Optimal Portfolio Allocation using Daily Correlation Modelling Charles S. Bos∗,a,c and Roman Kr¨ausslb,d a Tinbergen Institute, The Netherlands b Center for Financial Studies, Germany c Department of Econometrics & O.R., d Department of Finance, VU University Amsterdam, The Netherlands February 15, 2008
Preliminary and incomplete
Abstract Traditional mean-variance efficient portfolios do not capture the potential wealth creation opportunities provided by predictability of asset returns. This paper examines the benefits of actively managed portfolio diversification that accrue to a representative foreign investor who considers international investment opportunities among six major Euro-area countries. To do so, we specify two advanced models for time-varying mean, variances and correlations and compare their results with standard portfolio allocation strategies like the buy-and-hold and fixed weight strategies. Our empirical findings indicate that models incorporating stochastic correlation and volatility result in significantly higher returns than GARCH-based variants or naive portfolio optimisation. Keywords: International portfolio diversification; Optimal asset allocation; Dynamic correlation; Stochastic volatility. JEL classification: C32, C52, G11
Corresponding author: C.S. Bos, Department of Econometrics & O.R., VU University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands. Email
[email protected], tel/fax: +31 (0)20 598 60 23/+31 (0)20 598 60 20. ∗
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1
Introduction
Traditional mean-variance efficient portfolios do not capture the potential wealth creation opportunities provided by predictability of asset returns. However, trading strategies that apparently try to beat the market date back to the beginning of trading in financial markets. Most portfolio managers have investment mandates with relative return targets. They are typically constrained to hold assets in a well-defined number of asset classes and are frequently limited to little or no leverage. Their mandates are to meet or exceed the respective returns on their asset classes, which implies that they have to beat a simple buy-and-hold (indexing) strategy. International portfolio diversification has long been advocated as an effective way to achieve higher risk-adjusted returns than domestic investment alone. The main premise underlying this strategy is that international stocks tend to display lower levels of co-movement than stocks trading on the same market. Since national markets are subject to different shocks, international diversification facilitates risk sharing among global investors. Idiosyncratic shocks may be diversified away. Stock return cross-country correlations play a key role in international finance. Changes in these correlations affect the volatility of portfolios and asset prices. As these correlations increase, one expects that fewer domestic risks are internationally diversifiable, thus, portfolio volatility increases. Clearly, the less international markets are correlated with each other, the greater the benefits of international diversification in the form of risk reduction. Several empirical evidence supports the pre-eminence of international diversification strategies like Grubel (1968) and Levy and Sarnat (1970). More recent empirical papers find that these benefits are still present despite increasing integration across financial markets (see de Santis and Gerard (1997)) and in the face of time-varying correlations (see Ang and Bekaert (2002)). The proposition of international portfolio diversification, however, relies on the assumption that the required inputs to the classical mean-variance analysis are known with ’certainty’. Optimal portfolio optimisation a la Markowitz (1952) traditionally proceeds in two steps. First, the moments of the distribution of returns are estimated from a time series of historical returns; then the mean variance portfolio selection problem is solved separately, as if the estimates were the true parameters. Markowitz (1952) focuses on the second step, and assumes along with most of the practitioners today that expected returns and variances can be constructed from e.g. some period of historical data. The choice of the length of the period, the updating frequency, the treatment of ‘outliers’, all influence the final portfolio choice, and it remains unclear whether there is any one optimal setting. Contrary to the above mentioned approach, this research focuses on an alternative approach for forming expectations of future returns and variances, using recent developments in time series econometrics for tracking variance and correlation at close range. Most practitioners are by now familiar with the GARCH Bollerslev (1986) approach of filtering variances. Derived from these are similar filters capable of tracking a dynamic correlation measure Engle and Sheppard (2001) and Engle (2002). Even though such deterministic filters improve on choosing some historical period for estimating (co-)variance, the concept of assuming variance to vary as a deterministic function of past returns is rather counter-intuitive at the least. The alternative approach for volatility modelling is the strand of literature concerning stochastic volatility (see Harvey, Ruiz, and Shephard (1994) and Jacquier, Polson, and Rossi 2
