Bachelors Program in Accounting and Finance
Fundamentals of Financial Management
•
Bachelor
Risk, Return and Portfolio Theory
Introduction
Financial decisions involve uncertain future cash flows. Investors therefore evaluate not only how much return an investment may generate, but also the uncertainty surrounding that return. Risk–return analysis provides the quantitative foundation for security valuation, portfolio construction, capital budgeting and the determination of required returns.
A central insight of portfolio theory is that an asset should not be evaluated in isolation. Its contribution to portfolio risk depends on how its returns move relative to other assets. Diversification can eliminate much asset-specific risk, but it cannot eliminate economy-wide systematic risk. The Capital Asset Pricing Model (CAPM) consequently links required return to beta, a measure of systematic risk rather than total volatility.
Learning Objectives
After studying this chapter, students should be able to:
Calculate realized and expected returns.
Measure variance, standard deviation, covariance and correlation.
Explain the distinction between systematic and unsystematic risk.
calculate portfolio expected return and portfolio variance.
explain diversification, the efficient frontier and optimal portfolio choice.
Measure beta and apply the CAPM.
Interpret the Security Market Line and risk-adjusted performance.
Evaluate the assumptions and limitations of portfolio theory and CAPM.
Relate risk and required return to market efficiency.
1. Return and Investment Performance
1.1 Holding-period return
An investment return consists of income received and any change in the asset’s price. For one holding period:
R = \frac{P1 - P0 + D1}{P0}
where:
R = holding-period return, expressed as a decimal or percentage;
P0 = price at the beginning of the period, in currency units;
P1 = price at the end of the period, in currency units; and
D1 = cash income received during the period, such as a dividend.
This formula assumes that the income is received within the holding period and ignores taxes and transaction costs unless they are explicitly deducted.
Example. An investor buys a share for $50, receives a $2 dividend and sells it for $54. The return is:
R = \frac{54 - 50 + 2}{50} = 0.12 = 12\%
Returns must be measured over comparable time intervals. A monthly return should not be directly compared with an annual required return without appropriate compounding or annualization.
1.2 Expected return
Expected return is the probability-weighted average of possible returns:
E(R) = \sums=1S psRs
where ps is the probability of state s, Rs is the return in that state and \sum ps = 1. The model assumes that the possible states and their probabilities are meaningfully specified.
Expected return is not a guaranteed outcome. It is the probability-weighted centre of the return distribution.
2. Measuring Stand-Alone Risk
2.1 Variance and standard deviation
Variance measures the expected squared deviation of returns from their expected value:
σ2 = \sums=1Sps[Rs-E(R)]2
Standard deviation is the positive square root of variance:
σ = \sqrt{σ2}
Here, σ2 is measured in squared-return units, while σ has the same unit as return. Standard deviation is therefore easier to interpret. These measures treat positive and negative deviations symmetrically and provide a complete description of risk only under restrictive distributions, such as the normal distribution, or where investors care only about means and variances.
Using the returns in Worked Example 1:
State
Return
Deviation from 10%
Squared deviation
Probability-weighted value
Recession
−10%
−20%
0.0400
0.0080
Normal growth
10%
0%
0.0000
0.0000
Expansion
30%
20%
0.0400
0.0080
σ2 = 0.0080 + 0 + 0.0080 = 0.0160
σ = \sqrt{0.0160} = 0.1265 \approx 12.65\%
When historical observations are treated as a sample rather than a complete probability distribution, the usual unbiased sample variance uses denominator n-1:
s2 = \frac{\sumt=1n(Rt-\bar{R})2}{n-1}
The arithmetic mean and sample variance are estimates and may be unreliable when the sample is short or the return-generating process changes.
2.2 Coefficient of variation
For investments with different expected returns, relative risk may be compared through:
CV = \frac{σ}{E(R)}
The coefficient of variation measures standard deviation per unit of expected return. It is generally meaningful when expected return is positive and measured consistently. It becomes unstable when expected return is close to zero and is not a substitute for portfolio analysis.
2.3 Downside and non-normal risk
Variance does not distinguish undesirable losses from favourable surprises. Practitioners therefore also examine downside deviation, Value at Risk, expected shortfall, drawdowns and stress-test losses. These measures are particularly relevant when returns are skewed, have heavy tails or include illiquid assets. No single statistic captures every economically important form of risk.
