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I understand that when there is no uncertainty in future consumption future, i.e. Ct+1 = Ct. Then, the stochastic discount factor under no uncertainty is the same as the subjective discount factor.

But why exactly is the SDF key in determining asset prices?

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The SDF in the simple additively separable case is given by $$ M_{t+1} = \mathbb{E} \Big[\frac{\beta u^\prime(c_{t+1})}{u^\prime(c_t)} \Big]. $$

The SDF is not simply $\beta$ because in some states you might value a unit of consumption more than in another. Concave utility means that in states where you are poor you will value another marginal unit of consumption more. i.e $u^\prime(c_L) > u^\prime(c_H)$.

Now imagine you have two assets with the same expected payoff, but one pays off in the bad state when you have no labour income, and one pays off in the good state when you have lots of labour income. Well, because of concave utility the extra unit of consumption in the bad state is more valuable. Thus you will value the asset that pays in the bad state more than the other one. Again this is because simply $u^\prime(c_L) > u^\prime(c_H)$ due to concavity.

So valuation of the asset must take into account this differential that emerges from concave utility. So the way to take this into account is to 'weight' payoffs not just by their probability but also by whether they tend to payoff in good states or bad. The normalisation $\frac{u^\prime(c_{t+1})}{u^\prime(c_t)}$ is doing precisely this reweighting. And then the expectation is again adding probabilities to the relative weights.

This intuition generalises to more crazy SDFs like that emerging from Epstein-Zin, but then how you write marginal utilities in states can be more complicated than just $\frac{u^\prime(c_{t+1})}{u^\prime(c_t)}$. Hence, other utility functions will have different looking SDFs but the intuition is the same.

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