Why Diversification Cannot Remove Marketwide Risk

Why More Holdings Do Not Remove Every Risk

A portfolio of many companies can be less exposed to a problem at any one company. A product recall, a failed project, or the loss of a major customer may hurt one firm without affecting the others in the same way. Holding several firms can therefore reduce the effect of such company-specific shocks. But a broad decline in economic activity, a marketwide repricing of risk, or another shared movement can affect many firms at once. Adding more companies does not automatically cancel a shock they share.

The key distinction is between firm-specific risk, which differs across companies, and marketwide risk, which moves many companies together. The earlier material on portfolio variance and imperfect correlation provides the basic diversification idea: how much risk can be reduced depends on how returns move together. A simple factor model makes the shared-versus-company-specific distinction explicit.

Definition: In this tutorial, marketwide risk is the return variation caused by a common market movement affecting multiple assets. Firm-specific risk is return variation particular to an individual company. Diversification means spreading investment across assets so that the effect of differing firm-specific shocks can be reduced; it does not mean that every source of portfolio risk disappears.

A Simple Model of Shared and Firm-Specific Shocks

For clarity, consider the following one-factor model for the return on firm \(i\):

R_i = \mu_i + M + \varepsilon_i

Here \(R_i\) is the firm's return over the period, \(\mu_i\) is its expected-return component, \(M\) is a common market shock, and \(\varepsilon_i\) is the firm-specific shock. All terms are measured as returns over the same period. This version of the model assumes each firm has the same exposure to the common shock, so the market component enters with coefficient 1.

The assumptions matter. For the variance result derived below, assume \(M\) and every \(\varepsilon_i\) have mean zero, the firm-specific shocks have the same variance, each pair of distinct firm-specific shocks is uncorrelated, and \(M\) is uncorrelated with every firm-specific shock. These are simplifying conditions for analysis, not a claim that real-world company returns always satisfy them. In particular, the model does not represent firms with different sensitivities to market movements.

Under this model, a common shock is present in each asset's return. The firm-specific terms may differ: one may be positive while another is negative. Averaging across firms can make these differing terms offset. But the common term is still present in the portfolio return.

Worked Example: Separating a Shared Decline From Firm-Specific Results

A hypothetical equal-weight portfolio holds four companies. Their expected-return components are 2%, 1%, 0%, and 1%. During the period, the common market shock is −4%. Their firm-specific shocks are +3%, −2%, +1%, and −2%, respectively. Find each return and the portfolio return. Method: apply the model to each firm, then average the returns. Common mistake avoided: treating company-specific gains as if they remove the shared market shock from every holding.

Step 1: Calculate each company's return. Using \(R_i=\mu_i+M+\varepsilon_i\):

Company 1: \(2\%+(-4\%)+3\%=1\%\).
Company 2: \(1\%+(-4\%)+(-2\%)=-5\%\).
Company 3: \(0\%+(-4\%)+1\%=-3\%\).
Company 4: \(1\%+(-4\%)+(-2\%)=-5\%\).

Step 2: Average the returns. With four equal weights, the portfolio return is \((1\%-5\%-3\%-5\%)/4=-12\%/4=-3\%\).

Step 3: Check by separating the components. The average expected-return component is \((2\%+1\%+0\%+1\%)/4=1\%\). The average firm-specific shock is \((3\%-2\%+1\%-2\%)/4=0\%\). The portfolio return is therefore \(1\%-4\%+0\%=-3\%\), agreeing with the direct average.

The firm-specific shocks cancel in this particular period, but the common market shock does not. The portfolio still loses 3% because its 1% expected-return component is more than offset by the 4% market decline.

Why Firm-Specific Risk Shrinks in the Simple Model

Suppose \(n\) firms are held at equal weights, so each has weight \(1/n\). Applying the one-factor model to the portfolio return gives:

R_P = \frac{1}{n}\sum_{i=1}^n R_i = \frac{1}{n}\sum_{i=1}^n \mu_i + M + \frac{1}{n}\sum_{i=1}^n \varepsilon_i

The market shock remains \(M\), because the weights add to one: \(M/n\) from each of the \(n\) firms sums to \(M\). The firm-specific component is instead an average of the \(n\) company-specific shocks. Under the stated assumptions, its variance falls as more independent shocks are averaged.

