Rational Investing

TABLE 6.1 Historical performance of U.S. factors

These factors are constructed as discussed previously: by separating all stocks using percentiles of market capitalization and book-to-market and taking a long position in the portfolios with small stocks (high book-to-market) and short selling the portfolios with large stocks (low book-to-market).

The size and value factors provided an annualized average return (already in excess of the risk-free rate because these are long-short portfolios) of 3.50 percent and 4.95 percent with volatilities of 10.83 percent and 14.89 percent, yielding Sharpe ratios of 0.32 and 0.33, respectively. Performances were slightly less favorable over the shorter period from 1951 to 2015, with Sharpe ratios of 0.19 and 0.28, respectively.

We consider momentum as our fourth factor. Momentum is constructed by first sorting stocks on size and then on returns over the last twelve months, skipping the last month to avoid the return reversal effect, which results in six market capitalization–weighted portfolios (small and loser stocks, small and stable stocks, etc.). The last month is skipped because research has found that momentum strategies based on short horizons of one-week to one-month returns deliver contrarian profits over the next one week to one month. The factor is the average return of winner portfolios minus the average return of loser portfolios. Momentum is another empirically motivated factor that has had tremendous success, mainly because it offers superior risk-adjusted performance. During the years from 1951 to 2015, it has offered 9.01 percent annual excess return on average, higher than the market’s 7.12 percent, while having a lower annual volatility of 13.27 percent.

The final factor we consider is the low-minus-high beta factor. In a 2014 Journal of Financial Economics paper, Andrea Frazzini and Lasse Pedersen expand on the ideas in a publication from 1972 by Fischer Black (of the Black and Scholes option-pricing formula). They show that investors looking for higher returns but constrained by the amount of borrowing they can take will bid up the prices of risky high-beta stocks. In basic models in which investors are financially unconstrained, investors looking for higher returns can borrow to lever up the market portfolio. But in settings in which investors cannot borrow or can only borrow up to a certain limit, they are forced to buy riskier securities and shun safer ones. Their actions may make low-beta stocks more attractive in terms of risk-adjusted returns, and a factor that goes long low-beta stocks and that shorts high-beta stocks will offer positive risk-adjusted returns.

The low-beta factor has delivered a whopping 0.76 Sharpe ratio from 1931 to 2015 and 0.93 from 1951 to 2015, easily beating the performance of all other factors. We include the low-beta factor because it is a perfect example of what we should be looking for: a factor motivated by a simple, straightforward economic theory and supported by overwhelming empirical evidence. Frazzini and Pedersen illustrate the profitability of the low-minus-high beta in international equity markets, government and corporate bond markets, commodities, and currencies.

Of course, there are other factors supported by theory, for example, liquidity risk ⁵ and systematic return asymmetry.⁶ But as we discussed in chapter 5, many of these economically motivated factors are hard to measure and construct. Another reason why we focus on market, value, and momentum is that these factors are relatively easy to apply in different markets, a feature that will be useful later in this chapter when we examine multi-asset class portfolios.

All of these factors are also appealing because of the low correlations they have with the market index, as well as among themselves. Indeed, the cross-correlations reported in panels B and D for the two sample periods range from −0.33 to +0.33, meaning that they do not co-move much together. This is close to the Holy Grail in investing: assets that deliver positive risk-adjusted returns, and each tends to zig when others zag!

The only exception is the value-momentum pair whose low correlation is a mechanical result of their own construction. The value metric is constructed by dividing the accounting book value by the most recent market capitalization (e.g., dividing book value from December 2014 by the July 2015 market capitalization to obtain the August 2015 value ratio) instead of the market capitalization at the same point in time as the accounting information (e.g., dividing book value from December 2014 by the same month market capitalization to obtain the August 2015 value ratio).⁷ Hence a stock whose price has recently fallen will often end up in the long portion of the value factor and the short portfolio of the momentum factor, and a negative correlation will ensue.

This may seem too good to be true and to a certain extent it is. The cross-correlations in panels B and D are imperfect measures of dependence, meaning they hide extreme risks. These factors do not co-move much together in normal times, but they can be highly correlated during extreme periods,⁸ a feature that was highlighted during the so-called Quant meltdown in August 2007.

These factors by themselves may also present other forms of risk. The last row in panels A and C presents the sample skewness of monthly factor returns. Skewness is a measure of the asymmetry of returns; negative values indicate that an extreme negative return is more likely than an extreme positive return. Similarly, a positive skewness means that an asset is more likely to shoot up than to plummet. Table 6.1 shows that both the momentum and the low-beta factors exhibit strong and negative skewness. Therefore we can understand these factors’ high Sharpe ratios as a compensation for downside risk: they offer high returns on average because they sometimes crash dramatically, an aspect that investors tend to downplay when investing in normal times.

These factors are not always profitable. Historical performances are graphically presented in figure 6.1, which shows the rolling ten-year geometric returns (annualized) for each factor. Each point represents the average compounded return an investor would have experienced over the previous ten years for each factor. For example, in the mid-1970s, investors in the equity market would realize an overall return close to zero over the last ten years. Also, an investor who had invested in the stock market ten years before the 2008 financial crisis and who was forced to sell at that moment would realize a negative return. Clearly, there is no factor that dominates all the time. One notable fact: it appears that the market portfolio has dominated all other factors for the ten years leading to 2015, a situation not seen since the late 1950s.

This also explains why market-neutral products (those products that seek to eliminate market risk but remain exposed to other factors) have performed relatively poorly in recent years when compared to standard long-only balanced products. Furthermore, products that eliminate the market factor fall into the alternative industry segment, a sector known for its high fees. Hence from a fee point of view it is cheaper to invest in multifactor products that integrate the market factor than it is to invest in traditional products with an overlay of factor-based market-neutral strategies. Furthermore, because the correlation of the market factor to other factors is low, some portfolio construction efficiency is lost if all factors are not managed jointly within an integrated portfolio management process.

Figure 6.1 also serves as a warning against our natural tendency to favor strategies that have performed well recently while shunning ones that have not. It would have taken a considerable amount of confidence to tilt your portfolio toward value stocks in the early 1960s or in the late 1990s. But value has been profitable in many markets and over a long period of time. There will always be short-term differences in factor profitability. The least we can do is focus on factors for which we understand the economic motivation (we understand why somebody else is taking the other side of the trade) and that are supported by vast empirical evidence.

FIGURE 6.1 Ten-year rolling annualized geometric returns for U.S. equity factors

Now that we have established our investment universe of factors, we need to decide how to allocate among them.

Are Optimal Allocations Useful at All?

How do we allocate among assets? First, remember the importance of asset pricing theory in portfolio management. Investors form their portfolios according to their investment objectives and constraints, and prices form in financial markets. The result is a set of risk factors that offers positive average returns and possible mispricings to take advantage of.