(1994)). Stochastic correlation was developed at a more recent date, independently in Yu and Meyer (2006) and Bos and Gould (2007). The latter article compares the performance of the GARCH-based dynamic conditional correlation (DCC) models to the performance of the dynamic correlation stochastic volatility (DCSV) models in a bivariate setting. To the best of our knowledge, no other study has employed this innovative technique to study optimal portfolio allocation. This paper intends to push both DCC and DCSV models to a multivariate setting, such that they could serve for optimising larger portfolios. To do so, we develop a model of optimal asset allocation based on an utility framework. This applies to a more general context than the classical mean-variance framework since it can also account for the presence of constraints in the portfolio composition. The major problem with the use of daily returns across different countries is the nonsynchronous trading periods for different markets around the globe. Therefore, this article compares dynamic correlation models for the calculation of minimum variance optimal portfolios between European stock market indices. We focus our analysis on six major European countries who joined the Euro in January 1999 in order to concentrate solely at the construction of an optimal international portfolio. With this data, there is no need to take exchange rate risk into account, though such an extension is quite possible for different data series. Finding the optimal portfolio allocation requires not only knowledge of the variability of individual stock markets but also of the co-movements between the indices. For this purpose, we employ industry standard methods like the naive portfolio optimisation, more advanced GARCH techniques including estimating DCC models and alternatively through the use of unobserved components models. This last set comprises models with stochastically varying variances and/or correlations. In order to compare the performance of the different models in producing optimal portfolio allocations, the reduction in portfolio variance is examined. The empirical results suggest that the most important factor in reducing portfolio variance is the use of a flexible model for time-varying volatility, rather than capturing time variations in correlations as closely as possible. Our empirical findings indicate that models incorporating stochastic volatility result in significantly higher returns than GARCH-based variants or naive portfolio optimisation. The remainder of this article is organised as follows. Section 2 presents the data set and discusses the methodology. Section 3 reports our empirical findings. Section 4 summarises our results, draws conclusions and offers some suggestions for future research.
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Data and Methodology
Data For testing the viability of our approach we started off combining a data set of European stock market indices. We select those major national stock market indices of countries who joined the Euro in January 1999: Austria, Belgium, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg, Portugal, The Netherlands, and Spain. This enabled us to disregard any kind of exchange rate risk in the context of international portfolio diversification. From Datastream we collected the data at a daily frequency. We restrict our empirical analysis in the following to six major European stock markets to keep our analysis handy. Figure 1 displays the historical performance of the six European stock markets over the years 1999 to 2006. It clearly stands out that Austria’s ATX index outperformed all other major European stock markets since the beginning of 2003. The stock market indices of 3
France, Germany, Italy, The Netherlands, and Spain, on the contrary, seem to move more in tandem. 450 ATXINDX FRCAC40 DAXINDX ITMHIST AMSTEOE IBEX35I
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Figure 1: Stock indices for European countries, 1/1/1999–31/12/2007
Constructing a bivariate model In a first step, we have to construct at a daily frequency bivariate models for returns, volatilities and correlations, based on both the DCC and DCSV models including a component measuring the expected return. To relate of returns ri , rj observed over periods t = 1, .., T , we use a model expressed in unobserved components, as in Harvey (1989) and Durbin and (i,j) = (rit , rjt )′ , the model could be Koopman (2001). Combining the returns into a vector rt specified as (i,j)
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The unobserved expected return E(rt ) = µt evolves as a random walk, with supposedly a small but positive variance for each of the disturbance terms. Eventually, the unobserved components model could be extended, e.g. in the case of an international investment portfolio the exchange and interest rates should be taken into account as well. (i,j) For Σt , either a DCC specification or a DCSV specification is implemented. Details on these specifications are given in Appendix A. Both specifications allow for time variation in 2 σit ρijt σit σjt (i,j) (3) Σt = 2 ρijt σit σjt σjt The appendix describes quickly how, for the DCC models, the evolution of the covariance matrix is deterministic, and an adapted version of the Kalman filter equations can be used for constructing the likelihood function. This function is subsequently optimised for finding the maximum likelihood parameter estimates, for the DCC model. For the DCSV model, estimation is performed using a Bayesian approach, using the main ideas of Jacquier, Polson, and Rossi (1994) but further refined to allow for correlations as well. Part of the refinements have been applied earlier in Bos and Shephard (2006) to improve performance of the Markov chain Monte Carlo sampling scheme with longer stochastic 4
volatility series. A particle filter is applied using the posterior mode of the parameters to extract filtered expected returns E(µ), the volatilities σit and correlations ρijt .
