The Arithmetic vs. Harmonic Mean Explained
Averaging a fixed dollar amount buys shares at the harmonic mean of prices — always lower than the arithmetic mean. For a simple example: you invest $1,000/month when price is $50, $40, and $60 across three months. Simple (arithmetic) average price: ($50 + $40 + $60) / 3 = $50. Your actual average cost: $3,000 / (20 + 25 + 16.67 shares) = $3,000 / 61.67 = $48.65. DCA automatically saved you $1.35 per share.
DCA average cost evolution across 5 months of $1,000 monthly investment with price volatility
| Month | Price | Shares Purchased ($1,000) | Running Total Shares | Avg Cost/Share |
|---|---|---|---|---|
| Jan | $50.00 | 20.0 | 20.0 | $50.00 |
| Feb | $40.00 | 25.0 | 45.0 | $44.44 |
| Mar | $60.00 | 16.7 | 61.7 | $48.65 |
| Apr | $35.00 | 28.6 | 90.3 | $44.29 |
| May | $55.00 | 18.2 | 108.5 | $46.08 |
Volatility Is DCA’s Best Friend
Higher price volatility with the same average price always benefits DCA more than lower volatility. Compare: Stock A fluctuates $20-$80 around $50 average. Stock B stays exactly $50. After 12 months of $1,000/month DCA, the higher-volatility stock produces a lower effective average cost than $50 while the stable stock cost exactly $50. Volatility isn’t just risk — it’s DCA opportunity.
During the 2020 COVID crash, S&P 500 prices dropped from 3,380 to 2,237 (34%) then recovered to 3,756 by year-end. DCA investors who continued monthly purchases through the crash had effective average costs significantly below the year-end price — generating immediate paper profits on low-cost acquisitions.
When DCA Doesn’t Lower Average Cost
In a perfectly steadily rising market, DCA buys fewer shares each successive month (price always higher), so average cost equals the mathematical average of prices — no advantage over a lump sum. This is the market scenario where lump sum beats DCA, and it describes roughly two-thirds of market environments historically.
DCA average cost advantage by market pattern type
| Market Pattern | DCA Average Cost vs. Average Price | Outcome vs. Lump Sum |
|---|---|---|
| Steadily rising | Equal to average price | Lump sum wins (buys at lower start price) |
| Volatile (up and down) | Below average price | DCA competitive or wins |
| Crash then recovery | Well below average price | DCA typically wins |
| Steady decline | Above average price | Both lose; DCA buys more shares for recovery |
Model Your Average Cost Across Market Scenarios
Compare your DCA outcomes under different return and volatility assumptions to understand the real advantage.