m e , i { 1 , 2 , , N } , h { T + 1 , T + 2 , , H } and t { h , h 1 , , h T + 1 } (8)

Analogously to what was done in the estimate of ${\text{Σ}}_{i|h}$ we use EWMA to estimate ${\stackrel{^}{\alpha }}_{i|h}$ as follows:

${\stackrel{^}{\alpha }}_{i|h}=\lambda {\stackrel{^}{\alpha }}_{h,i|h}+\lambda \left(1-\lambda \right){\stackrel{^}{\alpha }}_{h-1,i|h}+\lambda {\left(1-\lambda \right)}^{2}{\stackrel{^}{\alpha }}_{h-2,i|h}+\cdots +\lambda {\left(1-\lambda \right)}^{\text{T}}{\stackrel{^}{\alpha }}_{h-T,i|h}$ (9)

where ${\stackrel{^}{\alpha }}_{i|h}$ estimates ${\alpha }_{i|h}$ and the value of $\lambda$ was chosen to minimize the estimate’s mean quadratic error.

4.1.1.4. Alpha Variance and Covariance Matrices

We now explain the steps involved in estimating the variance and covariance matrices of the alpha assets at each moment in time.

We intend to estimate:

${\text{Ω}}_{h}={E}_{h}\left[\begin{array}{ccc}\text{Var}\left({\alpha }_{1}\right)& \dots & \text{Cov}\left({\alpha }_{1}{\alpha }_{N}\right)\\ ⋮& \ddots & ⋮\\ \text{Cov}\left({\alpha }_{N}{\alpha }_{1}\right)& \dots & \text{Var}\left({\alpha }_{N}\right)\end{array}\right]$ (10)

Using the values of ${\stackrel{^}{\alpha }}_{it|h}$ which were estimated in the previous section we have that:

$\begin{array}{l}{\stackrel{^}{\Omega }}_{t|h}=\left[\left(\begin{array}{c}{\stackrel{^}{\alpha }}_{1t|h}-{\stackrel{¯}{\stackrel{^}{\alpha }}}_{1|h}\\ ⋮\\ {\stackrel{^}{\alpha }}_{Nt|h}-{\stackrel{¯}{\stackrel{^}{\alpha }}}_{N|h}\end{array}\right)\left(\begin{array}{ccc}{\stackrel{^}{\alpha }}_{1t|h}-{\stackrel{¯}{\stackrel{^}{\alpha }}}_{1|h}& \dots & {\stackrel{^}{\alpha }}_{Nt|h}-{\stackrel{¯}{\stackrel{^}{\alpha }}}_{N|h}\end{array}\right)\right],\\ \forall t\in \left\{h-T,\cdots ,h\right\}\end{array}$ (11)

where ${\stackrel{¯}{\stackrel{^}{\alpha }}}_{1|h}$ is the sample mean of ${\stackrel{^}{\alpha }}_{t|h}$ , $t\in \left\{h,h-1,\cdots ,h-T+1\right\}$ .

Now, as in the previous section, we employ EWMA to obtain the following estimate:

${\stackrel{^}{\Omega }}_{h}=\delta {\Omega }_{h|h}+\delta \left(1-\delta \right){\Omega }_{h-1|h}+\delta {\left(1-\delta \right)}^{2}{\Omega }_{h-2|h}+\cdots +\delta {\left(1-\delta \right)}^{\text{T}}{\Omega }_{h-T|h}$ (12)

where $\delta =0.96$ is the same decay exponent which we used to estimate ${\text{Σ ^}}_{i|h}$ .

4.1.2. Portfolios

Let us now explain the rules used to build the portfolios. We employed three methods to assemble the portfolios: the first one, called Naïve, is the simplest of the three; the other two, which we call Markowitz and Long-only Markowitz, are based on the Markowitz (1952) [1] methodology of optimal portfolios. The latter two differ from each other in terms of the restrictions imposed on each of them.

For each moment $h\in \left\{T+1,T+2,\cdots ,H\right\}$ we computed the portfolios of all three methodologies using the values obtained as described in section 4.1.

