In this post, I will explain the main ideas behind the implementation of one of the most efficient portfolios in terms of risk-return performance: the

**minimum-variance policies**.As shown in this blog, this policy attains better

**risk-adjusted returns**than the corresponding market index. For this reason, its popularity is growing in recent times.For instance, several investment companies, such as Acadian Asset Management, Martingale Asset Management, or Robeco, offer investment strategies based on these portfolios.

Finally, there are other sources of information on the Internet with more evidence supporting these strategies. Among them are the Falkenblog and the Quantivity blog.

The index provider MSCI offers several minimum

*-volatility indices*as benchmarks for financial institutions. And Standard & Poor’s has just announced the next lunch of an S&P500 low-volatility index. Moreover, iShares and Russell are planning to launch both low-volatility ETFs.Next, I will show some of the properties of these strategies. The main fact is the following:

*minimum-variance portfolios usually perform better out-of-sample than mean-variance portfolios, even when performance measures depend on both mean and variance, such as the Sharpe ratio*.In this previous post, you can find some financial explanations for this good performance. Next, you can find through an extensive experiment the statistical and financial evidence.

The next graph illustrates the statistical motivation for the minimum-variance portfolios.

The blue line represents the efficient frontier, in the Markowitz portfolios, when we know the true mean returns and the covariance matrix. We can observe the

**classical risk-return trade-off**: more risk is compensated by more return.But the red line represents the

**obtained when we estimate the mean returns and the covariance matrix using historical data. The***ex-post efficient frontier**risk-return trade-off does not hold anymore*. Or in other words, minimum-variance portfolios (ignoring estimates of mean returns) are**more efficient (ex-post)**than portfolios trying to optimize (ex-ante) the trade-off between risk and return, like the Markowitz classical ones.The main explanation is because in the financial markets, the estimation error in the expected returns is much larger than that in the covariance matrix. That is, the expected returns cannot be forecasted reasonably well and the associated classical portfolios

**suffer from the estimation risk**.Hence, the best is to focus on the estimation of the covariance matrix which is not as sensitive to the impact of the estimation risk. The

**minimum-variance portfolio**is defined as the solution to this problem:where

*w*denotes the vector of portfolio weights and ∑ denotes an estimate of the covariance matrix. The constraint simply denotes the weights must sum up to 1. In this formulation we allow for short sells, later we will see how to control the short-selling in an efficient way.Obviously, the performance of the minimum-variance portfolio depends crucially on the quality of the estimated covariance matrix, especially when the matrix is large.

Next, I will show some of the best estimation procedures for the covariance matrix.

- The classical estimation is simply
**the sample covariance matrix**. The associated portfolio will be denoted by ‘MinVar’. This procedure does not work very well in practice because it is quite unstable, especially when we have a large number of assets. The next procedures deal with this problem. **The factor model**. In this case, we assume the asset returns are a linear combination of the returns on a number of factors. In the presented experiment, we consider a single factor model (the CAPM) where the factor is the market index, but other factor models may be used, such as those proposed by Fama and French (1992). The associated portfolio will be denoted by ‘Factor’.**Shrinkage estimators**. These are my favorites. The idea is to compute a weighted combination between the sample covariance matrix (unbiased but with large variance) and a target matrix (biased but with low variance).

In particular, Ledoit and Wolf (2003) propose as the target matrix the single-index covariance matrix implied by the CAPM model. On the other hand, Ledoit and Wolf (2004b) propose the constant correlation matrix as the target matrix. Finally, Ledoit and Wolf (2004a) consider a scaled identity matrix as the target.

For the experiments, I have chosen the latter procedure because as the estimation risk is increased, the associated portfolio tends to the 1/N policy. The associated portfolio will be denoted by ‘LW’.

**Imposing short-sale constraints**. The idea is simply to avoid short sales, obtaining a long-only portfolio. Hence the formulation is

Jagannathan and Ma (2003) show that imposing the short-sale constraint is equivalent to shrinking the extreme elements of the covariance matrix, hence reducing the estimation risk. When the sample covariance matrix is used, the associated portfolio will be denoted by ‘Long MinVar’. When the LW covariance matrix is used, the associated portfolio will be denoted by ‘Long LW’.

The main advantage of this approach is that it deals with the estimation risk (indeed it is a shrinkage approach). Another advantage is that the associated portfolio is sparse, that is, it is only composed of a few proportion of assets, therefore suitable for a small investor. One (little) disadvantage is that this portfolio requires the use of an optimization solver (

**quadratic programming**).**Constraining the portfolio norm**. One of the disadvantages of the previous formulation is that it avoids short sales, hence we may miss some opportunities in terms of risk or returns. We could use the previous unconstrained LW approach, but the amount of short sales can be quite large, for instance go 200% long and 100% short.

