Portfolio Risk and Return - Part II
Learning Objectives Coverage
LO1: Describe the implications of combining a risk-free asset with a portfolio of risky assets
Core Concept
Combining a risk-free asset with a risky portfolio creates a linear risk-return relationship, enabling investors to achieve any desired risk level through leverage or deleveraging while maintaining optimal efficiency. This is a transformative insight: it allows separation of the investment decision (choosing the optimal risky portfolio, explored in Part I’s efficient frontier) from the financing decision (determining risk exposure). The two-fund separation theorem follows — all investors hold combinations of the same optimal risky portfolio and the risk-free asset, differing only in proportions based on their risk tolerance. risk-return
Mathematical Framework formula
Combined Portfolio Return:
E(Rc) = wf × Rf + (1 - wf) × E(Rp)
= Rf + (1 - wf)[E(Rp) - Rf]
Combined Portfolio Risk:
σc = (1 - wf) × σp
= |1 - wf| × σp (for leverage)
Where:
wf = weight in risk-free asset
1 - wf = weight in risky portfolio
Lending vs Borrowing Scenarios
Lending (wf > 0):
- Reduces risk and return
- 0 < wf < 1
- Example: 40% T-bills, 60% stocks
Borrowing (wf < 0):
- Increases risk and return
- wf < 0, (1 - wf) > 1
- Example: Borrow 50%, invest 150% in stocks
Practical Examples
-
Conservative Allocation (70% RF, 30% Risky):
Rf = 3%, E(Rp) = 12%, σp = 20% E(Rc) = 0.7(3%) + 0.3(12%) = 5.7% σc = 0.3 × 20% = 6% -
Leveraged Portfolio (Borrow 50%):
wf = -0.5, wp = 1.5 E(Rc) = 3% + 1.5(12% - 3%) = 16.5% σc = 1.5 × 20% = 30%
DeFi Application defi-application
In DeFi, the risk-free proxy is typically stablecoin lending — USDC in Aave (3—5% APY), the DAI Savings Rate (1—8% APY), or treasury-backed stablecoins. These serve the same role as T-bills in the traditional framework, anchoring the left end of the capital allocation line.
Leveraged DeFi strategies replicate the borrowing portfolio on the CAL. A user deposits ETH as collateral, borrows stablecoins at 5%, and uses those stablecoins to buy more ETH — amplifying both expected return and risk. This is conceptually identical to the leveraged portfolio example above, but with an added layer of liquidation risk that traditional leverage does not typically carry:
1. Deposit ETH as collateral
2. Borrow stablecoins at 5%
3. Buy more ETH (leverage)
4. Expected return amplified
5. Liquidation risk at high leverage
LO2: Explain the capital allocation line (CAL) and the capital market line (CML)
Core Concept exam-focus
The CAL represents all possible risk-return combinations from mixing a risk-free asset with any risky portfolio, while the CML is the special case where the market portfolio serves as the optimal risky portfolio. The CML represents the best possible CAL available to all investors, defining the efficient frontier when a risk-free asset exists and establishing the benchmark for portfolio performance. The CML dominates all other CALs because it is tangent to the Markowitz efficient frontier, and its slope equals the market’s Sharpe ratio. A key exam distinction: the CML applies only to efficient portfolios (total risk on x-axis), while the SML applies to all assets (beta on x-axis).
CAL Equation
General CAL:
E(Rp) = Rf + [(E(Ri) - Rf)/σi] × σp
Where:
- Slope = Sharpe ratio of risky portfolio
- Intercept = Risk-free rate
- Linear relationship
CML Equation
Capital Market Line:
E(Rp) = Rf + [(E(Rm) - Rf)/σm] × σp
Market Price of Risk:
Slope = [E(Rm) - Rf]/σm
Interpretation:
- Additional return per unit of risk
- Same for all efficient portfolios
Graphical Representation
Return
^
| CML (tangent to efficient frontier)
| /
| / Efficient Frontier
| / /
Rf |/__/
|
+-----------> Risk (σ)
σm
Numerical Examples
-
CML Construction:
Market: E(Rm) = 10%, σm = 16% Risk-free: Rf = 3% CML equation: E(Rp) = 3% + (7%/16%) × σp = 3% + 0.4375 × σp Portfolio with σp = 10%: E(Rp) = 3% + 0.4375(10%) = 7.375% -
CAL vs CML Comparison:
Suboptimal portfolio: E(R) = 9%, σ = 18% CAL slope = (9% - 3%)/18% = 0.333 Market portfolio: E(Rm) = 10%, σm = 16% CML slope = (10% - 3%)/16% = 0.4375 CML dominates (higher slope)
DeFi Application
- DeFi CML Construction:
DeFi Market Portfolio: - 40% BTC - 30% ETH - 20% DeFi tokens - 10% Stablecoins Expected return: 35% APY Volatility: 55% Risk-free (USDC): 4% DeFi CML slope = (35% - 4%)/55% = 0.564
LO3: Explain systematic and nonsystematic risk, including why an investor should not expect to receive additional return for bearing nonsystematic risk
Core Concept exam-focus
Systematic risk affects the entire market and cannot be diversified away, while nonsystematic risk is asset-specific and can be eliminated through diversification. Only systematic risk earns a risk premium. This distinction is one of the most important in all of finance: it explains why beta (not total volatility) determines expected returns via the CAPM, why diversification is crucial, and why investors are only compensated for risks they cannot avoid. The practical implication is that a well-diversified portfolio’s risk converges toward the systematic component, and the risk management focus should be on understanding and managing that exposure.
