Risk Measures
This sub‑topic covers the key risk measures used to evaluate portfolio managers in the NISM Series XXI‑A exam. Understanding these metrics helps you assess both total and systematic risk, and judge risk‑adjusted performance. The content links directly to the Performance Measurement chapter and is frequently tested in scenario‑based questions.
Learning Objectives
- 1Define and calculate standard deviation and variance as measures of total risk.
- 2Explain beta and its role in measuring systematic risk.
- 3Compute risk‑adjusted performance ratios such as Sharpe, Jensen’s alpha and Value at Risk.
- 4Interpret each risk measure for exam‑style portfolio evaluation.
Understanding Risk Measures
Risk measures quantify the uncertainty inherent in investment returns. In the Indian context, SEBI expects PMS distributors to disclose how much risk a portfolio may bear and how well the manager has compensated investors for that risk.
Two broad categories exist: total risk (variability of returns) and systematic risk (market‑related volatility). Total risk is captured by statistical dispersion measures, while systematic risk is measured by beta, which links the portfolio to the benchmark index.
Exam questions often present a set of returns or a beta value and ask you to compute a ratio or interpret the result. Missing the distinction between absolute risk (standard deviation) and risk‑adjusted return (Sharpe) is a common trap.
- Always identify whether the question asks for a raw risk figure or a performance ratio.
- Remember that SEBI’s guidelines require risk disclosures in percentage terms, not absolute rupee amounts.
Standard Deviation and Variance
Standard deviation (σ) measures the average distance of each period’s return from the mean return. It reflects the total volatility of a portfolio, encompassing both market‑related and idiosyncratic movements. Variance (σ²) is simply the square of standard deviation and is useful for algebraic manipulation, but investors usually interpret σ because it is expressed in the same units as returns.
In NISM exams, you will be given a series of periodic returns and asked to compute σ. Use the sample‑standard‑deviation formula (n‑1 in the denominator) unless the question explicitly states a population measure.
Why it matters: A higher σ indicates a riskier portfolio, which must be justified by higher returns for the portfolio manager to be deemed competent. The regulator looks at σ when evaluating the risk‑profile disclosure of a PMS.
- Standard deviation is expressed in percent per period (e.g., % per month or % per year).
- Variance is expressed in percent‑squared and rarely appears directly in exam answer choices.
Where:
R_{i}= Return in period i (in %)\bar{R}= Average return over n periods (in %)n= Number of return observations\sigma= Standard deviation of the portfolio returns (in %)Worked Example
Given five monthly returns: 8%, 10%, 6%, 12%, 9%. Step 1: Compute average \bar{R} = (8+10+6+12+9)/5 = 9%. Step 2: Compute squared deviations: (8-9)^2=1, (10-9)^2=1, (6-9)^2=9, (12-9)^2=9, (9-9)^2=0. Sum = 20. Step 3: σ = sqrt(20/(5-1)) = sqrt(5) = 2.236%. Verification: sqrt(20/4) = sqrt(5) = 2.236%.
If the question does not mention "population," assume the sample formula (divide by n‑1). Using n instead will give a lower σ and lead to an incorrect answer.
Beta – Systematic Risk
Beta (β) quantifies how a portfolio’s returns move relative to the market benchmark, usually the Nifty 50 or BSE Sensex. A β greater than 1 indicates higher sensitivity to market swings, while a β less than 1 suggests lower sensitivity. Beta is derived from the covariance of portfolio and market returns divided by the variance of market returns.
SEBI requires PMS disclosures to state the portfolio’s beta, enabling investors to gauge the degree of systematic risk they are taking. In exam scenarios, you may be given covariance and market variance, or you may need to compute beta from historical returns.
Interpretation matters: a high‑beta portfolio must deliver proportionately higher returns to justify the risk; otherwise, it may be flagged as under‑performing.
- Beta is a unit‑less ratio.
- Negative beta is possible for assets that move opposite to the market (e.g., gold).
Where:
\beta= Beta of the portfolio (unit‑less)\operatorname{Cov}(R_{p},R_{m})= Covariance between portfolio return R_p and market return R_m\operatorname{Var}(R_{m})= Variance of market returnsWorked Example
Given Cov(Rp,Rm)=0.012 and Var(Rm)=0.018: Step 1: β = 0.012 / 0.018 = 0.667. Verification: 0.012 ÷ 0.018 = 0.667.