We use the same asset allocation techniques to allocate among the risk factors that academics use to derive investors’ allocations and equilibrium prices. The only difference here is that we abstract from equilibrium consideration: we take the set of priced risk factors as given and try to combine them optimally.

Different investors have different preferences. A retail investor may want to accumulate wealth over a long horizon for his retirement and might prefer to hedge his labor income risk; a portfolio that provides a positive return and is liquid if the economy sours and he loses his job would be highly appealing to him. A pension fund will focus on maintaining its ability to fulfill pension promises made to retirees that are often linked to inflation. Mutual funds may focus on outperforming a benchmark index; hedge funds, their absolute performance. The latter two also care about outperforming their peers. In contrast, central banks may intervene in financial markets to influence interest rates and exchange rates, but care less about risk-adjusted performance.

All portfolio management techniques follow the same mold. They maximize a preference metric, subject to financial constraints and a model for how financial asset returns are generated. Portfolio management techniques range from the very simple to the very complicated, depending on the complexity of preferences, constraints, and asset return dynamics.

We begin with the simplest case, one that aims at maximizing the Sharpe ratio. We want to maximize the expected return of our portfolio (financial reward) while minimizing the amount of portfolio volatility (financial risk) with no consideration for anything else. While we use this as a starting point, this is not that far from what is done in reality. John Cochrane of the University of Chicago Booth School of Business notes, “Much of the money management industry amounts to selling one or another attempted solution to estimating and computing [the optimal Sharpe ratio portfolio], at fees commensurate with the challenge of the problem.”⁹

The ideal allocation for an investor who wants to optimize his portfolio’s Sharpe ratio and faces no constraints is well known and available in closed form.¹⁰ The optimal weight of an asset increases with its expected excess return, decreases with its volatility and cross-correlation to other assets. But this closed form solution is far from useful. In practice, investors face different allocation constraints, which means that the allocations will usually be obtained through a numerical optimization.

For example, imagine that you manage a U.S. equity fund and you rebalance your portfolio once a quarter by investing in the five U.S. equity factors. To capture the fact that volatilities and correlations in financial markets tend to change over time, you use a moving average of sample volatilities and correlations. At this point, we consider only one layer of sophistication: you use an exponentially weighted average to compute daily volatilities and correlations.¹¹ First, an exponentially weighted average places more weight on recent data, which is convenient because volatilities and correlations tend to be better forecasted by their recent levels than by longer averages. Second, you use daily returns even if your investment horizon is quarterly because having more returns for a given period gives you more precise risk estimates. This is in sharp contrast to estimating expected returns, in which having higher frequency returns does not give you more precise forecasts.¹²

Judging performances from historical simulations is always a difficult issue, and we need to work hard to make our experiment as realistic as possible. Every quarter, we use only the past histories of daily returns that would have been available at that point in time (for example, using daily returns up to June 30, 2015, to compute the allocation used on July 1, 2015). We also assume that factors are traded. But they involve thousands of stock positions. Of course, there is a high degree of netting out when they are combined (a stock’s short position in a factor may be canceled by its long position in another factor). We present here the results based on the factor returns, but we acknowledge that in reality an institutional investor would implement a sampling technique that would limit the number of stock positions, the number of transactions, or both.

We compare the performance of this fund with that of a 100 percent allocation to the market portfolio and limit the fund’s volatility to the benchmark’s level of volatility, which is around 15 percent over the whole period. For expected returns, we set all expected excess returns to the same level, removing the knowledge that one factor would perform better than another. This is equivalent to using a minimum variance portfolio, a point we will return to later.

The second column in table 6.2 shows the performance of the stock market portfolio as a benchmark, and the third column reports on our optimal allocation without use of leverage. The Sharpe ratio almost triples from 0.48 to 1.40. This is rather spectacular, especially given that the optimal weights reported in figure 6.2 do not vary that much. In fact, the optimal allocation resembles the equally weighted portfolio that we used as an example in chapter 5. This is not surprising, given a few factors: we have constrained the expected returns to be equal, the volatilities are not that different, and cross-correlations are low, in which case the two portfolios are close to each other.

TABLE 6.2 Performance of optimal portfolios of U.S. factors (1951–2015)

FIGURE 6.2 Allocation to U.S. equity risk factors in a Sharpe optimal portfolio

The Sharpe ratio is definitely impressive, but the average return is still slightly lower than the market portfolio. The increase in Sharpe ratio comes from a dramatic decrease in volatility due to diversification benefits. But what if you want more risk and more return? The fourth column shows the results by running the same portfolio with a limit of 200 percent on the portfolio weights. This allows a timid amount of leverage, especially given what sophisticated institutions can take on in real life.¹³ In this case, the portfolio achieves an average return of 12.30 percent versus 7.16 percent for the market portfolio, and a volatility of 8.72 percent versus 14.89 percent.

The efficiency of diversification across factors leads to portfolios with high Sharpe ratios but low volatility. This explains why managers of efficient factor-based products (such as equal risk premia products) will often use leverage. Their investors want a product with a high Sharpe ratio but still want the greater expected nominal returns that can only be obtained from leverage. However, it does not mean that significantly greater portfolio efficiency cannot be obtained without using leverage.

Table 6.2 also reports that the optimal portfolios have lower skewness than the market portfolio. This is bad; we want positive skewness, or at least not too negative skewness. How should we assess if going from −0.55 to −0.75 is acceptable? The answer is that we cannot. Our portfolio management technique is based on maximizing the Sharpe ratio as a preference metric, not on maximizing skewness. Achieving lower or even higher skewness with the optimal portfolio is done purely by chance. We therefore cannot use it as an assessment of our portfolio quality.

This is a subtle but important point. How we evaluate portfolios has to be in line with our preference metric. Here we aimed to maximize the Sharpe ratio, and this objective was achieved. If we also care about the asymmetry in returns, our preference metric should be different. Plenty of other ratios, such as the Sortino ratio, take into account asymmetry. In a similar vein, if an investor maximizes the Sortino ratio when deciding on his portfolio allocation, he should look at the realized Sortino ratio, not the Sharpe ratio or any other performance metrics, to decide whether he has been successful.

It is a crucial point to keep in mind when evaluating alternative portfolio solutions. For example, let’s say you find that an equally weighted allocation has realized a higher Sharpe ratio than the portfolio that tried to maximize it. You need to understand why it led to better performance or the results may have been obtained purely by chance and are not likely to be repeated. An equally weighted portfolio could be better, for instance, if expected returns, volatilities, and correlations are so hard to estimate that you are better off statistically to assume that they are all equal, in which case the equal weight allocation is the optimal Sharpe ratio allocation.

Even if there is netting out of long and short stock position when we allocate between the five factors, the optimal portfolio considered so far may involve some short selling of individual stocks. Further, the last line in table 6.2 reports the annualized tracking error, which is the volatility of the return differential between the portfolio and a benchmark, which in this case is the U.S. equity market index. A higher value implies that a large difference in return, positive or negative, is more likely to be experienced on any given month. With annualized tracking error of more than 12 percent, the optimal portfolio would undoubtedly behave very differently from the market portfolio.

