#### Momentum And Markowitz: A Golden Combination by SSRN

Flex Capital BV; VU University Amsterdam

BPG and Associates

QuantStrat TradeR

May 16, 2015

**Abstract: **

Mean-Variance Optimization (MVO) as introduced by Markowitz (1952) is often presented as an elegant but impractical theory. MVO is “an unstable and error-maximizing” procedure (Michaud 1989), and “is nearly always beaten by simple 1/N portfolios” (DeMiguel, 2007). And to quote Ang (2014): “Mean-variance weights perform horribly… The optimal mean-variance portfolio is a complex function of estimated means, volatilities, and correlations of asset returns. There are many parameters to estimate. Optimized mean-variance portfolios can blow up when there are tiny errors in any of these inputs…”.

In our opinion, MVO is a great concept, but previous studies were doomed to fail because they allowed for short-sales, and applied poorly specified estimation horizons. For example, Ang used a 60 month formation period for estimation of means and variances, while Asness (2012) clearly demonstrated that prices mean-revert at this time scale, where the best assets in the past often become the worst assets in the future.

In this paper we apply short look-back periods (maximum of 12 months) to estimate MVO parameters in order to best harvest the momentum factor. In addition, we will introduce common-sense constraints, such as long-only portfolio weights, to stabilize the optimization. We also introduce a public implementation of Markowitz’s Critical Line Algorithm (CLA) programmed in R to handle the case when the number of assets is much larger than the number of lookback periods.

We call our momentum-based, long-only MVO model Classical Asset Allocation (CAA) and compare its performance against the simple 1/N equal weighted portfolio using various global multi-asset universes over a century of data (Jan 1915-Dec 2014). At the risk of spoiling the ending, we demonstrate that CAA always beats the simple 1/N model by a wide margin.

### Momentum And Markowitz: A Golden Combination – Introduction

Imagine a high-school student, Harry, ignorant of both Markowitz and Lehman Brothers, enrolled in a class called “Finance for dummies”. At the end of august 2008 his teacher asks him to compute the so-called Efficient Frontier (return versus volatility) for a long-only mix of SPY (the SP500 ETF) and TLT (the long-term Treasury ETF) in order to determine the highest returning portfolio with target volatility (TV) of say, 10%.

Harry uses only monthly (total) return data for both assets and chooses to investigate the optimal portfolio over just the prior four months. He starts with a simple spreadsheet with only the four monthly returns (May – Aug 2008) for SPY and TLT, see fig. 1.

For each weight combination , Harry constructed the monthly portfolio returns, and observed the corresponding average portfolio returns and volatilities of rP over the 4 months, using simple spreadsheet functions. See fig. 2. Then he plotted both the average returns and volatilities for all 11 weight combinations as a so-called Efficient Frontier (EF). See fig. 3.

As Harry is aiming at a target volatility of 10%, he chooses the weight mix represented by the far right upper point (red dot) of the EF (with an average return of 6.7% and volatility of 9.1%) for his future portfolio. This point corresponds to wSPY=0% and wTLT=100%. At the end of September, when Harry examines the performance of his optimal portfolio, he observes that the portfolio generated positive returns, and proved resilient to the large crashes experienced by most asset classes in that month. In fact, if he had continued to follow this process at the end of each subsequent months, he would have ‘whistled past the graveyard’ of the 2008 crash with only TLT in his portfolio.

Let’s fast-forward 12 months from Harry’s original exercise to the end of august 2009. When Harry runs the same analysis with the same lookback of 4 months (May-Aug 2009) he observes the following returns and volatilities (see fig. 4 and 5).

Now with his target volatility of 10%, Harry chooses a portfolio of 90% SPY and 10% TLT (the red dot in fig 5.). This worked out well as Harry’s portfolio benefitted from the recovery in stocks.

The positive outcomes observed from Harry’s choices hints at a potential opportunity. But what has Harry done? Actually, Harry simply used Markowitz’s Mean Variance Optimization (MVO), as we will show later, but with a few constraints. Specifically, he only considers portfolios with positive weights (i.e. no short-sales), and he uses a short lookback period for parameter estimation. (He also imposes discrete weights, but this is a topic for another paper).

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