FRM Exam Part II · VaR Mapping

Beta Mapping and Equity Portfolio VaR Mapping

Updated 11 October 2026 · Fact-checked

Beta mapping replaces each stock with a position in a market index equal to its beta times its value. Systematic VaR is the index exposure times the index volatility times the z-value. Specific (residual) risk is added separately and shrinks as the portfolio is diversified across many stocks.

Understand Mapping Equity Portfolios and Beta Mapping

A bank may hold thousands of stocks. Modelling every stock and every correlation is slow and noisy. Mapping reduces the problem. You pick a small set of risk factors and express each position as an exposure to them.

The simplest factor is a market index. The single-index model says a stock's return is R_i = α_i + β_i × R_M + ε_i. The term β_i × R_M is systematic risk, driven by the market. The term ε_i is specific (residual) risk, unique to the firm. The residual is assumed uncorrelated with the market and with other stocks' residuals.

Beta mapping keeps only the systematic part. A position of value W_i in stock i is treated as a position of β_i × W_i in the index. For a portfolio, add these up: the index-equivalent exposure is Σ β_i × W_i. The portfolio's market VaR is then that exposure times the index volatility times the z-value for the confidence level.

Specific risk is what the mapping leaves out. For one stock it can be large. For a portfolio of N stocks with similar sizes, residual variances add up while weights shrink, so portfolio residual risk falls roughly with 1/√N. Systematic risk does not diversify away. In a well-diversified portfolio, the index VaR is close to the full VaR.

A multi-factor model extends this. It uses several factors, such as market, sector, size or style. Each stock gets a beta to each factor. VaR is computed from the factor exposures and the factor covariance matrix. This captures sector concentrations that a single index misses. Specific risk is still added separately.

Key formulas to remember

Single-index model: R_i = α_i + β_i × R_M + ε_i; ε_i is uncorrelated with R_M and with other stocks' residuals.
Beta: β_i = Cov(R_i, R_M) ÷ Var(R_M) = ρ_iM × σ_i ÷ σ_M; Measures sensitivity to the index, not total risk.
Portfolio beta exposure: Index-equivalent exposure = Σ β_i × W_i; Use value weights. Short positions have negative W_i.
Systematic (mapped) VaR: VaR_sys = z × σ_M × Σ β_i × W_i; σ_M must be for the same horizon as the VaR. Take the absolute value of the exposure.
Variance decomposition: σ_i² = β_i² × σ_M² + σ_ε,i²; Total variance equals systematic plus specific.
Portfolio specific risk: σ_ε,p = √(Σ W_i² × σ_ε,i²); Uses squared position values because residuals are uncorrelated.
Total VaR: VaR_total = z × √[(σ_M × Σ β_i W_i)² + Σ W_i² σ_ε,i²]; Combine systematic and specific by squares. Do not add the two VaRs.
Equal-position diversification: σ_ε,p = σ_ε × W_total ÷ √N; For N equal positions with equal residual volatility σ_ε.
Horizon scaling: σ_T = σ_daily × √T; Assumes independent, identically distributed returns.

How to solve Mapping Equity Portfolios and Beta Mapping questions

Use this order for any beta mapping or equity mapping question.

1Read the confidence level, horizon and the units of volatility (daily, annual, or already in currency).
2Convert each stock value to index-equivalent exposure: β_i × W_i. Keep the sign for shorts.
3Sum the exposures to get the net index position.
4Compute the index volatility over the VaR horizon and multiply by the net exposure and z (1.645 at 95%, 2.326 at 99%).
5If specific risk is asked for, compute √(Σ W_i² × σ_ε,i²) using full position values, not beta-adjusted ones.
6Combine systematic and specific risk as the square root of the sum of squares, then multiply by z.
7State the interpretation: what is covered, what is ignored, and how diversification changes the answer.

Quickest way: Net beta exposure first, then one multiplication

When to use it: Use when the question gives betas, values and index volatility and asks only for mapped (systematic) VaR.

Compute net exposure = Σ β × W in one line.
Multiply by σ_M for the horizon, then by z.
For specific risk with N equal positions, use σ_ε × W_total ÷ √N instead of summing squares.
Check whether the options differ by a factor of z or √T. This often removes two wrong choices at once.