Combining bivariate results For each combination of assets the above models are estimated. Even though this results in a large number of models, and a considerable computational effort, the restriction of only allowing for bivariate models simplifies the specification considerably. The bivariate results are combined into multivariate results, taking e.g. the expected return at time t of asset i to 1 P be E(rit ) = k−1 j6=i E(µi,t |Ri,t−1 , Rj,t−1 ), with Ri,t−1 = {ri,1 , .., ri,t−1 } assuming a set of k (1,..,k)
is constructed. Below, assets. In a similar fashion, a full k-variate covariance matrix Σt in the section highlighting the results, for France the factors constituting the final estimates of mean, variance and correlation are displayed for both the DCSV and DCC models.
Portfolio optimisation At this point, the article of Markowitz (1952) comes up, as it describes how a portfolio can be optimised for a standard utility function. A more recent reference is Bansal, Dahlquist, and Harvey (2004), who also describe concepts related to dynamic optimisation, and alternative hedging strategies. A comparison of the results to historical portfolios, and devising a trading strategy which limits the amount of switches of portfolio weights, is technically straightforward though still demanding when possible transaction costs are taken into account, see also Dumas and Luciano (1991). For ease of reference, we provide here our basic optimisation scheme which does not yet take transaction costs into account. Assume we have a vector of assets(-weights) w, with covariance matrix Σ and expected return µ, then the portfolio variance would be w′ Σw. The utility function could be c U = w′ µ − w′ Σw 2 for some risk aversion parameter c. This quantity we intend to minimise, under restrictions X wi = 1 (4) wi ≥ 0
(5)
If we forget for a moment the no-short selling restriction, then the Lagrangian function becomes c F (w, λ) = w′ µ − w′ Σw − λ(w′ ı − 1) 2 ∂F/∂w = µ − cΣw − λı ∂F/∂λ = −(w′ ı − 1)
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Solving these equations leads to 1 −1 Σ (µ − λı) c 1 λ ı′ w = 1 = ı′ Σ−1 µ − ı′ Σ−1 ı c c ı′ Σ−1 µ − c λ= ı′ Σ−1 ı 1 −1 ı′ Σ−1 µ − c ⇔w= Σ ı µ− c ı′ Σ−1 ı w=
Using constrained optimisation, the optimum can be found quickly. The above method of finding the minimum variance portfolio in a multivariate framework gives the optimum in a low number of deterministic steps, hence is perfectly viable for our situation. Changing the covariance daily, using DCC, DCSV or even the historic covariance matrix over the last N days, is not a problem.
3
Evaluating Model Performance
This section evaluates in detail the performance of the specified DCSV and DCC models and compares it to the results of the more classical asset allocation strategies. In particular, we will compare the different trading strategies for four different levels of risk aversions on the portfolio return, portfolio variance, the final utility, and the number of necessary portfolio changes (trades). We now compare the different investment strategies, i.e. DCSV (winner over DCC) against naive portfolio allocation strategies. First, a passive approach is used as a benchmark. The investor could hold an equally weighted index. Second, we examine the classical mean-variance portfolio strategy a la Markowitz. The investor chooses a portfolio on the efficient frontier, which is constructed on the basis of past sample estimates at each point in time. With riskfree lending at zero rate of interest, the investor picks the portfolio with the highest ratio of expected return to standard deviation, which lies on the tangent from the origin. This section evaluates in detail the performance of the specified models in four different settings and compares it with the results achieved by the more classical portfolio allocation strategies. In particular, we will compare the different strategies on the final utility, portfolio return, volatility and the number of necessary portfolio changes.
