A merchant ship clears Bristol for Jamaica in the autumn of 1738. An underwriter at Lloyd’s coffee house takes a line on her hull and cargo for a premium of a few per cent, and then, like everyone with money at risk, he waits. Weeks pass. The freshest intelligence is six weeks old: a passing ship had spoken her off Madeira, in good order, making reasonable way. By the reckoning of the season she is now overdue. The question that matters is the only one he cannot answer: is she late, or is she lost?
These are not two shadings of one answer. Late means she will make port, the premium was fair, and the voyage closes a modest profit. Lost means the hull is on the seabed off Hispaniola, the cargo, worth several thousand pounds, is gone, and the underwriter’s contingent liability becomes real the moment the loss is confirmed. The underwriter’s information does not separate these two worlds. His last sample of reality said afloat and well; everything since is inference laid across a gap. And the reassuring reading, late not lost, is not a blurrier version of the truth. It can be a sharp and confident picture of a ship that has been at the bottom of the sea for a month.
What kind of problem is this?
When the thing you are watching can change faster than you can look at it; when the intervals between observations are long enough for the state to turn over inside them; when no amount of care applied to the glimpses you do have will reconstruct what happened between them — that is a specific shape of problem. And the discipline that has thought most rigorously about how often you must look at something to know what it is doing is the theory of sampling.
So let’s borrow.
This is the multi-model move: recognise the shape of a problem, find the discipline that has thought rigorously about that shape, and import its frameworks deliberately rather than reinventing them from scratch.
Sampling theory was worked out by communications engineers. In 1928, at Bell Laboratories, Harry Nyquist established how fast a telegraph channel of a given bandwidth could carry distinct signals; two decades later Claude Shannon gave the result its modern form. It was built for engineered signals like telegraph pulses, a voice on a line, and an image broadcast through the air, all quantities with a well-defined highest frequency. Markets are messier: a price or a valuation has no fixed bandwidth, the process does not sit still, participants adapt, and observing can change what is observed. But the structural logic — that to reconstruct a band-limited signal, you must sample it at more than twice its highest frequency component, and that observing too slowly does not blur the truth but replaces it — transfers.
The Wheel That Turns Backwards
The statement is this. If a signal contains no frequency higher than B cycles per second, samples taken faster than 2B times a second determine it completely, and under ideal reconstruction the original can be recovered exactly, with nothing left out. The number 2B, twice the highest frequency present, is the signal’s Nyquist rate; half of whatever rate you actually sample at is that sampler’s Nyquist frequency.
The quantity B is the bandwidth: the highest frequency the signal contains, how fast it can change. A slow signal can be captured with leisurely samples; a fast one demands fast ones. The floor on how often you must observe is set by the signal, not by how much you care or how carefully you measure each sample.
The instructive part is what happens when you sample too slowly, below the Nyquist rate. The reconstruction does not merely degrade; it actively misleads.
Anyone who has watched old film of a stagecoach has seen this. The wheel turns forward, but on screen the spokes seem to crawl backwards, or turn at some lazy speed the wheel never had. The camera samples the scene a few dozen times a second, the spokes pass faster than that, and the eye, handed those sparse frames, builds the smoothest motion consistent with them: a slow backward turn that was never happening. Engineers call it aliasing. A frequency above half the sampling rate does not vanish; it folds down and reappears as, or contaminates, a lower-frequency component the original need not have contained.
This is the property that makes undersampling worse than honest ignorance. An undersampled signal is not blurry. It is crisp, plausible, and wrong. It offers a slow, smooth trend assembled out of fast events you never caught, with no internal sign that anything is amiss.
There is no cure after the fact. Once a signal has been sampled too slowly, the high-frequency information is not merely lost but disguised as something else, and from those samples alone no cleverness will bring it back. The remedy has to come before the sampling, never after it.
The Overdue Ship, Again
Now the overdue ship resolves into the practical analogue of an undersampling problem. Strictly, a voyage state that can jump from afloat to lost is not a band-limited signal, and has no finite Nyquist rate. But the failure mode is the one sampling theory teaches us to distrust: when the process can change materially inside the observation interval, the observer replaces an unseen transition with a smooth, plausible interpolation. Shipping intelligence arrived once every several weeks, so the underwriter was not holding a slightly stale but faithful picture, but a benign reconstruction laid across an interval in which the true state may already have changed: she is making slow way, the trades were light, she will be along. Late, not lost, was the spokes appearing to turn backwards.