Do not confuse a beta of 0.8 with low risk overall. It only means lower market‑related volatility; total risk could still be high if the portfolio has large idiosyncratic variance.
Sharpe Ratio – Risk‑Adjusted Return
The Sharpe ratio measures the excess return earned per unit of total risk (standard deviation). It is calculated by subtracting the risk‑free rate (Rf) from the portfolio’s average return (Rp) and dividing the result by the portfolio’s σ. A higher Sharpe indicates better risk‑adjusted performance.
In the Indian exam context, the risk‑free rate is often taken as the yield on a 10‑year government bond. The ratio is dimensionless, making it easy to compare portfolios of different sizes.
Typical exam question: you are given Rp, Rf, and σp; compute the Sharpe ratio and decide which of two portfolios is superior on a risk‑adjusted basis.
- Sharpe ratio > 1 is considered good in most Indian market conditions.
- Negative Sharpe indicates the portfolio underperforms the risk‑free asset.
Where:
S= Sharpe ratio (unit‑less)R_{p}= Average portfolio return (in % per annum)R_{f}= Risk‑free rate (in % per annum)\sigma_{p}= Standard deviation of portfolio returns (in % per annum)Worked Example
Given Rp = 12%, Rf = 6%, σp = 4%: Step 1: Excess return = 12% - 6% = 6%. Step 2: Sharpe = 6% / 4% = 1.5. Verification: (12-6)/4 = 1.5.
Jensen’s Alpha – Performance Attribution
Jensen’s alpha isolates the portion of portfolio return that cannot be explained by its beta exposure to the market. It is calculated as the actual portfolio return minus the expected return derived from the Capital Asset Pricing Model (CAPM). A positive alpha signifies outperformance after adjusting for systematic risk.
SEBI’s performance disclosure guidelines ask distributors to report Jensen’s alpha for each PMS, especially when the portfolio claims to be “actively managed.” The exam often tests your ability to compute alpha when given Rp, Rf, β, and market return Rm.
Remember: Jensen’s alpha is expressed in percentage points, not as a ratio.
- Alpha > 0 = manager added value.
- Alpha < 0 = manager under‑performed relative to market risk.
Where:
\alpha= Jensen’s alpha (in %)R_{p}= Portfolio return (in % per annum)R_{f}= Risk‑free rate (in % per annum)\beta= Portfolio beta (unit‑less)R_{m}= Market return (in % per annum)Worked Example
Given Rp = 14%, Rf = 5%, β = 1.2, Rm = 10%: Step 1: Expected return = 5% + 1.2 × (10% - 5%) = 5% + 1.2 × 5% = 5% + 6% = 11%. Step 2: Alpha = 14% - 11% = 3%. Verification: 14 - [5 + 1.2*(10-5)] = 3%.
Value at Risk (VaR) – Quantifying Potential Loss
Value at Risk estimates the maximum expected loss over a specified time horizon at a given confidence level. The parametric (variance‑covariance) method, which the NISM syllabus highlights, uses the portfolio’s standard deviation and the Z‑score corresponding to the confidence level (e.g., 1.65 for 95%).
For Indian PMS, VaR is often disclosed as a percentage of the portfolio value for a one‑day horizon. The regulator uses VaR to assess whether a PMS has adequate risk‑management controls.
Exam tip: remember the Z‑score values – 1.65 for 95%, 2.33 for 99%. Plug the numbers directly; no need for Monte‑Carlo simulation in the certification exam.
- VaR is expressed as a loss amount or as a percentage of the portfolio.
- VaR does NOT predict the worst‑case loss; it is a statistical estimate.
Where:
VaR= Value at Risk as % of portfolio valueZ_{\alpha}= Z‑score for confidence level (1.65 for 95%)\sigma_{p}= Standard deviation of daily portfolio returns (in %)Worked Example
Given daily σp = 2% and 95% confidence (Z=1.65): Step 1: VaR = 1.65 × 2% = 3.30%. Verification: 1.65*2 = 3.30.
