CFA Level 1 · Module 02 Quantitative Methods · Chapter 3
Two investors can hold the exact same fund over the exact same year and honestly report different returns. Neither is lying. The difference is that one of them added and withdrew money along the way, and those cash flows change what a simple return figure means. Measuring performance well, and comparing it against a fair benchmark, means being precise about whose performance you are measuring: the manager who picked the investments, or the investor who decided when to put money in and take it out.
This reading has two halves. The first builds two return measures that treat investor cash flows differently: the time-weighted return, which strips those cash flows out, and the money-weighted return, which lets them drive the result. The second half turns to the benchmark itself, the security market index, and shows how the choice of weighting scheme shapes the number an index reports. Both halves matter for the exam, and both are heavy on calculation, so work the examples with a pen.
Everything here builds on the return machinery from the first two readings of this module. Chain-linking, holding period returns, and the weighted average of returns all reappear, now put to work on the harder question of measuring performance fairly and choosing the benchmark to measure it against.
Imagine a manager who runs a fund brilliantly, earning a strong return every quarter. Now imagine an investor who, by bad luck, pours in a large amount of money just before the one weak quarter. The fund did well; the investor did poorly. If you compute a single return from the investor’s cash in and cash out, you measure the investor’s bad timing, not the manager’s skill. If instead you measure the return the fund earned on each rupee for each period it was invested, you measure the manager and ignore the timing.
This is the whole reason two measures exist. The time-weighted return removes the effect of when money entered or left, so it isolates the performance of the investments themselves. The money-weighted return keeps the effect of those flows, so it captures the actual experience of the investor who made them. Cash flows here means every external movement: the first purchase, later purchases, partial or full sales, and distributions taken out rather than reinvested. The word external matters, because income that is reinvested inside the portfolio is not a cash flow in this sense; only money crossing the boundary of the account counts.
Ask first whose performance you want to judge. To compare fund managers on a level field, use the time-weighted return, because it ignores cash-flow timing the manager does not control. To measure what an individual investor actually earned, use the money-weighted return, because their timing is part of their result.
The time-weighted return breaks the whole period into sub-periods at every point where money enters or leaves. Within each sub-period no external cash flow occurs, so a clean holding period return can be computed. Chain-link those sub-period returns, exactly as in the first reading, and the timing of the flows drops out, because each sub-period return is measured on whatever balance happened to be invested at the time.
where r1 through rn are the holding period returns of the sub-periods between cash flows, and n is the number of sub-periods. For a multi-year result, the annualized time-weighted return is found by taking the appropriate root, as with any geometric mean.
Setup. The Meridian Growth Fund starts a year with a balance of 50,000 rupees. Halfway through, after the balance has grown to 55,000, the investor withdraws 5,000, leaving 50,000. By year end the balance is 56,000. Find the time-weighted return for the year.
Answer: 23.2%. The 5,000 withdrawal split the year into two sub-periods but had no effect on the result, because each sub-period return was measured on the balance actually invested. That is the whole point: the manager is judged on the investments, not on the investor’s withdrawal.
In practice the time-weighted return is computed by valuing the portfolio every time a cash flow occurs, and modern systems value it daily, which makes the sub-periods short and the result precise. The more frequently the portfolio is valued, the closer the chained result comes to the true time-weighted return. This is also the measure behind global performance-reporting standards, so when a fund quotes its published return, it is almost always time-weighted.
The time-weighted return is the industry standard for reporting and comparing manager performance, and the exam expects you to know that. The mechanical task is almost always the same: find the sub-period returns around each cash flow, then chain-link. If the sub-periods span more than a year, remember to annualize with a root, not by dividing.
The money-weighted return, also called the dollar-weighted return, is the single rate that makes the present value of every cash flow, the amounts paid in and the amounts received, sum to zero. In other words, it is the internal rate of return on the investor’s own cash flows. Because larger amounts count for more, and amounts committed for longer count for more, this measure is sensitive to both the size and the timing of the flows.
where CFt is the net cash flow at time t (money paid in is negative, money received is positive), MWR is the money-weighted rate of return, and n is the number of periods. The MWR is the rate that makes the equation hold, that is, the internal rate of return.
Setup. An investor pays 10,000 rupees into a portfolio at the start. At the end of year 1 she adds another 5,000. At the end of year 2 the portfolio is worth 17,600, and she sells it all. Find the money-weighted return.
Answer: the money-weighted return is 10%. This is the rate the investor actually compounded her money at, given exactly how much she had invested and for how long. On the exam you would solve this with the internal-rate-of-return function rather than by hand.