The structure has not aged. It is the daily condition of any institution holding positions whose true value moves faster than it is observed. A private asset marked once a quarter has economics that do not pause politely between marks. A counterparty reviewed once a year can fail over a single weekend. In each case the interval between observations is not a gap in resolution that leaves the broad shape intact; it is room for an alias to form.
The clearest instance is the smoothness of illiquid returns. Getmansky, Lo and Makarov showed that the reported returns of assets which trade rarely or are valued by appraisal display artificially low volatility and high serial correlation, because each figure leans partly on the last and the true price is observed too seldom. David Geltner had documented the same smoothing in appraisal-based property indices. The mechanism here is not literal spectral aliasing but its close relatives, stale pricing and the moving-average smoothing of marks; the family resemblance is what matters. The low measured volatility of such a series says little about the asset and a great deal about the sampling: it is the calm the sampling invents.
An earlier post, When Right Still Looks Wrong, examined how one sets a threshold on a noisy but genuine signal, trading false positives against false negatives. Aliasing sits upstream of that decision: it corrupts the input before any threshold sees it. You can hold a perfectly calibrated judgement and still be confidently wrong, because what you are judging is not a faithful sample of reality but a smooth fiction the sampling produced.
Before The Gap Opens
If the diagnosis is a sampling problem, the remedies are the engineer’s, and there are only two.
The first is to raise the sampling rate. Observe more often: mark to market where a market exists, monitor counterparties continuously rather than at annual review, seek intelligence within the period, not only at its boundaries. This is worth doing, and it meets a hard floor. Some exposures are illiquid precisely because they cannot be observed or traded often; the observation cadence a quarterly-valued asset forces on you cannot be wished away, and pretending otherwise reinstates the original error in a more confident form.
The second remedy is the one institutions reach for too rarely. In engineering it is to band-limit the input before it reaches the sampler. The translation is awkward, because an institution usually cannot make an illiquid asset observable, and it cannot filter away the asset’s real economic jumps. What it can do is design the exposure before the gap opens: size it, collateralise it, margin it, hold liquidity in reserve against it, attach covenants, or refuse it outright when the unseen interval can contain a loss it could not survive. None of this recovers the path between two valuations, which is gone; it makes the unseen move survivable rather than fatal. The decision belongs at the front, in exposure design, not at the back, in reconstruction from a handful of stale marks after the loss has surfaced.
A corollary cuts against how risk is usually read. The smooth series always looks more trustworthy than the jagged one: lower volatility, steadier returns, fewer alarms read as quality and control. When the smoothness is genuine, that reading is correct. When it is a smoothing artefact of a process observed too slowly, the reading is exactly backwards, and the calmest line in the book conceals the most.
The Sampling Question
Knowing the theorem raises no one’s sampling rate by itself. It gives the underwriter no faster ship and does not make an illiquid asset liquid; understanding the dynamic does not exempt anyone from living inside it. What the theorem changes is the kind of attention one pays. It replaces a vague trust in sparse data with a specific suspicion: that the reassuring reading may be a confident reconstruction rather than a faithful one, and that the smoothness of a number can say as much about how often it was sampled as about the thing it measures.
The problem extends beyond investing. In medicine, a chronic condition monitored by a test every few months can flare and subside between visits, and the calm chart records a steadiness the patient never had. In opinion polling, sentiment sampled monthly can swing and return between surveys, so the smooth trend line is partly an artefact of when the questions were asked. In industrial monitoring, a sensor read too slowly reports a steady machine while a fast vibration builds unseen. In each case the cadence of observation, not the thing observed, sets what can be known.
The discipline is in asking: not whether the latest reading is accurate, but whether we are observing often enough for the smoothness we see to belong to the thing itself, rather than to the gaps between our glimpses of it?
References & Further Reading
Certain Topics in Telegraph Transmission Theory: Harry Nyquist
Communication in the Presence of Noise: Claude Shannon
The Origins of the Sampling Theorem: Hans Dieter Lüke
An Econometric Model of Serial Correlation and Illiquidity in Hedge Fund Returns: Mila Getmansky, Andrew W. Lo & Igor Makarov
Smoothing in Appraisal-Based Returns: David Geltner
A History of Lloyd’s: Charles Wright & C. Ernest Fayle