Performance Evaluation

Investment Theory

Lorenzo Naranjo

Fall 2026

Why Measure Performance?

  • Investors hand money to managers in the hope that they know something the market does not.
  • Pension funds hire dozens of managers; individuals choose among thousands of funds.
  • Each decision needs a comparison of the fund’s record with what the investor could have earned without the manager.
  • Questions:
    • How do we measure the return?
    • How do we adjust it for risk?
    • How precise is the answer?
    • Where did the return come from?

Time-Weighted and Dollar-Weighted Returns

  • The time-weighted return compounds the return of each period: 1 + \bar{r}_{\text{TW}} = \left[ (1 + r_{1}) (1 + r_{2}) \cdots (1 + r_{T}) \right]^{1/T}. \tag{1}
    • It measures what the manager did with each dollar.
  • The dollar-weighted return is the internal rate of return of the investor’s cash flows: C_{0} + \sum_{t = 1}^{T - 1} \frac{C_{t}}{(1 + \bar{r}_{\text{DW}})^{t}} = \frac{V_{T}}{(1 + \bar{r}_{\text{DW}})^{T}}. \tag{2}
    • It measures what investors earned, including how well they timed their purchases.

Example 1 (A Fund with Bad Timing) A fund starts with $100 million, returns 50\% in its first year, attracts $150 million of new money, and returns -20\% in its second year. Its time-weighted return is (1.5 \times 0.8)^{1/2} - 1 = 9.54\% a year. It ends with 0.8 \times 300 = \$240 million, so the dollar-weighted return solves 100 + 150/x = 240/x^{2}, which gives x = 0.9712, or -2.88\% a year. Investors put in $250 million and ended with $240 million.

The ARK Innovation ETF

Two stacked line charts sharing a time axis from late 2014 to mid 2026. The top chart shows one dollar growing slowly until 2019, jumping to about 7 dollars in early 2021, falling to about 2 dollars by late 2022, and recovering to about 4 to 5 dollars by 2026. The bottom chart shows total assets near zero until 2019, rising to a peak of about 25 billion dollars in 2021, then falling to between 6 and 9 billion dollars for the rest of the period.

Growth of $1 invested in the ARK Innovation ETF at its launch, on a log scale (top), and the fund’s total net assets in billions of dollars (bottom).

The Manager’s Record and the Investors’

  • Time-weighted: the fund compounded at 13.5% a year, November 2014 to June 2026.
  • Assets were tiny in its best years and peaked at $25.3 billion in June 2021, just before the collapse of 2022.
  • Dollar-weighted: investors as a group earned -5.5% a year.
  • Investors put in $10.6 billion more than they took out, and ended $4.1 billion below what they had invested.
  • Performance evaluation uses time-weighted returns, net of expenses for investors and gross of expenses to judge the manager.

Adjusting for Risk

  • Notation: a fund’s excess return R_{P} has mean \mu_{P}, volatility \sigma_{P}, beta \beta_{P} and residual volatility \sigma(e_{P}); the market has \mu_{M} and \sigma_{M}.
  • Sharpe ratio (Sharpe 1966) and M² (Modigliani and Modigliani 1997): \mathit{SR}_{P} = \frac{\mu_{P}}{\sigma_{P}} \qquad M^{2}_{P} = \frac{\sigma_{M}}{\sigma_{P}} \mu_{P} - \mu_{M} = \sigma_{M} (\mathit{SR}_{P} - \mathit{SR}_{M}).
    • M² levers or unlevers the fund to the market’s volatility and compares returns.
  • Treynor’s measure (1965) and Jensen’s alpha (1968): T_{P} = \frac{\mu_{P}}{\beta_{P}} \qquad \alpha_{P} = \mu_{P} - \beta_{P} \mu_{M}.
    • T_{P} - \mu_{M} = \alpha_{P} / \beta_{P}, so the two always agree on the sign.

The Information Ratio

  • The information ratio scales alpha by the residual risk that comes with it: \mathit{IR}_{P} = \frac{\alpha_{P}}{\sigma(e_{P})}.
  • From the Treynor-Black model, combining the fund with the market in the best proportions gives \mathit{SR}^{2} = \mathit{SR}_{M}^{2} + \mathit{IR}_{P}^{2}.
  • A large alpha with a lot of residual risk can be used only in small doses.