Extracted returns and volatilities Figure 2 shows exemplatory the constructed mean, volatility and correlations for France for the DCSV model. E.g. the left panel displays the mean returns µt as estimated for France in each of the bivariate estimation procedures. Clearly, the estimate for the mean return differs slightly whether the complementary series is either the German stock index or the Austrian one, but the overall impression of the expected return is very similar. Likewise, in the center panel the volatility estimates σt for France are combined. The rightmost panel of 2 displays the correlation of France over time with each of the other major indices. In general, the correlation between markets increased over time, with one market sticking out. The correlation running around the level 0.4–0.5 is the correlation 6
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Figure 2: Constructing mean, volatility and correlation for France of France with the Austrian stock index. Correlation with the other markets is far higher than with Austria, which clearly has its own market evolution. This special behaviour of the Austrian index can also be seen from Figure 3, which displays both the expected return µt (top panel) and the volatility σt (bottom panel) for all indices. Austria displayed a consistently higher expected return over the years 2001–2005, combined especially around 2003 with a lower volatility as well. After mid-2006, the market became less attractive, as returns went down while volatilities rose above levels of the other markets. Another market which has performed rather consistently over this time frame has been the Dutch stock market: Expected returns would lie at the bottom of the pack, with volatilities which were rather high, especially over the period 2002–2005. 0.2 ATXINDX FRCAC40 DAXINDX ITMHIST AMSTEOE IBEX35I
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Figure 3: Mean and volatility underlying portfolio decisions, using DCSV model Not reported in this article are similar plots for the DCC model. Using the deterministic DCC-GARCH filter, the expected returns, volatilities and correlations are roughly in line with the DCSV estimates, though far more erratic. This would in turn influence the stability of the optimal portfolio weights, as will be seen at the end of this section.
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Optimal portfolio weights The expected returns, volatilities and correlations serve as input for constructing the optimal portfolio, as explained in Section 2. Figure 4 displays the weights that result from such an optimisation, with the DCSV model providing the input, without any restrictions on the number of switches that can be made, and using a risk aversion of C = 0.5 in the utility function. The top panel of the figure relates to the case where no further restrictions on the portfolio are implemented, whereas the bottom panel restricts asset weights to lie between 0 and 1, effectively impeding any short selling. 5 DCSV, not restricted
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Figure 4: Weights for DCSV model, both unrestricted and restricted optimisation, using C = 0.5 risk aversion Without the short-selling restriction, the years until 2004 are relatively tranquil, with Austria taking a large part of the weight. In 2004, a year were Dutch returns and volatilities are relatively large, the investor would do better to go short on the Dutch stock market, investing more in countries like France (first half of 2004) or Spain (second half). At the end of 2005, as volatilities decrease and returns remain high, more extreme positions are taken, investing heavily (up to 450%) in the DAX, going short in the CAC40 and the Amsterdam EOE index. Imposing a restriction that an investor cannot go short makes the investment opportunities at least simpler. The bottom panel indicates that investing in Austria has been very smart until the beginning of 2004, and even until the end of 2005 up to 50% of the portfolio was invested in the ATX index. Starting 2006, the decisions are clear as well: First invest in the 8
IBEX, followed by a period of investments in Germany, which are only interrupted halfway 2007 by a few months worth of investments in The Netherlands.
Cumulative returns, volatility and utility Table 1 shows the cumulative results over the full sample period January 1, 1999 to December 31, 2007, of the four different trading strategies for the four different levels of risk aversion c = 0.5, 1, 2, and 5, respectively. Optimisation is unrestricted, which implies that an investor may go short. Moreover, we assume that markets are frictionless, so that the portfolio weights in the individual European stock markets can be arbitrarily scaled to obtain any level of profit. Table 1: Cumulative statistics 1999-2007, unrestricted optimisation c 0.5 1 2 5 Cumulative returns DCSV 0.130 0.097 0.080 0.071 DCC 0.060 0.049 0.043 0.039 Historical 0.046 0.042 0.040 0.038 Naive 0.023 0.023 0.023 0.023 Cumulative volatilities DCSV 0.494 0.422 0.401 0.393 DCC 0.676 0.587 0.565 0.560 Historical 0.807 0.704 0.673 0.661 Naive 1.148 1.148 1.148 1.148 Cumulative utilities DCSV 0.002 -0.119 -0.327 -0.925 DCC -0.110 -0.246 -0.524 -1.364 Historical -0.157 -0.311 -0.635 -1.618 Naive -0.264 -0.551 -1.125 -2.848 Cumulative results over the full sample period of the different strategies, for four levels of risk aversion. Optimisation is unrestricted, an investor may go short.
