Excess return on a portfolio of lognormal assets

Here's my attempt at "Stochastic Portfolio Theory in a Nutshell", for those who, like me, hadn't noticed a bug in Markowitz Theory.

Stochastic Portfolio Theory considers the decomposition of the instantaneous return on a continuously rebalanced portfolio into the instantaneous returns of constituent assets \begin{equation} \frac{ d Z_t} {Z_t} = \pi_t^\top \frac{ dX_t}{X_t} \label{portfolio} \end{equation} Here \(Z_t\) is the value of the portfolio, \(\pi_t\) is a vector of portfolio weights, the fractions indicate pointwise (i.e. pathwise) division and \(dX_t=(dX^1_t,...,dX^n_t) \) is a \(n x 1\) vector of stocks with lognormal dynamics: $$ d \left(log(X_t\right)_{n x 1} = \left( \gamma_t \right)_{n x 1} dt + \left( \xi \right)_{n x n} \left(dW_t\right)_{n x 1} $$ where \(dW_t\) is a vector and we've emphasized the dimensions throughout. Note that by Ito's Lemma we have $$ \frac{ dX_t}{X_t} = \left( \gamma_t + \frac{1}{2} diag \left( \xi \xi^{T} \right) \right) dt + \left( \xi \right)_{n x n} dW_t $$ where \(diag\) extracts a vector of diagonal entries. So if we mentally hit this on the left with the transpose of the portfolio weights \(\pi^\top\) we can see that the right hand side of our instantaneous return equation will have a Brownian term \(\pi_t^{T} \xi dW_t\) and a drift that we'll get back to momentarily. And if we are on the ball we'll remember that \( \frac{dZ}{Z} \) and \(d(\log Z)_t \) have the same Brownian terms. Thus \( d(\log Z)_t \) will also be driven by \(\pi_t^{T} \xi dW_t\) and can write for some as yet unspecified drift \(\gamma^{\pi}\) $$ d (\log Z_t) = \gamma^{\pi}_t dt + \pi_t^{T} \xi dW_t\ $$ To clean this up we apply Ito's Lemma again to retrieve \( Z_t = \exp(\log(Z_t))\) and thereby: $$ \frac{dZ_t}{Z_t} = \left\{ \gamma^{\pi}_t + \frac{1}{2} \pi^{T}_t \xi \xi^{T} \pi_t \right\} dt + \pi_t^{T} \xi dW_t $$ which reveals the drift term on the left hand side of the instantaneous return equations expressed in terms of a highly relevant quantity: the drift of the logarithm of portfolio wealth. Indeed we call \(\gamma^{\pi}_t\) the portfolio growth process. And we observe the important equality appearing in too few investment textbooks, if any beside Bob's: $$ \gamma^{\pi} = \pi_t \cdot \gamma_t + \underbrace{\frac{1}{2} \left( \pi_t \cdot diag(\xi \xi^{T}) - \overbrace{\pi^{T}_t \xi \xi^{T} \pi_t }^{portfolio\ variance} \right)}_{excess\ return\ process} $$ Thus in log space we might say that the portfolio growth is the linear combination of the growth in individual stocks plus the term involving curly braces. We refer to that additional kick as the excess growth rate. And we further observe that it decomposes into the difference between the weighted combination of stock variances and the portfolio variance process, denoted $$ \sigma^{\pi\pi}_t = \pi^{T}_t \xi \xi^{T} \pi_t $$ That's it for now. We note that if one is interested in the return on the logarithm of one's portfolio then the full decomposition into linear and excess return is obviously more pertinent than the linear term alone. And we see why minimizing portfolio variance subject to a known linear return is not, despite its significant popularity, the most relevant exercise.

The decomposition is useful independent of the investors utility function.

For more see Dr Fernholz's book on Amazon, of this summary paper.