Which Measure?

The fund is Use Because
The investor’s whole risky portfolio Sharpe ratio or M² All of its volatility is borne
One of many funds held together Treynor’s measure or alpha Residual risk diversifies away
An active portfolio added to an index fund Information ratio It sets the gain in the overall Sharpe ratio

Example 2 (One Fund, Three Verdicts) The market has \mu_{M} = 7\% and \sigma_{M} = 18\%, so \mathit{SR}_{M} = 0.389. A fund has \mu_{P} = 9\%, \sigma_{P} = 25\% and \beta_{P} = 1.2. Then \mathit{SR}_{P} = 0.36 and M^{2}_{P} = 18\% \times (0.36 - 0.389) = -0.52\%, but T_{P} = 7.5\% > 7\% and \alpha_{P} = 9\% - 1.2 \times 7\% = 0.6\%. Its residual volatility is \sqrt{0.25^{2} - 1.2^{2} \times 0.18^{2}} = 12.6\%, so \mathit{IR}_{P} = 0.048. As the only holding, it is worse than the market; as one of many funds, it is better; added to an index fund, it is nearly worthless.

Measures for Real Funds

Mean Vol. Sharpe M² Beta Treynor Alpha t IR
Vanguard 500 10.3 15.3 0.67 0.1 0.97 10.7 0.1 0.39 0.09
Magellan 9.5 17.9 0.53 -2.2 1.08 8.8 -1.9 -1.52 -0.35
Contrafund 12.0 15.5 0.77 1.6 0.93 12.9 2.2 1.87 0.43
Growth Stock 11.2 17.7 0.63 -0.6 1.05 10.7 0.2 0.11 0.03
Windsor 8.4 17.1 0.49 -2.8 1.02 8.2 -2.3 -1.76 -0.40
Dodge & Cox 8.7 17.6 0.49 -2.8 1.04 8.3 -2.3 -1.56 -0.36
Primecap 12.5 15.9 0.78 1.8 0.96 13.0 2.4 2.10 0.48
Sequoia 7.5 16.3 0.46 -3.2 0.85 8.8 -1.4 -0.68 -0.16
Market 10.5 15.8 0.67 0.0 1.00 10.5
  • Annualized, from net returns, July 2006 to June 2026. Mean, volatility, M², Treynor and alpha in percent.

Reading the Table

  • The rankings mostly agree: Primecap and Contrafund top every measure, with Sharpe ratios of 0.78 and 0.77 against 0.67 for the market.
  • Diversified funds with betas near one: volatility and beta convey nearly the same information.
  • The exception is Sequoia, a concentrated fund with beta 0.85.
    • Its Treynor measure ranks it with Magellan, its Sharpe ratio ranks it last.
  • The index fund’s alpha of 0.13% is indistinguishable from zero, but its beta is 0.97: the S&P 500 holds large firms only.

Multifactor Alphas

Alpha t MKT SMB HML RMW CMA UMD R^{2}
Vanguard 500 -0.33 -1.83 1.00 -0.12 0.03 0.05 0.02 -0.00 1.00
Magellan -2.28 -2.07 1.11 -0.09 -0.14 -0.02 -0.14 0.02 0.93
Contrafund 1.08 1.25 0.98 -0.09 -0.15 0.07 -0.18 0.09 0.94
Growth Stock -0.17 -0.18 1.07 -0.06 -0.29 -0.06 -0.27 0.01 0.95
Windsor -1.24 -1.37 0.96 0.09 0.26 0.01 0.11 -0.09 0.95
Dodge & Cox -1.29 -1.28 0.98 0.01 0.31 0.04 0.09 -0.11 0.94
Primecap 2.57 2.28 0.97 -0.05 0.05 -0.13 -0.03 0.04 0.91
Sequoia -1.14 -0.53 0.81 0.14 0.05 0.12 -0.15 -0.02 0.69
  • Growth funds load negatively on HML and CMA; Windsor and Dodge & Cox positively on HML.
  • Magellan’s alpha falls to -2.3% (t = -2.07). Primecap is the only fund with a significant positive alpha under both models.

How Precise Is a Measured Alpha?