Common mistakes in Mapping Equity Portfolios and Beta Mapping

Using total stock volatility instead of index volatility in the mapped VaR.
Stock volatility is often listed beside beta in the question.
Fix: After mapping, the position is an index position. Use σ_M only; stock-specific volatility belongs in the residual term.
Adding systematic VaR and specific VaR directly.
Students treat VaR figures as additive.
Fix: Residual risk is uncorrelated with the market, so combine variances. Total = √(sys² + spec²).
Using β_i × W_i in the specific-risk sum.
Students carry the mapped exposure through every step.
Fix: Specific risk is not scaled by beta. Use W_i² × σ_ε,i² with the full position value.
Saying diversification removes all risk in the portfolio.
Confusing specific risk with total risk.
Fix: Diversification shrinks specific risk, roughly with 1/√N. Systematic risk stays and scales with portfolio beta.
Mixing horizons, such as annual index volatility with a 10-day VaR.
Volatility is quoted in different units across questions.
Fix: Convert first: σ_10day = σ_annual × √(10 ÷ 252), or σ_daily × √10.
Ignoring signs on short positions or hedges.
Students sum absolute values of exposure.
Fix: Net exposure uses signed values. A short offsets long beta exposure. Take the absolute value only at the end.

Worked examples

Example 1

A portfolio holds ₹40 crore of stock A (β = 1.2) and ₹60 crore of stock B (β = 0.8). The index has daily volatility of 1.0%. Compute the one-day 99% systematic VaR using beta mapping (z = 2.326). Give the answer in ₹ crore.

Show the solution

Index-equivalent exposure of A = 1.2 × 40 = ₹48 crore.
Index-equivalent exposure of B = 0.8 × 60 = ₹48 crore.
Net exposure = 48 + 48 = ₹96 crore.
VaR = 2.326 × 0.01 × 96 = 2.2330 crore approximately.

Answer: About ₹2.23 crore. This is systematic VaR only; it excludes the specific risk of A and B.

Example 2

An investor holds 25 stocks, each worth USD 2 million (total USD 50 million). Each stock has residual volatility of 20% per year. Assume residuals are uncorrelated. What is the annual residual volatility of the portfolio in USD, and what happens if the number of equal positions rises to 100 with the same total value?

Show the solution

Residual volatility of one position = 0.20 × 2 million = USD 0.4 million.
Variance of 25 uncorrelated residuals = 25 × (0.4)² = 25 × 0.16 = 4 (USD million squared).
Portfolio residual volatility = √4 = USD 2 million.
Check with the shortcut: 0.20 × 50 ÷ √25 = 10 ÷ 5 = USD 2 million.
With 100 positions, each is USD 0.5 million. Shortcut: 0.20 × 50 ÷ √100 = 10 ÷ 10 = USD 1 million.

Answer: Residual volatility is USD 2 million with 25 stocks and falls to USD 1 million with 100 stocks. Quadrupling N halves specific risk. Systematic risk is unchanged.

Exam tips

Questions often ask which risk is removed by mapping and which remains. Mapping drops specific risk; diversification reduces it only when many names are held.
Check the horizon and units of volatility before any arithmetic. This is the most common trap.
Remember that a hedged portfolio with net beta near zero can still carry large specific risk. Expect a conceptual question on this.
If a question mentions sector concentration, the answer is likely a multi-factor model rather than a single index.
Use √(sum of squares) to combine systematic and specific risk. A direct sum is always a distractor.

Practice questions from VaR Mapping

Mapping Equity Portfolios and Beta Mapping: frequently asked questions

What is beta mapping in VaR?

It replaces each stock with a position in a market index equal to the stock's beta times its value. You then compute VaR on the net index position using index volatility. It captures systematic risk only.

Why is specific risk ignored in mapped VaR?

Specific risk is unique to each firm and uncorrelated across stocks. In a large diversified portfolio it becomes small relative to market risk. It can be added separately when the portfolio is concentrated.

How do I add specific risk to systematic VaR?

Compute the portfolio residual volatility as √(Σ W_i² × σ_ε,i²). Then combine it with the index risk using a square root of the sum of squares. Do not add the two VaR figures.

When should I use a multi-factor model instead of a single index?

Use it when the portfolio has sector, size, style or country tilts that one index cannot capture. Each stock gets a beta to each factor, and VaR uses the factor covariance matrix.