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I am designing an asset allocation strategy/fund which invests in four asset classes (via four independent sub-funds):

  1. Domestic equity
  2. International equity
  3. Domestic fixed income
  4. Foreign currencies

The strategy must be fully invested in these four funds at all times. I already have a default "strategic" allocation to the four funds based on our baseline views for the performance of the four asset classes. The managers of the strategy may, at times, have tactical views on relative value between

  • Domestic equity and fixed income
  • International and domestic equity
  • The broad direction of foreign currencies (relative to USD)

Managers express their tactical views discretely on a 5-point scale from strong sell to strong buy (where buy/sell refers to the first of the two asset classes in the pairings).

The catch is that the view on international vs. domestic equity relative value is best seen as a view on currency-hedged relative returns, whereas the international equity fund is not currency hedged. The currency fund's holdings do not correspond exactly to the currency exposures of the international fund, but they are close.

My job is to research how and how much to shift from one fund to another in response to the manager's views. For the purposes of this problem, it is reasonable to assume we have standard mean-variance preferences. I must also take into account transaction costs, which are rather large. Hence tactical moves must be relatively long-lived and stable.

Any ideas on how to approach this problem? How should I calculate the tradeoffs between various asset classes?

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Can you make the simplifying assumption that the currency exposures of the international fund are the same (not just close) to those of the the currency fund's holdings? –  Quant Guy Sep 9 '11 at 18:02
    
@Quant Guy of course you can, but that doesn't mean you should. The currency fund is actively managed independently of the international fund. In any case, an answer which depends on that simplifying assumption would still be useful as a first approximation. –  Tal Fishman Sep 9 '11 at 18:06
    
Agreed. Are there confidence levels or probabilities associated with the views? If yes, do you have a shrinkage target for your portfolio? Also, can we assume your utility function is plain vanilla mean-variance plus a penalty for transaction costs? –  Quant Guy Sep 9 '11 at 18:25
    
@Quant Guy I answered inside the question. –  Tal Fishman Sep 9 '11 at 18:33
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2 Answers 2

up vote 3 down vote accepted

There are a couple components to this problem:

  1. Construct a portfolio incorporating relative views where weights shrink towards a default policy 1a. Some views are in currency-hedged terms 1b. Relative views are on a 5-point scale
  2. Maximize a mean-variance utility function incorporating a penalty for transactions costs and requiring that weights sum to one.

Solve for 1: Construct a posterior that blends your prior (default strategic policy allocation) with the relative currency-hedge views. Black-Litterman is the right idea but it uses CAPM as the prior. You need a more flexible framework. I would suggest Meucci's Fully Flexible Views (2008) which uses entropy-pooling to blend an arbitrary number of views and confidences. Incidentally, Meucci has fully commented MATLAB code on his website www.symmys.com (I have a project to convert this to R for whoever is interested!)

Solve for 1a: To address that fact that views are based on currency-hedged returns whereas the component assets are not currency hedged, you need to translate the currency-hedged view into a non-currency hedged view. Then you can proceed to incorporate this view into Meucci's framework. (Side note: Black-Litterman will not work here because if you define a relative view in the Pick Matrix on the basis currency-hedged whereas the covariance matrix where these views are propogated are on the basis of non-currency hedged returns). Unlike Black-Litterman where you can only take views on return outcomes, you can also include views on an arbitrary number of factors which might include the currency-hedged returns. So you can express a view on the currency-hedged return in Meucci's framework as well.

Solve for 1b: Meucci's framework supports lax-views such as relative ranking and partial information views. So your view matrix (specifically, the inequality view matrix in Meucci's implementation) could express the idea that the returns from Strong Buys are greater than than returns from Buys, and so on.

Solve for 2: Now that you have your posterior, you need to perform an optimization.

I suggest incorporating transactions costs directly into your objective function. Your objective function would you have three quantities: arg(weights) max utility = posterior*optimal weight vector - lambda1*variance - lambda2*transactions costs.

