A model can price near-term payoffs accurately with disaster entry set to zero, then miss the long-run yield. Lu and Stachurski's own two-state illustration makes the point sharply. With a 1.75% annual chance of entering disaster, the disaster state's spectral component accounts for about 95% of the price of a fixed real payment due in 25 years, even when valuation begins in normal times.

This is pure theory. The economy follows a finite-state Markov chain in continuous time. Physical transitions enter a pricing generator, which also incorporates stochastic discounting and cash-flow growth. The states divide into communicating classes, sets whose members can all reach one another. Each class has its own pricing rate, the exponential rate at which prices of claims within that class rise or fall with maturity.

The structural result is crisp. Physical dynamics and pricing dynamics have identical classes and accessibility maps because a strictly positive growth-adjusted SDF cannot create or remove a path. Their rankings can differ. Recurrence determines the physical ordering, while class rates determine the pricing ordering. A class that the economy eventually leaves can still have the highest pricing rate.

The illustration uses a CRRA consumption model with a normal state and a disaster state. Its inputs are illustrative and, as the authors put it, "not a calibration." Nakamura et al. inform the growth, volatility and recovery assumptions. Entry frequencies are benchmarked to Barro's 1.5 to 2% a year. Recovery intensity is κ = 0.18, implying a mean disaster spell of 5.56 years.

The states visible to a claim

The paper's central object is the pricing corridor. It contains every class on some directed path from the current state's class to a class where the payoff is positive. The authors prove that removing everything beyond this corridor leaves the price unchanged at every maturity. The corridor rate, defined as the highest class rate within that set, determines the claim's long-run yield.

Consider risk aversion α = 4 and a fixed real payoff, with cash-flow exposure ϑ = 0. The normal class rate is λN = −0.1168, compared with λD = −0.0797 for the disaster class. Their gap is Δ = 0.0371 in favor of disaster. Once entry is switched off, normal times have no path into disaster, leaving that gap invisible to every claim valued from the normal state. A claim starting in disaster behaves differently. Even when payment occurs only after recovery, it inherits λD.

The source of that inheritance is strange. For a 50-year maturity, recoveries during the final ten years provide 36.8% of the recovery contribution to price. Conditional on recovery by maturity, their physical probability is only 0.062%.

The result depends on the payoff.

Under the baseline, with α ≥ ϑ, disaster dominance requires α − ϑ > 3.5369. Change the cash flow to consumption-proportional, so ϑ = 1, and the gap becomes Δ = −0.0391. Dominance then vanishes despite unchanged preferences and the same physical economy.

How soon does the crisis class take control?

Restore entry at order ε. The dominant class receives a loading of order ε^d, where d measures the rarity of the paths connecting the classes. Its higher rate compounds with maturity and eventually prevails. Under a separation condition stated explicitly by the authors, crossover maturity scales as (d/Δ) log(1/ε). They call the effect "a spectral peso problem": a class reached only rarely can barely register at ordinary maturities and still govern the long end.

The label fits.

The exact two-state results are stronger than the asymptotic description. With 1.75% annual entry, ε ≈ 0.01765 and the mean wait is 56.6 years. Starting from the normal state, crossover arrives at 4.34 years when κ = 0.14 and at 2.36 years when κ = 0.16. For κ = 0.18, the loading ωε is about 0.501, giving the disaster component more than half of price at maturity zero. By 25 years, its share reaches roughly 94.5%, 95.0% and 95.8% across the three cases.

Persistence works against the obvious intuition here. Lower κ means longer disasters and widens Δ to about 0.0771 at κ = 0.14, while reducing the loading. In this illustration, faster recovery moves the crossover forward.

The authors state the caveat directly: these entry rates make the asymptotic formulas inaccurate. At 1.75% entry with κ = 0.16, the first-order loading is 1.3473, versus an exact value of 0.4241. The coefficient-corrected crossover is negative, whereas the exact crossover is 2.36 years. Every plotted number therefore comes from exact formulas, resolving the accuracy issue for the figures.

The same comparison matters for calibration. At a 1.75% entry rate and κ = 0.16, entry remains too common for the log(1/ε) approximation to hold. Log scaling describes the argument's shape. Exact solutions must supply the magnitudes.

A second warning concerns interpretation. The dominant component's price share is a spectral component share. It differs from both a disaster probability and the price share attributable to paths that pass through disaster.

Disaster gets 8.9% of physical time, 71.9% of pricing mass

At κ = 0.18, disaster has a physical stationary probability of 8.9%. The long-run pricing law assigns it 71.9% of the mass. This law is the stationary distribution generated by the eigenfunction-tilted chain.

Rarer entry sends physical mass toward zero while pricing mass approaches one. The rare-entry and maturity limits also fail to commute. Taking maturity to infinity first produces a yield of −λD. Taking ε to zero first gives −λN.

Valuation without a trade

Long-dated real claims and dividend strips are the natural targets. We could not backtest any of it. Our data contain no observable consumption-based SDF or disaster-state series, and they lack a tradable universe of dividend strips or state-contingent claims against which prices could be tested. Using VIX as a crisis proxy and rotating ETFs would amount to testing a different strategy.

We did not find a fit to observed strip prices in the paper. The authors also caution that the 11.68% normal-state yield comes from a counterfactual economy without disaster entry.

Turning this valuation lens into a signal would require a credible estimate of ϑ for a real family of claims. The authors make the same point: quantitative use requires cash-flow exposure to be disciplined alongside transitions and discounting. The flip in ϑ shows the stakes. In the baseline illustration, disaster dominance requires α − ϑ above 3.5369. With α = 4, shifting ϑ from 0 to 1 changes Δ from 0.0371 to −0.0391. The paper offers no estimate of ϑ. Until dividends or credit have one, the paper's most usable result remains its warning about horizon: a rarely entered state can dominate claim value, with its weight determined by both the claim and its maturity.