4.1.2.1. The Naïve portfolios

The weigths of each assets alpha in the Naive portfolios are based solely on the signs of the expected alphas. The procedure involves three steps: 1) the assets were classified according to the signs of their expected alphas; 2) assets with positive alphas were assigned a portfolio weight of $2/{N}_{\left(+\right)}$ , where ${N}_{\left(+\right)}$ is the total number of assets with positive alphas; and 3) assets with negative alphas were assigned a portfolio weight of $-2/{N}_{\left(-\right)}$ , where ${N}_{\left(-\right)}$ is the total number of assets with negative alphas. This thereby produced portfolios with (equally- weighted) long positions in assets with positive alphas and (equally-weighted) short positions in assets with negative alphas in every period.

4.1.2.2. The Markowitz Portfolios

The Markowitz portfolios were designed to maximize the expected return conditional on a specific level of risk. The risk level metric used was the standard deviation of the market portfolio observed in the last 252 days relative to the reference date ( ${\sigma }_{\text{market}|h}$ ).

The Markowitz portfolios were therefore obtained by solving the following optimization problem:

$\begin{array}{l}\underset{{w}_{t}}{\mathrm{max}}\text{}{w}_{t}{\alpha }_{h|h}\\ s.t.\\ \text{}\begin{array}{l}{{w}^{\prime }}_{t}\iota =1;\hfill \\ {{w}^{\prime }}_{t}{\text{Ω}}_{t}{w}_{t}\le {\sigma }_{\text{market}|h};\hfill \\ {w}_{\mathrm{min}}\le {w}_{t,i}\le {w}_{\mathrm{max}};\hfill \end{array}\end{array}$ (13)

where ${w}_{t}$ is the ( $1×N$ ) weight vector; $\iota$ is an ( $N×1$ ) vector consisting of 1s; ${w}_{\mathrm{min}}$ is the minimum acceptable value of ${w}_{t}$ ; ${w}_{\mathrm{max}}$ is the maximum acceptable value of ${w}_{t};\text{}{\alpha }_{h|h}$ is the ( $N×1$ ) expected alphas vector computed previously; and ${\text{Ω}}_{t}$ is the ( $N×N$ ) alpha variance and covariance matrix.

4.1.2.3.The Long-Only Markowitz Portfolios

The long-only Markowitz portfolios differ from the plain Markowitz portfolios in that the former have an additional restriction on the portfolio weights: every weight must be positive, so that there are no short positions in individual assets.

In the long-only case the weights therefore follow from the solution of the following optimization problem:

$\begin{array}{l}\underset{{w}_{t}}{\mathrm{max}}\text{}{w}_{t}{\alpha }_{h|h}\\ s.t.\\ \text{}{{w}^{\prime }}_{t}\iota =1;\\ \text{}{{w}^{\prime }}_{t}{\text{Ω}}_{t}{w}_{t}\le {\sigma }_{\text{market}|h};\\ \text{}{w}_{t,i}\ge 0\end{array}$ (14)

4.1.2.4. Bounds on Leverage

The short positions allowed by the Markowitz portfolios imply the possibility of leverage, in the sense that the sum of the weights of long positions of the portfolio may exceed 100%.

We impose a lower bound ${w}_{\mathrm{min}}$ on the weights to limit the leverage of the portfolio. The goal was therefore to find a value for ${w}_{\mathrm{min}}$ that solved the following problem:

$\begin{array}{l}\underset{{w}_{\mathrm{min}}}{\mathrm{max}}\underset{i=1}{\overset{N}{\sum }}|{w}_{i}|\\ s.t.\\ \text{}\underset{i=1}{\overset{N}{\sum }}|{w}_{i}|\le L;\text{}\underset{i=1}{\overset{N}{\sum }}{w}_{i}=1;\text{}{w}_{t,i}\le {w}_{\mathrm{max}};\end{array}$ (15)

where $L$ is the maximum leverage. Here we use the sum operator (over all assets) instead of the vector $\iota$ to highlight that the total number of available assets affects the solution.

