Performance Measurement and Evaluation of Portfolio Managers: formula sheet

Key formulas

Absolute return

Absolute return (%) = (Ending value − Beginning value) ÷ Beginning value × 100

No time adjustment. Add income received if it is not already in the ending value.

CAGR

CAGR = (Ending value ÷ Beginning value)^(1/n) − 1

n is the number of years. Valid only when there are no interim cash flows.

Sub-period return

R = (Ending value of the sub-period before any flow − Beginning value of the sub-period) ÷ Beginning value of the sub-period, or R = Ending value before any flow ÷ Beginning value − 1

A sub-period ends at a cash flow or at the period end. Value the portfolio just before each cash flow, or at the period end for the last sub-period. Beginning value of the next sub-period includes the flow.

TWRR (linked return)

TWRR = [(1 + R1) × (1 + R2) × ... × (1 + Rn)] − 1

Annualise only if the total period is more than one year, using the CAGR step on the linked result.

XIRR

Σ [Cash flow(i) ÷ (1 + XIRR)^((date(i) − date(0)) ÷ 365)] = 0

Investments are negative, withdrawals and the final value are positive. Solved by spreadsheet or calculator, not by hand.

Standard deviation (σ)

σ = √[ Σ(R − average R)² ÷ (n − 1) ]

Uses sample data, so divide by n − 1. Higher σ means higher total risk. Annualising monthly σ: multiply by √12.

Beta (β)

β = Covariance(Rp, Rm) ÷ Variance(Rm)

Market beta is 1. β > 1 is aggressive, β < 1 is defensive. It measures only systematic risk.

Beta from correlation

β = ρ × (σp ÷ σm)

ρ is the correlation between portfolio and market returns.

Tracking error

TE = standard deviation of (Rp − Rb)

Rb is the benchmark return. It is not the simple difference of the two returns.

Active return

Active return = Rp − Rb

This is the series whose standard deviation gives tracking error.

Maximum drawdown

MDD = (Trough value − Peak value) ÷ Peak value

Shown as a negative percentage or as a fall in percent. Peak must come before the trough.

Sharpe ratio

Sharpe = (Rp − Rf) ÷ σp

σp is the standard deviation of portfolio returns. Uses total risk.

Treynor ratio

Treynor = (Rp − Rf) ÷ βp

Uses beta, so only market risk. Assumes a diversified portfolio.

CAPM expected return

E(Rp) = Rf + βp × (Rm − Rf)

Rm is the market return. Used inside Jensen's alpha.

Jensen's alpha

α = Rp − [Rf + βp × (Rm − Rf)]

Positive means outperformance after adjusting for beta. Zero means fair return for the risk.

Sortino ratio

Sortino = (Rp − Target) ÷ Downside deviation

The target is often the risk-free rate or a minimum acceptable return. Only returns below target enter the denominator.

Information ratio

IR = (Rp − Rb) ÷ Tracking error

Rb is the benchmark return. Tracking error is the standard deviation of (Rp − Rb).

Excess return (alpha over benchmark)

Excess return = Portfolio return − Benchmark return

Positive means the manager outperformed. Both returns must cover the same period and be calculated the same way.

Characteristics of a good benchmark

Appropriate • Investable • Transparent • Measurable • Specified in advance • Unambiguous

Remember these as a checklist. Exam options often swap in wrong traits such as 'highest return' or 'most popular'.

Fair comparison rule

Portfolio return (with dividends) ↔ Total return index (TRI)

A price index leaves out dividends, so it understates the benchmark's return.

Sub-period return

r = (Value at end of sub-period, before any new flow) ÷ (Value at start of sub-period) − 1

A new sub-period begins every time there is a deposit or withdrawal. The start value includes any flow made at that point.

TWRR (cumulative)

TWRR = [(1 + r₁) × (1 + r₂) × … × (1 + rₙ)] − 1

Sub-period returns are linked by multiplication, not added. Cash flows do not change the result.

Annualising a multi-year return

Annualised return = (1 + cumulative return)^(1 ÷ years) − 1

Used for periods longer than one year. For periods under a year, the return is generally shown as an absolute (non-annualised) figure. Confirm the exact convention in the workbook.

