Relative Returns

The relative return of two assets, {% X %} and {% Z %}, is defined to be

{% log(X(t)/Z(t)) %}

Given a set of assets {% X_i %} and a portfolio {% Z %}

{% Z(t) = \sum_i \pi_i(t) X_i(t) %}

the cross variation of the relative returns of two assets to the portfolio is given by

{% [log(X_i/Z), log(X_j/Z)] = [log(X_i) - log(Z), log(X_j) - log(Z)] %}

{% = [log(X_i), log(X_j)] - [log(X_i), log(Z)] \\ - [log(X_j), log(Z)] + [log(Z)] %}

Given the portfolio weights {% \pi_i(t) %}, define

{% \sigma_{i\pi}(t) = \sum_j \pi_j (t) \sigma_{ij}(t) %}

then

{% d[log X_i, log Z] = \sigma_{i \pi}(t)dt %}

Next, we define a matrix valued process

{% \tau_{ij}(t) = \sigma_{ij}(t) - \sigma_{i\pi} - \sigma_{j\pi} + \sigma_{z} %}

where

{% d[Z] = \sigma_z(t)dt %}

With these definitions we have

{% d[log(X_i/Z), log(X_j/Z)] = \tau_{ij}dt %}

and

{% d[log(X_i/Z)] = \tau_{ii}(t)dt %}

Given two portfolios, {% \pi %} and {% \eta %} we have

{% dlog(Z_{\pi}(t)/Z_{\eta}(t)) = dlog(Z_{\pi}(t)) - dlog(/Z_{\eta}(t)) %}

Given that

{% d log Z(t) = \sum \pi_i(t) d log X_i(t) + \alpha_{\pi}^*(t) dt %}

we then have

{% dlog(Z_{\pi}(t)/Z_{\eta}(t)) = \sum \pi_i(t) d log(X_i(t)/Z_{\eta}(t)) + \alpha_{\pi}^*(t)dt %}

The relative return versus the market portfolio

{% dlog(Z_{\pi}(t)/Z_{\mu}(t)) = \sum \pi_i(t) d log(\mu_i(t)) + \alpha_{\pi}^*(t)dt %}