62 Optimizing Optimization
There are two things to note: (1) both are expressed in terms relative to the
initial position and (2) both of these constraints are quadratic. By couching
the constraints in terms of initial positions, we draw an explicit link between
the concepts of robustness and turnover. An alternate way of thinking about
Equations (3.5) and (3.6) is as “ smart turnover ” constraints — they both limit
the optimizer’s ability to deviate from the initial position, except for the case
where there is a good reason (stable, historically accurate alpha) to do so.
Equation (3.5) is quadratic in order to penalize greater deviation from the
initial position for stocks with a large FE — in this way, it largely resembles
a quadratic penalty, levied as a constraint, rather than an objective function
term. Equation (3.6) is quadratic because that is the nature of a covariance
term, analogous to risk.
3.4.6 Preliminary results
Traditional methods used to stabilize portfolios, such as applying turnover
constraints, merely serve to limit the damage of incorrect data being input into
optimization techniques. These methods do nothing to differentiate between
the “ informative ” and “ noisy ” data that goes into the process.
Controlling FE exposure and variance improves portfolio stability more
than traditional methods. It directly addresses the main cause of instability:
the instability in returns expectations, by limiting the effect of the “ noisy ” and
allowing the “ informative ” to add value to the process.
To test this, we defined the following research project:
● US mid- and large-market capitalization equity universe limited as detailed below;
● Alphas from multifactor cross-sectional model (Value, Momentum);
● Limit universe to securities for which 30 months of data (returns, FE) are
available — in most cases, this limited the universe to about 500 stocks due to the
limited alpha universe during any given month.
● Backtest run on a monthly basis, starting January 1997 – November 2005;
● Portfolio weights were adjusted at the end of each month for changes in price and
optimized using one of two strategies using the market cap weighted universe as the
benchmark:
● Robust — constraining both M and S from Equations (3.5) and (3.6) to 1% each
and tracking error to 5%;
● Control risk and turnover — tracking error constrained to 5%, turnover con-
strained to realized turnover from robust strategy.
The resultant strategy performance is clearly dependent on how much it
costs to trade. Two sets of results follow:
- No cost to trade
- 30 bps per unit of currency turned over
First , the “ no cost to trade ” results ( Figure 3.1 ).
Now the 30 bps to trade results ( Figures 3.2 – 3.4 ).
From the “ net of control strategy ” chart, it is clear that for the alphas pro-
vided in this test, the robust strategy outperformed the control strategy over