What’s the point of maths in economics? I ask because of this:

Much of the mathematics used in economics is simply performance. Despite the sometimes undue attention given to mathematical methods, the actual mathematics economists use often contradicts their economic conclusions. The mathematics in economics is simply a tool to keep out outsiders: economists themselves do not take it that seriously.

I suspect this is sometimes (often?) true. But I want to give a counter-example. It comes from a source some of you don’t like, John Cochrane. It’s this equation (pdf) for predicting long-run returns on assets relative to returns on cash:

ER = RA x SDa x SDbg x corr (a, bg)

Where RA is a coefficient of risk aversion, SDa is the standard deviation of asset returns, SDbg is the volatility of background risk, and corr(a, bg) is the correlation between asset returns and background risk. The intuition here is simple. High expected returns should be compensation for higher risk or for a higher distaste for risk. And the risk that matters isn’t just the volatility of the asset, but its correlation with our background risks – the risks to our job or business. An asset that falls when we lose our job or business is much riskier than one that holds up well in bad times, and so must pay us higher returns on average to compensate us.

Now, the derivation of this is quite tricky: see the appendix to Cochrane’s paper. But once something has been derived once, it stays derived. Working through the derivations might be useful for torturing students, but the rest of us can skip them and get to the meat.

We can apply this equation to any asset. I suspect it has done a decent job of explaining long-run returns on UK housing, for example.

Let’s put some numbers on this for equities. The coefficient of risk aversion varies across individuals, across time, and from context to context, but let’s call it three (pdf), where one betokens indifference to risk. The standard deviation of annual equity returns has been 15% (or 0.15) since 1985. The volatility of background risk – how likely we are to lose our job or business – is hard to measure and varies from person to person, but let’s call it 0.1 for illustrative purposes. Corr (a, bg) also varies from person to person. For most of us, though, it’s positive: we know this because shares fall in recessions. Let’s call it 0.4.

We can then multiply these numbers through: 3 x 0.15 x 0.1 x 0.4 = 0.018. Which tells us we should expect equities to out-perform safe assets by 1.8 percentage points a year on average over the long-term.

My chart shows how this compares to reality. It shows that unless you had bought at the right time (such as after the bursting of the tech bubble or at the low point of the 2009 crisis) you’d have earned an equity premium of less than this on the All-share index. For the MSCI world index, however, the equity premium for pretty much any period in the 90s was very close to this prediction. Thanks to the boom in US equities, though, the premium has been more than this for most of this century.

Which represents a return to history. Back in 1985 Rajnish Mehra and Ed Prescott pointed out – based on this sort of thinking – that equities had done better (pdf) than they should for much of the 20^{th} century.

Why? Herein lies one virtue of this equation. It is sufficiently clear and parsimonious for us to ask: what does it leave out?

One thing is disaster risk. The (not so small) risk of a catastrophe (pdf) justified higher expected returns on equities for much of the 20^{th} century – especially for investors who were loss-averse. Another thing is market imperfections. Roger Farmer has pointed out (pdf) that many of the young are excluded from stock markets by borrowing constraints. That causes shares to be cheaper than they should be and riskier too – because such people cannot buy when prices fall. Which justifies higher returns.

I’d highlight several merits of this equation - ones which give us criteria by which to assess when maths is useful in economics.

First, it is simple enough to be useful. If we need Matlab to solve equations, they are too complicated for us to fully understand and manipulate. The maths then becomes “a tool to keep out outsiders”, and to prevent them following what’s going on.

Secondly, this equation allows reverse engineering. If people expect high returns, we can ask: what terms on the right hand side of the equation justify this? We need high expected returns on Bitcoin, for example, because the thing is so damned volatile and – as its fall last March showed – correlated with background risk.

Thirdly, maths can be useful without precise numbers - as my illustration showed. As the late Thomas Mayer said, there’s a trade-off between truth and precision. Equations are ways of organizing our thinking. They allow us to ask: what is it that we need to know? What might be missing from this equation? But it is only equations that are simple that allow us to do this.

Finally, this equation is useful for individuals. Economics should not be confined to academia, nor is it a subsidiary of politics. It should instead help people in their everyday life. Which is what this equation does. It draws our attention to an important fact. If you have low background risk – say because you’re in a safe job or have retired on a big final salary pension – you are better able to take on risks which are correlated with that risk than others. Assets that fall in recessions are less risky for you than they are for people who might lose their jobs in such recessions. You can therefore pick up a risk premium in normal times for taking on a risk that doesn’t bother you. Which means you’re getting something for nothing.

Now, imagine you retire from a job that you could lose in a downturn and take a final salary pension. Your background risk falls. So you’re better placed to take the recession risk which equities carry. Which means that you should own more shares when you are old than when you were younger. For you, the conventional financial advice that people should own fewer shares as they age is flat wrong. Which poses the question: if financial advisers are wrong on this, what else might they be wrong about?