3. Covariance, Correlation and Diversification
3.1 Covariance
Covariance measures whether two returns tend to move together:
Cov(RA,RB) = \sums=1Sps[RA,s-E(RA)][RB,s-E(RB)]
A positive covariance indicates a tendency to move in the same direction; a negative covariance indicates opposite movement. Covariance depends on the units and volatility of both variables, making direct comparisons difficult.
3.2 Correlation
Correlation standardizes covariance:
ρAB = \frac{Cov(RA,RB)}{σAσB}
Equivalently:
Cov(RA,RB) = ρABσAσB
Correlation lies between −1 and +1 when both standard deviations are positive. A correlation of +1 means perfect positive linear co-movement; −1 means perfect negative linear co-movement; and zero means no linear association. Zero correlation does not necessarily imply statistical independence.
3.3 Why diversification works
Diversification combines assets whose returns do not move perfectly together. Firm-specific adverse outcomes may then be offset by neutral or favourable outcomes elsewhere. The benefit depends on covariance, not merely on the number of securities held.
This is a conceptual decomposition of return variance under an appropriate factor model; standard deviations themselves are not generally additive.
Systematic risk arises from market-wide factors such as interest rates, inflation, recessions and geopolitical shocks. It cannot normally be eliminated through broad domestic equity diversification.
Unsystematic risk is specific to a company or industry, such as product failure, fraud or a factory fire. It can be substantially reduced through diversification.
Portfolio risk approaches systematic risk from above as the number of securities increases
Figure: Portfolio risk approaches systematic risk from above as the number of securities increases
Figure: As the number of suitably selected securities increases, portfolio risk approaches—but does not fall below—the systematic-risk floor.
Simply holding many highly correlated securities provides little protection. Effective diversification normally spans companies, industries, asset classes, countries and risk factors, while accounting for costs and currency exposure.
4. Portfolio Return and Risk
4.1 Expected portfolio return
For a portfolio containing n assets:
E(Rp) = \sumi=1nwiE(Ri)
where wi is the proportion of portfolio value invested in asset i and \sum wi=1. A negative weight represents a short position where short selling is permitted.
4.2 Portfolio variance
The general portfolio variance formula is:
σp2 = \sumi=1n\sumj=1nwiwjCov(Ri,Rj)
In matrix notation, this is σp2=wTΣw, where w is the weight vector and Σ is the covariance matrix.
For two assets, A and B:
σp2 = wA2σA2+wB2σB2+2wAwBρABσAσB
The weights must refer to the same portfolio and the volatilities and correlation must be estimated over consistent horizons.
Worked Example 2: A Two-Asset Portfolio
Suppose Asset A has an expected return of 8% and standard deviation of 10%. Asset B has an expected return of 14% and standard deviation of 20%. Their correlation is 0.25. A portfolio invests 60% in A and 40% in B.
The weighted average of the individual standard deviations is 14%. The portfolio’s 11.14% standard deviation is lower because correlation is below +1.
4.3 Two-asset minimum-variance weight
If weights sum to one and unrestricted borrowing or short selling is allowed, minimizing the two-asset variance gives:
wA^* = \frac{σB2-Cov(RA,RB)}{σA2+σB2-2Cov(RA,RB)}
and wB^*=1-wA^*. This solution is valid when the denominator is positive. Under long-only constraints, an infeasible negative or above-one weight must be replaced by the relevant boundary solution.
Thus, wB^*=0.125. The resulting variance is 0.009375, so standard deviation is approximately 9.68%. Its expected return is 8.75%.
5. Modern Portfolio Theory
Harry Markowitz’s mean–variance theory formalized the principle that investors should consider expected return, variance and covariance jointly.
5.1 Feasible set and efficient frontier
The feasible set contains all portfolios obtainable from the available assets and applicable constraints. A portfolio is mean–variance efficient if no feasible alternative:
offers a higher expected return for the same or lower risk; or
offers lower risk for the same or higher expected return.
The upper boundary of efficient risky portfolios is the efficient frontier. Portfolios below it are inefficient because another portfolio provides a better risk–return combination.
5.2 Investor choice
Portfolio theory usually assumes that investors prefer higher expected return and lower variance, other things equal. An investor chooses among efficient portfolios according to risk tolerance. More risk-averse investors select lower-risk portfolios; less risk-averse investors may choose portfolios with greater expected return and volatility.
5.3 Risk-free borrowing and lending
When a common risk-free asset is available and investors can borrow and lend at the risk-free rate, combinations of the risk-free asset and a risky portfolio lie on a straight capital allocation line. Under idealized CAPM assumptions, all investors hold the market portfolio of risky assets, combined with borrowing or lending according to risk tolerance. The resulting efficient line is the Capital Market Line (CML).
