Abstract
Buying steadily through a price collapse pushes your break-even far below where the price started, but it can't create a profit on its own. The outcome depends entirely on whether, and how far, the asset recovers.
This paper follows an investor who buys every week while the price falls along controlled paths: steady, geometric at 5% and 10% a week, a five-year slide with no bottom, and a crash followed by recovery. Silver at a nominal $100/oz is the worked example. Nothing in the math is specific to silver; it applies to stocks, indices, crypto, commodities, or anything else bought in installments.
- Cost basis compression. In a one-year 5%-per-week slide, buying the same dollars each week breaks even at $20.42; buying the same ounces breaks even at $35.79.
- Same-dollar buying always wins on cost. It produces the harmonic mean of prices, which can never exceed the arithmetic mean that same-quantity buying produces.
- The value ceiling. If the decline never ends, your holdings converge to a fixed value of C ÷ r, no matter how much you've put in.
- Recovery leverage. $5,200 bought weekly through a 93% crash becomes $25,461 on a full recovery (4.9×); the same money as a lump sum at the top only breaks even.
Definitions
Two buying rules are compared throughout, and the difference between them drives most of the results.
| Term | Meaning |
|---|---|
| Same amount (fixed quantity) | Buy the same number of units every week. Spending falls as the price falls. |
| Same dollars (dollar cost averaging) | Spend the same amount every week. Units bought rise as the price falls. |
| Lump sum | Invest the whole amount at the first week's price. |
| Break-even (cost basis) | Total invested ÷ units owned. The price at which you're exactly even. |
| Recovery needed | Break-even ÷ current price: how many times the price must multiply. |
Prices are deterministic, with no costs, taxes or interest on idle cash, unless you add a dealer premium in the simulator. Section 9 covers what the model leaves out.
The mathematics
Same-quantity buying gives you the arithmetic mean of prices. Same-dollar buying gives you the harmonic mean. Every other result follows from that.
Cost basis under each rule
By the arithmetic–harmonic mean inequality, the second is never larger than the first, on any price path, rising or falling. The gap grows as prices spread out. This is a property of averaging, not a forecast.
Closed forms for a geometric decline
With Pk = P0(1 − r)k, same-quantity break-even is:
For same-dollar buying with C per week, your holdings' value at the final price is a finite geometric series:
So the recovery needed is total invested ÷ value:
Finite declines
In every finite decline tested, same-dollar buyers need a much smaller rebound than same-quantity buyers, and both need far less than a return to the start.
| Path | Weeks | Final price | Same-amount break-even | Same-dollar break-even | Recovery needed (dollars) | |
|---|---|---|---|---|---|---|
| −5% a week | 52 | $7.31 | $35.79 | $20.42 | 2.8× | |
| −10% a week | 52 | $0.46 | $19.15 | $2.42 | 5.2× |
Same-quantity buying front-loads your money. In the 10% slide, about 65% of the year's spending happens in the first 10 weeks, at the highest prices.
Same-dollar buying back-loads your ownership. It owns fewer units until about week 24, then overtakes sharply, ending with roughly 4,100 units against 520 for the same money.
A deeper fall cuts both ways. It lowers the break-even in dollars, but leaves the price further below it, so the recovery needed rises.
The perpetual decline
If the decline never ends, steady buying can't outrun it. Your holdings level off at C ÷ r while what you've put in keeps climbing.
Think of a leaky bucket. Each week your existing holdings lose a fraction r of their value, and your new contribution C tops them up. They stop growing when the leak equals the refill: r × V = C.
Because invested capital grows in a straight line while value plateaus, the recovery you'd need grows by about n × r: roughly 2.6× more for every year at 5% a week. Steady buying doesn't reduce your exposure to an asset that never recovers; it increases it, one installment at a time.
Decline and recovery
If the asset does recover, buying through the crash turns it into leverage. The same $5,200 that only breaks even as a lump sum at the top grows to $25,461 when bought weekly on the way down.
| Strategy | Invested | Units | Break-even | Value at $100 | Multiple |
|---|---|---|---|---|---|
| Lump sum at $100, week 1 | $5,200 | 52 | $100.00 | $5,200 | 1.0× |
| $100/week during the crash, then hold | $5,200 | 255 | $20.42 | $25,461 | 4.9× |
| $100/week through crash and recovery | $10,400 | 500 | $20.78 | $50,038 | 4.8× |
The rebound doesn't have to be complete. Recovering to $50, which still looks like a 50% loss on a price chart, returns +145% to the crash-only buyer. Drag "Climbs back to" in the simulator to find the level where you break even.