Apparently arid mathematical economics can therefore sometimes undermine orthodoxy and vested interests.

My favourite twitter heterodox economists often claim that econometrics is a waste of time and if you can't see it by eyeballing the data it isn't there, and also that capital stock data and TFP estimates are made up numbers with no substance. If I had the time I would like to write a similar defence of those too.

Posted by: Paddy Carter | March 02, 2021 at 02:11 PM

Maths is a language. Economists are free to use languages other economists understand. But maths in economics go haywire sometimes.

My late friend the mathematician complained once that economists didn't understand what maths is. They used to ask him to put a mathematical formula of the idea in their theses - and when he had dome that, they believed that he had proved their ideas were correct.

Maths is however just a language. And you can tell a lie in any language.

The other trap is that maths is a very hermetic language. Many have deplored that Hyman Minsky's idea of financial crashes were not heeded until the crash had actually happened. But I tried once to read his book (Can it happen again) - and was taken aback by his mathematical language I didn't understand.

He would perhaps have been better understood if he had written in a language people - not least politicians - could read. Perhaps even the crash would have been averted.

Posted by: Jan Wiklund | March 02, 2021 at 04:17 PM

Finance equations are not the same as "Economics" equations. Finance is not the same as "Economics", even if many "sell-side" Economists seem to think that Finance something like that.

Posted by: Blissex | March 02, 2021 at 06:07 PM

Agree with Jan Wiklund and Blissex (on this one).

I wonder what the standard deviation of assets that measure volatility is? Options markets are larger in dollar volume than the underlying stock markets they derivatize. The idea behind options bets is that you can hedge and minimize the volatility of your returns. Is Cochrane even aware of tradable volatility products like VXX, SVXY, UVXY, and the options you can buy and sell on them?

It's also interesting that robinhood.com tries to make margin trading and fractional share purchases available for small retail investors, but regulators treat this as a threat ...

Posted by: rsm | March 03, 2021 at 12:12 AM

Maths sometimes reveals 'something funny going on'. A situation where the facts we think we know do not quite add up - think how Maxwell figured out electromagnetism or Einstein saw how time and space and gravity hung together. Looking around and recognising 'something funny going on' is the key to new discoveries and insights.

Perhaps there are plenty of practicing traders who never knew much formal economics or have forgotten it do notice 'something funny' or have a hunch about some market phenomenon. Linking them to someone who has the time and intuition and formal abilities - a formal learned society or the pub - which is more likely?

Posted by: theotherJim | March 03, 2021 at 09:49 AM

It strikes me that Cochrane's equation basically reduces to Expected Returns = Volatility. (The other terms are subjective, unobservable, or 1 if your job is trading.)

Volatility traders say everyone is trading volatility without necessarily knowing it, so why not trade VIX options directly? If you buy a S&P 500 ETF, you are short volatility so why not cut out the middle stock and just short the VIX?

Investopedia reports:

"VIX options are powerful instruments that traders can add to their arsenals. They isolate volatility, trade in a range, have high volatility of their own, and cannot go to zero. For those who are new to options trading, the VIX options are even more exciting. Most experienced professionals who focus on volatility trading are both buying and selling options. However, new traders often find that their brokerage firms do not allow them to sell options. By buying VIX calls, puts, or spreads, new traders gain access to a wider variety of volatility trades."

So Cochrane's equation reduces to something quite profound that I don't think he quite understands, because he doesn't trade options.

Posted by: rsm | March 04, 2021 at 09:08 AM

“This paper has suggested a simple model that can account for the key anomalies of the traditional monetary approach. It disaggregates the quantity of credit into a 'real' and a financial circulation. In time periods, when the ratio of credit in the financial circulation to credit in the real circulation rises, the simple quantity theory must be expected to disappoint, as it is a special case of the more general quantity theorem of disaggregated credit. In such time periods, a financial boom is likely, as asset prices are driven up by speculative borrowing on the back of collateralised assets. This explains why the traditional monetary quantity theory was not popular in the 1920s and 1930s, and again in the late 1980s and early 1990s. Then the traditionally defined velocity of money declines and excess credit creation can 'spill over' as foreign investment. However, during time periods such as the 1950s, when in many countries credit was mainly channeled into the real economy, asset prices remained stable and the traditional quantity theory could be expected to hold. The fact that the model can account for the major anomalies observed in many countries over many time periods demonstrates generality and robustness.

The empirical results for the Japanese case have been unambiguously supportive. The Japanese asset bubble of the 1980s was due to excess credit creation by banks for speculative purposes, largely in the real estate market. The apparent velocity decline is shown to be due to a rise in credit money employed for financial transactions, while the correctly defined velocity of the real circulation is found to be very stable“

https://eprints.soton.ac.uk/36569/1/KK_97_Disaggregated_Credit.pdf

Posted by: James Charles | March 06, 2021 at 09:38 AM