Modern Portfolio Theory 

Modern portfolio theory is the philosophical opposite of traditional stock picking. It is the creation of economists, who try to understand the market as a whole, rather than business analysts, who look for what makes each investment opportunity unique. Investments are described statistically, in terms of their expected long-term return rate and their expected short-term volatility. The volatility is equated with "risk", measuring how much worse than average an investment's bad years are likely to be. The goal is to identify your acceptable level of risk tolerance, and then to find a portfolio with the maximum expected return for that level of risk. This article covers the highlights of modern portfolio theory, describing how risk and its effects are measured, and how planning and asset allocation can help you do something about it.

Volatility and your Time Horizon

The volatility of an investment is measured by the standard deviation of its rate of return. (If your statistics is a little rusty, you can think of the standard deviation as measuring how far away from the average the return rate for any one year is likely to be. The greater the standard deviation, the more variable the rate of return.) This interactive graph shows how expected returns and expected fluctuations affect the likely outcomes of your investment. The graph uses a random process, so there is uncertainty built into the result - just like life! You choose an investment with a specified return and volatility, and the graph will produce a bell curve of possible outcomes. The area of the "negative return zone", shown in red, is proportional to the probability that you will lose money on your investment.


Here are some things you should try:

Set the time horizon to 5 years, and compare the results for the Stocks and Bonds portfolios. You'll find that the Stocks graph lies to the right of the Bonds graph, as you'd expect (the expected return is higher) but the "red zone" for Stocks is also larger. This is the classic risk/reward tradeoff. If two identical groups of people invested in portfolios like these, the average gain of the Stocks group would be greater than that of the Bonds group, but the Stocks group would have a larger number of investors who actually lost money.

Now try the Stocks portfolio with different time horizons. You'll find that the "red zone" decreases as the time horizon grows. This is why investment books always tell you that volatile investments may be appropriate if you don't plan to withdraw the money for decades; but if you're planning to start taking it out within the next few years, you should consider switching to a more stable portfolio, even if that means settling for a lower expected rate of return.

Try Stocks with a time horizon of 1 year. Now the graph doesn't look like a bell at all: over this short a time frame the stock market is "really" random, more like a game of chance than an investment.
This type of calculator is known as a Monte Carlo simulation, or MCS: that means it calculates many possible outcomes, to show you both your expected return and the risk that you'll do worse than that.

The Efficient Frontier and Portfolio Diversification

The graph on the previous page shows how volatility increases your risk of loss of principal, and how this risk worsens as your time horizon shrinks. So all other things being equal, you would like to minimize volatility in your portfolio.
Of course the problem is that there is another effect that works in the opposite direction: if you limit yourself to low-risk securities, you'll be limiting yourself to investments that tend to have low rates of return. So what you really want to do is include some higher growth, higher risk securities in your portfolio, but combine them in a smart way, so that some of their fluctuations cancel each other out. (In statistical terms, you're looking for a combined standard deviation that's low, relative to the standard deviations of the individual securities.) The result should give you a high average rate of return, with less of the harmful fluctuations.
The science of risk-efficient portfolios is associated with a couple of guys (a couple of Nobel laureates, actually) named Harry Markowitz and Bill Sharpe.
Suppose you have data for a collection of securities (like the S & P 500 stocks, for example), and you graph the return rates and standard deviations for these securities, and for all portfolios you can get by allocating among them. Markowitz showed that you get a region bounded by an upward-sloping curve, which he called the efficient frontier.

It's clear that for any given value of standard deviation, you would like to choose a portfolio that gives you the greatest possible rate of return; so you always want a portfolio that lies up along the efficient frontier, rather than lower down, in the interior of the region. This is the first important property of the efficient frontier: it's where the best portfolios are.
The second important property of the efficient frontier is that it's curved, not straight. This is actually significant -- in fact, it's the key to how diversification lets you improve your reward-to-risk ratio. To see why, imagine a 50/50 allocation between just two securities. Assuming that the year-to-year performance of these two securities is not perfectly in sync -- that is, assuming that the great years and the lousy years for Security 1 don't correspond perfectly to the great years and lousy years for Security 2, but that their cycles are at least a little off -- then the standard deviation of the 50/50 allocation will be less than the average of the standard deviations of the two securities separately. Graphically, this stretches the possible allocations to the left of the straight line joining the two securities.
In statistical terms, this effect is due to lack of covariance. The smaller the covariance between the two securities -- the more out of sync they are -- the smaller the standard deviation of a portfolio that combines them. The ultimate would be to find two securities with negative covariance (very out of sync: the best years of one happen during the worst years of the other, and vice versa).

