QM 3 Benchmarking Returns
The size, timing, and direction of cash flows into and out of a portfolio can move a measured return substantially, so the method used to calculate it matters. Consider a fund that receives EUR10,000 at the start of year one and EUR1,000 in each of the next two years, and loses 50 percent in that first year. Now compare it with a fund that receives EUR1,000 first, then EUR10,000, then EUR1,000, and suffers the same 50 percent loss in year one. The two funds take in the same EUR12,000 in total, yet the first ends year three worth EUR8,360 while the second ends at EUR13,805, purely because the large deposit met the loss in one case and avoided it in the other.
Two return measures handle this differently. The time-weighted rate of return breaks the investment life into subperiods at each cash flow event, computes the return within each subperiod, and links those subperiod returns geometrically. Because it strips out the effect of contributions and withdrawals, it captures the manager’s underlying skill between periods.
The money-weighted rate of return, also called the dollar-weighted return, is simply the internal rate of return (IRR) on the portfolio: the single discount rate that sets the present value of every cash flow to zero. Because it weights each subperiod by the amount invested at that time, it reflects the size and timing of the investor decisions.
A related pair of summary statistics is worth keeping straight. The geometric average return compounds the period returns, while the arithmetic average simply adds and divides. The geometric figure is the one that reconciles beginning and ending wealth.
Both funds above earn the same sequence of investment returns: minus 50 percent, then plus 10 percent, then plus 10 percent. What follows is the same table, split into a large-first-contribution scenario (Panel A) and a small-first-contribution scenario (Panel B).
| Year | Begin value | Contribution | Return (%) | End value |
|---|---|---|---|---|
| Panel A: year 1 | 10,000 | 0 | −50% | 5,000 |
| Panel A: year 2 | 5,000 | 1,000 | 10% | 6,600 |
| Panel A: year 3 | 6,600 | 1,000 | 10% | 8,360 |
| Panel B: year 1 | 1,000 | 0 | −50% | 500 |
| Panel B: year 2 | 500 | 10,000 | 10% | 11,550 |
| Panel B: year 3 | 11,550 | 1,000 | 10% | 13,805 |
The practical division of labor follows from this. The time-weighted return isolates what the manager controls, so the Global Investment Performance Standards (GIPS) require it for most public-market managers, who do not control investor deposits and withdrawals. The money-weighted return is permitted, and often preferred, where the manager does control the external cash flows, as in private equity with committed capital and capital calls. For the investor, the money-weighted return is the more relevant number, since it measures the actual return earned given their own timing.
The Albright Fund begins the year worth EUR100 million. Interim events fall at four-month intervals. On 30 April it reinvests EUR2 million of dividends and EUR10 million of realized capital gains, books EUR3 million of unrealized gains, and receives EUR20 million of new investor capital; its value is EUR115 million before that contribution and EUR135 million after. On 31 August it reinvests EUR3 million of dividends and is worth EUR130 million. On 31 December it receives EUR3 million of dividends that it does not reinvest, ending at EUR140 million.
In practice portfolios are valued daily, and a full-year time-weighted return links the daily holding-period returns. The day-count basis matters: at an average daily return of 0.05 percent, compounding over 365 calendar days gives (1.0005) to the 365th power minus one, about 20.02 percent, while compounding over 252 trading days gives (1.0005) to the 252nd power minus one, about 13.42 percent. Over multiple years, each annual return is linked and the geometric mean is taken.
A security market index is a single indicator that aggregates the performance of a chosen set of securities, whether a whole market, a sector, or an asset class. It is built as a portfolio of constituents, and its value is recalculated regularly from their observable market prices or estimated values. Indexes began as benchmarks: Charles Dow and Edward Jones published the first one in 1884 to track eleven transportation companies, and the Dow Jones Industrial Average followed in 1896 with twelve industrial firms. Today they do more than measure. They serve as the blueprint for index funds and exchange-traded funds that aim to replicate a benchmark, and custom indexes now tilt toward factors such as momentum, value, income, or minimum variance to blend passive tracking with active ideas.
Building and running an index involves a defined sequence: set the objective (which market or sector), select the constituents against stated criteria, choose a weighting methodology, fix an initial date and value for the calculation, and then maintain the index through rebalancing, reconstitution, and adjustments for corporate actions.
Two versions of the same index
The same constituents, identically weighted, can produce two different index series depending on how distributions are treated. A price return index counts only the prices of the constituents, so it captures capital appreciation alone. A total return index also reinvests all capital distributions, such as dividends and interest, so it captures appreciation plus the return on those reinvested distributions.
