Estimating Momentum Returns for Mean-Variance Optimization

# What is the expected return I should use for the momentum strategy in MV optimization framework?


# What is the expected return I should use for the momentum strategy in MV optimization framework?












As all research on the momentum strategies are focused on the indicator, i.e. the entry point, there seems not much discussion on its expected return? Though there are some discussions on the exit rules, it still doesn't clearly quantify the expected profit in advance. Not like a mean-reversion strategy, where its expected profit is clearly pre-determined: abs(entry price - mean). This caused a problem to me in the MV optimization framework, which requires input of expected return in advance.

What is the expected return I should use for the momentum strategy in MV optimization framework? Is there any classic reference for my question?

## Answer by Alexey Kalmykov (score 5, accepted)

https://quant.stackexchange.com/a/2991

If you don’t have any specific model which describes the behavior of the asset being traded, you can estimate the empirical distribution of returns by backtesting your momentum strategy. Then you can adjust this estimate during your strategy’s lifetime from your trading results. Additionally you can enhance this by accounting for different market regimes (e.g. “risk on/risk off”, volatility regime) or incorporate other external factors which influence the returns on your strategy (e.g. state of economy).

If you assume a model for the traded assets and your strategy exploits some properties of that model (e.g. autocorrelation), you can try to come up with a closed-form expression for the expectation of returns. In case the analytical solution is not feasible, you can use Monte-Carlo to approximate the expected value.

## Answer by John (score 3)

https://quant.stackexchange.com/a/3453

Another possible approach is taking views a la Black-Litterman. There is a 2006 paper "Incorporating Trading Strategies in the Black-Litterman Framework" that discusses the methodology in more detail.

There are several practical issues that one should consider when implementing a momentum strategy with optimization.

- I would pay careful attention to the time horizon of the optimization and the holding period of the momentum strategy. For instance, assume an optimization with a one-year time horizon, but with a momentum strategy that re-balances monthly. The Black-Litterman approach on an individual security basis would be more difficult in this case because your view isn't really over the whole one-year period. Rather it is only over the first month. The method that Alexey describes where one takes the momentum portfolio avoids this problem.

- The other model approach that Alexey describes is a model-based approach. There is a lot of latitude in what a model-based approach could consist of. I just want to note some practical issues with the Fama-Macbeth procedure as a modelling approach. This is a cross-sectional technique that is similar to constructing a momentum portfolio, but it also controls for other factors. This is not a problem if you want to calculate a momentum factor for 1000s of stocks, but it is more difficult if you want to calculate the returns to currency or asset class momentum since there are fewer entities to examine on a cross-sectional basis.

- It is also important to pay close attention to the trading costs in the optimization. For instance, if you allocate money to the momentum portfolio today, it means transaction costs today, but it also means transaction costs in the future. On the other hand, investments in the momentum portfolio may also be in the rest of the portfolio, so the overlap may help reduce trades. So it's not entirely clear to me what the best approach is to handle the transaction costs of a momentum portfolio, but I do know that you should pay close attention to it.

## Answer by Richard (score 0)

https://quant.stackexchange.com/a/7524

You could perhaps think about expected returns for a momentum stratgey in terms of relative expected returns across your assets i.e. the current price and your moving average (if that's how you define momentum) on each asset. or, more simplistically you could create expected returns in the form of a binary signal (+1 for long or -1 for short). i would then suggest you scale your +1/-1 alphas by the volatility of each asset (signal * vol). Neither gives you a precise expected return but the optimiser will still give you a set of weights that reflect your expectations on a relative basis....