# Category Archives: portfolio

## Strategy Allocation

I’m interested in allocating capital across a series of strategies or assets in a way that balances between maximizing return and minimizing drawdown.   I’ve not spent much time thinking about this previously, so this is a work in progress.

Mean-Variance Approach
It has long been traditional to use the mean-variance approach for portfolio allocation.    A problem with the mean-variance approach is that it penalizes excess returns on the “right” (positive) side of the distribution in addition to the “left” (negative).   This is one reason why many consider the Sharpe ratio to be a flawed measure with respect to how investors see risk.

The mean-variance approach penalizes for sections D and C of the distribution.   An investor is generally happy with excess returns (i.e. D), and unhappy with drawdowns in the distribution (namely areas A and C).  We would prefer to accept the reward of the whole right side of the distribution and attempt to minimize the left.

Lower-Moment Frameworks
To remedy this we can use an asymmetric measure of variance to focus on the part of the distribution that represents our risk.   The family of such moment functions are called the “Lower Moments”.    Lower moment functions allow us to focus on sections A and C in the distribution (ie the drawdowns) and not consider the right side as a contribution towards risk.

Lower moment functions are expressed as follows:

The idea, simply, is that the moment (degree m) is computed on the region of the distribution below some threshold (say 0).

For a risk measure, we may want to penalize returns < 0 or prehaps returns < risk free rate.   We would choose τ according to where we want to penalize.

Application to portfolio
Ok, so the lower-moment looks like a good measure of downside risk.  How do we incorporate it?  Let’s set up the problem:

1. let R represent a matrix of returns
One row for each period, number of columns = number of assets or strategies.
2. let w represent the vector of weights for each asset / strategy
3. let Σd represent the lower-moment covariance of returns with respect to the set of assets or strategies

Assumptions:

1. let us assume that past performance is indicative of future performance  (we can relax this later)
2. let us assume an eliptical (i.e. multivariate normal) distribution for negative  returns
3. let us assume that the drift grows in proportion to variance x time

Ok, those are pretty aggressive assumptions, but they allow a first look at putting this together.

We need to create a utility function that blends the upside and downside, subject to various constraints.  Here f(w) represents the portion of a convex utility function rewarding positive return, and g(w) a function describing the risk penalty.

Let’s formulate the utility as the expected return for the next period (given the prior returns) and the expected negative return component (our risk) over the period:

The first part of our utility E[r] represents the average observed return and the second part the negative drift (our downside risk).    E[r] can be computed in any number of ways, for instance:

• as a time weighted average of past returns
• as a regression model with seasonality
• from a stochastic model calibrated to past returns

The risk penalty is just ½ E[(τ - <w,r>)^2 | r < τ) · Δt, which approximates the expected downside drift over the period represented by Δt.

Now the above is a bit ad-hoc and I may be double counting the upside and downside with E[r].   I’ll have to think on the implications for mixing a lower moment measure and a full moment measure.

There may be some adjustments to make, but have outlined a first pass at penalizing returns and hopefully arriving at a more balanced weight selection.

A more generalized approach could be to compose the utility function from a parameterized sum of upper and lower moments (Farinelli and Tibiletti 2002).   I’ve adjusted this a bit to fit my problem:

The number of moments for the lower portion can be different for the upper portion (or simply let the coefficients of some moments be 0).    What I came up with in the prior section is essentially this function for 2 moments with coefficients {1,0} and {0,1/2}.

The utility function should have been (drift in proportion to sqrt), my bad:

Filed under portfolio, statistics, strategies

## CRP = CR*P?

Procrastinating on a hard problem, I decided to take a brief diversion to look at Constant Rebalanced Portfolios and Universal Portfolios, lured by Max Dama’s post on UP (Universal Portfolios).   I had read papers on these in the past but never explored them empirically.

CRP is the underpinning of Universal Portfolios, so will focus on CRPs.   Simply stated the CRP approach allocates a fixed % of capital to each asset in the portfolio.   The portfolio is rebalanced each period as some assets will have disproportionally increased or decreased in value.   As Max points out, this is basically a mean-reversion scheme.