Variance is determined by weights-transposed*covariance matrix*weights. Lambda1 is the usual mean-variance risk-aversion parameter. No surprises here - naturally this is mean-variance optimization.

We have an additional quantity. Lambda-2 is your transaction cost aversion parameter. Transactions costs are the cost involved for turning over your portfolio. The function returns the transactions costs given the current weight vector, the proposed weight vector, portfolio value, and some transaction cost assumption. A simple way is to take sum of (current weight vector minus the optimal weight vector) * portfolio value * cost per $ traded.

You are essentially creating a efficient frontier surface where your utility is maximized for various risk-aversions and transaction cost aversion levels.

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Thanks for answering. I will definitely look into Meucci's Fully Flexible Views. I agree with your view in principle that we should transform the view on international returns to conform to the international fund, but my manager's view is that we should do the opposite, stick with a view on currency-hedged returns and trade simultaneously in unhedged equities and in currencies. I settled on OLS regression for figuring out the right ratios, which turned out to be close enough to 50-50 in my case. –  Tal Fishman Sep 13 '11 at 20:09
    
Also, my manager is leaning towards partial equilibrium (solving the problem as multiple smaller sub-problems) rather than general equilibrium (Black-Litterman) to preserve transparency. In that case, @ProbablePattern's recommendation of running simulations is more representative of the ultimate process we will follow. –  Tal Fishman Sep 13 '11 at 20:12
    
I decided to split the points by giving you the check mark. –  Tal Fishman Sep 14 '11 at 0:29
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I would use something similar to Black Litterman where both confidence of manager views as well as dynamic correlations are used to re weight asset classes. For a good look at how transaction costs affect long term allocation decisions with changing parameters, you may be interested in Balduzzi and Lynch (1999).

Another approach to consider is to look at autocorrelation within each asset class to determine absolute magnitude of a weight change in a single period. Stock Market Mean Reversion and the Optimal Equity Allocation of a Long-Lived Investor might be useful.

Yet another way to approach the problem would be to run simulations to determine the TAA bands to place on each asset class to maximize long run returns incorporating transaction fees. This method would be sensitive to initial weights so a risk neutral portfolio framework may be a useful starting point. I use a block bootstrap to account for auto correlation in the return series and time varying correlations.

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1  
Thanks for the suggestions. I looked at both papers. The first talks about the cost of myopia, while the second tries to incorporate mean-reversion into the asset allocation problem. Both are far too theoretical for what I'm doing, and neither address my specific problem. I up-voted in any case to reward effort, but this, unfortunately, does nothing to actually answer my question. –  Tal Fishman Sep 9 '11 at 16:15
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The basic problem of allocation based on manager conviction can be solved with Black Litterman. To address the transaction cost constraint, you need to determine how good the managers are particularly under different asset class correlations. This is necessary to determine whether a given level of conviction is sufficient to overcome transaction costs. Without understanding the skill level of managers, you will not be able to find the magnitude of the trade. In studying the skill level of managers, you will need to determine the time horizon that a manager can accurately express views. –  ProbablePattern Sep 10 '11 at 12:30
    
You are right that the basic problem may be solved with Black-Litterman, and thank you for reminding me of that fact. However, the essence of my question is the method of calculation of tradeoffs (in other words, how to set up the problem so that it is amenable to B-L or any other solution method). B-L, btw, may be overkill for my application, and a partial equilibrium approach seems the most likely candidate at this time. –  Tal Fishman Sep 12 '11 at 14:10
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Another way to approach the problem would be to run simulations to determine the TAA bands to place on each asset class to maximize long run returns incorporating transaction fees. This method would be sensitive to initial weights so a risk neutral portfolio framework may be a useful starting point. I use a block bootstrap to account for auto correlation in the return series and time varying correlations. –  ProbablePattern Sep 13 '11 at 0:49
    
I like that idea much better than your original ideas. Thanks for the suggestion. Please consider incorporating it into the main answer body. –  Tal Fishman Sep 13 '11 at 20:05
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