Bryan Caplan  

Mean Reversion and the Permanent Income Hypothesis

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Here's a question from my last Ph.D. Microeconomics exam.  Post your answers in the comments, and I'll share my suggested answer tomorrow.

Suppose you - and you alone - discover that the stock market is mean-reverting.  

True, False, and Explain:  If you are rational, you will NOT obey the permanent income hypothesis.

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COMMENTS (18 to date)
AbsoluteZero writes:

"Suppose you - and you alone - discover that the stock market is mean-reverting."

What about the stock market is mean-reverting? Price? Earnings? A ratio like P/E, P/B, dividend yield? Volatility? Is it delta or absolute values? Which mean, arithmetic? Geometric?

And mean-reverting over what time scale? Seconds? Hours? Days? Months? Years?

And what is "the stock market"? The US? World? World minus US?

David Manheim writes:

I'm assuming the price of stocks is mean-reverting over the medium term.

I alone know about this - that means there is a ton of alpha waiting for me simply by keeping a routinely rebalanced equal-weighted portfolio - and I can borrow to invest. This doesn't change the permanent income hypothesis - I have some expected return I am making, it's just return based on volatility instead of beta.

So nothing changes, and the claim seems obviously false. (Which makes me worry I'm missing something...)

Eric writes:

Functional capital markets are necessary for PIH to hold. Functional capital markets is orthogonal to "stock market is mean-reverting".

Steve F writes:

False, I will obey the permanent income hypothesis whether or not I know stocks revert to the mean. Yet I will not change my consumption habits because this new knowledge is not new because investors already behave as if stocks revert to the mean.

Karl W Smith writes:

So you can go a lot of places with this thought experiment but on net I am going to pick TRUE.

A couple of reasons.

First, suppose that you are liquidity constrained to start, as most folks are. Then when the market falls below the long run mean, you will reduce your consumption in order to increase your purchases of stocks. This is despite the fact that your permanent income is actually rising.

As stock prices start to climb eventually you will shift to a point where liquidity is non-binding and your consumption will rise. This is despite the fact that in some sense your permanent income is likely flat and possibly falling depending on the timing of the upswing versus your expectation.

Second, suppose that you are not liquidity constrained. Then your permanent income is unbounded. Or at least bounded by constraints outside the guidelines of this thought experiment. Like for example, if you get too rich then folks are likely to just seize your assets through direct confiscation or special taxation.

However, you are likely to be so wealthy that your consumption doesn't rise with your permanent income because of cognitive constraints. IE figuring out how to spend all of that money.

Wallace writes:

I'm guessing the answer is "no."

The "permanent income hypothesis" seems to refer to the model in which people smooth their income - borrowing when income is expected and saving if not - due, I guess, to diminishing returns to gains to additional consumption in a given time period.

"Means reversion" may mean a model in which stock prices tend to revert to a historical rate of growth.

If stock prices were mean reverting, it would be worth it to invest some of your present income due to the expectation of outsized returns when prices were below trend. Conversely, it would be worth consuming more of your income that you might have saved if you expect prices to fall, rather than invest it for future consumption.

Not sure I've got it though. If you expect prices to fall, can't you short the market and still make more money in a future period? That would still seem to imply deferring some consumption.

Wallace writes:

Er so I guess I'm saying "True" you will not obey permanent income hypothesis.

Francisco G writes:

If financial markets are complete, then you WILL obey the permanent income hypothesis. The PIH does not depend on the distribution of returns, only on the availability of financial instruments to smooth consumption. In other words, consumption and investment decisions are independent in the presence of complete markets.

Without complete financial markets I imagine one could construct situations in which it is optimal to finance stock investment via low consumption.

Wallace writes:

Francisco G's answer seems to make better sense than my attempt.

Jon Murphy writes:

I'm going to put the answer I put on the exam because I really want to know if I got it right:

False. Since you realized that you, and you alone, know the market is mean-reverting and you can exploit that knowledge, you will adjust your expectations for long run income (ie your permanent income), thus obeying the PIH.