The Sharpe Ratio
The previous page showed that the efficient frontier is where the most risk-efficient portfolios are, for a given collection of securities. The Sharpe Ratio goes further: it actually helps you find the best possible proportion of these securities to use, in a portfolio that can also contain cash.
The definition of the Sharpe Ratio is:
S(x) = ( rx - Rf ) / StdDev(x)
where
x is some investment
rx is the average annual rate of return of x
Rf is the best available rate of return of a "risk-free" security (i.e. cash)
StdDev(x) is the standard deviation of rx
The Sharpe Ratio is a direct measure of reward-to-risk. To see how it helps you in creating a portfolio, consider the diagram of the efficient frontier again, this time with cash drawn in.

There are three important things to notice in this diagram:
1. If you take some investment like "x" and combine it with cash, the resulting portfolio will lie somewhere along the straight line joining cash with x. (This time it's a straight line, not a curve; cash is riskless, so there's no "damping out" effect between cash and x.)
2. Since you want the rate of return to be as great as possible, you want to select the x that gives you the line with the greatest possible slope (like we have done in the diagram).
3. The slope of this line is equal to the Sharpe Ratio of x.
Putting this all together gives you the method for finding the best possible portfolio from this collection of securities: First, find the investment with the highest possible Sharpe Ratio (this part requires a computer); Next, take whatever linear combination of this investment and cash will give you your desired value for standard deviation. The result will be the portfolio with the greatest possible rate of return.

Build a Portfolio: Asset Allocation with the Sharpe Ratio

Here are three things you should verify. For (1) and (2), start with a portfolio that includes stocks, bonds and cash.
1. If you lower your risk tolerance level, the allocation ratio of stocks-to-bonds will remain constant, and the amount of cash will increase. (Graphically, you're on the straight line joining cash to the Efficient Frontier, and moving to the left.)

2. If you decrease the covariance between stocks and bonds, you can allocate more money to stocks and bonds and less to cash, thus raising your rate of return. (This is taking advantage of the curved shape of the Efficient Frontier, stretching it further to the left and tilting the line up. By the way, this demo only lets you decrease covariance to zero, although negative covariance is possible, at least in theory. The size of the covariance will be on a scale roughly equal to the product of the two standard deviations; so for example, if the two investments have standard deviations of 15% and 7%, a large value for the covariance would be .15 x .07 = 0.0105.)
3. If you increase your risk tolerance to a high enough level, you'll get a zero-cash portfolio. This means you're up on the Efficient Frontier, but to the right of the point where it intersects the straight line. (In theory you could get up to the line even here if you are willing to hold a "negative" amount of cash, that is, to invest on margin.)

Index Funds and Optimal Portfolios
The portfolio demo was easy to use because it assumes that the investment universe consists only of two market securities, plus riskless cash. But of course the real investment universe is a lot bigger than that, with thousands of choices among U.S. stocks alone. In theory you could find the optimal point on the efficient frontier generated by this many securities, but doing that wouldn't be practical. For one thing, you'd have to calculate the covariance between every pair of securities: thousands of securities means millions of covariance calculations. But even if you could do all those calculations, you wouldn't really want to. That's because the efficient frontier is based on an idealized model of the way investments work; and when you apply a huge number of calculations to a model you tend to amplify the error between the model and reality, leaving you with more "noise" than anything else.
So as a practical matter, putting portfolio theory to work means reducing the problem to something about as simple as the portfolio demo, and investing in a small number of index funds rather than a huge number of individual stocks and bonds.
Index investing is where portfolio theory starts to rely on the efficient market hypothesis. When you buy an index you're allocating your money the same way the whole market is - which is a good thing if you believe the market has a plan. This is why portfolio theory really is a branch of economics rather than finance: instead of studying financial statements you study the aggregate behavior of investors, some of whom presumably have studied financial statements so that market valuations will reflect their due diligence.
(This viewpoint also gives rise to some bad blood that's pretty entertaining if you aren't involved. The economists see business analysts as necessary to market efficiency, but otherwise rather beneath them as a life form, like the bacteria that make yogurt: they're useful, but they're basically germs. The "germs" respond that the economists are delusional eggheads whose theories collapse whenever real money is involved.)
The pioneering result that helped popularize index investing was Tobin's "separation theorem", which Bill Sharpe summarized this way in an interview:
James Tobin ... in a 1958 paper said if you hold risky securities and are able to borrow - buying stocks on margin - or lend - buying risk-free assets - and you do so at the same rate, then the efficient frontier is a single portfolio of risky securities plus borrowing and lending....
Tobin's Separation Theorem says you can separate the problem into first finding that optimal combination of risky securities and then deciding whether to lend or borrow, depending on your attitude toward risk. It then showed that if there's only one portfolio plus borrowing and lending, it's got to be the market.
The reasoning behind this is easy to understand from the same kind of diagram we have already been looking at:

