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Equities as bets

Created:

10 July 2008

Written by:

Chris Dillow

What is an equity? This isn't an abstruse philosophical question. It might help us understand better the market's current troubles.

The standard answer is that a share price is the discounted present value of all future cashflows the shareholder can expect.

However, this view - the dividend discount model - doesn't get us far. It gives us a single equation with not one or two but three unknowns.

The model implies that the dividend yield on a share should be equal to the risk-free discount rate, plus a risk premium to compensate for the riskiness of future cashflows, minus expected annual growth in cashflows from now to infinity.

The trouble is, we cannot directly observe either the risk-free rate, or the risk premium, or expected cashflow growth.

You'd think the risk-free rate would be easy. It should be just the index-linked gilt yield. But here's the problem. This yield has fallen by two percentage points since early 1998. Other things equal, this should have cut the dividend yield on the All-Share by two percentage points, with growth stocks benefiting more than value stocks because they offer more cashflows in the distant future and so get a bigger kick from a fall in discount rates.

But neither of these things have happened. The All-Share's yield is almost two percentage points higher than it was in early 1998, and value stocks - despite their recent slump - have actually outperformed growth ones.

Now, we can reconcile these facts with the dividend discount model. Maybe shares are being priced off of a 'shadow' risk-free rate which has risen in the past 10 years. Or maybe the risk premium has risen four percentage points. Or expected growth fallen by four percentage points. Or some combination of the three.

However, none of these possibilities is directly observable.

What's more, when the market falls, it's often impossible to tell whether it's done so because expected cashflow growth has fallen, or because the risk premium has risen. But the distinction matters enormously. The latter implies that expected returns have risen, meaning there's a good chance of the market rising. The former doesn't.

Sure, it's a useful rule of thumb that big moves in shares - up or down - are usually more due to changes in expected returns. But rules of thumb aren't hard knowledge.

So, there's a lot wrong with the standard view. Luckily, though, there's an alternative. Don't think of shares as the discounted present value of future cashflows at all. Think of them instead as state-contingent securities that pay off different amounts in different states of the world.

The simplest state-contingent security is a bet. Financial services provider Paddy Power is offering seven-to-four on John McCain becoming the next US president. If you hold £1 of this asset, you get £2.75 in states of the world in which McCain becomes president, and nothing in other states.

You can think of the All-Share in a similar way, except - we hope - that zero pay-offs are less likely.

For example, imagine two states. In one, the All-Share yields 5 per cent, as it often did until the early 1990s. This implies an index value of around 2300. Think of this as a world in which economic volatility keeps shares cheaper than we have been used to in recent years. In the other state, the index yields 3 per cent - giving a level of just under 3900. This could be what we would get if the 'nice' decade returns.

You can now imagine the index's current value as signalling that we have a 70 per cent chance of entering state one, and a 30 per cent chance of entering state two. That is: (0.7 x 2300) + (0.3 x 3900) = 2780, which is roughly where we are as I write.

Now, imagine that the probability of entering state one - our bad state - were to rise by just one percentage point. Then, the price of the All-Share would become: (0.71 x 2300) + (0.29 x 3900) = 2764. This is a drop of 0.7 per cent.

In other words, tiny, reasonable changes in the probability we attach to differing but plausible states of the world can produce quite large moves in the index.

Regarding the market as a state-contingent security, therefore, shows that volatility is normal and rational. Indeed, David Meenagh of Cardiff Business School has shown that market volatility since 1963 is quite easily explained if we think of prices as reflecting varying probabilities of different states of the world. This contrasts to the dividend discount model which - as Yale University's Robert Shiller famously pointed out in 1981 - implies that volatility should be only around one-tenth what we actually see, and that only irrational investors can generate such high volatility.

Now, my example is over-simplified. In practice, there are countless different possible states: catastrophe, slump, recession, slowdown, normal growth, boom, exhilaration and Kurzweilian singularity, and all degrees in between.

Recognizing this renders otherwise bizarre behaviour explicable. Why does volatility rise in recession? Because investors begin to attach small and varying probabilities to catastrophe and slump. Why did we have a tech boom and bust earlier this decade? Because the probability of exhilarating growth rose and then fell.

Of course, it's not just the index that we should regard as a state-contingent security. We should regard sub-sets of the market, and individual shares, in the same way. Value stocks pay off badly in recession states but well in boom states. High-beta stocks pay off well in states in which investors have high appetite for risk, and badly in other states. Commodity stocks have a good pay-off in states where Bretton Woods II holds and the global economy grows nicely, but low pay-offs in other states.

And surely it makes more sense to regard Bradford & Bingley's price as depending upon the probability of its re-re-re-financing going well than it does upon a forecast for its earnings in 2011.

What this means is that most investors should change the way they think about shares. Stop thinking of them as having a 'fair value' of x, depending upon your forecast for earnings and assumptions about discount rates. This is just plucking numbers out of the sky.

Instead, ask the questions: in what states of the world would this pay off, and by how much? What reason do I have to think good states more likely than the market believes? Am I more or less worried than the market by the probability of bad states?

You might object that this method doesn't give us precise valuations, because we can't define possible states of the world, still less attach probabilities to them.

This is true. But such imprecision is an unavoidable consequence of the fact that the future is inherently unknowable. To hide from this fact, and pretend that future cashflows and discount rates can be predicted and shares precisely valued, is mere pseudo-science which gives us a false comfort that there's such a thing as 'value'.

But maybe it's false comfort you want.

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