We illustrate the issue with an example. Assume that we have 12 available assets and that ${w}_{\mathrm{max}}=2$ and $L=10$ . If ${w}_{\mathrm{min}}=-1$ then the maximum leverage vector ( ${w}_{{L}^{\mathrm{max}}}$ )3 would be given by:

${w}_{{L}^{\mathrm{max}}}=\left(-1,-1,-1,-1,-1,-1,-1,-1,+1,+2,+2,+2,+2\right)$

Note that all of the restrictions are satisfied except for the leverage bound, since the sum of the absolute values of the weights is equal to 13. This implies that the value of ${w}_{\mathrm{min}}$ must be set higher: in this example it must be equal to $-0.5$ , as explained below.

${w}_{{L}^{\mathrm{max}}}=\left(-0.5,-0.5,-0.5,-0.5,-0.5,-0.5,-0.5,-0.5,-0.5,+1.5,+2,+2\right)$

We use the following numerical procedure to find the required value of ${w}_{\mathrm{min}}$ . Given values for ${w}_{\mathrm{max}}$ , $N,$ and $L$ we obtain the vector of maximum leverage for a trial ${w}_{\mathrm{min}}$ by repeatedly (1) checking whether the restrictions were satisfied and then (2) changing the values of ${w}_{\mathrm{min}}$ until all of the restrictions were satisfied.

In this study we chose to specify a reasonable value for L and at the same time pick a value for ${w}_{\mathrm{max}}$ that is high enough that it does not constitute an active restriction for the solution of the problem.

4.1.3. Stoploss

We used a stoploss mechanism: the portfolio position was set to zero whenever the position accumulated a loss greater than or equal to 25%.

4.1.4. Alpha Portfolios

By following the previous steps, we have so far assembled a portfolio for each day of the period, except for the initial T early days, which were used only for the initial estimates. Thus, one can consolidate the 21 portfolios into a final portfolio as follows:

a) using the weights obtained with the portfolio assembled with the information available at the time T, we invest on date T + 1;

b) on date T + 2 we accumulate the return obtained in the previous step and assemble a new portfolio using the information available on T + 1;

d) we continue the procedure until we obtain 21 portfolios;

e) on date T + 21 + 1 we set to zero the position of the portfolio that we have obtained during the previous 21 days and invest its cumulative value on a portfolio assembled using the information available up to date T + 21 + 2;

f) we continue the procedure by every day discarding a portfolio and replacing it with a new one, so that we are always investing in a combination of 21 portfolios.

The end result is a portfolio consisting always of several portfolios with distinct maturities. The first portfolio was assembled using only the initial information and is invested in for only one day, while the oldest portfolio belongs to the portfolio beginning 21 days ago and will be discarded the following day. The performance of the alpha portfolio is given by the average performance of the 21 portfolios that constitute it. We choose to analyse the portfolios in this way, so our results were less sensitive to the data that the analysis begins.

4.1.5. Constant Weights or Quantities

We deal with various types of portfolios. Each alpha is a portfolio consisting of the asset and the market, which becomes more complex as we add more and more alpha assets. These in turn become even more elaborate when we consolidate portfolios of distinct maturities into a single portfolio.

As we accumulate the portfolio returns, this process raises the following question: what should be kept constant over time, the asset weights or the asset quantities?

We considered both approaches. We first kept weights constant―we call this the Equal Weight approach. Regardless of past performance all weights were kept fixed and equal to the originally calculated weights. This involves of course selling the outperforming assets and buying the underperforming assets so that so that they remain constant as portions of the whole portfolio.

In the second approach, called Value Weight, the asset weights vary depending on their past performance. Outperforming assets gain weight and underperforming assets lose weight in the portfolio. This is equivalent to keeping constant the initially purchased quantities: no rebalancing trades are made.

4.2. Performance Metrics Analysis

In this section we discuss the metrics used to assess portfolio performance. As explained previously, our methodology yields various portfolio types, charac- terized by their restrictions on the maximum and minimum weights, the maximum amount of leverage, and other criteria.

We apply a set of tests to compare the performance of each portfolio with that of the market portfolio. The metrics used in these comparisons are based on De Miguel, Garlappi and Uppal (2007) [10] and explained below.

