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FRM Part I · FRM Exam Part I

Modern Portfolio Theory and CAPM for FRM Part I

Modern Portfolio Theory shows how combining assets lowers risk through diversification and how to choose portfolios on the efficient frontier. CAPM says expected return equals the risk-free rate plus beta times the market risk premium. To solve questions, identify the formula, plug in the numbers carefully, and check units.

What this chapter covers

This chapter covers how investors combine risky assets and how markets price risk. You start with portfolio mean and variance, then see how correlation drives diversification. From there you build the Markowitz efficient frontier, add a risk-free asset to get the Capital Market Line, and arrive at CAPM and the Security Market Line.

The second half turns theory into tools. Performance measures such as the Sharpe ratio, Treynor ratio, Jensen's alpha and information ratio let you rank managers. The single-index model and market model use regression to estimate beta and split risk into systematic and idiosyncratic parts. The chapter ends with the limitations of CAPM and its extensions, which tests whether you understand the assumptions behind the formulas.

The chapter links to the rest of the paper. Quantitative Analysis supplies the variance, covariance and regression tools you use here. Valuation and Risk Models uses the same ideas in VaR and factor models. In Foundations of Risk Management, CAPM and the Sharpe ratio sit behind risk-adjusted performance and the idea that only systematic risk earns a premium.

Many questions in this chapter involve applying standard formulas, so careful practice turns directly into correct answers. Many candidates lose time on them because they mix up similar ratios or skip a unit check. The material also underpins later topics such as factor models, VaR and risk-adjusted performance, so effort here pays back across the whole exam. GARP publishes topic-level weightings in the Study Guide but not a question count for each chapter, so do not skip this chapter because it looks short.

Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM): topics in the order to study them

  1. 1Portfolio Risk and Return, DiversificationEverything else uses portfolio mean, variance and correlation, so master these first.
  2. 2Markowitz Efficient Frontier and Optimal PortfoliosIt builds on portfolio variance and shows which risky portfolios are worth holding.
  3. 3Capital Market Line and Two-Fund SeparationAdding a risk-free asset to the frontier gives the tangency portfolio and the CML.
  4. 4CAPM and the Security Market LineCAPM follows from the CML logic and moves from total risk to beta as the risk measure.
  5. 5Performance Measures: Sharpe, Treynor, Jensen, Information RatioThese ratios use the CML, SML and beta, so they only make sense after those.
  6. 6Single-Index Model, Market Model and RegressionIt shows how beta and alpha are estimated and how total risk splits into systematic and specific parts.
  7. 7CAPM Limitations and ExtensionsStudy it last, once you know the assumptions well enough to see where they fail.

How to prepare Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)

Treat this chapter as a short chain of ideas, each built on the one before. Learn the logic first, then drill the calculations until they are routine.

  1. Write out the two-asset portfolio variance formula: σp² = w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2. Practise it until you can do it quickly on a calculator, and take the square root at the end.
  2. Sketch the efficient frontier, CML and SML by hand. Label axes (standard deviation versus expected return for the first two, beta versus expected return for the SML) and note what each line represents.
  3. Memorise the core formulas: CAPM E(Ri) = Rf + βi × [E(Rm) − Rf]; Sharpe = (Rp − Rf) ÷ σp; Treynor = (Rp − Rf) ÷ βp; Jensen's alpha = Rp − [Rf + βp(Rm − Rf)].
  4. Do worked questions that ask which ratio to use. Sharpe suits a whole portfolio, Treynor suits a portfolio that is one part of a well-diversified holding, and the information ratio uses active return over tracking error.
  5. Practise regression questions: beta = Cov(Ri, Rm) ÷ σm², and total variance = β²σm² + residual variance. Check the R² reading as the share of variance explained by the market.
  6. Finish with conceptual questions on assumptions and limitations. Be able to say why each CAPM assumption matters and what happens when it fails.
  7. Mix timed questions from the whole chapter weekly so you stay fast and can tell the formulas apart under pressure.

Common mistakes in Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)

  • Forgetting to take the square root, so variance is reported as standard deviation.

    Fix: Write σp² on the page, then σp on a separate line, and check which one the question asks for.

  • Confusing the CML with the SML.

    Fix: Remember the axes: the CML plots against standard deviation and covers efficient portfolios; the SML plots against beta and covers any asset.

  • Using the Sharpe ratio when Treynor fits, or the reverse.

    Fix: Ask whether the portfolio is the investor's whole holding (Sharpe, total risk) or one part of a diversified holding (Treynor, systematic risk).

  • Using total return instead of excess return over the risk-free rate.

    Fix: Always subtract Rf first when calculating Sharpe, Treynor, alpha and the market risk premium.

  • Treating beta as total risk or saying a high-beta asset must have high volatility.

    Fix: Remember beta measures sensitivity to the market only. Volatility also includes idiosyncratic risk and depends on the correlation with the market.

  • Treating CAPM as proven or ignoring its assumptions.

    Fix: Learn the main assumptions, such as mean-variance investors, homogeneous expectations and borrowing and lending at the risk-free rate, and know how multi-factor models address their failure.

Last-day revision: Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)

  • Portfolio return is the weighted average of asset returns; portfolio variance also depends on covariances.
  • Two assets: σp² = w1²σ1² + w2²σ2² + 2w1w2ρ12σ1σ2.
  • Lower correlation means more diversification benefit; at ρ = 1 there is none.
  • Diversification removes idiosyncratic risk, not systematic risk.
  • The efficient frontier gives the highest expected return for each level of risk; the minimum-variance portfolio is its leftmost point.
  • CML: E(Rp) = Rf + [(E(Rm) − Rf) ÷ σm] × σp, and it applies to efficient portfolios only.
  • Two-fund separation: all investors hold the risk-free asset and the same tangency portfolio, in different mixes.
  • SML: E(Ri) = Rf + βi[E(Rm) − Rf], and it applies to any asset or portfolio, priced by beta.
  • Sharpe uses total risk (σ), Treynor uses beta, Jensen's alpha is the return above the CAPM prediction.
  • Information ratio = active return ÷ tracking error.
  • Beta = Cov(Ri, Rm) ÷ σm² = ρim × σi ÷ σm.
  • Single-index model: total variance = systematic (β²σm²) + idiosyncratic variance.

Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM) practice questions

Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM) in other exams

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

Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM): frequently asked questions

How many formulas do I need for MPT and CAPM in FRM Part I?

A small set covers most questions: portfolio return and variance, the CML, CAPM and the SML, beta, and the Sharpe, Treynor, alpha and information ratios. Learn them with their conditions, not just their shape. Then practise questions until you can pick the right one quickly.

Do I need a financial calculator for this chapter?

The questions involve arithmetic such as square roots, products and sums across several assets, so you need to be quick and accurate on a calculator. Check GARP's current policy on permitted calculators. Practise with the same model you will use in the exam.

What is the difference between the Sharpe ratio and Jensen's alpha?

The Sharpe ratio is excess return per unit of total risk and is a ratio. Jensen's alpha is the return above what CAPM predicts for the portfolio's beta and is a return difference. Sharpe ranks portfolios, while alpha shows whether a manager beat the benchmark given market risk.

Does GARP tell me how many questions will come from this chapter?

No. GARP publishes topic-level weightings in the annual Study Guide and Learning Objectives, but not a question count for each chapter. Check the current year's learning objectives for exact coverage and prepare every topic in this chapter.