Portfolios of Risky Assets

This chapter examines how combining risky assets shapes a portfolio’s risk and return. The central insight is that portfolio risk is not simply the average of its components’ risks: because asset returns do not move in lockstep, combining them can push risk below what any single asset offers on its own. This effect, known as diversification, is one of the most fundamental ideas in modern finance.

We begin with two risky assets to build intuition, deriving the investment opportunity set they generate and identifying the portfolio within it that achieves the lowest possible risk, the minimum-variance portfolio. We then examine how diversification behaves as more assets are added to a portfolio, and generalize the two-asset formulas to any number of risky assets using matrix notation. This will allow us to characterize the investment opportunity set achievable with many risky assets.

A Portfolio of Two Risky Assets

Suppose we have two risky assets A and B. The return of a portfolio P in which we invest 1 - w in A and w in B is r_{P} = (1 - w) r_{A} + w r_{B}. \tag{1} The expected return of the portfolio is \mu_{P} = (1 - w) \mu_{A} + w \mu_{B}, \tag{2} whereas its variance can be computed as \begin{aligned} \sigma_{P}^{2} & = \operatorname{V}((1 - w) r_{A} + w r_{B}) \\ & = (1 - w)^{2} \sigma_{A}^{2} + w^{2} \sigma_{B}^{2} + 2 w (1 - w) \sigma_{A, B}. \end{aligned} \tag{3} Here \sigma_{A, B} denotes the covariance of returns between A and B.

Example 1 (A Portfolio with Two Risky Assets) Consider two risky assets A and B for which you have the following information.

Asset Expected Return Standard Deviation
A 10% 20%
B 15% 35%

The correlation between the assets returns is 0.4. If you invest 40% in A and 60% in B, your portfolio will have an expected return of \mu = 0.4 \times 0.10 + 0.6 \times 0.15 = 13\%. Since the covariance of returns is \sigma_{AB} = 0.20 \times 0.35 \times 0.4 = 0.028, the standard deviation of the portfolio returns is \sigma = \sqrt{0.4^{2} \times 0.20^{2} + 0.6^{2} \times 0.35^{2} + 2 \times 0.4 \times 0.6 \times 0.028} = 25.29\%.

Suppose that now we want to vary w and see all the possible combinations of expected return (\mu_{P}) and standard deviation (\sigma_{P}) that we can obtain. In the previous example, we split the wealth into the two assets so that 0 \leq w \leq 1 and implies that 0 \leq 1 - w \leq 1. What if we wanted to invest more than 100% in asset B?

In financial markets, it is possible to borrow an asset and sell it immediately, a transaction known as short-selling the asset. There is typically no fixed date by which the asset must be returned to the lender; rather, it must be returned whenever the lender demands it.1

1 In practice, the lender (typically a broker) can recall the borrowed shares at their discretion, forcing the short-seller to buy them back on short notice; this is known as recall risk. Recall risk is distinct from short squeeze risk, in which a sharp rise in the asset’s price forces short-sellers to cover their positions at a loss, which can itself push the price up further. The short-seller must also compensate the lender for any dividends paid on the asset while the short position is open.

We assume throughout this class that short-selling is allowed. This means the weight w can be greater than one, in which case we short A to overinvest in B, or less than zero, in which case we short B to overinvest in A.

By varying w from -\infty to \infty, we obtain what is called the investment opportunity set generated by the two risky assets. If we plot \mu_{P} on the y-axis as a function of \sigma_{P} on the x-axis, the resulting figure is a hyperbola.

A risk-return chart with volatility on the x-axis and expected return on the y-axis. A curved frontier generated by two risky assets passes through points A and B, showing all portfolio combinations as the weight on one asset varies, including short-selling regions.
Figure 1: The figure shows the investment opportunity set generated by two risky assets A and B.

The Minimum Variance Portfolio (MVP)

As shown in the previous plot, there is a portfolio that has the minimum variance among all portfolios between A and B. To find its composition, we can use standard optimization techniques: \begin{aligned} \frac{d}{dw} \sigma_{P}^{2} & = \frac{d}{dw} (1 - w)^{2} \sigma_{A}^{2} + w^{2} \sigma_{B}^{2} + 2 w (1 - w) \sigma_{A, B} \\ & = - 2 (1 - w) \sigma_{A}^{2} + 2 w \sigma_{B}^{2} + 2 (1 - 2 w) \sigma_{A, B} = 0. \end{aligned} \tag{4} Thus, - (1 - w) \sigma_{A}^{2} + w \sigma_{B}^{2} + (1 - 2 w) \sigma_{A, B} = 0. or w_{MV} = \frac{\sigma_{A}^{2} - \sigma_{A, B}}{\sigma_{A}^{2} + \sigma_{B}^{2} - 2 \sigma_{A, B}}. \tag{5}

Property 1 (The Minimum Variance Portfolio) Given two risky assets A and B, there is a portfolio that has the minimum variance among all possible portfolios that can be built with A and B. The weights of the minimum variance portfolio are given by \begin{aligned} w_{A} & = \frac{\sigma_{B}^{2} - \sigma_{AB}}{\sigma_{A}^{2} + \sigma_{B}^{2} - 2 \sigma_{AB}}, \\ w_{B} & = \frac{\sigma_{A}^{2} - \sigma_{AB}}{\sigma_{A}^{2} + \sigma_{B}^{2} - 2 \sigma_{AB}}. \end{aligned} In the previous expression, \sigma_{A} and \sigma_{B} denote the standard deviation or volatility of returns. The term \sigma_{AB} denotes the covariance of A and B and is equal to \sigma_{AB} = \sigma_{A} \sigma_{B} \rho_{AB}, where \rho_{AB} is the correlation between the returns of A and B.