On the starting date the two versions are equal. Over time the total return index typically pulls ahead, because reinvested distributions compound, and the gap widens over longer horizons. A price return index suits non-dividend-paying stocks and short-term price-trend analysis; a total return index suits dividend payers, long-horizon growth, and any strategy where reinvested income is a meaningful part of the return.
The general index value
Whatever the weighting method, an index value at time t is the weighted sum of the constituent prices, where the weights depend on the method chosen.
The weighting method decides how each constituent’s price change flows through to the index, and it shapes the index behavior more than any other design choice. Four methods are common: market-capitalization weighting, fundamental-factor weighting, price weighting, and equal weighting. In every case an aggregate reference value is summed and then divided by a divisor, an arbitrary number chosen to set a convenient starting value such as 100 or 1,000. The examples below use the same five securities and set each index to 100 at inception.
Market-capitalization weighting
Also called value weighting, this method sets each weight in proportion to the constituent’s market capitalization, which is the number of shares outstanding times the price. Summing the market caps of every constituent gives the reference value, which the divisor then normalizes.
| Security | Shares (Q) | Price (P) | Market cap | Weight |
|---|---|---|---|---|
| A | 15,000 | 50 | 750,000 | 30.0% |
| B | 10,000 | 40 | 400,000 | 16.0% |
| C | 20,000 | 30 | 600,000 | 24.0% |
| D | 20,000 | 25 | 500,000 | 20.0% |
| E | 50,000 | 5 | 250,000 | 10.0% |
| Total (divisor 25,000) | 2,500,000 | 100.0% |
Security A holds 30.0 percent because its 750,000 market cap is that share of the 2,500,000 total. Dividing 2,500,000 by a divisor of 25,000 sets the index at 100. The chief advantage is that a security’s share of total market cap stays constant as prices move, so the index rebalances itself and needs little maintenance. The drawback is that it tilts toward the largest and fastest-appreciating names, which can crowd out smaller firms.
Float adjustment
When strategic owners hold shares they will not sell, only a fraction of the shares is actually available to the public. That fraction is the market float, and most capitalization-weighted indexes weight by float-adjusted market cap so the index reflects only tradable value.
| Security | Shares (Q) | Float (f) | Float shares | Price | Reference value | Weight |
|---|---|---|---|---|---|---|
| A | 15,000 | 100% | 15,000 | 50 | 750,000 | 39.9% |
| B | 10,000 | 90% | 9,000 | 40 | 360,000 | 19.2% |
| C | 20,000 | 80% | 16,000 | 30 | 480,000 | 25.6% |
| D | 20,000 | 20% | 4,000 | 25 | 100,000 | 5.3% |
| E | 50,000 | 75% | 37,500 | 5 | 187,500 | 10.0% |
| Total (divisor 18,775) | 1,877,500 | 100.0% |
Security D has a strategic owner, so its float drops to 20 percent, cutting its reference value to 100,000 and its weight from 20.0 percent to 5.3 percent. Freely traded Security A rises to 39.9 percent. The float shrinks total capitalization from 2,500,000 to 1,877,500, and the divisor falls to 18,775 to keep the starting value at 100. Indexes aimed at global investors go further, excluding shares foreign investors cannot buy, which gives a free-float-adjusted version.
Fundamental-factor weighting
Fundamental weighting sets weights from a business measure such as revenue, earnings, book value, dividends, or workforce size, aiming to reflect a company’s economic footprint rather than its market valuation. Each weight is the constituent’s fundamental value over the total across all constituents.
An earnings-weighted index, for example, favors companies with high earnings relative to price and drops companies with no earnings, which gives it a value tilt at the expense of growth names that plow earnings back into the business. Because prices drift away from fundamentals between updates, these indexes need periodic rebalancing to restore the fundamental proportions.
Price weighting
Price weighting is the simplest scheme and now rare, surviving mainly in the Dow Jones Industrial Average (30 constituents) and the Nikkei 225. Each weight is the security price over the sum of all prices, which is the same as holding exactly one share of each constituent.
For the five securities, the prices sum to 50 plus 40 plus 30 plus 25 plus 5, or 150, and a divisor of 1.5 sets the index at 100. The highest-priced name, Security A, takes the largest weight at 33.3 percent, so the index reflects high-priced shares disproportionately even though price alone says nothing about company size. Stock splits are the main headache here, as covered in the maintenance section.
Equal weighting
Equal weighting assigns every constituent the same weight, 1 divided by N, regardless of price or size, so no single large or high-priced name dominates. To place equal value in each name at inception, the index holds a different share count per security.
