CFA Level I Exam · Introduction to Risk Management

Measuring Risk: Drivers and Metrics for CFA Level I

Updated 7 October 2026 · Fact-checked

Risk measurement turns uncertainty into numbers. Standard deviation and beta measure equity risk, duration and convexity measure bond rate risk, and delta, gamma and vega measure option risk. VaR is the loss expected to be equaled or exceeded with a stated probability over a period. CVaR averages losses beyond VaR. Match the metric to the risk driver.

Understand Measuring Risk: Drivers and Metrics

Risk comes from drivers: the underlying things that move a value. For equities the driver is market movement. For bonds it is the interest rate. For options it is the price and volatility of the underlying. A risk metric is a number that tells you how much the value changes when a driver moves.

For equities, standard deviation measures total risk, meaning how widely returns spread around the mean. Beta measures only sensitivity to the market, so it captures systematic risk. For bonds, duration estimates the percentage price change for a small change in yield. Convexity corrects that estimate for large moves, because the price-yield curve is curved, not straight.

For derivatives, the Greeks measure sensitivity. Delta is the option price change per unit change in the underlying. Gamma is the change in delta per unit change in the underlying. Vega is the change in option value per change in volatility. Theta is time decay and rho is sensitivity to the interest rate.

Value at risk (VaR) is the loss amount that is expected to be equaled or exceeded with a stated probability over a period (the minimum loss in the worst x% of outcomes). For example, a one-day 5% VaR of $2 million means there is a 5% chance of losing at least $2 million in a day. It is not the worst case. CVaR (conditional VaR, or expected shortfall) is the average loss in the cases where the loss is at least as large as VaR, so it says more about the tail.

VaR can be found three ways: the parametric (variance-covariance) method assumes normal returns, historical simulation uses past returns, and Monte Carlo simulation uses generated scenarios. Each has limits. VaR says nothing about how bad losses are beyond the cutoff, it depends on its inputs and assumptions, and it can miss fat tails and liquidity problems. Stress tests and scenario analysis help fill those gaps.

Key formulas to remember

Parametric VaR (return form): VaR = [−E(R) + z × σ] × portfolio value; For a loss-side cutoff, z is 1.645 at 5% and 2.33 at 1% (one-tailed, normal). Use E(R) and σ for the same period.
Scaling VaR over time: σ(T days) = σ(1 day) × √T; Valid when daily returns are independent. With the mean set to zero, VaR scales with √T.
Duration price estimate: %ΔP ≈ −ModDur × ΔYield; Good for small yield changes. Use effective duration for bonds with embedded options.
Duration plus convexity: %ΔP ≈ −ModDur × ΔY + ½ × Convexity × (ΔY)²; Convexity adds a positive amount for both rises and falls in yield for an option-free bond.
Delta approximation for options: ΔOption ≈ Delta × ΔUnderlying; Call delta is between 0 and 1. Put delta is between −1 and 0.
Delta-gamma approximation: ΔOption ≈ Delta × ΔS + ½ × Gamma × (ΔS)²; Gamma improves the estimate for larger moves in the underlying.
Beta: β = Cov(Ri, Rm) ÷ Var(Rm); Beta of 1 means the asset moves with the market on average.

How to solve Measuring Risk: Drivers and Metrics questions

Use this order for any question on risk drivers or metrics.

1Identify the risk driver in the stem: market, interest rate, option underlying, volatility, or tail loss.
2Pick the matching metric: standard deviation or beta for equities, duration and convexity for bonds, a Greek for options, VaR or CVaR for loss limits.
3Check the units and period. Convert annual figures to the VaR horizon, using √T for volatility.
4Insert the numbers and keep the sign convention. A VaR is quoted as a positive loss amount.
5For bonds and options, compute the first-order effect, then add the second-order term (convexity or gamma) if the move is large or the stem gives it.
6Eliminate options that confuse the metrics, such as calling VaR a worst-case loss or CVaR a smaller number than VaR.
7Sanity check: CVaR should be at least VaR, and a long option position should have positive gamma.

Quickest way: Match metric to risk, then scale

When to use it: Use for fast VaR, duration and Greek questions when you have about 90 seconds.