Historical precedent
Real markets have fallen as far as the model, and taken anywhere from months to more than four decades to come back. Silver's 1980–1991 fall (−93%) nearly matches the one-year 5% scenario, only spread over eleven years.
| Episode | Peak → trough | Drawdown | Back to the old peak | |
|---|---|---|---|---|
| Silver, 1980 | $49.45 (Jan 1980) → $3.61 (Feb 1991) | −93% | About 45 years (Oct 2025) | |
| Silver, 2011 | $49.51 (Apr 2011) → $13.71 (Dec 2015) | −72% | About 14.5 years (Oct 2025) | |
| Nikkei 225 | 38,916 (Dec 1989) → about 7,000 (Mar 2009) | About −82% | 34 years 2 months (Feb 2024) | |
| Nasdaq Composite | Mar 2000 → 2002 | About −78%* | About 15 years (Apr 2015) | |
| Dow Jones Industrial Average | Sep 1929 → 1932 | About −89%* | About 25 years (late 1954) |
Sources: MintBuilder silver price history; MetalCharts, silver in the 2010s; Nippon.com on the Nikkei record; Deutsche Börse / pfp Advisory; Navellier on the Dow. *Approximate, from general market knowledge. "Simulate" draws a smooth path through each episode's peak, trough and recovery date; real prices zigzagged, and the indices are scaled to a $100 start.
Your clock is shorter than the index's. "Years to regain the peak" measures someone who bought everything at the top. A steady buyer breaks even when the price reaches their cost basis, far below the peak. Load the Nikkei episode and look at when the value line crosses the invested line.
Inflation lengthens the real clock. Silver's 1980 peak is estimated at about $185 in 2026 dollars, so in purchasing-power terms it has never been regained.
Beyond silver
The math only needs a price series. What differs between assets is how likely the price is to come back, and that's where the strategy succeeds or fails.
| Asset type | Can it go to zero? | Main risk for steady buyers |
|---|---|---|
| Broad stock index | Practically no | Very long flat stretches, like the Nikkei after 1989 |
| Single stock | Yes | Bankruptcy; averaging down into a failing company |
| Precious metal | Practically no | Decades of stagnation in real terms; no yield |
| Industrial commodity | Practically no | Lasting demand loss; roll costs on paper exposure |
| Cryptocurrency | Yes, for most tokens | Total loss or delisting; no cash flows |
| Real estate | Rarely | Leverage and local-market collapse |
An index renews itself: failing members are replaced. A single company, token or contract doesn't. Rename the asset in the simulator; the math won't change, but your judgment about the recovery should.
"It can't go to zero" isn't enough. An asset doesn't need to reach zero to destroy a steady buyer's capital. It only needs to keep falling for longer than you keep buying.
Frictions and behavior
Real costs hurt most where the strategy is strongest: at very low prices, where a fixed per-unit cost becomes a big share of each purchase.
Physical bullion sells above spot; in 2011, premiums on small silver products reportedly hit $5–$10 an ounce. In the one-year 5% scenario, a $5 premium raises the break-even from $20.42 to $27.55. Set the dealer premium in the simulator to see it on any path.
Other costs
Storage and insurance for metal; fund fees; selling spreads; the interest your cash could have earned elsewhere; and taxes. In the US, physical precious metals are generally taxed as collectibles at a higher maximum long-term rate than stocks (a general note, not tax advice).
The behavioral trap
The math assumes you keep buying. That is the assumption most likely to break. In the crash-and-recovery scenario, your holdings sit near $1,800 for months while you've put in over $5,000. Quitting at the bottom turns a recoverable paper loss into a permanent one. An automatic plan, sized so you can sustain it through the worst stretch, is the main defense.
Limitations
The paths are smooth; real prices are noisy, and volatility around a falling trend usually helps same-dollar buyers, so these results are conservative for that rule. The model shows what happens on each path, not how likely each path is. Figures are nominal. It assumes the cash is always there each week. And lump sums are only tested at the peak; in rising markets, investing all at once generally beats spreading it out more often than not, because the money is invested longer.
Conclusions
Buying through a decline is a cost-basis tool, not a source of return. It changes how much recovery you need; only the asset can supply the recovery.
- Break-even lands well below the starting price in every finite decline tested.
- Same-dollar buying always produces the lower break-even.
- In a never-ending decline, value caps at C ÷ r while the recovery needed keeps growing.
- If the asset rebounds, accumulating on the way down can multiply your money 4–5×, and even a partial rebound can double it.
- Whether to keep buying is a judgment about the asset and about your ability to stick with the plan. The arithmetic can't answer it.