Rare Disasters and Asset Markets in the Twentieth Century*

Let's start with a feeling every investor has had, even if they wouldn't describe it this way: stocks look too good for how scary they feel, short-term safe assets pay too little for comfort, and markets swing more wildly than standard models seem to allow. Robert Barro argues that those three puzzles are not three separate mistakes in our theories. They are one shadow, cast by the same mountain: rare, economy-wide disasters. If you fold a small probability of very large contractions into an otherwise standard Lucas-tree world with constant relative risk aversion and complete markets, the "excess" reward for stocks, the surprisingly low real bill rate, and the volatility of returns line up. The history that motivates this isn't subtle. Consider World War I, the Great Depression, and World War II. In the twentieth century, Barro counts enough big contractions to calibrate a disaster probability at roughly one and a half to two percent per year. When they hit, they're not mild: per-capita output drops by 15 percent at the low end and, in the worst cases, by more than 60 percent. If you look at long samples across the Group of Seven, real stocks average around seven point one percent per year while real bills hover near zero, minus a tenth of a percent, making the equity premium about seven points. Narrow the window to the postwar era, from nineteen fifty-four to two thousand four, and stocks come in near eight point seven percent real while bills are around one point seven percent. Yet the premium still sits right around seven. Add in the fact that growth is skewed to the downside and fat-tailed—Germany and Japan are particularly extreme—and you see why a model built on mild, symmetric shocks stumbles. Barro doesn't just point to the wars; he builds a distribution. Using Angus Maddison's reconstruction of per-capita GDP for twenty Organization for Economic Cooperation and Development countries, and including Latin America and Asia where the data run long enough, he tallies the big ones—declines of at least 15 percent measured over successive years. There are sixty such contractions across the core set, which works out to a baseline frequency of about one point seven percent per year for entering a 15 percent or worse event. Many go well beyond that threshold: twenty-four cases drop by 30 percent or more; ten fall by at least 45 percent. If you account for normal trend growth, about two and a half percent per year, the mean contraction measured relative to trend rises from roughly zero point two nine to zero point three five. That's a lot of mass in the left tail. Pre-nineteen fifty-four growth looks decidedly non-normal, with huge kurtosis—around forty-one for Germany and forty-nine for Japan—and pronounced negative skew. Post-nineteen fifty-four, things calm down: means look similar across countries, variances shrink, and standard tests stop rejecting normality. The point isn't that disasters are everywhere; it's that they're rare, large, and consequential enough to shape prices even when you're not in one. How do you get that into a model without throwing away the usual machinery? Barro takes a Lucas-tree economy and adds two simple twists. First, he lets depreciation jump up in a disaster, and a fraction of the physical trees get wiped out. Think of v sub t as a shock to depreciation so that delta t equals the usual rate plus v sub t, and with probability p each period, a disaster destroys a fraction b of the capital. Second, he makes the so-called risk-free bill risk-free in normal times, but subject to partial default when disasters hit. The key move is to separate the disaster exposure of the two assets: equities still claim what's left of the trees, while bills can lose a fraction of their promised payoff in bad states. In calibrations, the conditional default probability on bills given a disaster is about zero point four, and the loss rate d moves roughly one-for-one with the economic contraction b. That allows wartime contractions to pull down bill returns in a way that resembles what we see in the data, without assuming the equity markets default. Once you do that, the equity premium falls out in a clean expression. Say it out loud: the premium equals two pieces added together. One is the classic term: risk aversion theta times the variance of normal growth, sigma squared. The other is the disaster term, which is proportional to the probability of a disaster, p, times one minus q, where q is the chance the bill dodges default even if there is a disaster, scaled by a normalizing constant that comes from the model's preferences and technology. That second term is the hinge. If p is zero, you're back to Mehra and Prescott. If p is positive, even small, it starts to matter a lot. If firms use debt—if equity is levered by a constant ratio lambda—the premium on equity scales up by one plus lambda. Under the usual assumption of independent and identically distributed shocks, the term structure is flat: a three-year bill is just three one-year bills strung together in expected value, and the same form carries through to longer-horizon equity. Now ground it with numbers. Barro's baseline uses a trend growth gamma of zero point zero twenty-five, a standard deviation sigma of zero point zero two for the non-disaster growth shock, a subjective discount rate rho of zero point zero three, and relative risk aversion theta in the three to four range. The disaster probability p comes from the Maddison counting, about zero point zero seventeen per year, and the size distribution of b has a mean near zero point two nine, or zero point three five against trend. On the financial side, the conditional bill default probability is around zero point four, and in the United States, a debt-equity ratio lambda of about zero point five is a reasonable heuristic. With those values, the unlevered equity premium lands near zero point zero three six. Expected equity returns sit around seven point one percent, expected bill returns around three point seven percent, and the price to earnings ratio comes out near nineteen to twenty in discrete time, tending to about eighteen as you shrink the period length. Condition on not being in a disaster, and you nudge the premium and expected return up slightly; with the baseline parameters and lambda at zero point five, the levered premium rises by roughly zero point zero zero six. If you turn off disasters entirely, the contrast is stark. In the Mehra and Prescott world where p equals zero, the premium collapses to theta times sigma squared. Plug in theta of four and sigma of two percent, and you get zero point zero zero sixteen. That's nowhere near seven percent. Worse, the implied real bill rate shoots up to about zero point one two seven. To force the premium up to seven points with p stuck at zero, you'd need either volatility thirteen times larger than we observe or risk aversion in the