FRM Exam Part II · VaR and Risk Budgeting in Investment Management

Optimal Portfolio Allocation and Risk-Adjusted Performance

Updated 11 October 2026 · Fact-checked

Risk-adjusted measures divide return by risk. Sharpe ratio uses total excess return over total volatility. Information ratio uses active return over tracking error. The optimal allocation equates the ratio of excess return to marginal VaR across all positions. If any ratio is higher, shift risk budget towards it until they are equal.

Understand Optimal Portfolio Allocation and Risk-Adjusted Performance

Return alone cannot rank managers. A manager who earned 12% with huge risk may be worse than one who earned 8% with little risk. Risk-adjusted measures fix this by dividing return by a risk number.

The Sharpe ratio compares a portfolio with the risk-free rate. It is (Rp − Rf) ÷ σp. It suits a total portfolio or a stand-alone investment where the alternative is cash.

The information ratio (IR) compares a portfolio with a benchmark. It is the active return (Rp − Rb) divided by the tracking error, the standard deviation of active return. It suits an active manager judged against an index. It tells you how much active return you get per unit of active risk.

For allocation, think of risk as a budget. Each position adds risk, measured by its marginal VaR (how much portfolio VaR changes for a small increase in that position). The best portfolio is reached when every position gives the same excess return per unit of marginal VaR. If one position gives more, move risk into it and out of the others. This is the same idea as equalising marginal returns in any optimisation.

In the usual textbook form, the optimal condition is: (expected return of i − Rf) ÷ marginal VaR of i is the same for all i. At the optimum, this common ratio is also linked to the portfolio's overall Sharpe ratio.

Key formulas to remember

Sharpe ratio: SR = (Rp − Rf) ÷ σp; Rf is the risk-free rate; σp is the standard deviation of portfolio returns.
Information ratio: IR = (Rp − Rb) ÷ TE; TE is tracking error, the standard deviation of (Rp − Rb). Numerator is the mean active return (alpha against the benchmark).
Tracking error: TE = σ(Rp − Rb); Use the volatility of the active return, not the volatility of the portfolio.
Optimal risk budget condition: (Ri − Rf) ÷ MVaRi = same value for every position i; MVaRi is the marginal VaR of position i. If ratios differ, reallocate towards the higher ratio.
Marginal VaR link to beta: MVaRi = (VaRp ÷ Wp) × βi; βi is the beta of position i against the portfolio, and Wp is portfolio value. So the condition can be stated as excess return ÷ beta being equal.
Risk-adjusted value of a ratio: Annualised IR ≈ monthly IR × √12; Only valid if active returns are independent and identically distributed.

How to solve Optimal Portfolio Allocation and Risk-Adjusted Performance questions

Use this order for any question on risk-adjusted performance or optimal allocation.

1Identify what the question compares against: the risk-free rate (Sharpe) or a benchmark (information ratio).
2Write down the return numerator: excess return over Rf, or active return over the benchmark.
3Find the correct risk denominator: total volatility for Sharpe, tracking error for IR. Check the period matches the return.
4If tracking error is not given, compute it from active-return volatility, or from the variance of the portfolio, benchmark and their correlation.
5Calculate the ratio and compare it across managers or positions. Higher is better, given the same measure.
6For allocation questions, compute excess return ÷ marginal VaR (or ÷ beta) for each position.
7If the ratios differ, move risk budget towards the higher ratio. If they are equal, the allocation is optimal.
8State the interpretation in words: return per unit of total risk, or per unit of active risk.

Quickest way: Ratio-first shortcut

When to use it: Use when the question gives returns and risk figures and asks which manager or position is better, or whether the allocation is optimal.

Subtract Rf or the benchmark return from each return first.
Divide by the matching risk figure and compare the four options quickly.
For optimality, divide each excess return by its marginal VaR or beta; unequal values mean not optimal.
Add risk to the highest ratio, cut from the lowest.
Check units and periods (annual vs monthly) before choosing.