Some institutional investors, even if they have the operational capabilities, might prefer not to adopt such a portfolio. A fund manager could be worried about career risk; if he experiences a string of unfavorable returns compared to his benchmark, it may be the end of his career, regardless of whether he truly held a portfolio that can deliver a 1.5 Sharpe ratio in the long term.

The last column of table 6.2 shows the performance of an optimal Sharpe ratio portfolio, constraining the tracking error to be no more than 5 percent annually. Hence we examine what happens if we impose a limit to the deviations against our benchmark index. Our Sharpe ratio is 0.66 versus 0.48 for the benchmark. To put this 0.18 increase in perspective, it means that at equal volatility we would gain 2.68 percent (0.18 × 14.89 percent) annualized return by shifting from the benchmark to the optimized portfolio.

Building on what we have just seen, the next section will examine portfolio solutions offered in the market that exploit the benefit of factor investing but invest solely in long positions and without the use of leverage.

A Closer Look at Alternative Indices and Smart Beta Products

In the asset management industry, considerable effort is put on marketing new funds, products, and portfolio solutions. This is not necessarily a bad thing. If you find a new portfolio management technique that actually results in sustainable performance (net of fees), you offer something valuable to investors. We examine in this section the relative merits of alternative indices and so-called smart beta products (or alternative beta, exotic beta, or whatever marketing term is in vogue at the moment). We consider in all cases the longest history of returns available.

The Fundamental Index

The first example of these investment solutions has been introduced by Rob Arnott and popularized by his firm Research Affiliates. Arnott, with his colleagues Jason Hsu and John West, calls this solution the Fundamental Index and explains more about it in The Fundamental Index: A Better Way to Invest. Instead of using market capitalization to weigh securities, their portfolio is built by using a moving average of several non-price-based variables: dividends, sales, revenues, and book value.

The objective is to obtain a diversification of mispricing risk. Let’s say you believe that large stocks in the capitalization-weighted portfolio are more likely to be overpriced and stocks that have smaller allocations are more likely to be underpriced. A traditional capitalization-weighted index will consequently overweight overpriced securities and underweight underpriced securities: exactly the opposite of what you want! Assuming that market prices eventually revert to their fundamental values—meaning the mispricing of a stock is temporary—incorrect weighting can cause a performance drag.

Ideally, a stock’s importance in the market should be based on its fundamental value. But we only observe their market values, which will differ depending on how mispriced the stock is. Therefore a stock’s importance in a market is not determined by its fundamental value but by its market price. Hence its weighting will depend on how mispriced it is relative to other stocks’ mispricings. For example, assuming the entire market is overvalued by 10 percent on average, if one security within the market was overvalued by 5 percent and another by 15 percent, the former would be in fact underweighted in a capitalization-weighted index while the latter would be overweighted. A stock’s overweighting in market capitalization index relative to its true but unknown fundamental weight is highly correlated to its mispricing relative to other mispricings.

It’s now time to face the elephant in the room: we cannot know the true fundamental values of each security. An extreme solution would be to use an equal allocation for each stock, which would naturally break the link between allocation and mispricing. But in most cases, an equal weight portfolio is not satisfactory, as it may lead to unacceptable sector allocations or positions in small and illiquid stocks.

Research Affiliates’s solution is to use other metrics (dividends, sales, revenues, and book value) that are related to a company’s economic footprint but not determined in a stock market. We stress this last part because these variables are determined in other markets. For example, sales and revenues are driven by a company’s ability to sell its product in the consumer market and book value depends on the acquisition costs of its inputs, all of which can be impacted by people overpaying and bidding up prices. Regardless, the ingenuity of the RAFI portfolio is that it gives an allocation to a company based on its economic size and is therefore more intuitive and acceptable than an equally weighted portfolio.

TABLE 6.3 Performance analysis of alternative equity indices

The second column in table 6.3 contains the historical performance of the FTSE RAFI U.S. 1000 Index, which covers 1962 to 2015. Panel A reports that the portfolio generated an average annual return over more than fifty years of 12.30 percent with a volatility of 15.17 percent, which corresponds to a Sharpe ratio of 0.50. This compares favorably to the market index, which has had a Sharpe ratio of 0.37 over the same period.¹⁴

In panels B through D, we conduct factor analyses of the RAFI Index. We are using our return equation from chapter 2, putting RAFI’s excess returns on the left side, our factor returns on the right side, and estimating RAFI’s exposures to each factor using a regression analysis. Another way of looking at the results is to note that the combination of the factors on the right side weighted by their estimated exposures produces a portfolio that replicates the product on the left side. Viewed from this replicating portfolio perspective, α can be understood as the part of the returns that cannot be replicated by the factor portfolio. For example, a value of 0.5 percent means that we may replicate the RAFI’s variations in returns, but we are missing the 0.5 percent constant returns to generate the same level of average returns that the product has. In contrast, a value of −0.5 percent means that the replicating factor portfolio outperforms it by an average of 0.5 percent.

We start in panel B with a simple analysis, using only the market portfolio as a factor. RAFI’s market exposure of 0.94 indicates that the index is slightly less exposed to broad market movements than the market itself and that the best replicating portfolio consists of putting 94 percent in the market factor and 6 percent in the risk-free rate. More interesting is the estimated annualized alpha of 2.14 percent. The two stars indicate that this estimate is strongly statistically significant. So when compared only to the market portfolio, RAFI seems to generate a significant amount of value added.

Panel C reports the analysis results of RAFI Index versus the market, size, value, and momentum factors. Panel D then augments the set of factors with the low-beta strategy. In both cases, the RAFI Index is strongly exposed to the market portfolio and the value factor. It is also exposed to the low-beta factor. The negative exposure to the size factor should not cast any shadows on RAFI’s performance; recall that the size factor is built using all U.S. stocks whereas the RAFI U.S. 1000 uses only the largest one thousand U.S. securities. It is therefore tilted by construction to large-capitalization stocks, resulting in the negative exposure to the small-minus-big factor.

But most important is the small and insignificant α estimates (0.08 percent and −0.23 percent). We can therefore understand the Fundamental Index as a bundle of the market portfolio, the value, and the low-beta factors. The Fundamental Index concept has been criticized as being just a value tilt in disguise, and the factor analysis seems to confirm this point. RAFI’s argument stands on the premise that the economic variables used are better measures of the importance of a company than market prices. AQR co-founder Cliff Asness demonstrated that weighting by book value, for instance, is equivalent to tilting market capitalization weights using a company’s book-to-market ratio relative to the aggregate book-to-market ratio.¹⁵ He noted, “It’s a neat way to explain value to a layman.”