As usual you're trying to build an optimal portfolio for your risk tolerance; and as before, it will lie somewhere on the straight line joining the cash rate Rf to some optimal mix on the efficient frontier. We're specifically assuming what Sharpe said, that high risk investors can and will buy on margin, with money borrowed at the low rate Rf. That's why there is just one straight line in the picture, and one unique optimal mix on the efficient frontier; so the problem of building an optimal portfolio is "separated" into somehow finding the optimal mix and then combining it with cash to give you your desired risk tolerance.
Now for the part that's really interesting. Assume that everybody is facing the same efficient frontier that you are, and that the market is efficient in the specific sense that it behaves in the aggregate as if everybody is trying to build an efficient portfolio this way. That means it behaves as if everybody is on your straight line, with the same optimal mix as you. So the mix that the market is holding - the index - is guaranteed to be your own personal optimal mix.
That's an incredibly elegant result... but it requires you to accept some really strong hypotheses. (Two quick jabs: real investors can't afford to be so cavalier about the special risks of margin buying; and different tax brackets mean different people face different efficient frontiers. Goodbye single straight line; so long universal optimal mix.)
Probably due to problems like those, results about index investing have trended away from proofs that index funds are optimal toward statistical models confirming that index funds are hard to beat. That's a trend we'll be following on the next three pages, with CAPM (a theoretical model that looks like a statistical model) and the three factor model (a pure statistical model with a little theory suggested as an afterthought).

Capital Asset Pricing Model
Bill Sharpe made his first big breakthrough by taking the picture on the previous page and showing how the market must price individual securities in relation to their asset class (a.k.a. the index, or the "optimal mix" in the picture). The derivation isn't exactly a walk in the park (yikes!), but the result is a simple linear relationship known as the Capital Asset Pricing Model:
r = Rf + beta x ( Km - Rf )
where
r is the expected return rate on a security;
Rf is the rate of a "risk-free" investment, i.e. cash;
Km is the return rate of the appropriate asset class.
Beta measures the volatility of the security, relative to the asset class. The equation is saying that investors require higher levels of expected returns to compensate them for higher expected risk. You can think of the formula as predicting a security's behavior as a function of beta: CAPM says that if you know a security's beta then you know the value of r that investors expect it to have.

Naturally, somebody has to verify that this simple relationship actually holds true in the market. Part of the question is how few classes you can get away with: whether you can use a very coarse division into just "stocks" and "bonds", or whether you need to divide much further (into "domestic mid-cap value stocks", and so on). There are also ongoing attempts at "building better betas" that incorporate company debt and other traditional valuation measures, instead of relying solely on past volatility, to measure risk. All of this is a full-time job for academic modern portfolio theorists (and deriding the whole effort is a popular hobby for some traditional stock analysts: how could a magnificent company equal a mediocre one times beta? To them, CAPM seems like a very blunt instrument.)
CAPM has a lot of important consequences. For one thing it turns finding the efficient frontier into a doable task, because you only have to calculate the covariances of every pair of classes, instead of every pair of everything.
Another consequence is that CAPM implies that investing in individual stocks is pointless, because you can duplicate the reward and risk characteristics of any security just by using the right mix of cash with the appropriate asset class. This is why followers of MPT avoid stocks, and instead build portfolios out of low cost index funds.
(One point about that last paragraph. If you are trying to duplicate an expected return that's greater than that of the asset class, you have to hold "negative" cash, meaning you have to buy the index on margin. This is consistent with the big message of MPT - that trying to beat the index is inherently risky).