The money-weighted return is exactly the internal rate of return met in the time value of money and corporate finance: the single discount rate that equates the value put in with the value taken out. Because it solves a polynomial equation, an unusual cash-flow pattern can in principle produce more than one rate, but the ordinary shape of an initial investment, some additions or withdrawals, and a final value yields one sensible answer. The task on the exam is to lay out the cash flows with the correct signs and let the internal-rate-of-return function find the rate.
When there are no cash flows in or out during the period, the two measures agree. The moment an investor adds or removes money, they can diverge, and the direction of the gap tells a story. If an investor adds money just before a strong period, the money-weighted return rises above the time-weighted return, because more money was present to enjoy the good stretch. If an investor adds money just before a weak period, the money-weighted return falls below the time-weighted return, because more money was exposed to the loss.
Setup. A fund returns positive 25% in the first half of a year and negative 20% in the second half. An investor starts with 10,000 rupees, and at the mid-point, after the gain, adds another 7,500. Compare the time-weighted and money-weighted returns.
Answer: the time-weighted return is 0%, but the money-weighted return is about negative 5.6%. The fund was flat, yet the investor lost money, because she added 7,500 right before the 20% fall. The manager did not fail; the timing did.
| Feature | Time-weighted | Money-weighted |
|---|---|---|
| What it measures | Performance of the investments | Experience of the investor |
| Sensitive to cash-flow size and timing | No | Yes |
| How it is computed | Chain-link sub-period returns | Internal rate of return on cash flows |
| Best used for | Comparing managers on a level field | Judging what an investor actually earned |
The sign of the gap between the two returns is a quick read on cash-flow timing. Money-weighted above time-weighted means money went in before good periods; money-weighted below means money went in before bad ones. The size of the gap grows with the size of the flows relative to the account.
Judging a manager by the money-weighted return. A manager does not control when clients add or withdraw money, so a money-weighted figure can punish good management for bad client timing, or flatter poor management for lucky timing. Use the time-weighted return to compare managers, and reserve the money-weighted return for the investor’s own scorecard.
A portfolio grows from 20,000 to 22,000 in the first sub-period, then a deposit is made, and it grows from 30,000 to 31,500 in the second sub-period. What is the time-weighted return for the whole period?
Sub-period 1: (22,000 − 20,000) ÷ 20,000 = 10%. Sub-period 2: (31,500 − 30,000) ÷ 30,000 = 5%. Chain-link: (1.10)(1.05) − 1 = 1.155 − 1 = 15.5%. The deposit size does not enter the calculation.
An investor’s money-weighted return for the year is well below the fund’s time-weighted return. What does that tell you about the timing of their cash flows?
They had more money invested during the fund’s weaker stretches, almost certainly because they added money just before those stretches. Good timing would push the money-weighted return above the time-weighted return, so a shortfall points to poor timing, not poor management.
A security market index is a single number that represents the value of a defined group of securities, so that its change over time stands in for the performance of that market or segment. An index is the benchmark against which managers and portfolios are judged, and the return an index reports depends heavily on how its members are weighted, which is the focus of the rest of this reading.
The value of an index at any moment is built from the prices of its constituents, combined according to a weighting rule and scaled by a divisor that keeps the series continuous. The return of an index over a period is simply the percentage change in its value, and, as with any asset, that return comes in a price-only version and a total-return version. An index is defined by three choices: which securities are in it, how they are weighted, and how and when it is maintained. Change any one of those and the same underlying market can produce a different index number.
Indexes are not just scorekeeping. They do real work across the investment industry, and the exam expects you to recognize the main uses. First, an index is a benchmark: an active manager’s return is judged against the index for their market, so the index defines what beating the market even means. Second, an index is the blueprint for index funds and exchange-traded funds, which simply hold the constituents in their index weights, letting an investor buy an entire market cheaply and passively. Third, an index is a gauge of market sentiment and economic health, which is why a single number appears in the news each evening. Fourth, indexes serve as proxies for asset classes when setting a portfolio’s allocation, and as the underlying for derivatives such as index futures and options.
You are not asked to memorize the constituents of any particular index. You are asked to match a described purpose to the right idea: benchmarking active managers, building a passive fund, gauging a market, or serving as a derivative underlying. Knowing the uses, and knowing that the weighting scheme drives the reported return, is what the questions test.