To be fair, we cheat a little when interpreting the factor analysis results. We assume we could have known in 1962 that the market portfolio plus an exposure to the value and low-beta factors would give good results over the next fifty years. But the diversification of mispricing risk underlying the Fundamental Indexing approach was not a known concept either in 1962. On the other hand, we are not claiming that RAFI has no value; the point here is not that the Fundamental Index is of no use. On the contrary: it has delivered overperformance with respect to the market portfolio over a long period of time, has a low turnover, is sold at a low fee, and its construction is transparent to investors. Rather, the point is to show that it is a bundle of well-known and thoroughly studied factors and does not deliver added performance beyond its exposure to these factors. It should be understood as a smart, cost-effective packaging of factors, not as a new finance paradigm. Also, as the portfolio simulation that will be completed in a later section shows, we can achieve more balanced factor exposures.

The S&P 500 Equal Weight Index

As stated in the previous section, an equally weighted portfolio is one way to remove the potential bias toward overpriced securities in a market capitalization–weighted index. But there are other reasons why one could be interested in an equally weighted product. In a 2009 paper in the Review of Financial Studies, Victor DeMiguel, Lorenzo Garlappi, and Raman Uppal show that using a mean-variance optimal portfolio, as promoted by modern portfolio theory, fails to beat a simple equally weighted portfolio in several investment universes. This noteworthy paper organizes the agenda of future research; in many ways it is similar to the paper discussed in chapter 4 that showed that most predictors could not better forecast stock market returns than a sample average. Since the paper’s publication, other researchers have shown that optimal portfolios do perform better when, for instance, estimation risk is diversified by combining an optimal portfolio with the minimum variance portfolio and the equally weighted portfolio.¹⁶ In fact, the equally weighted portfolio can be seen as an extreme case of statistical robustness: we assume there is so much statistical noise that we implicitly set all expected returns, volatilities, and correlations respectively to the same value.

Let’s consider the historical performance of the S&P 500 Equal Weight Index. Over the period from 1990 to mid-2015, as reported in the third column of table 6.3, the equally weighted index had delivered an average annual return of 12.05 percent with a Sharpe ratio of 0.56, beating its market-capitalization index of all U.S. stocks with a Sharpe ratio of 0.49 for the same period. As shown by the regression results in panels B through D, the equally weighted index has a significant exposure to the market, value, and low-beta factors. In all cases, however, it fails to generate significant value added. Note that the factor exposures cannot be compared across products as they have different return histories.

The Maximum Diversification Portfolio

The firm TOBAM (Thinking Outside the Box Asset Management) uses an intuitive concept of maximum diversification, as epitomized in a 2008 paper in the Journal of Portfolio Management¹⁷ by Yves Choueifaty and Yves Coignard. By definition, the optimal Sharpe ratio portfolio considers both expected returns and portfolio volatility risk. Unfortunately, more often than not, these portfolios perform poorly in practice. Much of the blame falls on the difficulty in forecasting expected returns. The TOBAM portfolio instead focuses on the diversification of unpriced risk source of performance. It maximizes the ratio of the weighted average volatility divided by portfolio volatility. For example, if you combine assets that all have a volatility of 20 percent and you achieve a portfolio volatility of 10 percent because low cross-correlations lead to diversification benefits, the ratio is 2. If you only achieve a portfolio volatility of 20 percent (which happens if all assets are perfectly correlated), your ratio is 1. The first portfolio offers better diversification than the second.

From 1979 to 2015, the maximum diversification portfolio generated an average return of 13.27 percent and volatility of 15.17 percent per year. Its Sharpe ratio of 0.57 compares favorably to the market portfolio’s ratio of 0.48 for the same period. The portfolio delivers strongly significant exposures to the market, size, value, and low-beta factors. Even though its α is not significant in all regressions, it offers a well-balanced set of risk factor exposure, albeit without momentum.

Thinking about the maximum diversification portfolio brings us back to the need to have consistent preference and portfolio evaluation metrics. If the maximum diversification portfolio generates a higher Sharpe ratio than the optimal Sharpe ratio portfolio, it is because its construction relies on better implicit forecasts of expected returns, volatilities, and/or correlations and on better use of these forecasts. The maximum diversification portfolio is equivalent to the maximum Sharpe ratio portfolio if we assume that asset expected excess returns are proportional to their volatilities.¹⁸ Therefore for a given set of expected return, volatility, and correlation forecasts, the maximum diversification portfolio will lead to a better Sharpe ratio, if imposing this restriction leads to more robust forecasts of expected returns.

Current empirical evidence on whether expected excess stock returns are proportional to their volatility remains elusive. It is possible that this relation is too small to discern, but we cannot conclude for the moment that this is a better restriction to use than others. For example, the minimum variance portfolio assumes that all expected excess returns are equal across assets. Whether the maximum diversification portfolio is a better alternative portfolio solution than the minimum variance portfolio depends on which expected return restriction is better. We do not yet have a definitive answer to this question. Nonetheless, the maximum diversification portfolio does deliver an interesting balance of risk factor exposure.

A Full Portfolio

We now consider a portfolio with multiple asset classes. We limit ourselves to publicly traded asset classes: U.S. and international equities, government bonds, commodities, and currencies.

We use the market, size, value, momentum, and low-beta factors for three different regions of developed country equity markets: Europe, North America, and the Pacific.¹⁹ We look again to Fama and French, who recently demonstrated²⁰ the importance of the size, value, and momentum factors in international developed equity markets. Table 6.4 reports Sharpe ratios ranging from 0.13 to 0.89 for the value factors. The Sharpe ratios for the size factors in North America and the Pacific regions are 0.16 and 0.13, respectively, while size has been detrimental to investors’ wealth in Europe with a negative ratio of −0.22. Note that the performances reported for North America are different from the ones in table 6.1. First, we consider Canada as well as the United States. Second and most importantly, the period studied is different. This is a subtle reminder of the lesson from figure 6.1: risk factors are not always profitable. However, momentum and equity low-beta factors are profitable across all regions with Sharpe ratios ranging from 0.35 to 0.86.

Consider risk factors in the government debt market. We look only at U.S. Treasury bonds in this example, but this analysis could be extended to include other sovereigns and corporate bonds. We use Treasury bonds with maturities from one year to thirty years to build two risk factors: a market capitalization–weighted market portfolio of all bonds and a term risk factor constructed by buying a ten-year bond and short selling a one-year bond.²¹ The second factor captures the slope of the term structure of interest rate. We consider it a risk factor because it has been shown to be a reliable predictor of the state of the economy.²² Both government factors generated appealing Sharpe ratios over the period from 1989 to 2015 (0.77 and 0.44, respectively).

We use momentum and carry factors in the commodity and currency markets. Commodity returns are obtained by investing in twenty-four liquid futures contracts that are the closest to maturity.²³ Currency returns are obtained by trading spot exchange rates and one-month forward contracts for sixteen developed countries.²⁴ To construct these factors, we use the rank-based approach. First, we rank each month’s commodities or currencies based on the previous year’s return. The weight of a commodity (currency) in its momentum factor is its rank minus the median rank, which is then scaled such that the portfolio is one dollar long and one dollar short. Momentum in the commodity market has been profitable between 1989 and 2015, with a Sharpe ratio of 0.50, whereas currency momentum has been much less interesting.

