Risk-Free Asset, Preferences, and Optimization

Sukrit Mittal Franklin Templeton Investments

Outline

Why add a risk-free asset?
Portfolios with a risk-free security
Capital allocation line
Investor preferences and indifference curves
Optimal portfolio choice
Mathematical interlude: constrained optimization
Lagrange multipliers
Proofs and intuition

1. Why Add a Risk-Free Asset?

So far, all portfolios involved only risky assets.

That world is incomplete.

In reality, investors can always:

Park money safely
Borrow or lend at (approximately) risk-free rates

Ignoring this option distorts everything.

Conceptual Shift

With a risk-free asset:

Risk becomes optional
Leverage becomes possible
Portfolio choice separates into two problems

This separation is not an assumption.

It is a theorem.

2. Portfolio with One Risky Asset and One Risk-Free Asset

Let:

$R_f$ = risk-free return (constant)
$R$ = return on a risky portfolio
$w$ = fraction invested in risky asset

Portfolio return:

\[R_p = wR + (1-w)R_f\]

This is the simplest mixed portfolio.

Expected Return

\[\mathbb{E}[R_p] = w\mathbb{E}[R] + (1-w)R_f\]

Linear in $w$.

Nothing surprising yet.

Risk of the Portfolio

Since $R_f$ is constant:

\[\sigma_p = |w|\sigma\]

All risk comes from the risky asset.

The risk-free asset contributes none.

3. Capital Allocation Line (CAL)

Combining a risky portfolio with a risk-free asset traces a straight line in $(\sigma, \mu)$ space.

Equation:

\[\mathbb{E}[R_p] = R_f + \frac{\mathbb{E}[R] - R_f}{\sigma} , \sigma_p\]

This is the capital allocation line.

Interpretation of the CAL

Intercept: risk-free rate
Slope: Sharpe ratio

Steeper line = better risk-return trade-off.

Markets reward efficiency, not bravery.

4. Many Risky Assets + Risk-Free Asset

When many risky assets exist:

Investors first form an optimal risky portfolio
Then mix it with the risk-free asset

This is the two-fund separation theorem.

Preferences determine how much risk, not which risk.

5. Investor Preferences

To choose among portfolios, we need a model of preferences.

Finance borrows this from economics.

We assume:

More return is better
Less risk is better

Nothing exotic.

Mean–Variance Preferences

Preferences are represented by a utility function:

\[U(\mu, \sigma) = \mu - \frac{\gamma}{2}\sigma^2\]

Where $\gamma > 0$ measures risk aversion.

This is not psychology.

It is tractable mathematics.

6. Indifference Curves

An indifference curve consists of portfolios yielding the same utility.

Holding $U$ constant:

\[\mu = U + \frac{\gamma}{2}\sigma^2\]

This is a parabola in $(\sigma, \mu)$ space.

Properties of Indifference Curves

Upward sloping
Increasingly steep
Do not intersect

Higher curves correspond to higher utility.

Geometry replaces psychology.

7. Optimal Portfolio Choice

The investor chooses the portfolio where:

An indifference curve
Is tangent to the CAL

This tangency determines the optimal risk exposure.

Different investors, same risky portfolio.

Different mixing proportions.

Key Result

Risk tolerance affects:

How much to invest in risky assets

It does not affect:

Which risky portfolio to hold

This result is deep, old, and still misunderstood.

8. Mathematical Interlude

So far, we relied on geometry.

Next, we formalize this using constrained optimization.

This requires a new mathematical tool.

Enter Lagrange multipliers.

9. Motivating Example

Maximize:

\[f(x,y) = x^2 + y^2\]

Subject to:

\[x + y = 1\]

Unconstrained, this explodes.

The constraint changes everything.

Geometric Intuition

Level curves of $f$: circles
Constraint: straight line

The optimum occurs where:

A level curve is tangent to the constraint

Gradients align.

10. Lagrange Multiplier Method

Define the Lagrangian:

\[\mathcal{L}(x,y,\lambda) = f(x,y) + \lambda(g(x,y) - c)\]

First-order conditions:

\[\nabla f = -\lambda \nabla g\]

The multiplier enforces the constraint.

Economic Interpretation of $\lambda$

The Lagrange multiplier measures:

How much the objective improves if the constraint is relaxed.

In finance, this interpretation will matter.

A lot.

11. Constrained Extrema: General Case

Maximize $f(x)$ subject to $g(x)=0$.

Necessary condition:

\[\nabla f(x^*) = \lambda \nabla g(x^*)\]

This replaces unconstrained critical points.

Proof Sketch

At the optimum:

Movement along the constraint is allowed
Movement off the constraint is forbidden

Thus, directional derivatives along the constraint vanish.

Gradients must align.

That is the whole proof.

12. Why This Matters for Finance

Portfolio optimization is:

Maximization of expected utility
Subject to budget constraints

Every serious result ahead relies on this machinery.

Without Lagrange multipliers, modern finance collapses.

13. Exercises

Exercise 1

Derive the optimal weight $w$ for a portfolio combining a risky asset and a risk-free asset under mean–variance utility.

Exercise 2

Solve the constrained problem:

\[\max_{x,y} xy \quad \text{s.t.} \quad x^2 + y^2 = 1\]

Using Lagrange multipliers.

Exercise 3

Interpret the Lagrange multiplier in the context of a budget constraint.

Explain its economic meaning.

Final Takeaways

Adding a risk-free asset changes everything
Preferences are needed to choose among portfolios
Indifference curves formalize risk aversion
Optimal choice is a tangency condition
Lagrange multipliers are unavoidable

Next, we will unify all of this into full mean–variance optimization.