Lecture 1: Measuring Fund Performance

Institutional Investors

Author

Calvin J. Chiou

Published

October 18, 2026

Overview

How do we know whether a fund manager adds value? This lecture introduces the canonical returns-based framework for fund performance evaluation, following the survey by Wermers (2011). The central takeaway from Wermers: the Carhart (1997) four-factor model is the baseline model for evaluating actively managed equity portfolios.

We will:

  1. Motivate the need for multi-factor performance models
  2. Build from CAPM to Fama-French 3-factor to Carhart 4-factor, step by step
  3. Download price data for 10 popular US ETFs using tidyquant
  4. Obtain factor returns directly from Kenneth French’s Data Library using the frenchdata package
  5. Estimate and interpret Carhart alphas and factor loadings
  6. Visualize the results with interactive highcharter charts

1. Motivation: Why Not Just Compare Raw Returns?

A fund that returned 15% last year sounds impressive—until you learn it was a small-cap value fund during a year when small-cap value stocks returned 18%. The fund actually destroyed value relative to the passive alternative.

Wermers (2011) states the core principle clearly: “Managers should be rewarded for bets not easily replicated by uninformed investors.” In other words, we need to control for the passive risk factors that investors can access cheaply, and only then can we attribute the residual—alpha—to manager skill.

The Evolution of Baseline Models

Evolution of performance benchmarks
Year Model Factors Controlled
Jensen (1968) CAPM (1-factor) Market (systematic) risk
Fama and French (1993) Fama-French 3-factor Market + Size (SMB) + Value (HML)
Carhart (1997) 4-factor FF3 + Momentum (UMD)

Wermers (2011) (Section 3.1) notes that a regression of a diversified long-only portfolio on the Carhart model typically yields \(R^2 > 90\%\), meaning nearly all return variation is explained by these four systematic factors. This is precisely why it serves as the baseline: it leaves little room for data mining.

2. Setup

Code 
library(tidyverse)
library(tidyquant)
library(frenchdata)   # Kenneth French's Data Library
library(broom)
library(gt)
library(scales)
library(highcharter)  # interactive charts (replaces ggplot2 for all figures)

# Optional: apply one of your own highcharter themes to every chart, e.g.
# options(highcharter.theme = hc_theme_smpl())

3. The Ten US ETFs

We use ten ETFs spanning broad market, style (size/value), sector, and active strategies. This cross-section lets students observe how factor loadings vary systematically across fund types—a key pedagogical goal.

Ten US ETFs used in this lecture
Ticker Name Category Expected Factor Tilt
SPY SPDR S&P 500 Broad market β ≈ 1, others ≈ 0
QQQ Invesco Nasdaq-100 Tech/growth Negative HML
IWM iShares Russell 2000 Small cap Positive SMB
VTV Vanguard Value Large-cap value Positive HML
VUG Vanguard Growth Large-cap growth Negative HML
ARKK ARK Innovation Active growth High β, negative HML
XLE Energy Select SPDR Energy sector Positive HML (value)
XLF Financial Select SPDR Financial sector High β
VNQ Vanguard Real Estate REITs Positive HML
IWD iShares Russell 1000 Value Large-cap value Positive HML
Code 
etf_tickers <- c(
  "SPY",   # S&P 500
  "QQQ",   # Nasdaq-100
  "IWM",   # Russell 2000
  "VTV",   # Vanguard Value
  "VUG",   # Vanguard Growth
  "ARKK",  # ARK Innovation (active)
  "XLE",   # Energy sector
  "XLF",   # Financials sector
  "VNQ",   # Real estate
  "IWD"    # Russell 1000 Value
)

etf_labels <- tibble(
  symbol   = etf_tickers,
  name     = c("S&P 500", "Nasdaq-100", "Russell 2000",
               "Vanguard Value", "Vanguard Growth", "ARK Innovation",
               "Energy Sector", "Financials Sector",
               "Real Estate (REIT)", "Russell 1000 Value"),
  category = c("Broad market", "Tech/Growth", "Small cap",
               "Large value", "Large growth", "Active growth",
               "Sector", "Sector", "REIT", "Large value")
)

Download ETF Price Data

Code 
start_date <- "2015-01-01"
end_date   <- "2024-12-31"

prices_raw <- tq_get(
  etf_tickers,
  from = start_date,
  to   = end_date,
  get  = "stock.prices"
)

# Quick check
prices_raw |>
  select(symbol, date, adjusted) |>
  slice_head(n = 5)
# A tibble: 5 × 3
  symbol date       adjusted
  <chr>  <date>        <dbl>
1 SPY    2015-01-02     169.
2 SPY    2015-01-05     166.
3 SPY    2015-01-06     165.
4 SPY    2015-01-07     167.
5 SPY    2015-01-08     170.