Example 2 (Minimum Variance Portfolio) Using the data of Example 1, we find that the weights of the minimum variance portfolio are given by \begin{aligned} w_{A} & = \frac{0.35^{2} - 0.028}{0.20^{2} + 0.35^{2} - 2 \times 0.028} = 88.73\% \\ w_{B} & = \frac{0.20^{2} - 0.028}{0.20^{2} + 0.35^{2} - 2 \times 0.028} = 11.27\% \end{aligned} Thus, the expected return and volatility of the minimum variance portfolio are \begin{aligned} \mu & = 0.8873 \times 0.10 + 0.1127 \times 0.15 = 10.56\%, \\ \sigma & = \sqrt{0.8873^{2} \times 0.20^{2} + 0.1127^{2} \times 0.35^{2} + 2 \times 0.8873 \times 0.1127 \times 0.028} \\ & = 19.66\%. \end{aligned}

The MVP is the point farthest to the left of the investment opportunity set.

A risk-return frontier for two risky assets with points A and B marked, plus a point labeled MV at the leftmost part of the curve. The MV point identifies the portfolio with the smallest volatility among all combinations of the two assets.
Figure 2: The figure shows the minimum variance portfolio obtained from combining two risky assets.

An Equally-Weighted Portfolio

The minimum-variance frontier demonstrates that combining two risky assets can alter the risk-return profile of the original assets. This concept is known as portfolio diversification.

To understand this phenomenon better, say that we have N securities and we invest the same amount in each, so the return of this portfolio (that we call P) is r_{P} = \frac{1}{N} \sum_{i = 1}^{N} r_{i}. The variance of P is given by \sigma_{P}^{2} = \frac{1}{N^{2}} \sum_{i = 1}^{N} \sum_{j = 1}^{N} \operatorname{Cov}(r_{i}, r_{j}). \tag{6}

The previous expression looks complex, but it is possible to decompose it into two intuitive parts. First, note that the average variance of the securities is given by \text{Avg. Variance} = \frac{1}{N} \sum_{i = 1}^{N} \operatorname{V}(r_{i}). Furthermore, the average covariance of the securities can be computed as \text{Avg. Covariance} = \frac{1}{N (N - 1)} \sum_{i = 1}^{N} \sum_{\substack{j = 1 \\ i \neq j}}^{N} \operatorname{Cov}(r_{i}, r_{j}) Using these two quantities, we can write (6) as \sigma_{P}^{2} = \frac{1}{N} (\text{Average Variance}) + \frac{N - 1}{N} (\text{Average Covariance}). \tag{7}

As N increases in (7), we have that \sigma_{P}^{2} \xrightarrow[N \rightarrow \infty]{} \text{Average Covariance}. We can see that the average variance of the portfolio disappears! In other words, this is the variance that can be diversified away.

While diversification reduces risk, it is constrained by the degree to which assets move together. The average covariance between asset pairs sets a limit on the extent of risk reduction achievable through diversification.

This limitation became evident during the 2008 financial crisis, when correlations among mortgage-related assets that had appeared only loosely related rose sharply as the crisis unfolded, so portfolios that seemed well diversified suffered simultaneous losses. Even an equally-weighted portfolio spread across many holdings cannot eliminate risk that stems from the average covariance among them.

A Portfolio of Many Risky Assets

The two-asset formulas above are a special case of a more general framework. Suppose there are N risky assets with returns r_{1}, r_{2}, \ldots, r_{N}. Collect the returns and the portfolio weights into vectors \mathbf{r} = \begin{pmatrix} r_{1} \\ r_{2} \\ \vdots \\ r_{N} \end{pmatrix}, \qquad \pmb{w} = \begin{pmatrix} w_{1} \\ w_{2} \\ \vdots \\ w_{N} \end{pmatrix}, where w_{i} is the fraction of wealth invested in asset i. Since short-selling is allowed, each w_{i} can be any real number, but the weights must still add up to the investor’s total wealth, \pmb{w}^{\top} \mathbf{1} = 1, \tag{8} where \mathbf{1} denotes the N-dimensional vector of ones. The portfolio return is the linear combination r_{P} = \pmb{w}^{\top} \mathbf{r} = \sum_{i = 1}^{N} w_{i} r_{i}. \tag{9}

