TA Review - Risk and Return

TA: Dipesh Ghimire

Lecture 1, Autumn 2026.

Why Mean–Variance Optimization?

Ultimately, an investor wants to maximize expected utility of future wealth:

\[ \max_w \mathbb{E}[U(W_{t+1})]. \]

For a risk-averse investor,

\[ U'(W) > 0, \qquad U''(W) < 0. \]

So more wealth is preferred, but an additional dollar of wealth has diminishing marginal utility.

A Taylor expansion around a fixed reference wealth \(W\) gives

\[\begin{split} \begin{aligned} \mathbb{E}[U(W_{t+1})] \approx {}& U(W) + U'(W)\mathbb{E}[W_{t+1}-W] \\ &+ \frac{1}{2}U''(W)\mathbb{E}[(W_{t+1}-W)^2] \\ &+ \frac{1}{6}U'''(W)\mathbb{E}[(W_{t+1}-W)^3] + \cdots \end{aligned} \end{split}\]

The second-order approximation depends on the first two moments of wealth. Higher-order terms introduce the third and higher moments. The mean-variance objective below is a local approximation with risk aversion treated as constant; it is exact under additional assumptions such as normally distributed returns and exponential utility.

Thus the standard MV objective is

\[ \boxed{\max_w \quad w^\top \mu - \frac{\lambda}{2}w^\top\Sigma w} \]

where

\[ w^\top\mu = \mathbb{E}[R_p], \qquad w^\top\Sigma w = \operatorname{Var}(R_p). \]

\(\lambda > 0\) captures risk aversion.

It is also mathematically convenient:

• Expected return is linear in \(w\);

• Variance is quadratic in \(w\);

• \(\Sigma \succeq 0\), so variance is convex;

• In the basic cases, we have closed-form solutions.

Portfolio Risk And Correlation

For a portfolio consisting of two assets, the portfolio return is given by:

\[ r_p = w_1 r_1 + w_2 r_2. \]

The variance of the portfolio return is derived as follows:

\[\begin{split} \begin{aligned} \operatorname{Var}(r_p) &= \operatorname{Var}(w_1 r_1 + w_2 r_2) \\ &= w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \operatorname{Cov}(r_1, r_2). \end{aligned} \end{split}\]

Recalling that the covariance can be expressed in terms of the correlation coefficient \(\rho_{12}\) and standard deviations:

\[ \operatorname{Cov}(r_1, r_2) = \rho_{12} \sigma_1 \sigma_2, \]

we obtain the standard expression for portfolio variance:

\[ \boxed{\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2} \]

Meanwhile, the expected portfolio return remains linear in weights:

\[ \boxed{\mu_p = w_1 \mu_1 + w_2 \mu_2} \]

The Role of Correlation and Diversification

Portfolio variance is quadratic in weights. Both individual variances and covariances contribute.

To understand the impact of correlation, assume long-only weights, \(w_1,w_2 \geq 0\), and the budget constraint \(w_1 + w_2 = 1\). The strict diversification benefit applies when both weights are positive.

1. Perfect Positive Correlation (\(\rho_{12} = 1\)):
   The assets move perfectly together. There are no diversification benefits, and portfolio volatility lies on the straight line connecting the individual asset volatilities in mean-standard deviation space.

2. Imperfect Correlation (\(\rho_{12} < 1\)):
   The covariance term decreases relative to the \(\rho_{12}=1\) case, reducing portfolio variance. This causes the efficient frontier to “bend” to the left in mean-volatility space, creating a diversification benefit.

3. Impact of Decreasing Correlation:
   As \(\rho_{12}\) decreases, the curvature becomes more pronounced:

   \[ \rho_{12} \in \{0.8, 0.5, 0, -0.5\} \]
   Lower correlation shifts the opportunity set further to the left, allowing for lower risk at any given level of expected return.