Compute Monthly Returns

The Carhart model is estimated on monthly returns—the same frequency as the French factor data.

Code 
returns_monthly <- prices_raw |>
  group_by(symbol) |>
  tq_transmute(
    select     = adjusted,
    mutate_fun = periodReturn,
    period     = "monthly",
    type       = "arithmetic",   # simple returns to match French data convention
    col_rename = "ret"
  ) |>
  ungroup() |>
  left_join(etf_labels, by = "symbol")

4. Kenneth French’s Factor Data

Kenneth French maintains a publicly accessible data library at mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html. The frenchdata package downloads and parses these files directly.

The Four Factors

The Carhart model uses four factors:

\[R_{i,t} - R_{f,t} = \alpha_i + \beta_i \cdot RMRF_t + s_i \cdot SMB_t + h_i \cdot HML_t + u_i \cdot UMD_t + \varepsilon_{i,t} \tag{1}\]

Carhart four-factor model components
Factor Full name Economic rationale
\(RMRF\) Market excess return Compensation for bearing systematic risk
\(SMB\) Small-minus-big Size premium: small stocks earn higher returns on average (Fama and French 1993)
\(HML\) High-minus-low Value premium: high book-to-market (cheap) stocks outperform growth stocks (Fama and French 1993)
\(UMD\) Up-minus-down (momentum) Past winners continue to outperform past losers (Jegadeesh and Titman 1993; Carhart 1997)

Download and Prepare Factor Data

Code 
# Fama-French 3 factors: Mkt-RF, SMB, HML, RF (monthly)
ff3_raw <- download_french_data("Fama/French 3 Factors")

# Momentum factor: UMD (monthly)
mom_raw <- download_french_data("Momentum Factor (Mom)")

# Extract monthly tables and convert from % to decimal
ff3_monthly <- ff3_raw$subsets$data[[1]] |>
  mutate(
    date   = lubridate::ym(date),          # "YYYYMM" → date
    across(c(`Mkt-RF`, SMB, HML, RF), ~ . / 100)
  ) |>
  rename(RMRF = `Mkt-RF`) |>
  filter(date >= as.Date(start_date), date <= as.Date(end_date))

mom_monthly <- mom_raw$subsets$data[[1]] |>
  mutate(
    date = lubridate::ym(date),
    Mom  = Mom / 100
  ) |>
  rename(UMD = Mom) |>
  filter(date >= as.Date(start_date), date <= as.Date(end_date))

# Combine into a single factor tibble
factors <- ff3_monthly |>
  left_join(mom_monthly, by = "date")

factors |> slice_head(n = 6)
# A tibble: 6 × 6
  date          RMRF     SMB     HML    RF     UMD
  <date>       <dbl>   <dbl>   <dbl> <dbl>   <dbl>
1 2015-01-01 -0.031  -0.0061 -0.0348     0  0.0372
2 2015-02-01  0.0613  0.0064 -0.0175     0 -0.0287
3 2015-03-01 -0.0111  0.0306 -0.0038     0  0.0272
4 2015-04-01  0.0059 -0.0298  0.0181     0 -0.0726
5 2015-05-01  0.0137  0.0096 -0.011      0  0.0576
6 2015-06-01 -0.0152  0.0295 -0.008      0  0.0301

Align Dates and Compute Excess Returns

French data uses end-of-month dates; tq_transmute() returns the last trading day of each month. We align on year-month.

Code 
# Standardize ETF returns to year-month for merging
returns_ym <- returns_monthly |>
  mutate(ym = lubridate::floor_date(date, "month")) |>
  select(symbol, name, category, ym, ret)

factors_ym <- factors |>
  mutate(ym = lubridate::floor_date(date, "month")) |>
  select(ym, RMRF, SMB, HML, UMD, RF)

# Merge and compute excess return
data_merged <- returns_ym |>
  left_join(factors_ym, by = "ym") |>
  mutate(excess_ret = ret - RF) |>
  filter(!is.na(RMRF))   # drop months outside factor data range

5. Cumulative Return Plot

The chart is interactive: hover to compare all ETFs at a given month, click a legend entry to hide or show a series, and drag across the plot to zoom in on a sub-period.

