For the brave: portfolio theory and the rational investor
Writing about portfolio theory is probably even less fun than reading about it, but this part is important so stay with me if you are interested in the theory behind this book. You should know that the investing style and portfolio construction in this book is not something I have come up with. It is a practical implementation of the generally acknowledged most advanced theory on the subject. Implementing advanced portfolio theory in the real world gives us the best possible portfolio and is actually very easy and logical for someone that does not have an edge on the markets.
The foundation of portfolio theory is that you have a riskless investment that you can optimally combine with a series of other investments. These combinations of investments create different portfolios with a range of individual risk profiles. Because of the optimal investment combination you get the best expected return for any level of risk.
Imagine the simple scenario where you can choose between investing only in two different things: the riskless investment and investment A. Since we know what return we will get from the riskless investment (otherwise it would not be riskless) we can plot the two options on a simple chart (see Figure A.1).
The line between the riskless investment and A represents different proportions of the riskless investment and investment A: from 0–100% (all A) to 100–0% (all riskless).
In academic theory, the riskless investment was generally acknowledged to be short-term US government bonds. For practical purposes there was no risk that the US government would default overnight: thus ‘riskless’. Those short-term US bonds still provided a return and that return was appropriately called the riskless return. The presence of this riskless return suggested that if you were not willing to take any risk with your investments whatsoever there would be the choice of investing in short-term US government bonds. Because this was the lowest risk security it was also accepted that it should be expected to carry the lowest expected return: there is no such thing as a free lunch.
Figure A.1 A simple risk/return chart for two options
Of course in today’s world the term ‘riskless investment’ sounds almost like a contradiction in terms. At the time of writing the credit rating of the US, UK and French governments have been downgraded from the highest rating, the financial press is flooded with stories of government debt and deficits, and the number of issuers with the highest credit rating have dwindled since 2007. Granted that in the run up to 2007 investors acted as if everything was riskless, but that is another story.1
Investment A can be almost anything, but think of it as a share in Microsoft. Investment A carries a higher expected return than the riskless asset, but there is also some risk associated with that return. More risk for a higher expected return; again, no free lunch.
If you want no risk, you go all riskless, and if you want more risk, you buy investment A. If you want risk between the two, you combine them.
Adding assets
Now we introduce the possibility of another investment, B. Like A, B has its own expected risk/return profile. Importantly though, the movement in the price of A and B are not entirely independent of each other, measured by the correlation (relationship) between those stocks. Remember that word – correlation. It is one of the most important, yet overused, words in the world of finance.
Correlation gives an idea of how A and B move relative to each other. With a correlation of zero there is no relationship between the two, and a correlation of 1 suggests that they move in perfect tandem. If A goes up, so will B. At the danger of oversimplifying complex statistics, most stocks within a general stock market have correlations of roughly 0.5–0.9 with each other, although correlations can change a lot over time. This means that most stocks tend to move in the same direction. You would expect Microsoft and Apple to have a high correlation, while Microsoft and the price of wheat would be less correlated. The lower that two investments’ correlation is, the more there are diversifying benefits of lower risk from investing in both of them instead of just one.
Adding the possibility of investing in investment B to Figure A.1 gives the choices shown in Figure A.2.
Figure A.2 Adding in a third option
The line between A and B is not straight – this is because we assume A and B are not perfectly correlated and we benefit from the diversification of having two investments. Instead of a straight line, the curved line between A and B represents different proportions of A and B.
We can combine an investment in the riskless investment with any combination of A and B (any point on the curved line between A and B). As you see on Figure 3.5, if we draw a tangent line from the riskless investment point to the curved line then point T is where they meet. T is called the tangency point, and the line between the riskless investment and T is called the capital market line. Looking at the chart in this example, point T consists of roughly 40% of investment A and 60% of investment B (you can see point T is closer to B). You can also see from the chart that if you want the risk of point T or less, you get the highest expected return from combining point T and the riskless investment.