Finally, we consider factors similar to the term factor in the bond market; the basis factor for commodities and the carry factor for currencies. Basis is defined as the negative of the slope of the term structure of futures contracts.²⁵ The carry factor in currencies consists of buying currencies of countries with high interest rates and shorting currencies of countries with low interest rates.²⁶ These term structure factors have been shown to be profitable in the commodity market²⁷ and the currency market.²⁸ Indeed, their respective Sharpe ratios are 0.50 and 0.61, respectively.

We form minimum variance portfolios just as we did for our previous portfolio composed of U.S. equity factors. Looking at table 6.5, we compare its performance to the global equity market (the market capitalization portfolio of all developed country equity markets). In this case, the Sharpe ratio almost quintuples, going from 0.32 to 1.52. The leveraged portfolio, which allows factor weights to sum to 200 percent, results in an average annual return of 11.05 percent with a volatility of 7.07 percent (see note 13).

The dynamic allocations reported in figure 6.3 show that the portfolio is again close to an equally weighted portfolio of all factors (as they become available through time). Why is this simplistic portfolio so profitable? Of course, one should again be cautious when interpreting results from historical simulations. But here we used only liquid stocks in developed markets, U.S. government bonds, and liquid commodity and currency futures. Hence this portfolio is implementable by an institutional investor and its efficiency would not be significantly hampered by a need for portfolio sampling (in order to limit the number of positions) if it were needed.

TABLE 6.4 Historical performance of risk factors in multiple asset classes (1989–2015)

TABLE 6.5 Performance of multi-asset class optimal portfolios of factors (1989–2015)

The answer lies in the diversification potential obtained by investing across risk factors and across asset classes. We often hear that diversification does not work; it’s argued that diversifying by investing in different asset classes fails because these asset classes jointly generate poor performance during crisis periods exactly when their diversification properties are most needed.

But this reasoning is flawed. Asset pricing models tell us that we should diversify across priced risk factors, not across asset classes. Diversification across risk factors does work.²⁹ Table 6.4 reports that the average correlation of each factor with all other factors is close to zero in all cases. Figure 6.4 reports the three-year realized geometric return of the optimal portfolio without leverage and the global equity benchmark portfolio. Consider in particular the financial crisis of 2008. Though performance suffers, the optimal portfolio offers less volatile performances and remains in positive territory. Factor diversification really works.

FIGURE 6.3 Allocation to risk factors in a Sharpe optimal multi-asset class portfolio

FIGURE 6.4 Three-year annualized geometric return (1989–2015)

Portfolios for a Constrained Investor

Outperformance is relative. For instance, it is easy to say that a fund manager’s or product’s performance can be explained by momentum. But for many retail investors and smaller institutions, momentum is hard to integrate into a portfolio. It involves frequently trading a large number of stocks or applying sophisticated sampling techniques to limit the number of stocks traded.

In these cases, a fund (mutual fund, hedge fund, or exchange-traded fund) that provides an exposure to factors may be attractive to some investors even if it has a negative α. This is not controversial. If you are a retail investor who faces high transaction costs and you try to implement a momentum strategy, you will obtain its performance minus the cost that you incur, say 1.0 percent (which includes not just transaction costs, but also other costs such as the fact that you are not working on something else while you are trading). In this instance, a fund that gives you its performance but charges management fees of 0.50 percent is a pretty sweet deal.

With that in mind, we’ll now examine the performance of more constrained portfolios and consider the ability to obtain factor exposures through the exposures to traditional indices. We select indices that have liquid exchange-traded funds available. We use indices in historical simulations assuming that before these exchange traded funds were available the same exposure could have been obtained by investing in index funds.

We use indices to capture the market, size, and value factors in equity markets for three different regions: the United States, EAFE (Europe, Australasia, and the Far East), and emerging markets. We focus on market, size, and value because they’ve been around for a long time. There are indices that offer either an exposure to a market capitalization–weighted index or portfolio tilts related to size and value and funds that track them. We thus take into account whether an investor would have been able to invest in these indices at each point in time during our historical simulation (even now, relatively few ETFs offer pure exposure to momentum in the U.S. market, let alone in international markets).

We use a broad U.S. Treasury market index to get exposure to the government bond market factor and a U.S. long-term Treasury Index to obtain the tilt implied by the term slope factor. In commodities, we use only the S&P Goldman Sachs Commodity Index. The momentum and basis factors in the commodity market, as well as the carry trade factor in the currency market, are now available through ETFs, but given the short time series, we leave them aside.

Table 6.6 presents the starting date of each index (which varies across the indices), the ticker of one ETF that tracks this index as well as its current management fee, and the summary statistics of historical performance. The ETF management fees vary from as low as 0.10 percent per year (U.S. long-term Treasury Index) to as high as 0.75 percent (the commodity index). In the following sections, we present performances without incorporating management fees. This is because we want our results to be comparable to the ones in the previous sections for which we did not incorporate transaction and management costs. We also want our results to be indicative of potential performance, regardless of how these indices are traded. Our results can serve as benchmarks for retail investors who might buy these ETFs and institutions that might be able to invest in these indices at lower costs.

All Sharpe ratios are economically sizable, and all equity indices present negative historical skewness, whereas bond indices exhibit positive skewness. We now turn to long-only unlevered optimal allocations.

Until now, we have set all expected returns as equal, which is equivalent to using the minimum variance portfolio. We use this example to introduce one more layer of sophistication: using expected return predictions to find our optimal Sharpe ratio allocation. As in chapter 5, we set expected excess returns to the yield (computed over the last twelve months) without any further adjustment, which is a simplified version of a predictive model put forward by Miguel Ferreira and Pedro Santa-Clara from the Universidade Nova de Lisboa.³⁰ But even if this model looks oversimplified, it has had positive realized predictive power for all assets, considered over their respective histories.³¹

On one hand, we now face a common problem that plagues portfolio management: expected returns are notoriously hard to predict and portfolio weights are highly sensitive to their value. Small changes in these inputs can change allocation dramatically, and estimation errors make uncontrolled allocation perform poorly in real life. On the other hand, even if we have a perfect prediction model—meaning that we forecast expected returns, volatilities, and correlations with no estimation and model error—we may still be uncomfortable with the suggested allocation that maximizes the Sharpe ratio. Would you be comfortable with a 100 percent allocation to bonds? How about a 100 percent allocation to stocks? Of course, this is a consequence of choosing the Sharpe ratio as a preference metric, which does not accommodate our degree of comfort with the magnitudes of the individual allocations.

TABLE 6.6 Historical performance of indices in multiple asset classes

In reality, we generally impose constraints on portfolio weights. We can therefore define our preference as maximizing the Sharpe ratio with an allocation we are comfortable with. What kind of portfolio constraints should we impose? It turns out that there is a strong relation between the kinds of constraints that investors would like to impose to be comfortable with a portfolio allocation and the kinds of constraints that lead to more robust portfolio performance from a purely statistical point of view (i.e., the ones that reduce the impact of estimation errors).