Risk Decomposition formula
Total Risk = Systematic Risk + Nonsystematic Risk
σi² = βi²σm² + σε²
Where:
- σi² = Total variance
- βi²σm² = Systematic variance
- σε² = Nonsystematic (residual) variance
R-squared:
R² = Systematic Risk / Total Risk = βi²σm² / σi²
Sources of Risk
Systematic Risk Sources:
- Interest rate changes
- Inflation shocks
- Economic cycles
- Political uncertainty
- Global pandemics
- Currency movements
Nonsystematic Risk Sources:
- Management changes
- Product recalls
- Lawsuits
- Labor strikes
- Accounting fraud
- Competition
Diversification Effect
Portfolio with n stocks:
σp² = (1/n)σ̄² + ((n-1)/n)C̄ov
As n → ∞:
σp² → C̄ov (average covariance)
Nonsystematic risk → 0
Only systematic risk remains
Practical Examples
-
Single Stock Risk Breakdown:
Apple Stock: Total volatility = 25% Beta = 1.2 Market volatility = 16% Systematic risk = 1.2² × 16%² = 36.86% of variance Nonsystematic = 63.14% of variance R² = 0.3686 -
Portfolio Diversification Path:
1 stock: σ = 35% 5 stocks: σ = 25% 20 stocks: σ = 18% 100 stocks: σ = 16.5% Market index: σ = 16% Limit = systematic risk only
Why No Premium for Nonsystematic Risk
- Easily Eliminated: Simple diversification removes it
- No Market Impact: Doesn’t affect overall economy
- Arbitrage Argument: If priced, create diversified portfolio for free lunch
- Competitive Markets: Investors won’t pay for unnecessary risk
DeFi Application defi-application
The systematic vs. nonsystematic distinction maps directly onto DeFi. Systematic DeFi risks — crypto market cycles, regulatory changes, Ethereum gas fees, and stablecoin depegs — affect the entire ecosystem and cannot be diversified away within crypto. These are the risks for which DeFi yields compensate investors. Nonsystematic DeFi risks — smart contract bugs, protocol hacks, governance attacks, and protocol-specific liquidity crises — can be substantially reduced through multi-protocol, multi-chain diversification.
Systematic DeFi Risks: Nonsystematic DeFi Risks:
- Crypto market cycles - Smart contract bugs
- Regulatory changes - Protocol hacks
- Ethereum gas fees - Governance attacks
- Stablecoin depegs - Liquidity crises
Diversification Strategy:
- Multiple protocols → reduces smart contract risk
- Different chains → reduces infrastructure risk
- Various strategies → reduces strategy-specific risk
This framework explains why spreading capital across Aave, Compound, MakerDAO, and Curve reduces overall risk — the protocol-specific risks are imperfectly correlated and wash out, leaving only the systematic DeFi exposure that earns the risk premium.
LO4: Explain return generating models (including the market model) and their uses
Core Concept
- Definition: Return generating models are statistical representations that explain asset returns as functions of one or more systematic factors plus an idiosyncratic component, with the market model being the simplest single-factor version
- Why it matters: These models provide frameworks for understanding return drivers, estimating expected returns, measuring abnormal performance, and constructing factor-based portfolios
- Key models:
- Single-index (market) model
- Multi-factor models
- Macroeconomic factor models
- Fundamental factor models
- Statistical factor models
Market Model
Market Model Equation:
Ri,t = αi + βi × Rm,t + εi,t
Where:
- Ri,t = Return of asset i in period t
- αi = Intercept (alpha)
- βi = Sensitivity to market (beta)
- Rm,t = Market return in period t
- εi,t = Random error term
Assumptions:
E(εi) = 0
Cov(εi, Rm) = 0
Cov(εi, εj) = 0 for i ≠ j
Multi-Factor Models
General Form:
Ri = αi + βi,1F1 + βi,2F2 + ... + βi,kFk + εi
Fama-French Three-Factor:
Ri - Rf = αi + βi,MKT(Rm - Rf) + βi,SMB×SMB + βi,HML×HML + εi
Where:
- SMB = Small minus Big (size factor)
- HML = High minus Low (value factor)
Model Applications
-
Performance Attribution:
Decompose returns into: - Market contribution: β × Rm - Alpha (skill): α - Random (luck): ε -
Risk Estimation:
Systematic risk: β²σm² Total risk: β²σm² + σε² Tracking error: σε -
Expected Return Forecasting:
E(Ri) = αi + βi × E(Rm)
Estimation Example
Regression Output:
Ri = 0.002 + 1.3 × Rm + εi
R² = 0.65
Standard error = 0.04
Interpretation:
- Alpha = 0.2% per period
- Beta = 1.3 (30% more volatile)
- 65% of variance explained
- 35% is idiosyncratic
DeFi Application
- DeFi Factor Models:
DeFi Token Return Model: R = α + β1×BTC + β2×ETH + β3×TVL_growth + β4×Gas_fees + ε Where: - BTC/ETH = Crypto market factors - TVL_growth = Protocol adoption - Gas_fees = Network congestion
LO5: Calculate and interpret beta
Core Concept exam-focus capm
Beta measures an asset’s systematic risk by quantifying its sensitivity to market movements — it represents the expected percentage change in the asset’s return for a 1% change in market return. Beta is the key risk measure in the CAPM, determines required returns, guides portfolio construction decisions, and helps investors understand and manage market exposure. The calculation methods (covariance/variance, correlation formula, and regression) draw on Quantitative Methods skills.
