StudyNISM
10.7

Measures of Risk

This sub‑topic covers the quantitative measures used to assess risk in mutual fund portfolios. Understanding these measures helps you evaluate fund volatility, systematic exposure and risk‑adjusted returns – all of which are frequently asked in the NISM Series V‑A exam. The content links risk concepts to practical fund selection and regulatory language used by SEBI.

Learning Objectives

  • 1Define and calculate standard deviation, variance and coefficient of variation.
  • 2Explain systematic risk and compute beta for a mutual fund.
  • 3Interpret Sharpe and Treynor ratios for risk‑adjusted performance.
  • 4Compare different risk measures and choose the appropriate one for exam questions.

Understanding Risk in Mutual Funds

Risk represents the uncertainty about the future returns of a mutual fund. In the Indian context, SEBI defines risk as the possibility of a fund’s actual performance deviating from its expected performance, which can affect an investor’s wealth.

Two broad categories of risk are recognised: total risk (the overall variability of returns) and systematic risk (the portion linked to market movements). Total risk is measured by statistical dispersion, while systematic risk is captured by beta, which relates the fund’s returns to a benchmark such as the Nifty 50.

For the NISM exam, you will often be asked to identify the correct formula, interpret a given risk metric, or select the most suitable measure for a scenario. Remember that the exam focuses on the concepts as presented in the official NISM syllabus, not on advanced academic definitions.

Key Risk Measures

The most commonly tested risk measures are:

  • Standard Deviation – quantifies total risk by showing how much returns deviate from the mean.
  • Beta – captures systematic risk relative to a market index.
  • Sharpe Ratio – evaluates risk‑adjusted return using total risk.
  • Treynor Ratio – evaluates risk‑adjusted return using systematic risk.
  • Coefficient of Variation (CV) – standardises risk by relating it to the average return.

Each measure serves a distinct purpose. For example, a fund with a high standard deviation but low beta may have high total volatility but low market‑related risk, a nuance that exam questions often probe.

Standard Deviation & Variance

Standard deviation (σ) is the square root of variance and indicates the average distance of each period’s return from the mean return (μ). In mutual fund analysis, it is expressed as a percentage and is calculated on historical periodic returns (usually annualised).

Variance (σ²) is the average of the squared deviations. While variance is useful for statistical derivations, investors and exam takers focus on standard deviation because it is in the same unit as the returns.

Exam tip: Never confuse the formula for variance with that for standard deviation – the latter includes a square‑root sign. A common trap is to report variance as the risk measure, which will lead to loss of marks.

∑Formula: Standard Deviation
σ=∑i=1N(Ri−μ)2N\sigma = \sqrt{\frac{\sum_{i=1}^{N} (R_i - \mu)^2}{N}}

Where:

σ= Standard deviation of returns (percentage)
R_i= Return in period i (percentage)
μ= Mean (average) return over N periods (percentage)
N= Number of periods

Worked Example

Given five annual returns: 10%, 12%, 8%, 11%, 9%: Step 1: μ = (10+12+8+11+9)/5 = 10%. Step 2: Compute squared deviations: (0)^2, (2)^2, (-2)^2, (1)^2, (-1)^2 = 0,4,4,1,1. Step 3: Σ = 10. Step 4: Variance = 10/5 = 2 (%^2). Step 5: σ = sqrt(2) ≈ 1.41%. Verification: sqrt(10/5) = 1.41%.

⚠️Exam Trap – Variance vs. Standard Deviation

Students often write the variance formula as the answer for a standard deviation question. Remember that the standard deviation includes the square‑root sign; omitting it will be marked incorrect.

Beta – Measuring Systematic Risk

Beta (β) measures how a fund’s returns move in relation to a market benchmark. A β of 1 indicates that the fund’s returns move in line with the market; β > 1 implies higher sensitivity, while β < 1 indicates lower sensitivity.

Beta is calculated using the covariance of the fund’s returns with the market returns divided by the variance of the market returns. SEBI’s guidelines use beta to assess the systematic risk exposure of equity‑linked schemes.

For the exam, you may be given a set of returns for a fund and its benchmark and asked to compute beta, or you may need to interpret a given beta value to decide if a fund is aggressive or defensive.

∑Formula: Beta
β=Cov⁡(Rf,Rm)σm2\beta = \frac{\operatorname{Cov}(R_f,R_m)}{\sigma_m^2}

Where:

β= Beta of the fund
Cov(R_f,R_m)= Covariance between fund returns (R_f) and market returns (R_m)
σ_m^2= Variance of market returns

Worked Example

Fund returns: 12%, 14%, 10%, 13%, 11%. Market returns: 10%, 12%, 8%, 11%, 9%. Step 1: Mean fund = 12%, mean market = 10%. Step 2: Deviations fund: 0,2,-2,1,-1; market: 0,2,-2,1,-1. Step 3: Covariance = Σ(dev_f × dev_m)/N = (0+4+4+1+1)/5 = 2. Step 4: Market variance = Σ(dev_m^2)/N = 10/5 = 2. Step 5: β = 2 / 2 = 1. Verification: (2) / (2) = 1.

ℹ️Beta Interpretation Pitfall

A beta of 0.8 does NOT mean the fund will earn 80% of market returns. It means the fund’s return moves 0.8 times the market’s movement; absolute returns also depend on the risk‑free rate and alpha.

Risk‑Adjusted Performance Ratios

Risk‑adjusted ratios combine return and risk into a single metric, allowing investors to compare funds with different volatility profiles. The two ratios most frequently tested are the Sharpe ratio (uses total risk) and the Treynor ratio (uses systematic risk).

The Sharpe ratio answers: "How much excess return am I getting per unit of total risk?" The Treynor ratio answers: "How much excess return am I getting per unit of market‑related risk?" Both require the risk‑free rate (Rf), usually the yield on a 10‑year Indian government bond.