Holding 60, 75, 100, 120, and 600 shares of A through E puts EUR3,000 of value in each, for an aggregate of 15,000; a divisor of 150 sets the index at 100. The catch is drift: as prices move, the equal weights break, so the index needs frequent rebalancing, which limits its use for actual portfolios.
| Security | Price weighted | Equal weighted | Float-adjusted market cap |
|---|---|---|---|
| A | 33.3% | 20.0% | 39.9% |
| B | 26.7% | 20.0% | 19.2% |
| C | 20.0% | 20.0% | 25.6% |
| D | 16.7% | 20.0% | 5.3% |
| E | 3.3% | 20.0% | 10.0% |
| Initial index value | 100 | 100 | 100 |
Once the base value is set, the index is recomputed as prices change. Using the updated prices alone gives the price return; adding reinvested capital distributions gives the total return. The one-period return is just the new index value over the old, minus one, and it also equals the weighted sum of the constituent returns. Carry the same five securities forward one year, with the price moves and dividends below.
| Security | Old price | New price | Price return | Dividend | Total return |
|---|---|---|---|---|---|
| A | 50 | 55 | 10.0% | 2 | 14.0% |
| B | 40 | 50 | 25.0% | 3 | 32.5% |
| C | 30 | 35 | 16.7% | 0 | 16.7% |
| D | 25 | 15 | −40.0% | 1 | −36.0% |
| E | 5 | 10 | 100.0% | 0.5 | 110.0% |
Use the float-adjusted market-capitalization-weighted index from the previous section, with a divisor of 18,775 and a starting value of 100.
Take the same price and dividend data, but now value the price-weighted index (divisor 1.5) and the equal-weighted index (divisor 150), both starting at 100.
Putting the three methods side by side on identical holdings shows how much the weighting choice matters. The same five securities, over the same year, deliver quite different index returns.
| Weighting method | Price return | Total return |
|---|---|---|
| Price weighted | 10.0% | 14.3% |
| Equal weighted | 22.3% | 27.4% |
| Float-adjusted market cap | 20.9% | 25.2% |
The gap between each price return and its total return is the effect of reinvested and compounded distributions, and the spread across the three methods reflects the weighting alone. The price-weighted index leans on the highest-priced names, the equal-weighted index gives every name identical pull, and the market-cap index is driven by the largest companies.
Keeping an index true to its design involves two ongoing processes. Rebalancing restores the intended weights among the existing constituents. Reconstitution updates which securities are in the index at all. Indexes weighted by market capitalization or by price rebalance themselves as prices move, but equal-weighted and fundamental-factor indexes drift and must be rebalanced on a schedule, often quarterly.
Rebalancing an equal-weighted index
In an equal-weighted index, rebalancing pushes every weight back to 1 divided by N. That means selling down the securities that have risen and buying more of those that have fallen, a counter-trend, sell-high buy-low activity that generates turnover. In the worked example above the price-only equal-weighted index reached a value of 122.33, but by then Security E had run to a 32.7 percent weight while Security D had sunk to 9.8 percent, so restoring each to 20 percent requires trimming E and adding to A, C, and D. Rebalancing can call for fractional shares, and the divisor may be tweaked to keep the index series continuous.
Adjusting the divisor for a stock split
A stock split multiplies the share count and cuts the price proportionally, leaving market capitalization unchanged, so it should not move a price-weighted index. Because the split mechanically lowers the sum of prices, the divisor must be recomputed to hold the index value steady.
The price-weighted index of the five securities stands at 100, with prices 50, 40, 30, 25, 5 summing to 150 and a divisor of 1.5. Security A undergoes a two-for-one split, halving its price to 25.
Reconstitution
Reconstitution periodically refreshes the membership so the index still represents its target market. Providers rank securities, re-apply the inclusion criteria, then remove names that no longer qualify (including those lost to bankruptcy, delisting, or acquisition) and add names that now qualify. The new constituents and weights then drive the index going forward, and the divisor may be adjusted so values stay comparable across the change. Reconstitution forces turnover in tracking portfolios, and securities expected to join a major index often rise beforehand while those expected to leave tend to fall.
Because turnover is costly, providers manage the frequency and thresholds. The Russell indexes use a banding rule, moving an incumbent between the large-cap and small-cap segments only when its market cap crosses the breakpoint by more than 5 percent, and the Russell 2000 reconstitutes once a year. The FTSE 100 reviews quarterly: a company that falls to rank 111 or lower is removed automatically, and a company that rises to rank 90 or higher is added automatically, with borderline cases handled through the FTSE 250.
Many indexes have listed futures and options written on them, so a change in constituents ripples into the composition, value, and settlement of those derivatives. Index-tracking funds must trade to match the new membership too. The SPDR S&P 500 ETF Trust, or SPY, is a well-known case: it tracks the S&P 500 and routinely adjusts its holdings to follow that index’s scheduled rebalancing and reconstitution, keeping the fund aligned with the benchmark it is built to mirror.