layout: default title: Portfolios of Multiple Assets ————————————

Portfolios of Multiple Assets

Sukrit Mittal Franklin Templeton Investments

Outline

  1. From two assets to many
  2. Portfolio return and risk (general case)
  3. Geometry of diversification
  4. Three risky securities
  5. Minimum variance portfolio (MVP)
  6. Minimum variance set and line
  7. Introducing the market portfolio
  8. Exercises

1. From Two Assets to Many

The two-asset case taught us geometry.

The multi-asset case teaches us structure.

Nothing fundamentally new appears.

What changes is notation—and discipline.

Why This Matters

Real portfolios:

  • Contain many assets
  • Are constrained by budgets
  • Must be optimized systematically

Guesswork does not scale.

Linear algebra does.

2. Portfolio Return: General Case

Let:

  • $n$ risky assets
  • returns $R = (R_1, \dots, R_n)^\top$
  • weights $w = (w_1, \dots, w_n)^\top$

Portfolio return:

\[R_p = w^\top R\]

Budget constraint:

\[\sum_{i=1}^n w_i = 1\]

Expected Return

Let $\mu = \mathbb{E}[R]$.

Then:

\[\mathbb{E}[R_p] = w^\top \mu\]

Linearity survives dimension.

Risk will not be so polite.

3. Portfolio Risk: Variance

Let $\Sigma$ be the covariance matrix:

\[\Sigma_{ij} = \text{Cov}(R_i, R_j)\]

Portfolio variance:

\[\sigma_p^2 = w^\top \Sigma w\]

This single quadratic form drives modern portfolio theory.

Properties of the Covariance Matrix

  • Symmetric
  • Positive semi-definite

These are not technicalities.

They guarantee meaningful optimization problems.

4. Geometry of Diversification

Expected return is linear in $w$.

Variance is quadratic in $w$.

This asymmetry creates:

  • Curvature
  • Trade-offs
  • Efficient frontiers

Diversification is geometry in disguise.

5. Three Risky Securities

With three risky assets:

  • The weight space is two-dimensional
  • The attainable set fills an area

Yet the efficient set collapses to a curve.

More choice does not mean more freedom.

Return and Risk

For three assets:

\[R_p = w_1R_1 + w_2R_2 + w_3R_3\] \[\sigma_p^2 = w^\top \Sigma w\]

With $w_1 + w_2 + w_3 = 1$.

Same equations. Higher dimension.

6. Minimum Variance Portfolio (MVP)

The minimum variance portfolio solves:

\[\min_w ; w^\top \Sigma w\]

subject to:

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

Expected returns play no role here.

Risk comes first.

Solution via Lagrange Multipliers

Lagrangian:

\[\mathcal{L}(w,\lambda) = w^\top \Sigma w + \lambda(\mathbf{1}^\top w - 1)\]

First-order condition:

\[2\Sigma w + \lambda \mathbf{1} = 0\]

This is linear algebra, not magic.

Closed-Form Expression

Solving yields:

\[w^{\text{MVP}} = \frac{\Sigma^{-1}\mathbf{1}}{\mathbf{1}^\top \Sigma^{-1}\mathbf{1}}\]

This formula appears everywhere for a reason.

7. Minimum Variance Set and Line

As we vary expected return constraints:

\[w^\top \mu = \mu_p\]

we obtain a family of portfolios.

Their image in $(\sigma, \mu)$ space forms the minimum variance frontier.

Minimum Variance Line

Without a risk-free asset:

  • The efficient set is a curve
  • Only the upper branch is relevant

Below the MVP, portfolios are dominated.

Efficiency is selective.

8. Adding the Risk-Free Asset (Recap)

Once a risk-free asset is available:

  • The efficient frontier becomes a straight line
  • Only one risky portfolio matters

This line dominates all others.

Geometry collapses again.

9. Market Portfolio

The market portfolio is:

  • The tangency portfolio
  • The risky portfolio with the highest Sharpe ratio

It solves:

\[\max_w \frac{w^\top \mu - R_f}{\sqrt{w^\top \Sigma w}}\]

Subject to $\mathbf{1}^\top w = 1$.

Characterization of the Market Portfolio

The solution satisfies:

\[w^{\text{M}} \propto \Sigma^{-1}(\mu - R_f\mathbf{1})\]

Preferences disappear.

Markets choose the risky portfolio.

Interpretation

  • Everyone holds the same risky portfolio
  • Differences arise only through leverage

This result is fragile empirically.

But foundational theoretically.

10. Exercises

Exercise 1

Given three assets with returns $\mu$ and covariance matrix $\Sigma$:

  • Compute the minimum variance portfolio
  • Compute its variance

Exercise 2

Show that the MVP weights sum to one.

Explain why this matters economically.

Exercise 3

Given a risk-free rate $R_f$, derive the market portfolio.

Explain why it does not depend on risk aversion.

Final Takeaways

  • Multi-asset portfolios require linear algebra
  • Expected return is linear, risk is quadratic
  • The MVP anchors the efficient frontier
  • The risk-free asset collapses the frontier
  • The market portfolio emerges naturally

Next, we will turn these results into asset pricing.