Chapter 3: Mathematics of the Portfolio Frontier

Section 3.1: Second-Degree Stochastic Dominance (SSD)

Core Idea

• Asset A must offer at least the same expected return as Asset B
• But with less risk (measured by variance)

This leads us to portfolio selection: among all portfolios offering the same expected return, the dominant one is the one with the least variance.

What Is Second-Degree Stochastic Dominance (SSD)?

SSD is a decision rule for risk-averse investors.

Let \( F_A(x) \) and \( F_B(x) \) be cumulative distribution functions (CDFs) of the returns of assets A and B.

We say A dominates B in the second degree if:

...and strictly less for at least one \( x \). This reflects preference under concave utility.

Interpretation

• For risk-averse agents, less variance is better, holding expected return constant.
• If Asset A has equal mean but lower variance, then it’s preferred under SSD.

In Portfolio Context

Suppose you have many portfolios, all yielding \( E[R] = 8\% \).

SSD says: pick the one with the lowest variance — that’s the efficient frontier logic in Markowitz’s framework.

Visual

Risk (Variance) →→→

Portfolio A (Low Variance)
|
|     •
|
|
|          • Portfolio B (Higher Variance)
|
----------------------------
         Same Expected Return
  

Section 3.2: Mean-Variance Model and Investor Preferences

1. The Mean-Variance Model: Origins and Essence

This refers to the Modern Portfolio Theory (MPT) introduced by Harry Markowitz, which proposes that investors choose portfolios based on just two things:

• Expected Return: \( E[\tilde{R}] \)
• Variance (or standard deviation) of return: \( \text{Var}(\tilde{R}) \) or \( \sigma(\tilde{R}) \)

Core Assumptions

• Rationality: Investors are logical and consistent
• Risk Aversion: Investors dislike uncertainty
• Utility depends only on mean and variance: No concern for skewness or kurtosis
• Normally distributed returns: Ensures mean and variance fully define the distribution

Visualization: Efficient Frontier

Expected Return ↑
                • A
              •
            •      ← Efficient Frontier
          •
        •
      •
      ----------------------------→ Risk (Standard Deviation)
  

• Efficient Frontier: Set of portfolios that maximize expected return for a given level of risk
• Investors pick a point on the frontier, depending on their risk preference

2. Why Mean-Variance? Utility Theory Justification

The preferences assumed in MPT come naturally from standard utility theory:

• Monotonicity: More wealth → higher utility
• Strict concavity: Diminishing marginal utility of wealth → aversion to risk

Mathematically:

• \( u'(W) > 0 \): increasing
• \( u''(W) < 0 \): concave → risk aversion

These properties justify:

3. The Limitation: Only Works for Special Cases

This is a critical caveat. The mean and variance only capture the first two moments of a distribution. But:

• Real-world returns often have skewness (3rd moment) and kurtosis (4th moment)
• Some utility functions are not well approximated by a second-order Taylor expansion

Example

Consider two investments with the same mean and variance, but one has fat left tails (higher crash risk). MPT would treat them equally — but any real investor would prefer the safer one.

Visual: Two Distributions With Same Mean/Variance

Return →
            |              *
Probability |       *   *   *
            |    *           *     ← Fat right tail
            | *               *
            |------------------------→
                  Mean
  

Return →
            |     *       *
Probability |  *             *
            |*                 *   ← Fat left tail (crash risk)
            |------------------------→
                  Mean
  

These two have the same \( E[\tilde{R}] \) and \( \text{Var}(\tilde{R}) \), but very different risks.

4. Why Still Use It? Analytical Tractability and Empirical Power

Analytical Tractability

• Easy to compute: only need means, variances, and covariances
• Optimization becomes quadratic programming: solvable even for large portfolios
• Leads to closed-form solutions in many cases

Rich Empirical Implications

• Forms the basis for:
  • Capital Market Line (CML)
  • Capital Asset Pricing Model (CAPM)
  • Sharpe ratio, portfolio beta, etc.
• Despite its simplifications, MPT often gives good first approximations and predictive power in real markets

Summary Table

• Origin: Markowitz (1952), assumes decisions based on \( E[\tilde{R}], \text{Var}(\tilde{R}) \)
• Justification: Comes from monotonic and concave utility functions
• Limitation: Not valid for all distributions or utility functions
• Why Still Popular?: Simple math, strong predictive models (CAPM, efficient frontier)

Final Thoughts

This section reminds us that the mean-variance model is a practical approximation, not a universal truth. It's powerful because it's simple and useful, not because it's always accurate. Recognizing both its utility and its limits is key to using it wisely in both theoretical and applied finance.

Section 3.3: Utility Function Expansion Using Taylor Series

1. Context: Why Expand Utility Using Taylor Series?

Investors care about their expected utility, not just expected wealth. But computing E[u(~W)] for an arbitrary utility function u and wealth distribution ~W is hard.

To make it tractable, we approximate u(~W) by expanding it around its mean E[~W] using a Taylor series.

2. The Taylor Expansion

Let ~W be the random end-of-period wealth. The utility function is expanded as:

u(~W) = u(E[~W]) + u'(E[~W])(~W - E[~W]) 
+ (1/2)u''(E[~W])(~W - E[~W])² + R₃

Breakdown of Terms:

• u(E[~W]): Utility of the expected wealth — baseline
• u'(E[~W])(~W - E[~W]): Linear sensitivity — vanishes under expectation
• (1/2)u''(E[~W])(~W - E[~W])²: Penalty for risk (variance)
• R₃: Remainder term — captures skewness, kurtosis, etc.

