## **Learning Outcomes**

This article explains how modern portfolio theory describes the risk–return trade-off for CFA Level 1 candidates, focusing on constructing and evaluating efficient portfolios of risky and risk-free assets. It clarifies how diversification reduces portfolio risk through covariance and correlation, and how to compute and interpret portfolio variance and standard deviation for two-asset combinations. It details the construction and interpretation of the minimum-variance frontier, the efficient frontier, and the global minimum-variance portfolio, emphasizing which portfolios are rational choices for risk-averse investors. It examines the role of the capital allocation line (CAL) and the capital market line (CML) in combining a risky portfolio with a risk-free asset, and how their slopes represent the price of risk. It discusses how investor utility, risk aversion, and indifference curves determine the optimal risky portfolio and the final asset allocation along the CAL or CML. It also reinforces how changes in asset correlation shift the efficient frontier and influence optimal holdings across different investor risk profiles.

## **CFA Level 1 Syllabus**

For the CFA Level 1 exam, you are expected to understand how portfolio theory supports diversification and the analytical construction of the efficient frontier and to apply these concepts to investor choice and risk aversion, with a focus on the following syllabus points:

- Describe risk aversion and its implications for portfolio selection
- Calculate and interpret the mean, variance, and covariance (or correlation) of asset returns for historical data
- Calculate and interpret portfolio standard deviation and explain how diversification reduces risk
- Describe and interpret the minimum-variance and efficient frontiers of risky assets and the global minimum-variance portfolio
- Explain the selection of an optimal portfolio, given investor utility and the capital allocation line

## **Test Your Knowledge**

_Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision._

1. What is meant by the minimum-variance frontier in portfolio theory?
2. Why does adding an asset with low correlation to an existing portfolio reduce portfolio risk?
3. True or false? The optimal portfolio for each investor lies at the point where their highest indifference curve is tangent to the capital allocation line.
4. What is the capital market line and what does its slope represent?

## **Introduction**

Portfolio theory explains how investors can combine assets to construct portfolios that minimize risk for a given return, or maximize return for a given risk. The efficient frontier summarises the set of optimal portfolios of risky assets. By introducing a risk-free asset, investors can achieve better risk–return combinations, captured by the capital allocation line. The choice of the optimal portfolio depends on the investor's risk aversion and is graphically represented by the point of tangency between an indifference curve and the capital allocation line.

> **Key Term: efficient frontier**
> The set of portfolios that offers the highest expected return for each level of risk, or the lowest risk for each expected return, from the possible combinations of risky assets.
>
> **Key Term: capital allocation line**
> A straight line showing all possible combinations of a risky portfolio and a risk-free asset, representing the risk–return trade-off available to investors.

## THE EFFECT OF CORRELATION AND DIVERSIFICATION

Diversification reduces portfolio risk if the assets included are not perfectly correlated. Portfolio variance incorporates both the variance of each asset and the covariance between assets. As correlation decreases, the risk of the portfolio for a given return also decreases.

### **Worked Example 1.1**

**Suppose Asset A has volatility of 16%, Asset B has volatility of 12%, and their correlation is 0.7. What is the impact on portfolio risk if you invest 60% in A and 40% in B?**

> **Answer:**
> The portfolio standard deviation is:
>
> σₚ = √[ (0.6)²×(0.16)² + (0.4)²×(0.12)² + 2×0.6×0.4×0.7×0.16×0.12 ]  
> σₚ ≈ √[0.009216 + 0.002304 + 0.006451] ≈ √0.017971 ≈ 13.4%
>
> This is lower than if you held only the riskier asset alone.

## MINIMUM-VARIANCE AND EFFICIENT FRONTIER

The **minimum-variance frontier** is the set of portfolios with the lowest variance for each expected return, composed of different combinations of assets. The segment of this frontier that offers the highest expected return for a given level of risk forms the **efficient frontier**.

### **Worked Example 1.2**

**Three portfolios—A, B, and C—offer the same return, but portfolio C has lower risk than A or B. Which is on the minimum-variance frontier?**

> **Answer:**
> Portfolio C, because for the same return, it has the lowest risk.

## GLOBAL MINIMUM-VARIANCE PORTFOLIO AND EFFICIENT FRONTIER

Among all risky portfolios, the left-most point on the minimum-variance frontier is the **global minimum-variance portfolio**. All rational, risk-averse investors should choose only portfolios on or above this point—these are the efficient portfolios.

> **Key Term: global minimum-variance portfolio**
> The unique portfolio of risky assets with the lowest possible variance (risk) out of all possible asset combinations.

## OPTIMAL PORTFOLIO SELECTION

When both risky and risk-free assets are available, the optimal combination for an investor is determined by the capital allocation line (CAL). The CAL is tangent to the efficient frontier at the optimal risky portfolio, which then can be mixed with the risk-free asset to suit a given investor's risk preference.

> **Key Term: optimal risky portfolio**
> The risky portfolio that, when combined with the risk-free asset, yields the highest possible slope of the capital allocation line for investors.

Investor preferences are represented by indifference curves—curves along which an investor has equal utility. The most preferred allocation is where the investor's highest attainable indifference curve is tangential to the CAL.

> **Key Term: indifference curve**
> A curve on a graph of risk versus return representing combinations of portfolios between which an investor is indifferent.

### **Worked Example 1.3**

**Investor A is more risk-averse than Investor B. Both have access to the same CAL. Who will invest more in the risk-free asset?**

> **Answer:**
> Investor A, because greater risk aversion leads to a tangency point farther to the left on the CAL (higher risk-free holding, lower risky portfolio holding).

## CAPITAL MARKET LINE (CML)

The **capital market line** (CML) is a special case of the CAL where the risky portfolio is the market portfolio—all risky assets, weighted by market value. It shows all possible portfolios combining the market portfolio and the risk-free asset.

> **Key Term: capital market line**
> The line from the risk-free rate tangent to the efficient frontier when the market portfolio is used as the risky portfolio; its slope is the market price of risk.

## **Summary**

The efficient frontier represents the best combinations of risky assets. Including a risk-free asset allows for even better risk–return trade-offs, depicted by the capital allocation line or, in market equilibrium, the capital market line. The optimal investor portfolio depends on risk aversion and is found where an investor’s indifference curve is tangent to the CAL. Lower asset correlation means greater diversification benefits, lowering risk for a given return.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Portfolio risk is reduced by diversification—lower correlation between assets leads to lower overall risk.
- The minimum-variance frontier shows the lowest-risk portfolios for any return; the efficient frontier is the top part relevant for investors.
- The global minimum-variance portfolio is the overall lowest-risk risky portfolio.
- The capital allocation line represents new possible portfolios when a risk-free asset is included.
- The optimal risky portfolio is found at the tangency of the CAL and the efficient frontier.
- The optimal investor portfolio is at the tangency between the investor’s indifference curve and the CAL (or CML).
- The capital market line uses the market portfolio and risk-free asset in equilibrium.

## **Key Terms and Concepts**

- efficient frontier
- capital allocation line
- global minimum-variance portfolio
- optimal risky portfolio
- indifference curve
- capital market line

Portfolio theory and capm - Efficient frontier and optimal portfolio