## **Learning Outcomes**

This article explains the core elements of portfolio theory and the capital asset pricing model (CAPM), including:

- distinguishing between systematic (market) and unsystematic (diversifiable) risk and assessing how each contributes to total portfolio risk
- calculating and interpreting expected return, variance, standard deviation, covariance, and correlation for individual assets and portfolios
- evaluating the impact of diversification and correlation on portfolio risk and recognizing the limits of diversification in practice
- constructing portfolios that combine a risk-free asset with risky assets and interpreting the capital market line (CML)
- describing the assumptions and implications of capital market theory for the market portfolio and investor behavior
- using beta to measure an asset’s sensitivity to market movements and to decompose total risk into systematic and unsystematic components
- applying the CAPM formula to estimate required return and drawing inferences from mispricing relative to the security market line (SML)
- interpreting the position of assets and portfolios on the CML and SML to judge whether they offer adequate compensation for risk in an exam setting

## **CFA Level 1 Syllabus**

For the CFA Level 1 exam, you are expected to understand the foundations of portfolio theory and the capital asset pricing model (CAPM) and, before tackling exam questions, ensure you can do so with a focus on the following syllabus points:

- Explain risk aversion and investor utility and how these shape portfolio choices
- Calculate and interpret the expected return, variance, and correlation of asset and portfolio returns
- Describe the effect of diversification and correlation on portfolio risk and return
- Distinguish between systematic (market) and unsystematic (diversifiable) risk
- Apply the capital market line (CML) and security market line (SML) concepts
- Calculate and interpret beta and apply the CAPM formula to estimate expected return

## **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 type of risk can be eliminated by diversification: systematic or unsystematic?
2. How does correlation between assets affect overall portfolio risk?
3. In the CAPM, what does beta measure, and what is the required return on an asset with beta = 1 if the risk-free rate is 3% and the expected market return is 9%?
4. Suppose you combine a risk-free asset and the market portfolio. What relationship does the resulting risk-return line describe?

## **Introduction**

Investment professionals must balance risk and return to construct portfolios suited to their clients' preferences. Portfolio theory addresses this trade-off and the role of diversification, while the capital asset pricing model (CAPM) provides a mechanism to estimate an asset's required return given its market risk. Understanding these principles is critical for effective investment analysis and portfolio construction.

## The Risk–Return Trade-off and Risk Aversion

Investors are generally **risk averse**: they prefer higher return for the same risk, or lower risk for the same expected return. Their utility can be expressed as a function of expected return and risk (measured as variance or standard deviation). Indifference curves represent combinations of risk and return that provide equal utility to an investor.

> **Key Term: risk aversion**
> The preference for higher return at equal risk, or for lower risk at equal return; generally describes most investors' behavior.

A risk-averse investor will require higher expected return to accept more risk. In practical terms, risk is measured using variance or standard deviation of returns.

> **Key Term: variance**
> A measure of dispersion, calculated as the average squared deviation from the mean; in portfolio theory, represents investment risk.

## Portfolio Expected Return and Risk

The **expected return** of a portfolio is a weighted average of the expected returns of its constituent assets:

$$
E(R_P) = w_1 E(R_1) + w_2 E(R_2) + ... + w_n E(R_n)
$$

> **Key Term: expected return**
> The probability-weighted average return for a given asset or portfolio.

Portfolio **variance** depends not just on the variances of individual assets but also on the **covariance** (and thus the correlation) between asset returns:

$$
\sigma_P^2 = \sum w_i^2 \sigma_i^2 + \sum_{i \neq j} w_i w_j \text{Cov}(R_i, R_j)
$$

> **Key Term: correlation**
> A standardized measure (from -1 to +1) of how two asset returns move together.

If correlation between assets is less than +1, combining them can reduce total portfolio risk.

## Diversification and Correlation

When assets within a portfolio are not perfectly positively correlated, diversification can reduce portfolio risk below the weighted average of individual risks.

> **Key Term: diversification**
> The risk-reducing strategy of holding different assets to lower total portfolio risk.

A low or negative correlation between assets magnifies this benefit. The lower the average correlation, the lower the achievable portfolio risk. If correlation is exactly -1 (perfect negative correlation), it is possible to construct a riskless portfolio from risky assets.

