NISM Certifications · NISM-Series-X-A: Investment Adviser (Level 1)

Portfolio Performance Measurement and Evaluation: formula sheet

Full chapter guide

Key formulas

Holding period return (HPR): HPR = (Ending value − Beginning value + Income) ÷ Beginning value; Not annualised. Always include dividends or interest received.
Multi-period return: Cumulative return = (1 + R1) × (1 + R2) × ... × (1 + Rn) − 1; Compound the returns. Never add them.
Arithmetic mean: Arithmetic mean = (R1 + R2 + ... + Rn) ÷ n; Always greater than or equal to the geometric mean. Equal only when all returns are identical.
Geometric mean: Geometric mean = [(1 + R1) × (1 + R2) × ... × (1 + Rn)]^(1/n) − 1; Use it for the average compounded return over several periods.
CAGR: CAGR = (Ending value ÷ Beginning value)^(1/n) − 1; n is the number of years. Valid only with a single investment and no interim flows.
Annualising an HPR: Annualised return = (1 + HPR)^(365 ÷ days held) − 1; Use it to compare holding periods of different lengths.
Sub-period return for TWR: Sub-period return = Value just before the next cash flow ÷ Value just after the previous cash flow − 1; Split the period at each deposit or withdrawal. Then compound all sub-period returns.
XIRR (money-weighted return): Σ [Cash flow ÷ (1 + r)^(days from first date ÷ 365)] = 0; Solve for r. Investments are negative flows, withdrawals and final value are positive. Use a calculator or spreadsheet.
Standard deviation (sample): σ = √[ Σ(Rᵢ − R̄)² ÷ (n − 1) ]; Variance is σ². Use n instead of n − 1 if the question treats the data as the full population.
Beta: β = Cov(Rp, Rm) ÷ σm²; Also β = ρ × (σp ÷ σm), where ρ is the correlation with the market. Market beta is 1.
Tracking error: TE = standard deviation of (Rp − Rb); Active return = portfolio return − benchmark return. TE is not the average of the differences.
Downside deviation: DD = √[ Σ(min(Rᵢ − MAR, 0))² ÷ n ]; MAR is the minimum acceptable return. Returns above MAR count as zero but still stay in n.
Parametric VaR: VaR = z × σ × portfolio value; z is about 1.645 at 95% and 2.33 at 99% (one-tailed). Multi-day VaR = one-day VaR × √days.
Portfolio beta: βp = Σ(wᵢ × βᵢ); Weights are proportions of portfolio value.
Sharpe ratio: (Rp − Rf) ÷ σp; Rp = portfolio return, Rf = risk-free rate, σp = standard deviation of portfolio returns. Total risk.
Treynor ratio: (Rp − Rf) ÷ βp; βp = portfolio beta. Systematic risk only. Output is excess return per unit of beta.
Jensen's alpha: α = Rp − [Rf + βp × (Rm − Rf)]; Rm = market return. Result is in percentage points of return, not a ratio.
Sortino ratio: (Rp − target return) ÷ downside deviation; Target is often Rf or the minimum acceptable return. Only returns below the target enter the risk measure.
Information ratio: (Rp − Rb) ÷ tracking error; Rb = benchmark return. Tracking error = standard deviation of (Rp − Rb).
Interpretation rule: Higher ratio = better; α > 0 = outperformance; Compare only on the same period, same Rf and same benchmark.
Active return: Active return = Portfolio return − Benchmark return; Positive means outperformance. Compare like with like: TRI for TRI-based returns, same period, same basis.
Composite benchmark return: Σ (weight of asset class × return of its index); Weights must match the policy asset allocation and add up to 100%.
TRI vs price index: TRI return ≈ Price index return + reinvested income (dividend or interest); TRI return is higher than price return when the index constituents pay income.
Good benchmark checklist: Appropriate, unambiguous, investable, measurable, specified in advance, reflective of current opinions, accountable; Know these as a list; options often swap in a wrong quality.
Active return: Active return = Rp − Rb; Rp is total portfolio return and Rb is total benchmark return. This is the amount to be explained.
Asset allocation effect (sector i): (wp,i − wb,i) × (Rb,i − Rb); Weight difference times the sector's benchmark return relative to the total benchmark return. Using Rb,i alone gives the same total across sectors, but the relative form shows each sector's true contribution.
Security selection effect (sector i): wb,i × (Rp,i − Rb,i); Uses the benchmark weight and the return difference within the sector.
Interaction effect (sector i): (wp,i − wb,i) × (Rp,i − Rb,i); Weight difference times return difference. Often shown as a separate item.
Total check: Allocation + Selection + Interaction = Rp − Rb; Sum over all sectors. Use this to verify your answer.
Current weight of an asset class: Weight = Value of asset class ÷ Total portfolio value × 100; Use current market values, not the amount originally invested.
Drift: Drift = Current weight − Target weight; A positive drift means overweight, so sell. A negative drift means underweight, so buy.
Threshold band: Band = Target weight ± tolerance; Rebalance only when the current weight falls outside the band. At exactly the limit, follow the policy wording.
Amount to trade to reach target: Trade = (Target weight × Total portfolio value) − Current value of asset class; A positive result is a purchase. A negative result is a sale. Trades across classes net to zero if no money is added or withdrawn.
Return versus goal check: Required return vs. achieved return since start; If achieved return is below the return needed for the goal, discuss higher savings, a longer horizon or a changed goal. Do not simply raise risk.
Fair presentation test: Accurate + Complete + Not misleading + Disclosed basis; Use this to judge any reporting question. Failing one element makes the presentation unfair.
Time-weighted return (basis used for comparison): TWR = [(1 + R₁) × (1 + R₂) × … × (1 + Rₙ)] − 1; Removes the effect of client cash flows, which the manager does not control. GIPS generally requires it, but money-weighted returns are required for certain strategies such as private equity and closed-end funds. It is the usual basis for comparing managers.
Composite: Composite = group of portfolios managed under a similar mandate or strategy; GIPS: include all actual, discretionary portfolios managed under the same strategy. Fee-paying ones must be included; non-fee-paying ones may be included with disclosure. Do not drop poor performers.
Net return: Net return ≈ Gross return − Fees and expenses; Clients care about net-of-fee returns. State clearly which one is shown.
GIPS compliance claim: Claim only if ALL requirements are met; Partial compliance cannot be claimed. GIPS are voluntary and apply to firms.

