# Performance Evaluation Concept Card: [Your elective, fund or manager search] > A concept card for judging investment managers in an MBA investments elective: return measurement (time-weighted > and money-weighted), risk-adjusted measures (Sharpe, Treynor, Jensen's alpha, M-squared, information ratio and > appraisal ratio), factor-model alpha, market-timing tests, statistical significance and the role of the GIPS > standards. It builds on beta and the security market line in mba/mba-capm-wacc-card and on the efficient frontier in > mba/mba-portfolio-theory-card; it does not re-teach them. Attribution by allocation and selection is covered in depth > by finance/performance-attribution-report, which this card links to. It is for second-year students in an > investments or asset management elective, candidates for asset management, wealth and endowment roles, and committee > members who hire and fire managers. Worked example A ranks three managers for the Tallow Creek Medical Center > foundation, Worked example B (half solved) contrasts time-weighted and money-weighted returns and tests an alpha, and > Exercise C (unsolved) asks how long it takes to tell skill from luck. Source basis: Treynor (1965), Sharpe (1966), > Treynor and Mazuy (1966), Jensen (1968), Treynor and Black (1973), Henriksson and Merton (1981), Brinson, Hood and > Beebower (1986), Grinold (1989), Sortino and Price (1994), Carhart (1997), Modigliani and Modigliani (1997) and Fama > and French (2010), written up in original words; GIPS facts are dated (October 2026). Every return and risk figure is > an invented *Example* value. Replace every [bracketed] value and every row marked *Example*. This is a learning aid, > not investment advice. ## Document control | Field | Value | |---|---| | Elective, fund or manager search | [Investments elective, session 9 / manager search for a foundation] | | Learner or analyst | [Your name] | | Evaluation period | YYYY-MM-DD to YYYY-MM-DD | | Return frequency and number of observations | [Monthly, 60 observations] | | Benchmark for each mandate | [Named index, total return, currency] | | Risk-free rate series | [Named Treasury bill or overnight rate series, source] | | Fee basis | [Gross of fees / net of fees] | | GIPS status of each manager's figures | [Claims compliance, verified by whom, or not compliant] | | Reviewed by | [Study partner, tutor or investment committee member] | | Version | [0.1] | | Last updated | YYYY-MM-DD | ## Contents - [When to use it](#when-to-use-it) - [The idea in plain language](#the-idea-in-plain-language) - [Formulas and notation](#formulas-and-notation) - [Assumptions and limits](#assumptions-and-limits) - [GIPS standards (dated facts)](#gips-standards-dated-facts) - [Worked example A (fully solved): ranking three managers](#worked-example-a-fully-solved-ranking-three-managers) - [Worked example B (half solved): which return, and is the alpha real?](#worked-example-b-half-solved-which-return-and-is-the-alpha-real) - [Exercise C (unsolved): skill or luck](#exercise-c-unsolved-skill-or-luck) - [Process diagram: evaluating a manager](#process-diagram-evaluating-a-manager) - [How managers and committees use performance evaluation](#how-managers-and-committees-use-performance-evaluation) - [Connections to other cards](#connections-to-other-cards) - [Common mistakes and misconceptions](#common-mistakes-and-misconceptions) - [Self-test questions](#self-test-questions) - [My notes and next steps](#my-notes-and-next-steps) - [Sources and further reading](#sources-and-further-reading) - [Answer key](#answer-key) - [Revision history](#revision-history) ## When to use it **One-sentence rule:** measure returns time-weighted when the manager does not control cash flows, adjust them for the risk that matters to the owner (total risk if the portfolio is the owner's whole wealth, beta if it is one of many, tracking error if it is a benchmark mandate), and test whether the result is statistically distinguishable from luck before acting on it. | Use when | Use something else when | |---|---| | Comparing public-market managers or funds over a common period | Evaluating private equity or venture funds with committed capital (use IRR, multiples and public market equivalents, mba/mba-pe-fund-economics-card) | | Reporting to an investment committee or in a fund factsheet (finance/fund-factsheet) | Explaining where excess return came from by sector or decision (use attribution, finance/performance-attribution-report) | | Hiring, retaining or sizing an active manager | Measuring downside risk at a confidence level (use mba/mba-value-at-risk-card) | | Testing whether a strategy's alpha survives standard risk factors | Building the factor model itself (use mba/mba-factor-models-card) | ## The idea in plain language **Raw return is not enough.