Regression, Alpha, R-Squared
One use of CAPM is to analyze the performance of mutual funds and other portfolios - in particular, to make active fund managers look bad. The technique is to compare the historical risk-adjusted returns (that's the return minus the return of risk-free cash) of the fund against those of an appropriate index, and then use least-squares regression to fit a straight line through the data points:


Each data point in this graph shows the risk-adjusted return of the portfolio and that of the index over one time period in the past. (For example, you might make a graph like this with twenty data points, showing the annual returns for each of the past twenty years.)
The general equation of this type of line is
r - Rf = beta x ( Km - Rf ) + alpha
where r is the fund's return rate, Rf is the risk-free return rate, and Km is the return of the index.
Note that, except for alpha, this is the equation for CAPM - that is, the beta you get from Sharpe's derivation of equilibrium prices is essentially the same beta you get from doing a least-squares regression against the data. (Also note that alpha and beta are standard symbols that statisticians use all the time for this type of regression; Sharpe and his followers weren't trying to be obscure, as some people like to believe.)
Beta is the slope of this line. Alpha, the vertical intercept, tells you how much better the fund did than CAPM predicted (or maybe more typically, a negative alpha tells you how much worse it did, probably due to high management fees).
The quality of the fit is given by the statistical number r-squared. An r-squared of 1.0 would mean that the model fit the data perfectly, with the line going right through every data point. More realistically, with real data you'd get an r-squared of around .85. From that you would conclude that 85% of the fund's performance is explained by its risk exposure, as measured by beta. (Then you'd punch your fist in the air and say "And the other 15% is due to pure luck!" MPT never believes in investor skill: an investment's behavior equals that of its asset class, minus management fees, plus-or-minus unpredictable luck.)

Fama and French Three Factor Model
CAPM uses a single factor, beta, to compare a portfolio with the market as a whole. But more generally, you can add factors to a regression model to give a better r-squared fit. The best known approach like this is the three factor model developed by Gene Fama and Ken French.
Fama and French started with the observation that two classes of stocks have tended to do better than the market as a whole: (i) small caps and (ii) stocks with a high book-value-to-price ratio (customarily called "value" stocks; their opposites are called "growth" stocks). They then added two factors to CAPM to reflect a portfolio's exposure to these two classes:
r - Rf = beta3 x ( Km - Rf ) + bs x SMB + bv x HML + alpha
Here r is the portfolio's return rate, Rf is the risk-free return rate, and Km is the return of the whole stock market. The "three factor" beta is analogous to the classical beta but not equal to it, since there are now two additional factors to do some of the work. SMB and HML stand for "small [cap] minus big" and "high [book/price] minus low"; they measure the historic excess returns of small caps and "value" stocks over the market as a whole. By the way SMB and HML are defined, the corresponding coefficients bs and bv take values on a scale of roughly 0 to 1: bs = 1 would be a small cap portfolio, bs = 0 would be large cap, bv = 1 would be a portfolio with a high book/price ratio, etc.
One thing that's interesting is that Fama and French still see high returns as a reward for taking on high risk; in particular that means that if returns increase with book/price, then stocks with a high book/price ratio must be more risky than average - exactly the opposite of what a traditional business analyst would tell you. The difference comes from whether you believe in the efficient market theory. The business analyst doesn't believe it, so he would say high book/price indicates a buying opportunity: the stock looks cheap. But if you do believe in EMT then you believe cheap stocks can only be cheap for a good reason, namely that investors think they're risky...
Fama and French aren't particular about why book/price measures risk, although they and others have suggested some possible reasons. For example, high book/price could mean a stock is "distressed", temporarily selling low because future earnings look doubtful. Or, it could mean a stock is capital intensive, making it generally more vulnerable to low earnings during slow economic times. Those both sound plausible; but they seem to be describing completely different situations (and what happens when a company that isn't capital intensive becomes "distressed"?) It may be that the success of this model at explaining past performance isn't due to the significance of any of the three factors taken separately, but in their being different enough that taken together they do an effective job of "spanning the dimensions" of the market.
(There's actually another interpretation that's so much less cerebral that it's probably correct. The broad market index weights stocks according to their market capitalization, making it size-biased and valuation blind; so maybe the extra two factors in this model are just a couple of tweaks to adjust for these two problems. This also explains why momentum is sometimes used as yet another factor: market capitalization shows where the market has been putting its money for years, while momentum shows where it has been putting it lately; so if you want to take advantage of market efficiency you start with the index and then tweak it a little with momentum.)

Portfolio Analysis
Like CAPM, the Fama and French model is used to explain the performance of portfolios via linear regression; only now the two extra factors give you two additional axes, so instead of a simple line the regression is a big flat thing that lives in the fourth dimension.
Even though you can't visualize this regression, you can still solve for its coefficients in a spreadsheet. The result is typically a better fit to the data points than you get with CAPM, with an r-squared in the mid-ninety percent range instead of the mid eighties.