hundreds. Neither passes the smell test. The rare-disaster channel brings the premium and the safe rate back to earth with parameters that look like the ones we measure. You can also see how the knobs move things around. Keep theta at four and lambda at one-half. If the disaster probability rises modestly from zero point zero seventeen to zero point zero twenty-five, the levered premium climbs from about zero point zero five four to around zero point zero seven eight. If bills become a little riskier in bad states, q falls from zero point four to zero point three, and the same premium ticks up from roughly zero point zero five four to zero point zero six three. Think of those as two ways of saying the same thing: either bad times are a little more likely, or the asset that's supposed to be safe is a little less sheltering when they arrive. The size distribution matters too. Using the historical spread of b, hitting a seven-point levered premium with lambda at zero point five takes theta on the order of four point three. If disasters are systematically larger—say b clustered around zero point five—you could get there with theta near three point three. If they're smaller, b around zero point two five, you'd need theta around ten or, equivalently, an implausibly high p. That's the model telling you: it's not just the chance of a hit; it's how hard it lands. A natural worry is that all of this hinges on a particular clock speed or bond maturity. It doesn't, much. If you extend the model's period from a sliver to a few years or trend-adjust the disaster sizes, the premium ticks up modestly. For theta equal to four with trend-adjusted b, the unlevered premium sits around zero point zero six four, and the levered version ranges roughly from six to eight percent depending on the exact length. Swap the one-period bill for longer-term bonds, and you get similar equity premiums—more like six than seven—but the broad picture holds. That's a good sign. It means you're not secretly relying on the maturity structure for the punchline. What about the wartime windows themselves, when bond investors actually suffered? Here, the model's split between equities and bills in disasters earns its keep. In U.S. data, nominal returns in wars don't move much, but inflation does—so expected real returns on safe assets go low or negative. In nineteen seventeen, for example, the nominal return was about four point eight percent, expected inflation around seven point five percent, leaving an expected real return of roughly minus two point eight percent. In nineteen forty-two, nominal was essentially flat at zero point three percent, inflation near seven point two percent, and real returns fell to about minus six point eight percent. Even in the early two thousands, as geopolitical risk rose, ten-year real rates slid from roughly three point eight percent to two point two percent. In the rare-disaster lens, that's exactly what higher perceived p does: it pushes up precautionary saving but drags down the price investors are willing to pay for safety when safety itself can be clipped in the tail. Under the hood, adding capital formation—the A K production structure with trees as reproducible capital—doesn't rewrite the story. Output is A times capital; depreciation jumps in disasters; a one-period equity claim is a claim on next period's output; and the bill still pays unless a disaster strikes, in which case a fraction of its payoff can be lost. The premium formula stays in the same shape: the smooth-growth term theta sigma squared plus the disaster term proportional to p times one minus q. Investment shifts levels—returns and price to earnings ratios—but not the structure that pins the premium to disaster risk. And under independent and identically distributed shocks, the term structure remains flat. If you allow the disaster probability to vary over time, the curve would bend, but that's an extension rather than the baseline. It's worth pausing on the empirical texture that makes this credible. The twentieth century's fat tails are real, and they're mostly wars. Switzerland is one of the vanishing few in the core sample without a 15 percent contraction; the United Kingdom avoids one if you strip out immediate postwar adjustments. Data gaps exist—Greece around World War I, for example—but the broader pattern is clear across Europe, the Americas, and Asia. In the postwar era, growth distributions look tame, and tests don't scream "non-normal." That's not a contradiction. It's the model's point: you can go decades without a tail event and still have prices reflect them. So what follows from taking rare disasters seriously? First, you don't need extreme risk aversion or wild consumption volatility to explain why stocks earn a lot more than bills. A small probability of big, economy-wide hits does that job, and the numbers line up: p around one point seven percent, average contraction size around zero point two nine, and a conditional default chance on bills around zero point four. Second, leverage amplifies what's already there: when p is positive, a debt-equity ratio of about one-half roughly multiplies the unlevered premium by one and a half. Third, the behavior of safe assets in wars—nominals steady, inflation up, real returns down—looks exactly like a world in which the government bill is safe in good times but loses some of its shine when the house is actually on fire. There are limits, and Barro is clear about them. The calibration leans heavily on historical data; the jump process is assumed to be independent and identically distributed; markets are complete; investment frictions and trade are largely abstracted from. The qualitative implications survive reasonable tweaks—letting growth vary with the business cycle affects levels more than the premium—but the core mechanism is not a catch-all for every asset-pricing puzzle. If you want to push beyond calibration, there are tempting empirical levers. Options on broad stock indices embed prices for jump risk. As Xavier Gabaix has suggested, and as Pedro Santa-Clara and Shu Yan have explored with S and P options, you can back out implied probabilities of rare events and see how they move. If disaster risk varies across countries, you should see it in international pricing—think of the Swiss franc's reverse peso episodes where the safe haven earns less precisely because it protects against global tails. And if disaster risk truly drifts over time, the yield curve should echo that by bending, not staying flat. But let's keep the main thread tight. A small, well-measured chance of big, economy-wide damage—estimated from the history we actually lived—goes a long way toward reconciling why equities pay what they do and why so-called safe assets sometimes look awfully meager. That's not romantic. It's sober. And it reminds us that markets don't just price the weather. They price the storm you might not see in your lifetime but can't afford to ignore.