By linearity of expectation, the expected return is \mu_{P} = \sum_{i=1}^{N} w_{i} \mu_{i} = \pmb{w}^{\top} \pmb{\mu}, where \pmb{\mu} = (\mu_{1}, \ldots, \mu_{N})^{\top} and \mu_{i} = \operatorname{E}(r_{i}). The variance, however, involves every pairwise covariance among the assets: writing r_{P} - \mu_{P} = \sum_{i} w_{i} (r_{i} - \mu_{i}) and squaring, \sigma_{P}^{2} = \operatorname{E}\left[ \left( \sum_{i = 1}^{N} w_{i} (r_{i} - \mu_{i}) \right)^{2} \right] = \sum_{i = 1}^{N} \sum_{j = 1}^{N} w_{i} w_{j} \operatorname{Cov}(r_{i}, r_{j}), since expanding the square and taking expectations turns each cross term w_{i} w_{j} (r_{i} - \mu_{i})(r_{j} - \mu_{j}) into w_{i} w_{j} \operatorname{Cov}(r_{i}, r_{j}), and each i = j term into w_{i}^{2} \sigma_{i}^{2}. This double sum is the N-asset generalization of (3); keeping track of its N variances and N(N - 1)/2 distinct covariances quickly becomes unwieldy as N grows.

Matrix notation packages this double sum compactly. Denote by \pmb{\Sigma} the covariance matrix of \mathbf{r}, the N \times N symmetric matrix whose (i,j) entry is \sigma_{ij} = \operatorname{Cov}(r_{i}, r_{j}), with \sigma_{ii} = \sigma_{i}^{2} on the diagonal. The double sum above is exactly the quadratic form \pmb{w}^{\top} \pmb{\Sigma} \pmb{w}, so \mu_{P} = \pmb{w}^{\top} \pmb{\mu}, \qquad \sigma_{P}^{2} = \pmb{w}^{\top} \pmb{\Sigma} \pmb{w}. \tag{10} The same expand-and-take-expectations argument, applied to the covariance of two different portfolios P and Q with weight vectors \pmb{w}_{P} and \pmb{w}_{Q}, gives2 \operatorname{Cov}(r_{P}, r_{Q}) = \pmb{w}_{P}^{\top} \pmb{\Sigma} \pmb{w}_{Q}, \tag{11} which reduces to (10) when Q = P.

2 For any weight vectors \pmb{a} and \pmb{b}, \operatorname{Cov}(\pmb{a}^{\top} \mathbf{r}, \pmb{b}^{\top} \mathbf{r}) = \sum_{i = 1}^{N} \sum_{j = 1}^{N} a_{i} b_{j} \operatorname{Cov}(r_{i}, r_{j}) = \pmb{a}^{\top} \pmb{\Sigma} \pmb{b}, by the same argument used above for \sigma_{P}^{2}; setting \pmb{a} = \pmb{b} recovers the quadratic form for the variance.

3 With N = 2, set \pmb{w} = (1 - w, w)^{\top} and \pmb{\Sigma} = \begin{pmatrix} \sigma_{A}^{2} & \sigma_{A,B} \\ \sigma_{A,B} & \sigma_{B}^{2} \end{pmatrix}. Expanding the quadratic form gives \begin{aligned} \pmb{w}^{\top} \pmb{\Sigma} \pmb{w} & = \begin{pmatrix} 1 - w & w \end{pmatrix} \begin{pmatrix} \sigma_{A}^{2} & \sigma_{A,B} \\ \sigma_{A,B} & \sigma_{B}^{2} \end{pmatrix} \begin{pmatrix} 1 - w \\ w \end{pmatrix} \\ & = (1 - w)^{2} \sigma_{A}^{2} + w^{2} \sigma_{B}^{2} + 2 w (1 - w) \sigma_{A,B}, \end{aligned} which is exactly (3).

Equations (9) and (10) replace (2) and (3) for any number of assets, and collapse back to the two-asset formulas once N = 2.3

Example 3 (A Portfolio of Three Risky Assets) Consider three risky assets A, B, and C with the following expected returns and standard deviations.

Asset Expected Return Standard Deviation
A 8% 15%
B 12% 20%
C 15% 25%

The correlations between their returns are \rho_{AB} = 0.3, \rho_{AC} = 0.1, and \rho_{BC} = 0.4, so the covariance matrix is \pmb{\Sigma} = \begin{pmatrix} 0.0225 & 0.0090 & 0.00375 \\ 0.0090 & 0.0400 & 0.0200 \\ 0.00375 & 0.0200 & 0.0625 \end{pmatrix}.

If you invest \pmb{w} = (0.3, 0.3, 0.4)^{\top} in the three assets, (10) gives an expected return of \mu_{P} = \pmb{w}^{\top} \pmb{\mu} = 12.0\% and a variance of \sigma_{P}^{2} = \pmb{w}^{\top} \pmb{\Sigma} \pmb{w} = 0.0229, so that \sigma_{P} = 15.15\%. Computing \mu_{P} and \sigma_{P} only requires matrix multiplication, regardless of how many assets are in the portfolio.

