blackboard full of complicated graphs ans calculations

Modern Portfolio Theory: Part 1

I was recently contacted by a sophisticated investor who asked how I’d apply Modern Portfolio Theory (MPT) to their asset allocation process. They already had an MPT framework in place, and wanted to know how to improve it.

I thought it was an interesting question, so spent far longer than I should have done consulting my notes and hammering out a response.

If you’ve never heard of Modern Portfolio Theory before, and have no interest in learning about what it is or why it’s relevant for your investments, then look away now. I get that it’s not everyone’s cup of tea.

Personally, I find this stuff fascinating.

My next two posts, then, will be an edited version of my response. This first post will be about why I’m not a fan of using what’s known as mean-variance optimisation to apply MPT (more on that below), and the second part will be how I would choose incorporate MPT into portfolio construction.


What is Modern Portfolio Theory?


I’m not going to do a better job summarising MPT than Investopedia:

“The modern portfolio theory (MPT) is a practical method for selecting investments in order to maximize their overall returns within an acceptable level of risk.

American economist Harry Markowitz pioneered this theory in his paper “Portfolio Selection,” which was published in the Journal of Finance in 1952. He was later awarded a Nobel Prize for his work on modern portfolio theory.

A key component of the MPT theory is diversification. Most investments are either high risk and high return or low risk and low return. Markowitz argued that investors could achieve their best results by choosing an optimal mix of the two based on an assessment of their individual tolerance to risk.”

So MPT basically says two things:

  • You can’t get higher returns without taking on more risk, and
  • Any investment’s risk and return characteristics shouldn’t be viewed alone, but should be evaluated by how it affects the overall portfolio’s risk and return.

Sounds pretty straightforward.


What is Mean-Variance Optimisation?


The traditional method for applying MPT to portfolio construction is use what’s known as mean-variance optimisation (MVO).  

An investor picks a selection of funds/asset classes – this basket of funds could be anything from broad asset class funds (global stocks/bonds/property), to more granular funds (UK equities, German bonds), to individual companies’ stocks and bonds. The investor then chucks all these securities into an MVO calculation tool – usually a goliath Excel spreadsheet – which crunches the numbers.

The MVO calculator will analyse all the different possible combinations of the funds the investor has used as inputs based on their risk and return characteristics, and combine them to create a selection of ‘optimal’ portfolios which maximise returns for a given level of risk.

In picture form, it spits out something like this:

Efficient frontier 1

Each dot represents one particular combination of funds.

The combinations which sit on the black line are ‘optimal’. You can’t get more return for any given level of risk. Because these combinations maximise returns for the risk they’re taking, they form what’s called an ‘efficient frontier’.

Any dots not sitting on the black line are sub-optimal and aren’t worth investing in. For any dot below the line, there will be another dot on the efficient frontier which has a higher return for the same level of risk.

According to MVO, you should figure out how much risk you want to take, then select the combination of funds on the efficient frontier. This provides you with the highest level of risk-adjusted returns.


A real-world example


Here’s one I made earlier.

Let’s say you were constructing a global equity portfolio, and you wanted to figure out the optimal asset allocations to each region. The chart below shows the efficient frontier for a handful of regional asset classes: US large cap, US small cap, European stocks, Emerging Market stocks, and Pacific stocks. I haven’t cherry picked them to prove any sort of point, they’re just the default asset classes available on Portfolio Visualizer:

Efficient frontier 2.jpgSource: Portfolio Visualizer

I’ve asked the optimiser to find the combination of those assets which provides me with the highest Sharpe Ratio (i.e. the highest risk-adjusted return), so it’s highlighted that on the efficient frontier for me – about 15% standard deviation for about 11.5% return. Not bad!

You can see that because European stocks and Emerging Markets stocks both have lower returns than the portfolios on the efficient frontier for their given level of risk, they’re ‘inefficient’ portfolios, and not worth owning.

Seems pretty easy, right? All you need to do now is allocate to that maximum Sharpe Ratio portfolio, sit back and wait for the returns to come rolling in.

Unfortunately (and predictably), it’s not quite as easy as that.

Although MVO is an alluringly simple route to efficient asset allocation, it has several significant limitations which prevent it from being as useful as it first appears.


Limitations of MVO


1. “Garbage in, garbage out”. MVO analysis relies on several inputs for each fund/asset class, including return, risk, covariance and correlation as a starting point. The most glaring problem inherent in an MVO approach to MPT is a simple one: these figures are impossible to forecast.

2. Correlations change over time. The stock/bond correlation was positive for the majority of market history pre-2000, before turning and remaining negative for the majority of the last 20 years. This year’s flip back to a positive correlation between stocks and bonds has reminded investors how changeable this particular correlation can be – and it’s the most foundational negative correlation pair in markets. If the stock/bond correlation isn’t reliable, then intra-asset class correlations (e.g. between equity-market regions/sectors) and cross-asset class correlations (e.g. between stocks and real estate) certainly can’t be assumed to be constant.

