User:Quebec/Portfolio theory

Portfolio theory for dummies or DIY portfolio analysis

This page is a very simplified treatment of DIY portfolio analysis. Our goal is to introduce the reader to some of the basics of portfolio theory and portfolio behaviour. The key concept is diversification: we explore the theoretical benefits of mixing domestic (Canadian) stocks and bonds in portfolios, and diversifying between domestic and foreign (global) stocks. This forms the theoretical basis of standard asset allocation advice. We also look at diversification between domestic and foreign bonds, a less explored topic, especially from a Canadian point of view. Finally, just for fun, we add gold to the mix. Throughout the article, explanations on how to do this in a spreadsheet are given so that the reader can reproduce our results and perform further analysis independently.

We first remind the reader of, or introduce him/her to, four basic statistical and financial notions: annualized return, expected return, standard deviation, and correlation. The last three constitute the basic inputs of portfolio analysis. We then summarize some Canadian historical data to provide a reference point for what the future may hold, insisting mostly on the standard deviations and coefficient of correlations, rather than on the actual returns (future returns are expected to be lower). We will then be ready to construct hypothetical mixes of various asset classes, in particular bonds and stocks, to see how they behave. We start by mixing two asset classes, then move on to three. We conclude that:


 * portfolios behave differently than their constituent parts, sometimes surprisingly so
 * getting the stock/bond mix right is the most important decision in asset allocation
 * very conservative investors may want to add a little stock to their portfolios of cash and bonds: this theoretically reduces portfolio volatility and increases the return
 * short Canadian bonds are excellent portfolio diversifiers for stock-heavy portfolios
 * for stock-heavy portfolios, global diversification of stocks convincingly reduces portfolio volatility
 * for bond-heavy portfolios, global diversification of bonds also reduces portfolio volatility, but the effect is much less impressive
 * gold may, or may not be, a good portfolio diversifier, depending on assumptions

Annualized return
The annualized return is a constant return that, over a number of years, would smoothly bring a portfolio to the same final dollar amount as the actual, more much volatile, returns of risky assets. This is also known as the "compound annual growth rate" (CAGR), or the "compound return". The annualized return is a geometric average, as opposed to an arithmetic average. .

Example: If Johns buy a five year GIC with 3% interest, the arithmetic average and the geometric average will both be 3% after the five years. But for risky assets such as stocks and bonds, the geometric average (annualized return) will be less than the arithmetic average. If Linda puts money into a mutual fund and the returns over five years are +10%, -2%, +3%, +6%, +1%, the arithmetic average is 4%, but the annualized return is 3.5%. It’s the annualized return that pays the bills, so we don’t care about average returns.

In Excel, the annualized return of a series of five calendar year returns in cells A1 to A5 may be calculated as {=GEOMEAN(1+A1:A5)-1} (for details see this Gummy page).

Expected return
The expected return of an asset class is the future annualized return that we hope to get over several decades. Expected returns can be guesstimated based on knowledge of past history over several decades, and/or current market conditions (dividend yields, bond yields, etc.). Inflation-adjusted (or "real") returns subtract expected inflation, often 2%, from nominal returns.

For example, in The Intelligent Asset Allocator (2001), William Bernstein suggests the following inflation-adjusted expected returns for US investors:

Similarly, Rick Ferri provides the following 30 year inflation-adjusted estimates for US investors :

Standard deviation
There are multiple definitions of risk (e.g., see Risk (without charts or algebra)), but for studying portfolios, standard deviation (SD) is commonly used. SD is an indicator of the volatility of returns, assuming that the yearly returns of an asset class follow a normal distribution, commonly known as a bell curve. In reality this is not correct, and investors don't necessarily equate risk with volatility, but SD is a useful indicator of risk nonetheless, and is used widely. SD provides a number with which to compare the riskiness (volatility) of different asset classes, and helps evaluate which combination of assets (portfolio) will be the least risky for a given expected return, or the most profitable for a given amount of risk.

