Risk, Return and Portfolio Analytics

Introduction

Investment decisions require more than identifying securities with high expected returns. Investors must assess uncertainty, interactions among assets, portfolio constraints and whether performance adequately compensates for risk. Portfolio analytics converts price, income and benchmark data into measures that support asset allocation, security selection and risk governance.

This chapter develops the progression from individual-asset returns to covariance-based portfolio theory, the capital asset pricing model (CAPM), multifactor models, performance attribution and risk-adjusted evaluation. The models are useful but depend critically on data quality, assumptions and disciplined interpretation.

Learning Objectives

After completing this chapter, students should be able to:

1. Calculate and interpret simple, logarithmic and compounded returns.
2. Estimate volatility, covariance and correlation.
3. Explain how diversification affects portfolio risk.
4. Construct minimum-variance, efficient and tangency portfolios.
5. Apply CAPM and multifactor models to expected returns.
6. Attribute active performance to allocation and selection decisions.
7. Evaluate portfolios using risk-adjusted performance measures.
8. Recognize estimation, governance and model-risk limitations.

1. Measuring Investment Returns

1.1 Holding-period return

For one share held from period t-1 to t, the simple holding-period return is

where P_t-1 is the initial price, P_t is the final price and D_t is cash income attributable to the holding period. All monetary amounts must use the same currency and per-share basis.

The expression treats distributions as part of terminal wealth without additional reinvestment income. If a distribution occurs before the period end and is reinvested, the exact return should be calculated from the resulting number of shares or from a properly constructed total-return index.

Example. A share is purchased for $100, ends the year at $106 and pays a $2 year-end dividend:

1.2 Arithmetic and geometric returns

For T observations, the arithmetic mean is

It estimates an expected one-period return when observations are representative. It does not describe the investor’s compounded growth rate.

The geometric mean is

It is defined provided cumulative wealth remains positive. For returns of 20% and −10%, cumulative growth is 1.20×0.90=1.08, so

The arithmetic mean is 5%. The difference reflects volatility drag.

1.3 Logarithmic returns and cash-flow timing

A logarithmic return based on a total-return index I_t is

Here, I_t must incorporate distributions using an explicitly defined reinvestment convention. Log returns are additive across consecutive periods:

The shortcut \ln((P_t+D_t)/P_t-1) is exact only when the distribution is effectively received at the period end, or when D_t represents terminal value generated under a specified reinvestment convention. It is not generally valid for unspecified intra-period cash flows.

External deposits and withdrawals must not be treated as investment income. Time-weighted returns remove their effect by geometrically linking subperiod returns. Money-weighted return instead solves for the internal rate of return that equates dated cash flows to terminal value.

1.4 Annualization

If returns are independently and identically distributed with finite variance, a mean return measured over intervals of equal length may be annualized approximately as

where m is the number of periods per year. The volatility square-root rule is unreliable when returns exhibit serial correlation, changing volatility or irregular time intervals. Compounded annual return should instead be derived from cumulative wealth.

2. Risk, Volatility and Dependence

2.1 Variance and standard deviation

For a random return R, population variance is

where \mu=E(R). Standard deviation \sigma has the same return units as R.

Given a sample of T returns, an unbiased estimator under standard random-sampling conditions is

Volatility measures dispersion, not every dimension of risk. It treats gains and losses symmetrically and may understate exposure to skewness, illiquidity and extreme losses.

2.2 Covariance and correlation

Covariance measures whether two returns move together:

Its units are squared return units. Correlation standardizes covariance:

Correlation is dimensionless. A low or negative correlation creates diversification potential, although correlations may rise during market crises.

2.3 Tail and downside measures

Common supplementary measures include:

• Downside deviation: dispersion of returns below a target.
• Value at Risk (VaR): a loss threshold exceeded with a stated probability over a stated horizon.
• Expected shortfall: expected loss conditional on exceeding the VaR threshold.
• Maximum drawdown: largest peak-to-trough decline in observed wealth.

A statement such as “one-day 99% VaR is $2 million” means that, under the model, the probability of losing more than $2 million over one day is 1%. It does not specify the size of losses beyond that threshold.

3. Portfolio Return and Diversification

Let w_i be the proportion invested in asset i, with \sum_i w_i=1. Negative weights represent short positions when permitted.

Portfolio return is

Therefore,

where \boldsymbol{\mu} is the vector of expected returns.

Portfolio variance is

where \boldsymbol{\Sigma} is the covariance matrix. A valid covariance matrix must be symmetric and positive semidefinite, ensuring that calculated variance cannot be negative.

For two assets,

The final term explains why portfolio risk is generally not the weighted average of individual volatilities.