Thus, the fundamental condition for diversification benefits is:

\[ \boxed{\rho_{12} < 1} \]

Diversification with many assets

Portfolio Variance for \(N\) Securities

For a portfolio of \(N\) securities with weight vector \(w\) and covariance matrix \(\Sigma\), the portfolio variance is:

\[ \boxed{\sigma_p^2 = w^\top \Sigma w = \sum_{i=1}^N \sum_{j=1}^N w_i w_j \sigma_{ij}} \]

The Diversification Effect: Equally Weighted Portfolio

Consider an equally weighted portfolio where \(w_i = \frac{1}{N}\) for all \(i\). The variance expands to:

\[ \sigma_p^2 = \frac{1}{N^2} \sum_{i=1}^N \sigma_i^2 + \frac{1}{N^2} \sum_{i \neq j} \sigma_{ij} \]

Let \(\overline{\sigma_i^2}\) denote the average individual variance and \(\overline{\sigma_{ij}}\) denote the average pairwise covariance across all distinct pairs. We can rewrite the expression as:

\[ \boxed{\sigma_p^2 = \frac{1}{N} \overline{\sigma_i^2} + \frac{N-1}{N} \overline{\sigma_{ij}}} \]

If average individual variance remains bounded and average pairwise covariance converges, taking \(N \rightarrow \infty\) gives:

\[ \boxed{\lim_{N \to \infty} \sigma_p^2 = \overline{\sigma_{ij}}} \]

The average-variance term vanishes at rate \(1/N\) when average variance remains bounded. The limiting average covariance determines the remaining risk. An idiosyncratic-versus-systematic decomposition requires a factor model.

Special Case: Constant Correlation Model

Assume every asset has identical volatility \(\sigma\) and every pair shares a common correlation coefficient \(\rho\). For a valid \(N\)-asset covariance matrix, \(-1/(N-1) \leq \rho \leq 1\); a fixed correlation in the large-\(N\) limit must be nonnegative. Then \(\overline{\sigma_i^2} = \sigma^2\) and \(\overline{\sigma_{ij}} = \rho \sigma^2\). The portfolio variance becomes:

\[ \sigma_p^2 = \frac{\sigma^2}{N} + \frac{N-1}{N} \rho \sigma^2 \]

As \(N \rightarrow \infty\):

\[ \sigma_p^2 \longrightarrow \rho \sigma^2 \]

This demonstrates that diversification eliminates idiosyncratic risk but cannot eliminate common (systematic) risk driven by correlation.

Marginal Contribution to Risk

An asset’s standalone volatility is distinct from its contribution to portfolio risk. The marginal contribution of asset \(i\) to portfolio variance is derived via matrix calculus:

\[ \frac{\partial \sigma_p^2}{\partial w_i} = 2(\Sigma w)_i = 2 \operatorname{Cov}(r_i, r_p) \]

where \((\Sigma w)_i\) is the \(i\)-th element of the vector \(\Sigma w\).

This derivative varies one weight while holding the others fixed. A fully invested reallocation from asset \(j\) to asset \(i\) instead has variance derivative \(2[\operatorname{Cov}(r_i,r_p)-\operatorname{Cov}(r_j,r_p)]\). The marginal variance contribution is:

\[ \boxed{\text{Marginal Variance Contribution} \propto \operatorname{Cov}(r_i, r_p)} \]

Unconstrained Mean–Variance Portfolio

Unconstrained Mean-Variance Optimization

Assume \(\Sigma\) is positive definite and short positions are allowed. Here \(\mu\) denotes expected total returns. The unconstrained objective is:

\[ \boxed{\max_w \quad w^\top \mu - \frac{\lambda}{2} w^\top \Sigma w} \]

where \(\lambda > 0\) is the coefficient of risk aversion. The factor of \(\frac{1}{2}\) is included for algebraic convenience, as it cancels out during differentiation.

Derivation of Optimal Weights

To find the optimal portfolio weights \(w^*\), we take the gradient of the objective function with respect to \(w\):

\[ \nabla_w \left( w^\top \mu - \frac{\lambda}{2} w^\top \Sigma w \right) = \mu - \lambda \Sigma w \]

Setting the first-order condition (FOC) to zero:

\[ \mu - \lambda \Sigma w = 0 \]

Solving for \(w\):

\[ \lambda \Sigma w = \mu \implies w^* = \frac{1}{\lambda} \Sigma^{-1} \mu \]

Thus, the optimal unconstrained portfolio is:

\[ \boxed{w^* = \frac{1}{\lambda} \Sigma^{-1} \mu} \]

Interpretation

1. Portfolio Direction: The vector \(\Sigma^{-1} \mu\) determines the relative composition of risky assets. This direction is independent of the investor’s risk aversion \(\lambda\).