Bernstein lists the typical SD of various asset classes as follows (see also the Ferri estimates above):
 * US Money market: 2-3%
 * US short bonds: 3-5%
 * US long bonds: 6-8%
 * US stocks (conservative): 10-14%
 * US stocks (aggressive): 15-25%
 * Foreign (EAFE) stocks: 15-25%
 * Emerging market stocks: 25-35%

If, for the sake of illustration (again, this is not true), we assume that the returns of an asset are normally distributed, then two thirds of the time the returns will fall within 1 SD on either side of the mean, and 95% of the time they will fall within 2 SDs.

Example. An asset class or portfolio with an annualized return of 5% and a SD of 10% will have a yearly return between -5% and +15% two thirds of the time. This sounds reassuring. But it also means that the investor can expect to loose 5% or more every six years, and 15% or more every 44 years.

In Excel, the =STDEV function can be used to calculate the standard deviation of a series of yearly returns.

Correlation coefficient
Two asset classes are perfectly correlated if there is a linear relationship between their returns. In that case, the correlation coefficient is 1.0. If the two assets are perfectly negatively correlated, their coefficient is -1.0. For a coefficient of zero, there is no relationship. For diversification purposes, all else being equal, the lower the correlation coefficient, the better, as we’ll see below.

In Excel, the =CORREL function can be used to calculate the correlation coefficient between two columns of data representing the yearly returns of different asset classes.

If more than two asset classes are being studied, then their correlation matrix contains the correlation coefficients for each pair of variables. The correlation matrix can easily be produced in Excel using the Analysis Toolpack. An example of a correlation matrix for historical US data is simplified from Bernstein's table B-1:

Correlation of annual returns 1926-1998

This shows that over the compiled period, large and small US stocks were imperfectly, but quite highly, correlated (0.8), whereas US short bonds and US T-bills had approximately zero correlation with US stocks, making them excellent portfolio diversifiers for US investors.

Canadian data
We now look at some historical Canadian data, which unfortunately does not stretch as far back as the US data. We are mainly interested by the SDs and correlations, since the realized returns seem too high to use for future expectations.

1970-2013
Yearly returns are available for the period 1970-2013 at Libra investment management, for Canadian long bonds (former SCM Long index), Canadian equities (S&P/TSX Composite index and its predecessors), US equities (S&P500), MSCI EAFE equities and gold bullion, all expressed in Canadian dollars. Over this >40 years period, the returns and SDs were:

This can be more easily visualized on the risk (SD)-return graph:



The annualized returns of long Canadian bonds and various stock classes (TSX Comp, S&P500, EAFE) were about the same during this period – all about 10%. But the SD was much smaller for long bonds (10% vs. 17-22%). Therefore, the backward-looking optimal portfolio would have been highly concentrated in long bonds. For example, plugging Canadian T-bills, Canadian long bonds and the TSX Comp into a three-asset online mean-variance optimizer (MVO), with a typical coefficient of risk aversion of 3.4, appropriate for moderately risk-adverse investors, yields the following backward-looking optimal allocation: 0% T-bills, 78% long bonds, 22% TSX Comp. This goes strongly against the standard balanced portfolio advice of 50:50 or 40:60 bonds:stocks. But this tells us nothing about the future.

For the period 1970-2013, the correlations were:

1993-2013
We can get more asset classes by focussing on the period 1993-2013 in the Libra data. This adds yearly returns for Short Canadian bonds (former SCM Short index), All Canadian bonds (former SCM Universe index), Real-Return Bonds (RRBs), and MSCI Emerging Markets (EM), again all expressed in Canadian dollars. We can also obtain yearly returns for US aggregate bonds and emerging market bonds from the Bogleheads wiki. Both series are in US dollars, but any foreign bond exposure being contemplated should be hedged back to Canadian dollars, so there is no need to convert the returns to local currency. This yields the following table and graph:



We can see on this graph that:
 * The relationship between Canadian bonds (especially the short bonds and the “all” bonds points) and stocks (especially the TSX Comp and the S&P500) looks much more normal: more risk, more reward.
 * The EAFE index had significantly lower returns than the other stock classes, probably due in large part to Japan’s lost decades. This should not be seen as an argument to avoid Japanese or EAFE stocks: we don’t know what the future holds.
 * EM equities had a huge SD (a.k.a. a “wild ride”) but returned less than Canadian or US stocks. This is food for thought in terms of including or excluding this asset class in portfolios.
 * US bonds (hedged to the Canadian dollar) look similar to Canadian bonds in risk-return space, which is quite interesting in terms of their diversification potential.
 * EM bonds outpaced all stock classes with a lower SD. This may or may not happen again. It would be somewhat surprising if it did!

