Corporate Accounting and Financial Management · Security Analysis

Portfolio Theory and CAPM: Formulas, Beta and Numericals

Updated 11 October 2026 · Fact-checked

Portfolio theory shows how combining securities changes expected return and risk. Portfolio return is the weighted average of security returns. Risk depends on correlation, so diversification lowers it. CAPM says required return = risk-free rate + beta × (market return − risk-free rate). Plot this on the security market line to judge whether a share is fairly priced.

Understand Portfolio Theory and CAPM

A portfolio is a group of securities held together. Investors care about two things: the return they expect and the risk they bear. Risk here means variability of returns, measured by standard deviation.

Portfolio expected return is simply the weighted average of the expected returns of its securities. Risk is different. It is not a weighted average unless the securities move in perfect step. When securities do not move together, the ups of one offset the downs of another, and portfolio risk falls. This is diversification.

Total risk has two parts. Unsystematic risk is specific to a company or industry, such as a strike or a product failure. Diversification can remove it. Systematic risk comes from the whole market, such as interest rate changes or recession. It cannot be diversified away. So the market pays investors only for systematic risk.

Beta (β) measures how sensitive a security's return is to market return. A market portfolio has beta 1. A beta of 1.5 means the security tends to move 1.5 times as much as the market. A beta below 1 means a defensive security. The beta of a portfolio is the weighted average of the betas of its securities.

The Capital Asset Pricing Model (CAPM) turns beta into a required return. The line it draws, with beta on the x-axis and required return on the y-axis, is the security market line (SML). A security whose expected return is above the SML is underpriced (buy). One below the line is overpriced (sell or avoid).

Key rules to remember

Expected return of a portfolio: Rp = Σ (wi × Ri); wi is the proportion invested in security i. Weights must add up to 1.
Expected return of a security from probabilities: E(R) = Σ (Pi × Ri); Pi is the probability of outcome i.
Variance and standard deviation: σ² = Σ Pi × (Ri − E(R))²; σ = √σ²; Standard deviation is the usual measure of total risk.
Two-security portfolio risk: σp = √(w1²σ1² + w2²σ2² + 2 × w1 × w2 × ρ12 × σ1 × σ2); ρ12 is the correlation coefficient. Equivalent: replace ρ12σ1σ2 with covariance Cov12.
Correlation: ρ12 = Cov12 ÷ (σ1 × σ2); Lies between −1 and +1. At +1 there is no diversification benefit; at −1 risk can be reduced to zero with right weights.
Beta of a security: β = Cov(Ri, Rm) ÷ σm² = ρim × σi ÷ σm; σm² is the variance of market returns.
Portfolio beta: βp = Σ (wi × βi); A simple weighted average.
CAPM: Ke = Rf + β × (Rm − Rf); (Rm − Rf) is the market risk premium. Ke is the required return.
Security market line: Required return = Rf + β × (Rm − Rf); Same equation as CAPM, plotted against beta. Intercept is Rf; slope is the market risk premium.

How to solve Portfolio Theory and CAPM questions

Most questions fall into three types: portfolio return and risk, beta, and CAPM with valuation judgement. Use the same order each time.

1Read what is asked: return, risk, beta, required return, or whether to buy or sell.
2List the data: weights, returns, standard deviations, correlation or covariance, Rf and Rm. Convert percentages to decimals only when squaring.
3For portfolio return, multiply each weight by its return and add.
4For portfolio risk, compute variance first using the two-security formula, then take the square root at the end. Do not skip the covariance term.
5For beta, use Cov ÷ market variance, or ρ × σi ÷ σm. For a portfolio, take the weighted average of betas.
6For required return, put beta into Rf + β × (Rm − Rf).
7If asked for a decision, compare expected return with CAPM required return. Higher expected return means underpriced; lower means overpriced.
8Write a one-line conclusion with units in percent.

Quickest way: Weights, then formula, then compare

When to use it: Use this when time is short and the question gives clean numbers for a two-security portfolio or a CAPM check.

Write the weights and check they sum to 1.
Compute return and portfolio beta first. These are quick weighted averages.
Compute CAPM required return in one line.
For risk, calculate each term of the variance separately: w1²σ1², w2²σ2², and the covariance term. Add, then square root.
Compare expected return to required return and state buy, sell or hold.

