Finance-Theory · Unit 14 · Video 7 · Interactive Practice

Why Most Mutual Funds Trail the Index

IKey Formulas

Formula	Name	What it says
$R_{\text{fund}} - R_{\text{benchmark}}$	Fund excess return	Benchmark carries the fund's risk; measured net of fees. For U.S. equity funds, 1972–1991, the average was below zero
$\text{Risk premium} \equiv E[R_{it}] - r_f \;\Rightarrow\; r_f \approx 11.5\% - 8.3\% = 3.2\%$	Risk premium	Arithmetic averages of annual returns, 1926–2025; $3.2\%$ is the average T-bill rate
$\sigma_{\text{annual}} \approx \sigma_{\text{monthly}}\sqrt{12}$	Annualizing volatility	Average stock: $17\% \times \sqrt{12} \approx 59\%$ , roughly $60\%$ a year
$\dfrac{48}{19.4} \approx 2.5, \qquad \dfrac{80}{19.4} \approx 4.1$	VIX against its norm	Early November and late October 2008, against the VIX average of $19.4$ since 1990

Key Insight: The unit's ingredients for portfolio theory: an expected return with a premium of roughly $8\%$ a year, individual-stock risk $\sigma$ far above a portfolio's, and correlations $\rho$ that are positive but far from perfect.

IIChoosing a Fund Is Throwing a Dart

Without a way to spot the next Peter Lynch in advance, a random pick earns the histogram's average.

Fund fee Darts thrown 10

IIIWhat the Fee Costs Over Twenty Years

An active fund with no skill earns the market minus its fee, and the gap to an index fund compounds.

Active fund fee Year 5

💡 Over the 20 years to December 2025, about $92\%$ of actively managed U.S. equity funds trailed their benchmarks (S&P Dow Jones Indices): they returned less than the index, which is not the same as losing money.

IVHow Far Above Normal Was Late 2008?

The VIX is an annual volatility in percent; its 2008 readings dwarfed the long-run norm.

VIX level 48

VQuiz Questions

Problem 1 · Back Out the T-Bill Rate

Given: over 1926–2004 the arithmetic average annual return on U.S. stocks was $11.2\%$ and the average annual risk premium was $7.8\%$ . Find the average Treasury-bill rate these two figures imply.

✅ Correct! The premium is the return above T-bills, so the T-bill rate is what remains:

11.2\% - 7.8\% = 3.4\%

❌ Not quite. The premium is already part of the

11.2\%

; adding it counts it twice. Subtract it instead.

❌ Not quite.

7.8\%

is the premium itself, the return above T-bills, not the T-bill rate.

❌ Close, but…

10.3\% - 7.8\%

mixes a compounded return (1926–2025) with an arithmetic premium from a different period. Use the matching pair: both 1926–2004 arithmetic averages.

❌ Not quite. Rearrange

\text{Risk premium} = E[R] - r_f

r_f = E[R] - \text{Risk premium}

Show solution

The risk premium is the expected return minus the risk-free rate:

\text{Risk premium} \equiv E[R] - r_f \quad\Rightarrow\quad r_f = E[R] - \text{Risk premium}

With the 1926–2004 arithmetic averages:

r_f \approx 11.2\% - 7.8\% = 3.4\%

The same step on the 1926–2025 data gives $11.5\% - 8.3\% = 3.2\%$ . Both pairs are arithmetic averages of annual returns; the compounded $10.3\%$ cannot be paired with an arithmetic premium.

Problem 2 · Trailing Is Not Losing

Given: over 20 years a benchmark index compounds at $9.0\%$ a year (illustrative). A fictional active fund, Fund Q, trails that benchmark by $1.4$ percentage points a year. Which statement is true?

✅ Correct!

9.0\% - 1.4\% = 7.6\%

a year: the fund grew, just more slowly than the index it is measured against.

❌ Not quite. Trailing the benchmark means returning less than the index, not losing money. The

1.4

points is a shortfall relative to

9.0\%

, not a return below zero.

❌ Not quite. A fund that trails the benchmark earns less than it, so the shortfall is subtracted, not added.

❌ Not quite. Subtract the shortfall from the benchmark's return.

Show solution

"Trailing by $1.4$ points a year" is measured against the benchmark:

R_{Q} \approx 9.0\% - 1.4\% = 7.6\% \text{ a year}

That is a positive return, so Fund Q's investors made money, yet less than an index fund on the same benchmark would have earned. This is exactly how to read the finding that about $92\%$ of active U.S. equity funds trailed their benchmarks over the 20 years to December 2025.