  • Over T years, the standard error of alpha is about \sigma(e_{P}) / \sqrt{T}, so t(\hat{\alpha}_{P}) \approx \frac{\alpha_{P}}{\sigma(e_{P}) / \sqrt{T}} = \mathit{IR}_{P} \sqrt{T}. \tag{3}
  • A t of 2 needs T = (2 / \mathit{IR}_{P})^{2} years: 16 for an excellent \mathit{IR} of 0.5, 64 for 0.25.
  • Primecap: \mathit{IR} = 0.48 over 20 years gives t \approx 2.15; the regression gives 2.10.
  • Sharpe ratios are no more precise (Lo 2002): \operatorname{SE}(\widehat{\mathit{SR}}) = \sqrt{\frac{1 + \mathit{SR}^{2} / 2}{T}}.
    • The market’s 95% interval, 0.23 to 1.11, contains every fund.

Style Analysis

  • Sharpe (1992): find the mix of cheap index funds that tracks the fund most closely, \begin{gathered} r_{P,t} = \sum_{j = 1}^{n} w_{j} r_{j,t} + e_{P,t}, \\ \min_{w} \operatorname{V}(e_{P}) \quad \text{subject to} \quad \sum_{j} w_{j} = 1, \quad w_{j} \geq 0. \end{gathered}
  • The mix is the fund’s style; the residual is the return due to selection.
  • Here the asset classes are four Vanguard index funds (large and small, value and growth) and T-bills.

Style Weights

A horizontal stacked bar chart with one bar per fund, each bar summing to 100 percent and split among large value, large growth, small value, small growth and T-bills. The Vanguard 500 Index bar is half large value and half large growth. The Vanguard Small-Cap Index bar is roughly half small value and half small growth. Growth funds such as T. Rowe Price Growth Stock, Contrafund and Magellan are mostly large growth. Windsor and Dodge and Cox are mostly large value with about 30 percent small value. Sequoia is a mix of large growth, small value and about one fifth T-bills.

Style weights of seven active funds and two index funds, estimated on monthly returns of four Vanguard index funds and T-bills.

What Style Analysis Shows

  • The index funds check the method:
    • Vanguard 500: 49% large value, 50% large growth, R-squared 0.998.
    • Small-Cap Index: 54% small value, 46% small growth.
  • Selection returns, after costs, relative to a passive mix with the same style:
    • Primecap 2.4% a year (t = 2.19), Contrafund 1.3% (t = 1.53), Magellan -1.9% (t = -1.84).
  • The weights are chosen with hindsight. Sharpe recommended estimating the style from earlier returns only.

Market Timing

  • A manager who forecasts the market raises beta before good markets and lowers it before bad ones, making the fund’s return convex in the market’s.
  • Treynor and Mazuy (1966): R_{P,t} = \alpha_{P} + \beta_{P} R_{M,t} + \gamma_{P} R_{M,t}^{2} + e_{P,t}.
  • Henriksson and Merton (1981): R_{P,t} = \alpha_{P} + \beta_{P} R_{M,t} + c_{P} \max(-R_{M,t}, 0) + e_{P,t}.
    • \max(-R_{M}, 0) is the payoff of a put on the market: a good timer gives investors c_{P} puts for free.

Timing in the Data

  • Seven of the eight funds have a negative \gamma_{P}, and none of the positive estimates is significant.
  • The typical fund looks like a perverse timer, with more market exposure before low returns.
  • Ferson and Schadt (1996): much of this reflects public information.
    • Dividend yields and interest rates predict returns, and fund betas move with them for reasons unrelated to skill.
    • One mechanism: money flows in when the public expects high returns, and the extra cash lowers beta.
    • Conditional measures make perverse timing go away.
  • Convexity can also come from holding options, which the manager pays for through a lower alpha.

Performance Attribution

  • A sponsor sets a policy weight w_{B,j} for each asset class; the manager deviates through timing and selection (Brinson et al. 1986).
Index returns Actual returns
Policy weights I: \sum_{j} w_{B,j} r_{B,j} III: \sum_{j} w_{B,j} r_{P,j}
Actual weights II: \sum_{j} w_{P,j} r_{B,j} IV: \sum_{j} w_{P,j} r_{P,j}
  • Timing is II - I, selection is III - I, and the interaction is IV - III - II + I.

