Strategic Financial Management · Risks in Financial Market

Measuring Risk: Duration, Beta and Value at Risk

Updated 11 October 2026 · Fact-checked

Risk measures turn uncertainty into numbers. Standard deviation gives total risk, beta gives market risk, duration gives a bond's price sensitivity to interest rates, and Value at Risk gives the loss you should not exceed at a stated confidence level over a set period. Pick the measure the question names, then apply its formula.

Understand Measuring Risk: Duration, Beta and Value at Risk

Risk is the chance that actual returns differ from expected returns. To manage it, you first have to measure it. Each tool below measures a different kind of risk, so the first job is to match the tool to the risk.

Standard deviation measures total risk, meaning how widely returns scatter around their average. It counts both market-wide and company-specific movements. A higher value means a more volatile asset.

Beta measures only systematic (market) risk. It tells you how much a security moves when the market moves. A beta of 1.4 means the security tends to move 1.4% for every 1% move in the market. Beta below 1 means a defensive security. Diversification cannot remove this risk, which is why beta is the risk measure used in CAPM.

Duration measures interest rate risk in a bond. Macaulay duration is the weighted average time, in years, to receive the bond's cash flows, with present values as weights. Modified duration converts it into a price sensitivity: it gives the approximate percentage fall in price for a 1% rise in yield. Longer duration means more price risk. A lower coupon or a longer maturity raises duration.

Value at Risk (VaR) answers one question: what is the most I can lose over a given period, at a given confidence level, under normal market conditions? For example, a one-day 95% VaR of ₹1 crore means that on 95 days out of 100 you expect the loss not to exceed ₹1 crore. It does not say how bad the other 5 days can be. VaR can be found by the parametric (variance-covariance) method, historical simulation or Monte Carlo simulation. The exam mostly tests the parametric method.

Key rules to remember

Standard deviation (probability data): σ = √[Σ p × (R − E(R))²]; E(R) = Σ p × R. Variance is σ². For past return data, divide the sum of squared deviations by (n − 1) for a sample or n for a population, as the question states.
Beta: β = Cov(i, m) ÷ σm² = ρ(i, m) × σi ÷ σm; Cov is covariance of the security with the market, σm² is market variance and ρ is the correlation coefficient.
Portfolio beta: βp = Σ wi × βi; Weights are market-value proportions and must add to 1. Beta of the risk-free asset is 0.
Macaulay duration: D = Σ [t × PV(CFt)] ÷ Σ PV(CFt) = Σ [t × PV(CFt)] ÷ Bond price; Discount each cash flow at the bond's yield to maturity. The denominator equals the bond's price. Duration of a zero-coupon bond equals its maturity.
Modified duration: MD = D ÷ (1 + y); Use y per period. For annual coupons, y is the annual yield. For semi-annual coupons, convert carefully to years.
Price change from duration: ΔP ÷ P ≈ − MD × Δy; A linear estimate that works for small yield changes. For large changes, convexity matters.
Parametric VaR: VaR = z × σ × Portfolio value × √t; σ is the return standard deviation for one period, and t is the number of periods. Common one-tailed z values: 1.645 for 95% and 2.33 for 99%. Use the z value or normal table the question gives.

How to solve Measuring Risk: Duration, Beta and Value at Risk questions

Use this method for any question on measuring risk. It keeps you from mixing up the tools.

1Read what is asked: total risk (standard deviation), market risk (beta), bond price risk (duration) or maximum loss (VaR).
2List the data given: probabilities or returns, market data, cash flows and yield, or portfolio value, volatility, confidence level and holding period.
3Write the formula for the chosen measure before putting in numbers.
4Check units: percentage or decimal, annual or daily volatility, years or half-years.
5Compute step by step in a neat table, especially for duration and standard deviation, so you earn method marks.
6For VaR, scale volatility to the holding period with √t, then multiply by z and the portfolio value.
7State the answer with its unit (years, times, ₹) and a one-line interpretation.
8If the question asks for a view or recommendation, compare the measure with the investor's risk tolerance or the benchmark.

Quickest way: Duration table and VaR in three lines

When to use it: Use it when time is short, such as 14-mark questions with several parts or MCQs on risk measures.

For duration, build four columns: year t, cash flow, PV at yield, and t × PV. Total columns 3 and 4, then divide the second total by the first.
For a zero-coupon bond, skip the table. Duration equals years to maturity.
For a quick price impact, compute MD = D ÷ (1 + y) and multiply by the yield change.
For VaR, compute z × σ first, then scale for time, then multiply by the portfolio value.
For beta MCQs, check whether covariance and market variance are given, or correlation and both standard deviations. Use the matching form.
Sense-check: duration cannot exceed a bond's maturity for a coupon bond, and VaR must grow when time or confidence level rises.