If we want more risk in our portfolio than point T (so on the dotted line to the right of point T) the best solution is to add leverage and invest the additional capital in the combination of 40% of A and 60% of B in this example. By adding leverage and buying more of the 40/60% T combination2 we achieve a higher return than if we had allocated more to investment A to get more risk in the portfolio. You can see that the tangent line that continues on from point T is above the curved line where the combination consists increasingly of only investment A.
The optimised market – minimise risk and maximise returns
All of this material may seem abstract and theoretical, but I hope to show how the implications for the rational investor are simple and straightforward as the combined forces of the market have already done the work for you.
Extending the theory discussed above to the whole market, there are endless combinations. Instead of just A and B we can combine thousands of investments as illustrated in Figure A.3.
As shown in the chart, the new tangency portfolio T is no longer a combination of just two securities but a combination of the various combinations. By combining securities in different proportions we are able to create any risk/return portfolio in the shaded area. Since we want higher returns for a given level of risk we chose combinations of securities that get us to the bold curve. In portfolio theory this curve is called the efficient frontier. What the curve is telling us is that for each level of risk there is a combination of securities in the market that gives us the highest expected return. And by combining one of those points on the efficient frontier (point T) with the riskless asset we can create portfolios with the highest level of expected return for any level of risk.
I imagine some of my old economics professors would be aghast at the simplicity of the paragraphs above and the absence of long mathematical formulas. If you are interested in the maths and theory behind this summary of portfolio theory then Modern Portfolio Theory and Investment Analysis by Edwin Elton et al. (John Wiley & Sons, 2003) is a good textbook on the subject.
Bullshit in, bullshit out
One of the things I disliked about investment banking was building massive 50-page Excel models, outlining projections of companies and industries. We would often have little to go on in terms of projections other than short analyst reports which we would use to extrapolate all sorts of data to get 5–10-year projections with all the bells and whistles. We would call this ‘bullshit in, bullshit out’, suggesting that the financial models were only as good as the assumptions we put into them.
As with the large investment banking financial models, optimal portfolio theory is subject to getting our assumptions right. You probably noticed how casually the theory suggested that you input the expected risk and return for individual securities, and the correlation between them, and voilà, the efficient frontier and the tangency point are revealed. Or rather for about £50 you can buy software that will do that for you. But the world is obviously not that simple. Ask 10 market participants about the expected risk and return over the next year on Apple shares and its correlation with Microsoft, and you will get 10 different answers. Now ask the same people to do the same thing for all listed stocks and they will tell you that you are crazy – it’s not realistic to have this kind of expectation for more than a small portfolio of shares, and besides, risk and return expectations, and correlations, change all the time. It simply can’t be done.
The beautiful shortcut – follow the crowd
But here is the beautiful thing. If you generally believe in efficient markets, you don’t need to worry about the portfolio theory above or collecting millions of correlations and thousands of risk-return profiles. The market’s ‘invisible hand’ has already done all that for you. We don’t think we are able to reallocate between securities in such a way that we have a higher risk/return profile than what the aggregate knowledge of the market provides. Buying the entire market is essentially like buying the tangency point T.
To some people it will seem like too bold an assumption that capital has seamlessly flowed between countries and industries in such a way that world markets are efficiently allocated. But if we asked: which country/industry/company is it that you want to reallocate money to/from contrary to the combined information and analytical power of millions of investors allocating trillions of dollars, and why? Accepting that investing internationally gives us greater choice and diversification than only investing in one country, we need to figure out some way to allocate between those choices. If we picked the countries/industries/companies on anything other than their relative market sizes we would essentially be claiming we knew something more than the markets.3
You can of course disagree with all of this and make claims like ‘Microsoft will go up 20% next year regardless of the market and there is almost no risk that I am wrong’. Of course you might be right, but you are also clearly claiming an edge in knowing or seeing something that the rest of the market does not. Do that consistently and you’ll be rich.
Going back to the example of having only the choice of the riskless investment and investment A, that is essentially where we end up. If we replace investment A with the world equity portfolio, and replace the riskless investment with the minimal risk asset, we have moved on from the world of portfolio theory to the real world with investments we can actually implement. We will see later that the minimal risk asset depends on the base currency of your investments.
