Risk premium is expected return above cash. Separate expected compensation from realised excess return, and physical forecasts from risk-neutral pricing weights.
Risk premium
A risk premium is an asset’s expected total return above a risk-free return measured over the same horizon and in the same currency. It is the extra return anticipated for holding an uncertain payoff instead of a certain one. For a quant comparing a strategy with cash, this separates the compensation expected before investing from the gain or loss that eventually occurs.
For a known risk-free simple return Rf and an uncertain asset return R,
π=EP[R]−Rf.
The expectation uses physical probabilities, denoted P: a model of how likely the outcomes actually are, conditional on the information available. A premium is therefore an expectation with assumptions behind it. It is not a contractual payment or a promise that taking more risk will produce more profit. William Sharpe’s discussion of risk premia makes this expected-versus-realised distinction explicit.
An expected five points can become a realised loss
Consider an original, stipulated one-year USD model. A traded asset costs USD 100 today and makes a single terminal payment of either USD 80 or USD 140. Each outcome has physical probability one-half. There are no interim distributions, fees or taxes. A risk-free investment of USD 100 pays USD 105 at the same date: a 5% one-year simple return.
The expected premium is 10% − 5% = 5 percentage points. Equivalently, the expected terminal payment is USD 110, which exceeds the cash payment by USD 5 on USD 100 initially invested. Percentage points measure the difference between the two return rates; this is not a 5% relative increase in the cash return.
Once the year ends, the excess return is either −25 or +35 percentage points. In the lower outcome, the asset loses USD 20 outright and finishes USD 25 behind cash. Calling that outcome “earning the five-point risk premium” would confuse the expectation with the observation. Even a correctly specified positive premium permits a disappointing realised result.
A one-year USD asset costs 100 and pays 80 or 140. Physical weights one-half each imply expected payoff 110, expected return 10% and a 5 percentage point premium over cash. Pricing weights seven-twelfths and five-twelfths imply payoff 105; discounting at 5% returns price 100.
Original synthetic one-year USD model. The same terminal payoffs serve two calculations: physical expectation for returns, and pricing expectation for valuation. Cash earns a certain 5%; pp means percentage points.
Where the premium enters the price
An uncertain dollar is not automatically worth the same as a certain dollar due on the same date. Holding the physical payoff distribution fixed, a lower purchase price increases expected return. In the example, the stipulated price and physical expectation imply a 10% expected return, so the internally consistent risk-adjusted discounting calculation is
1+0.10EP[X]=1.10110=100,
where X is the terminal payment. This reconciles the given price with the assumed distribution; it does not independently establish which expected return investors should require.
Simply discounting the physical mean at the risk-free rate gives
1.05110≈104.7619,
which does not reproduce the stipulated traded price of USD 100. That calculation treats the uncertain physical mean as though it were a certain cash payment. A valuation needs a risk adjustment as well as a time adjustment.
There is another consistent representation. In this frictionless two-state market, trade in the asset and cash spans the two terminal states. The risk-neutral probabilities are the pricing weights that make discounted asset values consistent with those traded prices. If q is the weight on USD 140, then
The physical high-payoff probability remains one-half. Replacing it with 5/12 changes the purpose of the calculation: it incorporates risk into valuation weights. It does not revise the model’s forecast of how often the high payoff occurs. Expected excess return is zero under these pricing weights, while the physical expected premium remains five percentage points. Scott Sheffield’s MIT notes explain why a probability extracted from prices can differ from an ordinary outcome probability.
The full public sample The Binomial Model develops this connection through replication and then extends it to a recombining tree. Its interest-rate formulas use continuous compounding; this page uses a single one-year simple return.
Which risks get paid?
The relevant question is how an asset’s payoff interacts with investors’ other risks. A payoff that is especially low when aggregate income and wealth are already under pressure can require a positive premium: losses arrive when investors most need resources. A claim that pays particularly well in those states can provide insurance, commanding a higher price and a lower expected return. Its expected premium can be negative even though its own payoff is uncertain.
The capital asset pricing model gives a particular version of this idea. Under its assumptions, expected excess return depends on exposure to systematic risk, measured by market beta. Additional idiosyncratic risk earns no extra premium because investors can diversify it away. That is a model restriction, not a claim that every real investor can costlessly diversify every exposure.
Consequently, a volatile asset need not have a large positive premium. Neither a risk statistic nor the word “risky” supplies an expected-return estimate. An analyst’s forecast premium can also differ from the return required by an equilibrium model; the difference is a view about pricing, rather than proof of an arbitrage.
A historical average of realised excess returns is an estimate of a premium, subject to sampling uncertainty and changing conditions. A model-implied premium depends on the model and its inputs. State the horizon, currency, cash benchmark and treatment of distributions and costs before comparing either number across assets. A raw premium over cash also differs from alpha, which concerns return beyond a specified risk model or benchmark.
Sources
Edwin J. Elton, Martin J. Gruber, Stephen J. Brown and William N. Goetzmann, Modern Portfolio Theory and Investment Analysis, 9th edition (Wiley, 2014), Chapter 10 appendix, pp. 216–217: physical probabilities, pricing weights and state-dependent discount factors; Chapter 13, pp. 296–297: systematic risk and expected returns under CAPM.