The CML applies only to efficient portfolios and relates expected return to total standard deviation. The Security Market Line, introduced later, applies to individual securities and portfolios and relates required return to beta. The two lines must not be confused.
6. Systematic Risk and Beta
Beta measures an asset’s sensitivity to market returns:
βi = \frac{Cov(Ri,RM)}{Var(RM)}
where Ri is the asset return and RM is the market return. Beta is dimensionless because both numerator and denominator are in squared-return units.
Interpretation:
Beta
Typical interpretation
β=1
Same systematic sensitivity as the market
β>1
Greater market sensitivity
0<β<1
Positive but lower market sensitivity
β=0
No linear market sensitivity
β<0
Tends to move opposite to the market
If an asset’s covariance with the market is 0.018 and market variance is 0.012:
βi=\frac{0.018}{0.012}=1.50
Portfolio beta is the weighted average of component betas:
βp=\sumi=1nwiβi
Estimated beta depends on the selected market index, sampling frequency, estimation period and changing business leverage. It is therefore an estimate, not an enduring physical characteristic.
7. The Capital Asset Pricing Model
7.1 CAPM equation
The CAPM states that the required expected return on asset i is:
E(Ri)=Rf+βi[E(RM)-Rf]
where:
Rf = risk-free rate for the relevant horizon;
E(RM) = expected market return;
E(RM)-Rf = market risk premium; and
βi = systematic risk of asset i.
The risk-free rate, market return and required return must use consistent nominal or real terms, currency and investment horizons.
If Rf=4\%, E(RM)=10\% and βi=1.5:
E(Ri)=4\%+1.5(10\%-4\%)=13\%
CAPM compensates investors only for systematic risk because diversified investors can eliminate most unsystematic risk without sacrificing expected return.
7.2 Security Market Line
The Security Market Line (SML) plots required return against beta. Its intercept is the risk-free rate and its slope is the market risk premium.
Security Market Line showing required return increasing linearly with beta
Figure: Security Market Line showing required return increasing linearly with beta
Figure: Under CAPM, required return increases linearly with systematic risk measured by beta.
A security plotting above the SML offers a forecast return greater than the CAPM required return and may appear undervalued. A security below it may appear overvalued. This interpretation depends on the reliability of forecasts and the validity of CAPM.
Jensen’s alpha is the difference between realized or forecast return and the CAPM benchmark:
αi=Ri-[Rf+βi(RM-Rf)]
Positive historical alpha is not automatically evidence of skill; it may result from omitted risk factors, luck, data mining or inaccurate beta estimation.
7.3 CAPM assumptions and limitations
The standard CAPM relies on strong assumptions, including:
investors are rational mean–variance optimizers;
expectations are homogeneous;
markets have no taxes, transaction costs or significant restrictions;
investors can borrow and lend at a common risk-free rate;
all relevant assets are tradable and divisible; and
investors share a single investment horizon.
In practice, betas vary over time, borrowing rates exceed lending rates, markets are incomplete and expected returns are unobservable. Empirical research also finds that size, value, momentum, profitability and investment factors may help explain returns beyond market beta. Multifactor models can improve description, but they introduce additional estimation and interpretation challenges.
8. Market Efficiency and the Risk–Return Relationship
The Efficient Market Hypothesis (EMH) proposes that market prices reflect available information to varying degrees.
Form
Information reflected in prices
Main implication
Weak form
Historical prices and trading data
Past price patterns should not consistently generate abnormal returns after costs
Semi-strong form
All publicly available information
Public analysis should not consistently earn abnormal risk-adjusted returns
Strong form
All public and private information
Even private information would not provide an advantage; this form is generally considered unrealistic
Efficiency does not mean that prices are always correct or stable. It means that abnormal profits are difficult to earn consistently using the specified information set after adjusting for risk and costs.
Tests of efficiency face the joint-hypothesis problem: an observed abnormal return may indicate market inefficiency, or it may indicate that the asset-pricing model used to calculate normal return is wrong. CAPM and market efficiency are therefore related but distinct. CAPM specifies a risk–return benchmark; EMH concerns how information is incorporated into prices.
9. Illustrative Case: Portfolio Review at Northbridge Foundation
Background
Northbridge Foundation is a fictional charitable institution investing a long-term reserve. Its investment committee is considering a portfolio of three equity funds. Forecasts are annual and based on a common economic scenario model.
