Risk, Return and Portfolio Principles

Introduction

Every financial decision involves uncertainty. Investors commit funds today in exchange for uncertain future cash flows, while managers select projects whose future benefits and costs are not known with certainty. The central questions are therefore: What return is expected? How much risk is being accepted? Can risk be reduced by combining investments? What return should compensate an investor for bearing unavoidable risk?

This chapter develops the basic analytical tools used to answer these questions. It examines expected return, variance, standard deviation, covariance, correlation, beta, diversification, portfolio theory, the Capital Asset Pricing Model (CAPM), and the security market line. It also evaluates the assumptions and behavioral limitations of these models.

Learning Objectives

By the end of this chapter, students should be able to:

• calculate expected return, variance, and standard deviation;
• distinguish systematic from unsystematic risk;
• calculate covariance, correlation, portfolio risk, and beta;
• explain how diversification affects portfolio risk;
• apply the CAPM and interpret the security market line;
• assess the assumptions, practical uses, and limitations of portfolio models; and
• recognize behavioral and governance issues affecting risk decisions.

1. Return and Risk

Return

Holding-period return measures the gain or loss on an investment over a specified period. If an asset is purchased for P₀, sold for P₁, and produces income C₁, then:

Here, R is the holding-period return, P₀ is the beginning price, P₁ is the ending price, and C₁ is cash income such as dividends or interest. All monetary values must refer to the same period and currency. The formula assumes that income is received during the holding period and is not separately compounded.

For example, a share bought for $50, sold for $54, and paying a $1 dividend produces:

R = (54 − 50 + 1) ÷ 50 = 0.10 = 10%

Return may be historical (realized) or expected (forecast). A nominal return includes the effect of inflation; a real return measures purchasing-power growth. A commonly used approximation is:

The exact relationship is 1 + R_nominal = (1 + R_real)(1 + inflation).

Risk

Risk is the possibility that actual return differs from expected return. It includes both unfavorable outcomes and uncertainty more generally. A security with highly variable possible returns is usually considered riskier than one with stable returns, even if both have the same average return.

Risk analysis must identify the relevant probability distribution. A probability is a number from 0 to 1 representing the likelihood of an outcome. The probabilities of all mutually exclusive and exhaustive states must sum to 1.

2. Expected Return and Dispersion Measures

Expected return

For discrete possible returns, expected return is the probability-weighted average:

where p_i is the probability of state i, R_i is the return in that state, and n is the number of states. This is a forecast of the arithmetic mean, not a guarantee of the realized return.

Worked example. Suppose an investment has the following possible outcomes:

E(R) = 0.30(−10%) + 0.50(8%) + 0.20(20%) = 6%

The expected return is therefore 6%.

For equally likely historical observations, the sample arithmetic mean is ∑R_t ÷ T, where T is the number of observations. Arithmetic means are useful for estimating one-period expected returns. For multi-period investment planning, the geometric mean is often more informative because it reflects compounding:

Variance and standard deviation

Variance measures the probability-weighted squared deviation from expected return:

Standard deviation is the square root of variance:

Variance is expressed in squared return units, while standard deviation is expressed in return units, such as percentage points. Standard deviation is therefore easier to interpret. These measures are most appropriate when returns can reasonably be summarized by their dispersion around a mean and when extreme outcomes are not inadequately represented.

Using the preceding example, deviations from the 6% expected return are −16%, 2%, and 14%. Thus:

σ² = 0.30(−0.16)² + 0.50(0.02)² + 0.20(0.14)² = 0.00864

and:

σ = √0.00864 ≈ 0.09295 = 9.30%

A sample estimate based on historical data commonly uses T − 1 rather than T in the denominator:

The distinction is important: the probability formula describes a population distribution, whereas the sample formula estimates population variance from observations.

3. Covariance, Correlation, and Portfolio Risk

Covariance

Covariance measures how two asset returns move together:

Positive covariance indicates that returns tend to move in the same direction; negative covariance indicates opposite movement. Covariance is affected by the units and scale of measurement, so its magnitude is not easily interpreted by itself.

Correlation

Correlation standardizes covariance:

Correlation lies between −1 and +1. A value of +1 means perfect positive co-movement, 0 means no linear association, and −1 means perfect negative linear co-movement. Zero correlation does not necessarily mean that two assets are independent or that no nonlinear relationship exists.

Two-asset portfolio return

If weights w_A and w_B represent proportions invested in assets A and B, with w_A + w_B = 1, portfolio expected return is:

Weights may be negative when short selling is permitted, although practical and regulatory constraints may prohibit it.

Two-asset portfolio variance

The general two-asset variance equation is:

Using covariance expressed through correlation:

The covariance term is the mathematical source of diversification. Portfolio risk is not generally the weighted average of individual standard deviations.

