I agree with much of this and if I am understanding correctly the only real point of contention remaining is that I may have argued that the higher probability of catastrophic events over 30 or 50 years is somehow a function or extrapolation of the near normality of returns over 10 or 15 years. I agree that to some extent that might not be true and I have argued for the higher probability of catastrophe over the long run from two independent and unrelated arguments. It sounds like we agree with the second argument that the probability of volcano or asteroid or political collapse is higher over 30 years is higher than 5. But you don't buy that the normality of returns over 10 is an argument for normality over 30 and that the first argument is totally unrelated to the second.I think we are coming closer; but I still don't agree. The blue and yellow lines are based on about 10-15 independent periods, and yet the upswing and divergence of the solid from the dashed lines are what I would call consistent and significant.However, I don't need data to tell me the probability of a 90% loss is greater over 30 years than it is over 5. The probability of a major natural disaster or social/political upheaval event is greater over 30 years than over 5. Maybe it's not exactly as more likely as extrapolating a normal distribution of returns would theorize. But I think the overall point is very much intact. I do agree with your point that real assets are unlikely to evaporate and would have some mean reverting properties. So it's not perfectly normal. But I'd also speculate that while the partial/total collapse of the U.S. government (let's say it is replaced with some sort of violent populist/fascist/socialist government that doesn't respect private property rights) only has a 1-5% chance in the next 5 years, the probability over the next 30 years might be as high as 10%. The probability of an asteroid large enough to mostly destroy the economy might be .1% in 5 years but 1% in 50 years (or maybe it's 0% for both and this is a bad exaple). The risk of extreme events large enough to mostly or totally destroy the stock market certainly increases over longer horizons.
At a certain point it stops mattering though and this is where I have more of an issue with Samuelson's assumptions. I don't know that I care that much about a 99% loss vs 98% loss (or 100% loss).
Of course this doesn't tell us anything about very rare and very negative events. If we had enough time on hand, we could probably decompose the stock market returns into the effects of changes to risk-free rates, economic downturns, profit margins, valuation multiples / equity risk premia, and a few more; I'm sure others have done that already. This would probably make more sense than extrapolating a formula that never covered any single data point even close to such a scenario. Most if not all of these underlying parameters are "naturally" "soft" range bound and mean reverting. Then the question would become, think of a hypothetical scenario that can result in a drawdown, like > 90%, that the mean reverting characteristics and the growth of the exponential function cannot "heal" within 30 years.
Perhaps we can agree that if such an event happens at all, it would be a singular event during a 30 year period; the probability of two such events happening within 30 years is negligible (think of 2 independent asteroids in short sequence). Perhaps we can also agree that it would likely be an event with a root cause that is outside the normal economic and financial forces that until now governed the modern financial markets - a natural or man-made global disaster, or political upheaval; it would likely not be a slowly progressing or a series of several economic or valuation dislocations that would all go in the same direction, compound, and never end within a 30 year period; you can expand or compress things like valuation multiples, risk-free real rates, etc., only so much, until they either vehemently reverse or yield powerful carry returns.
So the question becomes with equal portfolio leverage, would you prefer a smaller chance of a singular catastrophic event happening during for example the next 10 years, with not enough time to recover by way of portfolio growth via exposure to the "regular" risk premia, or a somewhat bigger chance of the same catastrophic event happening during the next 30 years, with exponentially compounding growth during the remaining let's say 25-29 years.
I would tend to go with the second option, and I think the total loss after recovery from a singular catastrophic event would be less than it would be during a 10 year timeframe. Let's assume you start with $1M, ready to retire early with no future contributions expected, and let's ignore consumption for sake of simplicity; perhaps you have some semi-passive side income for living expenses, or perhaps it is a trust. Let's assume 4% equity risk premium (ignoring the asteroid), 5% annual geometric real return of mHFEA (ignoring the asteroid), and figuratively speaking an asteroid hitting and destroying 80% of Earth (substitute any other catastrophe of your imagination). Within 10 years you would end up with $1,000,000 * 1.05^10 * 0.2 -> $325,779. Within 30 years, you would end up with $1,000,000 * 1.05^30 * 0.2 -> $864,388 i.e. almost no loss. (Wait another 3 years or get some lazy side job, and you recovered your money, short of another asteroid hitting Earth.) Even though the likelihood of the same asteroid hitting was 3 times as high during 30 years as during 10 years, there would be no significant terminal portfolio loss in the 30 year scenario.