** A manager who levered up in a rising market looks brilliant; one who held cash looks dull. To compare them you must ask how much risk each took to earn its return, and which risk the owner cares about. **Sharpe ratio (total risk).** Sharpe (1966) proposed the excess return over the risk-free rate per unit of standard deviation, the "reward-to-variability" ratio. It is the slope of the line from the risk-free asset to the portfolio in mean-standard deviation space. If the fund is the owner's entire risky portfolio, total volatility is the right risk, and the highest Sharpe ratio wins, because the owner can lever it up or down with borrowing and lending. **M-squared (same ranking, in return units).** Modigliani and Modigliani (1997) turned the Sharpe ratio into a percentage: lever or de-lever the fund until its volatility equals the benchmark's, then read off the return. The difference from the benchmark return is easy for a committee to understand. **Treynor ratio and Jensen's alpha (systematic risk).** If the fund is one of many in a diversified portfolio, company-specific risk diversifies away, and only beta matters (mba/mba-capm-wacc-card). Treynor (1965) divided excess return by beta. Jensen (1968) measured the vertical distance from the security market line: the return the fund earned above what the CAPM requires for its beta. Jensen's study found that the average mutual fund did not beat the market after costs, a result that set the agenda for decades. **Appraisal ratio and information ratio (active risk).** Treynor and Black (1973) showed that an active manager's value in a portfolio depends on alpha per unit of residual (non-market) risk: the appraisal ratio. In practice, managers with a benchmark mandate are judged by the information ratio: excess return over the benchmark divided by tracking error, the volatility of that excess return. Grinold (1989) linked the information ratio to skill and breadth in the "fundamental law of active management": a small edge applied to many independent bets can produce a high ratio. **Factor alpha.** The CAPM alpha may simply reward exposure to known return factors such as size, value and momentum. Carhart (1997) added momentum to the Fama and French (1993) three-factor model and found that persistence in mutual fund performance was largely explained by momentum exposure and expenses. Fama and French (2015) added profitability and investment factors. The factor model chosen changes the alpha (mba/mba-factor-models-card). **Timing and selection.** Treynor and Mazuy (1966) tested market timing by adding a squared market-return term to the regression: a timer's beta is higher in up markets. Henriksson and Merton (1981) used an up-market dummy instead. Brinson, Hood and Beebower (1986) split a pension plan's return into policy (asset allocation), timing and selection and found that the policy mix explained most of the variation of returns over time. **Luck.** Returns are noisy. With an information ratio of 0.5, it takes about 16 years for the mean active return to reach a t-statistic of 2 ($t \approx IR \sqrt{T}$; the same logic with the appraisal ratio applies to alpha). Fama and French (2010) compared actual fund alphas with bootstrapped luck distributions and found little evidence of skill net of costs for the average fund. Treat short records with humility. ## Formulas and notation $$ S_p = \frac{\bar R_p - \bar r_f}{\sigma_p} \qquad T_p = \frac{\bar R_p - \bar r_f}{\beta_p} \qquad \alpha_p = \bar R_p - \left[ \bar r_f + \beta_p (\bar R_m - \bar r_f) \right] $$ $$ M^2_p = \bar r_f + S_p\, \sigma_m \qquad IR_p = \frac{\bar R_p - \bar R_b}{TE_p}, \quad TE_p = \sigma(R_p - R_b) \qquad AR_p = \frac{\alpha_p}{\sigma_{\varepsilon p}} $$ $$ \text{Single-index decomposition: } \sigma_p^2 = \beta_p^2 \sigma_m^2 + \sigma_{\varepsilon p}^2 \qquad TE_p^2 = (\beta_p - 1)^2 \sigma_m^2 + \sigma_{\varepsilon p}^2 \ \ (\text{benchmark} = \text{market}) $$ $$ \text{TWR} = \prod_{j=1}^{J} (1 + r_j) - 1, \quad r_j = \frac{V_j^{end}}{V_{j-1}^{end} + CF_{j-1}} - 1 \qquad \text{MWR: } \sum_k \frac{CF_k}{(1 + IRR)^{t_k}} = 0 $$ $$ \text{Treynor-Mazuy: } R_p - r_f = a + b (R_m - r_f) + c (R_m - r_f)^2 + e \qquad t_{\alpha} \approx AR \sqrt{T_{years}}, \ \ t_{active} \approx IR \sqrt{T_{years}} \qquad IR \approx IC \sqrt{BR} $$ | Symbol | Meaning | Unit | |---|---|---| | $\bar R_p$, $\bar R_m$, $\bar R_b$ | Average return of portfolio, market, benchmark over the period | Rate per year | | $\bar r_f$ | Average risk-free rate | Rate per year | | $\sigma_p$, $\sigma_m$ | Standard deviation of portfolio and market returns | Rate per year | | $\beta_p$ | Portfolio beta against the market | Number | | $\sigma_{\varepsilon p}$ | Residual (non-market) standard deviation | Rate per year | | $TE_p$ | Tracking error: standard deviation of the return difference from the benchmark | Rate per year | | $S_p$, $T_p$, $\alpha_p$, $M^2_p$ | Sharpe ratio, Treynor ratio, Jensen's alpha, M-squared | Number, rate, rate, rate | | $IR_p$, $AR_p$ | Information ratio, appraisal ratio | Number per year | | $r_j$, $V_j$, $CF_j$ | Sub-period return, value at end of sub-period, external cash flow | Rate, currency | | $t_{\alpha}$, $t_{active}$, $T_{years}$ | Approximate t-statistic of alpha and of the mean active return, length of record | Number, number, years | | $IC$, $BR$ | Information coefficient (correlation of forecasts with outcomes), breadth (independent bets per year) | Number | Annualising: multiply a monthly mean by 12 and a monthly standard deviation by $\sqrt{12}$, so a monthly Sharpe or information ratio is multiplied by $\sqrt{12}$. This assumes independent returns; serial correlation (common in smoothed or illiquid assets) makes the annualised ratio too high. ## Assumptions and limits | Assumption | Why it matters | When it fails and what to do | |---|---|---| | The benchmark is the right one | Alpha and the information ratio are defined against it | A small-cap value manager measured against a large-cap index looks skilled when markets favour small value; use a style-matched benchmark or a factor model | | Returns are roughly normal | Standard deviation captures risk | Option-like strategies with skewed payoffs look good on Sharpe until the crash; add downside measures such as Sortino and Price (1994) and maximum drawdown | | Beta is stable | Treynor and Jensen use a single beta | Market timers change beta; test with Treynor-Mazuy or Henriksson-Merton | | Prices are marked to market | Volatility reflects true risk | Appraisal-based or illiquid assets understate volatility and beta; unsmooth or use longer intervals | | No survivorship | The universe includes funds that closed | Databases that drop dead funds overstate average performance; ask for survivorship-free data | | Fees treated consistently | Gross and net figures are not comparable | State the fee basis for every figure; the owner earns net returns | ## GIPS standards (dated facts) As of October 2026, checked on the CFA Institute GIPS pages and in the text of the *2020 GIPS Standards for Firms* (errata version July 2020); check the current edition and guidance statements at [gipsstandards.org](https://www.gipsstandards.org/standards/gips-standards-for-firms/) before you rely on any provision: - The Global Investment Performance Standards (GIPS) are developed and maintained by CFA Institute and are voluntary. Separate provisions exist for firms, for asset owners and for verifiers. - The effective date of the 2020 edition is 1 January 2020; GIPS reports that include performance for periods ending on or after 31 December 2020 must follow the 2020 edition. - Compliance applies firm-wide; it cannot be claimed for a single composite, fund or portfolio. - A firm initially presents at least five years of GIPS-compliant annual performance (or since inception if shorter) and adds a year each year up to at least ten. - Firms must present time-weighted returns, unless the firm controls external cash flows and the composite or pooled fund is closed-end, fixed-life, fixed-commitment or holds significant illiquid investments; then money-weighted returns may be presented. - Verification by an independent third party is optional but widely expected by institutional buyers. - The GIPS Work in Process page listed exposure drafts (a guidance statement on trade errors, with comments open to 15 December 2026, and a best-practice guide for return attribution reporting), not a new edition of the standards for firms. Firms must also comply with the guidance statements in force. **Why it matters for analysis.