There is an assumption embedded in most lifetime income analysis. Often unexamined, it shapes almost every illustration and projection you'll encounter, and it is wrong in a way that matters enormously for how income systems should be designed.
The assumption is this: that what happens on average across many possible futures is a reliable guide to what will happen to you.
It isn't. And understanding why is critical.
Imagine a coin flip where heads doubles your money and tails cuts it in half. You start with $100.
Across a large group of people each flipping once, the average outcome is good: half double to $200, half drop to $50, so the average is $125. A 25% expected gain. Looks attractive.
Now imagine one person flipping that coin repeatedly over time. After two flips — one heads, one tails, in either order — they have $100. After ten flips, five each way, they have about $57. After twenty flips, ten each way, they have roughly $32.
Same coin. Same odds. Two completely different stories.
The first calculation is an ensemble average — what happens across many people at one moment. The second is a time average — what happens to one person across many moments. When outcomes compound through time, these two averages diverge. In the coin example, they diverge dramatically.
This divergence has a name in math: non-ergodicity. A system is ergodic when the time average equals the ensemble average — when one person's long-run experience matches the average across many people. Most financial systems involving compounding and withdrawals are not ergodic. The time average is lower than the ensemble average, often substantially, and the gap widens as time extends.
This isn't a technical footnote. It's the reason sequence of returns risk exists, the reason volatility drag is real, and the reason pooling can improve lived outcomes even when it doesn't change expected values. It's also the reason that probability-of-success metrics built on ensemble thinking can systematically mislead people making irreversible decisions along a single timeline.
Mainstream retirement analysis is almost entirely built on ensemble thinking. Monte Carlo simulations ask: across ten thousand simulated futures, in what fraction does the plan succeed? That's a useful question. But it's not the primary question for someone living one life.
You don't get to average across ten thousand futures. You live one sequence — one specific ordering of inflation, markets, health changes, work transitions, and timing. That sequence is your time average, and it's what determines your actual outcome.
The ensemble average might look fine. Your time average depends on the path.
This is not an argument against probabilistic thinking. It's an argument for being clear about what the probabilities describe. Ensemble probabilities describe the distribution of outcomes across many lives. They don't describe the distribution of outcomes across your life — because your life is one draw from that distribution, experienced sequentially, with each outcome affecting the base for the next.
Two people both rely on an invested asset pool for income. Same starting balance, same average long-run return, same withdrawal amount.
Person A experiences a significant market decline in the first three years, then a strong recovery.
Person B experiences the same strong returns first, then the same decline later.
Over the full horizon, the average annual return is identical. But the lived outcomes diverge substantially.
Person A is drawing income while the balance is depressed. Each withdrawal represents a larger fraction of a smaller pool, which permanently reduces the base available for future growth. When markets recover, they're recovering a smaller number. The income path is damaged in a way that the long-run average doesn't show.
Person B draws income during the strong years, builds a larger cushion, and absorbs the later decline from a position of strength. The same market sequence, arriving later, does far less damage.
Same average return. Same rules. Completely different lives. The difference isn't luck in any simple sense — it's the multiplicative, path-dependent structure of compounding combined with ongoing withdrawals. That structure is what non-ergodicity describes.
There is a way of putting this that requires no math at all.
The average depth of a river is four feet. That statement might be perfectly accurate. It is also potentially useless — because what matters when you're crossing is not the average depth but the specific depth at each step, including whether there's a ten-foot hole somewhere in the middle.
"On average, this should work" can be true. You can still step into the hole. Bad timing, bad sequence, an early shock — and the outcome changes permanently. The average doesn't drown. You do.
Lifetime income works the same way. An illustration that looks sound on average can still produce a path that fails — because the average is a property of the ensemble, and you're living the path.
Once you take time-average thinking seriously, the evaluation of any lifetime income claim changes.
The right questions are no longer primarily about expected outcomes. They're about path behavior: how does this claim hold up across the sequence you actually live, rather than the average sequence?
That reframe has direct structural implications.
Volatility isn't just noise around an expected return — it's a drag on the time average that compounds against you when withdrawals are live. Reducing volatility has more value in a time-average framework than it appears to have in an ensemble framework.
Sequence risk isn't a caveat or an edge case — it's a direct consequence of non-ergodicity, and it's most severe precisely when it's most consequential: early in the withdrawal period when the base is largest.
Pooling isn't just a way to share risk on paper — it's a way to change the time average directly, by smoothing the path rather than just improving the expected value. A pool that stabilizes income across many lives is doing something that self-funding fundamentally cannot do: it's converting individual path risk into collective path stability.
This is why pooling and time-average thinking belong in the same series. They're not separate topics. Pooling is the design response to the one life problem.
The distinction between ensemble averages and time averages in economics has been developed most rigorously by Ole Peters and colleagues at the London Mathematical Laboratory, building on earlier work in physics and information theory. The core argument — that non-ergodicity is pervasive in economic systems and that its implications have been systematically underappreciated — is both technically rigorous and almost entirely absent from consumer-facing financial writing.
You don't need the math to use the insight. The river is enough. But the research is there for anyone who wants to go deeper, and it is more consequential for how we think about lifetime income design than almost anything else in the academic literature. The Time-Average Growth topic in the Learn section is where the Longevity Standard framework engages with this work as a structural foundation.
If time averages are the right lens, the next question is practical: what does this mean for the specific problem of income continuity? Volatility interacts with withdrawals in a way that makes ruin not just an inconvenience but the binding constraint on design. That's where the series goes next.
On this Page
