Asset pricing models

[Alpha9]

I don't get them!
See early on in Chapter 18.

e.g. consumption-based asset pricing model:
p(t) = E[ß × u'(c(t+1))/u'(c(t)) × x(t+1)]
Whatever this actually means, it seems to me that everything is the same price unless ß is specific to the asset in question. So it must somehow be a measure of the asset's risk, but not (given we are told that ß<1) in the 'normal' beta way.
Can it price a risk-free asset?
And the formula apparently implies that we have
m = m(t+1) = ß × u'(c(t+1))/u'(c(t))
so m doesn't depend on time? Why not? Surely the price of an asset varies with time, even with the same payoff?

Then we get onto the CAPM.
p(t) = a + b Rw(t+1)
We are supposed to be able to ascertain a and b by considering, say, the risk-free rate and the wealth portfolio return (so presumably we have two simultaneous equations to solve in a and b). But that will surely give us a and b fixed for all assets, hence all assets are the same price: either that, or we have more than two variables in our two simultaneous equations, so we can't solve.
What gives?

I hope someone can help: the reading doesn't seem to be helping me! It might have something to do with the fact that I never had to do CT8, having been through a few transitions over the years!

Cheers, folks!

 

[Alpha9]

The light dawns...

I think I'm beginning to get it.

x(t+1) is a random variable - this is the fundamental point I was missing! And its characteristics are dependent on the asset we are pricing (so, for example, for a risk-free asset, x(t+1) is fixed). I think I was confused by the fact that (in the consumption-based model) c(t+1) is also a random variable.

ß is not asset-specific: it is a characteristic of the consumer, so constant for all assets.

So the consumption-based model makes sense to me now.

However I am still trying to reconcile why the two equations for the CAPM, viz:
E[Ri] = Rf + ßi × (E[Rm] - Rf)
and
pt = E[(a + b × Rw(t+1)) × x(t+1)]
are essentially the same.

As far as I can tell (from having read around a bit), Rw(t+1) is the same as Rm, i.e. the return on the market portfolio/weath portfolio, and is, I think, a random variable in both equations.

I think I can divide the second equation through by pt to get:
1 = E[(a + b × Rm) × Ri]
but I come a bit unstuck at that point.

Of course it probably doesn't matter, and I'm glad I've realised the main point. But if I do think of how this works, I'll post it in case anyone else is interested!

 

[Alpha9]

Bingo!

I'd simply missed that E[m(t+1)], in any asset pricing model, is 1/(1+rf).

Feed that through:

p(t) = E[m(t+1) × x(t+1)] = cov(m(t+1), x(t+1)) + E[m(t+1)]×E[x(t+1)]

Then divide by p(t), multiply by (1+rf) and rearrange to get:

E[R] - (1+rf) = -cov(m(t+1), R) × (1+rf)
= -b × cov(Rm, R) × (1+rf)
where R is (1+r), the return on the asset, i.e. x(t+1)/p(t)

i.e. E[R] - (1+rf) is proportional to ßi, which I think is all required. It'll do me, anyway!

 

[Alpha9]

Another error in original posting

For the CAPM, I suggested p(t) = a + b × Rw(t+1).
It isn't: that's m(t+1). And, as always, p(t) = E[m(t+1) × x(t+1)].
Sorry - hope I didn't lead anyone up the wrong path!