Common mistakes in Optimal Portfolio Allocation and Risk-Adjusted Performance

Using portfolio volatility instead of tracking error in the information ratio.
Both are called risk and Sharpe is more familiar.
Fix: For IR, the denominator is the standard deviation of the difference between portfolio and benchmark returns.
Using the benchmark return as Rf in the Sharpe ratio, or Rf as the benchmark in the IR.
Both subtract something from portfolio return.
Fix: Sharpe subtracts the risk-free rate. IR subtracts the benchmark return.
Comparing excess return to total VaR instead of marginal VaR for the optimal allocation.
Students forget that a position's contribution depends on correlation with the portfolio.
Fix: Use marginal VaR (or beta to the portfolio) in the denominator.
Mixing monthly returns with annual tracking error.
Data are given in different frequencies.
Fix: Convert both to the same period. Either annualise the monthly figures (monthly mean return × 12 and monthly volatility × √12) or de-annualise both. Multiplying a monthly mean return by 12 is an approximation that holds for arithmetic mean returns.
Assuming a higher Sharpe ratio always means a better manager for a benchmarked mandate.
Sharpe is treated as universal.
Fix: For active mandates against a benchmark, the information ratio shows skill relative to the benchmark; Sharpe measures total risk against cash.

Worked examples

Example 1

A fund returned 9% in a year with volatility of 12%. The risk-free rate was 3%. The benchmark returned 7%, and the tracking error was 4%. Calculate the Sharpe ratio and the information ratio.

Show the solution

Sharpe numerator: 9% − 3% = 6%.
Sharpe = 6% ÷ 12% = 0.50.
IR numerator: 9% − 7% = 2%.
IR = 2% ÷ 4% = 0.50.

Answer: Sharpe ratio = 0.50 and information ratio = 0.50. The Sharpe measures return per unit of total risk over cash, while the IR measures active return per unit of tracking error.

Example 2

A portfolio has two positions. Position A has expected return 8% and marginal VaR 0.020. Position B has expected return 6% and marginal VaR 0.010. The risk-free rate is 2%. Is the allocation optimal? If not, which way should risk be shifted?

Show the solution

Excess return of A = 8% − 2% = 6%.
Ratio for A = 0.06 ÷ 0.020 = 3.0.
Excess return of B = 6% − 2% = 4%.
Ratio for B = 0.04 ÷ 0.010 = 4.0.
The ratios are unequal, so the allocation is not optimal.
B gives more excess return per unit of marginal VaR.

Answer: Not optimal. Shift risk budget from A to B. As risk moves from A to B, B's marginal VaR rises and its ratio falls, while A's marginal VaR falls and its ratio rises. The shift continues until the two ratios are equal.

Exam tips

Read the denominator carefully: total volatility, tracking error or marginal VaR. Most wrong options use the wrong one.
Check the time unit. Annualise consistently before comparing.
For optimality questions, you only need to compare ratios; no full optimisation is required.
Remember an IR depends on the chosen benchmark, so a different benchmark can change the ranking.
Be ready to interpret: a higher ratio means more return per unit of risk, not a guarantee of future outperformance.

Practice questions from VaR and Risk Budgeting in Investment Management

Optimal Portfolio Allocation and Risk-Adjusted Performance in other exams

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

Optimal Portfolio Allocation and Risk-Adjusted Performance: frequently asked questions

What is the difference between the information ratio and the Sharpe ratio?

The Sharpe ratio divides excess return over the risk-free rate by total volatility. The information ratio divides active return over a benchmark by tracking error. Use Sharpe for total-portfolio risk against cash and IR to judge an active manager against a benchmark.

How do I compute the information ratio for an active manager?

Subtract the benchmark return from the portfolio return to get active return. Divide the mean active return by the standard deviation of active returns, which is the tracking error. Make sure both use the same period.

What is the optimal risk budget condition?

The ratio of excess return to marginal VaR should be the same for every position. If one position has a higher ratio, adding risk there raises return more per unit of risk, so the portfolio can improve.

Can the information ratio be negative?

Yes. If the portfolio underperforms the benchmark on average, active return is negative and so is the IR. A negative value means the active risk taken has been rewarded with underperformance.