An asset is a claim to future dollars. Its price today is the amount that makes investors indifferent between holding the claim and holding cash, once both time and uncertainty are accounted for.

Time is handled by the risk-free rate: even a guaranteed dollar in the future is worth less than a dollar today. Uncertainty enters because a dollar is not equally valuable in all future conditions. Dollars received when the overall economy is weak—when income, employment, and funding are constrained—are more valuable than dollars received when conditions are strong.

The stochastic discount factor formalizes this by assigning a single number to each possible future state. This number combines two effects: how far away the state is in time (time discounting) and how valuable resources are in that state (marginal utility). A weak economy two years from now gets a weight reflecting both the two-year time discount and the scarcity of resources in that state. When you multiply each state’s cash flow by its stochastic discount factor and sum across all states, you get the asset’s price directly. There is no separate discounting step—the weights already incorporate time value.

A concrete example: Consider two one-year claims, both with an expected payoff of $100. The risk-free rate is 2%. Claim A pays $100 in all states. Claim B pays $120 in good times and $80 in bad times, each with 50% probability. The stochastic discount factor assigns weights of 0.93 to the good state and 1.03 to the bad state, averaging to 0.98 (the one-year risk-free discount factor).

Claim A’s price: $100 × 0.93 × 50% + $100 × 1.03 × 50% = $98, yielding a 2% return.

Claim B’s price: $120 × 0.93 × 50% + $80 × 1.03 × 50% = $55.80 + $41.20 = $97, yielding a 3.09% return.

The extra 1.09% is the risk premium. It arises because Claim B pays more when the discount factor is low (good state) and less when it’s high (bad state).

For states multiple years away, the time component of the weight compounds annually while the state-specific adjustment varies by economic conditions. If the risk-free rate is 2% per year, the time discount is 0.98 for year one, 0.96 for year two, 0.94 for year three. A recession in year three with a marginal utility adjustment of 1.10 gets a total weight of 0.94 × 1.10 = 1.03.

Investors cannot identify every possible future state or know the exact weight for each. Instead, they approximate state-dependent valuation using discounted cash flow models with two inputs: projected cash flows and a required return.

The two inputs divide the work of capturing bad-state exposure. Cash flow projections handle risks that directly affect business output in downturns—demand declines, higher defaults, cyclical revenue. The required return handles timing risk—whether cash flows arrive when the broader economy is strong or weak. A cyclical retailer might see projected earnings fall 40% in recession (captured in cash flows). A utility might see earnings fall only 10%, but if those earnings arrive precisely when aggregate consumption is most constrained, the required return adjusts upward to reflect the covariance. Sound practice uses both channels without double-counting.

Investors use a single discount rate for all periods, set higher than the risk-free rate to reflect the asset’s tendency to pay off more in good times than bad times. This single rate, applied uniformly, produces the same price in expectation as the full state-by-state calculation. The required return equals the risk-free rate plus a risk premium calibrated to the asset’s covariance with aggregate economic conditions.

The risk premium reflects how much an asset’s returns move with the overall economy. Assets that deliver low returns when the market delivers low returns—when resources are already scarce—must offer higher expected returns to compensate. This relationship can be quantified: if an asset’s returns have a covariance of 0.04 with market returns, and the market pays 0.5% of extra return per 0.01 of covariance (the market price of risk), then this asset’s risk premium is 0.5 × 0.04 = 0.02, or 2%. This is identical to what the stochastic discount factor calculation produces, just expressed using observable market statistics rather than state-by-state weights.

Not all uncertainty commands a risk premium. The distinction is whether a risk can be eliminated when all investors diversify, or whether someone must necessarily bear it.

Idiosyncratic shocks—management turnover, facility fires, product recalls—cancel out entirely when all investors hold diversified portfolios. Firm A has bad news, Firm B has good news, and across all portfolios these average to zero. No one in aggregate bears this risk, so no compensation is needed. Because prices are set at the margin by diversified investors who don’t experience idiosyncratic risk in their portfolios, this uncertainty doesn’t affect prices.

Aggregate risks—recessions, credit stress, funding cost spikes—cannot be diversified away collectively. The total supply of risky assets is fixed: there are $X trillion of stocks and corporate bonds in the world, and in aggregate, investors must hold all of them. When a recession hits, these assets fall in value and someone is holding them when they fall. You can personally hedge by buying Treasuries or protection, but someone else must sell you that protection and accept the offsetting exposure. The aggregate risk doesn’t disappear—it just moves between investors. Because someone must bear aggregate risk, prices adjust to make them willing.