for the US. The other 40% is in stocks that have low-volatility and high-dividend yield (I like “boring” styles).

Part of the fun of investing is in discovering new facts about styles and drawing your own list of favourite styles. There is plenty to choose. Academic research, including my own, has established that some factors seem to be anomalies: stocks with exposure to these factors have historically produced higher returns than the market, even after adjusting for risk factors (those which are thought to compensate risks). While the returns to these factors do fluctuate in the short term, they show their true colors over time, thanks to the power of compounding.

There are enough anomalies to fill a Jurassic Park, and I haven’t the time to list all of them. Here are just some of the more prominent ones:

  • The Size Effect: small companies tend to have higher average returns than large companies after adjusting for betas.
  • The Value Effect: low priced stocks (relative to book value, earnings, dividends, etc.) also have higher risk-adjusted returns on average than high priced stocks like growth stocks.
  • The Profitability Effect: profitable firms have higher risk-adjusted returns on average than unprofitable companies even after accounting for their higher initial share prices.
  • Momentum: stocks that performed well in the last 12 months (past “winners”) on average continue to have superior returns in the next 12 to 24 months compared to past “losers”.
  • The Low Volatility Effect: low-volatility (boring”) stocks have higher average returns than high-volatility (“exciting”) stocks.

All the above anomalies have been extensively and carefully studied by the finest minds in finance. All are interesting, but the last one will blow you away for it says…High risk low return.jpgCan you believe this?  I’m sure a negative risk-return relationship is not what you learned in business school. If true, the low-volatility anomaly turns finance on its head!

Nice, but where’s the proof? The proof, ladies and gentlemen, is in a spreadsheet which I posted earlier in Blog #58. There, I presented the arithmetic mean (AM) and geometric mean (GM) returns of two volatility-sorted portfolios: low-volatility and high-volatility.  Each of portfolio consists of stocks on the NYSE-AMEX exchange. They are formed annually by sorting all eligible stocks based on the standard deviation of their daily returns

Several things jump out from the spreadsheet. First, the variance (the square of the standard deviation) of the high-vol portfolio is about 7 times higher than that of the low-vol portfolio, yet their arithmetic means are about the same.  Where is the reward for bearing risk? Second, look at the GM and you see that the GM of the two portfolios are oceans apart: 11.3% a year for low-vol versus 7% a year for high-vol. Since the GM formula (a) is a compounding formula and (b) already embeds volatility, the data is saying loud and clear: low risk stocks have higher average compound returns!

I can assure you that this data is not a fluke. Scores of researchers using US and non-US data as well as different sample periods have found the same thing. the low-volatility anomaly in other words, is robust.

How to exploit the low-volatility anomaly?  The easiest way is to look for Smart Beta ETFs that target low-volatility stocks. There are none on SGX at the moment but fortunately, there are quite a few in the US. For example, Blackrock, the world’s largest asset management firm, has four:

  • iShares Edge MSCI Min Vol USA ETF (USMV)
  • ishares Edge MSCI Min Vol EAFE ETF (EFAV)
  • ishares Edge MSCI Min Vol Emerging Markets ETF (EEMV)
  • ishares Edge MSCI Min Vol Global ETF (ACWV)
  • ishares Edge MSCI Min Vol Asia Ex-Japan (

The “Min Vol” label indicates that these ETFs are constructed using an in-house minimum volatility methodology that searches specifically for low-volatility stocks. Typical of ETFs, the expense ratios are low, ranging from 0.15% to 0.35%, with the last ETF having the highest expense ratio. The first four ETFs were started in October 2011 while the last, in June 2014. So it looks like the ETF industry is waking up to the findings of academic research.

The following graphic gives you some idea of the performance of the first four ETFs. The measurement period is from Nov 1, 2011 to Dec 31, 2017. The ETF returns assume that dividends are reinvested but excludes taxes. They (but not the market benchmarks) are net of expense ratios. In all but one case, the min-vol ETFs have higher annualized average returns with lower risk compared to market benchmarks.


Before you send cheques to any of the above ETFs, a word of caution is in order. First, I can’t say for sure whether this or any other anomalies will persist indefinitely. All I can say is most of these anomalies have been quite persistent over the last 50 years or so. The research evidence bears this out.