Code 
cumret_df <- data_merged |>
  arrange(symbol, ym) |>
  group_by(symbol, name) |>
  mutate(cum_ret_pct = (cumprod(1 + ret) - 1) * 100) |>   # in percent
  ungroup()

hchart(cumret_df, "line",
       hcaes(x = ym, y = cum_ret_pct, group = name)) |>
  hc_chart(zoomType = "x", height = 480) |>
  hc_title(text = "Cumulative Returns") |>
  hc_xAxis(title = list(text = NULL)) |>
  hc_yAxis(title  = list(text = "Cumulative return"),
           labels = list(format = "{value}%")) |>
  hc_tooltip(shared = TRUE, sort = TRUE,
             valueDecimals = 1, valueSuffix = "%") |>
  hc_plotOptions(series = list(lineWidth = 1.6,
                               marker    = list(enabled = FALSE))) |>
  hc_legend(layout = "horizontal", align = "center",
            verticalAlign = "bottom",
            itemStyle = list(fontSize = "11px")) |> 
  hc_add_theme(hc_theme_economist())
Figure 1: Cumulative simple returns of 10 US ETFs (2015–2024)

6. Step 1 — CAPM / Jensen’s Alpha (Single Factor)

Before moving to Carhart, we start with the CAPM as a baseline. Jensen (1968) showed that fund performance can be measured as the regression intercept \(\alpha\) from:

\[R_{i,t} - R_{f,t} = \alpha_i + \beta_i \cdot RMRF_t + \varepsilon_{i,t} \tag{2}\]

\(\alpha > 0\) means the fund earned more than its CAPM-implied expected return.

Code 
# Helper: significance stars from a p-value
sig_stars <- function(p) {
  case_when(
    p < 0.01 ~ "***",
    p < 0.05 ~ "**",
    p < 0.10 ~ "*",
    TRUE     ~ ""
  )
}

capm_results <- data_merged |>
  group_by(symbol, name) |>
  do({
    fit <- lm(excess_ret ~ RMRF, data = .)
    bind_cols(
      tidy(fit)  |> select(term, estimate, std.error, statistic, p.value),
      glance(fit)|> select(r.squared, adj.r.squared) |> slice(1)
    )
  }) |>
  select(symbol, name, term, estimate, std.error, statistic, p.value, r.squared) |>
  filter(term %in% c("(Intercept)", "RMRF")) |>
  mutate(term = recode(term, "(Intercept)" = "alpha", "RMRF" = "beta_mkt")) |>
  pivot_wider(
    names_from  = term,
    values_from = c(estimate, std.error, statistic, p.value),
    names_glue  = "{term}_{.value}"
  ) |>
  ungroup() |>
  mutate(
    alpha_ann      = alpha_estimate * 12,                 # annualized alpha
    alpha_stars    = sig_stars(alpha_p.value),            # new: stars for alpha
    beta_mkt_stars = sig_stars(beta_mkt_p.value)          # new: stars for beta
  )

capm_results |>
  arrange(desc(alpha_ann)) |>
  select(name,
         alpha_ann, alpha_stars, alpha_statistic, alpha_p.value,
         beta_mkt_estimate, beta_mkt_stars, beta_mkt_statistic,
         # kept (hidden by the merges below) so downstream chunks still work
         alpha_estimate, r.squared) |>
  gt() |>
  # numeric formatting
  fmt_number(
    columns  = c(alpha_ann, alpha_statistic, beta_mkt_estimate, beta_mkt_statistic, r.squared),
    decimals = 3
  ) |>
  fmt(columns = alpha_p.value, fns = \(x) sprintf("%.3f", x)) |>
  # attach stars to the coefficient they belong to, e.g. "0.042**"
  cols_merge(columns = c(alpha_ann, alpha_stars),       pattern = "{1}{2}") |>
  cols_merge(columns = c(beta_mkt_estimate, beta_mkt_stars), pattern = "{1}{2}") |>
  cols_hide(alpha_estimate) |>
  cols_label(
    name               = "ETF",
    alpha_ann          = "α (Ann.)",
    alpha_statistic    = "t(α)",
    alpha_p.value      = "p(α)",
    beta_mkt_estimate  = "Beta",
    beta_mkt_statistic = "t(β)",
    r.squared        = "R²"
  ) |>
  cols_align(align = "right", columns = -name) |>
  # shade rows where alpha is significant at the 10% level
  tab_style(
    style     = cell_fill(color = "#f0f4ff"),
    locations = cells_body(rows = alpha_p.value < 0.10)
  ) |>
  tab_header(
    title    = "CAPM / Jensen's Alpha",
    subtitle = "Monthly excess returns regressed on market factor only"
  ) |>
  tab_source_note(
    "Significance: *** p < 0.01, ** p < 0.05, * p < 0.10 (two-sided; OLS standard errors). Shaded rows: alpha significant at the 10% level."
  ) |>
  tab_source_note(
    "Stars on annualized alpha come from the monthly alpha t-test; multiplying by 12 changes the scale but not the t-statistic or p-value."
  )
CAPM / Jensen’s alpha with statistical significance
CAPM / Jensen's Alpha
Monthly excess returns regressed on market factor only
ETF α (Ann.) t(α) p(α) Beta t(β) R²
Nasdaq-100 0.045* 1.847 0.067 1.065*** 24.620 0.837
Vanguard Growth 0.023 1.200 0.232 1.058*** 31.505 0.894
S&P 500 0.005 1.019 0.311 0.957*** 100.684 0.988
Vanguard Value −0.013 −0.668 0.505 0.870*** 25.899 0.850
Financials Sector −0.015 −0.451 0.653 1.065*** 17.808 0.729
Russell 1000 Value −0.034* −1.886 0.062 0.928*** 28.983 0.877
ARK Innovation −0.042 −0.553 0.581 1.740*** 12.884 0.585
Real Estate (REIT) −0.054 −1.388 0.168 0.860*** 12.336 0.563
Russell 2000 −0.058* −1.962 0.052 1.173*** 22.241 0.807
Energy Sector −0.067 −0.892 0.374 1.200*** 8.963 0.405
Significance: *** p < 0.01, ** p < 0.05, * p < 0.10 (two-sided; OLS standard errors). Shaded rows: alpha significant at the 10% level.
Stars on annualized alpha come from the monthly alpha t-test; multiplying by 12 changes the scale but not the t-statistic or p-value.