Now you may say ok but a 95% to 100% wipeout instead of an 80% wipeout would leave residual value with similar utility in both 10-year and 30-year scenarios, but at 3x the likelihood in the 30-year scenario. True, but what is the likelihood of yourself surviving such a scenario? So you would have to condition the scenario analysis and probabilities on your own survival.
Furthermore you would want to condition your analysis on the scenarios where a less leveraged portfolio than the one under consideration, or an unleveraged portfolio, would have survived with meaningful residual value. The scenarios where everything or almost everything on Earth that your portfolio assets might represent simply ceases to exist, constitute risks that are inherent in the passage of time, and not inherent in a particular portfolio, asset allocation, or leverage ratio.
In any case I think the model needs adjustments for rare events, not simplistic extrapolation. A 1929-1931 scenario repeating somewhere between 1931 and 1959 before the market recovered, was less likely than the initial 1929-1931 scenario, or perhaps impossible under any sensical economic assumptions; likewise the rates-driven 1968-1982 scenario repeating in 1982-1996 and combining to a 28-year long multiples compression period would arguably have been impossible, as stock market multiples were already extreme in 1982.
Id emphasize again we don't need perfect normality or anything close to perfect normality to conclude that the probability of catastrophe rises over time. Even distributions with high auto correlation show increasing probability of catastrophe over time. Anything with autocorrelation significantly higher than -1 will have this property. It's more simply a result of the basic intuition that getting multiple bad years in a row is only possible when you actually have more years. Even if there is autocorrelation well below zero (bad years tending to be followed by good years). This is one reason why so many phenomenon tend to fall into near normal distributions even when you don't expect them to. The bell curve and increasing probability of catastrophe are more like artifacts of nearly any distribution than strictly normal ones. They're mathematical artifacts of any event with an element of chance. Only pre-determined causative sequences of events would seriously defy the bell curve or increasing probability of catastrophe with more events.
I'd also argue that the near normality over 10-15 years and my inclination to extend semi-normality to 30+ years is not entirely unrelated to the intuition that asteroid/volcanoes/political collapse are more likely over 30+ years that 10. While we didn't experience a major asteroid in the data, there were a few political and economic shocks and upheaval in the data, like the great depression and ww2. Having 2 great depressions in 30 years is certainly possible and more likely over 30 years than 5. There are some aspects of the great depression and ww2 that are likely mean reverting (primarily valuations) but there are other aspects that are not mean reverting. It's pretty clear that global gdp was set back by the great depression and ww2 and did not mean revert compared to a hypothetical where the great depression and ww2 did not happen.
Global gdp per capita increased a mere 25% from 1930 to 1950 whereas normally it increases about 60% every 20 years. There was some acceleration after ww2 globally and especially in the U.S. but not nearly enough to make up for the poor growth from 1930-1945. These are real shocks to the real economy that do show up in the actual historical data and are part of the reason that the data is near normal. Arguing for semi-normality of returns or a lack of perfect autocorrelation or a lack of perfect mean reversion is, in practice, arguing that 2 great depression and ww2 are more likely in 30 years than 10. You could certainly argue that people's propensity for stupid economic policy and blowing each other up is autocorrelating/mean reverting and there might even be some truth to it but the argument is certainly not strong enough to completely discount the probability. If the probability of a great depression every 10 years is 1 in in 20 then the probability of two in 10 years would be about 1 in 400 if there is no autocorrelation. The probability in 40 years will be close to 4 in 400 though substantially greater.
And I concede from an equity market perspective much of the great depression could be mean reverting (valuations) but much or most of the effect on gdp was not.
I take your argument about mean reversion and I think there is some truth to it. Many of the less catastrophic events that effect returns over 10 years may be mean reverting over 30 (ie valuations). Some of these events are probably not mean reverting and thus the probability of catastrophe rises over time. But I think the weakest of Samuelsons assumption by far is the assumption that we care bout losing 95% vs 90% as much as we care about losing 50%. We've both argued against this in various ways. At a certain point the result is so bad you stop caring about investing
Statistics: Posted by skierincolorado — Mon Oct 07, 2024 11:01 pm