2. Portfolio Scale: The scalar \(\frac{1}{\lambda}\) determines the total magnitude of the investment in the risky portfolio.

\[ \boxed{w^* \propto \Sigma^{-1} \mu} \]

Investors with the same \(\mu\) and \(\Sigma\) hold the same portfolio direction and scale it with risk tolerance. With a financing asset paying \(r_f\), replace \(\mu\) by expected excess returns \(\mu-r_f\mathbf{1}\).

• High Risk Aversion (\(\lambda \uparrow\)): \(\|w^*\| \downarrow\). The investor holds a smaller position in the risky portfolio.

• Low Risk Aversion (\(\lambda \downarrow\)): \(\|w^*\| \uparrow\). The investor leverages up the risky portfolio.

Limiting Behavior

As \(\lambda \to 0^+\), the penalty for variance becomes negligible. The optimal portfolio takes increasingly large positions in the direction of \(\Sigma^{-1} \mu\). At \(\lambda = 0\), the unconstrained problem is generally unbounded because the objective reduces to maximizing expected return without any risk constraint.

Tangency Portfolio Normalization

The unconstrained mean-variance solution establishes that the optimal risky portfolio lies in the direction of \(\Sigma^{-1} \mu\). To identify the specific Tangency Portfolio (\(w_T\)) - the portfolio of risky assets where weights sum to one - we normalize this direction to satisfy:

\[ \boxed{\mathbf{1}^\top w_T = 1} \]

where \(\mathbf{1}\) is an \(N \times 1\) vector of ones.

Normalization identifies a portfolio on this direction. Solving the mean-variance objective subject to a budget constraint is a different optimization problem. Assume \(\mathbf{1}^\top\Sigma^{-1}\mu>0\) for the upper origin-tangency portfolio.

Derivation

Since \(w_T\) must be proportional to the optimal direction, we can write:

\[ w_T = c \Sigma^{-1} \mu \]

for some scalar constant \(c\). To find \(c\), we substitute this into the budget constraint:

\[ \mathbf{1}^\top w_T = \mathbf{1}^\top (c \Sigma^{-1} \mu) = c (\mathbf{1}^\top \Sigma^{-1} \mu) = 1 \]

Solving for \(c\):

\[ c = \frac{1}{\mathbf{1}^\top \Sigma^{-1} \mu} \]

Substituting \(c\) back into the expression for \(w_T\), we obtain the closed-form solution for the Tangency Portfolio weights:

\[ \boxed{w_T = \frac{\Sigma^{-1} \mu}{\mathbf{1}^\top \Sigma^{-1} \mu}} \]

Interpretation

With total expected returns in \(\mu\), this portfolio maximizes expected return divided by volatility. It is the maximum-Sharpe portfolio when the risk-free rate is zero. For a nonzero risk-free rate, use excess expected returns as below.

Two Interpretations

Total-Return / Origin Tangency Portfolio

If we consider the tangency point from the origin in mean-standard deviation space (maximizing the ratio of expected return to volatility, \(\mu_p / \sigma_p\)), the weights are derived by normalizing the unconstrained direction \(\Sigma^{-1}\mu\):

\[ \boxed{w_T = \frac{\Sigma^{-1} \mu}{\mathbf{1}^\top \Sigma^{-1} \mu}} \]

This portfolio represents the most “efficient” combination of risky assets when no risk-free asset is explicitly considered in the optimization objective.