4.2.1. Sharpe Ratio Test

We begin by computing the Sharpe ratio of each of the portfolios. For each $i\in \left\{1,2,\cdots ,N\right\}$ we have:

$\stackrel{^}{S{R}_{\iota }}=\frac{\stackrel{^}{{\mu }_{\iota }}}{\stackrel{^}{{\sigma }_{\iota }}}$ (16)

where $\stackrel{^}{{\mu }_{\iota }}$ , $\stackrel{^}{{\sigma }_{\iota }}$ and $\stackrel{^}{S{R}_{i}}$ are respectively the mean return, the standard deviation, and the Sharpe ratio of portfolio $i$ .

We then use an approach suggested by Jobson and Korkie (1981) [11] and amended by Memmel (2003) [12] to test whether the differences among the Sharpe ratios are significant. Let

${H}_{0}:{\stackrel{^}{SR}}_{j}-{\stackrel{^}{SR}}_{M}=0$ (17)

where $j\in \left\{1,2,\cdots ,N\right\}/M$ and $M$ is the market portfolio. We then compute the ${\stackrel{^}{z}}_{iM}$ statistic, whose asymptotic distribution is the standard normal distribution4:

${\stackrel{^}{z}}_{iM}=\frac{{\stackrel{^}{\sigma }}_{i}{\stackrel{^}{\mu }}_{i}-{\stackrel{^}{\sigma }}_{M}{\stackrel{^}{\mu }}_{M}}{\sqrt{\stackrel{^}{\upsilon }}}$ (18)

such that

$\stackrel{^}{\upsilon }=\frac{1}{T}\left(2{\stackrel{^}{\sigma }}_{i}^{2}{\stackrel{^}{\sigma }}_{M}^{2}-2{\stackrel{^}{\sigma }}_{i}{\stackrel{^}{\sigma }}_{M}{\stackrel{^}{\sigma }}_{IM}+\frac{1}{2}{\stackrel{^}{\mu }}_{i}^{2}{\stackrel{^}{\sigma }}_{M}^{2}+\frac{1}{2}{\stackrel{^}{\mu }}_{M}^{2}{\stackrel{^}{\sigma }}_{i}^{2}-\frac{{\stackrel{^}{\mu }}_{i}{\stackrel{^}{\mu }}_{M}}{{\stackrel{^}{\sigma }}_{i}{\stackrel{^}{\sigma }}_{M}}{\sigma }_{iM}^{2}\right)$ (19)

Therefore, just as in a t test, we need only compare the value obtained above with the critical value given by the normal distribution: if the critical value is lower than the computed value then we reject the null hypothesis that the Sharpe ratios of the two portfolios are equal.

4.2.2. Certainty Equivalence Analysis

3The certainty equivalent is the increase in the risk-free rate that would render the agent indifferent between the risky investment and the risk-free investment. As was pointed out by De Miguel, Garlappi and Uppal (2007) [10] , Equation (20) corresponds not to the certainty equivalent but instead to the expected utility of a mean-variance investor, which is in turn approximated by the certainty equivalent of a quadratic-utility investor.

The certainty equivalent test goes largely as above. It also consists in calculating a performance metric for all portfolios followed by a difference test between the assembled portfolios and the market portfolio. In this new case the metric is the certainty equivalent3 of the portfolio, obtained as follows:

$EQ={\stackrel{^}{\mu }}_{i}-\frac{\gamma }{2}{\stackrel{^}{\sigma }}_{i}^{2}$ (20)

where $i\in \left\{1,2,\cdots ,N\right\}$ and $\gamma$ is the risk aversion parameter.

Having first computed the certainty equivalent for all portfolios using (20), we perform a difference test on the values so obtained.

