# The Capital Allocation Line – Introducing the Risk-free Asset

The discussion of diversification benefits focused on a portfolio consisting of risky assets; when a risk-free asset is incorporated, diversification is still prevalent but a linear trade-off between risk and return is established.

The introduction of a risk-free asset does not change the construct of the minimum variance frontier graphical structure (y-axis = expected return; x-axis = standard deviation or variance).

However, the introduction of the risk-free asset does shift the optimal risk-return trade off on the efficient frontier.

**Capital Allocation Line (CAL):** This line shows the highest returns for each level of risk in a portfolio containing a risky and risk-free asset.

**CAL Equation**

E(RP) = RF + ((E(RT) – RF)/σT)*σP

E(RP) = expected return

RF = rate of return on the risk free asset

E(RT) = expected return on CAL portfolio that is tangent to the minimum variance frontier

σT = standard deviation of the tangent portfolio

σP \= standard deviation of the portfolio analyzed

Note that when there is zero allocation to the risky asset, the portfolio’s return will equal the risk-free rate and the standard deviation will be zero; this is the y-intercept of the CAL.

**CAL Slope Coefficient** = **Sharpe Ratio**; This is the expected change in return for a given change in risk (where risk is defined as the standard deviation of returns).

Sharpe Ratio = (E(Rasset) – RF)/σasset

The Sharpe Ratio is a risk/return tradeoff measure; if two assets offer a similar expected return but different standard deviations, the asset with the higher Sharpe Ratio is considered superior.

**Optimal Portfolio:** Once the risk-free asset is introduced, the only optimal portfolio on the minimum variance frontier is the tangent portfolio.

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- CFA Level 2: Portfolio Management – Introduction
- Mean-Variance Analysis Assumptions
- Expected Return and Variance for a Two Asset Portfolio
- The Minimum Variance Frontier & Efficient Frontier
- Diversification Benefits
- The Capital Allocation Line – Introducing the Risk-free Asset
- The Capital Market Line
- CAPM & the SML
- Adding an Asset to a Portfolio – Improving the Minimum Variance Frontier
- The Market Model for a Security’s Returns
- Adjusted and Unadjusted Beta
- Multifactor Models
- Arbitrage Portfolio Theory (APT) – A Multifactor Macroeconomic Model
- Risk Factors and Tracking Portfolios
- Markowitz, MPT, and Market Efficiency
- International Capital Market Integration
- Domestic CAPM and Extended CAPM
- Changes in Real Exchange Rates
- International CAPM (ICAPM) - Beyond Extended CAPM
- Measuring Currency Exposure
- Company Stock Value Responses to Changes in Real Exchange Rates
- ICAPM vs. Domestic CAPM
- The J-Curve – Impact of Exchange Rate Changes on National Economies
- Moving Exchange Rates and Equity Markets
- Impacts of Market Segmentation on ICAPM
- Justifying Active Portfolio Management
- The Treynor-Black Model
- Portfolio Management Process
- The Investor Policy Statement

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