Using a single average figure to approximate such a changeable variable introduces considerable scope for error in the MVO outputs.

3. Correlations trend to 1 during a crash. Not only do correlations change over time, but there can be significant differences between average historic correlations and correlations during a crash. For example, if the average correlation over the last 50 years between emerging markets and the S&P was 0.4, but they both dropped together during crashes, diversification into EM isn’t going to diversify a portfolio when it’s needed. The use of an average correlation as an input into an MVO isn’t likely to paint an accurate picture of how that portfolio will perform in a crash.

4. Accurately forecasting asset class returns is impossible. Moving away from correlations and now looking at returns (another MVO input), the inescapable truth is that it’s impossible to forecast returns with any sort of accuracy.

Using historic returns to try and predict the future is one option, but this fails to incorporate any adjustments for current market conditions (valuations, interest rates, inflation, etc), and assumes past returns are reliable predictors of future returns – and we all know they aren’t. The SPIVA Persistence Scorecard shows how strong the iron law of mean reversion is. By using historic returns, you run the risk of optimising for an efficient frontier which is relevant to the past 10 years, but not the next 10.

This great chart from John Bogle’s Common Sense on Mutual Funds shows how much the efficient frontiers of two asset classes can move around between the decades – in this case it’s the efficient frontiers for the S&P 500 and the EAFE Index (Europe, Australasia, and the Middle East):

Efficient frontier 3.jpg

If you were trying to use any of those past 10-year time periods to create an efficient frontier for the next 10, you’re going to have a bad time.

On top of the difficulties of extrapolating past returns into the future, any historic return is going to be an average return. And using an average return introduces a host of other problems – chief of which is deciding on the time period used to calculate the average. Using as much historic data as possible seems a logical choice, but it brings with it the problems of what statistics nerds call ‘data stationarity’. If an investor uses data from 50 years ago in their average, that assumes the market is the same today as it was 50 years ago. If the market is structurally different today (as many argue it is), then the returns data from 50 years ago isn’t relevant and will produce an inaccurate forecast. If the investor only uses more recent data to calculate their historic average return, the data may well be more relevant to today’s market structure, but they’ll have to make a judgement call over where the ‘relevant’ data begins. Not only does this introduce subjective judgement, it runs the risk of extrapolating recent strong performance, and engaging in performance chasing.

Of all the MVO inputs, the return assumption has the greatest impact on the outputs, and given how unreliable return forecasts are, this seriously limits the optimisation’s usefulness.

5. Outputs are sensitive to small changes in inputs. Speaking of which, not only is it incredibly difficult to forecast returns, risk and correlations, but even small errors in those estimates can lead to wildly different efficient frontiers – especially when it comes to the return assumption.

6. Multiple assumptions need to be accurate. The likelihood that it’s possible to forecast returns, standard deviations, and correlations for multiple asset classes for a multi-year period with sufficient accuracy to produce an optimal portfolio is extremely low. If you were able to know all these things with any sort of accuracy, then you wouldn’t need an optimiser in the first place!

7. Constraints on outputs add further error. Without constraints, MVO also tends to optimise for highly concentrated portfolios in a subset of the available asset classes. It will, for example, suggest huge weightings to the asset class which happened to have the highest Sharpe ratio given the inputs. Alternatively, if it notices two highly correlated asset classes with different Sharpe ratios, it may well suggest a leveraged long-short portfolio to exploit the apparent opportunity.

Using our example from before, the maximum Sharpe ratio portfolio actually looked like this:

Efficient frontier 4.jpgSource: Portfolio Visualizer

It owned 99% in US large cap, and 1% in US small cap. So out of the 5 asset classes which went into the optimiser, the resulting recommended portfolio was to put everything into US large cap. This output makes sense when you consider that the optimiser simply takes past returns and extrapolates them into the future, given the heroic run that US large cap has had over the last 20 years. But I’m pretty certain it’s not going to be the same going forwards – certainly not confident enough to bet my entire portfolio on it.

MVO therefore requires constraints to produce useful outputs, including constraints over leverage, shorting, absolute asset class maximums, relative asset class maximums, liquidity, and others. The more constraints are used, the less the resulting frontiers reflect “optimal” portfolios, and the more they reflect the judgements used in the constraints. This adds additional levels of error, which are placed on top of the errors already inherent in the assumptions used as inputs to the model.

8. Returns aren’t normally distributed. MVO also assumes normally distributed returns, as it relies on standard deviation as an input. It’s been repeatedly shown that market returns aren’t approximated by the normal distribution, thanks to large crashes (“multi-sigma events”) occurring far more frequently than would be predicted by the normal distribution. Standard deviation is an easy horse to flog when criticising modern financial theories, because they all use it to one degree or another. But it doesn’t make it any less true – it’s still not a great proxy for market behaviour, and reduces the usefulness of the MVO’s outputs.

