Measuring risk
Measuring risk is a core skill for a research analyst. It helps you quantify the uncertainty of returns, compare investment options, and comply with SEBI guidelines. This sub‑topic explains the statistical tools used to capture risk and shows how they appear in NISM exam questions.
Learning Objectives
- 1Define systematic and unsystematic risk and their relevance.
- 2Calculate variance, standard deviation, beta, and coefficient of variation.
- 3Interpret downside risk measures such as VaR.
- 4Apply risk‑measurement techniques to Indian market data.
Understanding Risk
Risk, in the context of securities research, is the probability that actual returns will deviate from expected returns. It reflects the uncertainty inherent in market prices and can arise from macro‑economic factors, company‑specific events, or investor behaviour.
Two broad categories are recognised by SEBI and NISM: systematic risk, which affects the entire market (e.g., changes in interest rates, inflation, or political stability), and unsystematic risk, which is unique to a particular firm or sector (e.g., product recall, management change). Systematic risk cannot be eliminated through diversification, whereas unsystematic risk can be reduced by holding a well‑diversified portfolio.
For the exam, you must be able to identify which risk measure captures each type, and know why a research analyst reports both. Typical traps include treating beta as a total risk measure or ignoring the time‑horizon when discussing volatility.
Many candidates equate volatility with total risk. Remember that volatility (standard deviation) measures the dispersion of returns, while total risk also includes the possibility of extreme losses captured by downside measures.
Statistical Measures of Risk
Statistical risk measures start with the concept of variance, which quantifies the average squared deviation of returns from their mean. A higher variance indicates that returns are spread out over a wider range, implying greater uncertainty.
The square‑root of variance gives the standard deviation, expressed in the same units as returns (percentage points). Standard deviation is the most frequently used risk metric in the NISM syllabus because it is directly comparable across assets.
Both variance and standard deviation are calculated using historical return data, typically on a daily, weekly, or monthly basis. The exam may ask you to annualise a monthly standard deviation by multiplying by the square‑root of 12, so keep the conversion factor in mind.
Where:
R_{i}= Observed return in period i (in %)\mu= Mean (average) return over N periods (in %)N= Number of return observations\sigma^{2}= Variance of returns (percentage points squared)Worked Example
Given five monthly returns: 2%, 4%, -1%, 3%, 5%. Step 1: Compute mean \mu = (2+4-1+3+5)/5 = 2.6%. Step 2: Compute squared deviations: (2-2.6)^2=0.36, (4-2.6)^2=1.96, (-1-2.6)^2=12.96, (3-2.6)^2=0.16, (5-2.6)^2=5.76. Step 3: Sum = 21.20. Step 4: Variance = 21.20 / 5 = 4.24. Verification: \sigma^{2}=\frac{\sum (R_i-\mu)^2}{5}=4.24.
Where:
R_{i}= Observed return in period i (in %)\mu= Mean return over N periods (in %)N= Number of observations\sigma= Standard deviation of returns (in %)Worked Example
Using the variance 4.24 from the previous example: Step 1: \sigma = \sqrt{4.24}. Step 2: \sigma = 2.06%. Verification: \sqrt{4.24}=2.06.
Beta – Measure of Systematic Risk
Beta (β) captures the sensitivity of a security's returns to movements in the market index. It is derived from the Capital Asset Pricing Model (CAPM) and is a core concept for SEBI‑registered research analysts.
A beta greater than 1 indicates that the security is more volatile than the market (higher systematic risk), while a beta less than 1 signals lower systematic risk. A negative beta is rare but possible for assets that move opposite to the market, such as gold.
Exam questions often present a beta and ask you to interpret its implication for expected return or portfolio construction. Remember that beta reflects only systematic risk; it does not account for unsystematic components.
Where:
R_{i}= Return of the security in period i (in %)\bar{R}= Mean return of the security over N periods (in %)R_{m,i}= Return of the market index in period i (in %)\bar{R}_{m}= Mean market return over N periods (in %)N= Number of periods\beta= Beta coefficient (dimensionless)Worked Example
Suppose over 3 months the security returns are 4%, 6%, 5% and the market returns are 3%, 5%, 4%. Step 1: Compute means: \bar{R}= (4+6+5)/3 = 5%, \bar{R}_{m}= (3+5+4)/3 = 4%. Step 2: Compute numerator: (4-5)(3-4)+(6-5)(5-4)+(5-5)(4-4)=(-1)(-1)+(1)(1)+(0)(0)=1+1+0=2. Step 3: Compute denominator: (3-4)^2+(5-4)^2+(4-4)^2 = (-1)^2+1^2+0^2 = 1+1+0 = 2. Step 4: Beta = 2/2 = 1. Verification: \beta = 1.
Coefficient of Variation
The Coefficient of Variation (CV) expresses risk per unit of return. It is useful when comparing assets with different expected returns because it standardises the risk measure.
CV is calculated by dividing the standard deviation by the mean return. A lower CV indicates a more efficient investment – higher return for each unit of risk.
In NISM questions, you may be given two mutual fund schemes with differing returns and asked which offers a better risk‑adjusted performance. Compute CV for each and choose the lower value.