To see this, let each firm-specific shock have variance \(\sigma_{\varepsilon}^{2}\). The variance of their equal-weight average is:

\begin{aligned} \operatorname{Var}\left(\frac{1}{n}\sum_{i=1}^{n}\varepsilon_i\right) &= \frac{1}{n^2}\operatorname{Var}\left(\sum_{i=1}^{n}\varepsilon_i\right)\\ &= \frac{1}{n^2}\left(n\sigma_{\varepsilon}^{2}\right)\\ &= \frac{\sigma_{\varepsilon}^{2}}{n}. \end{aligned}

The second line follows because each of the \(n\) shocks contributes variance \(\sigma_{\varepsilon}^{2}\), while all pairwise covariances are zero under the model assumptions. Since \(M\) is uncorrelated with the firm-specific shocks, the portfolio variance is:

\operatorname{Var}(R_P) = \operatorname{Var}(M)+\frac{\sigma_{\varepsilon}^{2}}{n}

This formula gives the central result. Increasing \(n\) reduces the firm-specific variance term, which approaches zero as \(n\) grows. The marketwide variance term remains. In this model, the portfolio's variance approaches \(\operatorname{Var}(M)\), rather than zero. The corresponding standard deviation approaches the square root of that variance.

Theorem: With equal weights, a unit exposure to the same common shock for every asset, and independent, equal-variance firm-specific shocks that are uncorrelated with the common shock, portfolio variance is \(\operatorname{Var}(M)+\sigma_{\varepsilon}^{2}/n\). Diversification reduces the firm-specific term, but leaves the common market term unchanged.

Worked Example: Calculating the Risk That Remains

A hypothetical set of firms follows the simple model. The common market shock has standard deviation 2%, and each firm's independent firm-specific shock has standard deviation 5%. Assume both shocks have mean zero, and assume the market shock is independent of all firm-specific shocks. Find portfolio standard deviation for 1, 4, and 25 equally weighted firms, and find its limiting value as the number of firms grows. Method: convert the standard deviations to variances, apply the derived formula, and take square roots. Common mistake avoided: dividing the common market variance by the number of holdings.

Step 1: Convert each standard deviation to a variance. In decimal-return units, the common variance is \(0.02^2=0.0004\), and the firm-specific variance is \(0.05^2=0.0025\).

Step 2: Calculate the portfolio variance and standard deviation for one firm. For \(n=1\), the variance is \(0.0004+0.0025/1=0.0029\). The standard deviation is \(\sqrt{0.0029}=0.0538516\), or 5.385% to three decimals.

Step 3: Calculate for four firms. For \(n=4\), the variance is \(0.0004+0.0025/4=0.0004+0.000625=0.001025\). The standard deviation is \(\sqrt{0.001025}=0.0320156\), or 3.202% to three decimals.

Step 4: Calculate for 25 firms. For \(n=25\), the variance is \(0.0004+0.0025/25=0.0004+0.0001=0.0005\). The standard deviation is \(\sqrt{0.0005}=0.0223607\), or 2.236% to three decimals.

Step 5: Find the limit. As \(n\) grows, \(0.0025/n\) approaches zero, so variance approaches 0.0004 and standard deviation approaches \(\sqrt{0.0004}=0.02\), or 2%. Each result is at least 2%, as expected: this model retains the common market risk even when firm-specific risk becomes small.

Shared Exposure Does Not Require Identical Returns

A common market movement need not make all asset returns identical. Each firm can have a different expected-return component and a different firm-specific shock, yet the same common shock can still contribute to every return. The following calculation isolates those pieces. It also shows why one period in which firm-specific results happen to offset is not proof that the portfolio is protected against marketwide movements.

Worked Example: A Mixed Set of Firms During a Market Decline

A hypothetical portfolio allocates 50% to Firm A, 30% to Firm B, and 20% to Firm C. Their expected-return components are 1%, 2%, and 0.5%. The shared market shock is −4%. Their firm-specific shocks are +2%, −1%, and +1%, respectively. Find each return and the portfolio return. Method: calculate each return, then take the weighted average and check it by components. Common mistake avoided: assuming a positive firm-specific shock in one holding cancels the shared decline across the portfolio.