** Composites (all fee-paying discretionary portfolios managed to a strategy) prevent cherry-picking the best account. When a manager's presentation does not claim GIPS compliance, ask how the composite is defined, whether closed accounts are included and whether returns are gross or net. GIPS fixes how returns are presented; it does not say whether they were earned by skill. ## Worked example A (fully solved): ranking three managers **Setting** *(Example)*. The investment committee of the Tallow Creek Medical Center foundation reviews three US equity managers over five years (annualised figures). The benchmark is a broad market index, treated here as the market portfolio. Average risk-free rate 3.0%. | Portfolio | Return $\bar R$ | Std dev $\sigma$ | Beta $\beta$ | |---|---|---|---| | Benchmark *(Example)* | 9.0% | 16.0% | 1.00 | | Fund I *(Example)* | 10.5% | 18.0% | 1.10 | | Fund II *(Example)* | 9.8% | 13.0% | 0.75 | | Fund III *(Example)* | 11.8% | 24.0% | 1.35 | **Step 1: residual risk and tracking error.** From the single-index decomposition, Fund I's residual standard deviation is $\sqrt{18^2 - 1.10^2 \times 16^2} = \sqrt{14.24} = 3.7736\%$, and its tracking error is $\sqrt{(0.10)^2 \times 16^2 + 14.24} = 4.0988\%$. Fund II: residual 5.0000%, tracking error 6.4031%. Fund III: residual 10.4614%, tracking error 11.8659%. Excel: `=SQRT(18^2-1.1^2*16^2)`. With real data, residual risk is `=STEYX` of the regression of fund on market excess returns and tracking error is `=STDEV.S` of the monthly return differences times `SQRT(12)`. **Step 2: the measures.** | Measure | Benchmark | Fund I | Fund II | Fund III | Excel (Fund I) | |---|---|---|---|---|---| | Sharpe ratio | 0.3750 | 0.4167 | 0.5231 | 0.3667 | `=(10.5-3)/18` | | Treynor ratio (%) | 6.00 | 6.82 | 9.07 | 6.52 | `=(10.5-3)/1.1` | | CAPM required return | 9.0% | 9.60% | 7.50% | 11.10% | `=3+1.1*(9-3)` | | Jensen's alpha | 0 | +0.90% | +2.30% | +0.70% | `=10.5-(3+1.1*(9-3))` | | M-squared | 9.0% | 9.67% | 11.37% | 8.87% | `=3+(10.5-3)/18*16` | | M-squared minus benchmark | 0 | +0.67% | +2.37% | -0.13% | `=3+(10.5-3)/18*16-9` | | Information ratio | n/a | 0.3660 | 0.1249 | 0.2360 | `=(10.5-9)/4.0988` | | Appraisal ratio | n/a | 0.2385 | 0.4600 | 0.0669 | `=0.9/3.7736` | | t-statistic of alpha, 5 years | n/a | 0.53 | 1.03 | 0.15 | `=0.2385*SQRT(5)` | **Step 3: interpretation.** - **Fund III has the highest raw return (11.8%), beats the benchmark by 2.8 points and has a positive Jensen's alpha**, yet its Sharpe ratio (0.3667) is below the benchmark's (0.3750) and its M-squared is 0.13 points below the benchmark. Its large residual risk (10.46%) is rewarded with little alpha. If the foundation held Fund III alone, it would have done better holding a levered index position with the same volatility. - **Fund II leads on Sharpe, Treynor, alpha, M-squared and the appraisal ratio** but has the lowest information ratio (0.1249). The reason is its low beta: against a beta-1 benchmark, a 0.75-beta fund carries a large tracking error (the beta gap alone contributes a $0.25 \times 16\% = 4\%$ standard deviation, which is not stock-picking). The information ratio penalises a deliberate low-beta stance as if it were active risk. If the committee measures managers against the index, Fund I looks best; if it can adjust the beta at total-fund level, Fund II adds the most. - **No alpha is statistically significant.