Maximum-Sharpe Ratio Tangency Portfolio

In the presence of a risk-free asset with return \(r_f\), investors maximize the Sharpe Ratio. We instead work with excess expected returns:

\[ \tilde{\mu} = \mu - r_f \mathbf{1} \]

Assuming \(\mathbf{1}^\top\Sigma^{-1}\tilde{\mu}>0\), the fully invested maximum-Sharpe portfolio is:

\[ \boxed{w_T^{(rf)} = \frac{\Sigma^{-1} \tilde{\mu}}{\mathbf{1}^\top \Sigma^{-1} \tilde{\mu}}} \]

Geometric Interpretation: This portfolio \(w_T^{(rf)}\) is the point on the risky efficient frontier where the Capital Market Line (CML) - drawn from \((0, r_f)\) on the vertical axis - is tangent to the frontier. It represents the unique portfolio of risky assets that maximizes the Sharpe Ratio:

\[ \text{Sharpe Ratio} = \frac{w^\top \tilde{\mu}}{\sqrt{w^\top \Sigma w}} \]

Economic Intuition: What Drives Tangency Weights?

The direction \(\Sigma^{-1}\tilde{\mu}\) jointly depends on expected excess returns, variances, and covariances. Useful tendencies are:

1. Higher Mean Returns: Assets with larger components in \(\tilde{\mu}\) are favored.

2. Lower Volatility: Assets with smaller variances (diagonal elements of \(\Sigma\)) are favored.

3. Lower Covariance: Assets that have low covariance with the rest of the portfolio are favored.

The Role of Sharpe Ratios vs. Diversification

Points (1) and (2) relate directly to an asset’s individual Sharpe Ratio. A standalone Sharpe ratio alone does not determine an asset’s weight. The inverse covariance matrix and portfolio normalization matter; these tendencies are not general monotonic rules for normalized weights.

However, Point (3) highlights the critical role of covariance in mean-variance optimization. The term \(\Sigma^{-1}\) acts as a “diversification filter.”

• Diversification Benefit: An asset with a relatively poor standalone Sharpe ratio can still receive a significant weight if it has low (or negative) correlation with other assets in the universe.

• Correlation Penalty: Conversely, an asset with a high standalone Sharpe ratio may be downweighted or even shorted if it is highly correlated with other assets already held in the portfolio.

Thus, the tangency portfolio does not simply rank assets by their individual risk-adjusted returns; it optimizes the marginal contribution to the portfolio’s overall Sharpe ratio, balancing return potential against systemic redundancy.

Global Minimum-Variance Portfolio

The Global Minimum-Variance (GMV) portfolio is the fully invested portfolio that achieves the lowest possible variance, regardless of expected returns. It represents the leftmost point on the efficient frontier in mean-variance space.

Optimization Problem

We solve the following quadratic programming problem:

\[ \boxed{\min_w \quad \frac{1}{2} w^\top \Sigma w} \]

subject to the full-investment budget constraint:

\[ \boxed{\mathbf{1}^\top w = 1} \]

where \(\mathbf{1}\) is an \(N \times 1\) vector of ones. The factor of \(\frac{1}{2}\) is included for algebraic convenience.

Lagrangian Derivation

The Lagrangian for this constrained optimization problem is:

\[ \mathcal{L}(w, \gamma) = \frac{1}{2} w^\top \Sigma w - \gamma (\mathbf{1}^\top w - 1) \]

where \(\gamma\) is the Lagrange multiplier. Taking the first-order conditions (FOCs):

1. With respect to \(w\):

   \[ \frac{\partial \mathcal{L}}{\partial w} = \Sigma w - \gamma \mathbf{1} = 0 \implies w = \gamma \Sigma^{-1} \mathbf{1} \]

2. With respect to \(\gamma\):

   \[ \frac{\partial \mathcal{L}}{\partial \gamma} = -(\mathbf{1}^\top w - 1) = 0 \implies \mathbf{1}^\top w = 1 \]

Solving for Optimal Weights

Substitute the expression for \(w\) into the budget constraint:

\[ \mathbf{1}^\top (\gamma \Sigma^{-1} \mathbf{1}) = 1 \implies \gamma (\mathbf{1}^\top \Sigma^{-1} \mathbf{1}) = 1 \]

Define the scalar constant \(A\) as:

\[ \boxed{A \equiv \mathbf{1}^\top \Sigma^{-1} \mathbf{1}} \]

Solving for the Lagrange multiplier \(\gamma\):

\[ \gamma = \frac{1}{A} \]