The Investment Opportunity Set with Many Risky Assets

With two risky assets, tracing out (\sigma_{P}, \mu_{P}) as w ranges over all real numbers produces a single curve: each value of w pins down one portfolio, and there is exactly one free parameter to vary. With N risky assets, the weights \pmb{w} still have to satisfy the single budget constraint (8), which leaves N - 1 degrees of freedom. For N \geq 3, there are therefore many different portfolios that share the same expected return \mu_{P} but differ in variance \sigma_{P}^{2}. As a result, the set of achievable pairs (\sigma_{P}, \mu_{P}) — the investment opportunity set — is no longer a curve but a filled two-dimensional region.

That region still has a well-defined left boundary, however. For each attainable level of expected return m, exactly one portfolio achieves it with the smallest possible variance; every other portfolio with that same expected return lies strictly to its right. Collecting these minimum-variance points as m varies traces out a curve called the minimum-variance frontier, and every portfolio in the investment opportunity set lies on or to the right of it — which is exactly why the set looks like a frontier together with everything to its right, rather than just a curve.

We can find the frontier the same way we will find the minimum-variance portfolio below, by adding a second constraint. For a target expected return m, we minimize \min_{\pmb{w}} \; \pmb{w}^{\top} \pmb{\Sigma} \pmb{w} \quad \text{subject to} \quad \pmb{w}^{\top} \mathbf{1} = 1 \quad \text{and} \quad \pmb{w}^{\top} \pmb{\mu} = m. Introducing a Lagrange multiplier for each constraint and following the same steps as for the minimum-variance portfolio4 gives a frontier variance of \sigma_{P}^{2}(m) = \frac{C m^{2} - 2 A m + B}{D}, \tag{12} where A = \pmb{\mu}^{\top} \pmb{\Sigma}^{-1} \mathbf{1}, \qquad B = \pmb{\mu}^{\top} \pmb{\Sigma}^{-1} \pmb{\mu}, \qquad C = \mathbf{1}^{\top} \pmb{\Sigma}^{-1} \mathbf{1}, \qquad D = BC - A^{2}. Equation (12) is a parabola in (\sigma_{P}^{2}, \mu_{P})-space, which is exactly why the frontier traces out a hyperbola in (\sigma_{P}, \mu_{P})-space, generalizing the hyperbola we found earlier with only two assets. Its minimum over m occurs at m = A / C, with \sigma_{P}^{2} = 1 / C. This is exactly the minimum-variance portfolio derived in the next section, since C = \mathbf{1}^{\top} \pmb{\Sigma}^{-1} \mathbf{1} is the same quantity that appears there.

4 With two constraints, the Lagrangian is L(\pmb{w}, \lambda, \gamma) = \pmb{w}^{\top} \pmb{\Sigma} \pmb{w} - 2 \lambda (\pmb{w}^{\top} \pmb{\mu} - m) - 2 \gamma (\pmb{w}^{\top} \mathbf{1} - 1). The first-order condition gives \pmb{w} = \pmb{\Sigma}^{-1} (\lambda \pmb{\mu} + \gamma \mathbf{1}), and substituting into the two constraints yields the linear system \begin{pmatrix} B & A \\ A & C \end{pmatrix} \begin{pmatrix} \lambda \\ \gamma \end{pmatrix} = \begin{pmatrix} m \\ 1 \end{pmatrix}, so that \lambda = (Cm - A)/D and \gamma = (B - Am)/D. Plugging \pmb{w} back into \pmb{w}^{\top} \pmb{\Sigma} \pmb{w} = \lambda m + \gamma, using the two constraints, gives (12).

The figure below illustrates this for the three assets from Example 3. Each gray dot is a portfolio obtained by perturbing the equally-weighted portfolio with a random shock that is symmetric across the three assets and averages to zero, so the budget constraint continues to hold. Points near the frontier are denser than points far from it: moving away from a minimum-variance portfolio along a direction that keeps the expected return fixed raises the variance quadratically, so reaching a point deep in the interior, far from the frontier, requires substantially larger short and levered positions than reaching a point close to it.

A risk-return scatter plot showing a cloud of points representing simulated portfolios of three risky assets, denser near the minimum-variance frontier and thinning out farther into the interior. The minimum-variance frontier is a smooth curve that opens up to the right like a sideways parabola and bounds the cloud on the left. The three individual assets and the minimum-variance portfolio are marked separately.
Figure 3: The figure shows simulated portfolios of the three risky assets from Example 3, obtained by perturbing the equally-weighted portfolio with a random shock, together with the minimum-variance frontier and the minimum-variance portfolio.