Key interpretations:
- Beta = 1: Moves with market
- Beta > 1: Amplifies market moves (aggressive)
- Beta < 1: Dampens market moves (defensive)
- Beta = 0: Market neutral
- Beta < 0: Inverse relationship (hedging potential)
Beta Calculation Methods formula
Method 1: Covariance/Variance
βi = Cov(Ri, Rm) / Var(Rm)
= Cov(Ri, Rm) / σm²
Method 2: Correlation Formula
βi = ρi,m × (σi / σm)
Method 3: Regression
Run regression: Ri = α + βRm + ε
Beta = slope coefficient
Calculation Example
Given Data:
Asset returns: 15%, -5%, 20%, 10%, 5%
Market returns: 10%, -2%, 15%, 8%, 4%
Step 1: Calculate means
R̄i = 9%, R̄m = 7%
Step 2: Calculate covariance
Cov = Σ[(Ri - R̄i)(Rm - R̄m)] / (n-1)
= 54 / 4 = 13.5
Step 3: Calculate market variance
σm² = Σ(Rm - R̄m)² / (n-1)
= 40.8 / 4 = 10.2
Step 4: Calculate beta
β = 13.5 / 10.2 = 1.32
HP 12C Beta Calculation
Using linear regression function:
Clear: [f] [REG]
Enter data pairs:
10 [ENTER] 15 [Σ+]
-2 [ENTER] -5 [Σ+]
15 [ENTER] 20 [Σ+]
8 [ENTER] 10 [Σ+]
4 [ENTER] 5 [Σ+]
Calculate slope (beta):
0 [g] [y,r] → Beta
Beta Interpretation Examples
Technology Stock: β = 1.5
- 10% market rise → 15% expected rise
- 10% market fall → 15% expected fall
- High systematic risk
Utility Stock: β = 0.6
- 10% market rise → 6% expected rise
- Defensive, stable returns
Gold Mining: β = -0.2
- Slight negative correlation
- Potential hedge properties
Portfolio Beta
Portfolio Beta:
βp = Σwi × βi
Example:
40% Stock A (β = 1.2)
30% Stock B (β = 0.8)
30% Stock C (β = 1.5)
βp = 0.4(1.2) + 0.3(0.8) + 0.3(1.5)
= 0.48 + 0.24 + 0.45 = 1.17
DeFi Application
- DeFi Token Betas:
Relative to Crypto Market: BTC: β = 1.0 (market benchmark) ETH: β = 1.2-1.4 DeFi tokens: β = 1.5-2.5 Memecoins: β = 2.5-4.0 Stablecoins: β ≈ 0 High beta strategies: - Leveraged positions - Small-cap DeFi tokens - New protocol launches
LO6: Explain the capital asset pricing model (CAPM), including its assumptions, and the security market line (SML)
Core Concept exam-focus capm
The Capital Asset Pricing Model (CAPM) is an equilibrium model that describes the relationship between systematic risk (beta) and expected return. It establishes that expected return equals the risk-free rate plus a risk premium proportional to beta. CAPM provides the theoretical foundation for asset pricing, cost of capital estimation, performance evaluation, and understanding why only systematic risk is compensated in efficient markets. The Security Market Line (SML) is the graphical representation of CAPM — assets above the SML are undervalued (positive alpha, buy), assets below are overvalued (negative alpha, sell). This model, despite its restrictive assumptions, remains the workhorse of applied finance.