9. MVO is unlikely to find true diversifiers. In order for assets to be diversifiers, they need to have more than just correlations of less than 1. The diversification “free lunch” is only powerful if both assets have: a) similar levels of volatility, b) similar levels of return, and c) very low correlation.

a) If one asset is much less volatile than the other (e.g. short-term bonds and equities), then when one falls (equities) and the other rises (short-term bonds), the gains from the low-volatility asset class will be far smaller than the losses from high-volatility asset class. Owning the asset will help, but not much. The low-volatility asset class is really just reducing risk rather than providing diversification (as true diversification means lowering risk without lowering returns).

b) If one asset has much lower expected returns than the other, then the asset will drag down the overall portfolio’s expected return – which means it needs to provide extremely strong and reliable diversification effects for it to be worth including (and strong, reliable diversifiers are incredibly difficult to find). Again, the low-return asset (like short-term bonds) is likely to provide risk-reduction benefits, but not true diversification.

c) If the two assets don’t have very low correlations, then the diversification effect is weak. Even a correlation of zero doesn’t mean that when one rises, the other falls – it just means that sometimes when one rises the other falls. Sometimes they may well move together.

Finding asset pairs which display these three characteristics (similar volatility, similar return, and low correlation) robustly over long periods of time is extremely difficult. The main contender for a reliable diversifier to global equities is long-term bonds, as their higher volatility and higher returns compared to short-term bonds help provide diversification benefits. Various equity factors (or “risk premia”), such as size, value, and momentum have also been touted as being able to provide unique sources of risk – although the academic evidence is still disputed, as is their implementability. Due to the weaknesses of forecasting correlations mentioned above, MVO is unlikely to highlight any true asset class diversifiers.

10. MVO ignores real-world investor behaviour. Finally, the use of MVO encourages an engineering approach to investing – the belief that using advanced mathematical techniques will yield useful results. In reality, the ‘optimal’ portfolio is one the investor can continue to hold when markets are crashing and, in an ideal world, contribute more

Not only does MVO modelling output portfolios which can be incredibly difficult to stick with, it gives no consideration to the real-world behavioural difficulties faced by investors – including how we deal with crashes, the temptation to engage in performance-chasing, and how we react to underperforming a benchmark. One of the best portfolios you could’ve owned over the last 20 years would’ve been a leveraged bet on US tech. It might even have been ‘optimal’. But could any real human being have sat through the 90% drawdowns it would’ve suffered along the way? I certainly couldn’t.

11. Other approaches to MVO come with their own limitations. There have been several attempts to improve the usefulness of MVO (resampling using Monte-Carlo Simulations, Black-Litterman models, incorporating kurtosis/skewness, using an asymmetric definition of risk). But all come with their own limitations, and to some degree or another require accurate divinations of future returns, risk, and correlations between asset classes.

Overall, the likelihood that it’s possible to forecast returns, standard deviations, and correlations for multiple asset classes for a multi-year period with sufficient accuracy to produce an optimal portfolio is extremely low. Even if it were possible (an absolutely heroic “if”), the use of multiple subjective constraints necessary for creating real-world implementable portfolios adds additional levels of error, which even then cast doubts on the usefulness of the outputs. On top of that, MVO gives no consideration to the real-world behavioural difficulties faced by investors.

Implementing MPT using MVO isn’t likely to be a useful tool for asset allocation.

Perhaps most revealing is the fact that Harry Markowitz, the creator of MPT and MVO, didn’t use it for his own portfolio when investing his retirement account at the RAND Corporation (from Jason Zweig’s book Your Money or Your Brain):

“I should have computed the historical co-variances of the asset classes and drawn an efficient frontier. But I visualized my grief if the stock market went way up and I wasn’t in it — or if it went way down and I was completely in it. So I split my contributions 50/50 between stocks and bonds.”

So although MVO may not be a useful tool for implementing Modern Portfolio Theory, MPT itself still has some useful takeaways:

  1. It’s only the risk you cannot diversify away which leads to increased returns. The major prescription of MPT is to avoid idiosyncratic risk by diversifying as much as possible.
  2. The risk and return characteristics of any asset class by itself are irrelevant. The only thing which should matter is considering how the addition of an asset class impacts the risk and return of the entire portfolio.

While using optimisation to bring MPT to life might not be as useful as it first appears, these two broader core tenets of the theory are rock-solid.

This post has ended up going quite long, so I’ve chopped off what was going to be the next section, and that will form the basis for my next (shorter) post. It’ll be all about how we can apply these core MPT ideas to portfolio construction, and specifically how they lend themselves to adopting a predominantly passive approach.

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