Secondly, remember that dividends form part of total returns and US-sourced dividends are subject to a withholding tax of 30% for non-U.S. tax residents. A 30% withholding tax reduces every dollar of dividends to 70 cents. The tax effect is worse for high dividend yield stocks than low-yield stocks. This is something to keep in mind when you are investing in the US.

Finally, while expense ratios are lower for ETFs than for mutual funds, they do vary quite a bit across ETFs as shown earlier. As Cliff Asness, co-founder of AQR Capital Management puts it: “There is no investment product so great that a fee cannot make it bad.”

That’s all, folks. Time to say adieu. I hope you’ve enjoyed my ramblings and gotten some useful takeaways in how to manage your personal finance. I’ll keep the blog running for a while so that you can review earlier blogs (ideally, in sequential order).  And a big thanks to all who have sent in your comments and corrections to my typo and other mistakes. On this note, I wish you all, Many Happy Returns 🙂



Blog #62 Why You Should Read This

The move from active investing to passive has been a hot topic lately. Fund flows show that investors are voting with their feet. The news media has been all over the story. The Wall Street Journal has done a big spread on it; Bloomberg has covered it extensively as well.

Bill Miller, the legendary stock picker at Legg Mason Capital Management who beat the Standard & Poor’s 500 Index for 15 consecutive years, has an intriguing theory about why investors have been abandoning active investments. Although some people see passive investing as a form of active investing, he sees the precise opposite phenomenon:

Active fund managers are often nothing more than high-priced closet indexers.

Barry Ritholtz [1] recently spoke with Miller for a new episode of Masters in Business and exchanged e-mails during the past few days about the subject. Here’s a summary of Mr. Miller’s thoughts and insights:

No. 1. Most active investment management is too expensive
Miller suggests that about 70 percent of all active managers are really closet indexers because many of them pile into the same stocks as their benchmarks — just like an index fund. But a majority of them don’t produce net results as good as the average passive fund because their fees are so much higher.

So the choices are: You can buy cheap passive funds and match the market, or you can buy expensive active funds with the goal of beating the market. But buying expensive funds that are really closet indexers makes no sense. Miller points out that investors have begun to figure this out.

No. 2. Many closet indexers are doomed to underperform
Miller points out the technical reasons so many closet indexers’ underperform:

To minimize tracking error and to avoid large drawdowns relative to the market, most managers have limits on how much they can be overweight or underweight the various components of the benchmark (e.g., the sectors of the S&P 500).

So-called active-share percentage is where the risk and reward comes from, Miller says. Managers whose funds differ a lot from the index in their concentrations have greater risk of drawdowns and underperformance. However, without that risk, there is no chance of outperformance. Miller adds “their active share — the amount by which they differ from their benchmark — is very low and thus their ability to outperform is also low, even before expenses.”

No. 3. “Volatility is the price you pay for performance”
Samantha McLemore has been Miller’s co-manager at the Legg Mason Opportunity Trust since 2008. He quotes her as saying “volatility is the price you pay for performance.” As an example, he notes that the active-share percentage — how much a fund deviates from its benchmark in composition — in the Legg Mason Opportunity Trust fund is close to 100 percent. “That means my tracking error is also high, and drawdowns can be high as well,” he said. But Miller, let’s face it, is an exceptional stock picker. During the past five years, the fund has beaten the returns of 97 percent of its peers and outpaced the S&P 500 by an average of more than 4 percentage points annually in the same period.

No. 4. Job preservation is the reason for closet indexing
Hugging the benchmark is a form of job preservation. By guaranteeing a fund won’t deviate too far from the market, the manager gets to keep his job, even with mediocre performance. Closet indexing is a tribute to John Maynard Keynes’ famous observation that “Worldly wisdom teaches that it is better for reputation to fail conventionally than to succeed unconventionally.”

No. 5. We are only halfway through the shift from active to passive
Miller estimates that when the dust settles, about 70 percent of equity assets will be in some form of passive investment. Given that passive beats active net of fees most of the time, this move makes complete sense.