The returns on historical and on simple equally weighted portfolios of the six major European stock market indices provide natural benchmarks for judging the profitability of our switching strategies. Our empirical findings indicate that the DCSV model is for different risk aversion coefficients always the clear winner. The daily cumulative returns are in most of the cases more than double the one of the DCC model, and even more superior compared to the historical trading strategy, based on the previous annual risk-return realisations, and compared to the naive (1/n) investment strategy. The numbers presented are average daily percentage returns, so an all-years return of 0.130 (first panel, first column) means that the daily average of percentage returns is 0.130, implying an annual average return of say 250 × 0.130 ≈ 32% over the whole sample. The second panel of Table 1 shows that the cumulative volatilities of the DCSV trading model are also in every case lower than its counterparts of the DCC, historical and naive trading strategies. Even stronger are the results in case of the cumulative utilities. In every case our DCSV trading strategy performs better than its counterparts. It becomes clear that a model-based approach outperforms historical naive investment strategies. We can easily 9
identify an ordering of the four different trading strategies: the DCSV model is beating the historical asset allocation strategy, which outperforms the results of the DCC model. The naive portfolio strategy performs as expected the worst: over the years 1999 to 2007 it has the lowest cumulative returns, the highest cumulative variance and, thus, the lowest cumulative utility. This strong performance of the DCSV model might be due to the fact that we allow for short-selling, i.e. so far, we have assumed that an unlimited amount of capital can be invested. However, there may be restrictions placed on portfolio managers which prohibit them from exploiting our optimal trading strategy. Thus, we have a closer look at our model performance if we impose short-selling restrictions. Table 2 shows output similar to that of the previous table, now with the restriction of no short-selling imposed. Table 2: Cumulative statistics 1999-2007, restricted optimisation c 0.5 1 2 5 Cumulative returns DCSV 0.064 0.055 0.049 0.044 DCC 0.044 0.039 0.036 0.034 Historical 0.050 0.047 0.044 0.043 Naive 0.023 0.023 0.023 0.023 Cumulative volatilities DCSV 0.508 0.500 0.495 0.493 DCC 0.636 0.614 0.606 0.604 Historical 0.749 0.703 0.685 0.674 Naive 1.148 1.148 1.148 1.148 Cumulative utilities DCSV −0.064 −0.197 −0.449 −1.193 DCC −0.115 −0.268 −0.571 −1.478 Historical −0.138 −0.306 −0.642 −1.647 Naive −0.264 −0.551 −1.125 −2.848
See Table 1 for a description of the entries in the table. Here, the no shortselling restriction is imposed.
