I’ve been seeing a lot of comments lately about how the financial markets have gone completely wonky, efficient markets hypothesis looks crazy right now, etc. I don’t currently trade actively and haven’t run a lot of numbers, but just in terms of big-picture qualitative behavior, high stock prices make a lot of sense right now. This post is an informal explanation of why.

First, let’s forget about the efficient market price formula (i.e. price = expected sum of discounted future cash flows, $V_T=E\left[{\sum }_{t>T}{e}^{-R_T^t}{C}_t\right]$). I’ll talk about that a bit at the end, but it’s so widely and severely misunderstood that I’d need a whole post just to correct misconceptions. Instead, we’ll start from first principles: financial capital is a good, just like any other good. Its price is determined by supply and demand, just like any other good.

When stock prices are high, that means financial capital is cheap for companies: they can get a lot of capital by issuing a lot of stock. High stock price = cheap capital. Likewise with bonds: when bond prices are high, yields are low, meaning companies can borrow capital very cheaply.

What makes the cost of financial capital move? Well, the usual supply-and-demand reasoning:

• If people suddenly find themselves with lots of extra savings to invest, that means the supply of financial capital increases, and the cost of financial capital should fall (i.e. stock prices rise).
• If people expect lower returns in the future, they will want to invest less, so the supply of financial capital decreases, and the cost of financial capital should rise (i.e. stock prices fall).
• If there’s a credit crunch and companies suddenly need to borrow lots of money on short notice, then the demand for financial capital increases, so the cost of financial capital should rise (i.e. stock prices fall).
• If many companies are suddenly flush with cash, then the demand for financial capital decreases, so the cost of financial capital should fall (i.e. stock prices rise).

This should all be pretty intuitive, and you can probably brainstorm a few more examples along these lines.

Now, what’s been going on lately, and how does it fit into this picture?

Expectations of future earnings are generally down (although mostly just in the short term). Many companies suddenly need to borrow money in order to stay in business until the storm passes. On their own, these two factors should both push stock prices down: supply of financial capital should be low, and demand for financial capital should be high.

The size of both of these changes are big, but not too far out of line with a normal business cycle slowdown. They are significant, but not huge by historical standards. On the other hand, there has been one ridiculously huge delta which utterly dwarfs any fast change we’ve seen in economic fundamentals in the last seventy years:

That’s the personal savings rate - the amount people save, relative to their disposable income. Given how the modern financial system works, that’s basically the supply of financial capital. It quadrupled in a month.

Even if people were nervous enough about the recovery to allocate half as large a share of marginal savings to stocks as they were a year ago, even if real disposable income were down (it’s actually up, courtesy of stimulus payments), that would still be a near-2x increase in marginal savings allocated to stocks. That jump in the personal savings rate is ridiculously larger than any change in economic fundamentals in living memory; it shouldn’t be surprising if it completely dominates market behavior.

What About That Formula?

Warning: more math and jargon past this point.

Ok, now we’ve talked about first principles. Hopefully it all makes intuitive sense. How does it square with $V_T=E_t\left[{\sum }_{t>T}{e}^{-R_T^t}{C}_t\right]$?

The key questions are: what’s that discounting rate R, and what distribution is the expectation over?

Many people will say it’s the “risk-free rate”, i.e. yield on Treasury bonds, but those same people will outright admit that this does not actually work. It predicts prices far higher than actual stock prices, and says that people ought to sell treasuries in order to buy stock. Obviously people don’t do that, because we’re not risk-neutral (nor should we be). The whole notion of R being the risk-free rate is based on a dumb argument that nobody actually buys.

Some people who’ve seen some math finance may talk about the “risk-neutral distribution” and corresponding discount rate. These are great tools for pricing derivatives, but they’re severely underdetermined for problems like “determine stock price from fundamentals”. They just assert the existence of some distribution and discount rate which make the formula work; they say nothing at all about what the distribution and rate should be.

To get a proper foundation for the pricing formula, we need to go to financial economics. John Cochrane (aka The Grumpy Economist)  has a pretty decent book on the subject; he gives the economist’s simplest version of the pricing formula at the very beginning:

$V_t=E_t\left[\beta \frac{u^{\prime }\left(c_{t+1}\right)}{u^{\prime }\left(c_t\right)}{C}_{t+1}\right]$

Here the “discount rate” $e^{-R_t^{t+1}}$ for the timestep $t\to t+1$ is the magic expression $\beta \frac{u^{\prime }\left(c_{t+1}\right)}{u^{\prime }\left(c_t\right)}$. What is that?

• $c_t$ is the amount the investor consumes at time $t$ - i.e. if this is a retirement portfolio, it’s the amount taken out.
• $u$ is the investor’s single-time-step utility function, and $u^{\prime }$ is its derivative with respect to amount consumed.
• $\beta$ is the investor’s own discount factor, i.e. how much they value consumption tomorrow relative to today.

Note that I keep saying “the investor” here - this formula is for one single investor! We don’t need to assume that all investors are expected discounted utility maximizers. If any investor acts like an expected discounted utility maximizer, then the formula applies for that investor’s discount rate, utility function, and expectations. The formula comes directly from the investor's own utility-maximization condition.

(Side-note: I actually don’t like the formulation in which the investor has an explicit time-discount with consumption each time step; I prefer to have the investor just maximize expected utility at some far-future timestep with exogenous cash-flows along the way, as a more accurate model of something like e.g. a retirement fund. For current purposes, the results are quite similar. Take-away: the things we’re saying here are not too highly sensitive to the model setup.)

Now, if you followed that, you’re probably thinking “Huh?? That means prices have to satisfy different efficient pricing formulas for different investors. But they’re the same prices!”. That’s right. The trick is, each investor will adjust their portfolio and their consumption $c_t$ to make their equation hold. This formula isn’t for predicting prices, it’s for predicting how much of each asset the investor holds going into the next timestep.

If we want to use the formula to predict prices, then we have two options.

• The hard (but right) way: compute the whole equilibrium for all the investors.
• The easier (but unreliable) way: notice that the consumption, distribution and discount rate for most investors seem to follow pretty stable patterns, then assume that those patterns hold and ask what price that implies.

Most usages of the pricing formula are ultimately equivalent to the second, with various flavors of first-order corrections thrown in. That works as long as the investor fundamentals are stable, but if there’s a big change in investor characteristics - like, say, a giant jump in the savings rate (i.e. a drop in consumption) - then obviously it falls apart.