I think it's correct to say that stocks (and all assets) are less risky as T gets large under most useful definitions of risk. For example, as T gets large (assuming avg return on stocks is 6%) the probability of losing money goes to zero, as does the probability of losing to a lower-return asset. For most people this means its pretty low risk to invest in stocks long-term. Obviously the variance of total returns does grow (assuming no neg. autocorrelation), but the average annual returns converges to the 'true' expected annual return of the asset.

Put another way, with large enough T, assuming stocks return on average more than bonds, there will literally be zero cases where the bond returned more than the stock, despite the variance of total stock returns being much greater than that of bonds. Basically LLN just kicks in and avg return converges to expected return of each asset.

Dec 28, 2022·edited Dec 28, 2022Liked by Andy Preston

Just simulating some stuff, and I guess the whole LLN thing is a bit more complicated because of taking geometric means, but the point remains that with large T all that matters is expected annual return when choosing which asset to invest in. Annual fluctuations are irrelevant in the sense that they don't impact which asset will return more in the long-term. I'm thinking of T as years, but could be days or months or decades, same point goes through.

Yes this is very true, good point. I guess the practical issue is that T may only be 60 years or so for the average investor, and so not long enough for asymptotics in some sense.

Though the probability of stocks losing to bonds decreases with T, the severity of loss (when it does occur) also increases with T. This is widely discussed by Zvi Bodie, for example.

Yes, I guess that is technically true, as the variance explodes as T grows, but is there any reasonable model of investor preference that can rationalize someone choosing the asset of "30-year stock return" over "30-year bond return", again using historical returns to parameterize both of these? We already have trouble rationalizing annual stock returns and bond returns with standard models of risk aversion, let alone 30 years of that.

One way to think about risk here would be as follows: an investor with a high level of risk aversion might be indifferent between investing in stocks and bonds for 1 year. But they'll prefer stocks if they are investing for 2 or more years. Hence stocks become less risky as T grows, in this sense.

(I could be wrong about the risk-averse investor preferring 2-year stock rel. to bond returns to 1-year returns... makes sense intuitively to me though.)

Yeah, good point, I was referring to IID log-normal returns (GBM), as Bodie and finance academics do. I'm not sure what the real-world data say. I suspect real-world data has some mean-reversion, which may eliminate these worst-cast scenarios. But I haven't dug further!

Standard utility functions seem lacking. People get utility from more than just consumption. To get a realistic utility function, I'd propose the following changes.

The utility function should depend on savings rates, wealth, and work-based income in addition to consumption. People get utility from a high savings rates, especially at low wealth levels. And that diminishes as wealth increases. People get disutility from work (proxied by work-based income), and the disutility rises as wealth rises.

These change reflect real world preferences. Young people enjoy saving a lot of money because young people are unsure of their future earnings potential, and they're used to low consumption. Plus, consuming a lot before you've "earned" the right to consume a lot feels immoral. As people become wealthier, they are less concerned with saving money, and are more willing to spend. And as people become wealthier, they are less interested in paid work. What do you think?

I think it's correct to say that stocks (and all assets) are less risky as T gets large under most useful definitions of risk. For example, as T gets large (assuming avg return on stocks is 6%) the probability of losing money goes to zero, as does the probability of losing to a lower-return asset. For most people this means its pretty low risk to invest in stocks long-term. Obviously the variance of total returns does grow (assuming no neg. autocorrelation), but the average annual returns converges to the 'true' expected annual return of the asset.

Put another way, with large enough T, assuming stocks return on average more than bonds, there will literally be zero cases where the bond returned more than the stock, despite the variance of total stock returns being much greater than that of bonds. Basically LLN just kicks in and avg return converges to expected return of each asset.

edited Dec 28, 2022Just simulating some stuff, and I guess the whole LLN thing is a bit more complicated because of taking geometric means, but the point remains that with large T all that matters is expected annual return when choosing which asset to invest in. Annual fluctuations are irrelevant in the sense that they don't impact which asset will return more in the long-term. I'm thinking of T as years, but could be days or months or decades, same point goes through.

Yes this is very true, good point. I guess the practical issue is that T may only be 60 years or so for the average investor, and so not long enough for asymptotics in some sense.

I think 60 years would be long enough. Even 35 years seems to be sufficient, depends what you use for mean and variance of stocks/bonds, evidently.

Though the probability of stocks losing to bonds decreases with T, the severity of loss (when it does occur) also increases with T. This is widely discussed by Zvi Bodie, for example.

edited Jan 1, 2023Yes, I guess that is technically true, as the variance explodes as T grows, but is there any reasonable model of investor preference that can rationalize someone choosing the asset of "30-year stock return" over "30-year bond return", again using historical returns to parameterize both of these? We already have trouble rationalizing annual stock returns and bond returns with standard models of risk aversion, let alone 30 years of that.

One way to think about risk here would be as follows: an investor with a high level of risk aversion might be indifferent between investing in stocks and bonds for 1 year. But they'll prefer stocks if they are investing for 2 or more years. Hence stocks become less risky as T grows, in this sense.

(I could be wrong about the risk-averse investor preferring 2-year stock rel. to bond returns to 1-year returns... makes sense intuitively to me though.)

Yeah, good point, I was referring to IID log-normal returns (GBM), as Bodie and finance academics do. I'm not sure what the real-world data say. I suspect real-world data has some mean-reversion, which may eliminate these worst-cast scenarios. But I haven't dug further!

There is a classic paper by Samuelson on this idea which is a brilliant piece of trolling:

https://www.sciencedirect.com/science/article/pii/0378426679900232

Standard utility functions seem lacking. People get utility from more than just consumption. To get a realistic utility function, I'd propose the following changes.

The utility function should depend on savings rates, wealth, and work-based income in addition to consumption. People get utility from a high savings rates, especially at low wealth levels. And that diminishes as wealth increases. People get disutility from work (proxied by work-based income), and the disutility rises as wealth rises.

These change reflect real world preferences. Young people enjoy saving a lot of money because young people are unsure of their future earnings potential, and they're used to low consumption. Plus, consuming a lot before you've "earned" the right to consume a lot feels immoral. As people become wealthier, they are less concerned with saving money, and are more willing to spend. And as people become wealthier, they are less interested in paid work. What do you think?

To what degree does James Choi represent mainstream economic consensus?