5.2 Modern Portfolio Theory, CAPM, and Risk-Adjusted Metrics

Modern Portfolio Theory, CAPM, and Risk-Adjusted Metrics

Quick Answer: Modern Portfolio Theory (MPT), developed by Harry Markowitz, proves that a portfolio's total risk can be reduced without sacrificing expected return by combining assets with low or negative correlation ($\rho < +1.0$). The Capital Asset Pricing Model (CAPM) prices assets based solely on systematic risk (Beta, $\beta$). Fiduciaries evaluate managers using risk-adjusted metrics: the Sharpe Ratio measures excess return per unit of total risk ($\sigma$), the Treynor Ratio evaluates excess return per unit of systematic risk ($\beta$), Jensen's Alpha measures abnormal return relative to CAPM expectations, and the Sortino Ratio focuses exclusively on harmful downside volatility.

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1. Modern Portfolio Theory (MPT) & The Mathematics of Diversification

Prior to Harry Markowitz's 1952 breakthrough, investment analysis evaluated securities in isolation. Markowitz proved that an asset's individual volatility is far less important than how it co-moves with other assets in a portfolio.

Expected Return of a Portfolio

The expected return of a multi-asset portfolio, $E(R_p)$, is simply the weighted average of the expected returns of individual holdings:

$E(R_p) = \sum_{i=1}^{n} w_i E(R_i) = w_1 E(R_1) + w_2 E(R_2) + \dots + w_n E(R_n)$

Where $w_i$ is the portfolio weight of asset $i$, and $\sum w_i = 1.0$.

Portfolio Variance and Standard Deviation (Two-Asset Case)

Unlike return, portfolio variance ($\sigma_p^2$) is not a simple weighted average. It includes the covariance between the assets:

$\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \text{Cov}(1,2)$

Since covariance equals the product of individual standard deviations and the correlation coefficient ($\text{Cov}(1,2) = \sigma_1 \sigma_2 \rho_{1,2}$), the equation can be expressed as:

$\sigma_p = \sqrt{w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \sigma_1 \sigma_2 \rho_{1,2}}$

┌─────────────────────────────────────────────────────────────────────────────┐
│                     IMPACT OF THE CORRELATION COEFFICIENT                   │
├─────────────────┬───────────────────────────────────────────────────────────┤
│ Correlation (ρ) │ Diversification Benefit & Portfolio Volatility            │
├─────────────────┼───────────────────────────────────────────────────────────┤
│ ρ = +1.0        │ Perfectly Positively Correlated. Zero diversification     │
│ (Maximum Risk)  │ benefit; portfolio σ is the weighted average of assets.   │
├─────────────────┼───────────────────────────────────────────────────────────┤
│ ρ = 0.0         │ Uncorrelated. Substantial risk reduction; portfolio σ is  │
│ (Moderate Risk) │ significantly lower than the weighted average of assets.  │
├─────────────────┼───────────────────────────────────────────────────────────┤
│ ρ = -1.0        │ Perfectly Inversely Correlated. Complete risk elimination;│
│ (Minimum Risk)  │ portfolio σ can be driven to zero with optimal weights.   │
└─────────────────┴───────────────────────────────────────────────────────────┘

Step-by-Step Diversification Calculation

Consider a trust allocating 60% to Asset A ($E(R_A) = 10%, \sigma_A = 16%$) and 40% to Asset B ($E(R_B) = 6%, \sigma_B = 10%$):

1. Expected Portfolio Return: $E(R_p) = (0.60 \times 10\%) + (0.40 \times 6\%) = 6.0\% + 2.4\% = 8.4\%$
2. Portfolio Risk with Perfect Correlation ($\rho = +1.0$): $\sigma_p = (0.60 \times 16\%) + (0.40 \times 10\%) = 9.6\% + 4.0\% = 13.6\%$
3. Portfolio Risk with Low Correlation ($\rho = +0.20$): $\sigma_p^2 = (0.60^2 \times 16^2) + (0.40^2 \times 10^2) + (2 \times 0.60 \times 0.40 \times 16 \times 10 \times 0.20)$ $\sigma_p^2 = (0.36 \times 256) + (0.16 \times 100) + (0.48 \times 160 \times 0.20) = 92.16 + 16.0 + 15.36 = 123.52$ $\sigma_p = \sqrt{123.52} = 11.11\%$