In order to control for the amount of short sales, we can constraint the portfolio norm with the following formulation, see DeMiguel et al. (2009):

Here, the norm ||.|| represents the 1-norm, and δ represents the shortsale budget that

**can be freely distributed among all the assets**.For instance, if we want to construct a “130:30” portfolio (long 130% and short 30%), then δ should be equal to 1.6. In the presented experiment, we consider three such portfolios, denoted by ‘110:10’, ‘120:20’, and ‘130:30’, respectively.

As in the case of the constrained minimum-variance portfolio, this approach hedge against the estimation risk and also maintains the associated sparse properties, that is, it only invests on a few proportion of assets. Again, one (little) disadvantage is that these portfolios require the use of an optimization solver (

**quadratic programming**).**The experiment**.

In the following experiment, which extends the weekly portfolio recommendations in this blog, I will show that

**minimum-variance portfolios attain much better risk-adjusted returns**than broad indexes like the S&P 500.The experiment is as follows: every 4 months in the last years, I choose the components in the S&P500 index and compute the previous minimum-variance strategies using past weekly data to estimate each of the covariance matrices. Then, every week I compute the associated ex-post portfolio returns

**net of proportional transaction costs of 40bps**. Finally, I will show several performance measures: portfolio mean returns and volatilities, Sharpe rations, Value-at-risk, correlation with market index, etc.The results are the following. In the next graph, we can see the evolution of $1 over the last year for each of the minimum-variance portfolios. I have omitted the portfolio with the classical estimation (MinVar) because its performance is too bad due to the large instability incurred when inverting the covariance matrix.

The final returns (net of transaction costs) are roughly similar for all the strategies: around (13%). But we can observe the evolution or volatility for each strategy is different.

Hence, let us analyze the performance results in more detail. In the next table, the following performance measures for each strategy over the last year (52 weeks) are shown: annualized mean return (net of transaction costs), annualized volatility (vol), annualized Sharpe ratio (SR), and weekly 95%-VaR.

Factor | LW | Long MinVar | Long LW | 110:10 | 120:20 | 130:30 | S&P500 | |

Mean return | 0.117 | 0.144 | 0.146 | 0.147 | 0.147 | 0.144 | 0.141 | 0.127 |

vol | 0.092 | 0.092 | 0.120 | 0.110 | 0.103 | 0.098 | 0.095 | 0.167 |

SR | 1.276 | 1.561 | 1.217 | 1.333 | 1.420 | 1.471 | 1.486 | 0.759 |

95%-VaR | 0.021 | 0.026 | 0.026 | 0.026 | 0.025 | 0.027 | 0.028 | 0.042 |

We can observe that all minimum-variance strategies (except the factor-based portfolio) attain a better mean return than the index. And moreover, all the strategies attain a much better volatility than the index. This implies the Sharpe ratio obtained by the minimum-variance policies is roughly twice better than that of the S&P500. Finally, the Value-at-Risk achieved by the strategies is half that of the index.

From these results we can deduce the

*minimum-variance portfolios dominate the market index in terms of risk-adjusted returns*over the last year. Later we will see this is true over other past periods.One of the reasons these portfolios can achieve better returns is because they do not present a large correlation with the market index. For instance, over the last year, the correlation of the minimum-variance strategies respect to the S&P500 was around 70%. And the tracking error was around 12%.

Now, let us focus on which strategy is better. If we compare the factor model estimation with the shrinkage estimation of Ledoit and Wolf, we can see that clearly the latter is the best. Moreover, it seems it is the best of all portfolios. This is true, but it also requires a large amount of short-selling. For instance, sometimes this portfolio is long 200% and short 100%. Perhaps these positions are too extreme for some investors.

If we avoid short-selling, and focus on ‘Long MinVar’ and ‘Long LW’, we can see the latter is again the best. In other words, it seems shrinkage estimators are the best way to hedge against the estimation risk in the covariance matrix. And as commented previously, these long-only portfolios are sparse, that is, the portfolio composition has a few number of stocks, say around 30 stocks (out of the total 500) over the last year. Hence, these portfolios are suitable for small investors.

But what happens if we allow for some short sales (non extreme) in order to

**reduce the portfolio volatility without sacrificing the returns**. Then the best is to constraint the portfolio norm. For instance, with a 130:30 strategy, we can achieve a 9.5%-volatility portfolio with a mean return of 14%. As with the long-only portfolios, these strategies are also sparse. For instance, the portfolio composition for the 110:10 strategy had around 55 stocks (out of 500) on average over the last year.To better understanding the outperformance of the minimum-variance portfolios, the next graph shows the risk-return space for all the considered strategies over the last year.