For the period 1993-2013, the correlations were:

Some of the highlights from this table are:
 * Correlations between the TSX Composite and foreign stock classes expressed in Canadian dollars range from about 0.4 to 0.8
 * Short Canadian bonds have near-zero correlation with stocks, and low correlations with RRBs and EM bonds
 * RRBs have low to negative correlations with stocks, moderate correlations with EM bonds and All Canadian bonds
 * Gold correlates with nothing and even has negative correlations with stocks

Mixing two risky assets
The equations to calculate the return and SD of portfolios of two imperfectly correlated risky assets are given in Modern portfolio theory and by Sharpe. These equations are easily implemented in Excel and are used to plot risk-return graphs.

Historical US stock/bond mixes
We start with a well-known US example because it illustrates portfolio theory well and allows us to check that our spreadsheet works correctly. Over the period 1926-1998, five year US treasuries (“bonds”) returned 5.31% annualized with a SD of 5.71%, whereas large US stocks (“stocks”) returned 11.22% annualized with a SD of 20.26% (Table 2-1 in Bernstein ).

Note the much higher SD of the stocks compared to the bonds. Because of this investors, who act rationally (according to modern portfolio theory at least! ) demand a higher return for holding stocks, and the long term US data reflects that. However, the nearly 6% extra return of stocks over bonds seems unlikely to be repeated in the future (see expected returns above: something like 3% seems more plausible), and the 11%+ nominal return for stocks is also probably much higher than what we can expect.

Nevertheless, these particular numbers are used for illustrating portfolio theory because they allow us to recreate part of Bernstein’s fig. 4-2 using the annualized returns, the SDs, the correlation coefficient (0.11), and two equations, without needing the raw data. We construct mixes of bonds and stocks, using 5% allocation increments (100% bonds, then 95% bonds:5% stocks, and so on up to 100% stocks), and chart the SDs and annualized returns on a risk-return plot.



The place to be on such a risk-return plot in the upper left, where the returns are high and the volatilities are low. Unfortunately, this place is difficult to get to, and in general, we can only increase the return by increasing the risk of a portfolio.

If we start from the stocks end-member in the upper-right, and mix that with increasing proportions of bonds, we reduce both the expected return and the volatility, almost on straight line. For example, a 50%-50% mix would have returned 8.3% annualized with a SD of 10.8%. What is not seen on the graph is that if the correlation coefficient had been 1.0, instead of 0.11, the SD of a 50-50 mix would have been 13%, i.e. over 2% more. We come back to this idea in the next section.

The most intriguing portion of the curve is that with high bond allocations, in the lower left. Very conservative investors sometimes have portfolios consisting only of cash and fixed income securities: they avoid stocks because they are deemed “too risky”. But portfolios behave differently than their constituent parts. If we start from 100% bonds and move up two notches to 10% stocks, this improves the return by 0.6% while leaving the SD unchanged. Moving up another two notches to 20% stocks, we increase the return by a further 0.6% and the SD is only now slightly larger than that of pure bonds (6.4% vs. 5.7%). This explains the counter-intuitive advice that even the most conservative investors should have a small amount of stocks in their portfolios, if their time horizon is long enough. Adding risky assets to a portfolio dominated by “safer” ones can actually lower the volatility and/or increase the return.

Influence of correlations
Now let’s look at two hypothetical risky assets, which we’ll call B (for bonds) and S (for stocks), with the following characteristics:

The SDs are those of “All Canadian Bonds” and the TSX Composite (Canadian stocks), respectively, based on the 1993-2013 data (see above). The TSX Composite also had a 17% SD for the 1970-2013 period (no statements can be made about the “All Canadian Bonds” asset class for the full 1970-2013 period).

For expected returns, we just picked something plausible, and less than the 1993-2013 and 1970-2013 periods (see also the Ferri estimates above, with 2% added for inflation).