Worked Example: A Two-Asset Portfolio

Assume:

Correlation is 0.25, so covariance is

Expected portfolio return is

Variance is

Thus,

Although Asset B has 18% volatility, adding it does not raise portfolio volatility to the weighted average of 13.2%, because the assets are imperfectly correlated.

Systematic and idiosyncratic risk

Idiosyncratic risk is specific to an issuer or security and can be reduced through broad diversification. Systematic risk arises from economy-wide forces and cannot generally be eliminated by holding more securities exposed to the same factors. Diversification reduces concentration; it does not guarantee protection against market-wide losses.

4. Mean–Variance Portfolio Theory

Harry Markowitz’s framework selects portfolios using expected return, variance and covariance. Its basic assumptions include a single investment horizon, preferences based on expected return and variance, divisible assets and a consistent opportunity set. Normal returns or quadratic utility can justify focusing only on mean and variance, although real returns may be non-normal.

4.1 Global minimum-variance portfolio

The global minimum-variance portfolio solves

Without short-sale constraints and with invertible covariance matrix,

For two assets, the weight of Asset A is

This follows by substituting w_B=1-w_A into portfolio variance and setting its derivative with respect to w_A equal to zero.

Using the previous data:

Thus w_B=0.1647. The portfolio has an expected return of approximately 8.66% and volatility of 9.54%.

4.2 Efficient frontier

The minimum-variance portfolio for a target return \mu^* solves

subject to

The efficient frontier is the upper portion of the minimum-variance boundary: no other feasible portfolio offers a higher expected return for the same risk or lower risk for the same expected return.

4.3 Risk-free asset and tangency portfolio

With risk-free rate R_f, the risky portfolio maximizing the Sharpe ratio is the tangency portfolio. When unrestricted weights are allowed and \boldsymbol{\Sigma} is invertible,

The capital market line (CML) combines the risk-free asset and tangency portfolio:

This equation applies to efficient combinations of the risk-free asset and tangency portfolio. It does not price arbitrary individual securities.

Figure: The capital market line passes through the risk-free asset and is geometrically tangent to the efficient frontier at portfolio T.

In practice, optimization often includes long-only limits, asset-class ranges, liquidity requirements, transaction costs and regulatory capital constraints. These turn the smooth theoretical frontier into a constrained opportunity set.

5. CAPM and Security Pricing

The CAPM relates expected return to non-diversifiable market risk:

where R_M is market return and

Beta is dimensionless. A beta of 1.2 indicates that the asset historically moved about 1.2% for each 1% market movement within the fitted linear model, not that it will earn 1.2 times the market return.

The security market line relates expected return to beta. Unlike the CML, it applies to individual assets and portfolios if CAPM holds.

Example. If R_f=4\%, E(R_M)=9\% and \beta_i=1.2, CAPM required return is

If the asset earns 11%, its realized CAPM alpha for that period is 1%, subject to the limitations of using realized returns and estimated beta.

CAPM depends on restrictive assumptions, including mean–variance investors, frictionless markets, homogeneous expectations and a common investment horizon. The true market portfolio is also unobservable. CAPM therefore serves as a benchmark rather than an infallible forecasting rule.

6. Multifactor Models

A statistical or economic factor model expresses asset return as

where F_j is factor j, \beta_ij is exposure, and \varepsilon_i is idiosyncratic return. In a time-series regression, the intercept and factors must be defined consistently—often as excess returns.

Common factors include market, size, value, momentum, interest-rate, credit-spread, inflation and currency factors. For banks and insurers, duration, credit quality, liquidity and catastrophe exposure may be especially important.

If residuals are uncorrelated with factors, variance decomposes as

This decomposition supports hedging, risk budgeting and stress testing. Estimated factor relationships may nevertheless be unstable, correlated with one another or discovered through data mining.

7. Performance Attribution

Performance measurement asks what return occurred; attribution asks why it occurred. For sector i, let portfolio and benchmark weights be w_Pi and w_Bi, and returns be r_Pi and r_Bi. One arithmetic Brinson convention defines:

where R_B is total benchmark return. Under internally consistent single-period data, these effects sum to active return.

Worked Attribution

Portfolio return is 8.8%, benchmark return is 7.5%, and active return is 1.3%. The effects are:

Multiperiod attribution requires a linking method because arithmetic effects do not compound automatically. Organizations must document their attribution convention and maintain consistent benchmark classifications.

8. Risk-Adjusted Performance Measures

Sharpe ratio

It measures excess return per unit of total volatility and is appropriate when the evaluated portfolio represents the investor’s total risky investment.

