Why the sensitivity is so severe
In FV = PV(1 + r)^n the rate appears inside a base that is raised to a power. Changing r does not shift the result proportionally; it changes the base of an exponential. The longer the horizon, the more that difference is amplified.
Over one year, 6% against 7% is a $100 difference on $10,000. Over forty years it is a $47,000 difference on the same deposit. Nothing about the arithmetic changed — only the exponent.
| Years | 4% | 6% | 8% | 10% |
|---|---|---|---|---|
| 10 | $14,802 | $17,908 | $21,589 | $25,937 |
| 20 | $21,911 | $32,071 | $46,610 | $67,275 |
| 30 | $32,434 | $57,435 | $100,627 | $174,494 |
| 40 | $48,010 | $102,857 | $217,245 | $452,593 |
Annual compounding. Across 40 years, 10% produces nine times what 4% produces.
Contribution plans are less rate-sensitive
A lump sum experiences the full horizon at the assumed rate. A contribution plan does not: the deposit made in year 28 of a 30-year plan compounds for two years regardless of the rate. On average, contributions experience roughly half the horizon.
That makes savings plans meaningfully more robust to a wrong rate assumption than single-deposit projections, which is a useful property when the assumption is the least reliable input you have.
| Plan | At 4% | At 6% | Increase |
|---|---|---|---|
| $100,000 lump sum | $324,340 | $574,349 | +77% |
| $500 a month | $347,024 | $502,258 | +45% |
Monthly compounding for the contribution plan, annual for the lump sum.
Running scenarios instead of a single rate
The correct response to an unknowable input is not a better point estimate; it is a range. Running a projection at the base rate, two points below and two points above converts a false precision into a usable band.
The value of the exercise is in what it tells you about the plan's dependence. If the lower scenario still meets the goal, the plan is funded by contributions and is robust. If only the upper scenario reaches the target, the plan is a bet on returns, and it should be labelled as one.
The same arithmetic when you are the borrower
Rate sensitivity works identically against you on a loan, with an additional twist: the payment is fixed by the rate at origination, so a higher rate raises both the payment and the share of each payment consumed by interest.
On a $300,000 30-year loan, moving from 5% to 7% raises the monthly payment from $1,610 to $1,996 and total interest from $279,767 to $418,527 — an extra $138,760 for the same house.
| Rate | Monthly payment | Total interest |
|---|---|---|
| 4% | $1,432 | $215,609 |
| 5% | $1,610 | $279,767 |
| 6% | $1,799 | $347,515 |
| 7% | $1,996 | $418,527 |
Choosing a rate you can defend
A defensible rate is one you can state a reason for, matched to the account and the horizon. Cash goals earn cash rates. A thirty-year equity-heavy portfolio has historically earned more, with dispersion wide enough that any single number is a simplification.
Whatever you choose, use the same rate consistently across a comparison, and state it next to the result. A projection quoted without its rate assumption is not a projection; it is an assertion.
Fees are a rate change, not a charge
The most useful consequence of rate sensitivity is what it says about costs. A 1% annual fee is not a 1% reduction in your result; it is a one-point reduction in the compounding rate, and the exponent does the rest.
On $100,000 over thirty years, moving from 7% to 6% costs $186,876 — roughly a quarter of the ending balance, from a charge that appears as a single line item each year. The same logic applies to tax charged annually rather than deferred, and to the difference between a gross and a net quoted return.
This is why cost comparisons deserve the same scrutiny as return assumptions. They act on the same part of the equation, and they are the part you can actually control.
The asymmetry worth internalising
Rate sensitivity is not symmetric around the base case. Because the relationship is exponential, an equal move up and down does not produce equal changes: at 30 years on $100,000, moving from 6% to 8% adds $431,917, while moving from 6% to 4% removes $250,009.
That sounds like good news, and in one narrow sense it is. But it also means an optimistic assumption is rewarded more than a pessimistic one is punished on paper, which is precisely why projections drift upward when nobody is checking them.
The defensive habit is to make the lower scenario the one you plan against and treat the upper scenario as a bonus. A plan that only works in its best case is not a plan; it is a hope with a spreadsheet attached.
How to state a rate-sensitive result
A projection quoted as a single number invites more confidence than the arithmetic supports. A projection quoted as a range with its assumptions attached invites the right question, which is whether the plan survives the low end.
The useful format is short: the base case, the low case, and the assumption that separates them. "Around $500,000 at 6%, or $350,000 at 4%" tells a reader more in one line than a single figure carried to the dollar.
This is also the format that ages well. When returns turn out differently, a range that was stated honestly needs updating rather than defending, and the plan built against its lower end usually needs no change at all.
Educational tool. This article explains arithmetic, not personal circumstances. It is general educational content, not investment, tax or insurance advice, and every figure in it is a projection from stated assumptions rather than a prediction. Check any result against your own situation, and see the editorial policy for how this content is produced and corrected.