Deriving Portfolio Gains from Risky Asset Returns and the Risk-Free Rate
Summary
The note explains how a portfolio gain process can be written using undiscounted asset prices while still displaying excess returns over the risk-free rate. It starts from risky assets whose proportional returns contain drift and diffusion terms, then separates each drift into its excess return and the risk-free rate. The risk-free contributions sum to the portfolio’s total wealth times the rate when portfolio holdings represent wealth invested across the assets.
This algebra recovers the familiar form with excess-return gains, risk-free growth on wealth, and stochastic return exposure. The discussion is a conceptual derivation rather than a general treatment: it simplifies by setting dividends and certain other terms to zero and assuming a constant interest rate. Its key interpretive point is that an apparent mixture of discounted and undiscounted expressions can reflect a change in how the portfolio return is decomposed, not inconsistent accounting.
Key ideas
- Portfolio gains can be expressed through proportional changes in risky asset prices.
- Separating each asset drift into excess return and the risk-free rate isolates portfolio financing growth.
- If invested proportions sum to one, the risk-free components aggregate to the risk-free rate times portfolio wealth.
- The derivation assumes simplified dividend and interest-rate conditions.
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# Confusion about the formula for gain process in a financial market
# Confusion about the formula for gain process in a financial market
In this wikipedia page, we consider the following financial market
The formulas for the stocks are given here
And the gain process of a portfolio $\pi$ is defined such that
From what I understand, the first term of the formula of the gain process is due to the riskless asset, meaning that we consider undiscounted quantities (otherwise the riskless asset would not be considered in the expression of the gains I guess). But then the second term comes from the discounted formula of the risky assets. Hence it is a bit unclear for me what exactly are the computations behind this formula and whether we use discounted quantities or not.
I thought that the formula for gain process was roughly given by
\begin{equation} G(t) = \int_0^t \pi_r \frac{dS_r}{S_r} \end{equation} but this doesn't seem to correspond with Wikipedia.
I would be glad if someone could explain a bit more about it, particularly since it is hard to find any reference for this or gain processes in general. Thank you in advance.
## Answer by yrual (score 2, accepted)
https://quant.stackexchange.com/a/75530
@nbbo2 Thank you very much for providing this useful reference, I had a look into it and I think I understand now :) For simplicity, let's take $A \equiv 0$, $\delta \equiv 0$ and $r(s) \equiv r$ (it is not very important anyway). Using undiscounted expression of the price process, one has that
\begin{align} dG(t) &= \sum_i \pi_i(t) \frac{dS_i(t)}{S_i(t)} \\ &= \sum_i \pi_i(t) b_i(t) dt + \sum_i \pi_i(t) \sum_j \sigma_{ij}dW_j(t) \\ &= \sum_i \pi_i(t) (b_i(t) - r) dt + \sum_i \pi_i(t)rdt + \sum_i \pi_i(t) \sum_j \sigma_{ij}dW_j(t) \\ \end{align} But now since $\frac{\pi_i(t)}{G(t)}$ is the proportion of wealth invested in asset i at time t, it is clear that $\sum_i \frac{\pi_i(t)}{G(t)} = 1$ and thus \begin{align} \sum_i \pi_i(t)rdt &= \sum_i \frac{\pi_i(t)}{G(t)} G(t)rdt \\ &= G(t)rdt \end{align} Therefore we finally obtain \begin{align} dG(t) = \sum \pi_i(t) (b_i(t) - r) dt + G(t)rdt + \sum_i \pi_i(t) \sum_j \sigma_{ij}dW_j(t) \end{align} which is what is obtained in the Wikipedia article.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.