A price return index reflects only the prices of its constituents, so it captures capital appreciation alone. A total return index also assumes that dividends and other distributions are reinvested, so it captures both price change and income. The gap between the two versions of the same index grows over time as reinvested income compounds, which is why a long-run total return index sits well above its price-return twin. When someone quotes an index level, it is usually the price version; when they compare long-run performance across markets, the total return version is the fair one to use.
where V0 and V1 are the index values at the start and end of the period, and Inc is the income from the constituents expressed in index points. The price version omits Inc; the total return version includes it.
The heart of index construction is the weighting rule, the recipe that decides how much each constituent contributes to the index. Three schemes dominate, and each answers the question “how much should each company count?” differently. A price-weighted index counts each company in proportion to its share price. A market-capitalization-weighted index counts each in proportion to its total market value. An equal-weighted index counts every company the same, regardless of size or price. The three can be built from the identical set of stocks and still report different returns, because they emphasize different members.
A price-weighted index adds up the prices of its constituents and divides by a number called the divisor. Because the weight of each stock is just its price relative to the sum of prices, a high-priced share carries more weight than a low-priced one, even if the high-priced company is the smaller business. This is the defining quirk, and limitation, of price weighting.
where P1 through PN are the prices of the N constituents and D is the divisor, initially set to the number of constituents and later adjusted to keep the index continuous.
Setup. The Meridian 3, a price-weighted index, holds three shares priced at 100, 60, and 20 rupees, with a divisor of 3. Over the period the first rises 10% to 110, the second rises 10% to 66, and the third falls 5% to 19. Find the index value before and after, and the index return.
Answer: the index rose from 60 to 65, a return of 8.33%. Notice that the 100-rupee share and the 60-rupee share, both up 10%, drove the index far more than the 20-rupee share that fell 5%, purely because their prices are higher. That is price weighting at work.
Assuming the biggest company moves a price-weighted index the most. It is the highest-priced share, not the largest company, that carries the most weight. A small firm with a high share price can outweigh a giant with a low share price, which is exactly why price weighting is considered a weak representation of a market.
A market-capitalization-weighted index counts each company by its total market value, price multiplied by shares outstanding, so larger companies carry more weight. This is the most widely used scheme, because it reflects the actual size of each company in the market and mirrors how much money is invested in each. The weight of each constituent is its market cap divided by the total market cap of the index.
where wi is the weight of constituent i, Pi is its price, Qi is its shares outstanding, and the denominator is the total market capitalization of all constituents. The index return is then the weighted average of the constituent returns, using these weights.
Many providers use a float-adjusted version, counting only the shares actually available to trade rather than those locked away with founders, families, or governments. The logic is unchanged; only the share count in the weight is trimmed to what the market can really buy, which gives a more realistic picture of investable value.
| Company | Price | Shares | Market cap | Weight |
|---|---|---|---|---|
| Northbank | 50 | 10,000,000 | 500m | 33.3% |
| Corvus | 20 | 30,000,000 | 600m | 40.0% |
| Saraswati | 100 | 4,000,000 | 400m | 26.7% |
| Total | – | – | 1,500m | 100% |
Setup. Using the Kaveri Index constituents in Exhibit 2, suppose Northbank returns positive 10%, Corvus positive 5%, and Saraswati negative 4% over the period. Find the market-cap-weighted index return.
Answer: about 4.27%. Corvus, the largest by market value even though it has the lowest price, carried the most weight, which is why market-cap weighting is regarded as the truest reflection of a market’s overall movement.
A market-cap-weighted index is self-adjusting. As a company’s price rises its market cap and its weight rise automatically, so the index never needs manual reweighting to keep pace with size. The trade-off is concentration: a few very large companies can come to dominate, so the index quietly tilts toward whatever is currently most valuable in aggregate.
An equal-weighted index gives every constituent the same weight, one divided by the number of members, regardless of price or size. Its return is therefore the simple average of the constituent returns. This sounds neutral, and in a sense it is, but it has a hidden cost: as prices move, the weights drift away from equal, so the index must be rebalanced regularly by trimming the winners and topping up the laggards to restore the equal split. That rebalancing creates turnover, which in a real fund means trading costs the other schemes largely avoid.
where r1 through rN are the returns of the N constituents, each given the weight 1 ÷ N. The equal-weighted index return is their simple average.
Setup. Take the same three companies as the Kaveri Index, Northbank, Corvus, and Saraswati, returning positive 10%, positive 5%, and negative 4%, but now weight them equally. Find the equal-weighted index return, and compare it with the market-cap result.
Answer: 3.67%, below the 4.27% market-cap figure. Equal weighting gave the small negative-return company as much say as the large positive-return company, which pulled the average down. Equal weighting tilts an index toward its smaller members.
A price-weighted index holds two shares priced at 80 and 40, with a divisor of 2. The first falls to 76 and the second rises to 48. What is the index return?
