There are two extensions/amendments to the formulation presented above that can be made:
• the first extension relates to designing a portfolio that gives a constant excess return, return over and above the market index. This is therefore a model for a relative return portfolio, enhanced indexation (return above the index).
• the second extension relates to designing a portfolio with mixed characteristics, so a portfolio that is a mix of absolute and relative return.
We deal with each of these extensions below.
3.3.1
Enhanced indexation (relative return) portfolio
One argument that can be advanced against ARPs is that in good times (when the market/index is rising) it is a poor investment strategy to aim for an absolute return. Rather one should aim to do better than the index and produce a relative return portfolio, enhanced indexation. Due to the flexibility of our model we can easily amend it to produce portfolios that are designed to out-perform an index. For simplicity we shall continue to call the portfolio produced an ARP, rather than an enhanced indexation portfolio.
Suppose we regress the excess return of our chosen ARP (so return over and above index return) against time over the period we are considering. Our approach to enhanced indexation is to say that, ideally, this regression would have a slope of zero. This equates to a portfolio that (over time) has a constant (expected) excess return per time period (that return being given by the regression intercept).
Now (see notation above) ˆA and ˆB are the least-squares regression intercept and slope when the returns from the index (Rt) are regressed against time. Hence we have that
when the excess return from the ARP (PN
i=1wirit− Rt) is regressed against time it will
have regression intercept (PN
i=1wiαˆi− ˆA) and regression slope (
PN
Our first stage optimisation for the excess return regression slope is to try and achieve a regression slope that is, in absolute value terms, as close to zero as possible. This is therefore minimise |PN
i=1wiβˆi− ˆB|, which can be linearised to:
minimise E (3.22) subject to (3.1)-(3.5), (3.7)-(3.10), (3.15), (3.21) and: E ≥ N X i=1 wiβˆi− ˆB (3.23) E ≥ −( N X i=1 wiβˆi− ˆB) (3.24)
Our second stage optimisation for the regression intercept, to try and achieve a re- gression intercept that is as large as possible, has an objective function that is maximise
PN
i=1wiαˆi − ˆA. In this objective ˆA is a constant and so can be ignored. Hence we have
that the second stage objective here is precisely the same as the second stage objective given above, Equation (3.16), where this objective is optimised subject to (3.1)-(3.5), (3.7)-(3.10), (3.15), (3.17), (3.21) and (3.23)-(3.24).
The third stage follows in a similar fashion as for the ARP-RT model given above. Here the objective is to optimise Equation (3.18) subject to (3.1)-(3.5), (3.7)-(3.10), (3.15), (3.17), (3.19), (3.21) and (3.23)-(3.24).
We refer to the model presented here as the ARP based on the regression of Excess Return against Time, ARP-ERT.
3.3.2
Mixed portfolio
In ARP-RT as presented above we have a pure absolute return model, whereas in ARP- ERT as presented above we have a pure enhanced indexation (relative return) model. It is possible to combine both models to produce portfolios with mixed characteristics - so a combined absolute return/relative return portfolio. Again for simplicity we shall continue to call the portfolio produced an ARP.
Let λ ≥ 0 represent the weight that we attach to relative return as compared to absolute return. In the first stage optimisation for the regression slope we minimise max[|PN
i=1wiβˆi|, λ|
PN
i=1wiβˆi − ˆB|], so minimise the maximum absolute value of both
regression slopes (for the regressions of return against time and excess return against time) considered individually. Here we have introduced λ as a weighting for the regression slope associated with the relative return component of the objective. Again this is nonlinear but can be linearised as:
minimise E (3.25) subject to (3.1)-(3.5), (3.7)-(3.10), (3.13)-(3.15), (3.21) and: E ≥ λ( N X i=1 wiβˆi− ˆB) (3.26) E ≥ −λ( N X i=1 wiβˆi− ˆB) (3.27)
During the second stage optimisation, we maximise [minPN
i=1wiαˆi, λ(
PN
i=1wiαˆi− ˆA)],
so maximise the minimum value of both regression intercepts considered individually. Although this is a nonlinear objective as ˆA and λ are both constants we can simplify it to
maximise PN
i=1wiαˆi. Hence we have that the second stage objective here is precisely the
same as the second stage objective given above, Equation (3.16), where this objective is optimised subject to subject to (3.1)-(3.5), (3.7)-(3.10), (3.13)-(3.15), (3.17), (3.21) and (3.26)-(3.27).
The third stage follows in a similar fashion as for the ARP-RT model given above. Here the objective is to optimise Equation (3.18) subject to (3.1)-(3.5), (3.7)-(3.10), (3.13)- (3.15), (3.17), (3.19), (3.21) and (3.26)-(3.27).
We refer to the model presented here as the ARP based on the regression of Return and Excess Return against Time, ARP-RERT.