Fund
Expected return
Standard deviation
Beta
Current weight
A: Defensive
7%
9%
0.60
40%
B: Broad market
11%
16%
1.10
35%
C: Growth
16%
28%
1.70
25%
The correlations are ρAB=0.30, ρAC=0.15 and ρBC=0.55. The risk-free rate is 3.5%, and the expected market risk premium is 6%. Governance policy limits portfolio beta to 1.00.
Problem and analysis
The current expected return is:
E(Rp)=0.40(7\%)+0.35(11\%)+0.25(16\%)=10.65\%
Portfolio beta is:
βp=0.40(0.60)+0.35(1.10)+0.25(1.70)=1.05
The portfolio therefore breaches the beta limit.
The covariances are 0.00432 for A–B, 0.00378 for A–C and 0.02464 for B–C. Applying the full portfolio variance formula gives variance of 0.0156096 and standard deviation of approximately 12.49%. This is below the weighted average standard deviation of 16.2%, demonstrating diversification.
CAPM requires:
E(Rp)=3.5\%+1.05(6\%)=9.80\%
The forecast return exceeds the CAPM benchmark by 0.85 percentage points, but this forecast alpha is uncertain.
Approach and solution
The committee transfers five percentage points from Growth Fund C to Defensive Fund A. Revised weights are 45%, 35% and 20%.
The revised portfolio has:
expected return of 10.20%;
beta of 0.995;
variance of approximately 0.0134031; and
standard deviation of approximately 11.58%.
Its CAPM required return is:
3.5\%+0.995(6\%)=9.47\%
Outcome and learning points
The revised allocation meets the beta policy and reduces estimated total risk, although its expected return also declines. The decision illustrates that governance constraints, covariance and downside capacity must be considered alongside return forecasts. It also shows that beta and standard deviation answer different questions: beta measures market exposure, whereas standard deviation measures total volatility.
10. Professional Practice, Governance and Contemporary Developments
Sound risk–return analysis requires controls over both models and data. Professionals should:
use consistent horizons, currencies and nominal or real assumptions;
document data sources, estimation windows and benchmark choices;
conduct sensitivity analysis for returns, correlations and betas;
stress-test recessions, interest-rate shocks and liquidity constraints;
account for transaction costs, taxes and rebalancing;
avoid presenting forecasts as guaranteed outcomes; and
monitor concentration by issuer, industry, geography and risk factor.
Ethical concerns arise when analysts selectively choose estimation periods, conceal unfavourable scenarios or overstate model precision. Insider trading, market manipulation and misleading performance reporting violate legal or professional standards. Governance should include independent model review, clear investment mandates and transparent reporting.
Technology enables rapid covariance estimation, optimization and scenario simulation. Machine-learning systems can detect nonlinear patterns, but they also create risks of overfitting, opaque decisions and unstable relationships. Human oversight remains necessary. Contemporary portfolio management increasingly incorporates climate transition risk, cybersecurity, geopolitical fragmentation and environmental, social and governance information. These factors should be analyzed as financially relevant exposures rather than mechanically assumed to increase or reduce returns.
Chapter Summary
Risk–return analysis begins with holding-period and expected returns. Variance and standard deviation measure stand-alone volatility, while covariance and correlation measure co-movement. Portfolio expected return is a weighted average, but portfolio variance depends on all variances, covariances and weights.
Diversification reduces unsystematic risk when assets are not perfectly positively correlated, but systematic risk remains. Markowitz portfolio theory identifies efficient portfolios offering the greatest expected return for a given risk level. Beta measures exposure to market movements, and CAPM links beta to required return through the Security Market Line. CAPM is useful but relies on restrictive assumptions and imperfect estimates.
Market efficiency concerns how information enters prices, while asset-pricing models determine benchmark returns. Professional application requires reliable data, consistent assumptions, stress testing, ethical communication and recognition of model limitations.
Key Terms
Alpha: Return above or below an asset-pricing benchmark.
Beta: Sensitivity of an asset’s return to market return.
Capital Asset Pricing Model: Model relating required return to beta and the market risk premium.
Correlation: Standardized measure of linear co-movement between two variables.
Covariance: Measure of the joint movement of two returns.
Diversification: Combining investments to reduce portfolio risk.
Efficient frontier: Set of portfolios offering the highest expected return for each risk level.