Kevin Dick writes:

If you alone know the stock market is mean reverting (and this situation persists), you have a long term edge on the market--you go long after the market goes down and you go short when the market goes up (this works at whatever timescale at which it mean reverts).

Using this edge, you can apply the Kelly Criterion to grow your expected wealth faster than the market grows. In fact, you would eventually own almost the entire market.

If you're rational, you should optimize your expected discounted consumption stream. Since your wealth will be growing (in expectation) faster than everyone else's, you could consume an ever greater fraction of total output. Unless you have a very high consumption discount rate, this will lead to an optimal consumption path that grows over time.

I think.

MikeP writes:

If you are rational, you will NOT obey the permanent income hypothesis.


You can buy calls and puts for a basket of future dates to take advantage of any variance of the market from the mean. So you will certainly make money, and you will likely make a lot of money.

But the amount of money you can make depends on the unknown future variance of the market. Your expected income growth will be very high, and your actual income growth will probably be higher -- but it might not be. So you will be careful not to overconsume over your honestly expected future income. I.e., you will obey the permanent income hypothesis.

ShriS writes:

Depends on your utility function. With log utility, C/W is constant even if world time varying. Else, not (typically).

Basil H writes:

I apologize for sounding crazy, but I'm not sure that this question is well-formed.

Case 1: lending markets are perfect
- In this case, you borrow and borrow money until you alone arbitrage away the inefficiency and capture all of the profit
- Then the market inefficiency is erased, the question is sort of nonsensical, and the PIH holds

Case 2: lending markets are imperfect (liquidity constraints exist, for whatever reason)
- In this case, the PIH doesn't hold! Or at least, certainly not in its strictest form.
- Quoting from the New Palgrave:

"Another alternative to the standard PIH asset market structure is limiting the amount one can borrow against future income. The inability to borrow implies that eq. (4) may not hold. When constrained, a consumer may be forced to adjust consumption in response to a transitory or predictable shock to income. For example, if an individual receives a temporary income decline, the inability to borrow against future income may necessitate that consumption moves with contemporaneous income. Zeldes (1989) argues that liquidity constraints do bind for a significant fraction of consumers. Moreover, the inability to borrow presents consumers with the risk that a series of negative income shocks may force consumption down to extremely low levels. To mitigate this risk, potentially constrained consumers build up a ‘buffer stock’ of savings. See precautionary saving and precautionary wealth for a discussion of the accumulation of wealth for precautionary reasons."

William B writes:


Without making an unreasonable number of assumptions about what exactly you are allowed to do vis-a-vis borrowing money and selling short, your opportunity to make money in the stock market is the same whether the market is above or below its trend line.

So now you have a stock trading job where you make money at an unpredictable rate. Congratulations. Even so, at present you have some estimate of how volatile the stock market will be in future, which you can use to estimate your lifetime wealth. Adjust your present spending to achieve smooth marginal utility of consumption, just like the chumps with normal jobs are doing.

mariorossi writes:

As William B above the knowledge that the equity market is mean-reverting is a valuable asset you can exploit for permanent income.

In general, if you know that a market is mean-reverting with a certain speed you know you can make money by trading 2 opposite delta hedging strategies with different intervals. In a mean-reverting market, the strategy with wider hedging intervals will always generate less money then the strategy with tighter hedging intervals. You don't need to touch options, you can trade one option replicating strategy vs the other.

Depending on the mean reversion rate (the higher the better), you are going to make money in a pretty systematic way. If the mean reversion is very slow (months or years), than it's a lot less valuable.

The return from this knowledge wouldn't be infinite, given that the amount you can extract is a function of the volumne of trading. You can only profit when people that don't know markets mean-revert decide to trade with you.

Given that PIH requires complete markets, I would imagine that the statement is false. In a PIH world, you could borrow without constraint to finance your equity trading strategy and your consumption. If you were capital contrained to start with, then PIH coudln't hold in your world in general.

Rich Berger writes:

False. You may know that the market is mean reverting, but you don't know when.

rtd writes:

It must be True - but only because Noah Smith told me so :)

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