Each point represents the mean return and volatility for each portfolio over the last 52 past weeks. Clearly, all the minimum-variance portfolios (except perhaps the one based on factor models) dominate the market index.

I have computed the same yearly graph but using data for the last 5 years and the results are consistent: the minimum-variance portfolios dominate the market index 70% of the time. That is, their

**volatilities are always less**than that of the S&P500 and their returns are better 70% of the time.For instance, over the last 5 years (including the financial meltdown) the Long LW and the 110:10 strategies attained annual mean returns of 6.5% and 6.3%, respectively. The S&P500 attained 3.5%. Regarding volatilities, the Long LW and the 110:10 strategies obtained 15.5% and 14.1%, respectively. The S&P500 attained 22%. This implies Sharpe ratios for the minimum-variance portfolios almost three times better than that of the market index.

As a summary, this post shows how we can develop efficient portfolio strategies that hedge against the estimation risk, and therefore attaining consistently better risk-adjusted returns than market indexes.

**Basic references.**

DeMiguel, V., L. Garlappi, F. J. Nogales, and R. Uppal (2009). A Generalized Approach to Portfolio Optimization: Improving Performance By Constraining Portfolio Norms. Management Science 55, 798–812.

Fama, E. and F. French (1992). The Cross-Section of Expected Stock Returns. Journal of Finance 47, 427–465.

Ledoit, O. and M. Wolf (2003). Improved Estimation of the Covariance Matrix of Stock Returns with an Application to Portfolio Selection. Journal of Empirical Finance 10, 603–621.

Ledoit, O. and M. Wolf (2004a). A well-conditioned estimaor for large-dimensional covariance matrices. Journal of Multivariate Analysis 88, 365–411.

Ledoit, O. and M. Wolf (2004b). Honey, I Shrunk the Sample Covariance Matrix. Journal of Portfolio Management 30, 110–119.

Not very clear how you include portfolio norm constraints into quad programming...is there a way you can include code (looks like R code)?

ReplyDeleteI was also wondering whether you compared 1/N portfolios to min-variance.

Great blog,

Paolo

pcavatore,

ReplyDeletegood point: indeed, because we use a 1-norm, we can reformulate the norm constraints as linear constraints, hence maintaining the quad prog solver.

About the comparison with 1/N, the two policies are complementary. That is, 1/N obtains better returns but also worse volatilities. In terms of SRs, minimum-variance portfolios seem to perform better. Perhaps in the future I’ll write a post with a detailed comparison between these two portfolios.

Thanks for the comments.

Here's an Excel spreadsheet to calculate portfolios on the efficient frontier

ReplyDeleteAbout the 130/30 thing:

ReplyDelete"good point: indeed, because we use a 1-norm, we can reformulate the norm constraints as linear constraints, hence maintaining the quad prog solver."

How exactly do you do that? a norm-1 constraint means that, depending on where you are in the search region, your constraint will change. The only way I could think of to solve this would be to alter the objective function, to penalize departure from the desires norm 1 (eg 1.6).

Could you please clarify this?

Nice post!

The efficient frontier excel spreadsheet mentioned earlier has moved

ReplyDeleteThanks Javier for an interesting post! However, I'm not certain how the "proportional transaction costs of 40bps" is estimated. From what I understand, you adjust the weights on S&P 500 components every 4 months (17 weeks). Does that mean you rebalance 500 positions every 4 months? If so, isn't doing so extremely expensive to execute in practice?

ReplyDeleteBest,

Wilson

Sam,

ReplyDeleteThanks for the comment.

You are right, in theory you need to rebalance 500 positions every 4 months. But in practice note the portfolio contains roughly a 5% of those positions. That is, it only invests in 20-25 stocks (out of the 500). Hence, you only need to rebalance 25 stocks every 4 months. And the transaction costs only apply to these stocks.

Best.

I want to address Dr. Nickel's comment about controlling 1-norm constraint in the 130/30 optimization. I recently wrote a post, accompanied by R code, which shows a few methods to control 1-norm constraint, please have a look at

ReplyDeletehttp://systematicinvestor.wordpress.com/2011/10/18/13030-porfolio-construction/

In a nutshell, if assets in your universe are sufficiently volatile, you can impose following linear constraints:

# To create 130:30

# -v.i <= x.i <= v.i, v.i>0

# SUM(v.i) = 1.6

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