The Minimum-Variance Portfolio with Many Risky Assets

With many risky assets, finding the minimum-variance portfolio means solving \min_{\pmb{w}} \; \pmb{w}^{\top} \pmb{\Sigma} \pmb{w} \quad \text{subject to} \quad \pmb{w}^{\top} \mathbf{1} = 1, i.e., minimizing (10) subject to the budget constraint (8). Introducing a Lagrange multiplier \lambda for the constraint, define the Lagrangian L(\pmb{w}, \lambda) = \pmb{w}^{\top} \pmb{\Sigma} \pmb{w} - 2 \lambda \left( \pmb{w}^{\top} \mathbf{1} - 1 \right), where the factor of 2 is just a convenient normalization for \lambda. The first-order condition with respect to \pmb{w} is5 2 \pmb{\Sigma} \pmb{w} - 2 \lambda \mathbf{1} = \mathbf{0} \quad \Longrightarrow \quad \pmb{w} = \lambda \pmb{\Sigma}^{-1} \mathbf{1}, where \pmb{\Sigma}^{-1} exists because covariance matrices are positive definite, as noted in Probability Basics. Substituting into the budget constraint pins down the multiplier, \lambda \, \mathbf{1}^{\top} \pmb{\Sigma}^{-1} \mathbf{1} = 1 \quad \Longrightarrow \quad \lambda = \frac{1}{\mathbf{1}^{\top} \pmb{\Sigma}^{-1} \mathbf{1}}, which, substituted back, gives the following result.

5 Differentiating \pmb{w}^{\top} \pmb{\Sigma} \pmb{w} with respect to \pmb{w} gives 2 \pmb{\Sigma} \pmb{w} because \pmb{\Sigma} is symmetric, and differentiating \pmb{w}^{\top} \mathbf{1} gives \mathbf{1}.

Property 2 (The Minimum-Variance Portfolio with Many Assets) Given N risky assets with covariance matrix \pmb{\Sigma}, the minimum-variance portfolio has weights \pmb{w}_{MV} = \frac{\pmb{\Sigma}^{-1} \mathbf{1}}{\mathbf{1}^{\top} \pmb{\Sigma}^{-1} \mathbf{1}} \tag{13} and variance \sigma_{MV}^{2} = \frac{1}{\mathbf{1}^{\top} \pmb{\Sigma}^{-1} \mathbf{1}}. \tag{14}

To see why the variance takes this simple form, plug \pmb{w}_{MV} = \lambda \pmb{\Sigma}^{-1} \mathbf{1} into (10). Since \pmb{\Sigma} \pmb{w}_{MV} = \lambda \mathbf{1}, \sigma_{MV}^{2} = \pmb{w}_{MV}^{\top} \pmb{\Sigma} \pmb{w}_{MV} = \pmb{w}_{MV}^{\top} (\lambda \mathbf{1}) = \lambda \left( \pmb{w}_{MV}^{\top} \mathbf{1} \right) = \lambda, using the budget constraint \pmb{w}_{MV}^{\top} \mathbf{1} = 1 in the last step, which is exactly (14).

Equation (13) is the general version of (5): with N = 2 it reduces exactly to the weights w_{A} and w_{B} from Property 1.6

6 With N = 2 and \pmb{\Sigma} = \begin{pmatrix} \sigma_{A}^{2} & \sigma_{AB} \\ \sigma_{AB} & \sigma_{B}^{2} \end{pmatrix}, the formula for the inverse of a 2 \times 2 matrix gives \pmb{\Sigma}^{-1} \mathbf{1} = \frac{1}{\sigma_{A}^{2} \sigma_{B}^{2} - \sigma_{AB}^{2}} \begin{pmatrix} \sigma_{B}^{2} - \sigma_{AB} \\ \sigma_{A}^{2} - \sigma_{AB} \end{pmatrix}, so that \mathbf{1}^{\top} \pmb{\Sigma}^{-1} \mathbf{1} = \dfrac{\sigma_{A}^{2} + \sigma_{B}^{2} - 2 \sigma_{AB}}{\sigma_{A}^{2} \sigma_{B}^{2} - \sigma_{AB}^{2}}, and dividing each entry of \pmb{\Sigma}^{-1} \mathbf{1} by this quantity recovers exactly w_{A} and w_{B} from Property 1.

Example 4 (The Minimum-Variance Portfolio with Three Assets) Using the covariance matrix \pmb{\Sigma} from Example 3, (13) gives minimum-variance weights of 64.07\% in A, 19.05\% in B, and 16.88\% in C. This portfolio has an expected return of \mu_{MV} = 9.94\% and, by (14), a variance of \sigma_{MV}^{2} = 0.0168, i.e., a standard deviation of \sigma_{MV} = 12.95\%. Notice that \sigma_{MV} \approx 12.95\% is lower than the volatility of any of the three individual assets, illustrating diversification with more than two risky assets.

With many assets, the orthogonality property of the minimum-variance portfolio is considerably more interesting than in the two-asset case: it holds not just for portfolios on the line connecting two assets, but for every portfolio in the entire investment universe, however many assets it holds. Let \pmb{w}_{P} be the weights of an arbitrary portfolio P, so that \pmb{w}_{P}^{\top} \mathbf{1} = 1. By (11) and \pmb{w}_{MV} = \lambda \pmb{\Sigma}^{-1} \mathbf{1}, \operatorname{Cov}(r_{P}, r_{MV}) = \pmb{w}_{P}^{\top} \pmb{\Sigma} \pmb{w}_{MV} = \pmb{w}_{P}^{\top} \pmb{\Sigma} \left( \lambda \pmb{\Sigma}^{-1} \mathbf{1} \right) = \lambda \left( \pmb{w}_{P}^{\top} \mathbf{1} \right) = \lambda = \sigma_{MV}^{2}, using the budget constraint \pmb{w}_{P}^{\top} \mathbf{1} = 1 and (14) in the last two steps. The covariance of the MVP with any other portfolio is always the same and equal to its own variance! This also implies that the correlation of the MVP with any other portfolio is always positive, regardless of how many assets are in the investment universe.