The bottom line, as Miller sees it, is that the shift from active to passive hasn’t been properly framed. It is simply switching from expensive passive investing to inexpensive passive investing.

[1] Barry Ritholtz is an American author, newspaper columnist, blogger, equities analyst, CIO of Ritholtz Wealth Management.

Article source: Shift From Active to Passive Isn’t What It Seems

Blog #61 The Law of Active Fund Management

I like mutual funds (unit trusts, as they called in Singapore). I like them not because most are great performers, nor because they offer affordable diversification, but because we know what mutual fund managers do everyday. Broadly speaking, they manage their portfolios actively, as opposed to an exchange traded fund (ETF) which is essentially passive. Not only that, we also know that mutual funds are expected to be almost fully invested at all times because investors don’t pay mutual fund managers to hoard cash. Combining these two facts, we have a good picture of what mutual funds do – they spend most of their time picking securities such as stocks (equities funds), bonds (income funds) or both (balanced funds). To be brief, I will focus on equities funds for the rest of this blog.

Stock picking isn’t at all easy (see my previous blog). But since fund managers are supposed to be ‘experts’ in this area, mutual fund investors naturally have high hopes that this expertise will translate to returns that are superior to those obtained from ETFs. To use the jargon, investors expect to alphas. Sadly, most investors end up disappointed.

Since 2002, Standard & Poor’s Dow Jones Indices LLC, a division of the S&P Global has been tracking the relative performance of equity and bond mutual funds in the US and other markets such as Australia, Canada, Europe, India, Japan, Latin America, South Africa. I will focus on US funds since America has the lion’s share of global mutual funds by numbers and assets under management. Also, the performance patterns of non-U.S. funds are broadly similar to those in the US.

S&P summarizes the relative performance of different types of equity funds using what they call a SPIVA Scorecard (SPIVA stands for S&P Inactive versus Active). The SPIVA Scorecard compares the net-of-fees returns of actively managed mutual funds against their appropriate benchmarks on a semiannual basis (a report is produced each mid-year and at year-end). For example, the year-end 2016 Scorecard reports the percentage of funds in a specific category that either outperforms or underperforms their respective benchmarks over a 15-year period. Results for shorter periods of 1, 3, 5, and 10 years are also reported.

Because market conditions can impact managers’ performance from year to year, it makes more sense to look at longer-term returns. Hence, I will focus on rolling 3-year relative performance instead of yearly numbers. All the data are extracted from the 2016 SPIVA US Scorecard report which is freely available from S&P’s website. The raw data for compiling the US SPIVA Scorecard comes form the Chicago Research for Securities Prices (CRSP) US Mutual Funds Database. This is a comprehensive database containing data on more than 10,000 US mutual funds. Importantly, it is free of survivorship bias.


Let’s now zoom in on the main question: do most equity funds beat the index? A “yes” would inspire confidence about active fund management. A “no” suggests that true stock selection skills are in short supply in the mutual funds industry.

There is a lot of data to look at, enough to give you a migraine. To minimize this risk, I will present graphics rather than numbers.Let’s start with the following chart showing the relative performance of “domestic funds”, those that invest solely in US stocks.


What story would you like this graph to tell? What do you think is an appropriate caption for this chart?

Answer to the caption question: The percentage of domestic equity funds beaten by the benchmark index based on rolling 3-year returns.

The chart shows that in most years from 2002 through 2016, the index has outperformed active funds, not the other way around since the performance line is generally above the 50% mar. Making things worse, this line has risen ominously over the years indicating that relatively more and more domestic funds have trailed the index!  For the full 15-year period, a whopping 82% of domestic fund managers have been “underwater” so to speak.

Before I go on, spend a minute to reflect on what this evidence means to you as an investor.

Let’s proceed to look at other equity styles. The next six charts show the relative performance of large-cap versus mid-cap funds. Each of these categories are in turn subdivided into growth funds, value funds and those which are neutral to these styles (core). SPIVA2.jpg

Again, the basic story is the same. Years in which the market beat funds outnumber years in which funds beat the market (by ‘market’ I mean the relevant benchmark index). Many investors gravitate towards styles like “growth” or “value”. The evidence shows that investors are generally better off sending their cheques to a growth or value ETF than to a mutual fund. This is not to say that investors don’t get lucky with mutual funds. But unlike true skills which is consistent, luck is not. What the evidence says is that the odds of a mutual fund investor outperforming the index on a consistent basis is less than 50%, often much less.