Let’s consider a few examples. Few of us have the ability or the willingness to keep short positions in our personal portfolio. As it happens, imposing positive weights as a constraint when obtaining our optimal portfolio is equivalent to controlling for estimation errors in the covariance matrix of returns, which leads to more robust realized performances.³² Alternatively, we could impose a limit on the concentration in our portfolio, measuring the concentration as the sum of squared portfolio weights. For example, the equally weighted portfolio has the lowest amount of concentration (1/N where N is the number of assets in the portfolio) and a portfolio fully invested in one asset has a concentration of 1 or 100 percent. In one of their many insightful contributions to the portfolio management literature, Victor DeMiguel, Lorenzo Garlappi, Francisco Nogales, and Raman Uppal show that imposing such a constraint is equivalent to shrinking the covariance matrix, which reduces the most extreme covariance estimates.³³ Shrinking covariance matrices toward a target matrix was popularized by Olivier Ledoit and Michael Wolf,³⁴ who now offer their services on covariance matrix estimation through their firm Studdridge International.

What kind of limit should we use? Even if the parameter can be calibrated according to different objectives, we instead use a simple and intuitive way to impose a global constraint. As stated previously, the concentration of an equally weighted portfolio is 1/N. Therefore we can impose that our portfolio is at least as diversified as an equally weighted portfolio of, say, five assets by constraining its concentration to be lower than 1/5.

We could instead impose lower and upper allocation limits to each of the assets, for example, 0 percent to 40 percent for the U.S. equity market, 0 percent to 15 percent for the commodity index, and so on. This is subject to debate. While these individual limits can be imposed, we rely on a more holistic approach by imposing a limit on concentration of our portfolio. Next, we’ll examine realized portfolio performances.

The Value of Expected Return Predictions

Table 6.7 presents the results from three different portfolios. First, we use the MSCI All Country World Index as a benchmark. Without any particular views or expertise, an investor can at least invest in the global market capitalization–weighted portfolio, which is available at a low cost. For example, the ETF SPDR MSCI ACWI IMI has a gross expense ratio of 0.25 percent.

Second, we present the performance of the optimal Sharpe ratio portfolio that uses average excess return to predict future average excess returns. This is the most naïve prediction you can make, which makes it a good benchmark to see the value added by expected return predictions. Every quarter, we use the average excess return of each asset up to the previous quarter as a prediction for the next quarter’s expected excess return. As we progress through time, we have more and more returns from which we can estimate average returns of each asset. However, by taking such a long view, we omit information that may be useful about current market conditions. Finally, we report the performance of our optimal Sharpe ratio portfolio, using yields as expected excess return predictions.

The last portfolio has realized a Sharpe ratio of 0.73 for the period of January 1985 to June 2015, which compares favorably to the 0.54 for the sample-average-based portfolio and the 0.39 for the MSCI ACWI benchmark. At equal levels of volatility, the value of using our simple expected return prediction technique is 2.28 percent per year ((0.73 − 0.54) × 12.01 percent), which is substantial.

Perhaps surprisingly, the value added is reflected in lower realized volatility (8.48 percent versus 12.01 percent), not higher average excess return (6.22 percent versus 6.49 percent). Empirically, an asset with a low payout ratio (low cash payout or high price) tends to experience lower returns going forward, and an asset that falls usually experiences an increase in volatility. An allocation based on the current payout yield would advantageously allocate less to this asset than a model based only on the current level of volatility. After the asset has fallen, it will have a high payout ratio (high cash payout compared to low prices) and high volatility. Hence an allocation using this information will advantageously allocate more to this asset than an allocation based solely on the current high volatility level. This issue is also addressed by having a risk model that incorporates mean reversion, a topic we will discuss in the next section.

TABLE 6.7 Performance of multi-asset class optimal portfolios of indices (1985–2015)

Globally, these effects lead to more stable portfolio allocations than a model based only on risk measures or relying only on historical average returns. Figure 6.5 presents dynamic portfolio allocations across time, which vary depending on which assets are available at each point in time and their relative volatility and correlations, as well as their payout ratio.

FIGURE 6.5 Optimal allocation to equity, bond, and commodity indices (1985–2015)

Is this really a good deal for a retail investor? Absolutely. The ETF fees paid, weighted by the average allocation, are 0.13 percent per year, compared to the 0.25 percent expense ratio on the MSCI ACWI ETF. This means that the increase in performance remains even when taking the ETF management fee into account. This conclusion remains even when considering another ETF as a benchmark. For example, the Vanguard Total Stock Market ETF, which tracks all U.S. stocks exactly as our U.S. market factor at the beginning of this chapter, charges 0.05 percent per year. This 0.08 percent return advantage does not adversely affect the sizable increase in risk-adjusted performance. Having established the value of using our expected excess return predictions, we now turn to the economic value of risk models.

The Value of Risk Model Sophistication

Do we really need a more sophisticated risk model than the simple one we have been using so far? To see why the answer is a resounding yes, let’s examine the most ubiquitous patterns of risk in financial markets.

Volatilities and correlations vary greatly through time, and their variations are persistent; this means that when volatility is high it tends to stay high some time, and vice versa. Our exponentially weighted average, by putting more weight on recent returns, satisfactorily captures this fact. So far, so good.

However, a simple methodology based on an exponential weighted average fails in one regard: its volatility and correlation predictions are constant, regardless of the forecasting horizon. If the market has been particularly turbulent recently, I forecast it will remain volatile over the next day, month, quarter, or even year. In reality, volatilities and correlations mean revert, meaning that following episodes of high values, they tend to slowly come back down to average levels.

This mean reversion is important to capture. Let’s say equity market volatility has been especially high recently, for instance, 40 percent on an annual basis. When deciding on your allocation to equity for the next year, the allocation based on a 40 percent volatility forecast would be too low: you would tend to keep a defensive position for too long. If you rightfully think that volatility is expected to slowly come back down to an average level of around 15 percent, your forecast for the next year might be something in the middle, say 20 percent or 30 percent, depending on the speed of mean reversion. This more precise forecast avoids keeping a defensive position for too long. The opposite is also true: if the market has recently been unusually calm, you would take a too aggressive position and could suffer as volatility increases back to its average level.