Example 5 (Computing a Correlation with the MVP) Consider an investment opportunity set where the MVP has a standard deviation of returns of 20%. An asset A has a standard deviation of returns equal to 30%. Thus, \sigma_{A} \sigma_{MV} \rho_{A, MV} = \sigma_{A, MV} = \sigma_{MV}^{2}, or \rho_{A, MV} = \frac{\sigma_{MV}}{\sigma_{A}} = \frac{2}{3}.

Equations (5) and (13) demonstrate that the weights of the MVP depend solely on variances and covariances, which can typically be estimated with high precision from the data. In contrast, constructing any other portfolio within the investment opportunity set requires estimating expected returns, which are generally more challenging to estimate accurately. This is the reason why some practitioners like to use the MVP for their asset allocation. Moreover, imposing short-sale constraints to the estimation of the minimum-variance portfolio makes the model more robust. Jagannathan and Ma (2003) show that with no short-sale constraints in place, the sample covariance matrix performs as well as covariance matrix estimates based on factor models, shrinkage estimators, and daily data.

Jagannathan, Ravi, and Tongshu Ma. 2003. “Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps.” Journal of Finance 58 (4): 1651–83.

Practice Problems

These problems give you a chance to practice the concepts introduced in this chapter. Try to solve each one on your own before expanding the solution.

Problem 1 (Expected Return and Volatility of a Two-Asset Portfolio) Suppose you are choosing between two mutual funds, the Dynamic Growth Fund (D) and the Emerging Opportunities Fund (E). Fund D is the steadier of the pair, with an expected return of \mu_{D} = 12\% and a volatility of \sigma_{D} = 18\%, while Fund E offers a higher expected return of \mu_{E} = 18\% but comes with substantially more risk, \sigma_{E} = 30\%. The two funds’ returns are only mildly related, with a correlation of \rho_{DE} = 0.25. You decide to build a portfolio that puts 60% of your wealth in D and the remaining 40% in E. What expected return and volatility should you expect from this portfolio?

Solution Using (2), the expected return is \mu_{P} = 0.6 \times 0.12 + 0.4 \times 0.18 = 14.4\%. The covariance of returns is \sigma_{DE} = 0.18 \times 0.30 \times 0.25 = 0.0135, so using (3) the volatility is \sigma_{P} = \sqrt{0.6^{2} \times 0.18^{2} + 0.4^{2} \times 0.30^{2} + 2 \times 0.6 \times 0.4 \times 0.0135} = 18.04\%.

Problem 2 (The Minimum-Variance Portfolio with Negatively Correlated Assets) You are examining two stocks, Anchor Steel Co. (STL) and Voltaic Motors Inc. (VLT), that behave quite differently from one another: STL has a volatility of \sigma_{STL} = 25\%, while VLT is considerably more volatile, with \sigma_{VLT} = 40\%. What makes this pair especially interesting is that their returns tend to move in opposite directions, with a correlation of \rho_{STL, VLT} = -0.20. You wonder just how much of this offsetting behavior you can exploit. Find the weights of the minimum-variance portfolio, and compute its volatility. Compare this volatility to \sigma_{STL} and \sigma_{VLT}.

Solution The covariance is \sigma_{STL, VLT} = 0.25 \times 0.40 \times (-0.20) = -0.02. Using Property 1, w_{STL} = \frac{\sigma_{VLT}^{2} - \sigma_{STL, VLT}}{\sigma_{STL}^{2} + \sigma_{VLT}^{2} - 2 \sigma_{STL, VLT}} = \frac{0.16 - (-0.02)}{0.0625 + 0.16 - 2 \times (-0.02)} = 68.57\%, \qquad w_{VLT} = 31.43\%. The variance of this portfolio is \sigma_{MV}^{2} = 0.6857^{2} \times 0.0625 + 0.3143^{2} \times 0.16 + 2 \times 0.6857 \times 0.3143 \times (-0.02) = 0.0366, so \sigma_{MV} = 19.12\%. This is lower than both \sigma_{STL} = 25\% and \sigma_{VLT} = 40\%. The negative correlation between STL and VLT allows diversification to eliminate more risk than either asset carries on its own.

Problem 3 (Diversification with Many Equally-Weighted Assets) A portfolio manager has access to a large universe of stocks, each with a variance of returns of 0.36 (i.e., a volatility of 60\%) and an average covariance between any two stocks of 0.09. If the manager forms an equally-weighted portfolio of N = 25 of these stocks, what is the portfolio’s volatility? What volatility does the portfolio approach as N \to \infty, and what does this limit represent?