Table 2 indicates that for every level of risk aversion the DCSV model outperforms the other strategies. Moreover, its volatility is significantly lower so that its cumulative utility is superior to the other models. The numbers presented are average daily percentage returns, so an all-years return of 0.064 (first panel, first column) means that the daily average of percentage returns is 0.064, implying an annual average return of say 250 × 0.064 ≈ 16% over the whole sample. This result is significantly better than the performance of the simple equal-weighted asset allocation strategy which cumulates to an annual average return of 250 × 0.023 ≈ 6%. Despite the better performance, the DCSV shows also a lower cumulative volatility and a better cumulative utility than its counterparts. Figure 5 displays the cumulative returns for different starting dates for the DCSV, DCC, historical and naive (equal-fractions) trading strategies using c = 0.5 as the level of risk aversion for the restricted optimisation, i.e. investors cannot go short in other markets. It becomes clear that at the end of the period, the DCSV trading strategy is beating significantly the other optimisation strategies. One can also see that until the starting date of 2004, the historical trading strategy performs sometimes for short intervals better than the DCSV. 10
However, from the beginning of the year 2005 on, DCSV outperforms the other strategies at any point in time. This indicates that the strong empirical findings do not depend on the investment horizon chosen or whether we start our optimal portfolio allocation are in bull or in a bear market. Figure 6 displays the changes in utility ∆u to be gained by changing the portfolio allocation versus the change in portfolio ∆w. The top panel displays results for the DCSV model with unrestricted optimisation for a risk aversion coefficient of c = 0.5. The bottom panel covers the same model and risk aversion coefficient but imposes the no short-selling restriction. The figure indicates for instance that most portfolio changes are small: On average only around 6% of the portfolio is changed. There is quite a difference though in the amount of improvement in expected utility such a change in portfolio can bring. It might be worthwhile to limit trading to only those occasions where a change ∆w indeed brings a clear increase in ∆u, or to wait otherwise.
Considering trading costs The question arises how practical our empirical findings are. So far, we have assumed transactions are costless. Although this approach is clearly unrealistic, it provides a useful benchmark for a more detailed trading strategy evaluation. We have seen that if there are no transaction costs, then the assets should always be invested in this portfolio which has the highest conditional expected return. However, we know transaction costs have a substantial effect on portfolio returns. In order to calculate realistic returns, direct tradings costs has to be considered. Our empirical findings indicate that a dynamic trading strategy that switches portfolios to try to capturing any return differences generates large excess returns compared to the traditional buy-and-hold strategy but substantial losses when transaction costs are included. First preliminary results indicate that our dynamic strategies that wait for larger expected return differences between the European stock markets do much better after incurring transaction costs and also still beat the simple buy-and-hold strategy. The introduction of trading costs will effectively limit the amount of switches in the portfolio. Hence, a strategy should be devised to only trade when a sufficient gain in expected utility can be made, without too high costs involved. Assume the position at day t − 1 is described by portfolio weights wt−1 , and the model established that an optimal portfolio for day t would be wt∗ . Expected utility on t would P day ∗ −w be U (wt−1 ) if the old portfolio is left untouched, or U (wt∗ ) if a fraction ∆w ≡ 21 |wit it−1 | of the portfolio is traded. A possible strategy could be: • If
U (wt∗ ) − U (wt−1 ) > αQ ∆w
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Figure 6: Change in utility ∆u to be gained by changing portfolio, versus change in portfolio ∆w, using the DCSV model. Table 3 describes the outcome of such a strategy, both for the unrestricted and restricted optimisation, for the DCSV and DCC models. For comparison, the outcome of the historical trading strategy are provided as well. The columns indicated the fraction of days on which the portfolio moves, the total turnover of the portfolio over the period 1999–2007, and the average daily return, volatility, and utility over the full investment period. P Both for the unrestricted and the restricted optimisation it is found that the turnover ∆w, directly related to the incurred trading costs, drops strongly even when limiting trades to occur only if the gain in utility is twice Q, the median gain in utility per trade. The average returns, volatility and utility are affected, of course, but only slightly so. E.g. the average daily return drops from 0.130 for the daily trader to 0.118 for the trader with α = 10, who trades on average every 12th day, using unrestricted optimisation and the DCSV model. The active trading strategies, using either DCSV or DCC, still have far higher turnover then the passive historical trading strategy. The returns, volatility and utility of the DCSV model are far better than of the other strategies, but the DCC model only manages to lower volatility somewhat compared to the passive strategy.