The correlation of yearly returns between All Canadian Bonds and the TSX Composite was 0.07 over the period 1993-2013, but let’s see what happens when we hypothetically vary the correlation coefficients between B and S from -1.0 to 1.0, for illustration purposes. This time we use 10% allocation increments to simplify the graph.



The green curve, with a correlation is 1.0, is the least interesting: adding progressively larger amounts of B, i.e. moving from right to left on the graph, just dilutes the performance of S. When assets are perfectly correlated, both the portfolio return and SD are simply weighted averages of the returns and SDs of the components. For example, a 50%-50% mix of B and S would give a 5.5% annualized return with a SD of 11%.

But as the correlation coefficients decrease, the curves shift towards the upper left corner of the diagram (pink, then blue curves). This nicely illustrates the effect of diversification, and the importance of picking assets with low correlations when designing portfolios. Canadian bonds, with low volatilities and correlations typically near zero relative to stocks (especially for short bonds), are therefore excellent portfolio diversifiers. For example, with a correlation coefficient of zero, a 50%-50% mix of B and S would give the same 5.5% annualized return but with a SD of 8.5% instead of 11%. Again, portfolios behave differently than their constituent parts.

With perfect negative correlation (black curve), the SD of the portfolio goes to zero, but in the real world an investor will be challenged to find two assets with positive expected returns and a correlation of -1.0. In most correlation matrices of historical data, the lowest values are only slightly negative to slightly positive, typically involving correlations between stocks, on one hand, and assets such as T-bills, short bonds or gold, on the other hand. One exception is the strongly negative S&P500 (US equities) to gold bullion correlation of -0.63 for the 1993-2013 period.

Mixing three risky assets
Now that we are familiar with risk-return plots for portfolios of bonds and stocks (two asset classes), we can add a third asset class, such as global stocks or foreign bonds. The equations to calculate the return and SD of portfolios of three imperfectly correlated risky assets are given in Wikipedia.

Adding global stocks
We construct hypothetical portfolios of domestic bonds (asset class B), domestic stocks (asset class S1) and global stocks (asset class S2). The properties of B and S1 are the same as above, and S2 has the same expected return and SD as S1. We have not chosen a different return or SD for S1 versus S2 in order to avoid sterile discussions about which particular S1-S2 (domestic-foreign stock) mix is the best. We just can’t know in advance in what the best proportion will be, so we’ll use an equal mix of S1 and S2.

The two bond-stock pairs (B-S1 and B-S2) have hypothetical correlation coefficients of zero, whereas the two stock classes have hypothetical correlations ranging from 0.4 to 0.8 (see the 1993-2013 and 1970-2013 data above). In this section we concentrate on stock-heavy portfolios (70%), which will benefit the most from global diversification, but even at 30-40% stocks global diversification seems warranted. The black curve is a mix of B and one stock class (S1 or S2), whereas the coloured curves include the two stock classes, equally weighted, with variable coefficients of correlation.



Let’s look at the black curve first. This represents the equivalent of a two-fund portfolio of Canadian bonds and Canadian stocks only (or Canadian bonds and global stocks only). A portfolio of 70% stocks (30% bonds) has an expected return of 6.1% with a relatively high SD of 12.0%. Recall that if the distribution of returns follows a normal distribution, then one year out of six, the portfolio will loose about 6% or more. Every 44 years, the portfolio will loose 18% or more. This seems quite benign but in real life the distributions are not normal and extreme events occur more frequently. So any potential reduction in volatility is worth investigating.

The blue, green and red curves add global diversification to the picture. They represent mixes of bonds (B) and stocks, but the stocks are now divided equally between S1 and S2 (domestic and global). At 70% stocks, this global diversification reduces the SD by 0.6% for a 0.8 correlation between S1 and S2, by 1.2% for a 0.6 correlation, and by 1.9% for a 0.4 correlation, bringing the SD down to 10.1% instead of 12.0% for the two-asset portfolio. The expected return, meanwhile, remains unchanged at 6.1%.