Treynor ratio

It uses systematic risk and is most meaningful for a well-diversified portfolio held as part of a broader portfolio.

Jensen’s alpha

It measures return relative to CAPM’s prediction.

Sortino ratio

Here R_T is the target return and \sigma_D is downside deviation measured relative to a clearly defined target.

Information ratio

Tracking error TE is the standard deviation of active returns R_p-R_B. The ratio assesses consistency of active management relative to a benchmark.

These ratios must use returns, risk measures and risk-free rates over consistent horizons. Comparisons based on short samples may be statistically unreliable.

9. Illustrative Case: Asset Allocation for Meridian Bank

Background

This fictional case concerns Meridian Bank’s employee pension reserve. The investment committee is considering Asset A, a diversified bond portfolio, and Asset B, a public-equity portfolio. Their expected returns, volatilities and correlation are those used earlier: 8%, 10%, 12%, 18% and 0.25, respectively. The risk-free rate is 3%.

Problem

The committee seeks an 8% expected return while maximizing diversification. Short selling is prohibited, but investment in the risk-free asset is permitted.

Analysis and approach

Using the covariance matrix and excess returns, the tangency-portfolio formula produces:

Its expected return is

Its variance is approximately 0.0103316, giving volatility of 10.16%. Its Sharpe ratio is approximately

Let y be the proportion invested in the tangency portfolio. To achieve the 8% target:

Therefore y=0.77778. Total allocations become:

• Asset A: 0.77778×0.642857=50.00\%
• Asset B: 0.77778×0.357143=27.78\%
• Risk-free asset: 22.22%

Portfolio volatility is

Outcome

The allocation reaches the 8% expected-return target with lower volatility than investing solely in either risky asset. However, the committee also subjects the portfolio to equity-crash, interest-rate and liquidity stress scenarios. It adopts allocation ranges rather than treating estimated weights as exact truths.

Learning points

1. Asset allocation depends on covariance as well as expected return.
2. Combining a tangency portfolio with a risk-free asset can match different risk tolerances.
3. Optimized weights are highly sensitive to estimated expected returns.
4. Stress tests and governance constraints complement mathematical optimization.

10. Implementation, Technology and Governance

Modern portfolio systems estimate covariance matrices, solve constrained optimizations and monitor factor exposures in near real time. Machine-learning methods can model nonlinear patterns and alternative data, but they create risks of overfitting, opacity and regime dependence.

Sound professional practice includes:

• adjusting prices correctly for splits, distributions and corporate actions;
• separating external cash flows from investment returns;
• testing covariance matrices for numerical validity;
• using shrinkage or factor-based estimates when samples are limited;
• conducting backtests without look-ahead or survivorship bias;
• reporting turnover, transaction costs and liquidity effects;
• validating models independently and documenting overrides;
• stress-testing correlations, volatility and factor exposures;
• maintaining benchmark and performance-calculation governance.

Banks and insurers must also consider applicable capital, liquidity, concentration and prudent-investment requirements. Ethical concerns arise when analysts selectively choose benchmarks, conceal model limitations or present uncertain forecasts as guaranteed outcomes. AI-assisted decisions should remain explainable, auditable and subject to human accountability.

11. Limitations and Contemporary Developments

Historical estimates may be poor forecasts because expected returns, volatilities and correlations change over time. Sample covariance matrices become unstable when the number of assets is large relative to the number of observations. Mean–variance optimization can therefore produce extreme and fragile allocations.

Contemporary responses include covariance shrinkage, Bayesian expected returns, robust optimization, resampling, volatility targeting and risk-parity approaches. Climate transition, physical climate risk, cyber risk and geopolitical fragmentation are increasingly incorporated through scenarios and factor exposures. These methods improve decision support but do not eliminate model uncertainty.

Chapter Summary

Investment return must be measured using internally consistent prices, income and cash-flow timing. Log returns are additive when based on a properly constructed total-return series. Volatility describes dispersion, while covariance and correlation determine how individual risks combine in a portfolio.

Markowitz theory identifies minimum-variance and efficient portfolios. Introducing a risk-free asset produces a tangency portfolio and capital market line. CAPM links expected return to market beta, while multifactor models provide broader explanations of systematic exposure. Attribution separates active return into allocation, selection and interaction effects. Sharpe, Treynor, Sortino, Jensen alpha and information ratios evaluate performance relative to different concepts of risk.

Practical asset allocation must supplement optimization with constraints, transaction costs, stress testing, data controls and model governance. Quantitative results are decision aids rather than guarantees.

Key Terms

Holding-period return: Total price change and investment income relative to initial value.

Total-return index: Index incorporating price changes and reinvested distributions under a stated convention.

