The Market Portfolio and Its Proxy Problem

Why the Market Portfolio Is More Than a Stock Index

In CAPM, the market portfolio is not simply whichever market index is easiest to look up. It is a theoretical portfolio representing all risky assets in the economy or investment universe specified by the model. This distinction matters whenever an analyst uses an observable index to estimate beta or a required return: the estimate is then relative to the chosen index, which may leave out important kinds of risky assets.

An equity index can be a useful and convenient benchmark. But its constituents may represent only publicly traded shares, while investors’ risky wealth can also include corporate bonds, private businesses, real estate, and other assets. If those omitted assets behave differently from the index, an asset’s beta against the index need not equal its beta against the theoretical market portfolio.

Definition: The theoretical market portfolio is the portfolio containing all risky assets in the model’s investment universe, with each asset weighted in proportion to its market value. A market proxy is an observable portfolio or index used in practice as an approximation to that theoretical portfolio. The proxy is not automatically the market portfolio.

The universe must be understood in context. If the model includes only a specified set of tradable risky assets, the market portfolio covers all risky assets in that set. A broader conception of aggregate investor wealth can also raise difficult questions about assets such as privately held businesses and human capital. In either case, the defining idea is completeness relative to the model—not popularity, size, or ease of observation.

A risk-free asset is not one of the risky assets in this portfolio. In the CAPM setting, investors may combine the risky market portfolio with risk-free borrowing or lending, but that does not make the risk-free asset a component of the market portfolio itself.

Why Market Values Determine the Weights

Suppose the model’s risky assets are indexed by $j=1,\ldots,n$. Let $V_j$ be the market value of asset $j$ at the start of the period, and let $R_j$ be its return over that period. The total starting value is $\sum_{j=1}^{n}V_j$, so the market-value weight of asset $j$ is

w_j=\frac{V_j}{\sum_{k=1}^{n}V_k}.

These weights add to one because the sum of the asset values is the total value of the risky-asset portfolio. The portfolio return follows directly from the change in the total value of its holdings:

R_M =\frac{\sum_{j=1}^{n} V_j(1+R_j)-\sum_{j=1}^{n}V_j}{\sum_{j=1}^{n}V_j} =\sum_{j=1}^{n}w_jR_j.

This identity explains why the theoretical market portfolio is value-weighted. Holding each risky asset in proportion to its market value gives the return on the aggregate portfolio of those assets. It also shows why leaving out a category can change the portfolio return: the missing category’s value and return are absent from the weighted sum.

Worked Example: Constructing a Small Theoretical Market Portfolio

Method. Calculate each asset category’s share of total risky-asset value, then use those shares to calculate the portfolio return. Assume a hypothetical investment universe contains only the four risky categories listed below, with values measured at the start of one year.

Public shares are valued at $500 million, private business interests at $200 million, commercial property at $200 million, and corporate bonds at $100 million. Their returns over the year are respectively 8%, 5%, 4%, and 2%. Assume these values and returns are measured consistently for the same one-year period, and that none of the categories is risk-free.

Step 1: Find total risky-asset value.

500+200+200+100=1{,}000\text{ million dollars}.

Step 2: Find the market-value weights. The weights are $500/1{,}000=0.50$ for public shares, $200/1{,}000=0.20$ for private businesses, $200/1{,}000=0.20$ for commercial property, and $100/1{,}000=0.10$ for corporate bonds. They sum to $0.50+0.20+0.20+0.10=1.00$.

Step 3: Apply the weighted-return identity.

$$ R_M=(0.50)(0.08)+(0.20)(0.05)+(0.20)(0.04)+(0.10)(0.02) =0.040+0.010+0.008+0.002=0.060. $$

The hypothetical market portfolio return is therefore 6.0%. The public-share return alone is 8.0%, so using that category alone would not reproduce the return on this complete four-category portfolio. The check is that the weighted contributions sum to 0.060 and the weights sum to one.