4
Conclusions
When actively managing a portfolio of assets, existing theory explains very well what the optimal portfolio would be given a set of estimates of expected mean and variance. No clear guidance is provided however on how to obtain those estimates, and many a time a historical approach is taken to come up with these estimates. Such a historical approach however forgoes fully the activity in the market: An optimal portfolio should not be related to last years returns and variances, but to today’s estimates of these quantities. 13
α=0 α=2 α=5 α = 10
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Table 3: Limiting the number of trades Unrestricted optimisation Restricted optimisation DCSV P P moves ∆w rt r2t ut moves ∆w rt r 2t ut 1.000 159.932 0.130 0.494 0.002 0.917 72.321 0.064 0.508 −0.064 0.417 140.670 0.130 0.507 −0.001 0.368 61.641 0.064 0.513 −0.066 0.181 111.819 0.127 0.525 −0.009 0.159 48.919 0.064 0.523 −0.068 0.082 85.956 0.118 0.564 −0.027 0.078 41.620 0.061 0.539 −0.075 DCC P P 2 moves ∆w rt rt ut moves ∆w rt r 2t ut 1.000 407.092 0.060 0.676 −0.110 0.990 168.814 0.044 0.636 −0.115 0.310 248.070 0.058 0.667 −0.109 0.334 121.119 0.044 0.635 −0.115 0.101 129.141 0.053 0.662 −0.113 0.127 73.403 0.044 0.633 −0.115 0.035 54.374 0.035 0.645 −0.127 0.047 42.085 0.039 0.639 −0.121 Historical P P moves ∆w rt r2t ut moves ∆w rt r 2t ut 0.003 7.915 0.046 0.807 −0.157 0.003 2.619 0.050 0.749 −0.138
P The table reports the fraction of moves, turnover ∆w, and average return r t , volatility r2t , and utility ut for the DCSV, DCC and historical trading strategies, either unrestricted or without short selling, limiting trading through different levels of α. Results are all using a risk aversion of c = 0.5.
For this purpose, this article introduces both the Dynamic Correlation Stochastic Volatility (DCSV) and Dynamic Conditional Correlation (DCC) models, with an extra component measuring expected returns. The models are capable of providing a filtered estimate of today’s mean and variance, given the state of the markets. Combining bivariate results, even the mean and variance of a large portfolio can be extracted. We show how the model-based approach allows for an active trading strategy, with the portfolio following closely expected returns and volatilities in each of the markets under consideration. The DCSV model is found to provide more consistent evaluations of the optimal portfolio, with less of a bounce-back effect of switching between different positions. Even so, trading every day is probably suboptimal if trading costs are taken into account. Therefore we continue the analysis by trading only if the gain from a change in portfolio is large enough. Such an active trading strategy leads to returns, volatilities and utilities which seem far superior than any of the other strategies considered, with a relatively low turnover of the portfolio over the sample. In devising such an active strategy, it is essential to allow for stochastically time varying correlation and variance instead of using the more common DCC-GARCH approach, as the latter is too sensitive to one large shock disturbing the portfolio. Our empirical findings have strong implications for portfolio managers. We find that actively managed portfolios can significantly improve the classical mean-variance trade-off. In contrast to popular advice, we show that a buy-and-hold strategy should be avoided. The present application takes 6 major European indices into the portfolio. There is no limit to the size of the portfolio considered, apart from a slight increase in computational load for evaluating the models and portfolios, of course. But the method is equally applicable, with ease, to portfolios considering various alternative asset classes (bonds, hedge funds, private 14
equity traded funds), to assets traded on different U.S. stock markets, or even to world-wide assets if an adaptation of the model to incorporate currency risk is made. Furthermore, the model and utility function at present do not explicitly distinguish between upside and downside risk. Further improvements could be made in this direction as well.
A
Modelling details
This appendix provides additional details on the models that are used, and the estimation method applied to them. Repeating the basic model (1)–(2), (i,j)
ǫt ∼ N (0, Σt
rt = µ t + ǫ t ,
ηi,t ∼
µi,t+1 = φi µi,t + ηi,t ,
),
(1)
