Distinguish the CAPM portfolio of all risky wealth from an observable stock-index proxy, calculate a proxy return and interpret benchmark-dependent beta and alpha.
Market portfolio
The market portfolio in the capital asset pricing model (CAPM) contains all risky assets in the model, held in proportion to their market values. It is the aggregate risky wealth investors collectively own. A stock index can serve as an observable proxy, but choosing a capitalisation-weighted index does not make that index the theoretical market portfolio. That distinction matters whenever a quant estimates beta, evaluates alpha or tests an asset-pricing model.
What “all” means
Write Vi for the current market value of risky asset i, and U for the model's complete risky-asset universe. The market weights are
wiM=∑j∈UVjVi,RM=i∈U∑wiMRi.
The return equation uses beginning-of-period weights and one-period simple total returns, measured over the same horizon in a common currency, without external flows or rebalancing during the period. The denominator is total risky market value across the model's universe, rather than just the value of an index's constituents.
In a broad wealth interpretation, the universe extends beyond quoted shares to risky debt, private businesses, real estate and human capital. These claims are difficult to value and many cannot be freely traded. The standard textbook CAPM makes the deliberately strong assumption that all assets, including human capital, are marketable. Mentioning human capital describes that idealisation; it does not suggest that a worker can sell a traded claim on their entire future earnings.
The risk-free asset is combined separately with the risky market portfolio. Holding cash alongside a stock index therefore neither completes the missing risky-asset universe nor turns the resulting account into M.
Why the market appears in CAPM
Under the standard model's assumptions, investors share the same expected returns and covariances, optimise using mean and variance over the same horizon, and can borrow or lend at the same risk-free rate. They therefore choose the same tangency portfolio of risky assets, adjusting their overall risk by how much they lend or borrow.
Market clearing supplies the next step: collectively, investors must hold the outstanding supply of each risky asset. Their common risky portfolio must therefore have the market-value weights of the whole risky universe. The model identifies the tangency portfolio with M through an equilibrium argument. Capitalisation weighting alone does not establish that a portfolio is mean-variance efficient.
for a fixed one-period risk-free return Rf and positive market-return variance. The market's beta to itself is one. This is a claim about expected returns under the model, not a promise that a realised stock return equals beta times a realised market return in each period.
A proxy with a perfectly valid return calculation
Consider a fictional public-equity universe containing only stocks A and B, with beginning market values of USD 60 million and USD 40 million. A capitalisation-weighted portfolio of these shares assigns weights of 60% and 40%. Suppose their one-period total returns are +10% and −5%, with no fees, taxes or external flows.
RB=0.60(10%)+0.40(−5%)=6%−2%=4%.
Here the subscript B denotes the benchmark proxy. On an initial USD 100 account, A's USD 60 becomes USD 66 and B's USD 40 becomes USD 38. The account ends at USD 104. Each holding's distributions remain invested in that holding for this total-return illustration.
Synthetic one-period public-equity proxy: A has weight 60% and total return +10%; B has weight 40% and total return −5%. An initial USD 100 becomes USD 104, a 4% return. A separate panel describes the wider theoretical CAPM universe, including risky debt, private businesses, real estate and idealised marketable human capital.
Original synthetic example in USD. A contributes +6 percentage points and B contributes −2 percentage points to the proxy’s 4% return. Panel sizes do not represent measured shares of global wealth.
The arithmetic is complete for the specified proxy. It says nothing about returns on omitted risky assets. A larger stock index may provide much broader equity coverage, yet still exclude important components of aggregate risky wealth. Float adjustments, eligibility rules and regional boundaries can further distinguish an investable index from full market-value holdings.
The benchmark changes the measurement
For an observed benchmark B, let xi=Ri−Rf and xB=RB−Rf be matched excess returns. A population linear regression with an intercept has
assuming finite second moments and positive benchmark variance. In data, the corresponding estimates use sample moments. Change the benchmark and its covariance, variance and mean can change, so measured market beta and alpha can change too. The return frequency, currency, risk-free series and estimation window belong in the specification. An intercept relative to one equity index is not a benchmark-independent measure of skill.
Roll's critique identifies the underlying testing problem. The linear return–beta relation is tied mathematically to the mean-variance efficiency of the portfolio used to compute beta. A test using an incomplete market proxy therefore cannot unambiguously establish or reject the claim that the true aggregate market portfolio is efficient. A strong fit against a stock index does not, by itself, validate the theoretical market's composition or the CAPM's equilibrium assumptions.
The owned lesson Empirical Tests of the CAPM and Roll's Critique develops return–beta tests and the proxy problem in more detail. Its public page provides a preview; the full lesson requires membership.
For practical work, name the series precisely: for example, “beta to this equity total-return benchmark over this sample.” The proxy remains useful for exposure measurement and performance comparison, while the broader market portfolio remains the economic object the theory describes.
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 13, pp. 291–293 on marketability, separation and the market portfolio; Chapter 15, pp. 356–357 on the market-proxy testing problem. The numerical illustration and figure are original.
Eugene F. Fama and Kenneth R. French, “The Capital Asset Pricing Model: Theory and Evidence”, Journal of Economic Perspectives 18(3), 2004, pp. 25–46; abstract on the proxy problem in both empirical tests and applications.