Worked example. Asset A has expected return 10% and standard deviation 12%; asset B has expected return 6% and standard deviation 8%. An investor allocates 60% to A and 40% to B. Their correlation is 0.25.

Expected return:

E(R_p) = 0.60(10%) + 0.40(6%) = 8.4%

Variance:

σ_p² = (0.60)²(0.12)² + (0.40)²(0.08)² + 2(0.60)(0.40)(0.25)(0.12)(0.08)

σ_p² = 0.00864 + 0.001024 + 0.001152 = 0.010816

Therefore:

σ_p = √0.010816 ≈ 10.40%

The portfolio return is 8.4% and its standard deviation is approximately 10.4%.

Multiple-asset portfolios

For N assets, portfolio variance is:

In matrix notation:

where w is the vector of portfolio weights, Σ is the covariance matrix, and w′ is the transpose of the weight vector. This formulation includes each asset’s variance and every pairwise covariance.

4. Diversification and Types of Risk

Diversification means combining investments whose returns do not move perfectly together. As the number of securities increases, asset-specific fluctuations tend to offset one another. Diversification can reduce unsystematic risk but cannot eliminate economy-wide risk.

• Unsystematic risk, also called diversifiable, firm-specific, or idiosyncratic risk, arises from events affecting an individual company or narrow industry: product failure, fraud, labor disputes, or loss of a major customer.
• Systematic risk, also called market or nondiversifiable risk, arises from broad forces such as recessions, interest-rate changes, inflation surprises, wars, or economy-wide regulatory shocks.
• Total risk is commonly represented by variance or standard deviation and includes both components.

A well-diversified investor should generally not expect additional compensation for bearing avoidable, firm-specific risk. However, diversification has costs: transaction fees, taxes, research time, monitoring costs, and possible dilution of a concentrated information advantage.

5. Portfolio Theory and the Efficient Frontier

Modern Portfolio Theory, associated particularly with Harry Markowitz, treats investment selection as a joint risk–return problem. A portfolio is efficient if no other feasible portfolio offers a higher expected return for the same risk or lower risk for the same expected return. The set of efficient portfolios forms the efficient frontier.

The frontier depends on expected returns, variances, covariances, investment constraints, and the investor’s planning horizon. An investor with a risk-averse utility function may select a point on the frontier rather than simply choose the portfolio with the highest return. In a simplified mean–variance framework, investors prefer higher expected return and lower variance, all else equal.

The minimum-variance portfolio is the feasible portfolio with the lowest variance. It is not necessarily the portfolio with the lowest expected return or the portfolio selected by every investor. Optimization results are highly sensitive to input estimates, especially expected returns and correlations. Small estimation errors can produce large changes in recommended weights.

When a risk-free asset is available, combinations of the risk-free asset and a tangency portfolio may produce a capital allocation line. The slope of such a line is the reward-to-risk ratio:

where R_f is the risk-free rate. This expression is a Sharpe ratio when applied to an investment or portfolio and the risk measure is standard deviation.

6. Beta and the Capital Asset Pricing Model

Beta

Beta measures an asset’s sensitivity to movements in a market portfolio:

where R_i is the asset return, R_M is the market return, and σ_M² is market variance. Beta is dimensionless. A beta of 1 indicates market-like systematic sensitivity; a beta above 1 indicates greater sensitivity; a beta below 1 indicates lower sensitivity. A negative beta implies that the asset tends, on average, to move opposite the market.

Beta is a slope estimated from a regression of asset returns on market returns. It is not the same as standard deviation: standard deviation measures total volatility, whereas beta measures systematic sensitivity.

CAPM

The Capital Asset Pricing Model states that the required or expected return on asset i is:

The market risk premium is E(R_M) − R_f. The model assumes, in its standard form, that investors are rational and risk averse, have homogeneous expectations, can borrow and lend at a common risk-free rate, face no taxes or transaction costs, can trade divisible assets, and have access to the same information. It also assumes a single-period framework and that the market portfolio is mean–variance efficient.

Worked CAPM example. If the risk-free rate is 4%, expected market return is 10%, and an asset’s beta is 1.3:

E(R_i) = 4% + 1.3(10% − 4%) = 4% + 7.8% = 11.8%

An asset offering a return above its CAPM required return may appear underpriced, while one offering less may appear overpriced. This interpretation depends on the reliability of beta, the risk-free rate, and the market-risk-premium estimate.

7. The Security Market Line

The security market line (SML) graphs the CAPM relationship between beta and required return. Its vertical intercept is the risk-free rate, and its slope is the market risk premium. Each point on the line represents the return required for a particular level of systematic risk.

A security plotted above the SML offers more return than CAPM requires for its beta and may be considered positively mispriced. A security below the SML offers less than required and may be considered negatively mispriced. The SML differs from the capital market line: the SML applies to individual securities and portfolios using beta, while the capital market line applies to efficient portfolios using total standard deviation.

