Modeling Dollar-Cost Averaging with a Geometric Brownian Motion

Summary

This document formulates continuous dollar-cost averaging when the stock price follows geometric Brownian motion. With a constant contribution rate, each contribution buys shares at the prevailing price; the final portfolio value can be represented as the contribution rate multiplied by an integral of the stock’s relative growth from each purchase time to the investment horizon. The source cites a derivation by Milevsky and Posner and gives the integral representation, framing questions about deriving it from the stochastic differential equation and characterizing its distribution.

The document does not provide answers to those distribution questions, numerical results, or an approximation for quantiles. It asks how to obtain quartiles and a 95% confidence interval, and whether stochastic volatility or other richer price models are needed. Thus, it is a useful setup for analyzing dollar-cost averaging under a simplified model, but not a finished method for computing portfolio outcomes. Its assumptions include continuous contributions, a constant contribution rate, and GBM price dynamics.

Key ideas

• Under GBM, continuous fixed contributions lead to a final portfolio value expressed as an integral over purchase times.
• Each contribution buys a quantity of shares determined by the stock price at that time.
• The document cites prior work for the integral representation but does not derive it in detail.
• It leaves the distribution and quantile calculation unresolved and raises model complexity as a limitation.

Tags

# Performance of dollar cost averaging


# Performance of dollar cost averaging












If we're investing money into a stock $S$ at a continuous rate, $C$, what is the probability distribution of the amount we have invested?

For example, modelling a stock as GBM without contributions, $ \mathrm d S = \mu S \mathrm d t + \sigma S \mathrm d B $, gives us a lognormal distribution, $\mathrm{Lognormal(\mu t, \sigma^2t)}$. If we add contributions, $ \mathrm d S = (\mu S + C) \mathrm d t + \sigma S \mathrm d B $, what do we get?

Milevsky & Posner (2003) derive the following:

$$ \mathrm d S_t/S_t = \mu \mathrm d t + \sigma \mathrm d B_t \iff S_t = S_0 \exp\left[ (\mu - \frac 1 2 \sigma^2)t + \sigma B_t \right] \tag{2} $$

$$ P_T = C S_T \int_0^T \frac{\mathrm dt}{S_t} \tag{6} $$

Where $S_t$ is the stock price and $1/S_t$ is the amount of stock we can buy with $1.

$$ P_T = C\int_0^T \hat S_\tau \mathrm d \tau = C\int_0^T \exp\left[\mu \tau - \frac 1 2 \sigma^2 \tau + \sigma \hat B_\tau \right] \mathrm d \tau \tag{8} $$

where $\hat S_\tau \sim S_t$ and $\hat B_\tau \sim B_t$.

I understand (6), but how do we derive it from the differential equation?

How can we solve this integral in terms of the normal/lognormal distribution?

How can we find, or approximate, the quintile function? I want to use this to get quartiles & 95% CI for this investment strategy.

Are there any problems with modelling the problem this way, or is it important to use more complicated models such as stochastic volatility?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.