CAPM Equation formula exam-focus
CAPM Formula:
E(Ri) = Rf + βi[E(Rm) - Rf]
Where:
- E(Ri) = Expected return of asset i
- Rf = Risk-free rate
- βi = Beta of asset i
- E(Rm) = Expected market return
- [E(Rm) - Rf] = Market risk premium
CAPM Assumptions
1. Investor Assumptions:
- Risk-averse
- Utility maximizers
- Rational (mean-variance)
2. Market Assumptions:
- No transaction costs
- No taxes
- Unlimited borrowing/lending at Rf
- Perfect competition
3. Security Assumptions:
- Infinitely divisible
- No restrictions on short selling
4. Information Assumptions:
- Homogeneous expectations
- Same one-period horizon
- All information freely available
Security Market Line (SML)
SML Properties:
- X-axis: Beta (systematic risk)
- Y-axis: Expected return
- Intercept: Risk-free rate
- Slope: Market risk premium
SML Equation:
E(R) = Rf + β[E(Rm) - Rf]
Position Analysis:
- On SML: Fairly priced
- Above SML: Undervalued (buy)
- Below SML: Overvalued (sell)
CAPM Applications
-
Cost of Equity:
Company with β = 1.3 Rf = 3%, E(Rm) = 10% Cost of equity = 3% + 1.3(7%) = 12.1% -
Project Evaluation:
Project beta = 1.5 Expected return = 14% CAPM required = 3% + 1.5(7%) = 13.5% Decision: Accept (14% > 13.5%) -
Security Valuation:
Stock beta = 0.8 Expected return = 8% CAPM required = 3% + 0.8(7%) = 8.6% Alpha = 8% - 8.6% = -0.6% Decision: Overvalued (sell)
Graphical Example
Return
^
15% -| . (β=1.5, Undervalued)
| /
10% -| /Market (β=1, E(R)=10%)
| /
8% -| /. (β=0.8, Overvalued)
|/
3% -+-----------> Beta
0 0.8 1.0 1.5
DeFi Application defi-application capm
Applying CAPM to DeFi requires defining a crypto risk-free rate (stablecoin lending at approximately 4%) and a DeFi market return (the broad crypto market return, roughly 40% in bull markets). The resulting market premium of 36% is far higher than traditional equity premiums, reflecting the extreme volatility and novel risks of the space. A protocol with beta of 2.0 would require a 76% return just to compensate for systematic risk, and anything above that represents genuine alpha.
DeFi Risk-free: Stablecoin yield = 4%
DeFi Market return: 40%
DeFi Market premium: 36%
Protocol with β = 2.0:
E(R) = 4% + 2.0(36%) = 76% required return
Actual yield = 100%
Alpha = 100% - 76% = 24% (attractive)
This framework helps distinguish between DeFi protocols that genuinely generate alpha and those that merely compensate for the high beta inherent in crypto. Many behavioral biases — particularly overconfidence and anchoring to headline APYs — cause investors to confuse beta compensation for alpha.
LO7: Calculate and interpret the expected return of an asset using the CAPM
Core Concept
- Definition: The CAPM expected return calculation determines the minimum return an investor should require for bearing an asset’s systematic risk, serving as a benchmark for investment decisions
- Why it matters: This calculation provides the hurdle rate for investment decisions, the discount rate for valuation, and the benchmark for measuring abnormal returns (alpha)
- Key applications:
- Investment selection
- Capital budgeting
- Performance evaluation
- Fair value determination
Step-by-Step Calculation
Steps:
1. Identify risk-free rate (Rf)
2. Determine market return E(Rm)
3. Calculate market premium [E(Rm) - Rf]
4. Obtain or calculate beta (β)
5. Apply CAPM: E(R) = Rf + β[E(Rm) - Rf]
Calculation Examples
-
Example 1: Single Stock:
Given: - Rf = 2.5% - E(Rm) = 9.5% - Stock beta = 1.25 E(R) = 2.5% + 1.25(9.5% - 2.5%) = 2.5% + 1.25(7%) = 2.5% + 8.75% = 11.25% -
Example 2: Portfolio:
Portfolio composition: - 50% Stock A (β = 1.4) - 30% Stock B (β = 0.9) - 20% Stock C (β = 0.5) Portfolio beta = 0.5(1.4) + 0.3(0.9) + 0.2(0.5) = 0.7 + 0.27 + 0.1 = 1.07 E(Rp) = 2.5% + 1.07(7%) = 10.0%
HP 12C CAPM Calculation
Calculate E(R) for β = 1.3:
0.095 [ENTER] (Market return)
0.025 [-] (Less Rf = Premium)
1.3 [×] (Times beta)
0.025 [+] (Plus Rf)
Result: 0.116 or 11.6%
Sensitivity Analysis
Impact of Beta Changes:
Base case: Rf = 3%, MRP = 7%
β = 0.5: E(R) = 3% + 0.5(7%) = 6.5%
β = 1.0: E(R) = 3% + 1.0(7%) = 10.0%