Key Takeaway: By combining assets with a correlation of $+0.20$, the portfolio standard deviation drops from $13.60%$ to $11.11%$ (an 18.3% risk reduction) with zero loss in expected return (8.4%). This is the mathematical core of Modern Portfolio Theory.

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2. The Efficient Frontier and Capital Market Line (CML)

Expected
Return E(R)
     ▲                  Capital Market Line (CML)
     │                   / (Slope = Sharpe Ratio)
     │                  / 
     │                 /• Tangency (Market) Portfolio
     │             .--'---
     │           .'   •   '--.  Efficient Frontier
     │          /             \
     │         │   Feasible    │
     │         │  Portfolios   │
     │Rf •────/─────────────────
     │       /
     └──────┴────────────────────────► Total Risk (σ)

• Efficient Frontier: The set of optimal portfolios in risk-return space that offer the maximum expected return for a specified level of risk, or the minimum risk for a specified expected return.
• Capital Allocation Line / Capital Market Line (CML): When a risk-free asset ($R_f$, such as U.S. Treasury bills) is introduced, investors can combine $R_f$ with the optimal risky "Market Portfolio" (the tangency point on the Efficient Frontier). The straight line extending from $R_f$ through the Market Portfolio dominates all other risky portfolios on the frontier.
• CML Equation: $E(R_p) = R_f + \left( \frac{E(R_m) - R_f}{\sigma_m} \right) \sigma_p$ Note: The CML applies only to efficient, fully diversified portfolios because the horizontal axis represents total risk ($\sigma_p$).

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3. Systematic vs. Unsystematic Risk

Portfolio
Risk (σ)
   ▲
   │ █
   │ ███  UNSYSTEMATIC RISK (Idiosyncratic / Diversifiable)
   │ █████   • Company lawsuits, product failures, CEO scandals
   │ ███████ • Can be eliminated through 20–30+ holdings
   │ ────────────────────────────────────────────────────────────
   │ ░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░
   │ ░░░░ SYSTEMATIC RISK (Market / Undiversifiable / Beta) ░░░░░
   │ ░░░░   • Interest rates, inflation, recessions, wars   ░░░░░
   │ ░░░░   • Cannot be diversified away; priced by CAPM    ░░░░░
   └────────────────────────────────────────────────────────► Number of Stocks
     1       5       10      15      20      25      30+

1. Systematic Risk (Market / Non-Diversifiable Risk): Macroeconomic factors affecting the broad financial system (interest rate changes, inflation shocks, recessions, geopolitical crises). Measured by Beta ($\beta$). Fiduciary portfolios cannot eliminate systematic risk through diversification; investors receive a risk premium for bearing it.
2. Unsystematic Risk (Specific / Diversifiable / Idiosyncratic Risk): Microeconomic risks unique to an individual company or narrow industry (patent expirations, management fraud, strikes, earnings misses). Holding 25 to 30+ non-correlated stocks eliminates over 90% of unsystematic risk.

UPIA § 3 Exam Mandate: Because the market provides no risk premium or extra return for bearing unsystematic risk, a fiduciary who fails to diversify a concentrated stock holding violates the prudent investor rule unless special circumstances (e.g., family enterprise, tax concentration drag) explicitly justify retention.

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4. The Capital Asset Pricing Model (CAPM) and Security Market Line (SML)

The Capital Asset Pricing Model (CAPM) determines the theoretical expected rate of return for any individual asset or portfolio based exclusively on its systematic risk (Beta).