Stanford economist, William Sharpe, achieved fame for developing the Capital Asset Pricing Model or CAPM. He also wrote a highly interesting article with the title “The Arithmetic of Active Management” (Financial Analysts’ Journal 1991). Below is a summary of the gist of that article:


Sharpe’s logic is presented as an equation (above) based on a simple but powerful logic. An investor is either passive or active. Hence, the market is collectively made up of passive and active investors. Passive investors buy index funds and achieve market-like returns both before and after fees (this is due to their low fees). For the equation to add up, active investors as a group must also earn market-like returns before active fees. The punchline is that after fees, their net returns will be inferior to those of passive investors. This is true at all times and for all investment styles. Putting it bluntly, Sharpe’s “law” of active fund management implies that the average active investor is doomed to fail or lag behind the index. As shown earlier, mutual funds provide exactly the kind of evidence that confirm Sharpe’s logic.

Please understand that while Sharpe’s law is mathematically true, it does not offer deeper explanations why passive investment outperform active investment.  The high fees that mutual funds charge is certainly one reason. Perhaps the average fund manager isn’t smart enough in picking the right stocks or if he did, lacked the holding power to realize their full potential (again, refer to Blog #60). Or the mutual fund manager was basically hugging a market-like portfolio (heavens forbid), and hence is not deviating enough from the index to make a difference to the fund’s returns. Whatever the case, the data strongly supports Sharpe’s observations. That is enough to sway me towards passive investing.

Let me conclude by showing a few more charts. The following three charts show mutual funds that target small firms. Like their large-cap counterparts, these funds are also not beating their respective benchmarks. In all cases, the relative performance line is well above the 50% mark for most years.


How about funds with a more international focus?  On the left, we have global equity funds, and on the right, emerging market funds. Same story. I rest my case.



Blog #60 “I Could Have”

I’ve not talked much about stock picking so far. This is partly because I have no expertise in picking stocks, which explains why I prefer a ‘boring’ diversification strategy to active stock selection. Over the years, diversification has served me well as a risk management tool. It has certainly relieved me of the pressure of constantly trying to find winning stocks like those of Apple or Amazon. Of course, by not picking stocks, I miss out the highs and honestly, I do feel stupid when friends tell me that this or that stock they bought beat the market hands down.

By and large, I’ve come to terms with such scintillating stories, reasoning that luck may have a big part to play in their out-sized returns; after all, how many Warren Buffetts or Jim Simons do we know? Perseverance in the face of losses is crucial too. What is the use of saying that you have the foresight but lack the temperament to ride through the volatility? Runaway successes always look easier on hindsight.

Back to stock picking, to see the immense challenge of being a great stock picker, consider the story of Apple, surely one of the great success stories in modern times. Indeed, Apple is good example of a baby boomer’s retirement dream come true if he or she had:

(a) the good luck to have gotten the shares at IPO at a split-adjusted of 40 US cents
(b) the foresight that Apple back then would transform into the tech giant it is today
(c) the guts to ride through rough patches in Apple’s stock price.

The story goes back to December 1980 when Apple went public, raising about $100 million by selling its stock for US$22 a piece (40 cents after adjusting for stock splits). By June 1983, the stock price has almost doubled. It then tumbled to 46 cents the following year, foreshadowing more bad news to come.

In September 1985, Steve Jobs, then Apple’s VP and General Manager for the Mac department dropped a bombshell by leaving the company after months of tension between him and CEO John Scully. According to informed sources, Jobs spent the year in a midlife crisis, deciding what he wanted to do with his life and flirting with all kinds of possibilities from entering politics to becoming an astronaut. Jobs eventually founded Pixar Animations and computer firm NEXT, which produced a computer that was impressively powerful but too pricey for the market.