β = 1.5: E(R) = 3% + 1.5(7%) = 13.5%
β = 2.0: E(R) = 3% + 2.0(7%) = 17.0%
Each 0.1 beta = 0.7% return change
Real-World Application
- Tech Company Valuation:
Apple Inc: - Beta = 1.2 (Bloomberg estimate) - Rf = 4% (10-year Treasury) - Market premium = 5.5% (historical) Required return = 4% + 1.2(5.5%) = 10.6% Use for: - DCF discount rate - Option pricing - Performance benchmark
DeFi Application
- DeFi Protocol Required Returns:
Conservative (Aave): β = 1.5 E(R) = 4% + 1.5(36%) = 58% Moderate (Uniswap): β = 2.0 E(R) = 4% + 2.0(36%) = 76% Aggressive (New DeFi): β = 3.0 E(R) = 4% + 3.0(36%) = 112% Compare to actual yields for alpha
LO8: Describe and demonstrate applications of the CAPM and the SML
Core Concept
- Definition: CAPM and SML applications span investment analysis, corporate finance, and portfolio management, providing frameworks for pricing assets, evaluating investments, and measuring performance
- Why it matters: These applications transform theoretical concepts into practical tools for making investment decisions, setting hurdle rates, and identifying mispriced securities
- Key applications:
- Security selection
- Capital budgeting
- Performance evaluation
- Risk-adjusted pricing
- Portfolio optimization
Security Selection Application
Screening Process:
1. Calculate required return via CAPM
2. Compare to expected/forecast return
3. Calculate alpha (excess return)
4. Make investment decision
Example:
Stock A: E(R) = 15%, β = 1.3
CAPM: R = 3% + 1.3(8%) = 13.4%
Alpha = 15% - 13.4% = 1.6%
Decision: Buy (positive alpha)
Capital Budgeting Application
Project Evaluation:
1. Estimate project beta
2. Calculate required return
3. Use as discount rate for NPV
4. Accept if NPV > 0
Example:
New factory project:
- Beta = 1.4 (similar to company)
- Required return = 3% + 1.4(7%) = 12.8%
- NPV at 12.8% = $2.3M
- Decision: Accept
Portfolio Construction
Optimization Process:
1. Start with market portfolio
2. Identify positive alpha securities
3. Overweight undervalued assets
4. Underweight overvalued assets
5. Maintain target beta
Example Portfolio Tilt:
Market weight: 2%
Positive alpha: Overweight to 3%
Negative alpha: Underweight to 1%
Performance Attribution
Return Decomposition:
Total Return = Rf + β(Rm - Rf) + α
Example:
Portfolio return = 14%
Rf = 3%, Rm = 10%, β = 1.2
Expected = 3% + 1.2(7%) = 11.4%
Alpha = 14% - 11.4% = 2.6%
Attribution:
- Risk-free: 3%
- Market risk: 8.4%
- Alpha (skill): 2.6%
SML Analysis Example
Multiple Securities:
Stock Beta Expected Required Alpha Action
A 0.8 9% 9.4% -0.4% Sell
B 1.2 12% 11.2% 0.8% Buy
C 1.5 14% 13.5% 0.5% Buy
D 0.6 7% 7.8% -0.8% Sell
Real Options Valuation
Using CAPM for Real Options:
Project with embedded options:
- Base project beta = 1.3
- Option increases value volatility
- Adjusted beta = 1.5
- Higher required return = 13.5%
DeFi Application
- DeFi Protocol Evaluation:
Yield Farming Opportunity: - Protocol APY: 45% - Protocol beta: 2.5 - Required: 4% + 2.5(36%) = 94% Alpha = 45% - 94% = -49% Decision: Skip (negative alpha) Better Alternative: - Blue-chip staking: 15% APY - Beta: 1.2 - Required: 4% + 1.2(36%) = 47.2% - Alpha = 15% - 47.2% = -32.2% - Still negative but better risk-adjusted
LO9: Calculate and interpret the Sharpe ratio, Treynor ratio, M-squared, and Jensen’s alpha
Core Concept exam-focus
These four performance measures evaluate portfolio returns adjusted for risk, each suited to a different context. The Sharpe ratio uses total risk (standard deviation) as the denominator — appropriate for evaluating a standalone portfolio. The Treynor ratio uses systematic risk (beta) — better for evaluating a portfolio that is one component of a larger allocation. M-squared provides an intuitive percentage comparison by leveraging or deleveraging a portfolio to match market risk. Jensen’s alpha measures excess return relative to CAPM, directly quantifying active management skill. Rankings can differ across measures, which is why the finance exam tests when to use each one. These measures connect to the CAPM and to Part I’s risk calculations.