2 N (0, σi,η ).
(2) (i,j)
this is combined with either a DCC-GARCH specification for the covariance Σt , or with a DCSV specification. Both need their specialised estimation procedure. DCC-GARCH specification and estimation The Dynamic Conditional Correlation model (Engle and Sheppard 2001; Engle 2002, DCC) concentrates on the elements of covariance matrix ρt σ1 t σ2 t σ12 t , (3’) Σt = ρt σ1 t σ2 t σ22 t with the elements σi2t = ωi + δi σi2t−1 + αi Et−1 (ǫ2i t−1 ), Et−1 (ǫi t−1 ) Et−1 (ǫj t−1 ) + λqij t−1 , qij t = (1 − λ) σit−1 σjt−1 q12 t ρt = √ . q11 t q22 t
(6) (7) (8)
Note how the GARCH equation usually depend on the observed ri,t−1 , but now the error term of the return equation should drive the GARCH processes. As the level µi,t−1 is not observed with certainty, the error term and its square need to be replaced by their expectations given all available information at the respective time period. These expectations are extracted from running a step of the Kalman filter (Anderson and Moore 1979; Harvey 1989), adapting the variances for the next period, and taking the next step with the Kalman filter. A side product of the Kalman filter equations is the prediction-error decomposition of the data series, with variances. These can be combined into a likelihood value for the data conditional on a set of parameters. A standard maximisation procedure optimises the likelihood over the free parameters, and filtered expectations, variances, correlations are extracted for the maximum likelihood estimates. Stochastic specification using DCSV Instead of the deterministic filter for the variance as applied in the DCC model above, now the variances and correlations in (3) are allowed to
15
evolve randomly over time. The specification for these quantities is exp(qt ) − 1 , exp(qt ) + 1 = qt + ηt ,
ρt ≡ qt+1 σi2t
(9) ηt ∼ N (0, ση2 ),
= exp(hi t ),
hi t+1 = γi + φ(hi t − γi ) + ξi t ,
ξt = N (0, Σξ ) .
(10) (11) (12)
With such a specification for the covariance matrix, the likelihood function of the DCSV model is only observable conditional on this unobserved matrix. To get rid of the unobserved components, both hi,t , ρt have to be integrated out of the likelihood function, which is not an easy task. Instead of literally integrating these quantities out (an virtually impossible task, as the dimension of the integral would be 3 × T ) and optimising the resulting likelihood function directly, a Markov chain Monte Carlo method is used with data augmentation. The algorithm proceeds, after initialising the parameters θ = (φi , γi , σi ξ , ση ), i = 1, 2 and states, by iterating over the following steps: i Drawing a new set of mean returns µt from the simulation smoother (De Jong and Shephard 1995; Durbin and Koopman 2002), conditional on the variances and correlations ii Sampling a new vector of ρ, by successively sampling ρt |ρt−1 , ρt+1 , rt , µt , ht , θ, for t = 1, .., T . As this density is not available in closed form, we use a random walk MetropolisHastings (MH) step to sample from the posterior density P (qt |qt−1 , qt+1 , rt , µt , ht , θ) ∝ P (qt |qt−1 , qt+1 , ση ) × L(rt ; µt , ht , ρt (qt )). Both densities are simple Gaussian, such that sampling a new value of qt is not difficult. After sampling qt , it is transformed back to ρt ; iii Sampling two new vectors of h jointly, from the density of ht |ht−1 , ht+1 , rt , µt , ρt , θ. Again, the density is not easily tractable in closed form, but the posterior is again a combination of the density of ht |ht−1 , ht+1 , θ and the likelihood of the present observation L(rt |mut , ht , ρt , θ). iv Assuming a prior π(ση ) ∼ IG-1(αη , βη ), the posterior of ση is !−1 T 2 X (q − q ) 1 T − 1 t t−1 + αη , β = + P (ση |ρ) ∼ IG-1 α = 2 2 βη t=2 v The parameters γi |hi follow a simple normal density, assuming a normal prior with mean µγ and variance σγ2 ; vi The remaining parameters φi , σi ξ are sampled per asset i using another MH step, with a random walk normal candidate density. After performing a sufficient number of iterations this algorithm results in a sample from the posterior density of the parameters θ and states h, ρ. The posterior mode of the parameters θ estimated over the sample is used as input for a particle filter (Pitt and Shephard 1999), to 16
extract filtered estimates of the states ρt , ht , µt |r1 , .., rt , conditioning only on past and present information. This gives a more fair comparison than using the output of the MCMC chain, which describes the distribution of the states ρt , ht |y1 , .., yT conditional on the full data set. To be included: Remarks on priors and parameter restrictions.
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