Economic intuition behind pricing cash flow

Question

I read the book of Skiadas Asset Pricing Theory 2009. I don't quite understand what does mean pricing cash flow. In the book it's written:

$\textbf{Definition 2.9}$ A cash flow $x^*$ is a pricing cash flow if $x^* \in X$ and there exist constants $\rho$ and $q$ such that $q \neq 0$ and:

$x(0) + \rho\mathbb{E}[x(1)] + q\mathbb{Cov}[x^*(1),x(1)] = 0 \qquad \forall x \in X$

$X$ - market of cash flows

$x(0)$ price of cash flow

$x(1)$ payoff at different states of nature $K$

I'm looking for some geometric and/or economic interpretation. Thanks in advance!

Answer 1

All that's going on here are essentially consequences of a linear pricing function.

That asset prices should be linear in their payoffs makes intuitive economic sense: the value of a basket of payoffs is the sum of the basket contents. An assumption that the pricing function is linear is sometimes referred to as the law of one price.

Quick review

Let $f$ be a pricing function which gives you the current price $X_0$ of a future, stochastic payoff $X_1$. If $f$ is linear, $f(a X_1 + b Y_1) = a f(X_1) + b f(Y_1)$, then $f$ can be written as the inner product with some stochastic discount factor.

$$ f(X_1) = \mathbb{E}[MX_1]$$

Let $X^*$ be the projection of $M$ onto the space of payoffs $\underline{X}$. $X^* \in \underline{X}$ will also work as the discount factor for $X_1 \in \underline{X}$.

$$ f(X_1) = \mathbb{E}[ X^* X_1] $$

Now we can just do some algebra:

$$ X_0 = \mathbb{E}[X^*X_1]$$ $$ X_0 = \mathbb{Cov}[X^*, X_1] + \mathbb{E}[X^*] \mathbb[X_1]$$

I'm following how John Cochrane defines $X^*$ in his book Asset Pricing. The book here appears to call any scalar multiple of $X^*$ a pricing cash flow?

Anyway, you can manipulate these equations to bring out classic regression beta models and the mean variance frontier.