Over at Apple, things didn’t get much better after Jobs left. The company went through three CEOs, made a raft of business mistakes such as licensing its operating system and making computers that fell further and further behind Microsoft’s Windows machines. An investor who held Apple’s stock from IPO through 1996 would would have pocketed $1.66 for every dollar invested. The annual compound return over this period? A paltry 3.2% or barely above the risk-free rate. Meanwhile, over the same period, the market (the S&P 500 index) multiplied more than five times.

Jobs made a spectacular comeback to Apple in July 1997 and was its CEO until his death in 2011. This was the defining period of Apple’s glory. First came the iMAC (in 1998), then the iPod in the fall of 2001 and the first iPhone in June 2007. The rest, as they say is history. From June 1996 to December 2017, Apple’s stock was up 300 times versus 3 times for the S&P 500. In terms of average annual compounded return, it was 31% versus 5.4%.

Many questions swirled in my mind as I recounted the history of Apple:

  • Who had the tenacity to hold on to Apple’s share after Jobs resigned?
  • Who had the foresight to know that Jobs will return to Apple and make it great?
  • Who figured out that the US$150 million that Microsoft invested in Apple in 1997 in exchange for Apple dropping a long-running lawsuit which alleged Microsoft copied the look and feel of the Mac OS for Windows, would be the lifeline that would breathe new life into a struggling Silicon Valley firm?
  • Who had the guts to keep Apple’s stock after the carnage following the bursting of the tech stock bubble burst in 2000?
  • Who could have foreseen that the design of every Apple product will be the company’s great calling card?

Nobody – that’s who.

I doubt anyone had the foresight to see through the fog before Apple achieved greatness. Gary Kasparov, considered one of the greatest chess grand players of all time, said he could only look three to five moves ahead in a typical game [1]. Predicting the trajectory of a stock is infinitely more difficult than predicting your chess opponent’s moves (stock investment is a game a million opponents!)

As I mentioned earlier, foresight is not enough for financial success; you need to overcome the primal emotion of fear and loss-aversion. Without perseverance, it is easy to throw in the towel. How many lucky Apple IPO investors held on to their investment after Jobs left the company? How many after the management turmoil that followed?  I doubt there are more than a handful of such intrepid investors. The following chart provides a partial explanation.


This chart shows that the average holding period of US investors has fallen sharply from a peak of 8 years in 1950 to less than 2 years over the last three decades. Investors in other markets for which there is data show a broadly similar trend. So, investors everywhere are becoming more impatient. Which raises the question: how does one reap Apple-like returns when our holding period is just 2 years?

The story of Apple is exhilarating but not unique. The stocks of many other firms go through periods of famine and feasts. The lesson here is not that nobody held the stock long enough to prosper from it, but that such an investor is luckier and more patient than the rest of us.

1. Kasparov, G.K., and M. Greengard (2007), How Life Imitates Chess: Making the Right Moves from the Board to the Boardroom, NY: Bloomsbury.






Blog #59 Dollar Average Compounding

I hope you find the formula given in the previous blog interesting. You can use the formula to estimate your median future wealth after investing $X for a specified number of years. In the example given, X is $100,000 and the geometric mean return is 4.88% per year. After 30 years, $100,000 grows to about about $420,000.  Remember that this is your median wealth. Due to uncertainty, your actual wealth can be higher or lower than the median. That said, the median wealth is still a useful number because it corresponds to the midpoint of the wealth range.

While a sum of $420,000 is pretty inspiring, most people don’t have $100,000 to plonk into the stock market at one go.  Instead, most of us invest bit by bit, say $10,000 every year or $834 each month. The bad news is that the previous formula doesn’t work in this case. The good news: there is another formula which does!

The new formula applies to an investor who invests a constant $Y every year or grows this amount by a constant rate each year. Throughout his investment journey, the stock market is assumed to earn a specified average rate of return with a specified level of risk. The question the investor wants to know is: “after N years, how much wealth can I expect to have?”.

The new formula isn’t as simple as the previous formula, so I won’t show it here. Instead, to make things user-friendly, I’ve coded it in Excel as a wealth calculator which you can download here.

