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Tuesday 26 May 2009

Markowitz's Portfolio Theory and the Efficient Frontier

Managing Risk

You’d have thought that the management of risk in respect of stockmarket investment would have a long and reputable history. After all, the very idea of a share is all about allowing individuals to spread their capital and risks across multiple, partial investments.

This is not so, stockmarket risk management only really started with Harry Markowitz’s seminal paper Portfolio Selection in 1952. Typically the industry then ignored his ideas for twenty years before belatedly getting around to using them for, well, everything. And then some, leading eventually to the invention of the index tracker.

What is Risk?

The early seventies coincided with one of the worst periods for stockmarket investors in the last century, ending the long bull run that had commenced in the wake of the Second World War. The investment industry, complacent after twenty years of easy returns, wasn’t prepared for the ensuing carnage and in a search for tools to manage the new reality turned to Markowitz’s ideas.

Harry Markowitz had originally been conducting research into something called linear programming, a branch of mathematics that’s all about minimising or maximising one variable while keeping another constant. Think about meeting the nutritional needs of a diet at minimum cost.

When he turned his attention to stockmarket investment he realised that he could apply linear programming to portfolios of stocks – aiming to maximise returns while keeping risk constant or, of course, to minimise risk while keeping returns constant. However, to make this work, Markowitz had to turn risk from something that everyone recognised when they saw it (usually after the event) to something you could quantify. He had to turn it into a number. He did this by equating risk to variance – something most of us will be more familiar with as volatility.

Volatility and Correlation

Volatility is – simplistically – the amount by which an investment’s value varies over given time periods. A share whose price ranges between $1.50 and $4.50 is more volatile than one that moves between $2.50 and $3.50 even though their average value is the same. Markowitz reasoned that the sensible investor would prefer the second stock over the latter – which makes sense if you’re worried about the possibility of big losses. If your investment’s value varies a lot over time then there’s a real risk you may need to liquidate it at exactly the wrong time. All other things being equal a lower volatility is a good thing.

Using linear programming Markowitz was able to show that a portfolio of stocks, even if they individually exhibited large levels of volatility, could offer lower volatility than the underlying stocks themselves. This turned standard theory on its head – before this individual stocks had been selected, on gut feel, to provide the maximum return for the minimum risk. Markowitz’s work showed that you could get a less risky portfolio by selecting portfolios of more volatile stocks – as long as those stocks tended to not move together.

Think about two stocks, each of which tend to move together but in opposite directions and by the same amount. If one moves up $1 then the other moves down $1. The net effect is no change to your portfolio value. This is, in highly simplified form, the idea that underlies portfolio theory. The degree to which the stocks move together (or not) is called correlation.

The Efficient Frontier, CAPM and VaR

Under portfolio theory you can select the return you want and the maths will find you the set of the least risky portfolios that will provide that return (or you can select the level of risk and it’ll find you portfolios that give you the maximum return). These portfolios constitute the set of efficient portfolios and lie on the so-called Efficient Frontier.

From this spawned a whole new branch of the investing industry leading to the Capital Asset Pricing Model (CAPM) which outputs the alphas and betas that get thrown around by professional and amateur investors alike. Beyond that in the future lay Value at Risk, VaR, a measure we’ve seen at work on a grand scale in the last year or so.

Pernicious People

Underlying all of this superstructure, though, is Harry Markowitz’s model and the assumptions it’s based on. He tried to remove humanity from the investing equation but no matter how hard we try we can’t help interfering in our own best laid plans.

Firstly the model relies on historical data in order to construct its efficient portfolios – what other way have we of assessing the way in which different stocks move in conjunction with each other? This, of course, isn’t a given. History may be a guide to the past but it’s definitely not a set of rules to be followed prescriptively.

Secondly, underpinning the concepts of portfolio theory is the idea that humans will act rationally to maximise their returns at minimum risk. If humans don’t generally behave like this and are apt to, say, sell all their stocks in a panic during a crash or start buying wildly overvalued shares during a boom then no amount of theory is going to save them from getting horribly burnt.

Thirdly, risk is not always volatility. If you buy a house and intend to live in it for the rest of your life then the short-term changes in its value are of no importance to you whatsoever. The same is generally true of the stockmarket, at least at the index level. A long term, buy and hold, passive investor won’t view risk in quite the same way as a day trader.

Fourthly, there’s an argument that the Efficient Frontier is actually unobtainable unless you’re able to go short in the portfolio – something not possible for long-only investment funds. We’ll come back to that in the future.


So count me all the ways that humans misbehave – they can all interact to undermine the ideas underlying portfolio theory. Yet, at root, it constitutes a set of basic tools that are critical for investors whether they’re active or passive.

Understanding how a diversified portfolio of stocks can be constructed to minimise the risks of volatility is important: simply buying a random selection of shares in companies does not guarantee diversification. Also recognising that even a well diversified portfolio can fall prey to systemic problems, when all stocks or all asset classes suddenly line up and move together is critical to an investor’s mental health – being caught unaware by this can lead to all sorts of stupid investing behaviour.

Portfolio Theory, CAPM or VaR aren’t panaceas for anyone involved in investing. Markowitz's idea led, eventually, to the creation of the index tracker - for the most part investing remains an easier game if you remove the fear of short term volatility entirely by ignoring it and focusing on the long term. Otherwise, for active investors, they’re tools and they need to be used as such - with care and understanding.

Diversification and the Endowment Effect

Moreover investors need to think carefully about how they diversify their portfolios geographically: the evidence suggests quite clearly that investors favour their own markets and even stocks in companies they know well over foreign markets and unfamiliar investments (French and Poterba, 1991). The psychological effects that cause this behaviour are well understood. The Familiarity Effect leads people to buy what they know and the Endowment Effect means they’re reluctant to part with it, regardless of logic or portfolio theory.

What Markowitz did was put a number on risk to allow it to be managed. The first danger for investors is in not understanding the importance of Portfolio Theory for risk management of stockmarket investments. The second is in believing that it can explain everything. People don’t get programmed linearly – we come with randomness built in, not as an optional extra. Thank goodness.

Related Posts: Sir Hugh Invents the Share and Gets Lost, Alpha and Beta - Beware Gift Bearing Greeks, Risky Bankers Need Swiss Cheese, Not VaR

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