Sharpe Ratio formula
Formula:
Sharpe Ratio = [E(Rp) - Rf] / σp
Interpretation:
- Excess return per unit of total risk
- Higher is better
- Negative means underperformed Rf
- Compare to benchmark Sharpe
Example:
Portfolio: R = 12%, σ = 18%
Risk-free: 3%
Sharpe = (12% - 3%) / 18% = 0.50
Market Sharpe = (10% - 3%) / 15% = 0.47
Conclusion: Outperformed (0.50 > 0.47)
Treynor Ratio
Formula:
Treynor Ratio = [E(Rp) - Rf] / βp
Interpretation:
- Excess return per unit of systematic risk
- For well-diversified portfolios
- Higher is better
- Assumes nonsystematic risk eliminated
Example:
Portfolio: R = 14%, β = 1.3
Risk-free: 3%
Treynor = (14% - 3%) / 1.3 = 8.46%
Market Treynor = (10% - 3%) / 1.0 = 7%
Conclusion: Outperformed
M-Squared (M²)
Formula:
M² = (Rp - Rf) × (σm/σp) + Rf
= Sharpe_p × σm + Rf
Interpretation:
- Risk-adjusted return in percentage terms
- Leverages/deleverages to market risk
- Direct comparison with market return
- Intuitive percentage interpretation
Example:
Portfolio: R = 15%, σ = 25%
Market: Rm = 10%, σm = 15%
Rf = 3%
M² = (15% - 3%) × (15%/25%) + 3%
= 12% × 0.6 + 3% = 10.2%
Compare: M² = 10.2% vs Market = 10%
Outperformed by 0.2% risk-adjusted
Jensen’s Alpha
Formula:
αp = Rp - [Rf + βp(Rm - Rf)]
Interpretation:
- Actual return minus CAPM expected
- Positive = outperformance
- Negative = underperformance
- Measures active management skill
Example:
Portfolio: Actual return = 16%
Beta = 1.2, Rf = 3%, Rm = 10%
Expected = 3% + 1.2(10% - 3%) = 11.4%
Alpha = 16% - 11.4% = 4.6%
Significant positive alpha
Comparative Analysis
Portfolio Performance Metrics:
A B C
Return 18% 14% 11%
Std Dev 22% 16% 10%
Beta 1.4 1.1 0.7
Rf = 3%, Rm = 10%, σm = 15%
Sharpe: 0.68 0.69 0.80
Treynor: 10.7% 10.0% 11.4%
M²: 13.2% 13.3% 15.0%
Alpha: 4.2% 1.3% 1.1%
Rankings vary by measure!
HP 12C Calculations
Sharpe Ratio:
0.12 [ENTER] 0.03 [-] → 0.09
0.18 [÷] → 0.50
Jensen's Alpha:
0.16 [ENTER] (Actual return)
0.10 [ENTER] 0.03 [-] (Market premium)
1.2 [×] (Times beta)
0.03 [+] (Plus Rf)
[-] → 0.046 (Alpha)
Practical Example
- Mutual Fund Evaluation:
Fund Data: - 3-year return: 13.5% - Volatility: 19% - Beta: 1.15 - Benchmark: 11%, σ = 16% - T-bills: 2.5% Sharpe = (13.5% - 2.5%) / 19% = 0.58 Treynor = (13.5% - 2.5%) / 1.15 = 9.57% M² = 0.58 × 16% + 2.5% = 11.78% Alpha = 13.5% - [2.5% + 1.15(8.5%)] = 1.23% All measures show outperformance
DeFi Application defi-application
Applying all four performance measures to a DeFi yield farming portfolio reveals how different risk lenses can tell different stories. A portfolio with 85% returns and 120% volatility looks attractive on a Sharpe basis (0.675), but the negative Jensen’s alpha (-9%) reveals that the returns are not sufficient to compensate for the extreme beta exposure. The investor would have been better served by levering up a lower-beta strategy to achieve the same return. This analysis is particularly valuable for evaluating DeFi yield sources — many high-APY strategies simply embed high beta rather than generating genuine alpha.
Yield Farming Portfolio:
- Annual return: 85%
- Volatility: 120%
- Beta to crypto: 2.5
- Crypto market: 40%, σ = 60%
- Stable yield: 4%
Sharpe = (85% - 4%) / 120% = 0.675
Treynor = (85% - 4%) / 2.5 = 32.4%
M² = 0.675 × 60% + 4% = 44.5%
Alpha = 85% - [4% + 2.5(36%)] = -9%
Good Sharpe but negative alpha
High risk not fully compensated
Core Concepts Summary (80/20 Principle)
Essential Knowledge (20% that explains 80%)
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Capital Market Line (CML):
- Combines risk-free asset with market portfolio
- E(Rp) = Rf + [(E(Rm) - Rf)/σm] × σp
- Best available risk-return combinations
- All investors use same risky portfolio
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CAPM Equation:
- E(Ri) = Rf + βi[E(Rm) - Rf]
- Only systematic risk is priced
- Linear relationship with beta
- Foundation for required returns
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Systematic vs Nonsystematic Risk:
- Systematic: Market-wide, non-diversifiable, priced
- Nonsystematic: Firm-specific, diversifiable, not priced
- Total variance = β²σm² + σε²
- Diversification eliminates only nonsystematic
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Beta Interpretation:
- Measures systematic risk exposure
- β = Cov(Ri,Rm)/σm²