Solution Using (7) with N = 25, \sigma_{P}^{2} = \frac{1}{25} \times 0.36 + \frac{24}{25} \times 0.09 = 0.1008, so \sigma_{P} = 31.75\%. As N \to \infty, the first term vanishes and \sigma_{P}^{2} \to 0.09, i.e., \sigma_{P} \to 30\%. This limit is the average covariance between the stocks: it represents the risk that cannot be diversified away, no matter how many stocks are added to the portfolio.

Problem 4 (A Three-Asset Portfolio Using the Covariance Matrix) You are building a portfolio out of three risky assets, P, Q, and R, that span a range of risk and return. Asset P is the most conservative of the three, with an expected return of \mu_{P} = 9\% and a volatility of \sigma_{P} = 12\%. Asset Q sits in the middle, offering \mu_{Q} = 14\% at a volatility of \sigma_{Q} = 22\%. Asset R is the most aggressive, with \mu_{R} = 20\% and \sigma_{R} = 30\%. The three assets are not equally related to one another: P and Q have a modest correlation of \rho_{PQ} = 0.2, P and R are uncorrelated, \rho_{PR} = 0, and Q and R move together more closely, with \rho_{QR} = 0.5. You settle on the weights \pmb{w} = (0.5, 0.3, 0.2)^{\top}. Compute the expected return and volatility of the resulting portfolio.

Solution The expected return is \mu_{P} = 0.5 \times 0.09 + 0.3 \times 0.14 + 0.2 \times 0.20 = 12.7\%. The covariance matrix is \pmb{\Sigma} = \begin{pmatrix} 0.0144 & 0.00528 & 0 \\ 0.00528 & 0.0484 & 0.033 \\ 0 & 0.033 & 0.09 \end{pmatrix}. Using (10), the variance is \sigma_{P}^{2} = \pmb{w}^{\top} \pmb{\Sigma} \pmb{w} = 0.0171, so \sigma_{P} = 13.08\%.

Problem 5 (Correlation with the Minimum-Variance Portfolio) In a large investment universe, the minimum-variance portfolio has a volatility of 15\%. An asset B in this universe has a volatility of 45\%. What is the correlation between the returns of B and the returns of the minimum-variance portfolio?

Solution Since \sigma_{B, MV} = \sigma_{MV}^{2} for every asset or portfolio in the universe, \rho_{B, MV} = \frac{\sigma_{B, MV}}{\sigma_{B} \sigma_{MV}} = \frac{\sigma_{MV}^{2}}{\sigma_{B} \sigma_{MV}} = \frac{\sigma_{MV}}{\sigma_{B}} = \frac{0.15}{0.45} = 33.33\%.

Problem 6 (Comparative Statics: Correlation and Minimum-Variance Risk) Consider two pairs of risky assets, both with \sigma_{A} = 20\% and \sigma_{B} = 30\%. In Pair 1 the correlation is \rho_{1} = 0.8; in Pair 2 it is \rho_{2} = -0.3. Without computing exact weights, predict which pair’s minimum-variance portfolio has lower volatility, and why. Then verify your prediction by computing the minimum-variance weights and volatility for each pair.

Solution

Lower correlation means the two assets move together less, so combining them offsets more risk; we should expect Pair 2’s minimum-variance portfolio to have the lower volatility.

For Pair 1, \sigma_{AB} = 0.20 \times 0.30 \times 0.8 = 0.048, so w_{A} = \frac{0.09 - 0.048}{0.04 + 0.09 - 2 \times 0.048} = 123.53\%, \qquad w_{B} = -23.53\%, and \sigma_{MV}^{2} = 1.2353^{2} \times 0.04 + (-0.2353)^{2} \times 0.09 + 2 \times 1.2353 \times (-0.2353) \times 0.048 = 0.0381, so \sigma_{MV} = 19.52\%. Note that the weight on B is negative: with such high correlation, the minimum-variance portfolio requires short-selling B to hedge A.

For Pair 2, \sigma_{AB} = 0.20 \times 0.30 \times (-0.3) = -0.018, so w_{A} = \frac{0.09 - (-0.018)}{0.04 + 0.09 - 2 \times (-0.018)} = 65.06\%, \qquad w_{B} = 34.94\%, and \sigma_{MV}^{2} = 0.6506^{2} \times 0.04 + 0.3494^{2} \times 0.09 + 2 \times 0.6506 \times 0.3494 \times (-0.018) = 0.0197, so \sigma_{MV} = 14.05\%.

As predicted, Pair 2’s minimum-variance portfolio (14.05\%) has substantially lower volatility than Pair 1’s (19.52\%), confirming that lower (or more negative) correlation between two assets improves the diversification benefit available from combining them.

Problem 7 (A Pension Fund’s Minimum-Variance Portfolio) A pension fund manager is considering two mutual funds: a stock fund and a long-term government and corporate bond fund. The probability distribution of the funds is as follows:

Fund Expected Return Standard Deviation
Stock fund (S) 20% 30%
Bond fund (B) 12% 15%

The correlation between the fund returns is 0.10.