Expected return: Probability-weighted average of possible returns.
Holding-period return: Total income and price change relative to initial investment.
Market portfolio: Portfolio containing all risky assets in market-value proportions in the theoretical CAPM.
Market risk premium: Expected market return minus the risk-free rate.
Portfolio variance: Total squared volatility determined by weights, variances and covariances.
Required return: Minimum expected return justified by an investment’s risk.
Risk-free rate: Return on an asset assumed to have no default or reinvestment uncertainty over the relevant horizon.
Security Market Line: CAPM relationship between required return and beta.
Standard deviation: Square root of variance and a measure of total volatility.
Systematic risk: Market-wide risk that cannot be eliminated through ordinary diversification.
Unsystematic risk: Company- or industry-specific risk that can be diversified.
Variance: Expected squared deviation of returns from expected return.
Review Questions
Short Answer Questions
Distinguish between realized return and expected return.
Why is standard deviation easier to interpret than variance?
What does a negative covariance indicate?
Distinguish systematic risk from unsystematic risk.
Why does CAPM use beta rather than standard deviation?
Long Answer Questions
Explain how covariance and correlation determine the benefits of diversification.
Discuss the efficient frontier and how investor risk aversion affects portfolio choice.
Derive and interpret the two-asset portfolio variance formula.
Explain the assumptions, uses and limitations of CAPM.
Critically evaluate the relationship between market efficiency and asset-pricing models.
Analytical / Application Questions
An asset has possible returns of −5%, 8% and 20% with probabilities of 0.25, 0.50 and 0.25. Calculate its expected return, variance and standard deviation.
A portfolio invests 30% in an asset with standard deviation 12% and 70% in an asset with standard deviation 18%. Their correlation is −0.20. Calculate portfolio variance and standard deviation.
A share has beta 1.30. If the risk-free rate is 4% and the expected market return is 9%, calculate its CAPM required return and assess a forecast return of 11%.
Multiple Choice Questions with Answers
Question 1
An investment has returns of −10%, 10% and 30% with probabilities of 0.20, 0.60 and 0.20. What is its expected return?
A. 8%
B. 10%
C. 12%
D. 14%
Correct Answer: B. 10%
Question 2
Using an expected return of 10% for the distribution −10%, 10% and 30% with probabilities 0.20, 0.60 and 0.20, what is the variance?
A. 0.0100
B. 0.0160
C. 0.0200
D. 0.0256
Correct Answer: B. 0.0160
Question 3
Which measure expresses the strength and direction of linear co-movement on a scale from −1 to +1?
A. Variance
B. Beta
C. Correlation
D. Alpha
Correct Answer: C. Correlation
Question 4
A portfolio invests 60% in an asset expected to return 8% and 40% in an asset expected to return 14%. What is the portfolio’s expected return?
A. 9.2%
B. 10.0%
C. 10.4%
D. 11.6%
Correct Answer: C. 10.4%
Question 5
Which risk can normally be reduced substantially through broad diversification?
A. Systematic risk
B. Interest-rate risk affecting the whole market
C. Economy-wide recession risk
D. Company-specific risk
Correct Answer: D. Company-specific risk
Question 6
An asset’s covariance with the market is 0.018 and market variance is 0.012. What is its beta?
A. 0.67
B. 1.20
C. 1.50
D. 2.16
Correct Answer: C. 1.50
Question 7
If the risk-free rate is 4%, the expected market return is 10% and a share’s beta is 1.5, what return does CAPM require?
A. 10%
B. 11%
C. 13%
D. 15%
Correct Answer: C. 13%
Question 8
Which statement correctly distinguishes the CML from the SML?
A. The CML uses beta, while the SML uses standard deviation.
B. The CML applies to efficient portfolios, while the SML relates required return to beta.
C. The CML applies only to individual shares, while the SML applies only to risk-free assets.
D. The CML and SML are identical under all market conditions.
Correct Answer: B. The CML applies to efficient portfolios, while the SML relates required return to beta.
Question 9
Under semi-strong market efficiency, prices reflect which information set?
A. Historical prices only
B. Publicly available information
C. Private information only
D. No accounting information
Correct Answer: B. Publicly available information
Question 10
A portfolio has weights of 40%, 35% and 25% in assets with betas of 0.60, 1.10 and 1.70. What is the portfolio beta?
A. 0.95
B. 1.00
C. 1.05
D. 1.20
Correct Answer: C. 1.05
Fundamentals of Financial Management
Chapter 5
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