The input section of the calculator looks like this:


All you have to do is key in the  inputs in the gray cells. They are: AM is the arithmetic mean rate of return (in percent per year), Volatility is the standard deviation of annual returns, Contributions refer how much you wish to invest each year starting now and Annual Increase is the rate at which you want to grow your annual investments. The wealth calculator will automatically project your future median wealth for various periods in the future from 10 to 35 years as shown. That’s it!

In the example shown here, you think that the long-term arithmetic mean return of the stock market is 8% and the volatility is 20%. With these inputs, the geometric mean is automatically calculated for you and then modified to project the future median wealth (details in the spreadsheet). The output (black cells) shows that if you invest $10,000 (starting now) for say 10 years, you can expect a median wealth of $156,438. If you double your investment horizon, the median wealth is more than double to $432,659 and so on.

Blog #58 The Shape of Stock Returns II

I can’t tell you for sure where the stock market is heading next week, but I can show you a formula on what return you can expect will earn if you follow a buy and hold strategy over the long term. There two key phrases in my sentence: “buy-and-hold” and “long-term”. The formula estimates the average return on your investment assuming you are a patient investor expecting a long-term reward from the stock market. It is not a speculator’s formula.

Here is the formula:

g ≅ µ – (1/2)σ²

On the left-hand side, g is the average compound rate of return on a buy-and-hold investment (say a stock portfolio). g is also known as the geometric mean rate of return.

The word “compound” implies that one simply allow short-term (say, monthly) returns to snowball into something bigger. The “snowball effect” is aided by your investment’s simple average return, also known as the arithmetic mean. The arithmetic mean is denoted by mu (µ). All things equal, a high µ means a high g. Unfortunately, the second term (1/2)σ² spoils the party. σ² is the variance of the stock portfolio. It is defined as the square of standard deviation; both are measures of risk. More volatile stocks have higher variance while stable stocks have lower variance.

The wavy symbol ≅ means that g is approximately equal to µ – (1/2)σ². Thus, the formula gives an approximation for g, not the exact solution. Still, approximations are useful because it saves us a lot of manual calculations! More importantly, the formula says that there is “tug-of-war” between µ and (1/2)σ². Everybody likes a stock that has high µ and low σ² but in general, this isn’t possible because risk and return go hand in hand (I have more to say about this interesting topic but let’s save it for another day).

Now let’s answer two important questions: what use is the formula and how to check whether it “works”?

If the formula is accurate, it should give us a close approximation of the geometric mean return earned on a buy-and-hold investment. To see if this is so, I collected a long time series of monthly stock returns to calculate (a) the exact g and (b) the approximate g given by the formula. If the formula is good, (a) should be close to (b).

My monthly returns pertain to two portfolios consisting of stocks traded in the US from 1962 to 2013. The portfolios are “Low-Vol” and “High-Vol”.  Low-Vol primarily consists of low volatility stocks. These are stocks whose prices don’y jump around too much from month to month.  High-Vol on the other hand comprises mainly high-volatility stocks. This portfolio has a disproportionate share of firms in sectors like semiconductors, computer software, digital equipment, aerospace, and precision manufacturing among others. You can download the spreadsheet here.

For each portfolio, I compute and present the following statistics: arithmetic mean, variance, exact g and approximate g. Here are the results:GM_result.jpg

Focus on the last two rows. You see that for each portfolio, the approximate g is pretty close to the exact g, which confirms that the formula does indeed work well. Best of all, it achieved this feat using just two variables, µ and σ². In physics, it is often said that the most powerful equations are the simplest (think of Einstein’s E=mc²). We don’t have many simple yet powerful equations in finance, but I’m proud to say that g ≅ µ – (1/2)σ² is one of them.