3. Taking Expectation

Now take the expectation:

E[u(~W)] = u(E[~W]) + (1/2)u''(E[~W])Var(~W) + E[R₃]

• First-order term vanishes: E[~W - E[~W]] = 0

Intuition:

• Second-order term reflects penalty for variance
• If u'' < 0, more variance reduces utility
• Remainder term captures higher-moment effects

4. What Is the Remainder Term R₃?

The remainder term is:

R₃ = ∑ (from n=3 to ∞) [1/n!] × u⁽ⁿ⁾(E[~W]) × (~W - E[~W])ⁿ

E[R₃] = ∑ (from n=3 to ∞) [1/n!] × u⁽ⁿ⁾(E[~W]) × m⁽ⁿ⁾(~W)

• u⁽ⁿ⁾: n-th derivative of utility
• m⁽ⁿ⁾(~W): n-th central moment of wealth

Central Moments Summary:

• 1st: 0
• 2nd: Variance
• 3rd: Skewness
• 4th: Kurtosis

5. Interpretation of Equation

E[u(~W)] ≈ u(E[~W]) + (1/2)u''(E[~W])Var(~W)

• Utility increases with expected wealth
• Utility decreases with variance if u'' < 0

Implication:

• Preference for mean
• Aversion to variance

This justifies mean-variance analysis in portfolio theory.

6. But... Expected Utility Is Not Always Mean-Variance Based

• If higher moments matter, mean-variance is insufficient
• Full distribution is relevant for general utility functions

Example:

• Asset A: positive skew
• Asset B: negative skew

Same mean and variance, but different investor preferences—mean-variance analysis fails to distinguish.

7. Visualizing the Concepts

Utility Curve and Risk Aversion

Utility ↑
        •
     •     ← u(E[W])
  •
•
---------------------→ Wealth
         ↑
       E[W]

• Concave curve → diminishing marginal utility
• Variance reduces expected utility

Skewed Distributions With Same Mean/Variance

        •       •
      •   •   •   •
    •       •       •
------------------------→ Return

Mean and variance are the same, but tail behavior differs.

8. Final Takeaways

• Taylor expansion makes expected utility manageable
• Mean and variance are not always sufficient
• Higher-order moments like skewness and kurtosis often matter
• Mean-variance preference is a simplification, not a general truth

🔷 Section 3.4: Mean-Variance from Quadratic Utility

🧠 The Idea

The Taylor series remainder term E[R₃] from Section 3.3 vanishes if:

• u⁽ⁿ⁾(W) = 0 for n ≥ 3

This holds when the utility function is quadratic:

u(W) = aW - (b/2)W², with b > 0

Then: u'(W) = a - bW, u''(W) = -b, and u⁽ⁿ⁾(W) = 0 for n ≥ 3

✅ Result

Since all higher-order derivatives are zero:

E[u(Ẇ)] = u(E[Ẇ]) + (1/2)u''(E[Ẇ])Var(Ẇ)

This expression is exact.

🔍 Rewriting Expected Utility

Given: u(Ẇ) = Ẇ - (b/2)Ẇ²

So: E[u(Ẇ)] = E[Ẇ] - (b/2)E[Ẇ²]

But E[Ẇ²] = (E[Ẇ])² + Var(Ẇ) = μ² + σ²

Therefore: E[u(Ẇ)] = μ - (b/2)(μ² + σ²)

🎯 Implication

Utility depends only on the mean and variance of wealth Ẇ. Mean-variance analysis is therefore exact under quadratic utility.

⚠️ Problems with Quadratic Utility

1. Satiation

• Utility becomes decreasing for large W
• u'(W) = a - bW < 0 when W > a/b
• Implies more wealth leads to less utility, which contradicts rational investor behavior

2. Increasing Absolute Risk Aversion (IARA)

• ARA: A(W) = -u''(W)/u'(W) = b / (a - bW)
• Since the denominator decreases with W, A(W) increases with wealth
• This contradicts the usual assumption that richer individuals are less risk-averse

❌ Counterintuitive Implications

• Richer individuals may avoid risk entirely
• Beyond a point, people may prefer losing wealth
• Risky assets could appear as inferior goods

📊 Visual: Quadratic Utility Function

u(W)
│
│        *
│     *     *   ← Utility increases initially
│   *         *
│ *             *  ← Peaks at satiation point
│
└──────────────────→ W
                 ↑
            Max utility point

🔶 Section 3.5: Mean-Variance from Multivariate Normality

🧠 Key Idea

Even with an arbitrary utility function, if returns are normally distributed, mean-variance analysis can still be valid.

📌 Why?

• The normal distribution is fully determined by mean and variance
• Higher moments are functions of just μ and σ²
• Hence, the Taylor remainder term E[R₃] depends only on μ and σ²

🧮 Portfolio Implication

If: Ṙ = (R₁, R₂, ..., Rₙ) ~ MVN(μ, Σ)

Then: Any linear combination Ẇ = wᵗṘ is also normally distributed

So utility maximization reduces to optimization over mean and variance

✅ Advantages of Normality

• Simplicity: Optimization is over μ and Σ
• Stability: Sums of normals remain normal
• Compatibility: Applies to many utility functions if E[u(Ẇ)] exists

⚠️ Problems with Normality Assumption

1. Unbounded Below

• Normal returns allow Ẇ < 0 with positive probability
• For log utility u(z) = ln(z), ln(z) is undefined if z ≤ 0
• Negative wealth implies negative consumption, which is unrealistic

2. Violates Limited Liability

• In reality, wealth cannot fall below zero
• But normal models assume returns can go to negative infinity

📘 Lognormal Alternative?

• Lognormal returns (Ẇ = exp(Z), Z ~ N(μ, σ²)) ensure Ẇ > 0
• But sums of lognormals are not lognormal
• This complicates portfolio construction

📌 Summary Table

🔁 Final Synthesis

Last updated: August 04, 2025