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](https://en.wikipedia.org/wiki/Projection_(linear_algebra)) 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.