Specifically, letting ${f}_{iM}\left(v\right)=\left[{\mu }_{i}-\frac{\gamma }{2}{\sigma }_{i}^{2}\right]-\left[{\mu }_{M}-\frac{\gamma }{2}{\sigma }_{M}^{2}\right]$ , we test the null hypothesis ${H}_{0}:f\left(v\right)=0$ , knowing that:

${\text{Ψ ^}}_{iM}=\sqrt{T}\left({f}_{iM}\left(\stackrel{^}{v}\right)-{f}_{iM}\left(v\right)\right)~N\left(0,{\frac{\partial f}{\partial v}}^{\text{T}}\Theta \frac{\partial f}{\partial v}\right)$ (21)

where

$\Theta =\left[\begin{array}{cc}\begin{array}{cc}{\sigma }_{i}^{2}& {\sigma }_{iM}\\ {\sigma }_{Mi}& {\sigma }_{M}^{2}\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}2{\sigma }_{i}^{4}& 2{\sigma }_{iM}^{2}\\ 2{\sigma }_{Mi}^{2}& 2{\sigma }_{M}^{4}\end{array}\end{array}\right]$ (22)

We analyse the results as before: if the computed value is higher than the critical value then we reject the null hypothesis. Note that we are setting the risk aversion parameter $\gamma$ equal to 1.

5. Results

We now discuss the performance of the portfolios. The analysis is organized as follows: we first compare the returns of the alpha portfolios with the return of the market portfolio over the whole period; we then compare the alpha and market portfolios by using the performance metrics described in section 4.2.

The various types of portfolios involved in this paper are summarized in Table 1, which specifies for each portfolio the weighting convention (equal weight or value weight), the abbreviated names, and some explanatory remarks.

The first portfolio is the market portfolio, whose performance is used as the benchmark. The portfolios are classified into four types, each containing two portfolios: one with equal weights and one with value weights as explained in section 4.1.5.

The first of the four types are the Markowitz long-only portfolios, which consist of portfolios without any short positions, as explained in section 4.1.2.3. Second, the naive portfolios follow the simpler procedure described in section 4.1.2.1. Third, there are the (restricted-weight) Markowitz portfolios whose methodology has been described in section 4.1.2.2 and whose weights are restricted to a specific range, displayed in Table 1 as going from −X to Y. Finally, the bounded leverage portfolios, which were described in section 4.1.2.4, have leverage limited to 10.

5.1. Alpha Portfolio Analysis

We now compare the returns of the alpha and market portfolios. Table 2 displays the cumulative average returns for each portfolio and the market portfolio

Table 1. Analysed portfolios.

The first column displays the name of the portfolio, the second column indicates whether the portfolio was assembled with constant weights (Equal) or weights varying according to past performance (Value), and the third column includes some explanatory remarks.

Table 2. Mean return rates.

Average per-day (% p.d.) rates of cumulative returns of the alpha and market portfolios. The rates are grouped into two-year intervals. The “Full period” column displays the average per-day rates over the entire period.

in terms of daily return (% p. d.) and the values were grouped into two-year intervals.

Note that the returns of the alpha portfolios are often much higher than those of the market portfolio. This is not true, however, between 2004 and 2009, when alpha returns generally tie with or are lower than the market returns.

The results indicate a partial success in assembling portfolios that beat the market. Though the alpha returns are higher over the whole period, this does not occur consistently over time: there are fairly long periods during which the strategies do not accurately predict the asset alphas.

It follows that inertial behavior of the asset alphas―which was used as a hypothesis in this work―cannot be fully confirmed. On the other hand, this inertia did generate significant gains when the entire period is considered. As we shall see shortly, the results do show that the resulting gains justify the additional risk incurred. We now check this using the metrics which were discussed in sections 2.3 and 2.4.

5.2. Sharpe Ratio Analysis

We now turn our attention to comparing the Sharpe ratio of the portfolios. The results are summed up in Table 3. The first column names the portfolios, the

Table 3. Sharpe ratio difference tests.

Given each portfolio we tested the hypothesis ${H}_{0}:{\stackrel{^}{\mu }}_{i}/{\stackrel{^}{\sigma }}_{i}-{\stackrel{^}{\mu }}_{M}/{\stackrel{^}{\sigma }}_{M}=0$ , where $\mu$ is the mean portfolio return, $\sigma$ is its standard deviation, $i$ is the index of the alpha portfolio and $M$ is the market portfolio. Significance levels: ***1%, **5%, *10%

second column lists their Sharpe ratios, and the third column shows the ${\stackrel{^}{z}}_{iM}$ statistic regarding the difference test between the Sharpe ratios of the alpha and market portfolios that was discussed in section 4.2.1. Finally, the last column reports the test p-value.