Common mistakes in Measuring Risk: Duration, Beta and Value at Risk

Treating beta and standard deviation as the same thing.
Both are called measures of risk, so students use them interchangeably.
Fix: Standard deviation is total risk. Beta is only systematic risk relative to the market. A security can have a high standard deviation and a low beta.
Forgetting to divide by the bond price when computing duration.
Students total t × PV and stop, or divide by face value.
Fix: Divide the total of t × PV by the sum of PVs, which is the price at the same yield.
Using modified duration without the (1 + y) adjustment, or applying it with the wrong sign.
Macaulay duration and modified duration are both just called duration.
Fix: Compute MD = D ÷ (1 + y). Bond prices move opposite to yields, so the price change carries a negative sign.
Not scaling VaR for the holding period.
Students use daily σ directly for a 10-day VaR.
Fix: Multiply by √t when volatility is for one period and the holding period is t periods.
Reading 95% VaR as the maximum possible loss.
The word 'maximum' in the definition hides the confidence level.
Fix: Say that the loss is expected to exceed VaR on about 5% of occasions. VaR says nothing about the size of those losses.
Mixing percentages and decimals or annual and daily figures.
Data come in different forms within the same question.
Fix: Convert everything to one form before you start. Write 1.2% as 0.012 and keep yield and period consistent.

Worked examples

Example 1

A 3-year bond of face value ₹1,000 pays an annual coupon of 10%. Its yield to maturity is 12%. Compute (a) the price, (b) Macaulay duration, (c) modified duration, and (d) the approximate fall in price if the yield rises by 0.5 percentage points.

Show the solution

Cash flows: ₹100 in year 1, ₹100 in year 2, ₹1,100 in year 3.
Present values at 12%: year 1 = 100 ÷ 1.12 = ₹89.29. Year 2 = 100 ÷ 1.2544 = ₹79.72. Year 3 = 1,100 ÷ 1.404928 = ₹782.96.
Price = 89.29 + 79.72 + 782.96 = ₹951.96 (approximately).
t × PV: 1 × 89.29 = 89.29. 2 × 79.72 = 159.44. 3 × 782.96 = 2,348.87. Total = 2,597.60.
Macaulay duration = 2,597.60 ÷ 951.96 = 2.73 years (approximately).
Modified duration = 2.73 ÷ 1.12 = 2.44 (approximately).
Price change ≈ − 2.44 × 0.5% = − 1.22%. In rupees, 1.22% of ₹951.96 is about ₹11.6.

Answer: Price ≈ ₹951.96; Macaulay duration ≈ 2.73 years; modified duration ≈ 2.44; a 0.5 percentage point rise in yield lowers the price by about 1.22%, or roughly ₹11.6.

Example 2

An Indian mutual fund holds an equity portfolio worth ₹50 crore. Daily return volatility is 1.2%. Using the parametric method, compute the one-day VaR at 95% confidence (z = 1.645) and the 10-day VaR. Interpret the result.

Show the solution

Daily volatility in decimals = 0.012.
One-day VaR = 1.645 × 0.012 × ₹50 crore = 0.01974 × 50 = ₹0.987 crore, or about ₹98.7 lakh.
10-day VaR = one-day VaR × √10 = 0.987 × 3.1623 = ₹3.12 crore (approximately).
Interpretation: under normal market conditions, the fund expects its one-day loss to exceed ₹98.7 lakh on only about 5 days in 100. Over 10 days, the loss is expected to exceed ₹3.12 crore on about 5% of occasions.
Note that VaR does not show how large the loss can be on those exceptional days.

Answer: One-day 95% VaR ≈ ₹98.7 lakh; 10-day 95% VaR ≈ ₹3.12 crore.

Exam tips

Read the confidence level and holding period first. Many VaR errors come from missing them.
Show the duration table in full. Examiners award marks for the PV and t × PV columns even if the final division is slightly off.
In MCQs, check the direction of change: when yields rise, bond prices fall, and higher duration means a bigger fall.
When a case asks for a recommendation, link the measure to a decision, such as choosing a lower-duration bond if rates are expected to rise.
Learn the limits of each tool. A short note on why VaR ignores losses beyond the cut-off, or why beta captures only market risk, often earns a mark.

Practice questions from Risks in Financial Market

Measuring Risk: Duration, Beta and Value at Risk in other exams

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

FRM Part IValue at Risk (VaR) Basics

Measuring Risk: Duration, Beta and Value at Risk: frequently asked questions

What is Value at Risk in simple words?

VaR is the loss amount that you expect not to exceed over a set period at a chosen confidence level. A one-day 95% VaR of ₹1 crore means the loss should be below ₹1 crore on about 95 days out of 100. It does not tell you how large the loss can be on the remaining days.

How do I calculate the duration of a bond for CMA Final?

Discount each cash flow at the yield to maturity and multiply each present value by its year. Add those products and divide by the sum of present values, which is the bond price. The result is Macaulay duration in years. Divide by (1 + y) to get modified duration.

What is the difference between beta and standard deviation?

Standard deviation measures total risk, including company-specific risk that diversification can remove. Beta measures only market risk, which cannot be diversified away. Use beta for CAPM and well-diversified portfolios, and standard deviation for stand-alone risk.

Which VaR method is asked in the exam?

The parametric method, using a z value, volatility, portfolio value and holding period, is the usual numerical form. You should also know in words that historical simulation and Monte Carlo simulation are alternative methods. Use the z value that the question gives.