How a Proxy Enters CAPM

In the CAPM model, let $R_i$ be the return on asset $i$, $R_f$ the risk-free return, and $R_M$ the return on the theoretical market portfolio. Under the CAPM assumptions—including a single investment period, mean-variance portfolio choice, common expectations, frictionless markets, and access to borrowing or lending at the risk-free rate—the model relates expected return to beta against the market portfolio:

E[R_i]=R_f+\beta_{iM}\bigl(E[R_M]-R_f\bigr), \qquad \beta_{iM}=\frac{\operatorname{Cov}(R_i,R_M)}{\operatorname{Var}(R_M)}.

The beta in this model is specifically a covariance relationship with $R_M$, not with any convenient index return. In practice, $R_M$ is not observed as a complete return series. Analysts therefore often substitute an index return, say $R_P$, and calculate a proxy beta:

\beta_{iP}=\frac{\operatorname{Cov}(R_i,R_P)}{\operatorname{Var}(R_P)}.

This proxy beta describes the asset’s relationship with the proxy. It equals the theoretical market beta only if the proxy represents the relevant market portfolio sufficiently well for the calculation at hand. The equality is not guaranteed merely because the proxy is broad, familiar, or widely used.

Worked Example: An Omitted Asset Category Changes Beta

Method. First compute the variance and target-asset covariance for the complete market portfolio. Then compare the resulting beta with the beta against an equity-only proxy. Assume all returns and covariances refer to the same annual period, the figures are population moments for this hypothetical calculation, and the target asset is separate from the two market components.

Suppose the complete risky market consists of an equity component $E$, weighted 0.75, and other risky assets $O$, weighted 0.25. Their return variances are $\operatorname{Var}(R_E)=0.032$ and $\operatorname{Var}(R_O)=0.008$, and their covariance is $\operatorname{Cov}(R_E,R_O)=0.006$. The target asset has covariance 0.020 with $E$ and 0.004 with $O$.

Step 1: Calculate the complete market variance. For two components, the variance of the weighted return includes both individual variances and their covariance:

\begin{aligned} \operatorname{Var}(R_M) &=(0.75)^2(0.032)+(0.25)^2(0.008) +2(0.75)(0.25)(0.006)\\ &=0.018+0.0005+0.00225\\ &=0.02075. \end{aligned}

Step 2: Calculate covariance with the complete market. Covariance with a weighted sum is the same weighted sum of the covariances:

\operatorname{Cov}(R_i,R_M) =(0.75)(0.020)+(0.25)(0.004) =0.015+0.001 =0.016.

Step 3: Calculate both betas. Against the complete market, beta is $0.016/0.02075=0.7711$, rounded to four decimal places. If equity alone is used as the proxy, beta is $0.020/0.032=0.6250$. These are different because both the covariance and variance in the beta calculation depend on the benchmark.

To show the possible required-return consequence, suppose the risk-free rate is 3.0% and the theoretical market risk premium is 6.0%. These rates are annual and expressed on the same basis. The CAPM estimate using the complete-market beta is $3.0\%+(0.7711)(6.0\%)=7.63\%$, rounded to two decimal places. If an analyst incorrectly treats the equity-proxy beta as the theoretical beta while retaining that same premium, the calculation is $3.0\%+(0.6250)(6.0\%)=6.75\%$. The latter is a proxy-based calculation, not the model’s required return using the stated complete-market beta.

Check. The complete-market variance includes the covariance contribution 0.00225, and the weighted covariance contributions sum to 0.016. The example’s lesson is not that a proxy always lowers beta, but that omission can change the inputs and therefore the result.

Why the Complete Portfolio Cannot Be Observed Directly

A complete market portfolio is an idealized aggregate, not a quoted security with a single observable price. Several practical obstacles prevent direct measurement:

Incomplete coverage. Common indexes cover selected securities or asset classes, rather than every risky asset in the model’s universe.
Uncertain values. Private businesses and other infrequently traded assets do not have continuous market prices. Valuations may be estimates, updated with a delay, or based on transactions that are not comparable.
Different measurement conventions. Asset classes can have different return frequencies, valuation methods, and treatment of income. Combining them requires consistent definitions and time periods.
Changing composition. Assets enter, leave, or change value over time. A proxy’s rules for inclusion and weighting may not track the complete aggregate.

As a result, the chosen proxy is a substantive modeling decision. A broad public-equity index may be appropriate for one analysis, while a proxy including several traded asset classes may be more relevant for another. Neither choice becomes the theoretical market portfolio solely by being used in a calculation.