Log return: Natural logarithm of a total-return wealth relative.

Geometric mean: Constant periodic return producing the observed cumulative growth.

Volatility: Standard deviation of returns.

Covariance: Measure of joint variation between two returns.

Correlation: Standardized covariance bounded between −1 and 1.

Diversification: Reduction of concentration and idiosyncratic risk by combining exposures.

Covariance matrix: Matrix containing asset variances and pairwise covariances.

Global minimum-variance portfolio: Fully invested portfolio with the lowest attainable variance.

Efficient frontier: Portfolios offering the highest expected return for each risk level.

Tangency portfolio: Risky portfolio with the highest Sharpe ratio relative to a risk-free asset.

Capital market line: Risk–return relationship for combinations of the risk-free asset and tangency portfolio.

Beta: Sensitivity of an asset’s return to market return in a linear model.

CAPM: Model relating expected return to market beta.

Factor model: Model explaining returns through systematic factors and idiosyncratic residuals.

Performance attribution: Decomposition of active return into sources such as allocation and selection.

Tracking error: Standard deviation of portfolio return relative to benchmark return.

Review Questions

Short Answer Questions

1. Distinguish between arithmetic and geometric mean returns.
2. Why must cash-flow timing be specified when calculating total and log returns?
3. What does a correlation of zero imply about two asset returns?
4. Distinguish between the capital market line and security market line.
5. What does tracking error measure?

Long Answer Questions

1. Explain how covariance creates diversification benefits in a two-asset portfolio.
2. Derive the two-asset global minimum-variance weight.
3. Discuss the assumptions, applications and limitations of CAPM.
4. Compare CAPM with multifactor return models.
5. Evaluate the strengths and weaknesses of major risk-adjusted performance measures.

Analytical / Application Questions

1. An asset rises from $50 to $53 and pays a $1 end-period dividend. Calculate its simple and logarithmic holding-period returns.
2. Construct the expected return and variance of a portfolio invested 30% and 70% in assets with returns of 6% and 11%, volatilities of 8% and 16%, and correlation of −0.20.
3. Design a governance process for an AI-supported portfolio optimization system used by an insurance company.

Multiple Choice Questions with Answers

Question 1

A share rises from $80 to $84 and pays a $2 end-period dividend. What is its holding-period return?

A. 5.0%

B. 7.5%

C. 10.0%

D. 2.5%

Correct Answer: B. 7.5%

Question 2

Which condition makes \ln((P_t+D_t)/P_t-1) an exact one-period log-return expression?

A. The distribution is effectively received at period end under the stated wealth convention

B. The asset has zero volatility

C. The distribution is an external investor deposit

D. Prices are normally distributed

Correct Answer: A. The distribution is effectively received at period end under the stated wealth convention

Question 3

Assets A and B have volatilities of 10% and 20% and correlation 0.30. What is their covariance?

A. 0.006

B. 0.060

C. 0.015

D. 0.030

Correct Answer: A. 0.006

Question 4

A portfolio invests 40% in an asset returning 5% and 60% in an asset returning 10%. What is its expected return?

A. 6.0%

B. 7.0%

C. 8.0%

D. 9.0%

Correct Answer: C. 8.0%

Question 5

Which portfolio lies at the point where the capital market line touches the efficient frontier?

A. Market-neutral portfolio

B. Tangency portfolio

C. Global maximum-return portfolio

D. Zero-beta portfolio

Correct Answer: B. Tangency portfolio

Question 6

Under CAPM, the risk-free rate is 3%, expected market return is 8%, and an asset's beta is 1.4. What is its required return?

A. 7.0%

B. 8.0%

C. 10.0%

D. 12.2%

Correct Answer: C. 10.0%

Question 7

Which risk is most directly reduced through broad security diversification?

A. Systematic market risk

B. Risk-free-rate risk

C. Idiosyncratic risk

D. Economy-wide inflation risk

Correct Answer: C. Idiosyncratic risk

Question 8

In Brinson attribution, which effect primarily measures the impact of overweighting or underweighting sectors relative to a benchmark?

A. Allocation effect

B. Selection effect

C. Compounding effect

D. Residual volatility

Correct Answer: A. Allocation effect

Question 9

A portfolio earns 11%, the risk-free rate is 3%, and portfolio volatility is 16%. What is its Sharpe ratio?

A. 0.25

B. 0.50

C. 0.69

D. 0.88

Correct Answer: B. 0.50

Question 10

Which measure equals average active return divided by tracking error?

A. Treynor ratio

B. Sortino ratio

C. Information ratio

D. Jensen beta

Correct Answer: C. Information ratio