Unfortunately, this is a “blind” mean-reversion scheme in the sense that there is no measure of the likelihood of mean-reversion or the period over which it will take place.   The implicit assumption is that mean-reversion will occur between rebalancing periods.    The more worrying aspect is that money in “winning” assets will be diverted to “losing” assets (where by losing, refer to assets that trend downward with little MR in the upward direction).

Examples
The classic (and absurd) example where CRP does phenomenally well, is of a pair of assets where one asset is constant and the other asset appreciates and depreciates on alternating periods (in this case up 25% and down 25% repeatedly).   Needless to say, this provides exponential growth (I could only plot the first 100 days without obscuring the other detail):

Empirical Tests
I did not find that CRP did much better than the equivalent Buy & Hold portfolio with the same weightings.  Indeed, depending on transaction costs, could do significantly worse.     There are some asset sets, undoubtedly, that would do much better than Buy & Hold over specific periods, but would be few and far between given the fragility of CRP assumptions.

Making the Concept Work
The CRP concept is one of moving money away from assets we expect will mean-revert (in the negative direction) and increasing money in assets that will mean-revert (in the positive direction).   This is done in a rigid fashion and makes no observation as to whether a devaluing asset is likely to mean-revert in the next period.

It had occurred to me that modifying the rebalancing to take into account the likelihood of mean-reversion or more generally, the likelihood of appreciation or depreciation in the next period would be a better guide in rebalancing the portfolio.    Of course, the performance of such a scheme depends on the degree of accuracy of such measures.

Not surprisingly, there has been work in this area, for instance the ANTICOR algorithm described by (Borodin, El-Taniv, and Gogan) in “Can We Learn to Beat the Best Stock”.    Their approach is to use the autocorrelation and cross-correlation of prior periods to adjust the portfolio weighting in a CRP portfolio.

The fragility in this approach is two-fold:

1. relies on fixed windows as a means to determine correlation or anti-correlation
I would expect that mean-reversion cycle periods would differ for different assets.   That said, they need a way to compare across assets, so may be a reasonable compromise.
2. correlation is a coarse measure
There are other measures that may be more effective in determining future mean-reversion or direction.

That said, the approach is parsimonious and seems to have performed quite well in the empirical tests.

Pattern / Sequence Learning Approaches
There have been a number of papers describing approaches where the choice of weightings for the portfolio in the next period is determined based on finding prior patterns that match the current local pattern in past data.   I.e. the prior K returns are converted to a series of symbols and compared to historical sequences.   One attempts to locate the approximate sequence one or more more times in past history and observe the optimal portfolio weights that maximized cumulative return in the past.    This is repeated with varying length and discretization “fuzziness”.

The observed weights are blended based on the performance of the past weightings and degree of match with the current sequence.    The authors point to excellent results on empirical data.   It is surprising that there is a reasonable amount of information in the daily returns, I had thought would be more dominated by noise.

One Final Note
Although I have been “disparaging”  CRP, it can be shown that some weighting of CRP will yield the most optimal portfolio provided returns are i.i.d.   There lies the rub.

Filed under machine-learning, portfolio

## Shannon’s Investment Strategy

Was reading the book ‘Fortune’s Formula’, which I highly recommend. Claude Shannon, the genius of Information theory fame, came up with an approach to investing in the market using a interesting variant of Kelly’s betting approach.

Assuming a market with constant mean (no drift / trend over time):

1. Invest 1/2 of your capital in an asset
2. Periodically rebalance
3. If the market went up, sell enough units of the asset to have exactly 1/2 of your capital invested
4. If the market goes down, buy enough units of the asset to maintain 1/2 investment

This is an effective scheme (assuming no transaction costs). Why?

1. rebalancing implicitly executes a mean reversion strategy
2. losses reduce the capital in the market
3. wins increase the capital in the market

In effect, this is a ratcheting investment approach. As was pointed out, most assets are not constant mean over time. This would imply a strategy that trades mean reversion around a longer term drift in the mean. How might such a strategy work?

1. since drift might be upwards or downwards, fundamental position should be long or short
2. rebalancing should take into account the expected movement of the mean so that the ratio of cash to position will depend on this

This is referred to as a Constant Rebalanced Portfolio (CRP). Thomas Covers, later extended on this concept with non-even distribution of allocations with his Constant Universal Portfolio (CUP).