Start: (80 + 40) ÷ 2 = 60. End: (76 + 48) ÷ 2 = 62. Return: (62 − 60) ÷ 60 = 3.33%. The higher-priced share fell and the lower-priced share rose by the same 4 rupees, so they offset in price terms, but the index still rose because the totals moved up by 4.
No weighting scheme is neutral. Each embeds a bias, and choosing an index means choosing which bias you can live with. Price weighting overweights high-priced shares and is distorted by share splits. Market-cap weighting overweights the largest companies and can become concentrated in a handful of giants, so a rally or slump in a few mega-caps can swing the whole index. Equal weighting overweights small companies and demands frequent rebalancing, which adds turnover and cost. Knowing these tendencies lets you read any index return correctly rather than treating it as an objective truth about the market.
| Scheme | Weight is based on | Built-in bias | Rebalancing need |
|---|---|---|---|
| Price-weighted | Share price | Toward high-priced shares | Divisor changes for splits |
| Market-cap-weighted | Total market value | Toward the largest companies | Low, self-adjusting |
| Equal-weighted | Equal for all | Toward smaller companies | High, frequent rebalancing |
Why does an equal-weighted index require frequent rebalancing while a market-cap-weighted index does not?
In an equal-weighted index the winners grow to more than their target weight and the losers shrink below it, so the manager must sell winners and buy losers to restore equal weights. In a market-cap-weighted index the weights move with prices automatically, so a rising company’s higher weight is exactly what the scheme wants, and no manual reset is needed.
An index is not a fixed list. Two maintenance actions keep it representative. Rebalancing resets the weights back to the scheme’s target, most often in an equal-weighted index whose weights have drifted. Reconstitution changes the membership itself, adding companies that now qualify and removing those that no longer do, usually on a scheduled review. Both actions must be done without creating a false jump in the index value, and that is the job of the divisor.
Whenever something changes that is not a genuine market move, a share split, a company entering or leaving, the divisor is recomputed so that the index value is the same immediately before and immediately after the change. Only real price movements should move the index; mechanical changes should not. Reconstitution also matters for anyone tracking the index, because a fund built on it has to trade to match the new membership, which is a known source of turnover around review dates.
Setup. Return to the Meridian 3 price-weighted index, valued at 65 with constituent prices 110, 66, and 19 and a divisor of 3. The first company does a two-for-one share split, halving its price to 55. Find the new divisor that keeps the index at 65.
Answer: the divisor falls from 3 to about 2.1538. The split cut the share price in half, which is not a loss of value for investors, so the divisor absorbs the change and the index holds at 65. Without this adjustment the index would drop sharply for no real reason.
Which return measure, time-weighted or money-weighted, would a mutual fund use to report its performance to the public, and why?
The time-weighted return. The fund cannot control when investors buy and sell its units, so it reports the return the portfolio itself earned, which is comparable across funds. A money-weighted figure would reflect the timing of one particular investor’s flows, which is not a fair basis for comparison.
The time-weighted return measures how the investments performed, ignoring when money was added or removed. The money-weighted return measures what the investor actually earned, giving full weight to the size and timing of their cash flows. One judges the manager, the other judges the investor.
Because a manager does not control when clients deposit or withdraw money. The time-weighted return strips that timing out, so two managers can be compared on the performance of their portfolios alone, which is the only thing they actually control.
When the investor happened to have more money invested during the stronger periods, usually because they added funds just before a good stretch. If instead they added money just before a weak stretch, the money-weighted return comes in below the time-weighted return.
Because the weight of each stock is its price divided by the sum of all prices. A share priced at 200 counts ten times as much as one priced at 20, regardless of which company is larger. This is why price weighting is considered a poor reflection of a market’s true size.
Because each stock’s weight moves automatically with its price. When a company’s shares rise, its market value and therefore its index weight rise on their own, with no trade required. The scheme keeps up with company size without manual rebalancing.
Because as prices move the weights drift away from equal, the winners becoming overweight and the losers underweight. Restoring the equal split means selling winners and buying losers on a regular schedule, which adds turnover and cost that the other schemes avoid.
A split lowers a share price without changing the company’s value, so leaving the index formula unchanged would make the index drop for no real reason. The divisor is recomputed so the index reads the same right before and right after the split, letting only genuine price moves affect it.
Through direct calculation: chain-linking sub-period returns for a time-weighted figure, setting up the internal rate of return for a money-weighted figure, and computing index values and returns under each weighting scheme. Expect at least one question asking which measure or which scheme fits a described situation.
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