Step 1: Calculate the three returns.
Firm A: \(1\%-4\%+2\%=-1\%\).
Firm B: \(2\%-4\%-1\%=-3\%\).
Firm C: \(0.5\%-4\%+1\%=-2.5\%\).

Step 2: Calculate the weighted portfolio return. The weighted contributions are \(0.50(-1\%)=-0.50\%\), \(0.30(-3\%)=-0.90\%\), and \(0.20(-2.5\%)=-0.50\%\). Thus the portfolio return is \(-0.50\%-0.90\%-0.50\%=-1.90\%\).

Step 3: Check by components. The weighted expected-return component is \(0.50(1\%)+0.30(2\%)+0.20(0.5\%)=0.50\%+0.60\%+0.10\%=1.20\%\). The common shock contributes \(-4\%\), since the weights sum to \(0.50+0.30+0.20=1\). The weighted firm-specific component is \(0.50(2\%)+0.30(-1\%)+0.20(1\%)=1.00\%-0.30\%+0.20\%=0.90\%\). Combining them gives \(1.20\%-4\%+0.90\%=-1.90\%\), matching the direct calculation.

Without the common shock, the other two components would total 2.10%. The market decline reduces that result by 4 percentage points. Diversifying across these three firms does not remove the common movement.

What the Result Does—and Does Not—Say

The formula is a result under particular assumptions, not a universal rule that every additional asset lowers risk in the same amount. If firm-specific shocks are correlated, their average may retain some variation. If assets have different exposures to the common movement, the shared component of the portfolio depends on those exposures and weights. The model used here sets each common exposure to one precisely to make the mechanism transparent; it is not a general model for all portfolios.

Nor does the result mean that marketwide risk can never be changed. It means that adding more firms with the same common exposure does not eliminate that shared component. Changing the portfolio's total exposure to the common movement, for example through a hedge or a different asset allocation, is a different action from merely spreading an investment across more companies. Such actions have their own risks and are outside this simple diversification calculation.

A common exam trap is to reason that a sufficiently large number of holdings must make portfolio risk zero. The derivation shows exactly why that is wrong: only the average of independent firm-specific shocks shrinks with \(n\); the common shock is carried through the weighted portfolio. Another trap is to apply the \(\sigma_{\varepsilon}^{2}/n\) term when shocks are correlated. That step relies on the covariance terms being zero, so it must not be used without the independence or uncorrelatedness assumption.

Key takeaway: In a portfolio of firms exposed to a shared market movement, diversification can average out differing firm-specific shocks. It cannot, by itself, remove the common component that moves across the holdings.

Check Your Understanding

For each question, use the assumptions stated in that question and keep return units consistent.

Five equally weighted firms have an average expected-return component of 0.8%. Their common market shock is −3%, and their firm-specific shocks are −2%, 1%, 0%, 3%, and −2%. Find the average firm-specific shock and the portfolio return. (Answer: average firm-specific shock 0%; portfolio return −2.2%.)
A portfolio holds nine equally weighted firms. The common market shock has standard deviation 1.5%, and each independent firm-specific shock has standard deviation 6%. Assume mean-zero shocks and zero covariance between the common and firm-specific shocks. Calculate portfolio variance and standard deviation. (Answer: variance 0.000625; standard deviation 2.5%.)
Four equal-weight firms have an average expected-return component of 1.25%, a common market shock of +2%, and firm-specific shocks of −1%, 0%, +2%, and +3%. Find the average firm-specific shock and portfolio return. (Answer: average firm-specific shock 1%; portfolio return 4.25%.)
A portfolio holds two firms at weights 70% and 30%. Their weighted expected-return component is 2%, the shared market shock is −5%, and their firm-specific shocks are +4% and −4%, respectively. Calculate the weighted firm-specific shock and portfolio return. (Answer: weighted firm-specific shock 1.6%; portfolio return −1.4%.)
Under the simple model, the common market shock has variance 0.0009 and each independent firm-specific shock has variance 0.0016. For 16 equal-weight firms, calculate portfolio variance and standard deviation, then give the limiting standard deviation as the number of firms grows. (Answer: variance 0.001; standard deviation approximately 3.162%; limiting standard deviation 3%.)

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