Limitation of CAPM alpha: If IWM (small cap) has \(\alpha > 0\) in the CAPM, it might simply be capturing the size premium that any investor could access by buying small stocks passively—not manager skill. This is why we need additional factors.

7. Step 2 — Fama-French 3-Factor Model

Fama and French (1993) document that two additional risk factors—size (SMB) and value (HML)—explain cross-sectional variation in average returns beyond the market. Adding these factors purges the CAPM alpha of style-related biases:

\[R_{i,t} - R_{f,t} = \alpha_i + \beta_i \cdot RMRF_t + s_i \cdot SMB_t + h_i \cdot HML_t + \varepsilon_{i,t} \tag{3}\]

Code 
ff3_results <- data_merged |>
  group_by(symbol, name) |>
  do({
    fit <- lm(excess_ret ~ RMRF + SMB + HML, data = .)
    bind_cols(
      tidy(fit)  |> select(term, estimate, std.error, statistic, p.value),
      glance(fit)|> select(r.squared, adj.r.squared) |> slice(1)
    )
  }) |>
  filter(term %in% c("(Intercept)", "RMRF", "SMB", "HML")) |>
  mutate(term = recode(term,
                       "(Intercept)" = "alpha", "RMRF" = "b_mkt",
                       "SMB" = "b_smb", "HML" = "b_hml"
  )) |>
  pivot_wider(
    names_from  = term,
    values_from = c(estimate, std.error, statistic, p.value),
    names_glue  = "{term}_{.value}"
  ) |>
  mutate(alpha_ann = alpha_estimate * 12) |>
  ungroup()

ff3_results |>
  arrange(desc(alpha_ann)) |>
  select(name, alpha_ann, alpha_statistic, alpha_p.value,
         b_mkt_estimate, b_smb_estimate, b_hml_estimate, r.squared) |>
  gt() |>
  fmt_number(columns = where(is.numeric), decimals = 3) |>
  cols_label(
    name             = "ETF",
    alpha_ann        = "α (Ann.)",
    alpha_statistic  = "t(α)",
    alpha_p.value    = "p(α)",
    b_mkt_estimate   = "β(MKT)",
    b_smb_estimate   = "β(SMB)",
    b_hml_estimate   = "β(HML)",
    r.squared        = "R²"
  ) |>
  tab_header(
    title    = "Fama-French 3-Factor Model",
    subtitle = "Controlling for market, size, and value"
  ) |>
  tab_style(
    style     = cell_fill(color = "#f0f4ff"),
    locations = cells_body(rows = alpha_p.value < 0.10)
  )
Fama-French 3-Factor Model
Controlling for market, size, and value
ETF α (Ann.) t(α) p(α) β(MKT) β(SMB) β(HML) R²
Nasdaq-100 0.029 1.772 0.079 1.103 −0.150 −0.397 0.926
Vanguard Growth 0.009 0.846 0.400 1.091 −0.132 −0.341 0.965
Financials Sector 0.001 0.028 0.978 1.045 0.021 0.599 0.887
S&P 500 0.000 0.150 0.881 0.982 −0.142 0.012 0.996
Vanguard Value −0.008 −0.677 0.500 0.880 −0.108 0.357 0.948
ARK Innovation −0.018 −0.309 0.758 1.540 1.236 −0.867 0.761
Russell 2000 −0.021 −2.523 0.013 1.012 0.859 0.226 0.986
Russell 1000 Value −0.026 −2.448 0.016 0.922 −0.015 0.345 0.959
Energy Sector −0.032 −0.571 0.569 1.142 0.152 1.162 0.671
Real Estate (REIT) −0.048 −1.228 0.222 0.836 0.127 0.046 0.569

Notice how alpha estimates change relative to CAPM—this is the style-correction at work. IWM’s alpha shrinks once we control for SMB; VTV’s alpha changes once we control for HML.