Investment A in the chart, therefore, consists of many thousands of underlying equities from all over the world in the portfolio (see later). By combining the minimal risk asset and A (world equities) in various proportions we choose various risk/return levels in the most efficient way, from minimal risk to the risk of the world equity markets, or greater than that if we borrow money. Point T is already the tangency point, or optimal portfolio, and we don’t think we can reallocate money between the many securities in such a way that we end up with better risk/return characteristics (see Figure A.4).
Later, when we add other government and corporate bonds, we will see that this is akin to when we added the possibility of investment B earlier. While adding a bit of complexity to the portfolio, the other government and corporate bonds enhance the risk/return profile of the whole portfolio.
Figure A.4 Combining the minimal risk asset and world equities
The best theoretical and actual portfolio
The rational portfolio is a compromise: a compromise between what we would like to create in a theoretical world and what is available practically. In an ideal (theoretical) world we should own a small slice of all of the world’s assets to maximise diversification and returns. This clearly is not possible in reality, but the rational portfolio is a very good simplification that we can actually implement. Because the asset classes of the rational portfolio have active and liquid markets for the pricing of thousands of individual securities, we don’t need any specific insight to select securities in those markets. And because government bonds, equities and corporate bonds give a very good representation of the world’s assets, a portfolio representing those asset classes is a very good simplification of what we should ideally be striving for in a portfolio. We can accept the premise that market forces have set a price on individual securities and the aggregate market at a level that is consistent with the risk/return characteristics of that asset class. Because equities are riskier, we get higher expected returns, etc. For other investments left out of the rational portfolio there is typically not a liquid and efficient market to set prices for the individual investments, so someone without an edge is unable to simply buy into the whole asset class and expect to get its overall risk/return.
So there is no theoretical inconsistency in being a rational investor – on the contrary. We don’t think we know any better than the market about the risk/return profiles of individual securities or how they move relative to one another. By pricing securities, the market effectively incorporates the views of thousands of investors and presents us with the results of the market as it currently stands.
So what is ‘the market’?
In equities, the market has often meant your local stock market. And if you invest in your local index in a cheap way you are doing better than by picking individual stocks or active mutual funds, but not as well as you could be. You could be picking a much broader geographical range of the world stock markets in proportion to their values.
So only publicly listed equities?
In order to keep things simple, we typically refer to the markets as the listed equity markets. If you invest in a combination of the minimal risk asset and broad-listed equity markets in a cheap and tax efficient way, you are doing better than most. If you are willing to add a bit of complexity there’s a lot of merit in adding other government and corporate bonds to your portfolio (see later).
Summary
- The ‘invisible hand’ of the markets has optimised the values of the investments available. We should celebrate this simplicity and buy the whole market. We don’t think we can reallocate between securities to get a better risk/return profile.
- Combine the world equity markets with investing in the minimal risk asset to get to the kind of risk profile you want.
- The markets could mean for you only a broad range of equities, but adding other government and corporate bonds has a lot of merit.
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1 An important point on portfolio theory and the minimal risk asset that will be discussed later. In accepting that the minimal risk asset is not entirely without risk the line between it and investment A is theoretically not straight, but a curve. Depending on how the minimal risk asset and investment A move relative to each other you might actually end up in a situation where the lowest risk combination of the two is not the 100% minimal risk asset, but instead has some investment A in it. For the purpose of simplicity, I have ignored this possibility.
2 This leveraged portfolio assumes that we can borrow money at the riskless rate to invest in more of the T combination, which we can’t. In reality, the cost of borrowing would be higher and the line would be flatter to the right of T than it would be to the left of T, to reflect this. If you were unwilling or unable to borrow money, the optimal portfolio at higher risk levels than T would consist increasingly of A and would be represented graphically on the curved line from T to A.
3 We are here assuming that capital can flow easily between countries and industries, which is increasingly a fair assumption.