Calculating risk adjusted returns
This sub‑topic covers the calculation of risk‑adjusted returns, a core concept for research analysts. Understanding how to adjust returns for risk helps you compare securities on a like‑for‑like basis and is frequently tested in the NISM Series XV exam. The formulas you learn here are directly applicable to mutual fund analysis, equity research, and portfolio performance reporting.
Learning Objectives
- 1Define risk‑adjusted return and why it matters for investment evaluation.
- 2Calculate the Sharpe Ratio, Treynor Ratio, and Jensen's Alpha using the official NISM formulas.
- 3Interpret the meaning of each ratio and identify the most appropriate ratio for a given scenario.
- 4Avoid common calculation mistakes that appear in NISM exam questions.
Understanding Risk‑Adjusted Returns
Risk‑adjusted return measures the reward earned per unit of risk taken. It converts a raw return figure into a comparable metric by incorporating a risk proxy such as standard deviation, beta, or market excess return.
In the Indian context, SEBI requires research analysts to disclose risk‑adjusted performance when presenting mutual fund or portfolio returns. This ensures investors can judge whether higher returns are simply compensation for higher volatility.
For the NISM exam, you will often be asked to compute or interpret three standard ratios – Sharpe, Treynor, and Jensen’s Alpha – each using a slightly different risk measure. Knowing which denominator to use is crucial for scoring marks.
- Risk‑adjusted returns enable apples‑to‑apples comparison across asset classes.
- They are a key part of the performance attribution framework taught in the syllabus.
Students often substitute standard deviation for beta (or vice‑versa) when the question explicitly asks for Sharpe or Treynor. Remember: Sharpe uses total volatility (σ), Treynor uses systematic risk (β).
Sharpe Ratio
The Sharpe Ratio, introduced by William Sharpe, evaluates how much excess return a portfolio generates per unit of total risk, where total risk is measured by the standard deviation of portfolio returns.
Formulaically, it is the difference between the portfolio return (Rp) and the risk‑free rate (Rf) divided by the portfolio's standard deviation (σp). A higher Sharpe indicates better risk‑adjusted performance.
In NISM questions, the risk‑free rate is usually the yield on a 10‑year Indian government bond. The standard deviation is expressed in the same time‑basis as the returns (typically annualised).
Where:
R_{p}= Annualised portfolio return in percentR_{f}= Annual risk‑free rate in percent (e.g., 10‑year G‑Sec yield)\sigma_{p}= Annual standard deviation of portfolio returns in percentWorked Example
Given Rp = 12%, Rf = 6%, σp = 15%: Step 1: Sharpe = (12 - 6) / 15 Step 2: Sharpe = 6 / 15 = 0.40 Verification: (12 - 6) / 15 = 0.40.
Treynor Ratio
The Treynor Ratio focuses on systematic risk, measured by the portfolio's beta (βp) relative to the market. It answers the question: how much excess return is earned per unit of market risk?
Because beta isolates market‑related volatility, the Treynor Ratio is especially useful when evaluating diversified portfolios where unsystematic risk is minimal.
In the exam, beta values are often provided directly, or you may need to compute beta from regression outputs. Remember to keep the numerator identical to the Sharpe Ratio – the excess return over the risk‑free rate.
- Higher Treynor values indicate superior performance on a market‑risk basis.
- Do not confuse beta (a unit‑less coefficient) with standard deviation (a percentage).
Where:
R_{p}= Annualised portfolio return in percentR_{f}= Annual risk‑free rate in percent\beta_{p}= Portfolio beta (dimensionless)Worked Example
Given Rp = 12%, Rf = 6%, \beta_{p} = 1.2: Step 1: Treynor = (12 - 6) / 1.2 Step 2: Treynor = 6 / 1.2 = 5.00 Verification: (12 - 6) / 1.2 = 5.00.
Jensen’s Alpha
Jensen’s Alpha measures the abnormal return of a portfolio relative to the return predicted by the Capital Asset Pricing Model (CAPM). It captures the manager's skill after accounting for market risk.
The formula subtracts the CAPM‑expected return (Rf + βp(Rm‑Rf)) from the actual portfolio return. A positive alpha means the portfolio outperformed its risk‑adjusted benchmark.
For NISM, the market return (Rm) is typically the BSE Sensex or NSE Nifty annual return. The alpha is expressed in percentage points, making it easy to compare across funds.
- Alpha isolates manager skill; Sharpe and Treynor focus on risk‑adjusted efficiency.
- Beware of sign errors – a negative alpha indicates under‑performance.
When computing Jensen’s Alpha, many candidates omit the (Rm‑Rf) market risk premium. Ensure you multiply beta by the full market premium before adding the risk‑free rate.
