FRM Exam Part I · The Building Blocks of Risk Management

Risk Return Tradeoff and Risk-Adjusted Performance Measures

Updated 11 October 2026 · Fact-checked

The risk return tradeoff says higher expected return usually requires accepting more risk. Risk-adjusted measures compare return per unit of risk. Sharpe = (Rp − Rf) ÷ σp, Treynor = (Rp − Rf) ÷ β, and RAROC = risk-adjusted return ÷ economic capital. Compute each, then rank: higher is better.

Understand Risk Return Tradeoff and Risk-Adjusted Performance

Investors and banks are paid for taking risk. If two investments offer the same return, you prefer the one with less risk. If they carry the same risk, you prefer the higher return. The risk return tradeoff says that, in a competitive market, you cannot reliably earn higher expected returns without accepting more risk. A higher return is a reward for bearing risk, not a free gain.

Raw returns are therefore misleading. A fund that returned 15% with wild swings may be worse than one that returned 10% with small swings. Risk-adjusted performance measures fix this by dividing excess return (return above the risk-free rate) by a measure of risk.

The choice of risk measure matters. The Sharpe ratio uses total risk (standard deviation), so it suits a portfolio that is your whole investment. The Treynor ratio uses systematic risk (beta), so it suits a portfolio that is one part of a well-diversified holding. Jensen's alpha measures return above what CAPM predicts for the portfolio's beta. The information ratio divides active return over a benchmark by tracking error.

Banks use a different lens. RAROC (risk-adjusted return on capital) divides risk-adjusted profit by the economic capital a business unit needs to absorb unexpected losses. Revenues less costs and expected loss, plus any return on the capital, form the numerator. A unit is judged by comparing RAROC with a hurdle rate, usually the cost of equity. Above the hurdle, the unit creates value.

The exam tests two skills: calculating each ratio correctly and choosing the right one for the situation. Know what risk each ratio uses, and you can answer most questions.

Key formulas to remember

Sharpe ratio: Sharpe = (Rp − Rf) ÷ σp; σp is the standard deviation of portfolio returns (total risk). Use for stand-alone portfolios.
Treynor ratio: Treynor = (Rp − Rf) ÷ βp; Uses beta (systematic risk). Use for a portfolio that is part of a diversified whole. Be careful if beta is negative or near zero.
Jensen's alpha: α = Rp − [Rf + βp × (Rm − Rf)]; Positive alpha means the portfolio beat the CAPM required return.
Information ratio: IR = (Rp − Rb) ÷ tracking error; Tracking error is the standard deviation of (Rp − Rb). Rb is the benchmark return.
RAROC: RAROC = (Revenues − Costs − Expected loss) ÷ Economic capital; Some versions add income earned on the capital. Compare with the hurdle rate (cost of equity).
CAPM required return: E(R) = Rf + β × (E(Rm) − Rf); Used to build Jensen's alpha and to judge hurdle rates.

How to solve Risk Return Tradeoff and Risk-Adjusted Performance questions

Use this method for any risk-adjusted performance question. It keeps you from mixing up the measures.

1Read what is asked: a ratio value, a ranking, or which measure suits the situation.
2List the inputs: portfolio return, risk-free rate, standard deviation, beta, benchmark return, economic capital, costs and expected loss.
3Check the units. Put all returns and rates in the same period, for example annual, and in decimals or percentages consistently.
4Subtract the risk-free rate (or benchmark, or costs and expected loss for RAROC) to get the excess or risk-adjusted numerator.
5Divide by the correct risk measure: σ for Sharpe, β for Treynor, tracking error for information ratio, economic capital for RAROC.
6Compare results. Higher is better. For RAROC, compare with the hurdle rate.
7Sanity check: if the question says the portfolio is diversified, favour Treynor; if it is the whole portfolio, favour Sharpe.

Quickest way: Numerator, denominator, compare

When to use it: Use when you have about a minute and the question gives all inputs directly.

Write the numerator first: return minus the correct benchmark (risk-free, benchmark or costs and expected loss).
Pick the denominator by the name: Sharpe = σ, Treynor = β, IR = tracking error, RAROC = economic capital.
Divide once on your calculator and compare with the answer options.
If options are close, check whether you subtracted Rf and whether the units match.