8. Step 3 — Carhart 4-Factor Model (Baseline)

Carhart (1997) adds the momentum factor (UMD) to the Fama-French model. Stocks that have outperformed over the prior 12 months tend to continue outperforming over the next few months (Jegadeesh and Titman 1993). If a fund merely holds past winners, its FF3 alpha is overstated—Carhart strips this out.

\[\boxed{R_{i,t} - R_{f,t} = \alpha_i + \beta_i \cdot RMRF_t + s_i \cdot SMB_t + h_i \cdot HML_t + u_i \cdot UMD_t + \varepsilon_{i,t}} \tag{4}\]

As Wermers (2011) (p. 542) states: “The returns-based model most widely used among academics in analyzing equity managers is the four-factor model of Carhart (1997).”

Code 
carhart_results <- data_merged |>
  group_by(symbol, name) |>
  do({
    fit <- lm(excess_ret ~ RMRF + SMB + HML + UMD, data = .)
    bind_cols(
      tidy(fit)  |> select(term, estimate, std.error, statistic, p.value),
      glance(fit)|> select(r.squared, adj.r.squared) |> slice(1)
    )
  }) |>
  filter(term %in% c("(Intercept)", "RMRF", "SMB", "HML", "UMD")) |>
  mutate(term = recode(term,
    "(Intercept)" = "alpha", "RMRF" = "b_mkt",
    "SMB" = "b_smb", "HML" = "b_hml", "UMD" = "b_umd"
  )) |>
  pivot_wider(
    names_from  = term,
    values_from = c(estimate, std.error, statistic, p.value),
    names_glue  = "{term}_{.value}"
  ) |>
  mutate(alpha_ann = alpha_estimate * 12) |>
  ungroup()

carhart_results |>
  arrange(desc(alpha_ann)) |>
  select(name,
         alpha_ann, alpha_statistic, alpha_p.value,
         b_mkt_estimate, b_smb_estimate,
         b_hml_estimate, b_umd_estimate,
         r.squared) |>
  gt() |>
  fmt_number(columns = c(alpha_ann, alpha_statistic,
                          b_mkt_estimate, b_smb_estimate,
                          b_hml_estimate, b_umd_estimate,
                          r.squared),
             decimals = 3) |>
  fmt(columns = alpha_p.value, fns = \(x) sprintf("%.3f", x)) |>
  cols_label(
    name             = "ETF",
    alpha_ann        = "α (Ann.)",
    alpha_statistic  = "t(α)",
    alpha_p.value    = "p(α)",
    b_mkt_estimate   = "β(MKT)",
    b_smb_estimate   = "β(SMB)",
    b_hml_estimate   = "β(HML)",
    b_umd_estimate   = "β(UMD)",
    r.squared        = "R²"
  ) |>
  tab_header(
    title    = "Carhart Four-Factor Model",
    subtitle = "Wermers (2011) baseline model — sorted by annualized alpha"
  ) |>
  tab_style(
    style     = cell_fill(color = "#f0f4ff"),
    locations = cells_body(rows = alpha_p.value < 0.10)
  )
Carhart Four-Factor Model
Wermers (2011) baseline model — sorted by annualized alpha
ETF α (Ann.) t(α) p(α) β(MKT) β(SMB) β(HML) β(UMD) R²
Nasdaq-100 0.033 1.980 0.050 1.082 −0.174 −0.416 −0.070 0.928
Vanguard Growth 0.012 1.117 0.266 1.074 −0.152 −0.358 −0.059 0.967
S&P 500 0.001 0.214 0.831 0.981 −0.144 0.011 −0.004 0.996
Financials Sector −0.002 −0.088 0.930 1.061 0.040 0.615 0.053 0.888
Vanguard Value −0.009 −0.768 0.444 0.886 −0.100 0.364 0.023 0.948
ARK Innovation −0.014 −0.245 0.807 1.518 1.210 −0.888 −0.075 0.761
Russell 2000 −0.024 −3.049 0.003 1.032 0.883 0.245 0.068 0.987
Energy Sector −0.024 −0.424 0.672 1.093 0.094 1.114 −0.171 0.675
Russell 1000 Value −0.025 −2.327 0.022 0.915 −0.023 0.339 −0.023 0.959
Real Estate (REIT) −0.052 −1.306 0.194 0.857 0.151 0.066 0.072 0.571

Interpreting the Results

Factor loadings tell us about passive style exposures:

  • \(\beta(MKT) > 1\): more market-sensitive than the index (e.g., ARKK)
  • \(\beta(SMB) > 0\): tilts toward small-cap stocks (e.g., IWM)
  • \(\beta(HML) > 0\): value tilt (e.g., IWD, VTV, XLE); \(\beta(HML) < 0\): growth tilt (e.g., QQQ, VUG)
  • \(\beta(UMD) > 0\): momentum tilt; \(< 0\): contrarian or mean-reverting strategy

Alpha after controlling for all four factors measures what is not explained by these passive risk exposures. For passive index ETFs, we expect:

\[\alpha \approx 0 - \text{expense ratio}\]

A passive ETF like SPY should produce \(\alpha \approx -0.09\%\)/year (its expense ratio) once we properly control for all factors.