Core reporting rules

Method = TWRR; Data repository = APMI; Comparison = against benchmark

Returns must follow the prescribed method and period. Past performance does not guarantee future returns.

Active return

Active return = Portfolio return − Benchmark return

The total value added or lost versus the benchmark. Attribution effects add up to this.

Allocation effect (per segment)

(Wp − Wb) × (Rb_segment − Rb_total)

Wp and Wb are portfolio and benchmark weights. Rb_segment is benchmark return of the segment. Some versions use (Wp − Wb) × Rb_segment; both are used, so follow the data given.

Selection effect (per segment)

Wb × (Rp_segment − Rb_segment)

Uses the benchmark weight. Rp_segment is the portfolio's return in that segment.

Interaction effect (per segment)

(Wp − Wb) × (Rp_segment − Rb_segment)

Combined effect of different weight and different selection.

Total attribution

Allocation + Selection + Interaction = Active return

Use this to check your arithmetic.

Quick revision

• Absolute return = (Ending value − Beginning value) ÷ Beginning value; it ignores time.
• CAGR = (Ending value ÷ Beginning value)^(1 ÷ n) − 1, where n is years; it suits a single investment with no interim flows.
• XIRR handles irregular cash flows on specific dates; it reflects the investor's own timing of money.
• TWRR removes the effect of cash flows, so it measures the manager's skill rather than the client's timing.
• Standard deviation measures total risk, meaning the volatility of returns.
• Beta measures sensitivity to the market; a beta above 1 means the portfolio moves more than the market.
• Tracking error is the standard deviation of the difference between portfolio and benchmark returns.
• Sharpe ratio = (Portfolio return − Risk-free return) ÷ Standard deviation; it uses total risk.
• Treynor ratio = (Portfolio return − Risk-free return) ÷ Beta; it uses market risk only.
• Alpha is the excess return over what beta and the market explain; Sortino counts only downside deviation.
• Choose a benchmark that matches the strategy and compare managers with similar mandates.
• Past performance does not guarantee future returns; never promise returns to clients.

Common mistakes

• Using CAGR for a portfolio with additions or withdrawals Fix: Check for interim flows first. If they exist, use XIRR for the client's return and TWRR for the manager.
• Including the cash flow in the sub-period gain Fix: Take the value just before the flow as the end of that sub-period. The deposit starts the next one.
• Saying beta measures total risk. Fix: Beta is only market (systematic) risk. Standard deviation is total risk.
• Calling tracking error the difference between portfolio and benchmark return. Fix: That gap is active return. Tracking error is the standard deviation of that active return.
• Using total return instead of excess return in Sharpe or Treynor. Fix: Always subtract Rf in the numerator. Only the information ratio uses the benchmark instead.
• Mixing up standard deviation and beta in Sharpe and Treynor. Fix: Remember: Sharpe = Standard deviation (both start with S). Treynor = beta (T for 'the market's beta').
• Comparing a mid-cap or small-cap PMS with the Nifty 50. Fix: Match the index to the strategy. Use a mid-cap or small-cap index for those strategies, or a broad index for multi-cap.
• Using a price index when a TRI is available. Fix: Remember that TRI includes reinvested dividends. It is the like-for-like comparison.
• Adding sub-period returns instead of multiplying them. Fix: Always link with (1 + r) factors. 10% and 20% give 1.10 × 1.20 − 1 = 32%, not 30%.
• Using the closing value minus total money invested as the return when cash flows occurred. Fix: Remember that TWRR removes the effect of client flows. Split the period at each flow and link the sub-period returns.

Exam tips

• Look for the keywords: 'manager skill' or 'remove effect of cash flows' points to TWRR; 'client's experience' or 'dated cash flows' points to XIRR.
• Do not annualise returns for periods of one year or less unless the question says so.
• Expect conceptual MCQs: which measure is affected by cash flow timing, and which is preferred to compare managers.
• NISM-Series-XXI-A has negative marking of 10% of the marks for each question, so avoid blind guesses.
• Compute the growth factors first. Many options differ only by adding rather than compounding returns.
• Expect definition questions that ask which measure fits a situation. Learn the keyword matches.
• Watch for options that swap beta and standard deviation, or tracking error and active return.
• Remember the benchmark beta is 1, and use that to judge aggressive or defensive portfolios.