In NISM questions, you may be given portfolio return, beta, standard deviation and Rf, and asked to compute one of these ratios or to select the better‑performing fund based on the higher ratio.

∑Formula: Sharpe Ratio
S=Rp−RfσpS = \frac{R_p - R_f}{\sigma_p}

Where:

S= Sharpe ratio (dimensionless)
R_p= Average portfolio return (percentage)
R_f= Risk‑free rate (percentage)
σ_p= Standard deviation of portfolio returns (percentage)

Worked Example

Given R_p = 12%, R_f = 6%, σ_p = 1.41%: Step 1: Excess return = 12% - 6% = 6%. Step 2: S = 6% / 1.41% ≈ 4.26. Verification: 6 / 1.41 = 4.26.

∑Formula: Treynor Ratio
T=Rp−RfβpT = \frac{R_p - R_f}{\beta_p}

Where:

T= Treynor ratio (dimensionless)
R_p= Average portfolio return (percentage)
R_f= Risk‑free rate (percentage)
β_p= Beta of the portfolio (dimensionless)

Worked Example

Using R_p = 12%, R_f = 6%, β_p = 1: Step 1: Excess return = 6%. Step 2: T = 6% / 1 = 6. Verification: 6 / 1 = 6.

Coefficient of Variation

The Coefficient of Variation (CV) standardises risk by expressing standard deviation as a proportion of the mean return. It is useful when comparing funds with different average returns.

A lower CV indicates a more efficient fund – it delivers higher returns per unit of risk. The formula is simple, making it a favourite for quick exam calculations.

Remember that CV is expressed as a decimal or percentage; many candidates mistakenly treat it as a raw standard deviation value, leading to wrong conclusions.

∑Formula: Coefficient of Variation
CV=σμCV = \frac{\sigma}{\mu}

Where:

CV= Coefficient of variation (decimal)
σ= Standard deviation of returns (percentage)
μ= Mean (average) return (percentage)

Worked Example

With σ = 1.41% and μ = 12%: Step 1: CV = 1.41 / 12 = 0.1175. Step 2: Expressed as 11.75%. Verification: 1.41 ÷ 12 = 0.1175.

Comparative View of Risk Measures

Summary of Common Risk Measures for Mutual Funds

MeasureFormula (simplified)CapturesTypical Use in Exams
Standard Deviationσ = sqrt( Σ(Ri-μ)² / N )Total risk (volatility)Calculate from historical returns
Betaβ = Cov(Rf,Rm) / σm²Systematic (market) riskCompare fund’s market sensitivity
Sharpe Ratio(Rp - Rf) / σpRisk‑adjusted return using total riskIdentify best‑performing fund
Treynor Ratio(Rp - Rf) / βpRisk‑adjusted return using systematic riskAssess performance of funds with different betas
Coefficient of Variationσ / μRisk per unit of returnCompare efficiency across funds with different returns

Typical Risk Metrics for Different Fund Types (Illustrative)

Example: Choosing a Fund Based on Risk Measures

Scenario

Rohan, a 35‑year‑old investor, wants to invest Rs. 1,00,000 in a mutual fund for the next 5 years. He is risk‑averse but wants better returns than a debt fund. He shortlists an Equity Fund (β = 1.1, σ = 18%) and a Hybrid Fund (β = 0.6, σ = 9%). The current risk‑free rate is 6% per annum. Both funds have an average annual return of 12% over the past five years.

Solution

Step 1: Compute Sharpe ratio for each fund. Equity: (12‑6)/18 = 6/18 = 0.33. Hybrid: (12‑6)/9 = 6/9 = 0.67. Step 2: Compute Treynor ratio. Equity: (12‑6)/1.1 = 6/1.1 ≈ 5.45. Hybrid: (12‑6)/0.6 = 6/0.6 = 10. Step 3: Compare. The Hybrid Fund has a higher Sharpe and Treynor ratio, indicating better risk‑adjusted performance despite the same average return. Hence, for a risk‑averse investor, the Hybrid Fund is the more suitable choice.

Conclusion

Rohan should prefer the Hybrid Fund because it delivers higher excess return per unit of both total and systematic risk, a conclusion directly supported by Sharpe and Treynor calculations.

⭐Exam Takeaways

  • Standard deviation measures total volatility; always apply the square‑root to the variance formula.
  • Beta quantifies systematic risk; compute it as covariance with the market divided by market variance.
  • Sharpe ratio uses total risk (σ) while Treynor ratio uses systematic risk (β); choose based on the question’s focus.
  • Coefficient of Variation allows comparison of funds with different average returns; lower CV means more efficient risk‑return trade‑off.
  • When interpreting risk metrics, remember that a higher beta implies higher market sensitivity, not higher absolute returns.

Practice Questions

8 questions on Measures of Risk

1

Which of the following best describes systematic risk in a mutual fund?

2

What is the formula used to calculate the standard deviation (σ) of a set of returns?

3

Given the five annual returns 10%, 12%, 8%, 11% and 9%, what is the standard deviation (σ) of these returns?

4

Using the fund and market returns in the example, what is the beta (β) of the fund?

5

Rohan compares an Equity Fund (β=1.1, σ=18%) with a Hybrid Fund (β=0.6, σ=9%). Both have Rp=12% and Rf=6%. Which fund offers the higher risk‑adjusted performance on both Sharpe and Treynor ratios?

6

An investor wants to compare a debt fund (average return 8%, σ 5%) with an equity fund (average return 12%, σ 20%). Which risk measure should be used to assess risk per unit of return?

7

A mutual fund has a beta of 0.8. Which statement correctly interprets this beta value?

8

According to the material, a lower Coefficient of Variation (CV) indicates:

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