- β > 1: Aggressive, β < 1: Defensive
- Determines required return in CAPM
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Performance Measures:
- Sharpe: (R - Rf)/σ (total risk)
- Treynor: (R - Rf)/β (systematic risk)
- Jensen’s α: R - [Rf + β(Rm - Rf)]
- M²: Risk-adjusted to market level
Critical Relationships
- Risk-Return Trade-off: Higher systematic risk requires higher return
- Two-Fund Separation: Everyone holds combinations of risk-free and market portfolio
- SML vs CML: SML for all assets, CML for efficient portfolios only
- Diversification Limit: Can only eliminate nonsystematic risk
Comprehensive Formula Sheet
Risk and Return
Portfolio with Risk-Free Asset:
E(Rc) = wf × Rf + (1 - wf) × E(Rp)
σc = |1 - wf| × σp
Capital Market Line
CML: E(Rp) = Rf + [(E(Rm) - Rf)/σm] × σp
Slope (Market Price of Risk) = [E(Rm) - Rf]/σm
Beta Calculations
Beta = Cov(Ri,Rm) / σm²
Beta = ρi,m × (σi/σm)
Portfolio Beta: βp = Σwi × βi
CAPM
E(Ri) = Rf + βi[E(Rm) - Rf]
SML: Same equation, x-axis = β
Risk Decomposition
Total Variance: σi² = βi²σm² + σε²
R-squared: R² = βi²σm² / σi²
Systematic Risk = βi²σm²
Nonsystematic Risk = σε²
Performance Measures
Sharpe Ratio = [E(Rp) - Rf] / σp
Treynor Ratio = [E(Rp) - Rf] / βp
M² = Sharpe_p × σm + Rf
Jensen's Alpha = Rp - [Rf + βp(Rm - Rf)]
Market Model
Ri,t = αi + βi × Rm,t + εi,t
E(εi) = 0, Cov(εi,Rm) = 0
HP 12C Calculator Sequences
Beta Calculation (Regression)
Clear: [f] [REG]
Enter x,y pairs: Rm [ENTER] Ri [Σ+]
Calculate beta: 0 [g] [y,r]
CAPM Expected Return
Market premium: Rm [ENTER] Rf [-]
Risk premium: β [×]
Expected return: Rf [+]
Sharpe Ratio
Excess return: Rp [ENTER] Rf [-]
Sharpe: σp [÷]
Jensen’s Alpha
Actual return: Rp [ENTER]
Market premium: Rm [ENTER] Rf [-]
Beta premium: β [×]
Expected return: Rf [+]
Alpha: [-]
M-Squared
Sharpe ratio: [Calculate as above]
M²: σm [×] Rf [+]
Practice Problems
Basic Level
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CML Calculation: Given Rf = 3%, E(Rm) = 11%, σm = 20%, find expected return for portfolio with σp = 15%.
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Beta Interpretation: A stock has β = 1.4. If market rises 10%, what is expected stock return change?
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CAPM Return: Calculate required return for stock with β = 0.8, given Rf = 4%, E(Rm) = 12%.
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Risk Decomposition: Stock has total variance of 0.09, beta of 1.2, market variance of 0.04. Calculate systematic and nonsystematic risk.
Intermediate Level
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Performance Measures: Portfolio earned 15% with σ = 22% and β = 1.3. Market returned 12% with σ = 18%. Rf = 3%. Calculate all four performance measures.
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SML Analysis: Three stocks:
- A: E(R) = 14%, β = 1.2
- B: E(R) = 10%, β = 0.8
- C: E(R) = 16%, β = 1.5 Given Rf = 3%, Rm = 11%, which are undervalued?
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Leveraged Portfolio: Create portfolio with E(R) = 15% using Rf = 3% and market portfolio with E(Rm) = 10%, σm = 16%. Calculate required leverage and resulting σp.
Advanced Level
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Multi-Factor Attribution: Portfolio returned 18% with β = 1.4. Market returned 12%, Rf = 3%. Decompose return into components.
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Optimal Portfolio Selection: Compare two portfolios:
- A: E(R) = 13%, σ = 20%, β = 1.1
- B: E(R) = 11%, σ = 15%, β = 0.9 Market: E(Rm) = 10%, σm = 16%, Rf = 3% Which is superior and why?
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Risk Budget Allocation: Construct portfolio with target β = 1.2 using:
- Stock A: β = 1.5
- Stock B: β = 0.8
- Stock C: β = 1.0 Find optimal weights.
DeFi Problems
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DeFi Performance Analysis: Calculate performance measures for:
- DeFi yield farm: 95% APY, 140% volatility, β = 3.0
- Crypto market: 45% return, 70% volatility
- Stable rate: 5%
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Protocol Comparison: Using CAPM, evaluate:
- Protocol A: 60% APY, β = 2.0
- Protocol B: 40% APY, β = 1.5
- Protocol C: 80% APY, β = 2.5 Given crypto market premium = 40%.