** The best t-statistic is about 1.03; at an appraisal ratio of 0.46, Fund II would need about $(2 / 0.46)^2 = 18.9$ years for a t-statistic of 2. The right measure follows the role: Sharpe or M-squared if the fund is the whole risky portfolio; Treynor, alpha and the appraisal ratio if it is one sleeve among many; information ratio for a benchmark-constrained mandate. **Chart: risk and return against the capital market line** (from the table above; the line runs from the risk-free rate through the benchmark). ```vega-lite { "$schema": "https://vega.github.io/schema/vega-lite/v5.json", "description": "Annualised return against standard deviation for three funds and the benchmark, with the capital market line (fictional Example)", "width": "container", "height": 260, "layer": [ { "data": {"values": [{"sd": 0, "ret": 3.0}, {"sd": 26, "ret": 12.75}]}, "mark": {"type": "line", "strokeDash": [4, 4]}, "encoding": { "x": {"field": "sd", "type": "quantitative", "title": "Standard deviation (% a year)"}, "y": {"field": "ret", "type": "quantitative", "title": "Average return (% a year)", "scale": {"zero": true}} } }, { "data": { "values": [ {"sd": 0, "ret": 3.0, "name": "Risk-free"}, {"sd": 16, "ret": 9.0, "name": "Benchmark"}, {"sd": 18, "ret": 10.5, "name": "Fund I"}, {"sd": 13, "ret": 9.8, "name": "Fund II"}, {"sd": 24, "ret": 11.8, "name": "Fund III"} ] }, "layer": [ { "mark": {"type": "point", "filled": true, "size": 80}, "encoding": { "x": {"field": "sd", "type": "quantitative"}, "y": {"field": "ret", "type": "quantitative"}, "color": {"field": "name", "type": "nominal", "title": null} } }, { "mark": {"type": "text", "dx": 8, "dy": -8, "align": "left"}, "encoding": { "x": {"field": "sd", "type": "quantitative"}, "y": {"field": "ret", "type": "quantitative"}, "text": {"field": "name", "type": "nominal"} } } ] } ] } ``` Funds above the dashed line (Fund I, Fund II) beat the benchmark's reward per unit of total risk; Fund III sits just below it. ## Worked example B (half solved): which return, and is the alpha real? **Part 1: time-weighted versus money-weighted** *(Example)*. The foundation's account with Fund II starts 2025-01-01 at USD 50.0m. On 2025-07-01 it is worth USD 54.0m, and the foundation adds USD 20.0m (so USD 74.0m is invested). On 2025-12-31 it is worth USD 70.3m. 1. First half return: $54.0 / 50.0 - 1 = 8.00\%$. 2. Second half return: $70.3 / 74.0 - 1 =$ [ ]. 3. Time-weighted return for the year: $(1.08)(1 + r_2) - 1 =$ [ ]. Excel: `=(54/50)*(70.3/74)-1`. 4. Money-weighted return: the half-year IRR of $\{-50, -20, +70.3\}$ is 0.2497% (`=IRR({-50,-20,70.3})`), so the annual rate is [ ] (`=(1+IRR({-50,-20,70.3}))^2-1`). With dates: `=XIRR({-50,-20,70.3},{DATE(2025,1,1),DATE(2025,7,1),DATE(2025,12,31)})` = 0.5010%. 5. Modified Dietz check: gain $70.3 - 50.0 - 20.0 = 0.30$; weighted capital $50.0 + 0.5027 \times 20.0$; return = [ ]. 6. Which number belongs in the manager's GIPS report and which tells the foundation what its money earned? [ ] **Part 2: is Fund II's alpha real?** A regression of 60 monthly excess returns of Fund II on market excess returns gives an intercept (monthly alpha) of 0.25% with a standard error of 0.15% *(Example)*. 7. t-statistic: $0.25 / 0.15 =$ [ ]. Two-sided p-value with 58 degrees of freedom: 0.1010 (`=T.DIST.2T(1.6667,58)`). 8. Critical value at 5%: [ ] (`=T.INV.2T(0.05,58)`). Is the alpha significant? [ ] 9. Annualised alpha: [ ]% a year. Name one factor exposure that could explain it if you ran a Carhart (1997) four-factor regression instead: [ ] ## Exercise C (unsolved): skill or luck **Setting** *(Example)*. The Corvane Robotics employee retirement plan committee is choosing between two active equity managers with 36 months of history against the same benchmark. | Manager | Mean monthly excess return over risk-free | Monthly std dev | Mean monthly return over benchmark | Monthly tracking error | |---|---|---|---|---| | Manager P *(Example)* | 0.70% | 4.2% | 0.20% | 1.1% | | Manager Q *(Example)* | 0.55% | 2.9% | 0.12% | 0.5% | 1. Compute each manager's monthly and annualised Sharpe ratio. 2. Compute each manager's monthly and annualised information ratio. 3. Compute the t-statistic of each manager's mean active return over 36 months ($IR_{monthly} \times \sqrt{36}$). 4. How many years of data would each need for a t-statistic of 2 at the same information ratio? 5. Manager P's marketing material shows only gross-of-fee returns for "representative accounts". What do you ask for, with reference to the GIPS standards? 6. Recommend which manager to shortlist and state one qualitative check you would add. ## Process diagram: evaluating a manager The order matters: fix the data and the benchmark first, then measure, then test significance, and only then decide. ```mermaid flowchart TD A["Define the mandate and the owner's role for it"] --> B["Agree the benchmark and risk-free series"] B --> C{"Manager controls cash flows?"