There is another, perhaps more dangerous, way to look at this. We can to travel upward along a vertical line of constant SD, for example 12%. With global diversification of the stocks, we can increase the expected return to between 6.2% and 6.5% at the same SD (up from an expected return of 6.1% with one stock class only). However – and this is not always obvious in discussions on portfolio theory – this means increasing the stock proportion to between 74% and 84% (up from 70%) depending on the assumed coefficient of correlation between S1 and S2. Because the SD of the portfolio is unchanged, in theory the risk is the same even with more stocks in the mix. But in practice, it would seem prudent to keep the initial stock/bond mix to 70% stocks (or whatever is selected as acceptable by the investor) and accept that the main advantage of global diversification is a lower volatility of returns, i.e. we prefer to travel leftward on the graph, not upwards.

At stock proportions of less than about 25%, global diversification of stocks does not change significantly the expected return and SD on the graph: one stock class is essentially the same as two. Instead, at these low stock proportions, the investor may have to turn to foreign bonds or real-return bonds, for diversification.

Adding foreign bonds
This time we will mix two types of bonds (B1, B2) with one type of stock (S). The S component, assumed to be domestic (but it does not matter for our story, it could be foreign), is as above (6% return, 17% SD). The two bond classes, assumed to be respectively domestic and foreign, have the same returns and SDs as those used before (4% return, 6% SD). The B1-B2 correlation is set arbitrarily at 0.8, based on the range of 1993-2013 correlations between different types of Canadian bonds and US bonds. We compare bond-stock mixes using one bond class only (B1 or B2) with mixes that integrate the two bond classes, equally weighted.



At first glance, the difference between the black curve (one type of bonds only) and the red curve (B1+B2) is modest, and non-existent above 30-40% stocks. Zooming-in on bond-heavy portfolios, there is a small reduction in SD at stock allocations ranging from 0% to 30%. For example, at 10% stocks (i.e. 90% bonds), the SD decreases by 0.3% (from 5.7% to 5.4%). Other types of developed market foreign bonds may have lower correlations with Canadian bonds, but we don’t have the data to test this. Readers will have to decide if such a small potential diversification benefit is worth it, considering hedging costs, etc. There are other reasons, such as avoiding single country risk, to consider foreign developed market bonds in bond-heavy portfolios (see Foreign bonds).

Adding gold
How about some shiny yellow metal in your portfolio? Gold has low correlations with everything, so it’s the perfect portfolio diversifier, right? Well, no so fast! Is the expected return any good? Let’s look at two scenarios, a pessimistic one and an optimistic one: (1) gold has an expected return in line with inflation, i.e. 2%; (2) gold has an expected nominal return of 5.4%, like in the 1993-2013 Canadian data. In both scenarios we assume a SD of 21% for G (the average of the 1970-2013 and 1993-2013 periods). We model portfolios with 0%, 5%, 10% and 25% G (25% being the proportion of gold in the Permanent Portfolio), and variable proportions of bonds (B) and stocks (S). We suppose that G has zero correlation with B and a -0.3 correlation with S. To make space for G in the B:S mix, we economize on the S.

If gold only increases in line with inflation (2%), then adding the classic 5-10% gold to your portfolio will do you no good (the 5% G line is not plotted to simplify the graph).



The results are not pretty. Recall that we wish to travel towards the upper left portion of the diagram, where the returns are high and the SDs low. Because gold has such a low expected return and a high SD in this simulation, it simply pulls the curves towards it, in the lower right portion of the diagram, exactly the opposite of what we wished to achieve. Oops…

Now if gold returns 5.4%, the story is completely different:



Suddenly, adding 5-10% gold to a bond-stock portfolio makes a lot of sense: the coloured (green and blue) curves are nicely shifted to the left. Even the 25% gold (red) curve is quite intriguing. In summary, negative correlations are not enough for gold to make a good portfolio diversifier: the expected returns also have to be high enough, otherwise gold is out. What does your crystal ball tell you about the future annualized returns of the shiny yellow metal over several decades?

Other scenarios
The reader now has to tools to investigate other scenarios such as adding Canadian REITs, adding RRBs, adding EM bonds, adding EM stocks, etc. to simple portfolios of Canadian nominal bonds and stocks, to see if adding these additional asset classes can indeed provide theoretical diversification benefits, and under what conditions.

Rebalancing
All simulations assume that the portfolio is rebalanced every year to its target asset allocation.