  1. What are the investment proportions in the minimum-variance portfolio of the two funds?

  2. What is the expected value and standard deviation of the minimum-variance portfolio’s rate of return?

  3. Compute the expected return and standard deviation of the following portfolios.

    Proportion in S Proportion in B Expected Return Standard Deviation
    25% 75%
    50% 50%
    75% 25%
Solution

  1. The covariance between the funds is \sigma_{SB} = 0.30 \times 0.15 \times 0.10 = 0.0045. Using Property 1, w_{S} = \frac{\sigma_{B}^{2} - \sigma_{SB}}{\sigma_{S}^{2} + \sigma_{B}^{2} - 2 \sigma_{SB}} = \frac{0.0225 - 0.0045}{0.09 + 0.0225 - 2 \times 0.0045} = 17.39\%, \qquad w_{B} = 82.61\%.

  2. Using (2) and (3) with these weights, \mu_{MV} = 0.1739 \times 0.20 + 0.8261 \times 0.12 = 13.39\%, \sigma_{MV} = \sqrt{0.1739^{2} \times 0.09 + 0.8261^{2} \times 0.0225 + 2 \times 0.1739 \times 0.8261 \times 0.0045} = 13.92\%. Note that \sigma_{MV} = 13.92\% is below the volatility of both the stock fund (30\%) and the bond fund (15\%), illustrating the diversification benefit from combining two imperfectly correlated funds.

  3. Applying (2) and (3) to each pair of weights gives

    Proportion in S Proportion in B Expected Return Standard Deviation
    25% 75% 14% 14.13%
    50% 50% 16% 17.43%
    75% 25% 18% 23.18%

    For example, with 25\% in S and 75\% in B, \mu_{P} = 0.25 \times 0.20 + 0.75 \times 0.12 = 14\%, \sigma_{P} = \sqrt{0.25^{2} \times 0.09 + 0.75^{2} \times 0.0225 + 2 \times 0.25 \times 0.75 \times 0.0045} = 14.13\%, and analogously for the other two rows. As the weight shifts from B toward S, both the expected return and the volatility of the portfolio rise, since S has both a higher expected return and a higher standard deviation than B.

Problem 8 (Adding a Concentrated Stock Position) Daniel Reyes manages a $4,000,000 fully diversified portfolio for a client. The client unexpectedly receives a bonus of $1,000,000 in vested shares of her employer, Meridian Corp. Her financial adviser has produced the following forecasts:

Asset Expected Return Standard Deviation
Original Portfolio 8.5% 10%
Meridian Corp 18% 22%

The correlation coefficient between Meridian Corp and the original portfolio is 0.30.

  1. The bonus changes the client’s overall portfolio, and she is deciding whether to keep the Meridian shares. Assuming she keeps them, calculate the expected return and standard deviation of her new $5,000,000 portfolio, which includes the Meridian shares.
  2. If she sells the Meridian shares, she will invest the proceeds in risk-free Treasury bills yielding 4.5%. Assuming she sells the shares and replaces them with the Treasury bills, calculate the expected return and standard deviation of her new portfolio.
  3. A colleague suggests that instead of buying Treasury bills, she sell the Meridian shares and use the $1,000,000 to buy shares of Atlas Corp, which has the same expected return and standard deviation as Meridian Corp. The colleague argues, “It doesn’t matter whether she keeps the Meridian shares or switches to Atlas instead.” State whether the colleague’s comment is correct or incorrect, and justify your answer briefly.
Solution

  1. The client now holds 80\% of her wealth in the original portfolio and 20\% in Meridian Corp. The covariance between the two is \sigma_{OM} = 0.10 \times 0.22 \times 0.30 = 0.0066, so using (2) and (3), \mu_{P} = 0.8 \times 0.085 + 0.2 \times 0.18 = 10.4\%, \sigma_{P} = \sqrt{0.8^{2} \times 0.10^{2} + 0.2^{2} \times 0.22^{2} + 2 \times 0.8 \times 0.2 \times 0.0066} = 10.22\%.

  2. Replacing Meridian Corp with a risk-free asset removes all covariance risk, so the portfolio is simply an 80/20 mix of the original portfolio and the risk-free rate, \mu_{P} = 0.8 \times 0.085 + 0.2 \times 0.045 = 7.7\%, \qquad \sigma_{P} = 0.8 \times 0.10 = 8\%.

  3. The colleague’s comment is incorrect. Matching Meridian Corp’s expected return and standard deviation only pins down Atlas Corp’s own risk and return; it says nothing about how Atlas Corp moves together with the rest of the portfolio. The standard deviation of the combined position depends on the covariance — equivalently, the correlation — between the new stock and the original portfolio, not just on the new stock’s standalone volatility. Unless Atlas Corp also happens to share Meridian Corp’s 0.30 correlation with the original portfolio, switching into it will change \sigma_{P} even though \mu_{P} stays at 10.4\%.