} C -- No --> D["Time-weighted returns, GIPS composite"] C -- "Yes: closed-end or illiquid" --> E["Money-weighted returns and multiples"] D --> F{"Role in the owner's portfolio"} E --> F F -- "Whole risky portfolio" --> G["Sharpe and M-squared"] F -- "One sleeve of many" --> H["Treynor, alpha, appraisal ratio"] F -- "Benchmark mandate" --> I["Information ratio and tracking error"] G --> J["Factor regression and timing tests"] H --> J I --> J J --> K{"t-statistic of alpha above about 2?"} K -- Yes --> L["Attribution and qualitative due diligence"] K -- No --> M["Treat as unproven, extend record or reduce weight"] L --> N["Committee decision and monitoring plan"] M --> N ``` ## How managers and committees use performance evaluation - **Investment committees** of endowments, foundations and pension plans set the benchmark and the measures in the investment policy statement (finance/investment-policy-statement-institutional) before any manager is hired, so the test cannot be changed after the results arrive. - **Consultants and manager-research teams** run factor regressions and holdings-based analysis to separate skill from style, and check survivorship-free peer universes. - **Asset managers** report Sharpe, information ratio and tracking error in factsheets (finance/fund-factsheet) and explain excess return with attribution (finance/performance-attribution-report). - **Wealth advisers** use the money-weighted return to tell a client what their money earned and the time-weighted return to judge the manager (finance/client-portfolio-review). - **Private markets investors** use IRR, multiples and public market equivalents rather than Sharpe ratios, because marks are smoothed and cash flows are controlled by the general partner (finance/lp-quarterly-report, mba/mba-pe-fund-economics-card). - **Chief investment officers** use the fundamental law to set realistic targets: an information ratio of about 0.5 sustained over many years is generally regarded as strong. ## Connections to other cards | Card or template | Connection | |---|---| | mba/mba-capm-wacc-card | Beta and the security market line behind Treynor and Jensen | | mba/mba-portfolio-theory-card | The capital market line behind the Sharpe ratio and M-squared | | mba/mba-factor-models-card | Multi-factor alpha: Fama-French and Carhart regressions | | mba/mba-regression-for-managers-card | Reading a regression intercept, its standard error and t-statistic | | mba/mba-fixed-income-card | Bond managers: duration and spread positions drive their excess return | | mba/mba-value-at-risk-card | Downside risk measures alongside volatility | | mba/mba-pe-fund-economics-card | IRR, multiples and public market equivalents for private funds | | finance/performance-attribution-report | Allocation, selection and interaction effects | | mba/mba-investments-guide | The elective roadmap that places this card | ## Common mistakes and misconceptions 1. **Ranking on raw return.** Fund III had the highest return and the worst risk-adjusted record on Sharpe and M-squared. 2. **Using one measure for every question.** Sharpe for a sleeve of a diversified portfolio overstates the cost of diversifiable risk; the information ratio for a fund with a deliberate low beta penalises a choice the owner may want. 3. **Judging a manager on a money-weighted return.** It rewards or punishes the owner's timing of contributions, which the manager did not control. 4. **Annualising naively.** Multiplying a monthly Sharpe ratio by $\sqrt{12}$ assumes independent returns; smoothed or illiquid returns make the result too high. 5. **Calling a three-year alpha "skill".** With an information ratio of 0.5 you need about 16 years for a t-statistic of 2; short records are mostly noise. 6. **Picking the benchmark after the fact.** A benchmark chosen to flatter the manager produces alpha that is only style exposure. Agree it in the investment policy statement. 7. **Ignoring factor exposures.** A positive CAPM alpha can vanish once size, value and momentum are included. 8. **Mixing gross and net of fees.** The owner earns net returns; compare like with like. 9. **Trusting survivorship-biased peer rankings.