TITLE
A Theoretical Analysis of Dollar-Cost Averaging Under Different Index Price Trajectories — Mathematical Conditions, Price Trajectories, Volatility, Recovery, and Investor Behavior
AUTHOR(S)
D.L. Shi
ABSTRACT
Dollar-cost averaging (DCA) is the practice of investing a fixed dollar amount at regular intervals regardless of the current market price. This paper develops a simple theoretical framework for DCA under different index price trajectories. The analysis begins with a break-even condition: for a completed DCA program to be profitable at liquidation, the final asset price must exceed the investor’s average cost per share. For equal-dollar contributions, this average cost is the harmonic mean of the purchase prices. The paper then examines linear and exponential growth, temporary declines followed by recovery, irregular market paths, and permanent impairment. A key clarification is made between profit as a percentage of cumulative contributions and an annualized money-weighted return. Under continuous equal-rate investing into an index that rises linearly from 100 to 110 over one year, ending wealth is approximately 1.0484 times total contributions, a 4.84% gain on contributed capital; this should not be called a 4.84% annualized return because the contributions were invested for different lengths of time. The framework further shows why DCA can benefit from temporary price weakness if recovery occurs, while permanent deterioration remains destructive. The practical implication is that long-term DCA is most coherent when applied to a broadly diversified asset with durable expected growth, low probability of permanent loss, and sufficient investor conviction to maintain the strategy through severe drawdowns.
DOI
https://doi.org/10.62252/NSS.2026.1038
PAGES: 676-685
How to cite this article
Shi, D. L. 2026. A theoretical analysis of dollar-cost averaging under different index price trajectories–Mathematical conditions, price trajectories, volatility, recovery, and investor behavior. Naturalis Scientias, 3: 676-685. DOI: https://doi.org/10.62252/NSS.2026.1038. www.naturalisscientias.com.