The CAPM Equation

$E(R_i) = R_f + \beta_i \left[ E(R_m) - R_f \right]$

Where:

• $E(R_i)$ = Expected return on asset $i$
• $R_f$ = Risk-free rate of return (e.g., 3-month Treasury bill yield)
• $\beta_i$ = Beta of asset $i$ (sensitivity to market movements)
• $E(R_m)$ = Expected return of the overall market (e.g., S&P 500)
• $[E(R_m) - R_f]$ = Market Risk Premium (Equity Risk Premium)

Beta ($\beta$) Mechanics

$\beta_i = \frac{\text{Cov}(R_i, R_m)}{\sigma_m^2} = \rho_{i,m} \left( \frac{\sigma_i}{\sigma_m} \right)$

• $\beta = 1.0$: Asset moves in lockstep with the market.
• $\beta > 1.0$: High sensitivity / aggressive asset (e.g., $\beta = 1.35$ implies the asset is expected to gain 13.5% when the market gains 10%, but drop 13.5% when the market drops 10%).
• $\beta < 1.0$: Low sensitivity / defensive asset (e.g., $\beta = 0.70$ implies utility stocks).
• $\beta = 0.0$: Uncorrelated with the market (Risk-free asset).

Worked CAPM Calculation

A trust portfolio manager considers an equity holding with $\beta = 1.25$. The 3-month Treasury bill rate ($R_f$) is $4.0%$ and the expected market return ($E(R_m)$) is $10.0%$:

1. Market Risk Premium = $10.0% - 4.0% = 6.0%$
2. Expected Return: $E(R_i) = 4.0% + 1.25 \times (6.0%) = 4.0% + 7.5% = 11.5%$

CML vs. SML Comparison

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5. Risk-Adjusted Performance Measures

To evaluate investment managers under UPIA § 9 and OCC Regulation 9, fiduciaries must analyze risk-adjusted performance using specific mathematical metrics:

┌─────────────────────────────────────────────────────────────────────────────┐
│               RISK-ADJUSTED PERFORMANCE METRICS COMPARATIVE MATRIX          │
├──────────────┬───────────────────────────────┬──────────────┬───────────────┤
│ Metric       │ Formula                       │ Risk Measure │ Ideal Use     │
├──────────────┼───────────────────────────────┼──────────────┼───────────────┤
│ **Sharpe**   │ (Rp - Rf) / σp                │ Total Risk   │ Entire/Total  │
│ **Ratio**    │                               │ (Std Dev, σ) │ Portfolios    │
├──────────────┼───────────────────────────────┼──────────────┼───────────────┤
│ **Treynor**  │ (Rp - Rf) / βp                │ Systematic   │ Diversified   │
│ **Ratio**    │                               │ Risk (Beta, β│ Sub-Managers  │
├──────────────┼───────────────────────────────┼──────────────┼───────────────┤
│ **Jensen's** │ Rp - [Rf + βp(Rm - Rf)]       │ Systematic   │ Value-Add /   │
│ **Alpha (α)**│                               │ Risk (Beta, β│ Excess Return │
├──────────────┼───────────────────────────────┼──────────────┼───────────────┤
│ **Sortino**  │ (Rp - Target) / Semi-Dev      │ Downside     │ Conservative /│
│ **Ratio**    │                               │ Risk (σ_d)   │ Income Trusts │
├──────────────┼───────────────────────────────┼──────────────┼───────────────┤
│ **Info**     │ (Rp - Rb) / Tracking Error    │ Active Risk  │ Active vs.    │
│ **Ratio (IR)│                               │ (σ_(Rp - Rb))│ Benchmark     │
└──────────────┴───────────────────────────────┴──────────────┴───────────────┘

1. Sharpe Ratio

$S = \frac{R_p - R_f}{\sigma_p}$ Measures the excess return per unit of total risk. Because it uses standard deviation ($\sigma_p$), it penalizes the portfolio for both systematic and unsystematic risk. It is the primary metric when evaluating an investor's entire wealth portfolio.