Example 3 (Attributing a Balanced Fund’s Return) Policy: 60% stocks, 30% bonds, 10% cash; index returns 8%, 3% and 1%. The fund held 70%, 25% and 5%, and earned 9%, 2.5% and 1% in each class. Then I = 5.8%, II = 6.4%, III = 6.25% and IV = 6.975%. The fund beat its policy by 1.175\%: timing 0.6\%, selection 0.45\% and interaction 0.125\%.

What 93.6% Means

  • In 91 large pension plans, 1974 to 1983, the policy explained on average 93.6% of the variation of each plan’s quarterly returns over time.
  • It does not say that asset allocation explains 94% of the differences in returns across funds.
  • Active management cost the plans about 1.1% a year: 9.01% against 10.11% for their policies.

Benchmarks from Holdings

  • Daniel et al. (1997) match each stock a fund holds with one of 125 portfolios formed on size, book-to-market and past return.
    • Characteristic selectivity: stock returns minus their matched portfolios.
    • Characteristic timing: shifting toward characteristics before they pay off.
    • Average style: the return of the fund’s average characteristics.
  • Over 2,500 funds, 1975 to 1994: some selectivity, especially in aggressive-growth funds, no characteristic timing, and gains no larger than the extra fees of active funds.

Gaming the Measures

  • Goetzmann et al. (2007): changing leverage or trading options, with no information, can raise the Sharpe ratio, alpha and timing measures.
  • A measure that cannot be gamed this way must be a certainty equivalent: \hat{\Theta} = \frac{1}{(1 - \rho) \Delta t} \ln \left[ \frac{1}{T} \sum_{t = 1}^{T} \left( \frac{1 + r_{t}}{1 + r_{f,t}} \right)^{1 - \rho} \right].
  • Example: monthly returns of -10\%, 5\%, 17\% and -2\% with T-bills at 1% give \hat{\Theta} = 6.6\% with \rho = 2 and 1.2\% with \rho = 3.
  • Morningstar’s star ratings use nearly the same measure, with \rho = 3.

Takeaways

  • Time-weighted returns judge the manager; dollar-weighted returns describe the investors.
  • Sharpe, Treynor, alpha and the information ratio answer different questions.
  • Multifactor alphas separate skill from cheap style tilts.
  • Alphas and Sharpe ratios are imprecise: decades of data are needed.
  • Style analysis, timing regressions and attribution show where a return came from.
  • A good measure must be hard to game.

References

Brinson, Gary P., L. Randolph Hood, and Gilbert L. Beebower. 1986. “Determinants of Portfolio Performance.” Financial Analysts Journal 42 (4): 39–44.
Daniel, Kent, Mark Grinblatt, Sheridan Titman, and Russ Wermers. 1997. “Measuring Mutual Fund Performance with Characteristic-Based Benchmarks.” Journal of Finance 52 (3): 1035–58.
Ferson, Wayne E., and Rudi W. Schadt. 1996. “Measuring Fund Strategy and Performance in Changing Economic Conditions.” Journal of Finance 51 (2): 425–61.
Goetzmann, William, Jonathan Ingersoll, Matthew Spiegel, and Ivo Welch. 2007. “Portfolio Performance Manipulation and Manipulation-Proof Performance Measures.” Review of Financial Studies 20 (5): 1503–46.
Henriksson, Roy D., and Robert C. Merton. 1981. “On Market Timing and Investment Performance. II. Statistical Procedures for Evaluating Forecasting Skills.” Journal of Business 54 (4): 513–33.
Lo, Andrew W. 2002. “The Statistics of Sharpe Ratios.” Financial Analysts Journal 58 (4): 36–52.
Modigliani, Franco, and Leah Modigliani. 1997. “Risk-Adjusted Performance.” Journal of Portfolio Management 23 (2): 45–54.
Sharpe, William F. 1966. “Mutual Fund Performance.” Journal of Business 39 (1): 119–38.
Sharpe, William F. 1992. “Asset Allocation: Management Style and Performance Measurement.” Journal of Portfolio Management 18 (2): 7–19.
Treynor, Jack L., and Kay K. Mazuy. 1966. “Can Mutual Funds Outguess the Market?” Harvard Business Review 44: 131–36.