Finally, substituting \(\gamma\) back into the expression for \(w\), we obtain the closed-form solution for the GMV portfolio weights:

\[ \boxed{w_{GMV} = \frac{\Sigma^{-1} \mathbf{1}}{\mathbf{1}^\top \Sigma^{-1} \mathbf{1}} = \frac{1}{A} \Sigma^{-1} \mathbf{1}} \]

Interpretation

Notice that \(w_{GMV}\) depends only on the covariance matrix \(\Sigma\) and not on the expected returns \(\mu\). The GMV portfolio avoids estimating expected returns, but remains sensitive to covariance estimation error and an ill-conditioned covariance matrix.

Two-Fund Separation

The GMV and origin-tangency portfolios span the minimum-variance frontier when the two portfolios are distinct and well-defined. Here \(\mu\) again denotes total expected returns.

Consider the fully invested mean–variance problem for a target portfolio return \(\mu_p\):

\[ \min_w \quad \frac{1}{2} w^\top \Sigma w \]

subject to:

\[ w^\top \mu = \mu_p, \qquad w^\top \mathbf{1} = 1 \]

Define the following scalar constants derived from the inverse covariance matrix:

\[ A = \mathbf{1}^\top \Sigma^{-1} \mathbf{1}, \quad B = \mathbf{1}^\top \Sigma^{-1} \mu, \quad C = \mu^\top \Sigma^{-1} \mu \]

Cauchy-Schwarz gives \(D \geq 0\). Assume expected returns are not identical across assets, so \(D>0\), and assume \(B>0\) for the upper origin-tangency portfolio:

\[ \boxed{D = AC - B^2} \]

Solving the Lagrangian yields the optimal weights as a function of the target return \(\mu_p\):

\[ \boxed{w^*(\mu_p) = \frac{A\mu_p - B}{D} \Sigma^{-1} \mu + \frac{C - B\mu_p}{D} \Sigma^{-1} \mathbf{1}} \]

The Two-Fund Structure

The critical observation is that every MV-efficient portfolio is a linear combination of only two fundamental directions: \(\Sigma^{-1} \mu\) and \(\Sigma^{-1} \mathbf{1}\). These correspond precisely to our previously derived special portfolios:

\[ w_T = \frac{\Sigma^{-1} \mu}{B}, \qquad w_{GMV} = \frac{\Sigma^{-1} \mathbf{1}}{A} \]

Therefore, every portfolio on the mean–variance frontier can be expressed as an affine combination (convex only when \(0 \leq \delta \leq 1\)):

\[ \boxed{w^*(\mu_p) = \delta \, w_T + (1 - \delta) w_{GMV}} \]

This result is known as Two-Fund Separation.

Solving for the Mixing Parameter \(\delta\)

Recall the expected returns of the two basis portfolios:

\[ \mu_{GMV} = \frac{B}{A}, \qquad \mu_T = \frac{C}{B} \]

Since expected return is linear in weights, the target return satisfies:

\[ \mu_p = \delta \mu_T + (1 - \delta) \mu_{GMV} \]

Solving for \(\delta\):

\[ \boxed{\delta = \frac{\mu_p - \mu_{GMV}}{\mu_T - \mu_{GMV}}} \]

Equivalently, in terms of the scalar constants:

\[ \boxed{\delta = \frac{B(A\mu_p - B)}{D}} \]

Interpretation of \(\delta\)

Under the assumptions above, \(\delta\) determines the location on the minimum-variance frontier:

• \(\delta = 0\): \(w^* = w_{GMV}\) (Global Minimum Variance portfolio).

• \(\delta = 1\): \(w^* = w_T\) (Origin-tangency portfolio; maximum Sharpe when \(r_f=0\)).

• \(0 < \delta < 1\): Portfolio lies on the efficient frontier segment between GMV and Tangency.

• \(\delta > 1\): Investor leverages beyond the tangency portfolio (long \(>100\%\) of \(w_T\), short \(w_{GMV}\)) to achieve higher expected return.

• \(\delta < 0\): Portfolio lies on the inefficient portion of the frontier below GMV.