DeFi Applications & Real-World Examples
Traditional Finance Applications
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Equity Fund Analysis:
Growth Fund Performance: - 5-year return: 14.5% annually - Volatility: 21% - Beta: 1.25 - Benchmark: S&P 500 (10%, σ = 16%) - T-bills: 2% Sharpe = 0.595 Treynor = 10% Alpha = 2% M² = 11.52% -
Corporate Project Evaluation:
Tech Expansion Project: - Industry beta: 1.4 - Rf = 3%, Market premium = 7% - Required return = 3% + 1.4(7%) = 12.8% - Project IRR = 15% - Positive NPV, accept project
DeFi Portfolio Strategies
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Risk-Parity DeFi Portfolio:
Equal Risk Contribution: - Stablecoins: 60% (low vol) - Blue-chip crypto: 25% (medium vol) - DeFi tokens: 10% (high vol) - Yield farming: 5% (very high vol) Each contributes ~25% to risk -
Beta-Neutral DeFi Strategy:
Long positions: - ETH staking: β = 1.2 - AAVE lending: β = 1.5 Short positions: - Perp shorts: β = -1.35 Net beta ≈ 0 Market-neutral returns -
Smart Beta DeFi:
Factor tilts: - TVL momentum (high TVL growth) - Low volatility (stable protocols) - Quality (audited, established) - Value (low P/TVL ratios)
Risk Management Applications
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Beta Hedging:
Portfolio beta = 1.5 Want to reduce to 1.0 Hedge ratio = (1.5 - 1.0) / 1.0 = 0.5 Short 50% in market index -
Volatility Targeting:
Target vol = 12% Strategy vol = 20% Leverage = 12% / 20% = 0.6 Allocate 60% to strategy 40% to cash/stables
Common Pitfalls & Exam Tips
Conceptual Pitfalls
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CML vs SML Confusion:
- CML: Efficient portfolios only, σ on x-axis
- SML: All securities, β on x-axis
- CML slope > SML slope at market point
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Risk Measure Matching:
- Sharpe for undiversified portfolios
- Treynor for diversified portfolios
- Don’t mix risk measures
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Beta Interpretation:
- Beta ≠ correlation
- Beta can exceed 1 or be negative
- Portfolio beta is weighted average
Calculation Errors
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Percentage vs Decimal:
- Always check units
- Convert consistently
- Final answer in requested format
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Risk-Free Rate Inclusion:
- Add Rf back for total return
- Subtract Rf for excess return
- Don’t double-count
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Sign Conventions:
- Negative alpha = underperformance
- Negative beta = inverse relationship
- Check economic intuition
Exam Strategy
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Quick Checks:
- CAPM return > Rf if β > 0
- Higher beta → higher required return
- Sharpe ratio dimensionless
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Time Savers:
- Memorize CAPM formula
- Know beta = 1 means market return
- Remember Rf is y-intercept
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Common Questions:
- Calculate required returns
- Identify over/undervalued securities
- Compare performance measures
- Interpret beta values
Key Takeaways
Core Principles
- Only systematic risk earns a premium: Diversifiable risk is not compensated
- CAPM provides equilibrium pricing: Expected return linear in beta
- Market portfolio is optimal for all: Two-fund separation theorem
- Performance measures serve different purposes: Choose based on context
- Beta drives required returns: Not total volatility
Practical Applications
- Use CAPM for cost of capital: Standard in corporate finance
- SML identifies mispricing: Securities above/below for buy/sell signals
- Sharpe ratio for standalone evaluation: Treynor for portfolio components
- Jensen’s alpha measures skill: Positive alpha indicates outperformance
- CML represents efficient frontier with risk-free asset: Dominates risky-only frontier
DeFi Innovations
- 24/7 markets change dynamics: Continuous price discovery affects beta
- Protocol risks beyond traditional model: Smart contract risk not captured by beta
- Extreme volatility challenges assumptions: Normal distribution assumptions violated
- Composability creates new factors: TVL, governance, cross-chain effects
- Regulatory uncertainty adds systematic risk: Affects entire crypto market
Cross-References & Additional Resources
Related Finance Topics
- Portfolio Risk and Return Part I — Efficient frontier, diversification, portfolio variance
- Equity Investments — Using CAPM for discount rates and valuation
- Fixed Income — Systematic interest rate risk, duration
- Risk Management — Risk measurement, VaR, stress testing
- Corporate Issuers — Cost of capital, project evaluation
Key Academic Papers
- Sharpe (1964): “Capital Asset Prices: A Theory of Market Equilibrium”
- Lintner (1965): “The Valuation of Risk Assets”
- Black (1972): “Capital Market Equilibrium with Restricted Borrowing”
- Roll (1977): “A Critique of the Asset Pricing Theory’s Tests”
- Fama & French (1992): “The Cross-Section of Expected Stock Returns”
DeFi Resources
- DeFi Pulse: Market metrics and TVL
- Dune Analytics: On-chain beta calculations
- CoinMetrics: Correlation matrices
- Glassnode: Market indicators
- Token Terminal: Protocol fundamentals
Tools and Platforms
- Bloomberg Terminal: Beta estimates
- FactSet: Risk model analytics
- Morningstar Direct: Performance attribution
- Portfolio Visualizer: Online CAPM tools
- Python/R packages: Statistical analysis
Review Checklist
Conceptual Understanding
- Explain two-fund separation theorem
- Distinguish systematic from nonsystematic risk
- Understand CAPM assumptions and limitations
- Know when to use each performance measure
- Interpret SML and security positions
Calculation Proficiency
- Calculate expected returns using CAPM
- Compute beta from data
- Determine portfolio beta
- Calculate all performance measures
- Apply CML formula
Application Skills
- Identify mispriced securities
- Evaluate investment opportunities
- Decompose returns into components
- Select appropriate risk measures
- Apply to real-world scenarios
Exam Readiness
- Complete all practice problems
- Review formula sheet
- Practice HP 12C sequences
- Understand common question types
- Time management strategy ready