### **Worked Example 1.1**

**You hold 50% in Asset X (expected return 8%, risk 16%) and 50% in Asset Y (expected return 12%, risk 20%), with a correlation coefficient of 0.**

> **Answer:**
> The expected return is 10%. Portfolio risk is:
>
> $$
> \sqrt{0.5^2(0.16^2) + 0.5^2(0.20^2) + 2(0.5)(0.5)(0)(0.16)(0.20)} = \sqrt{0.0128 + 0.02} = \sqrt{0.0328} \approx 18.1\%
> $$
>
> Diversification reduces risk below the average of 16% and 20%.

### **Exam Warning**

> When calculating portfolio risk, do not simply average standard deviations. Always use the full variance-covariance formula, including correlation.

## Systematic and Unsystematic Risk

Portfolio risk consists of two components:

- **Systematic risk:** Also called market risk; affects all securities and cannot be eliminated by diversification (e.g. macroeconomic factors).
- **Unsystematic risk:** Also called specific or diversifiable risk; unique to a company or industry and can be eliminated by holding a sufficiently diversified portfolio.

> **Key Term: systematic risk**
> The portion of total investment risk that cannot be diversified away, also called market risk.
>
> **Key Term: unsystematic risk**
> The portion of total risk unique to a company or sector, removable by diversification.

Unsystematic risk approaches zero as the number of portfolio holdings increases, but systematic risk remains.

### **Worked Example 1.2**

**If you buy 20 randomly selected shares across different industries, which risk(s) are you still exposed to?**

> **Answer:**  
> You have essentially eliminated unsystematic risk, but remain exposed to systematic (market) risk.

## Capital Market Theory

If all investors are rational, risk-averse, and have homogeneous expectations, they will combine:

- A risk-free asset (with zero volatility), and
- The optimal risky portfolio (the "market" portfolio).

Plotting all possible investor combinations of the risk-free asset and the market portfolio produces the **capital market line (CML)**.

> **Key Term: capital market line (CML)**
> The set of expected return and total risk (standard deviation) combinations for portfolios formed from the risk-free asset and the market portfolio.

The tangency point between the investor's highest attainable indifference curve and the CML indicates the optimal combination given their risk aversion.

## The Capital Asset Pricing Model (CAPM) and Beta

CAPM provides a linear relationship between an asset’s expected return and its systematic risk (beta):

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

Where $R_f$ is the risk-free rate, $E(R_m)$ the expected market return, and $\beta_i$ the sensitivity of asset $i$ to market movements.

> **Key Term: beta**
> A measure of an asset's market risk—the sensitivity of its return to the market return; the slope coefficient in the CAPM.

- If beta = 1, the asset has average market risk.
- If beta < 1, less volatile than the market.
- If beta > 1, more volatile than the market.

Only systematic risk (beta) is priced in the CAPM. Diversifiable (unsystematic) risk is not.

### **Worked Example 1.3**

**A stock has $\beta = 1.2$, the risk-free rate is 2%, the market return is 7%. Using CAPM, what is its required return?**

> **Answer:**
>
> $$
> E(R) = 2\% + 1.2 \times (7\% - 2\%) = 2\% + 6\% = 8\%
> $$

## The Security Market Line (SML)

The SML is a graphical representation of the CAPM. Every correctly priced asset or portfolio should fall on the SML, with expected return as a function of beta.

> **Key Term: security market line (SML)**
> The linear relationship between expected return and beta as defined by the CAPM; plots the required return for any given beta.

Portfolios above the SML are undervalued (offering higher return for risk), while those below are overvalued.

## Diversification in Practice

Perfect diversification is not always achievable—some assets may remain correlated, and liquidity, sector, or geographical exposures may limit results. Nevertheless, holding asset classes with low or negative correlation remains the most effective way to decrease risk without sacrificing expected return.

### **Revision Tip**

> Invest in assets with low or negative correlations to maximize diversification benefits; never depend solely on increasing the number of holdings if correlations are high.

## **Summary**

Effective portfolios balance risk and return through diversification, minimizing unsystematic risk. Only systematic risk remains and is priced by the market according to the CAPM, with return proportional to beta. Understanding CML and SML concepts is essential for CFA exam success.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Explain risk aversion and its influence on portfolio construction
- Calculate portfolio expected return and risk, including the effect of correlation
- Distinguish between systematic (market) and unsystematic (diversifiable) risk
- Understand and apply the concepts of CML and SML
- Use beta and CAPM to estimate expected return
- Recognize the limits and benefits of diversification

## **Key Terms and Concepts**

- risk aversion
- variance
- expected return
- correlation
- diversification
- systematic risk
- unsystematic risk
- capital market line (CML)
- beta
- security market line (SML)

Portfolio theory and capm - Risk return and diversification