Key Risk Measures – Formula and Interpretation
| Risk Measure | Formula (simplified) | Interpretation |
|---|---|---|
| Standard Deviation (σ) | √[ Σ(Ri‑R̄)² / (n‑1) ] | Total volatility; higher σ = higher total risk |
| Beta (β) | Cov(Rp,Rm) / Var(Rm) | Systematic risk relative to market; β>1 = more volatile than market |
| Sharpe Ratio (S) | (Rp‑Rf) / σp | Excess return per unit of total risk; higher S = better risk‑adjusted performance |
| Jensen’s Alpha (α) | Rp – [Rf + β(Rm‑Rf)] | Return above (or below) CAPM expectation; α>0 = manager added value |
| VaR (95%) | 1.65 × σp | Maximum expected loss over one day at 95% confidence; expressed as % of portfolio |
Sharpe Ratio Comparison of Three Portfolio Managers
Scenario
An investor is evaluating two PMS offerings. PMS‑X has Rp = 13%, σp = 5%, β = 0.9, and VaR (95%) = 8%. PMS‑Y has Rp = 15%, σp = 9%, β = 1.4, and VaR (95%) = 15%. The risk‑free rate is 6% and the market return is 10%. Which PMS is more suitable for a risk‑averse client?
Solution
First compute Sharpe ratios: <br/>PMS‑X: (13‑6)/5 = 7/5 = 1.40. <br/>PMS‑Y: (15‑6)/9 = 9/9 = 1.00. Sharpe X > Sharpe Y, indicating better risk‑adjusted return. Next compute Jensen’s alpha: <br/>Alpha X = 13 – [6 + 0.9×(10‑6)] = 13 – [6 + 0.9×4] = 13 – [6 + 3.6] = 13 – 9.6 = 3.4%. <br/>Alpha Y = 15 – [6 + 1.4×(10‑6)] = 15 – [6 + 1.4×4] = 15 – [6 + 5.6] = 15 – 11.6 = 3.4% (same). Both managers add the same alpha, but PMS‑Y has higher β and higher VaR, meaning more market‑related risk and larger potential loss. For a risk‑averse client, PMS‑X is preferable due to lower σ, lower β, lower VaR, and higher Sharpe.
Conclusion
The exam expects you to weigh total risk (σ, VaR) and systematic risk (β) against risk‑adjusted returns (Sharpe, Alpha). PMS‑X wins for a conservative investor.
Never insert variance (σ²) into a Sharpe or VaR formula. The formulas require the standard deviation (σ), not its square.
⭐Exam Takeaways
- Standard deviation quantifies total risk; use the sample formula (divide by n‑1) unless stated otherwise.
- Beta measures systematic risk; β > 1 means higher market sensitivity, β < 1 means lower sensitivity.
- Sharpe ratio = (Rp‑Rf) ÷ σp; a higher value indicates superior risk‑adjusted performance.
- Jensen’s alpha isolates manager skill after adjusting for beta; positive α signals outperformance.
- Parametric VaR = Zα × σp; remember Z‑scores 1.65 (95%) and 2.33 (99%).
- Always compare both absolute risk (σ, VaR) and risk‑adjusted returns (Sharpe, Alpha) in scenario questions.
- SEBI requires risk disclosures in percentage terms; keep units consistent throughout calculations.
Practice Questions
8 questions on Risk Measures
What does the standard deviation (σ) measure in portfolio risk assessment?
If a portfolio has a beta of 0.8, which statement is true?
Given five monthly returns of 5%, 7%, 9%, 6% and 8%, what is the sample standard deviation (σ) of the portfolio?
A portfolio has an average return of 11% per annum, a risk‑free rate of 5% and a standard deviation of 3%. What is its Sharpe ratio?
For a portfolio with Rp = 13%, Rf = 4%, β = 1.1 and market return Rm = 9%, what is Jensen’s alpha?
Using the parametric VaR method, what is the 99% one‑day VaR for a portfolio with daily σ = 1.8%? (Z‑score for 99% = 2.33)
An investor who is risk‑averse must choose between Portfolio A (σ=4%, Sharpe=1.2, β=0.9, VaR=3%) and Portfolio B (σ=7%, Sharpe=1.0, β=1.3, VaR=6%). Which portfolio is more suitable?
Which Z‑score is used for a 95% confidence level when calculating VaR?