This is well and good, but can we use the formula to predict future returns?  My qualified answer is yes, provided you get your µ and σ² forecasts right. How then does one get good forecasts for µ and σ²? It doesn’t hurt to start with historical estimates of µ and σ² covering a long sample period (to prevent cheery picking particular stretches of stock market history that one fancies). You can then shade these historical estimates higher or lower if you think that past performance is unlikely to faithfully copy the past (this is admittedly a tricky exercise that calls for some expert knowledge in economics). To give you some broad perspective, historically, stock markets have rewarded investors with inflation-adjusted or real arithmetic mean returns of between 4 and 6 percent annually. If you add back the historical global inflation of 4% a year, this implies a historical range of 8% to 10% for nominal average returns (see Blog #30 Why I Love History). Some experts believe that going forward, a globally diversified equity portfolio will probably deliver average returns in the range of 5 to 7% as the low-hanging fruits have mostly been picked, and the world faces challenges due to aging populations, declining fertility, and slower productivity growth. Using a middle rate of 6% and assuming an annual standard deviation of 15%, our little formula implies that you can expect an annual compounded rate g of 4.88% in nominal terms. Even at this modest rate of return, every $100,000 invested over 30 years will compound to a median wealth of nearly $420,000. Such is the power of compounding over long horizons.

Blog #57 The Shape of Stock Returns I

If you have been following this blog lately, I have basically been saying that (a) investors try too hard to beat the market by attempting to predict it (b) despite their best efforts, most are doomed to fail.

Some of you may think that I’m too negative about active management (I am). Also, since I used he words “noisy” and “unpredictable” on a number of occasions to describe the zigs and zags of the stock market, you may also have the impression that I view the stock market as nothing more than a casino. That would be incorrect though. In this and the next blog, I will show why I have a high opinion of the stock market as a long-term investment tool. Instead of writing a three-page piece to bring my point across, I will show simply you a picture (below). The picture has an important message. You “read” the story by starting in the top row, left to right, then the bottom row, left to right.


What is this all about?  Why is it important?

The picture shows how wealth invested in a stock is created if you compound its returns over time. It’s a bit like how you roll over your time deposit after it matures because you want to earn “interest on interest”. Except your time deposit interest rate is fixed but stock returns bump around randomly.

How to model this bumping around random nature of stock prices? Here’s where we turn to statisticians for help. Statisticians use probability distributions, including some clever complicated ones to fit or model stock returns. Fear not. I can tell you what you really need to know using the simplest model.

This is the elegant binomial model. “Bi” means two. So, I will ask you to imagine a stock whose price evolves each period with just two outcomes: either its price goes up or it comes down compared to the price in the previous period. Since up (u) is one outcome and down (d) is another, we have just modelled a stock using the binomial model! A useful way to think about this model is to imagine the stock price as an expanding tree. At each node of the tree, the stock price splits into an up branch with a certain probability (say p) and a down branch with probability 1-p (these probabilities must add up to one if you recall from high school :))

The picture that you saw simulates a binomial stock price going forward from 2 nodes, to 5 nodes, to 50 nodes and finally 500 nodes. Using the tree analogy, it “grows” the tree from a skinny one with just 2 nodes to a bushy one with 500 nodes. More importantly, the 2-node tree represents the short-term (no compounding yet) while the 500-node tree represents the long-term after many periods of compounding. Instead of plotting all the branches sticking out (ugly!), the picture tallies all the end-period wealth and presents their distribution as a nice, smooth curve. Notice this curve has a long right tail. So it is not quite the normal or bell-shaped curve that most people are familiar with, but its long right-tailed cousin (the exact name for it is the log-normal distribution).


What’s the big deal about the log-normal distribution? Three things. First, it shows your mean and median wealth after many periods of compounding (they are indicated by the vertical lines). Half of all your possible wealth will be below the median and half above it. So, the median wealth gives you a “feel” of where you are likely to end up after years of investing. That’s good to know.

Second, the median wealth is lower than the mean. This is a bit unfortunate but is always true of right-tailed distributions.

Third, making up for the fact that the median is less than the mean is the long tail. This feature implies that a long-term investor can look forward to “bonus” lucky draws of very handsome returns. This is good news. Despite all the talk about “black swans”, the reality is that long-term investors are more likely to meet “good” outliers or “white eagles”!

Studies using actual data shows that the log-normal distribution closely model the distribution of long-term wealth derived from the stock market. Hence, we can be confident about its predictions regarding the shape of long-term returns. Now you can see why I do not think that the stock market is merely a casino for speculators. All gambling is a negative sum game (i.e., the average net payoff is negative).  True investment is different. In the long run, stocks generally pace the economy and trend upwards. It is most assuredly, a positive-sum game!. See “The Shape of Stock Returns II” coming up next..