Figure: The security market line relates required return to systematic risk measured by beta.

8. Illustrative Case: Northstar Manufacturing Pension Portfolio

Background. Northstar Manufacturing has a defined-contribution pension plan. Employees are concerned that the existing portfolio is concentrated in Northstar shares and therefore exposed to firm-specific risk.

Problem. The committee must decide whether to retain the concentration, diversify, and determine whether the proposed assets offer adequate compensation for systematic risk.

Analysis. The committee estimates that Northstar shares have a beta of 1.4 and a standard deviation of 32%, while a broad equity fund has a beta of 1.0 and a standard deviation of 18%. The risk-free rate is 3% and the expected market return is 9%. CAPM indicates required returns of:

• Northstar: 3% + 1.4(9% − 3%) = 11.4%;
• broad equity fund: 3% + 1.0(9% − 3%) = 9%.

Although Northstar may have attractive expected returns, its high standard deviation includes substantial firm-specific risk. Employees’ human capital is also tied to Northstar, because poor company performance could affect both wages and share value. This creates a concentration risk not captured by examining the stock alone.

Approach and outcome. The committee reduces company stock to a limited allocation, introduces broad equity and bond funds, establishes rebalancing ranges, and communicates that diversification does not eliminate market losses. It documents assumptions, fees, liquidity needs, and conflicts of interest. The revised portfolio has lower firm-specific exposure and a more transparent risk budget.

Learning points. Diversification is especially important when financial wealth and employment income depend on the same company. CAPM can provide a benchmark for required return, but it should not replace scenario analysis, liquidity analysis, or fiduciary judgment.

9. Limitations, Behavioral Influences, and Governance

Model limitations

Historical volatility and correlation are not stable laws. Correlations often rise during crises, precisely when diversification is most valuable. Normal-distribution assumptions can underestimate fat tails and extreme losses. Beta varies with leverage, business mix, operating conditions, and the chosen estimation period. The true market portfolio is unobservable because it should include financial assets, private assets, real estate, and human capital.

CAPM is consequently best viewed as a disciplined benchmark rather than a complete description of actual markets. Multifactor models add variables such as size, value, profitability, investment, momentum, liquidity, or interest-rate exposure, but they also introduce estimation and interpretation challenges.

Behavioral influences

Investors may depart from mean–variance rationality. Overconfidence can lead to excessive trading; familiarity can create home-country or employer-stock bias; loss aversion can cause investors to hold losing positions or avoid appropriate risk; and recency bias can cause recent returns to dominate expectations. Herding and panic can increase correlations and market volatility.

Behavioral awareness improves practice. Investment committees should use written policies, pre-specified rebalancing rules, independent review, scenario analysis, and decision records. These controls reduce the influence of emotion without pretending that judgment can be eliminated.

Ethics and governance

Risk reporting should not hide illiquidity, leverage, model uncertainty, or downside exposure behind a single standard-deviation number. Advisers should disclose conflicts of interest, fees, assumptions, and the basis for forecasts. Fiduciaries should align portfolios with the client’s objectives, risk capacity, time horizon, liquidity requirements, tax position, and legal constraints. Artificial intelligence may assist with data analysis, stress testing, and anomaly detection, but outputs require validation, explainability, data-quality controls, and human accountability. A model’s sophistication does not guarantee that its forecasts are reliable.

Best-Practice Workflow for Risk Analysis

1. Define the investor’s objective, horizon, liquidity needs, and constraints.
2. Gather consistent return data and document the measurement period.
3. Estimate expected returns, variances, covariances, correlations, and betas.
4. Test sensitivity to alternative assumptions and crisis scenarios.
5. Evaluate both total risk and systematic risk.
6. Consider taxes, fees, liquidity, concentration, and behavioral risks.
7. Select a diversified portfolio consistent with risk capacity, not merely risk preference.
8. Monitor exposures and rebalance according to a documented policy.
9. Review whether the assumptions and objectives remain appropriate.

Chapter Summary

Return measures the gain or loss from an investment, while risk describes uncertainty surrounding that return. Expected return is a probability-weighted average; variance and standard deviation measure dispersion. Covariance and correlation describe co-movement and determine whether combining assets reduces portfolio risk. Diversification primarily removes unsystematic risk, whereas systematic risk remains.

Modern Portfolio Theory identifies efficient portfolios that offer the best feasible return for a given risk. Beta measures systematic sensitivity to the market. CAPM links required return to the risk-free rate plus beta multiplied by the market risk premium. The security market line represents this relationship graphically. These models are useful benchmarks but depend on uncertain estimates and restrictive assumptions. Effective professional practice combines quantitative models with diversification, scenario analysis, governance, ethical disclosure, and awareness of behavioral influences.