Investing for the Future
Analysing the past is a job for academics; most people are more interested in investing intelligently for the future. Here the approach is to use software tools and/or professional advice to find the exposure to the three factors that's appropriate for you, and then to invest in special index funds that are designed to deliver that level of the factors. You can try this tool on another website. When you try it you should note that it in fact collapses all risk to the single factor of volatility, which is typical, and which brings up the earlier question of whether the additional factors really are measures of risk. In this case, how would you ever help somebody figure out what their "value tolerance" was? As a matter of fact, what would that question even mean?

Conclusions
There are two separate messages to take away from this. First, the three factors together account for practically all of a portfolio's behavior; that's the strongest evidence yet that mutual funds can't beat indexes. Second, history indicates that small value "just happens" to deliver higher returns and higher volatility than the stock market as a whole. Assuming the trend holds, then that's the practical message for investors. In particular, it improves what felt like a flaw in the Tobin argument: where Tobin said high-risk investors should buy the total stock market index on margin, Fama and French offer the saner alternative of just adding some small value to your portfolio.

(Also see the portfolio guidelines page in the index funds article for a calculator that uses Fama and French performance data.)

Insurance Analogy
Modern portfolio theory can be unintuitive, but an analogy to the insurance industry can make things clearer. So imagine you are in the business of selling fire insurance policies. Your long term profits are a function of the premiums you collect minus the claims you pay out. Your short term risk is the expectation that a given year might have an abnormally high number of claims to pay out. In this business you have a well-defined risk/reward tradeoff: the total amount of business you will be allowed to do depends on your ability raise capital to survive a really bad year.
Let's look at the risks in more detail. First of all, some years (like the hot, dry ones) will be bad for the fire insurance market as a whole. This is "systemic risk", that you can't do anything about. But in the bad years some policies (like those on wooden houses) will do even worse than average. You'll probably try to set high enough premiums on these "high beta" policies so that they'll be more profitable, long term, than average; otherwise, it isn't worth risking your capital base to underwrite them.
Now let's look at what you can do about risk. You could ignore risk, and concentrate only on long term profits. This is the approach Warren Buffett takes with his ultra-volatile, ultra-profitable "supercatastrophe insurance" business. Buffett defines his time line as "forever", and cheerfully acknowledges that he will have some truly horrific years with awesome claims to pay out. Of course he can only do this because he has deep pockets; no matter how bad a year it has, his company has enough capital to survive.
Or you could decrease risk, through diversification. For starters, you wouldn't concentrate all your business in one housing development or condominium tower, because the chance of a neighborhood fire would be too great, making your risk of a bad year far worse than the industry average. So you would spread the policies out geographically, decreasing your risk until it approached the systemic risk of the fire insurance business as a whole. This is how most people think of diversification, as "not putting all your eggs in one basket."
But you can do even better if you are willing to diversify into other segments of the market. For example, you'd probably find that the bad years for floods aren't the same as the hot and dry years that are bad for fires. Diversifying into flood insurance would really let you take advantage of the curved shape of the efficient frontier: the risk of your diversified fire and flood insurance company would be less than the average of the risks of the two types separately. So you'd be able to sell more policies, and make more profits.

This analogy seems like a pretty good one, because portfolio theorists think about market returns in the same "random-normal" way that insurance agents think about natural disasters. Buffett himself doesn't believe in MPT, which is disturbing since he's an expert in both sides of the analogy. For their part, MPT advocates tend to write Buffett off as a statistical outlier (that means he's a freak from the outer limits of the bell curve) or even as "the exception that proves the rule": out of all the millions of investors in the world you can only identify this one guy - in Omaha, Nebraska for heaven sakes - who seems to be able to beat the market.

MPT – Conclusions
This article has presented a rapid overview of modern portfolio theory. How much of it you can actually use depends on your own investment philosophy. Everybody can benefit from the first two points:

1. The short term dangers of volatility are real; even an excellent long term investment can be a disaster for you if your time horizon is short.

2. Diversification reduces volatility more efficiently than most people understand: the volatility of a diversified portfolio is less than the average of the volatilities of its component parts. If you're a dedicated stock picker then that's probably as far as you'll go. But if you lean toward mutual funds, you can use the third point:

3. A scientific way to attain a diversified portfolio is by using simple tools to choose a small number of low cost index funds. The obvious starting point would be a total stock market fund, possibly with a smaller portion (preferably within a tax-sheltered account like an IRA) in a small value fund.

One other point: MPT critics like to emphasize a few occasions where people have stretched theory way beyond common sense and lost a ton of money. But that shouldn't distract you from the "core" message of MPT, that volatility can be planned for, and that diversification and index investing let average people participate intelligently in the market.