Where:
\sigma= Standard deviation of returns (in %)\mu= Mean return (in %)CV= Coefficient of variation (dimensionless)Worked Example
Assume Scheme A has \mu = 12% and \sigma = 6%. Step 1: CV = 6 / 12 = 0.50. Verification: CV = 0.50.
Downside Risk Measures
While variance treats upside and downside deviations equally, investors are often more concerned with losses. Downside deviation and semi‑variance consider only returns that fall below a target (usually zero or a minimum acceptable return).
Value at Risk (VaR) is a regulatory‑focused metric that estimates the maximum expected loss over a given horizon at a specified confidence level (e.g., 95%). VaR is widely used in Indian asset‑management firms to satisfy SEBI risk‑management guidelines.
Exam questions may present a simple VaR calculation using the variance‑covariance method. Remember that VaR = Z_{α} × σ × √t, where Z_{α} is the standard normal critical value for the confidence level and t is the time horizon in years.
Comparison of Common Risk Measures
| Measure | Captures | Units | Typical Use in NISM |
|---|---|---|---|
| Variance | Average squared deviation (both sides) | (% )^2 | Foundation for SD |
| Standard Deviation | Dispersion of returns | % | Primary risk metric |
| Beta | Systematic risk relative to market | Dimensionless | CAPM & portfolio beta |
| Coefficient of Variation | Risk per unit of return | Dimensionless | Risk‑adjusted performance |
| VaR (95%) | Maximum expected loss at 95% confidence | Rupees or % | Regulatory risk reporting |
Risk Measure Values for Three Sample Indian Mutual Funds (Annualised)
When the syllabus provides monthly standard deviation, multiply by √12 to convert to an annual figure before comparing with annual beta or VaR.
Scenario
An Indian investor is evaluating Scheme X (average return 10%, SD 6%) and Scheme Y (average return 14%, SD 10%). The market beta for X is 0.9 and for Y is 1.3. The investor wants the better risk‑adjusted option.
Solution
Step 1: Compute CV for each scheme. CV_X = 6/10 = 0.60. CV_Y = 10/14 ≈ 0.71. Lower CV suggests Scheme X offers better return per unit of risk. Step 2: Assess systematic risk via beta. Scheme X's beta 0.9 < 1 indicates lower market sensitivity, while Scheme Y's beta 1.3 > 1 shows higher systematic risk. Combining both, Scheme X is preferable for a risk‑averse investor.
Conclusion
The example illustrates how NISM expects you to use CV and beta together to justify a recommendation.
Practical Steps for Risk Measurement
Begin by gathering historical price data for the security and the relevant market index. SEBI recommends a minimum of 250 daily observations for reliable variance estimates.
Use spreadsheet functions such as =VAR.P() for population variance and =STDEV.P() for standard deviation. For beta, compute covariance with =COVARIANCE.P() and divide by the market variance.
After calculating raw figures, annualise them if the exam provides monthly or weekly data. Finally, document the source, frequency, and any assumptions – this aligns with SEBI's risk‑management disclosure requirements.
A common mistake is to add individual variances directly. Always incorporate the covariance term; otherwise you overstate portfolio risk.
Scenario
An analyst creates a two‑stock portfolio: Stock A (weight 60%, SD 10%) and Stock B (weight 40%, SD 15%). The correlation coefficient between the stocks is 0.3.
Solution
Step 1: Convert correlation to covariance: Cov_{AB} = \rho \times \sigma_A \times \sigma_B = 0.3 \times 10 \times 15 = 45 (percentage points squared). Step 2: Portfolio variance = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B Cov_{AB}. Compute: w_A^2 \sigma_A^2 = 0.6^2 \times 100 = 36; w_B^2 \sigma_B^2 = 0.4^2 \times 225 = 36; 2 w_A w_B Cov_{AB} = 2 \times 0.6 \times 0.4 \times 45 = 21.6. Total variance = 36 + 36 + 21.6 = 93.6. Step 3: Portfolio SD = \sqrt{93.6} ≈ 9.68%. Verification: \sqrt{93.6}=9.68%.
Conclusion
The portfolio's risk is lower than a weighted average of individual risks, highlighting the diversification benefit.
⭐Exam Takeaways
- Risk is the probability of deviation from expected returns; systematic risk cannot be diversified away.
- Variance measures average squared deviation; standard deviation is its square‑root and the primary risk metric.
- Beta quantifies systematic risk relative to the market; interpret values >1, <1, and negative correctly.
- Coefficient of Variation (CV = σ/μ) enables risk‑adjusted comparison across assets with different returns.
- Downside risk measures (downside deviation, VaR) focus on potential losses; VaR uses a confidence level and time horizon.
Practice Questions
8 questions on Measuring risk
Systematic risk is best described as:
What is the formula for the variance of returns?
Given five monthly returns of 2%, 4%, -1%, 3% and 5%, what is the variance of these returns?
A beta of 1.2 for a security indicates that:
An analyst forms a two‑stock portfolio: Stock A (weight 60%, SD 10%) and Stock B (weight 40%, SD 15%) with a correlation of 0.3. What is the portfolio's standard deviation?
An investor compares Scheme X (average return 10%, SD 6%) with Scheme Y (average return 14%, SD 10%). Which scheme offers a better risk‑adjusted performance based on the Coefficient of Variation?
To annualise a monthly standard deviation, you should multiply the monthly figure by:
Which risk measure captures only systematic risk relative to the market?