9. Comparing Alpha Across the Three Models

A key lesson: alpha estimates are model-dependent. Adding factors changes (and usually shrinks) alpha.

Code 
alpha_compare <- capm_results |>
  select(name, capm_alpha = alpha_ann) |>
  left_join(
    ff3_results |> select(name, ff3_alpha = alpha_ann),
    by = "name"
  ) |>
  left_join(
    carhart_results |> select(name, carhart_alpha = alpha_ann),
    by = "name"
  ) |>
  arrange(desc(carhart_alpha))

alpha_compare |>
  gt() |>
  fmt_number(columns = c(capm_alpha, ff3_alpha, carhart_alpha), decimals = 3) |>
  cols_label(
    name           = "ETF",
    capm_alpha     = "CAPM α",
    ff3_alpha      = "FF3 α",
    carhart_alpha  = "Carhart α"
  ) |>
  tab_header(
    title    = "Alpha Comparison Across Models",
    subtitle = "Annualized; adding factors changes—usually shrinks—measured alpha"
  ) |>
  tab_spanner(
    label   = "Annualized Alpha",
    columns = c(capm_alpha, ff3_alpha, carhart_alpha)
  ) |>
  data_color(
    columns = carhart_alpha,
    fn      = scales::col_numeric(
      palette = c("tomato", "white", "steelblue"),
      domain  = NULL
    )
  )
Table 1: Alpha (annualized) across three models
Alpha Comparison Across Models
Annualized; adding factors changes—usually shrinks—measured alpha
ETF
Annualized Alpha
CAPM α FF3 α Carhart α
Nasdaq-100 0.045 0.029 0.033
Vanguard Growth 0.023 0.009 0.012
S&P 500 0.005 0.000 0.001
Financials Sector −0.015 0.001 −0.002
Vanguard Value −0.013 −0.008 −0.009
ARK Innovation −0.042 −0.018 −0.014
Russell 2000 −0.058 −0.021 −0.024
Energy Sector −0.067 −0.032 −0.024
Russell 1000 Value −0.034 −0.026 −0.025
Real Estate (REIT) −0.054 −0.048 −0.052

10. Visualizing Factor Loadings

highcharter has no direct equivalent of facet_wrap(), so we build one bar chart per factor and arrange the four panels in a grid with hw_grid(). All panels use the same ETF ordering (by median loading across the four factors), so you can read across a row to compare one ETF’s exposures.

Code 
loadings_long <- carhart_results |>
  select(name, b_mkt_estimate, b_smb_estimate,
         b_hml_estimate, b_umd_estimate) |>
  pivot_longer(
    cols      = -name,
    names_to  = "factor",
    values_to = "loading"
  ) |>
  mutate(factor = recode(factor,
                         "b_mkt_estimate" = "MKT",
                         "b_smb_estimate" = "SMB",
                         "b_hml_estimate" = "HML",
                         "b_umd_estimate" = "UMD"
  ))

# One common ETF order for all four charts (highest median loading at the top)
etf_order <- loadings_long |>
  group_by(name) |>
  summarise(med = median(loading), .groups = "drop") |>
  arrange(desc(med)) |>
  pull(name)

Market (MKT)

Code 
d_mkt <- loadings_long |>
  filter(factor == "MKT") |>
  arrange(match(name, etf_order))

highchart() |>
  hc_chart(type = "bar", height = 360) |>
  hc_title(text = "Carhart Loadings: Market (MKT)") |>
  hc_xAxis(categories = etf_order) |>
  hc_yAxis(title = list(text = "Loading"),
           plotLines = list(list(value = 0, width = 1,
                                 color = "#333333", zIndex = 3))) |>
  hc_add_series(name = "MKT", data = round(d_mkt$loading, 4),
                color = "#4E79A7") |>
  hc_legend(enabled = FALSE) |>
  hc_tooltip(pointFormat = "Loading: <b>{point.y:.3f}</b>")

Size (SMB)