Problem 3 · Calm and Turbulent Months, per Year

Given: the standard deviation of the market's monthly returns has ranged from roughly $2\%$ to $20\%$ a month. Annualize both ends with the square-root-of-time scaling (independent months). Find the annual $\sigma$ at the calm end, and the turbulent end's annual $\sigma$ as a multiple of the VIX's average of about $19.4$ since 1990.

Annual σ at the calm end (2% a month)

Turbulent end (20% a month), annualized, as a multiple of 19.4

✅ Correct!

2\% \times \sqrt{12} \approx 6.9\%

a year at the calm end;

20\% \times \sqrt{12} \approx 69.3\%

at the turbulent end, about

3.6

times the VIX's

19.4

average and in the range of its late-2008 readings.

❌ Not quite. Multiplying by

12

is the rule for the mean. Standard deviation grows with the square root of time: multiply by

\sqrt{12} \approx 3.46

❌ Not quite. Going from a month to a year makes

\sigma

larger: multiply by

\sqrt{12}

, don't divide.

❌ Not quite.

2^2 = 4

is the monthly variance in percent squared, not an annual standard deviation.

❌ Not quite.

20/19.4

compares a monthly

\sigma

with the VIX, which is an annual volatility. Annualize the

20\%

first.

❌ Not quite.

20 \times 12 = 240

scales the standard deviation like a mean. Use

\sqrt{12}

, not

12

❌ Not quite. That is

19.4

as a fraction of the annualized turbulent

\sigma

; the multiple asked for is the turbulent

\sigma

over

19.4

❌ Check the annual σ. Scale the monthly standard deviation by

\sqrt{12}

❌ Check the multiple. Annualize

20\%

a month, then divide by

19.4

Show solution

Step 1: Annualize each end (independent months, so $\sigma$ scales with $\sqrt{12} \approx 3.464$ ):

\sigma_{\text{calm}} \approx 2\% \times \sqrt{12} \approx 6.9\% \text{ a year}, \qquad \sigma_{\text{turbulent}} \approx 20\% \times \sqrt{12} \approx 69.3\% \text{ a year}

Step 2: Compare with the VIX average. The VIX is itself an annual volatility in percent, so the two are on the same footing:

\frac{69.3}{19.4} \approx 3.6

A calm month's volatility, stated per year, is well under the $16$ – $20\%$ norm; a turbulent month's is more than three times the VIX's long-run average, the territory of its $80$ -plus readings in late 2008. Market volatility changes over time.

Problem 4 · Twenty Years of Fee Drag

Given: a benchmark index returns $8.0\%$ every year before fees (illustrative). An index fund on it charges $0.05\%$ a year, so it grows at $7.95\%$ ; a fictional active fund with no skill charges $0.9\%$ a year, so it grows at $7.1\%$ . Equal sums are invested in each and left for 20 years. Find how far the active fund's final value falls below the index fund's, as a percentage of the index fund's.

✅ Correct!

(1.071/1.0795)^{20} \approx 0.854

, so the active fund ends about

14.6\%

below the index fund.

❌ Close, but…

20 \times 0.85 = 17

adds up the yearly fee gaps as if each were charged on the original sum. Growth compounds: compare

(1.071)^{20}

with

(1.0795)^{20}

❌ Not quite.

0.85

points is the gap in one year's growth rate. Over 20 years the gap compounds.

❌ Close, but…

1 - (1.071/1.08)^{20} \approx 15.4\%

compares the active fund with holding the stocks at no cost. The index fund pays its

0.05\%

fee too: divide by

1.0795

❌ Not quite. Grow each fund at its net rate for 20 years and compare the two final values.

Show solution

Step 1: Net growth rates. Index fund $8.0\% - 0.05\% = 7.95\%$ ; active fund $8.0\% - 0.9\% = 7.1\%$ a year.

Step 2: Ratio of final values (the invested sum cancels):

\left(\frac{1.071}{1.0795}\right)^{20} \approx 0.99213^{20} \approx 0.854

Step 3: The shortfall.

1 - 0.854 \approx 14.6\%

A fund with no skill earns the market minus its fee, so a fee gap of under one point a year costs about a seventh of the final value over 20 years: part of why the average active fund trails the passive alternative.

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