Using models that capture this mean reversion brings us to a crucial issue in portfolio management. Simply put: our expected return and risk predictions should be in line with our rebalancing frequency. It makes little sense to determine our rebalancing frequency separately from our allocation policy. Remember that all portfolio management techniques fall in the same mold: they optimize a preference metric subject to the investor’s financial constraints and the statistical behavior of asset prices. The issue of a rebalancing policy concerns all three elements: it can be influenced by our preference metric (we can have a long investment horizon or we can simply be averse to high portfolio turnover), our financial constraints (such as high transaction costs, which make frequent rebalancing unprofitable), and returns’ behavior (no momentum makes frequent rebalancing more profitable than if returns exhibit momentum). Therefore factors that determine the rebalancing policy should be included in the portfolio optimization problem from the start, not examined afterward. For instance, in this chapter we assume that we rebalance every quarter,³⁵ either because of our preference for lower turnover or due to financial constraints. Therefore we need to project volatilities and correlations’ expected mean reversion over the next quarter. The conclusion is quite simple: an investor who is going to trade every day should not have the same allocation as an investor who is going to rebalance every year.

Another stylized pattern in financial markets is that risk is not captured only by volatilities and correlations. Volatility and correlation perfectly describe risk if returns follow a normal distribution, which is a function that describes how probable different returns are. Normal distributions are common in finance. Historically, they were used because they are easy to handle analytically (meaning that many formulas can be derived). Under some assumptions, the central limit theorem states that many probability distributions will converge to normal distributions as random shocks are aggregated. But overwhelming empirical evidence shows that returns are not normally distributed. This is not a failure of theory: the normal distribution is simply not as realistic as we could hope.

Financial econometricians work with more flexible distributions that capture asymmetries in returns (that is, skewness) and accommodate for the fact that extreme events happen more often than implied by a normal distribution. Most asset returns display more downside risk than upside potential, which is not entirely captured with volatility. Similarly, asset returns usually exhibit more joint downside risk, meaning that they plummet at the same time during a crisis period, than joint upside potential, which cannot be fully captured by correlations. Having a model with time-varying volatilities and correlations does not change these facts.³⁶

While incorporating these asymmetries is done in advanced risk models, predicting time variations in return asymmetries remains difficult. This is not surprising from an economic point of view. Volatilities concern the magnitude of returns, whereas skewness is about the magnitude and the sign of returns. Just as forecasting the expected return is hard, forecasting skewness is even trickier. Competition among investors makes it hard to forecast the direction of returns, and the same argument applies to the problem of forecasting asymmetry risk. Short of predicting when assets are going to have more or less downside risk, the current compromise is to attempt to capture which type of assets tend to have more downside risk than others.

Peter Christoffersen of the University of Toronto and one of us (Langlois) compute the value added by a more realistic risk model for a fund that allocates across the U.S. equity market, size, value, and momentum factors.³⁷ This experiment is similar to those at the beginning of this chapter, but without the low-beta factor. The major difference is that they consider investors with different preference metrics than only the Sharpe ratio. The added value can reach up to 1.17 percent per year simply by switching from a basic risk model to one that is more realistic. In the highly competitive asset management industry, such an advantage can be significant. Of course, estimating these risk models requires more expertise, but using them does lead to higher risk-adjusted performance. This is good news for financial econometricians and risk managers!

These results are impressive because they are based on preferences that give little weight to downside risk. Hence the value added would be even higher for an investor who cares about the risk of large but rare negative returns or who has a value-at-risk constraint. Even when trying to maximize the Sharpe ratio, a more realistic risk model still has value because it leads to more precise volatility and correlation estimates.

This is only the starting point. Including more information into optimal portfolio allocations is a growing and promising stream of research. The risk models discussed so far are all return based, in the sense that the risk predictions are all derived using only past returns. State-of-the-art risk models also incorporate other variables such as valuation ratios and macroeconomic conditions.

Another promising and useful set of portfolio management techniques predicts portfolio weights instead of expected returns and risks. The standard approach is to separately determine which model and variables can best predict expected returns and risk measures. But the typical investor doesn’t actually care about predicting each of these values. Rather, he cares about using information to predict the portfolio weights that are going to perform best. In a series of papers written with co-authors, Michael Brandt of the Fuqua School of Business at Duke University shows how directly modeling portfolio weights as a function of firm characteristics and economic variables can side step many of the estimation problems that afflict portfolio management.³⁸ For example, a Journal of Finance paper co-written with Yacine Ait-Sahalia of Princeton University³⁹ shows that the most important variables for an investor depend on his investment horizon. Momentum is important for a short-term investor, dividend yield is key for a long-term investor, and all investors, regardless of preferences, should pay attention to the slope of the term structure of interest rates.

Again, all of the previous results are only the starting point.

Other Concerns?

In this chapter, we’ve mainly focused on Sharpe ratio–maximizing investors with a quarterly investment horizon. But there are many other concerns that can affect optimal portfolio allocations.

Let’s consider liquidity. Investors can rebalance every year (in which case they can invest more significantly in illiquid assets) or they can have short horizon (in which case liquidity and market structure matter a lot). We use middle-of-the-road examples in this chapter: investors who rebalance every quarter in all traded assets but the highly illiquid. We have not touched on allocation techniques for investors who invest in both liquid and illiquid assets.⁴⁰ For an excellent treatment on the role of liquidity in asset management, see Andrew Ang’s Asset Management.⁴¹

Another issue in asset management is whether one should hedge currency risk. A position in an international asset can be seen as a long position in the asset denominated in its local currency and a long position in the foreign currency. Nothing fundamentally changes from what we have seen so far. The portfolio optimal allocation can be derived using all assets denominated in the investor’s home currency. Outright positions in foreign currencies can be used to improve the preference metric of the portfolio, just as any other asset can be added to achieve such improvement.

For example, an allocation to international equities held by a Canadian investor will be different from the allocation of a U.S. investor. This difference is driven by the unique behavior of each currency. The Canadian dollar is pro-cyclical, meaning that it tends to increase when the world equity market increases and decrease when the world market falls. Hence hedging the implied currency exposure is not necessarily appealing because currency fluctuations act as a hedge for the international equity exposure of a Canadian investor. When the world market portfolio falls in value, the price of foreign assets in Canadian dollars increases (i.e., the Canadian dollar depreciates), which counteracts the fall in equity value. Hedging can increase total risk from the point of view of a Canadian investor.

In contrast, the U.S. dollar is a counter-cyclical currency. It tends to act as a safe haven by gaining in value when the world market falls. The losses on international investments are amplified for a U.S. investor because the currency in which the foreign assets are denominated fall in terms of U.S. dollars as the same time that the asset values fall. Therefore a U.S.-based investor would have a greater preference for hedging his foreign currency exposure to reduce risk.

One should realize that beyond the obvious exposure to the foreign currency in which the foreign asset is denominated, we are also exposed to other currencies through the nature of the asset (for example, a firm’s sales in other countries). Precisely understanding our foreign currency exposures is a daunting task, and it is therefore not possible to measure all currency exposures. The big decision is whether you take positions in foreign currencies in general. If you do, the correlations with other assets in your investment universe will determine your currency hedging needs.