Where:
R_{p}= Annualised portfolio return in percentR_{f}= Annual risk‑free rate in percent\beta_{p}= Portfolio beta (dimensionless)R_{m}= Annual market return in percent (e.g., Sensex)Worked Example
Given Rp = 12%, Rf = 6%, \beta_{p} = 1.2, Rm = 10%: Step 1: Expected CAPM return = 6 + 1.2\times(10 - 6) = 6 + 1.2\times4 = 6 + 4.8 = 10.8% Step 2: Alpha = 12 - 10.8 = 1.2% Verification: 12 - [6 + 1.2(10 - 6)] = 1.2%.
Comparing the Three Ratios
Key Differences Between Sharpe, Treynor, and Jensen’s Alpha
| Metric | Risk Measure | Interpretation | Best Used When |
|---|---|---|---|
| Sharpe Ratio | Total volatility (σ) | Excess return per unit of total risk | Portfolio contains both systematic and unsystematic risk |
| Treynor Ratio | Systematic risk (β) | Excess return per unit of market risk | Well‑diversified portfolio where unsystematic risk is negligible |
| Jensen’s Alpha | CAPM‑expected return | Absolute abnormal return (manager skill) | Assessing active management performance |
Risk‑Adjusted Return Metrics for Three Sample Portfolios
Scenario
An investor is choosing between two equity mutual funds. Fund X has an annualised return of 14%, standard deviation of 18%, beta of 1.1. Fund Y returns 12% with a standard deviation of 12% and beta of 0.9. The 10‑year G‑Sec yield is 6% and the market (Sensex) returned 10% during the same period.
Solution
First compute Sharpe for both funds. Sharpe X = (14‑6)/18 = 0.44. Sharpe Y = (12‑6)/12 = 0.50, so Y has a higher risk‑adjusted return on a total‑risk basis. Next compute Treynor. Treynor X = (14‑6)/1.1 = 7.27. Treynor Y = (12‑6)/0.9 = 6.67, indicating Fund X performs better on a systematic‑risk basis. Finally compute Jensen’s Alpha. Alpha X = 14‑[6+1.1(10‑6)] = 14‑[6+4.4] = 3.6%. Alpha Y = 12‑[6+0.9(10‑6)] = 12‑[6+3.6] = 2.4%. The investor must decide which risk perspective matters most; if the portfolio is already well‑diversified, Sharpe is preferred, pointing to Fund Y.
Conclusion
The example shows how the same two funds can rank differently depending on the chosen risk‑adjusted metric. NISM questions often test your ability to pick the appropriate ratio for the given context.
⭐Exam Takeaways
- Risk‑adjusted return converts raw returns into a comparable metric by accounting for risk.
- Sharpe Ratio = (Rp‑Rf) ÷ σp; uses total volatility and is ideal for portfolios with unsystematic risk.
- Treynor Ratio = (Rp‑Rf) ÷ βp; uses systematic risk and suits well‑diversified portfolios.
- Jensen’s Alpha = Rp – [Rf + βp(Rm‑Rf)]; measures abnormal return after adjusting for market risk.
- Never mix σ and β in the denominator – each ratio has a specific risk measure.
- For exam calculations, always express returns and risk measures in the same time‑basis (usually annual).
- Positive Sharpe, Treynor, and Alpha values indicate outperformance; negative values signal underperformance.
- Use the ratio that aligns with the question’s focus: total risk (Sharpe), market risk (Treynor), or manager skill (Alpha).
Practice Questions
9 questions on Calculating risk adjusted returns
Risk‑adjusted return measures the reward earned per unit of what?
Which risk measure appears in the denominator of the Sharpe Ratio?
A portfolio has an annualised return of 12%, a risk‑free rate of 5% and a standard deviation of 14%. What is its Sharpe Ratio?
If a portfolio’s return is 15%, the risk‑free rate is 6% and its beta is 1.5, what is the Treynor Ratio?
Using Jensen’s Alpha formula, compute alpha for a portfolio with Rp=13%, Rf=4%, βp=1.0 and market return Rm=9%.
Which risk‑adjusted metric is most appropriate for evaluating a well‑diversified portfolio where unsystematic risk is negligible?
An examinee calculates a Sharpe Ratio but mistakenly uses the portfolio’s beta in the denominator. Which error does this represent?
In Jensen’s Alpha formula, what term represents the market risk premium?
Three portfolios have Sharpe ratios of 0.40, 0.55 and 0.30 respectively. Which portfolio shows the highest risk‑adjusted performance on a total‑risk basis?