## Wednesday 15 July 2015

### Equilibrium asset pricing models - a quickie (a bit technical)

In my usual daily trawl through FT Alphaville's further reading (worth looking at, registration required but free - it isn't behind the FT's paywall) I came across this little gem.

"Actually I think there has been historically quite a lot of interest in models based on macro factors that would lend themselves relatively naturally to interpretation in terms of equilibrium valuation models. The problem is that they don't tend to work very well empirically... those models just didn't do a good job of explaining a substantial fraction of asset returns... Plus they usually seem to have unobservable or hard-to-observe components. For example, any model of interest rates is going to have to have some place for a market price of interest rate risk.... Even a small error in that estimate is going to produce a huge discrepancy to observed yields."

To be clear we're talking about models of the form P = f (a,b,c....) where P is the 'correct' asset price, and a,b,c are factors like for example the market price of interest rate risk and other such. To make these models work you also need to know the factor loadings; if the model was linear these loadings would just be the coefficients on the various factors in a pricing equation.

If this sounds like mumbo jumbo, bear in mind you probably already know about one such model, which is the CAPM (if you don't know what the CAPM is then it's still going to sound like mumbo jumbo and this post probably isn't for you). In the CAPM the factor is the market return and the loadings is the much maligned beta (covariance with the market). The CAPM equation is specified in excess returns rather than expected values, but it's easy to switch between the two when we note that the correct value is equal to todays price plus the expected return. If we add in the expected speed of reversion to the correct value then we can make the two approaches equivalent.

Now I've spent a fair bit of my own time in the past messing around with these models, and I have a few thoughts and ideas to share. To make this post more tractable I'll refer to a specific examples: where we are trying to estimate the equity risk premium.

### Choice of factor

There is an unfortunate habit in the literature of these models to choose factors which have nice economic meaning, regardless of whether they are observable. So for example in the equity risk premium world it usually makes more sense to use expected real interest rates. However it's difficult to get inflation expectations for some long horizon. In some countries you can use inflation linked securities, but biases mean it's not easy to extract inflation expectations from them. Survey measures are better, but will rarely go back  (for a longer and better discussion of this see Expected Returns).

It's tempting to fall back on creating your own model to forecast inflation. A reasonable model would include an equilibrium (based on a moving average of inflation), a rate of change (recent changes in inflation) and survey and asset price measures if available. Note however we've just exploded the number of coefficients we have to estimate. Also if the model is to forecast a long time in the future (say 20 years) it's going to consume a lot of data history, and waste the last two decades worth.

An unpleasant class of models is where the loadings are specified, and you find the factors. This is done in parts of the bond pricing literature. One of the few nice things about the macro factor models is that the factors have some kind of meaning. But rather than have intuitive interest rate factors like the level of rates, and the steepness of the curve, we have a factor which is based on a weird function of forward rates.

### Reversion to equilibrium

An obvious point is that these models are no good if they don't get the equilibrium value correct. As the quote notes a slight difference in the equilibrium can make a big difference. For example the equilibrium PE ratio of stocks (which we can extract from the equity risk premium) under one model might be 20, and with slightly different assumptions it might be 15. Give the normal range of stock PE ratios this will make a huge difference to performance.

 source: investopedia

### The opposition - kitchen sink

You can think of a macro factor model as a big bunch of variables in a regression; on some of which the coefficients or relative coefficients are specified (like the current real interest rate is the nominal interest rate minus current inflation); and others (the loadings) which we have to estimate.

Given all the caveats above you might conclude the best approach is to just throw all the observable factors you have into a giant kitchen sink regression and let them fight it out. So rather than trying to cobble together an inflation forecasting mini model as above, you just chuck the underlying observable variables into the regression with everything else.

But the big advantage of having some pre-existing structure in your data is it reduces the degrees of freedom, and if you are right about the structure will give you a more robust model.

(You could also take a Bayesian view that you think your imposed structure is correct, unless proven statistically false).

### The opposition - simple mean reversion

There is an even simpler method, which is to do a simple mean reversion model on one or more valuation factors. So in stocks you would just take the history of PE ratios, and then take some kind of slow moving average to measure the equilibrium. If the PE dips below that equilibrium then you buy; otherwise you sell.

Mild improvements can be made to this by incorporating more sensible variables, as is done in the CAPE model with real earnings, as long as you don't put yourself into a position when you have to start estimating coefficients again.

### Mean reversion isn't, usually, strong enough

These macro models should have an advantage over a kitchen sink approach where the same kind and number of underlying variables are regressed without any structure being imposed. However they're unlikely to beat a relatively simple mean reverting model. Worst still all such mean reverting models only perform well at relatively slow time scales, useless to any active investor.

If for example you are forecasting at a horizon of a few months, and you add a 'momentum' factor to your model, you'll find it dominates all your other predictive variables.

What kind of valuation model works at sensible time scales?* No, not a macro model but a micro, relative value, model with asset specific factors in it. In this case we don't need to worry if the equilibrium is correct or not; it's implicit in the fact that we have no net exposure, only relative value bets.

* though having said that relative momentum is also important in these kinds of models.

### Good at telling you where you are... not where you're going

I have similar feelings about these kinds of models as I do about "big data" (bear with me). Big data, it strikes me, is very good at modelling the current behaviour of for example spending by loyalty card shoppers*. Each of those shoppers is relatively similar (they all go to the same supermarket for a start), and importantly they are not interacting with each other, nor do their shopping habits change much when for example the US non farm release comes out. Unless their behaviour changes radically, which it tends not to, we can make some reasonable predictions about how they will behave.

However big data and market forecasting is more difficult. The prices are a result of a large number of relatively heterogenous participants interacting, and reacting to exogenous news and shocks, in a way that supermarket shoppers hardly do. At best you can overfit to the extreme and tell some interesting stories. Just don't try using it to trade.

Similarly well specified macro models can tell you some very interesting stories about the past. So it's possible to disaggregate the equity risk premium (without incidentally needing to estimate any coefficients), and conclude that much of the excess return of equities in the last 40 years is due to a secular fall in inflation.

The only useful prediction I can make out of that is that equity returns are likely to be lower in the future. That's useful, but it barely qualifies as a forecast and its no use whatsoever for any kind of dynamic trading.

### Conclusion

It's fun playing with macro models, and they're intellectually interesting, but they have no place in the armory of any investor or trader.