Code 
d_smb <- loadings_long |>
  filter(factor == "SMB") |>
  arrange(match(name, etf_order))

highchart() |>
  hc_chart(type = "bar", height = 360) |>
  hc_title(text = "Carhart Loadings: Size (SMB)") |>
  hc_xAxis(categories = etf_order) |>
  hc_yAxis(title = list(text = "Loading"),
           plotLines = list(list(value = 0, width = 1,
                                 color = "#333333", zIndex = 3))) |>
  hc_add_series(name = "SMB", data = round(d_smb$loading, 4),
                color = "#F28E2B") |>
  hc_legend(enabled = FALSE) |>
  hc_tooltip(pointFormat = "Loading: <b>{point.y:.3f}</b>")

Value (HML)

Code 
d_hml <- loadings_long |>
  filter(factor == "HML") |>
  arrange(match(name, etf_order))

highchart() |>
  hc_chart(type = "bar", height = 360) |>
  hc_title(text = "Carhart Loadings: Value (HML)") |>
  hc_xAxis(categories = etf_order) |>
  hc_yAxis(title = list(text = "Loading"),
           plotLines = list(list(value = 0, width = 1,
                                 color = "#333333", zIndex = 3))) |>
  hc_add_series(name = "HML", data = round(d_hml$loading, 4),
                color = "#59A14F") |>
  hc_legend(enabled = FALSE) |>
  hc_tooltip(pointFormat = "Loading: <b>{point.y:.3f}</b>")

Momentum (UMD)

Code 
d_umd <- loadings_long |>
  filter(factor == "UMD") |>
  arrange(match(name, etf_order))

highchart() |>
  hc_chart(type = "bar", height = 360) |>
  hc_title(text = "Carhart Loadings: Momentum (UMD)") |>
  hc_xAxis(categories = etf_order) |>
  hc_yAxis(title = list(text = "Loading"),
           plotLines = list(list(value = 0, width = 1,
                                 color = "#333333", zIndex = 3))) |>
  hc_add_series(name = "UMD", data = round(d_umd$loading, 4),
                color = "#B07AA1") |>
  hc_legend(enabled = FALSE) |>
  hc_tooltip(pointFormat = "Loading: <b>{point.y:.3f}</b>")

11. R² as a Diagnostic

Wermers (2011) notes that managed long-only portfolios typically have \(R^2 > 90\%\) under the Carhart model, precisely because these four factors capture most passive variation. Low \(R^2\) signals idiosyncratic strategies not captured by the standard factors.

Code 
r2_df <- carhart_results |>
  select(name, r.squared) |>
  arrange(desc(r.squared))     # highest R² at the top of the bar chart

highchart() |>
  hc_chart(type = "bar", height = 380) |>
  hc_title(text = "R² from Carhart 4-Factor Model") |>
  hc_xAxis(categories = r2_df$name) |>
  hc_yAxis(min = 0, max = 105, tickInterval = 20, endOnTick = FALSE,
           title  = list(text = "R-squared"),
           labels = list(format = "{value}%")) |>
  hc_add_series(name = "R²",
                data = round(r2_df$r.squared * 100, 1),
                color = "#4682B4",
                dataLabels = list(enabled = TRUE, format = "{y:.1f}%")) |>
  hc_legend(enabled = FALSE) |>
  hc_tooltip(pointFormat = "R²: <b>{point.y:.1f}%</b>")
Figure 2: R² from Carhart model — how much return variation is explained by the four factors?
Note

ARKK typically has a lower R² than passive ETFs—consistent with its active, concentrated strategy in disruptive-technology stocks. Wermers (2011) discusses this in the context of hedge funds (Section 6.3): funds with lower R² tend to take more idiosyncratic bets, which may or may not translate into outperformance.

12. What Alpha Means for Passive ETFs

For passive index ETFs like SPY and IWM, alpha should be:

\[\alpha \approx -\text{expense ratio} \approx 0\]

Why? If markets are competitive and these factors are the right benchmark, a well-implemented passive fund should have zero alpha before costs. Any measured positive alpha from a passive fund is a red flag—it may indicate benchmark misspecification.

For actively managed funds like ARKK, we are genuinely asking: did the manager add value beyond passive factor exposure? The answer is in \(\hat{\alpha}\) and its \(t\)-statistic.

Important

A positive (or negative) alpha that is statistically insignificant should be interpreted cautiously. With 10 years of monthly data (T ≈ 120), the power to detect even a 2% annual alpha at conventional significance levels is limited. Wermers (2011) (Section 3.3) discusses how the Kosowski et al. (2006) bootstrap method addresses this problem for large cross-sections of funds.