Key Terms

Expected return: Probability-weighted average of possible returns.

Holding-period return: Gain or loss, including income, relative to beginning investment value.

Variance: Probability-weighted squared dispersion around expected return.

Standard deviation: Square root of variance and a measure of total volatility.

Covariance: Measure of the joint movement of two variables.

Correlation: Standardized covariance ranging from −1 to +1.

Portfolio: Combination of assets held by an investor.

Diversification: Combining assets to reduce avoidable risk.

Systematic risk: Economy-wide risk that cannot be diversified away.

Unsystematic risk: Firm-specific risk that diversification can reduce.

Beta: Sensitivity of an asset’s return to market-return movements.

Market risk premium: Expected market return minus the risk-free rate.

CAPM: Model linking required return to risk-free return and systematic risk.

Security market line: Graph of CAPM-required return against beta.

Efficient portfolio: Portfolio offering the highest return for a specified risk or lowest risk for a specified return.

Efficient frontier: Set of all efficient portfolios.

Risk-free rate: Return on an assumed investment with no default or uncertainty over the relevant period.

Sharpe ratio: Excess portfolio return per unit of standard deviation.

Correlation risk: Risk that asset relationships change, especially during crises.

Mean–variance analysis: Portfolio analysis based on expected return and variance or standard deviation.

Review Questions

Short Answer Questions

1. What is the difference between expected return and realized return?
2. Why is standard deviation usually easier to interpret than variance?
3. Distinguish systematic risk from unsystematic risk.
4. What does a correlation of −1 imply for two assets?
5. Why is beta not a measure of total risk?

Long Answer Questions

1. Explain how covariance and correlation affect the risk of a two-asset portfolio.
2. Describe the assumptions underlying the Capital Asset Pricing Model and discuss their practical relevance.
3. Compare the efficient frontier, the capital market line, and the security market line.
4. Explain why diversification cannot eliminate systematic risk.
5. Evaluate the usefulness and limitations of historical standard deviation and beta in investment decisions.

Analytical / Application Questions

1. An asset has possible returns of −15%, 5%, and 25%, with probabilities of 0.2, 0.5, and 0.3. Calculate its expected return and standard deviation.
2. A portfolio invests 70% in Asset A and 30% in Asset B. Their standard deviations are 14% and 9%, and their correlation is −0.20. Derive the portfolio standard deviation and explain the effect of the negative correlation.
3. The risk-free rate is 3%, the expected market return is 8%, and a stock has beta 1.2. Calculate its CAPM-required return. If analysts expect the stock to return 10.5%, interpret its position relative to the security market line and identify limitations of that conclusion.

Multiple Choice Questions with Answers

Question 1

An investment is purchased for $40, sold for $44, and pays a $2 dividend. What is its holding-period return?

A. 5%

B. 10%

C. 15%

D. 20%

Correct Answer: C. 15%

Question 2

An asset has returns of 0% and 20% with equal probabilities. What is its expected return?

A. 5%

B. 10%

C. 15%

D. 20%

Correct Answer: B. 10%

Question 3

Which measure is expressed in the same units as investment returns?

A. Variance

B. Covariance

C. Standard deviation

D. Squared beta

Correct Answer: C. Standard deviation

Question 4

A correlation coefficient of −1 indicates that two assets have:

A. No relationship

B. Perfect positive linear movement

C. Perfect negative linear movement

D. Equal expected returns

Correct Answer: C. Perfect negative linear movement

Question 5

Which type of risk can generally be reduced through diversification?

A. Systematic risk

B. Unsystematic risk

C. Inflation risk affecting all assets

D. Market-wide recession risk

Correct Answer: B. Unsystematic risk

Question 6

A portfolio invests 50% in an asset with a 12% expected return and 50% in an asset with a 4% expected return. What is the portfolio’s expected return?

A. 6%

B. 8%

C. 10%

D. 16%

Correct Answer: B. 8%

Question 7

Beta primarily measures:

A. Total volatility of an individual asset

B. Sensitivity to market movements

C. The probability of default

D. The asset’s dividend yield

Correct Answer: B. Sensitivity to market movements

Question 8

If the risk-free rate is 4%, the expected market return is 10%, and beta is 0.5, CAPM implies a required return of:

A. 5%

B. 6%

C. 7%

D. 10%

Correct Answer: C. 7%

Question 9

The slope of the security market line is:

A. The risk-free rate

B. The market risk premium

C. The asset’s standard deviation

D. The portfolio weight

Correct Answer: B. The market risk premium

Question 10

Which statement best describes a major limitation of CAPM?

A. It ignores all forms of return

B. It assumes the true market portfolio and stable relationships can be identified

C. It cannot use beta

D. It applies only to bank deposits

Correct Answer: B. It assumes the true market portfolio and stable relationships can be identified