Common mistakes in Portfolio Theory and CAPM

Taking portfolio standard deviation as the weighted average of individual standard deviations.
Return is a weighted average, so students assume risk is too.
Fix: Use the full formula with correlation. A weighted average of risks is correct only when correlation is +1.
Forgetting to take the square root, and giving variance as risk.
Students stop after adding the three terms under the root.
Fix: Label the sum as σp² and then compute σp = √σp².
Using market return instead of the market risk premium in CAPM.
The formula looks like Rf + β × Rm.
Fix: Always subtract: use (Rm − Rf). Write it as a separate line before multiplying by beta.
Saying diversification removes all risk.
Students remember that risk falls with more securities.
Fix: Say it removes unsystematic risk only. Systematic risk remains and is measured by beta.
Mixing up correlation and covariance in the formula.
Both measure co-movement and both appear in the same formula.
Fix: If covariance is given, use 2 × w1 × w2 × Cov. If correlation is given, use 2 × w1 × w2 × ρ × σ1 × σ2. Never multiply by both.
Reading the buy or sell signal the wrong way round.
Students do not link the SML to price.
Fix: Expected return above CAPM return means the share offers more than needed, so it is underpriced and a buy. Below means overpriced.

Worked examples

Example 1

An investor puts ₹6,00,000 in share A and ₹4,00,000 in share B. Expected returns are 14% for A and 10% for B. Standard deviations are 20% for A and 12% for B. The correlation between A and B is 0.5. Find the expected return and standard deviation of the portfolio.

Show the solution

Total investment = ₹10,00,000. wA = 0.6 and wB = 0.4.
Portfolio return = 0.6 × 14% + 0.4 × 10% = 8.4% + 4% = 12.4%.
Term 1: wA²σA² = 0.36 × 400 = 144 (using σ in percent, so σA² = 400).
Term 2: wB²σB² = 0.16 × 144 = 23.04.
Term 3: 2 × 0.6 × 0.4 × 0.5 × 20 × 12 = 0.24 × 240 = 57.6.
Variance = 144 + 23.04 + 57.6 = 224.64.
σp = √224.64 = 14.99%, approximately 15%.

Answer: Expected return is 12.4% and portfolio standard deviation is about 15%. This is below the weighted average of the two risks (0.6 × 20 + 0.4 × 12 = 16.8%), showing the benefit of diversification.

Example 2

The risk-free rate is 7% and the expected market return is 13%. Share X has beta 1.4 and an expected return of 16%. Share Y has beta 0.8 and an expected return of 11%. Using CAPM, state whether each share is underpriced or overpriced. Also find the beta of a portfolio with 50% in X and 50% in Y.

Show the solution

Market risk premium = 13% − 7% = 6%.
Required return of X = 7% + 1.4 × 6% = 7% + 8.4% = 15.4%.
Expected return of X is 16%, which is above 15.4%. X plots above the SML, so it is underpriced.
Required return of Y = 7% + 0.8 × 6% = 7% + 4.8% = 11.8%.
Expected return of Y is 11%, which is below 11.8%. Y plots below the SML, so it is overpriced.
Portfolio beta = 0.5 × 1.4 + 0.5 × 0.8 = 0.7 + 0.4 = 1.1.

Answer: X is underpriced (16% against required 15.4%), so buy. Y is overpriced (11% against required 11.8%), so avoid or sell. The portfolio beta is 1.1.

Exam tips

Show the formula first, then substitute. Marks are awarded for method even if arithmetic slips.
Keep weights, returns and standard deviations in a small table before computing. It prevents mixing up securities.
In theory questions, always separate systematic and unsystematic risk and say which one beta measures.
End every CAPM numerical with a decision line: underpriced, overpriced or fairly priced, with the two rates compared.
Be ready to explain the SML in words: intercept Rf, slope equal to the market risk premium, and securities above or below the line.

Practice questions from Security Analysis

Portfolio Theory and CAPM in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Portfolio Theory and CAPM: frequently asked questions

What is beta in security analysis?

Beta measures how much a security's return moves when the market return moves. A beta of 1 means it moves with the market, above 1 means more volatile, and below 1 means less volatile. It measures systematic risk only.

What is the CAPM formula?

Required return = Rf + β × (Rm − Rf). Here Rf is the risk-free rate, Rm the expected market return and β the security's beta. Use it to find the return an investor should demand for the risk taken.

How does diversification reduce portfolio risk?

When securities do not move perfectly together, their gains and losses partly offset. This reduces unsystematic risk. The lower the correlation between securities, the greater the reduction.

What does the security market line show?

It shows the required return for each level of beta. Securities above the line give more than required and are underpriced. Securities below the line give less and are overpriced.