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NISM-Series-X-A: Investment Adviser (Level 1) · Portfolio Performance Measurement and Evaluation

Risk Measures: Standard Deviation, Beta and Tracking Error

Updated 11 October 2026 · Fact-checked

Risk measures show how much a portfolio's returns vary. Standard deviation measures total risk. Beta measures sensitivity to the market. Downside deviation counts only bad outcomes. Tracking error is the standard deviation of active returns versus a benchmark. Value at risk estimates a loss limit at a confidence level.

Understand Risk Measures: Standard Deviation, Beta and Tracking Error

Risk in portfolio evaluation means uncertainty of returns. You cannot judge a return without knowing the risk taken to earn it. The measures in this topic put a number on that risk.

Standard deviation is a total risk measure. It tells you how widely a portfolio's returns scatter around their average. It includes both market risk and risk specific to the securities held. A higher standard deviation means more volatile returns. It treats upside and downside surprises alike.

Beta is a relative (market) risk measure. It shows how much the portfolio moves when the market benchmark moves. A beta of 1 means it moves with the market. Above 1 means it swings more than the market. Below 1 means it swings less. Beta captures only systematic risk. A well-diversified portfolio can have a low standard deviation gap from its beta-implied risk, but a concentrated one can have high standard deviation even with a beta of 1.

Downside deviation fixes a weakness of standard deviation. Investors do not mind returns above target. So downside deviation measures dispersion only of returns below a chosen minimum acceptable return. It is the base of the Sortino ratio.

Tracking error measures how closely a portfolio follows its benchmark. It is the standard deviation of the difference between portfolio return and benchmark return (the active return). An index fund aims for a very low tracking error. An active fund has a higher one.

Value at risk (VaR) states the maximum loss expected over a set period at a set confidence level, under normal conditions. For example, a one-day 95% VaR of ₹2 lakh means a loss above ₹2 lakh is expected on only about 5% of days. VaR does not say how bad the loss can be beyond that limit.

Key formulas to remember

Standard deviation (sample)
σ = √[ Σ(Rᵢ − R̄)² ÷ (n − 1) ]
Variance is σ². Use n instead of n − 1 if the question treats the data as the full population.
Beta
β = Cov(Rp, Rm) ÷ σm²
Also β = ρ × (σp ÷ σm), where ρ is the correlation with the market. Market beta is 1.
Tracking error
TE = standard deviation of (Rp − Rb)
Active return = portfolio return − benchmark return. TE is not the average of the differences.
Downside deviation
DD = √[ Σ(min(Rᵢ − MAR, 0))² ÷ n ]
MAR is the minimum acceptable return. Returns above MAR count as zero but still stay in n.
Parametric VaR
VaR = z × σ × portfolio value
z is about 1.645 at 95% and 2.33 at 99% (one-tailed). Multi-day VaR = one-day VaR × √days.
Portfolio beta
βp = Σ(wᵢ × βᵢ)
Weights are proportions of portfolio value.

How to solve Risk Measures: Standard Deviation, Beta and Tracking Error questions

Use this method for any question on risk measures, whether it asks for a calculation or a concept.

  1. 1Identify what is asked: total risk, market risk, downside risk, benchmark-relative risk or loss limit.
  2. 2Match the measure: standard deviation for total, beta for market, downside deviation for below-target, tracking error for benchmark gap, VaR for loss limit.
  3. 3For calculations, list the data and note whether it is a sample or population.
  4. 4For tracking error, first compute the active return for each period, then take the standard deviation of those values.
  5. 5Apply the formula carefully: square deviations, sum, divide, then take the square root.
  6. 6For VaR, multiply z by σ by value, and scale by √time if the period is longer.
  7. 7Check the answer for sense: beta near 1 for a diversified equity fund, tracking error small for an index fund.
  8. 8Read all options and reject those that confuse total and relative risk.

Quickest way: Match the measure by keyword

When to use it: Use this for definition-type or comparison questions where no heavy arithmetic is needed.

  1. Spot the keyword: 'total' means standard deviation, 'market' or 'systematic' means beta.
  2. 'Benchmark', 'index fund' or 'active return' means tracking error.
  3. 'Below target' or 'Sortino' means downside deviation.
  4. 'Maximum loss', 'confidence level' or 'time horizon' means VaR.
  5. For a VaR scaling question, multiply by the square root of days, not by days.
  6. For portfolio beta, take the weighted average of betas.

Common mistakes in Risk Measures: Standard Deviation, Beta and Tracking Error

  • Treating standard deviation and beta as the same risk

    Both are called volatility measures.

    Fix: Standard deviation is total risk. Beta is only market-related risk measured against a benchmark.

  • Calculating tracking error as the average of return differences

    Students confuse it with average active return.

    Fix: Tracking error is the standard deviation of the active returns, not their mean.

  • Dropping above-target returns from the count in downside deviation

    Students think they are ignored.

    Fix: Treat them as zero deviation but keep them in the number of observations.

  • Scaling VaR by the number of days directly

    Linear scaling feels natural.

    Fix: Under the standard assumption, multiply by the square root of the number of days.

  • Reading VaR as the worst possible loss

    The word 'maximum' is misleading.

    Fix: VaR is the loss expected not to be exceeded at the stated confidence. Losses beyond it can still occur.

  • Using n instead of n − 1 for a sample, or the reverse

    Students forget to read the data type.

    Fix: Use n − 1 for sample data unless the question says population.

Worked examples

Example 1

A fund's annual returns over four years were 10%, 14%, 6% and 10%. Find the sample standard deviation.

Show the solution
  1. Mean = (10 + 14 + 6 + 10) ÷ 4 = 10%.
  2. Deviations: 0, 4, −4, 0.
  3. Squares: 0, 16, 16, 0. Sum = 32.
  4. Divide by n − 1 = 3: 32 ÷ 3 = 10.67.
  5. Square root: √10.67 ≈ 3.27%.

Answer: The sample standard deviation is about 3.27%.

Example 2

A portfolio has a value of ₹50,00,000 and a daily standard deviation of 1%. Using z = 1.645, find the one-day 95% parametric VaR.

Show the solution
  1. VaR = z × σ × value.
  2. Compute 1.645 × 0.01 = 0.01645.
  3. Multiply by ₹50,00,000: 0.01645 × 50,00,000 = ₹82,250.

Answer: The one-day 95% VaR is ₹82,250. A loss above this is expected on about 5% of days.

Exam tips

  • Expect conceptual questions that ask you to pick between total and market risk.
  • Remember that beta ignores unsystematic risk, so it can mislead for undiversified portfolios.
  • For a high tracking error, think active management. For a low one, think index fund.
  • Watch for the square root of time in VaR questions.
  • Negative marking applies, so skip only if you cannot remove at least two options.

Practice questions from Portfolio Performance Measurement and Evaluation

Risk Measures: Standard Deviation, Beta and Tracking Error in other exams

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

Risk Measures: Standard Deviation, Beta and Tracking Error: frequently asked questions

What is the difference between standard deviation and beta?

Standard deviation measures total risk, covering both market and security-specific risk. Beta measures only sensitivity to market movements. A concentrated portfolio can have a high standard deviation but a beta near 1.

How do you calculate tracking error?

Subtract the benchmark return from the portfolio return for each period to get active returns. Then compute the standard deviation of those active returns. That figure is the tracking error.

What is downside deviation?

It measures the dispersion of returns that fall below a minimum acceptable return. Returns above that level are not treated as risk. It is used in the Sortino ratio.

What does value at risk tell an adviser?

It gives a loss level that should not be exceeded over a set period at a stated confidence level. It does not show the size of losses beyond that level.