As shown in the Table, the market portfolio had a Sharpe ratio of 0.015, the lowest among all analysed portfolios. Here are the remaining portfolios in decreasing order of performance as measured by the Sharpe ratio: m-10.30.E, m-10.30.V, m-lvrg.E, m-lvrg.V m-10.10.E, m-20.20.E, m-20.20.V, m-20.30.E, m-20.30.V, m-10.10.V, naive.E, naive.V, m-long.V and m-long.E. Their Sharpe ratios ranged from 0.028 to 0.061.

Although all of the alpha portfolios have higher Sharpe ratios than the market portfolio, only the top four were significantly higher, all four at the 10% significance level. Note also that there was little difference in performance between the equal-weighted and value-weighted portfolios. As one can see the performance was essentially the same regardless of weight used.

5.3. Certainty Equivalent Analysis

We now analyse the alpha portfolios via the certainty equivalent differences tests discussed in section 4.2.2. As previously explained, these tests compare the certainty equivalent of each alpha portfolio with the market portfolio. We set out the results in Table 4, whose columns display the following information from left to right: abbreviated portfolio name, value of the certainty equivalent, value of the ${\text{Ψ}}_{iM}$ statistic, and the test p-value.

Table 4. Certainty equivalent difference tests.

Given each portfolio we tested the hypothesis ${H}_{0}:{\text{Ψ ^}}_{iM}=\sqrt{T}\left({f}_{iM}\left(\stackrel{^}{v}\right)-{f}_{iM}\left(v\right)\right)=0$ , where ${f}_{iM}\left(v\right)=\left[{\mu }_{i}-\frac{\gamma }{2}{\sigma }_{i}^{2}\right]-\left[{\mu }_{M}-\frac{\gamma }{2}{\sigma }_{M}^{2}\right]$ , $\mu$ is the mean portfolio return, $\sigma$ is the standard deviation, $i$ is the index of the alpha portfolio, $M$ is the market portfolio, and $\gamma$ is the risk aversion parameter, set at $\gamma =1$ . Significance levels: ***1%, **5%, *10%.

As in the previous test, the market portfolio had the worst result among all portfolios. The values for the alpha portfolios, however, are not always statistically higher than the market portfolio.

Here are the remaining portfolios in decreasing order of performance as measured by the certainty equivalent: m-20.30.E, m-20.20.E, m-10.30.E, m-20.30.V, m-20.20.V, m-10:30.V, m-lvrg.E, m-lvrg.V, m-10.10.E, m-10.10.V, naive.E, naive.V, m-long.V and m-long.E. Nine of the top alpha portfolios values have certainty equivalents statistically above the market portfolio at a 10% significance level.

The certainty equivalent test results are in general very similar to the Sharpe ratio test results. The lists of portfolios ordered by decreasing performance are similar for the different metrics used. There is moreover little difference in performance, again as measured by the various metrics, between equally-weighted and value-weighted portfolios of otherwise the same types.

6. Conclusions

We evaluated an investment strategy by betting on pricing errors made by the CAPM model, under the hypothesis that these errors are non-zero and persist over time. In order to do so, we first computed for each asset its expected pricing error (the asset alpha), defined as the expected return when one takes a long position in the asset while shorting the market with weight equivalent to the asset risk (beta). We essentially built optimal Markowitz (1952) [1] portfolios, replacing the expected returns vector by the expected error vector.

Our results show that over the entire period of analysis, all of the alpha portfolios obtained above-market returns. These results were not significantly affected by the choice of equal-valued or value-weighted portfolios. Except over some shorter periods, our hypothesis was fully confirmed, that is, the performance of our alpha portfolios was significantly better than that of the market portfolio. Furthermore, two metrics were used to assess portfolio performance, the Sharpe ratio and the certainty equivalent. Both metrics confirmed that the performances of the alpha portfolios were better than that was observed for the market portfolio. In other words, the pricing error of the CAPM model has an inertial component that allows one to obtain returns that are significantly above market returns.

Conflicts of Interest

The authors declare no conflicts of interest.

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