In their third article on risk management, Edhec's Noël Amenc and Lionel Martellini consider the usefulness of optimisation methods.
In the third of a series of articles on risk management techniques in asset management, we provide simple examples of
the usefulness of optimisation methods. We first present an exercise with a focus on minimising the portfolio variance, and then turn to a focus on extreme risk management. Our goal is not to introduce a fully fledged state-of-the-art optimisation model, but instead to present evidence that even a basic and simple procedure can lead to substantial efficiency gains.
The most widely quoted quantitative model in the strategic allocation literature is, of course, Markowitz’s (1952) optimisation model. The input data are the means and the variances, estimated for each asset class, and the covariances between the asset classes. The model then provides the optimal percentage to assign to each asset class to obtain the highest return for a given level of risk, measured by portfolio volatility.
Drawbacks of Markowitz
The main drawback of the Markowitz model is that the optimal proportions are very sensitive to the estimates of expected return values. What is more, the statistical estimates of expected returns are very noisy (see Merton (1980). As a result, the model often allocates the most significant proportion to the asset class with the largest estimation error. Therefore, the efficient frontiers are extremely difficult to obtain in practice and sophisticated methods, such as the one advocated by Black and Litterman (1992), are needed to generate meaningful portfolio decisions.
Even in situations where one is not in a position to rely on active views for the return on various asset classes, there is a pragmatic approach that avoids the problems without abandoning the model. This approach consists of focusing on the only portfolio on the efficient frontier for which the estimation of mean returns is not necessary, namely the minimum variance portfolio. Since the future returns of assets are always difficult to estimate precisely, it is preferable to obtain an efficient portfolio by minimising the risk rather than by optimising the risk/return combination. (For more details on the minimum variance approach, see, for example, Chan, Karceski and Lakonishok (1999) or Amenc and Martellini (2003). Though this approach avoids the problem of estimation risk for the expected returns, it is still faced with the estimation risk for the covariance matrix, a problem that can be addressed by using some of the techniques described above. In the illustration we provide below, for simplicity, our forecasts of the covariance matrix are simply derived from the sample estimates. In practice, an investor may choose to implement the noise dressing techniques referred to above. On the other hand, we choose to deal with the problem of estimation risk, not by imposing a model, but by imposing a maximum constraint of 20% for the weight of a given sector. Imposing constraints has been shown as a useful option as this increases the performance of asset allocation that uses the sample covariances compared to more sophisticated approaches of modelling the covariance matrix (see Jagannathan and Ma, 2003) . This means that our conclusions apply more generally rather than being limited to a certain type of model used for covariance forecasts.
In order to assess the performance of minimum variance portfolios, we run the following tests:
- The data used are daily returns for the broad market indices and for the sector indices. The daily returns are useful in order to improve the precision of the covariance estimates.
- We compute the minimum variance portfolio based on a calibration period of the daily returns observations for the past year. We obtain the optimal weights and hold this portfolio for the following three months. Then, we redo the analysis, rolling the sample three months forward and holding the new optimal portfolio for the following three months. Therefore, our analysis is purely out of sample.
In this way, we obtain the time series of returns for the strategy over the period 10/1996 to 09/2005 (the total period minus the calibration period). The resulting performance statistics of the minimum variance (“min var”) portfolios are given in table 1, which also reports the performance statistics of the corresponding broad market indices (“cap weighted”).
The table indicates the performance statistics over the out-of-sample period 10/1996 to 09/2005. It is based on the daily observations for the returns of sector indices obtained from Datastream Thomson Financial. No results are available for the period 10/1995 to 09/1996, since this data was needed for the initial calibration period.
Minimum variance portfolios
The volatility of the minimum variance portfolio is always significantly lower than that of the corresponding market index. This dominance is not achieved by construction and the portfolios can actually be obtained ex ante by an investor. What may perhaps be more surprising is that the lower risk of the minimum variance portfolio does not lead to a lower expected return for five out of six indices. This is only the case for the minimum variance portfolio of sectors composing the S&P 500 index. All other minimum variance portfolios also have higher expected returns than the corresponding index. Consequently, the Sharpe ratios show strong improvements compared to the market index, except for the S&P 500. The table below summarises the results.
This illustration summarises information from the table with the performance statistics. The right hand column indicates the difference in Sharpe Ratio between the capitalisation-weighted index and the sector allocation strategy in the minimum variance portfolio. It can be seen that in all cases, except for the case of the S&P 500, the minimum variance portfolio obtains a higher Sharpe Ratio then the capitalisation weighted index. It should be noted that for the S&P 500, the average return of the capitalisation weighted index is higher but the volatility is also higher than that of the minimum variance portfolio.
While the minimum variance approach leads to a reduction of average risk, a focus on the reduction of extreme risks is more suitable when considering the avoidance of extreme losses. For this reason, we now present an attempt to minimise portfolio risk, where we choose Value-at-Risk (VaR) or Conditional Value-at-Risk (CVaR) as opposed to portfolio return variance, as the risk measure.
We use weekly data on the DJ Stoxx Euro sector indices (bank, construction, energy, health, insurance, media, telecom, technology, utility) as well as the DJ Stoxx Euro global index for the period extending from January 1992 to December 2005. Using a one-year rolling window sample analysis, we estimate second moments and co-moments (volatilities and correlations), as well as third-order moments and co-moments (co-skewness and and co-kurtosis). Every six months, we optimise the portfolio allocation by minimising the portfolio CVaR, without any constraint on expected returns. We record the out-of-sample performance of these portfolios and compare it to the performance of the DJ Stoxx Euro global index. In table 2, we show the out-of-sample performance of the optimised portfolio and compare it to the performance of the benchmark.
As can be seen from this illustration, the out-of-sample measures of extreme risks (Var and CVaR) appear to have been significantly reduced in the case of the optimally designed portfolios when compared to the benchmark, which suggests that a sound ex-ante focus on risk management at the portfolio construction level is likely to have an ex-post impact at the performance level.
We have also performed a similar experiment in the context of a bond portfolio. The opportunity set is made up of the MSCI ECI Government three to five year index, the MSCI ECI Government seven to ten year index, the MSCI ECI Corp AAA index, as well as the MSCI ECI Corp. BBB index. We design the minimum CVaR portfolio based on a six-month rolling window analysis using three years’ worth of weekly data in the calibration phase, with data ranging from July 1997 through June 2006. We impose a minimum 50% investment in Treasury bonds (and a minimum of 10% in each of its maturity sub-indices) with a maximum 20% investment in high yield bonds, and compare the out-of-sample performance of the optimised portfolio to that of the MSCI ECI overall index used as a benchmark.
These results show the out-of-sample benefits of asset allocation techniques implemented with a focus on risk management.
* Annualised statistics are given
** Risk free rate and MAR are fixed at 2%
*** Non-annualised 5%-quantiles are estimated
This article is based on research included in the EDHEC publication, “The Impact of IFRS and Solvency II on Asset-Liability Management and Asset Management of Insurance Companies,” by Noël Amenc, Philippe Foulquier, Lionel Martellini and Samuel Sender, November 2006. This research was sponsored by AXA Investment Managers.© fe August 2007