13. Summary Statistics

Code 
summary_tbl <- data_merged |>
  group_by(symbol, name, category) |>
  summarise(
    ann_ret    = (prod(1 + ret))^(12 / n()) - 1,
    ann_vol    = sd(ret) * sqrt(12),
    sharpe     = (mean(ret - RF) / sd(ret)) * sqrt(12),
    .groups    = "drop"
  ) |>
  left_join(
    carhart_results |>
      select(name, alpha_ann, b_mkt_estimate, r.squared),
    by = "name"
  ) |>
  arrange(desc(sharpe))

summary_tbl |>
  select(name, category, ann_ret, ann_vol, sharpe, alpha_ann, b_mkt_estimate, r.squared) |>
  gt() |>
  fmt_percent(columns = c(ann_ret, ann_vol), decimals = 1) |>
  fmt_number(columns = c(sharpe, alpha_ann, b_mkt_estimate, r.squared), decimals = 3) |>
  cols_label(
    name             = "ETF",
    category         = "Category",
    ann_ret          = "Ann. Return",
    ann_vol          = "Ann. Vol.",
    sharpe           = "Sharpe",
    alpha_ann        = "Carhart α",
    b_mkt_estimate   = "Beta",
    r.squared        = "R²"
  ) |>
  tab_header(
    title    = "Performance Summary",
    subtitle = paste("Sample:", start_date, "to", end_date)
  )
Table 2: Fund performance summary: annualized statistics
Performance Summary
Sample: 2015-01-01 to 2024-12-31
ETF Category Ann. Return Ann. Vol. Sharpe Carhart α Beta R²
Nasdaq-100 Tech/Growth 18.4% 18.5% 0.919 0.033 1.082 0.928
Vanguard Growth Large growth 15.9% 17.8% 0.826 0.012 1.074 0.967
S&P 500 Broad market 13.0% 15.3% 0.771 0.001 0.981 0.996
Vanguard Value Large value 10.0% 15.0% 0.600 −0.009 0.886 0.948
Financials Sector Sector 11.3% 19.8% 0.556 −0.002 1.061 0.888
Russell 1000 Value Large value 8.3% 15.8% 0.479 −0.025 0.915 0.959
ARK Innovation Active growth 12.2% 36.2% 0.450 −0.014 1.518 0.761
Russell 2000 Small cap 7.8% 20.7% 0.386 −0.024 1.032 0.987
Real Estate (REIT) REIT 4.8% 18.2% 0.258 −0.052 0.857 0.571
Energy Sector Sector 4.8% 29.9% 0.248 −0.024 1.093 0.675

14. Key Takeaways

  1. The Carhart four-factor model is the academic standard for evaluating equity fund performance (Wermers 2011). It controls for market, size, value, and momentum exposures—the four main dimensions of passive strategy that any uninformed investor can exploit.

  2. Alpha is model-dependent. CAPM alpha, FF3 alpha, and Carhart alpha can differ substantially. A fund that looks good in CAPM may simply be loading on the size or value premium, which requires no skill.

  3. For passive ETFs, \(\alpha \approx 0\). The expense ratio creates a small negative alpha. Any significant positive alpha from a passive ETF signals factor model misspecification.

  4. Factor loadings reveal investment style. Negative HML loading → growth/tech; Positive SMB → small cap; Positive UMD → momentum strategy.

  5. R² near or above 90% is expected for diversified long-only portfolios. Low R² (like ARKK) indicates concentrated, idiosyncratic bets—which may or may not reflect skill (Wermers 2011, sec. 3.3).

  6. Statistical power is limited with 10 years of monthly data. A 2% annual alpha may not be detectable at the 5% level with only 120 observations.

Discussion Questions

  1. SPY tracks the S&P 500. Theoretically, what should its Carhart alpha be? If you find a positive alpha, what are three possible explanations?

  2. IWM (Russell 2000) will have a large positive SMB loading. Does this mean IWM outperforms? Or that it merely holds small stocks? What is the difference?

  3. ARKK is actively managed. Suppose its Carhart alpha is positive but statistically insignificant (\(p = 0.15\)). How should an investor interpret this?

  4. Compare the CAPM alpha and Carhart alpha of QQQ. Why do they differ? What factor explains most of the difference?

  5. Wermers (2011) writes that the Carhart model regression of a managed portfolio typically yields \(R^2 > 90\%\). Which ETF in our sample has the lowest R²? Why does this make intuitive sense given that fund’s investment strategy?

References

Carhart, Mark M. 1997. “On Persistence in Mutual Fund Performance.” Journal of Finance 52 (1): 57–82.
Fama, Eugene F., and Kenneth R. French. 1993. “Common Risk Factors in the Returns on Stocks and Bonds.” Journal of Financial Economics 33 (1): 3–56.
Jegadeesh, Narasimhan, and Sheridan Titman. 1993. “Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency.” Journal of Finance 48 (1): 65–91.
Jensen, Michael C. 1968. “The Performance of Mutual Funds in the Period 1945–1964.” Journal of Finance 23 (2): 389–416.
Wermers, Russ. 2011. “Performance Measurement of Mutual Funds, Hedge Funds, and Institutional Accounts.” Annual Review of Financial Economics 3: 537–74.
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