Performance Measurement in Finance

282 Performance Measurement in Finance

is the potential danger of correlation between the processes of making deci-

sions about individual asset weights.

One also notes an asymptotic structure in the orthant probability-based

forecast estimation which is not present in the ex post estimation. This struc-

ture is a direct result of the arcsine of trading model correlation present in

equation (10.16). First, the risk forecast tends to be reduced at low correlation

between trading models. When combining the risk of two assets, their individ-

ual risk is added in quadrature in the conventional way. However, a third term

in the summation which involves correlation between assets is also dependent

on correlation between asset weighting decisions. This latter term tends to be

reduced at lower trading model decision correlation, a possibility that doesn’t

exist in the standard formulation (equation (10.12)). Second, at higher lev-

els of correlation between trading model decisions, risk forecasts tend to be

boosted as the non-linear effect of the arcsine again becomes apparent.

10.5 CONCLUSIONS

The volatility characteristics of investment portfolios have been re-examined

from the perspective of long/short investment strategies. An orthant

probability-based formulation for the description of portfolio risk proposed by

Acar and Satchell (1998) was discussed. The standard and commonly accepted

variance/covariance framework accounts for correlation between assets. How-

ever, the new formulation extends standard portfolio risk estimation in order

also to account for potential correlation between the asset selection and

weighting decisions which may periodically occur. The differences between

the two approaches suggest that conventional estimation methods may not

apply fully to the absolute risk of long/short strategies or the benchmark rela-

tive risk of long-only strategies, even under normal distribution assumptions.

Within this framework, an instantaneous estimation of association is intro-

duced as a gauge of the similarity between asset weighting decisions at the

time of portfolio construction or adjustment. For a given investment subpe-

riod, this approach is viewed as more accurate than historical measures of

association which estimate an average correlation over a previous, often arbi-

trarily selected, time period. In addition, a generalized, multivariate-orthant

probability structure has been developed for variance estimation of portfo-

lios containing any number of assets. This was achieved by circumventing

the 2

N

orthant sections framework and reducing the problem to a pairwise

asset consideration of order N

2

. This is similar to the treatment prescribed

by the conventional variance estimate of multivariate linear combinations. To

date, statistical science has provided an exact solution only for bivariate and

trivariate orthant probabilities and the multivariate step is viewed as a critical

necessity in view of the investment portfolio application.