2. Treynor Ratio

$T = \frac{R_p - R_f}{\beta_p}$ Measures the excess return per unit of systematic risk (Beta). It assumes unsystematic risk has already been diversified away. It is the preferred metric when evaluating a specialized manager or asset-class sleeve that will be combined into a broadly diversified total portfolio.

3. Jensen's Alpha ($\alpha$)

$\alpha = R_p - \left[ R_f + \beta_p (R_m - R_f) \right]$ Calculates the absolute rate of return generated by the manager over and above the return predicted by CAPM for the portfolio's level of Beta.

• $\alpha > 0$: Manager generated superior risk-adjusted return (positive value-add).
• $\alpha = 0$: Manager performed exactly in line with CAPM expectations.
• $\alpha < 0$: Manager underperformed after accounting for systematic risk.

4. Information Ratio (IR)

$\text{IR} = \frac{R_p - R_b}{\text{Tracking Error}} = \frac{R_p - R_b}{\sigma_{(R_p - R_b)}}$ Measures the consistency of active manager excess return over a benchmark ($R_b$) per unit of active risk (Tracking Error). An $\text{IR} \ge 0.50$ is considered good; $\text{IR} \ge 1.0$ is exceptional.

5. Sortino Ratio

$\text{Sortino} = \frac{R_p - \text{MAR}}{\sigma_d}$ Where $\text{MAR}$ is the Minimum Acceptable Return (or $R_f$) and $\sigma_d$ is the downside semi-deviation (volatility of negative returns only). Unlike the Sharpe Ratio, Sortino does not penalize a manager for large upside volatility.

6. Coefficient of Determination ($R^2$)

$R^2$ measures the percentage of a portfolio's return movements explained by movements in the benchmark index (ranging from 0.0 to 1.0, or 0% to 100%).

Exam Rule: If $R^2 < 0.70$ (70%), the manager's Beta is statistically unreliable. In such cases, Treynor Ratio and Jensen's Alpha are invalid, and the fiduciary must rely on the Sharpe Ratio to evaluate risk-adjusted performance.

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6. Macroeconomic and Market Concepts (Blueprint Domain 5)

Fiduciary asset allocation never operates in a vacuum; the Asset Types and Management domain explicitly tests the macro environment:

• Economic Cycles: economies rotate through expansion, peak, contraction, and trough. Sector leadership rotates with the cycle — defensives (utilities, consumer staples, health care) outperform late-cycle and in contraction, while cyclicals (industrials, discretionary, financials) lead early recoveries.
• Monetary Policy: the Federal Reserve steers short-term rates through the federal funds target, open-market operations, quantitative easing (QE) — balance-sheet asset purchases that suppress longer yields — and quantitative tightening (QT) runoff. Easing supports equity multiples and rewards longer bond duration; tightening does the reverse.
• Inflation: tracked by the Consumer Price Index (headline vs. core). Because inflation erodes real returns, real return ≈ nominal return − inflation; TIPS, short-duration credit, real assets, and farmland/real estate hedge purchasing power — connect this directly to the UPIA § 2(c) duty to weigh the expected effect of inflation or deflation.
• Fiscal Policy: government spending and taxation decisions (deficits, marginal-rate changes, stimulus programs) shape aggregate demand, sector outcomes, and interest-rate pressure.
• Gross Domestic Product (GDP): the total market value of finished goods and services produced in a period; real GDP measures output net of inflation, and a rule-of-thumb recession signal is two consecutive quarters of negative real GDP growth (the NBER Business Cycle Dating Committee makes the official determination).
• International Currency Markets: global portfolios carry FX exposure — a strengthening U.S. dollar reduces the dollar-translated value of foreign holdings and compresses multinationals' repatriated earnings; fiduciaries weigh hedged versus unhedged international mandates as a distinct risk decision layered on asset-class risk.