Quick revision

HPR = (Ending value − Beginning value + Income) ÷ Beginning value.
CAGR = (Ending value ÷ Beginning value)^(1 ÷ n) − 1.
XIRR is used when cash flows happen on irregular dates.
Time-weighted return removes the effect of the timing and size of client cash flows, so it suits judging a manager.
Standard deviation measures total risk; beta measures market (systematic) risk.
Tracking error is the standard deviation of the difference between portfolio and benchmark returns.
Sharpe = (Rp − Rf) ÷ σp; Treynor = (Rp − Rf) ÷ β.
Sortino uses downside deviation instead of total standard deviation.
Positive alpha means return above what the portfolio's risk would justify.
A benchmark must match the portfolio's asset class, style and investable universe.
Attribution explains the active return, commonly through asset allocation and security selection effects.
Rebalancing returns a portfolio to its target allocation after drift.

Common mistakes

Using the arithmetic average to describe multi-year growth. Fix: Use the geometric mean or CAGR. Remember +50% then -50% gives a 25% loss, not zero.
Adding yearly returns to get the total return. Fix: Multiply the growth factors (1 + R) and subtract 1. Returns of 10% and 10% give 21%, not 20%.
Treating standard deviation and beta as the same risk Fix: Standard deviation is total risk. Beta is only market-related risk measured against a benchmark.
Calculating tracking error as the average of return differences Fix: Tracking error is the standard deviation of the active returns, not their mean.
Using beta in the Sharpe ratio or standard deviation in the Treynor ratio. Fix: Remember S for Sharpe and Standard deviation; T for Treynor and bTa (beta).
Calling Jensen's alpha a ratio or forgetting to compute the CAPM expected return first. Fix: Always compute Rf + β(Rm − Rf) first. Alpha is then Rp minus that figure.
Comparing a fund's return with a price index instead of the TRI. Fix: The fund earns dividends, so the fair comparison is the TRI, which includes reinvested income.
Choosing a broad market index for a sector or mid-cap fund. Fix: Choose the index that mirrors the fund's mandate and segment.
Using the portfolio weight instead of the benchmark weight in the selection effect. Fix: In the standard Brinson split, selection uses the benchmark weight. The extra part from the weight gap goes into interaction.
Measuring the allocation effect against the sector's return only, not relative to the total benchmark return. Fix: Use (Rb,i − Rb). A sector with a positive return can still hurt if it lags the total benchmark and you overweighted it.

Exam tips

Expect conceptual questions on which measure suits which purpose. Learn the pairing: manager evaluation with TWR, investor outcome with money-weighted return or XIRR.
Negative marking applies in this paper, and caselet questions can carry more marks. If two options look close, recheck whether the question wants arithmetic or geometric, total or annual.
Know that CAGR is a smoothed rate. It hides volatility and says nothing about the path of returns between the start and the end.
In TWR calculations, the sub-period must end just before each cash flow. A common trap option uses the end value without adding the deposit.
Remember the inequality: geometric mean is never above arithmetic mean. A question that offers both can often be settled by this rule.
Expect conceptual questions that ask you to pick between total and market risk.
Remember that beta ignores unsystematic risk, so it can mislead for undiversified portfolios.
For a high tracking error, think active management. For a low one, think index fund.