** Closed and merged funds drop out of many databases, raising the average of those left. 10. **Reading GIPS compliance as a quality stamp.** It ensures fair presentation; it says nothing about future skill. ## Self-test questions 1. When is the Sharpe ratio the right measure and when is the Treynor ratio? 2. A fund returned 12% with a standard deviation of 20% and a beta of 1.25; the risk-free rate is 4%. Compute its Sharpe and Treynor ratios. 3. For the fund in question 2, the market returned 10%. What is Jensen's alpha? 4. A fund's Sharpe ratio is 0.40 and the market's standard deviation is 15%; the risk-free rate is 4%. What is its M-squared return? 5. A manager has an excess return over the benchmark of 1.5% a year with a tracking error of 3%. What is the information ratio and roughly how many years are needed for a t-statistic of 2? 6. An account returns +10% in the first half and -5% in the second half. What is the time-weighted return for the year? 7. Under the GIPS standards (2020 edition), when may a firm present money-weighted instead of time-weighted returns? 8. What is survivorship bias and in which direction does it push average fund performance? 9. A manager has an information coefficient of 0.05 and makes 100 independent bets a year. What information ratio does the fundamental law predict? 10. How would you test whether a manager times the market? 11. Name three properties of a good benchmark. ## My notes and next steps | Item | My entry | |---|---| | The measure I would choose for my own retirement fund, and why | [ ] | | The formula I mixed up | [ ] | | A real fund factsheet I will recompute (Sharpe, IR) with source and date | [ ] | | Questions for the next class or office hours | [ ] | | Next card | [mba/mba-factor-models-card or mba/mba-value-at-risk-card] | | Review dates (2 days, 1 week, 1 month) | YYYY-MM-DD, YYYY-MM-DD, YYYY-MM-DD | ## Sources and further reading - Treynor, J. L. (1965). How to rate management of investment funds. *Harvard Business Review*, 43(1). - Sharpe, W. F. (1966). Mutual fund performance. *The Journal of Business*, 39(1), 119-138. - Treynor, J. L., and Mazuy, K. K. (1966). Can mutual funds outguess the market? *Harvard Business Review*, 44(4). - Jensen, M. C. (1968). The performance of mutual funds in the period 1945-1964. *The Journal of Finance*, 23(2), 389-416. - Treynor, J. L., and Black, F. (1973). How to use security analysis to improve portfolio selection. *The Journal of Business*, 46(1), 66-86. - Henriksson, R. D., and Merton, R. C. (1981). On market timing and investment performance. II. Statistical procedures for evaluating forecasting skills. *The Journal of Business*, 54(4), 513-533. - Brinson, G. P., Hood, L. R., and Beebower, G. L. (1986). Determinants of portfolio performance. *Financial Analysts Journal*, 42(4), 39-44. - Grinold, R. C. (1989). The fundamental law of active management. *The Journal of Portfolio Management*, 15(3), 30-37. - Fama, E. F., and French, K. R. (1993). Common risk factors in the returns on stocks and bonds. *Journal of Financial Economics*, 33(1), 3-56. - Sortino, F. A., and Price, L. N. (1994). Performance measurement in a downside risk framework. *The Journal of Investing*, 3(3), 59-64. - Carhart, M. M. (1997). On persistence in mutual fund performance. *The Journal of Finance*, 52(1), 57-82. - Modigliani, F., and Modigliani, L. (1997). Risk-adjusted performance. *The Journal of Portfolio Management*, 23(2), 45-54. - Fama, E. F., and French, K. R. (2010). Luck versus skill in the cross-section of mutual fund returns. *The Journal of Finance*, 65(5), 1915-1947. - Fama, E. F., and French, K. R. (2015). A five-factor asset pricing model. *Journal of Financial Economics*, 116(1), 1-22. - CFA Institute (2019, errata July 2020). *Global Investment Performance Standards (GIPS) for Firms*, 2020 edition. gipsstandards.org (checked October 2026). - Textbooks (by name only): Bodie, Z., Kane, A., and Marcus, A. J., *Investments*; Grinold, R. C., and Kahn, R. N., *Active Portfolio Management*; Bacon, C. R., *Practical Portfolio Performance Measurement and Attribution*. - Open resource: OpenStax, *Principles of Finance* (CC BY-NC-SA 4.0), chapters on risk and return and on portfolios. ## Answer key **Worked example B.