Read the last sentence first to see which metric is asked.
For VaR, write z × σ − mean, scale to the period, then multiply by value.
For duration, multiply −duration by the yield change in decimals, then by price or value.
Add ½ × convexity × (ΔY)² only if convexity is given.
Remove the options that break a rule: negative convexity for an option-free bond, CVaR below VaR, delta above 1 for a call.

Common mistakes in Measuring Risk: Drivers and Metrics

Treating VaR as the maximum possible loss.
The word 'risk' suggests a worst case.
Fix: Say VaR is the loss expected to be equaled or exceeded with the stated probability. Losses can be much larger.
Forgetting to scale volatility to the VaR horizon.
Annual volatility is given but the question asks for a daily or monthly figure.
Fix: Divide the annual σ by √(periods per year), using the same basis the question gives, such as 250 days.
Using the wrong z-value.
Confusing two-tailed and one-tailed values.
Fix: VaR is one-tailed: 1.645 for 5% and 2.33 for 1%.
Mixing up duration direction.
Students forget the minus sign.
Fix: Yields up means prices down. Put the minus sign in front of duration × ΔY.
Confusing delta with gamma.
Both are described as sensitivities to the underlying.
Fix: Delta is the first derivative with respect to the underlying price. Gamma is how fast delta itself changes.
Saying beta measures total risk.
Beta and standard deviation are both called risk measures.
Fix: Beta captures only market (systematic) risk. Standard deviation captures total risk.

Worked examples

Example 1

A portfolio is worth $50 million. Its expected daily return is 0.04% and its daily standard deviation is 1.2%. Assuming normal returns, what is the one-day 5% parametric VaR? A. $0.74 million B. $0.97 million C. $1.01 million

Show the solution

Use VaR = (z × σ − E(R)) × value with z = 1.645.
z × σ = 1.645 × 1.2% = 1.974%.
Subtract the mean: 1.974% − 0.04% = 1.934%.
Multiply: 1.934% × $50,000,000 = $967,000, or about $0.97 million.

Answer: B. $0.97 million

Example 2

An option-free bond has a modified duration of 6.0 and convexity of 50. Yield rises by 100 bps. Estimate the percentage price change. A. −6.25% B. −6.00% C. −5.75%

Show the solution

ΔY = 0.01.
Duration effect = −6.0 × 0.01 = −0.06 = −6.00%.
Convexity effect = ½ × 50 × (0.01)² = 0.5 × 50 × 0.0001 = 0.0025 = +0.25%.
Total = −6.00% + 0.25% = −5.75%.

Answer: C. −5.75%. Convexity reduces the loss from a rise in yield.

Exam tips

Know the direction of every metric: duration lowers prices as yields rise, and convexity adds value on both sides for option-free bonds.
Be ready for conceptual items on VaR limits: it ignores tail loss size, depends on method and inputs, and can be hard to compare across firms.
Remember CVaR is at least as large as VaR and describes the average tail loss.
Match each Greek with its driver: delta with price, gamma with delta change, vega with volatility, theta with time, rho with rates.
On numerical VaR items, check the time scaling first. It is the most common trap.

Practice questions from Introduction to Risk Management

Measuring Risk: Drivers and Metrics: frequently asked questions

What is the difference between VaR and CVaR?

VaR is the loss amount expected to be equaled or exceeded with a given probability over a period. CVaR is the average loss when losses are at or beyond the VaR level. CVaR is therefore never smaller than VaR and shows more about the tail.

How do I calculate parametric VaR for CFA Level I?

Take the expected return and standard deviation for the period. Compute z × σ − mean, using 1.645 for 5% or 2.33 for 1%. Multiply by the portfolio value.

What are the main limitations of value at risk?

VaR does not show how large losses can be beyond the cutoff. Results depend on the method, data and assumptions, such as normal returns. It can understate fat tails and liquidity risk, so firms add stress tests and scenario analysis.

Which method is used to estimate VaR?

The three methods are parametric (variance-covariance), historical simulation and Monte Carlo simulation. Parametric is fast but assumes a distribution. Historical uses past data. Monte Carlo is flexible but needs a model and more computing.