Information Ratio

An active manager is paid to deviate from the benchmark. Every deviation is a risk: it might beat the index or trail it. The Information Ratio asks whether those deviations paid off consistently.

• The numerator is active return — portfolio return minus benchmark return.
• The denominator is active risk — the standard deviation of those differences, which the workbook notes is simply the portfolio's tracking error.

A manager who beats the Nifty by 2% every single year has a very high IR. A manager who beats it by 10% one year and trails by 6% the next has the same average edge but a much lower IR. The workbook frames this as telling skill from chance.

The XXI-B workbook presents the Information Ratio twice, from two directions.

Ex post — Chapter 20, section 20.4.5. Compute it from realised returns. The workbook's table:

IR = 5 ÷ 2.92 = 1.715.

Ex ante — Chapter 18, section 18.3. Grinold and Kahn's Fundamental Law of Active Management predicts it from skill and opportunity: IR = IC × √Breadth. Managers with higher information ratios "deliver more active return for a given level of active risk", which makes IR the natural yardstick for comparing active managers.

Ex post:  IR = (Rp − Rb) ÷ Stdev(Rp − Rb)
            = Active return ÷ Tracking error

Ex ante:  IR = IC × √Breadth     (Fundamental Law of Active Management)

Illustrative figures. A large-cap PMS reports five years of returns against the Nifty 50 TRI.

Mean active return = (3 − 1 + 2 + 2 + 2) ÷ 5 = 1.6%
Deviations from 1.6: 1.4, −2.6, 0.4, 0.4, 0.4
Sum of squares = 1.96 + 6.76 + 0.16 + 0.16 + 0.16 = 9.20
Sample variance = 9.20 ÷ 4 = 2.30
Tracking error = √2.30 = 1.52%
IR = 1.6 ÷ 1.52 = 1.05

(Sample standard deviation, dividing by n − 1, reproduces the workbook's own 2.92.)

On a ₹1 crore account, a 1.6% average edge is about ₹1.6 lakh a year, delivered with a year-to-year wobble of about ₹1.5 lakh. An IR near 1 says the edge was steady relative to its noise — which is the question a client choosing between two PMS managers with similar headline returns should be asking.

Chapter 20 (Performance Measurement and Evaluation of Portfolio Managers), section 20.4.5, computes IR from the return table above and links it to tracking error; a sample question describes it as "average differential return over the benchmark per unit of variability of differential returns". Chapter 18 (Equity Portfolio Management Strategies), section 18.3, derives it from IC and breadth, and its caselets compute IR = 0.575 × √80 = 5.14.

• Benchmark, not risk-free rate. Sharpe subtracts Rf; IR subtracts the benchmark return.
• The denominator is tracking error (standard deviation of return differences), not the portfolio's standard deviation.
• "Appraisal ratio" is the workbook's alternative name for the same measure.
• The workbook's illustration of the Fundamental Law uses IC = 4, giving IR = 4 × √50 = 28.28 and 4 × √100 = 40. The √2 point is valid, but an IC of 4 is outside the −1 to +1 range the same chapter gives for IC. Real information ratios are far smaller.
• Doubling breadth raises IR by only √2 (1.414×); doubling IC doubles IR. IR is linear in IC and non-linear in breadth.
• The workbook calls the denominator a measure of residual (unsystematic) risk. That is the workbook's description; for calculation, it is the standard deviation of active returns.