** (2) $70.3 / 74.0 - 1 = -5.00\%$. (3) TWR $= 1.08 \times 0.95 - 1 = 2.60\%$. (4) MWR $= 1.002497^2 - 1 = 0.50\%$ (0.5001%). (5) Modified Dietz $= 0.30 / (50.0 + 0.5027 \times 20.0) = 0.4995\%$, close to the IRR. (6) The manager earned 8% on 50m, then lost 5% on a larger 74m after the foundation's contribution; the 2.60% time-weighted return removes the effect of that timing and belongs in the GIPS report and in the manager's evaluation. The 0.50% money-weighted return is what the foundation's money actually earned, and the poor timing of the contribution was the foundation's decision. (7) $t = 1.6667$. (8) Critical value 2.0017; $1.67 < 2.00$, so the alpha is not significant at 5% (p = 0.10). (9) $0.25\% \times 12 = 3.00\%$ a year; a value or momentum tilt (or small-size exposure) could explain part of it. **Exercise C.** (1) Manager P: monthly Sharpe $0.70/4.2 = 0.1667$, annualised $0.1667 \times \sqrt{12} = 0.5774$; Manager Q: $0.55/2.9 = 0.1897$, annualised 0.6570. (2) P: monthly IR $0.20/1.1 = 0.1818$, annualised 0.6298; Q: $0.12/0.5 = 0.2400$, annualised 0.8314. Excel: `=0.2/1.1*SQRT(12)`. (3) P: $0.1818 \times 6 = 1.09$; Q: $0.24 \times 6 = 1.44$; neither reaches 2. (4) P: $(2/0.6298)^2 = 10.1$ years; Q: $(2/0.8314)^2 = 5.8$ years. (5) Ask for a GIPS-compliant composite report: all fee-paying discretionary accounts in the strategy, net-of-fee as well as gross returns, at least five years (or since inception), the composite's dispersion and size, and whether the firm is verified. (6) Shortlist Manager Q: higher Sharpe and information ratio with lower total and active risk, though its record is still not statistically conclusive; add a factor regression and a holdings-based style check, and confirm fees and capacity. **Self-test.** 1. Sharpe when the portfolio is the owner's whole risky portfolio (total risk matters); Treynor when it is one of many holdings in a diversified portfolio (only beta matters). 2. Sharpe $= (12 - 4)/20 = 0.40$; Treynor $= (12 - 4)/1.25 = 6.40\%$. 3. $\alpha = 12 - [4 + 1.25 \times (10 - 4)] = 0.50\%$. 4. $M^2 = 4\% + 0.40 \times 15\% = 10.00\%$. 5. $IR = 1.5/3 = 0.50$; years for $t = 2$: $(2/0.5)^2 = 16$. 6. $1.10 \times 0.95 - 1 = 4.50\%$. 7. Only if the firm controls the external cash flows and the composite or pooled fund is closed-end, fixed-life, fixed-commitment or has illiquid investments as a significant part of its strategy (check the current text). 8. Databases that drop funds that closed or merged (usually poor performers) leave the survivors; the average of the survivors is biased upward. 9. $IR \approx 0.05 \times \sqrt{100} = 0.50$. 10. Regress fund excess returns on market excess returns plus a squared market term (Treynor and Mazuy 1966) or an up-market term (Henriksson and Merton 1981); a significantly positive coefficient on the extra term indicates timing. 11. Any three of: specified in advance, investable, measurable, unambiguous, appropriate to the manager's style, reflective of the manager's current views and owned by the manager as the yardstick. ## Revision history | Version | Date | Author | Change summary | |---|---|---|---| | 0.1 | YYYY-MM-DD | [Your name] | First pass through the card | | 0.2 | YYYY-MM-DD | [Your name] | *Example:* Replaced Example returns with a real manager's GIPS composite data | ## Standard and licence note Based on the performance measures and tests in the primary works listed above (Treynor 1965, Sharpe 1966, Treynor and Mazuy 1966, Jensen 1968, Treynor and Black 1973, Henriksson and Merton 1981, Brinson, Hood and Beebower 1986, Grinold 1989, Sortino and Price 1994, Carhart 1997, Modigliani and Modigliani 1997, Fama and French 1993, 2010 and 2015), and structure aligned with the GIPS Standards for Firms (2020 edition) as checked in October 2026; GIPS provisions are summarised in original words, not reproduced. The text, examples, numbers and questions are original; no textbook text is copied or paraphrased, and Tallow Creek Medical Center, Corvane Robotics, the funds and the managers are fictional. All returns are *Example* values, not market data. The template is independent and not affiliated with or endorsed by any business school, publisher, CFA Institute or data provider. Check the current official version of the GIPS standards and current data before you rely on a figure. Template licence: MIT.