Worked Example: Comparing Two Plausible Proxies

Method. Compute each proxy beta from its own covariance and variance, then apply the corresponding stipulated premium. This comparison illustrates sensitivity to the proxy; it does not establish that either proxy is the complete market portfolio. Assume the covariances and variances are hypothetical annual moments and that all rates are annual.

For a target asset, an equity index has variance 0.025 and covariance with the target of 0.018. A broader traded-asset index has variance 0.020 and covariance with the target of 0.016. Suppose the risk-free rate is 2.8%, the equity-index expected premium is 5.5%, and the broader-index expected premium is 5.0%.

Step 1: Calculate the equity-index beta.

\beta_{i,\mathrm{equity}}=\frac{0.018}{0.025}=0.72.

Step 2: Calculate the broader-index beta.

\beta_{i,\mathrm{broad}}=\frac{0.016}{0.020}=0.80.

Step 3: Calculate the two proxy-based return estimates. Using the equity index gives $2.8\%+(0.72)(5.5\%)=2.8\%+3.96\%=6.76\%$. Using the broader index gives $2.8\%+(0.80)(5.0\%)=2.8\%+4.00\%=6.80\%$. Both calculations are internally consistent with their stated inputs, but the difference in estimates reflects both the proxy beta and the premium used.

Check. The two estimates differ by $6.80\%-6.76\%=0.04$ percentage points. Their closeness does not prove that either index matches the theoretical market portfolio; it only shows that these particular hypothetical inputs produce similar estimates.

Common Proxy Mistakes

The most important exam trap is to label an index beta as the theoretical market beta without qualifying the substitution. For example, in the omitted-category example, the incorrect calculation $3.0\%+(0.6250)(6.0\%)=6.75\%$ treats the equity-proxy beta as if it were $\beta_{iM}$. The correction is to distinguish the quantities: the proxy calculation uses $\beta_{iP}=0.6250$, while the stated complete-market inputs give $\beta_{iM}=0.7711$ and a CAPM estimate of 7.63% under the specified premium.

Other common errors include assuming a broad index contains every risky asset; treating assets with difficult valuations as risk-free or irrelevant; and calculating a portfolio variance while omitting the covariance terms between components. A correct calculation can still be misleading if it uses a proxy that does not match the model’s intended market universe.

Key takeaway: CAPM’s market portfolio is the value-weighted portfolio of all risky assets in the specified model. An observable index can serve as a practical proxy, but its beta and premium are proxy-specific inputs. State the proxy clearly, use consistent return periods and units, and do not claim that a proxy-based estimate is identical to the theoretical CAPM result without justification.

Check Your Understanding

Work each problem before reading its final answer. Treat all returns, variances, and covariances as hypothetical and measured consistently over the stated period.

A hypothetical risky-asset universe has market values of $300 million, $500 million, and $200 million. The corresponding annual returns are 4%, 10%, and −2%. Calculate the market-value weights and the return on the theoretical market portfolio. (Answer: weights 0.30, 0.50, and 0.20; return 5.8%.)
A two-component market has weights 0.60 and 0.40, component variances 0.025 and 0.010, and covariance 0.005. Calculate the variance of the market return, including the covariance term. (Answer: $0.60^2(0.025)+0.40^2(0.010)+2(0.60)(0.40)(0.005)=0.0130$.)
A target asset has covariance 0.014 with a selected index, whose variance is 0.028. If the risk-free rate is 2.5% and the index’s expected premium is 5.5%, calculate the index-proxy beta and the corresponding proxy-based CAPM estimate. (Answer: beta $0.014/0.028=0.50$; estimate $2.5\%+0.50(5.5\%)=5.25\%$.)
A complete market has two components with weights 0.70 and 0.30, variances 0.040 and 0.010, and covariance 0.008. A separate target asset has covariances 0.018 and 0.006 with the respective components. Calculate the target’s beta against the complete market. (Answer: market variance $0.70^2(0.040)+0.30^2(0.010)+2(0.70)(0.30)(0.008)=0.02386$; target covariance with market $0.70(0.018)+0.30(0.006)=0.0144$; beta $0.0144/0.02386=0.6035$, to four decimals.)

What You'll Learn