Sequence of Returns Risk

The synthetic deterministic three-scenario model behind the Sequence of Returns Risk Calculator — how the compensating return is constructed, the withdrawal-before-growth order, and why this is not a Monte Carlo or historical simulation.

What the calculator isolates

The Sequence of Returns Risk Calculator answers one narrow question: holding the total return and the withdrawals fixed, how much does the order of the return years change the outcome? To isolate order, it builds three deterministic return sequences that all deliver the same long-run geometric return over the same horizon with the same annual withdrawal, then runs each through an identical year-by-year simulation. All of its math runs server-side in RetireCalc.Engine.

This is a synthetic deterministic model. It does not draw random return paths (no Monte Carlo), does not replay actual market history (no historical backtesting — use Portfolio Lab for that), and does not produce a probability of success. It shows the mechanical effect of return order on a single fixed set of assumptions.

Inputs

P — starting portfolio value.
W — annual withdrawal. It is a flat nominal amount: the same dollar figure every year, with no inflation escalation and no inflation input.
n — number of years modeled.
avg — the average annual return.
bear — the return during the bear-market stretch.
b — the number of bear years. The remaining years are g = n − b.

The compensating return

The good years do not use the avg rate. Instead the calculator solves for a single compensating return comp so that a run of b bear years followed by g compensating years produces exactly the same cumulative growth factor as n years at avg:

(1 + avg)ⁿ = (1 + bear)^b × (1 + comp)^g
⇒ comp = ( (1 + avg)ⁿ ÷ (1 + bear)^b )^1/g − 1 (for g > 0)
comp = avg (when g = 0, i.e. every year is a bear year)

The fractional exponent (an n-th root) cannot be expressed in exact decimal arithmetic, so that one step is computed in double precision and then converted to a decimal for the money simulation. If the solved comp exceeds 20%, the result is flagged with a warning that the inputs force an implausibly large recovery return.

The three return sequences

Flat — avg repeated for all n years.
Early bear — bear for the first b years, then comp for the remaining g years.
Late bear — comp for the first g years, then bear for the last b years.

Early bear and late bear contain the identical set of annual returns — b copies of bear and g copies of comp — in the opposite order. Because the multiset is identical, the two sequences have the same arithmetic mean and the same geometric return; only the ordering differs. By construction the compensating return also gives the Flat sequence the same long-run geometric return as the other two. Any difference in the ending balances is therefore attributable to return order alone.

The year-by-year simulation

Each sequence is run through the same loop. In every year the withdrawal is taken before that year’s return is applied:

Balance_y = ( Balance_y−1 − W ) × ( 1 + Return_y )

The portfolio is recorded as depleted in the first year the balance reaches zero or below. The loop continues, but every displayed balance from that point on — and the final value — is clamped at $0 (it is never shown as negative).
The comparison table reports balances at 5-year checkpoints plus the final year.
The headline “difference” percentage is round( |Flat_final − EarlyBear_final| ÷ Flat_final × 100 ) — the early-bear ending balance measured against the flat ending balance.

Limitations

Deterministic and synthetic — three hand-built sequences, not sampled or historical returns. It shows the direction and rough magnitude of sequence risk, not its probability.
Withdrawals are flat nominal, with no inflation, taxes, fees, Social Security, or other income.
A single bear block of a single fixed return — real bad runs are irregular. For actual historical sequences use Portfolio Lab; for a fixed-rate survival check use the Safe Withdrawal Rate Calculator.

Sequence of Returns Risk Calculator →Safe Withdrawal Rate Calculator →Portfolio Lab →Retirement Calculator →