To illustrate this point, consider a Canadian investor. Table 6.8 reproduces the experiment of table 6.7, in which we examined the value added by using yield-based expected return predictions on optimal portfolios. The difference here is that we compute the optimal allocation using asset returns denominated in Canadian dollars. We also make two changes to the investment universe. First, we add the MSCI Canada Index, the MSCI Small Cap Index, and the MSCI Value Canada Index to capture the fact that Canadian investors may have a home bias. Second, we change the two government bond indices to Canadian government bond indices.⁴²

The conclusion reached in the previous section remains true. Our optimal CAD-denominated portfolio beats the one using sample average returns and the Canadian equity market used as a reference.

In figure 6.6, we report the portfolio betas (i.e., the risk exposures) of three different currencies,⁴³ computed using our optimal portfolio (the last column in tables 6.7 and 6.8). For the U.S.-based portfolio we use the Canadian dollar, the Euro (constructed using European currencies before its introduction), and the Japanese Yen. For the Canadian-based portfolio, the Canadian dollar is replaced with the U.S. dollar.

TABLE 6.8 Performance of multi-asset class optimal portfolios of indices in Canadian dollars (1987–2015)

FIGURE 6.6 Beta of major currencies with U.S. and Canadian portfolios (1992–2015)

Currency betas are largely positive for the U.S. portfolio, which implies that these currencies should be short sold to minimize the portfolio risk. This is in line with the intuition given earlier. The portfolio holds foreign assets and is by construction exposed to foreign currencies. By shorting these currencies, we reduce (or hedge) the exposure.

These betas are in contrast with the betas found in the bottom graph for the Canadian-based portfolio in which the majority is negative. Hence to minimize the volatility of the portfolio, these currencies should be bought. We should increase the currency positions already taken by investing in foreign assets. This would actually be quite expensive, as all three currencies have depreciated against the Canadian dollar over this period, but it would have dampened volatility.

Inflation could be another concern. A pension fund might have payments that are indexed to inflation promised to its retirees, while an individual investor saving for retirement will care about protecting his purchasing power.

Unfortunately, inflation hedging is elusive. Correlations between inflation and our predictions for good inflation hedges are not high. Gold is a poor inflation hedge,⁴⁴ the evidence for real estate and commodities is mixed, and even inflation-protected bonds, such as Treasury Inflation-Protected Securities in the United States or Real Return Bonds in Canada, are not that effective due to their illiquidity and the volatility of the real rate of return, and they may be too expensive for their hedging ability. Short-term government bills, on the other hand, seem to be good inflation hedges, but the expected return is also quite low.

Just as we added constraints on portfolio weights, we can adjust our portfolio management approach in different ways to reflect inflation concerns. First, we can, of course, ensure that we have inflation-hedging assets in our universe of possible assets, just as we added currencies in the case of currency hedging. Next, we can integrate our inflation concern into our portfolio management technique either by adjusting our preference metric or by imposing a new constraint. The former is not obvious; we would need to combine our basic preference metric (e.g., Sharpe ratio) with the notion of inflation-adjusted returns. Alternatively, one can impose a constraint such that our portfolio should have a positive correlation with inflation, meaning that certain assets with inflation-hedging abilities will be favored in the optimal allocation. Still, incorporating inflation concerns remains a challenge.

A Note on the Use of Derivative Products in Portfolio Management

A derivative product is a financial contract whose value is derived from the value of another value, such as a stock or bond price, the weather, or the value of a commodity. The distinction between stocks, bonds, and derivatives is elusive. After all, a stock is also a contract whose value is derived from the income-producing capacity of a firm’s assets (i.e., real assets); this capacity depends on the state of the economy and other variables. And so on.

In this chapter, we have focused on portfolios composed of basic assets (stocks, bonds, and futures). Nothing prevents us from using other derivatives in building portfolios. Nothing fundamentally changes either.

The point to understand is that once the payoff of the portfolio has been chosen by the investor, there are many ways of engineering this payoff. Some may involve only stocks and bonds and some may involve other derivatives.

For example, consider two institutions investing in a U.S. stock with the same objective in terms of payoff: one linked to the performance of this company. One institution may have a better knowledge of how to value and trade options, and it can use this expertise to make a better choice between buying the stock or an option on the stock. If it is right, it achieves the desired payoff at lower cost (i.e., a higher Sharpe ratio).

Now imagine you want to obtain the return on the S&P 500 Index over the next quarter, but do not want to lose more than 3 percent. You can buy a put option on the index and invest the rest of your wealth in the index. Alternatively, you can solve the dynamic portfolio management problem that will generate this payoff. This is much more complicated than the one-period Sharpe ratio–maximizing portfolio we have considered in this chapter. You need to model how the index varies each minute, hour, or day and compute the optimal position in the stock market to hold at each point in time to generate the desired payoff at the end of the quarter. If you succeed, you will find that the optimal portfolio looks like the replicating portfolio of a long position in the market plus the same put option. Indeed, valuing a put option or finding the dynamic portfolio that generates the desired protection is equivalent. In fact, there is very little difference between the techniques used in the literature on dynamic portfolio management and on option pricing. For a retail investor who may also have other preoccupations, buying a put option is certainly cheaper than creating the put option through frequent dynamic trading.

Still, considering options does not alter our discussion of portfolio management set so far in this book. Appropriately considering our three sources of sustainable performance can be achieved by incorporating options in a portfolio. Some risk factors may only be extracted by trading options, but we do not cover this topic in this book.

Concluding Remarks

At one extreme, we have examined the issue of institutional investors concerned with owning complex portfolios of long and short positions across asset classes and with the use of reasonable leverage. In doing so, we have used much of our current understanding of finance literature and evidence from decades of portfolio management experience. We also considered the situation of investors who are more constrained, more preoccupied with tracking error (i.e., looking different) and career risk. Although retail investors do not have access to the full range and efficiency of factor-based long-short portfolios, there are many reasonably priced investment products that offer appropriate factor tilts. Furthermore, they are available at levels of overall MERs as low as those of many large institutional investors.

We have shown that all categories of investors could achieve higher levels of Sharpe ratios on their portfolios. Nevertheless, it may be optimistic to believe Sharpe ratios as high as 1.5 could be reached. Such portfolios require investing across portfolios of factors using short positions and gross leverage. Many institutional investors would hesitate to implement such portfolios because of short-term business risk and internal political reasons. However, our personal experience in designing efficient portfolios illustrates that it is possible to achieve Sharpe ratios close to one on unlevered balanced products subject to allocation policy constraints that make full use of relevant factor tilts and more sophisticated risk management and predictive methodologies. There is a middle ground, but we’ll explain in chapter 7 why it remains a challenge to push for products that are built on the principles we just discussed.

To a large extent, we have contributed to democratize the asset management process. Performances of active managers and of smart beta products can largely be explained by the underlying diversification processes used consciously or unconsciously by these managers and products, which lead to very specific risk factor tilts. Once we understand that a significant part of our skill requirement is to have an understanding of relevant risk factors and of methodologies to efficiently exploit and integrate these factors within a portfolio, the sources of alpha become more transparent and high fees are more difficult to justify.