Common mistakes in Risk Return Tradeoff and Risk-Adjusted Performance

Forgetting to subtract the risk-free rate in Sharpe or Treynor.
Students divide the raw return by risk because it feels simpler.
Fix: Always write (Rp − Rf) first. If Rf is given, it is meant to be used.
Using beta in the Sharpe ratio or standard deviation in the Treynor ratio.
The two formulas look alike and the names get swapped.
Fix: Remember: Sharpe uses Standard deviation (S for S); Treynor uses beta (systematic risk).
Forgetting to deduct expected loss in RAROC.
Students focus on revenue and costs and treat loss as capital's job.
Fix: Expected loss is a cost of doing business and is deducted from the numerator. Economic capital covers unexpected loss in the denominator.
Claiming that Sharpe and Treynor always give the same ranking.
Both divide excess return by a risk measure.
Fix: They agree only when portfolios are well diversified so total risk is mostly systematic. Otherwise rankings can differ.
Comparing a RAROC with the wrong hurdle or ignoring it.
Students stop at computing the ratio.
Fix: Value is created only when RAROC exceeds the hurdle rate, usually the cost of equity capital.
Mixing monthly returns with annual risk-free rates or volatilities.
Data in the question come in different periods.
Fix: Convert everything to the same horizon before dividing.

Worked examples

Example 1

Portfolio A returned 12% with a standard deviation of 18% and a beta of 1.2. Portfolio B returned 9% with a standard deviation of 10% and a beta of 0.8. The risk-free rate is 3%. Which has the higher Sharpe ratio, and what is it?

Show the solution

Sharpe A = (12% − 3%) ÷ 18% = 9 ÷ 18 = 0.50.
Sharpe B = (9% − 3%) ÷ 10% = 6 ÷ 10 = 0.60.
Compare: 0.60 > 0.50, so B earns more excess return per unit of total risk.

Answer: Portfolio B, with a Sharpe ratio of 0.60 (A is 0.50).

Example 2

A bank's lending unit earns revenues of USD 50 million, has costs of USD 20 million and expected credit losses of USD 10 million. Economic capital is USD 100 million and the hurdle rate is 15%. What is the RAROC and does the unit create value?

Show the solution

Numerator = 50 − 20 − 10 = USD 20 million.
RAROC = 20 ÷ 100 = 20%.
Compare with the hurdle rate: 20% > 15%.

Answer: RAROC is 20%, above the 15% hurdle, so the unit creates value.

Exam tips

Identify the risk measure first. The question wording (total risk, systematic risk, diversified, benchmark) tells you which ratio to use.
Expect conceptual questions on when Sharpe and Treynor disagree, not only calculations.
For RAROC, check whether the question gives income on capital and whether expected loss is separate from economic capital.
Keep the calculator in percentage mode consistently and avoid mixing decimals with percentages.
Remember that a higher ratio is better, but a negative excess return makes a ratio comparison unreliable, especially for Treynor with negative beta.

Practice questions from The Building Blocks of Risk Management

Risk Return Tradeoff and Risk-Adjusted Performance in other exams

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

NISM CertificationsRisk-Return Tradeoff and Risk-Adjusted Performance

Risk Return Tradeoff and Risk-Adjusted Performance: frequently asked questions

What is the difference between the Sharpe ratio and the Treynor ratio?

Both measure excess return per unit of risk. Sharpe divides by standard deviation, which is total risk. Treynor divides by beta, which is systematic risk only. Use Sharpe for a stand-alone portfolio and Treynor for a portfolio within a diversified holding.

How do I calculate RAROC in the FRM exam?

Take revenues, subtract costs and expected loss, then divide by economic capital. If the question gives income on capital, add it to the numerator. Compare the result with the hurdle rate to decide whether value is created.

What does the risk return tradeoff mean?

It means higher expected returns generally come with higher risk. You are paid for bearing risk that cannot be diversified away. It does not guarantee higher returns, only that higher expected returns need higher risk.

Is a higher Sharpe ratio always better?

Generally yes, for comparing similar portfolios over the same period. But it uses standard deviation, so it can understate risks such as fat tails or skewness. Treat it as a useful measure, not a complete one.