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You want to construct an optimal portfolio.

Let's say you have an alpha signal that arrives with some period (say quarterly). The alpha signal predicts arithmetic returns one-year ahead. You have risk estimates that are updated daily.

The optimizer your utility function is maximizing is expected returns from the quarterly alpha signal (associated with a confidence level), minimizing risk from the daily risk model, and minimizing transaction costs. On Day 1 determining the optimal single-period strategy weights is easy -- just turn the crank on the optimizer.

However, let's say a week passes and you have an opportunity to re-balance. Security prices have changed partly reflecting noise and partly new information. Your alpha forecasts are increasingly stale until the next quarterly alpha signal.

Question - what is the optimal optimal re-balancing procdedure?

There are a couple approaches and it is not obvious which is best - a Bayesian update of alpha signal, shrinkage towards a prior as the alpha signal is increasingly stale, or some other rule-of-thumb re-balancing rule. (Some form of simplification is necessary here since truly the optimal re-balancing would be a multi-period dynamic programming problem that is not practically solvable.)

Here are some approaches:

  1. Do nothing - ignore re-balancing opportunities. Cons: Security may have achieve the effective price targets or portfolio may have deviated from optimality (i.e. marginal returns per unit of risk is no longer balanced).

  2. Naive case - At the next re-balancing period, optimize with the the beginning of period alpha forecasts and the new risk estimates. Drawback is that we have less confidence in the alpha forecast as time passes. Imagine a security that has impounded severe negative news -- your optimizer would load up on this security if you used your beginning-of-period alpha signal. This approach treats all price changes as non-informative.

  3. Ratchet down the confidence of the alpha signal at each re-balancing period (until the next quarterly refresh) and allow the optimizer to shrink towards a prior such as the minimum variance portfolio.

  4. Low the confidence in the alpha signal AND re-calibrate the alpha signal to account for security price changes. For example, if the original alpha forecast for a security is 8% annualized and the position is +10% already then your re-calibrated forecast would be -2% (short). Con: Alpha signals are never so precise. They are most effective at ranking opportunities and this approach might lead to scenarios where you are shorting your strongest candidates on an alpha signal sorted basis.

  5. Somehow treat the price changes as informative and use Bayesian updating to adjust your alpha forecast after observing actual performance. Con: Lots of hand-waving here.

  6. Do not use the optimizer in subsequent re-balancing -- just use your risk model. Specifically, sell securities that have a higher marginal contribution to risk, and buy securities that have a low marginal contribution to risk.

Ultimately this seems like an empirical question that has to be tested. The best answer would cite empirical research on asynchronous optimal re-balancing if it exists.

Thanks!

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Quant guy, you know I generally like your contributions to the site but I'm not sure these long research questions are a good fit for this format. I think the standard is that a question should be answerable by an informed expert in the topic in 15 minutes or less. This question has so many embedded sub-questions. Do you think you could refocus the question on something less broad, taking just one of the sub-questions at a time? I see no reason not to turn a post such as this into a series of questions that ultimately help you arrive at a satisfactory answer to the broader question. –  Tal Fishman Sep 26 '11 at 16:14
    
Good feedback. I removed the sub-questions. –  Quant Guy Sep 26 '11 at 16:30
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2 Answers 2

I haven't completely followed your question. Are you asking about the optimal rebalance frequency in the presence of t-costs and a changing alpha signal?

Usually you would include the t-cost estimate as another term in your optimization (to constrain the weights). This acts to limit the trading "aggressiveness". In other words, you can think of the problem as (1) changing your rebalance frequency because trading too often will result in excess t-costs or (2) changing your trading aggressiveness, where aggressiveness simply denotes how far you trade toward your unconstrained portfolio. They have equivalent effects, except that strictly rebalancing less frequently has the advantage/disadvantage of making you sensitive to end points (e.g. if you rebalance on month-end, then your results will be sensitive to behavior at that specific time).

So I would suggest that you consider instead moving to a higher-frequency rebalance but adding a t-cost term into the portfolio optimization. This is tricky, but you can look at these recent papers:

  1. Garleanu and Pedersen "Dynamic Trading with Predictable Returns and Transaction Costs" 2009. This paper provides a closed form solution to the optimization by assuming quadratic t-costs. One major function of the paper is the additional focus on the alpha decay of the strategy, and the optimal combination of strategies with different alpha decay horizons.
  2. Skaf and Boyd "Multi-Period Portfolio Optimization with Constraints and Transaction Costs" 2009. This is more directly related to the problem as you have described it as a multi-period optimization, except they add additional constraints and solve it without dynamic programming as a convex optimization problem.
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Hi Shane - thank you for the links I will check them out. The scenario here is where the re-balancing period is more frequent (let's say weekly), than your alpha signal (let's say that arrives quarterly). My utility function does directly include transaction costs as a term in the optimization. You can re-balance weekly, however, I'm positing that some adjustment must be made to the alpha signal since it has decayed somewhat from initial portfolio construction. Hope that helps clarify! Thanks again for the links! –  Quant Guy Nov 20 '11 at 19:51
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I think this entire complicated-sounding problem can be shoe-horned into a traditional mean-variance optimization. However, there are multiple embedded sub-problems, each worthy of specific attention (this is why I recommend you split the question up further into multiple smaller questions).

Your expected returns can and should be updated as frequently as possible/feasible. If the signal itself is only observable quarterly, you should use properties of the signal or relationships with higher-frequency variables to forecast the signal. By the way, done naively you will get jumps when the new signal arrives. There are multiple methods of dealing with this, and it is worthy of a separate question.

The re-balancing frequency should be optimized depending on the magnitude of changes in risk and expected return as well as transaction costs. This should include the "nuisance cost" of frequent rebalancing in an otherwise low-frequency strategy. Essentially, this is classic utility maximization, where rebalancing has a fixed up-front monetary cost and a utility benefit in terms of expected risk/return. Changing confidence in the signal can be incorporated directly into the utility function (proper design of this utility function is another worthy sub-question). Any other approach is an ad hoc approximation or rule of thumb.

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