Quantitative Methods · Reading 3

Benchmarking Returns

CFA Level I · Quantitative Methods · Reading 3 · about 49 min

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What you'll learn

LOS 3.a Calculate and compare money-weighted and time-weighted rates of return and judge which one fits a given performance-evaluation situation.
LOS 3.b Describe index construction choices and weighting methods (price, equal, market-capitalization, float-adjusted, fundamental), and calculate and interpret index values, divisors, weights, and price and total returns, including rebalancing and reconstitution.

Module 3.1

Time-Weighted and Money-Weighted Returns

This reading compares the money-weighted return, the IRR of a portfolio's cash flows, with the time-weighted return, which links subperiod returns and is unaffected by deposits and withdrawals. It then covers how a security market index is built and maintained, how price, equal, market-capitalization and fundamental weighting differ, and how to calculate an index's value, price return and total return.

LOS 3.a — Money-weighted vs. time-weighted returns

When money moves into or out of a portfolio during the evaluation period, "the portfolio's return" can mean two different things: the growth of the investor's money, given when it was put in, or the growth that the manager's decisions produced on each dollar invested. The money-weighted rate of return measures the first and the time-weighted rate of return the second.

1. Money-weighted rate of return (MWR)

The money-weighted return (also called the dollar-weighted return) is the internal rate of return (IRR) of the portfolio's cash flows. It is the single discount rate that gives all the flows a net present value (NPV) of zero:

Key concept

t = 0 \sum T \frac{C F _{t}}{( 1 + M W R ) ^{t}} = 0

Steps:

Choose a period length, usually one year, such that every deposit and withdrawal falls at the start or end of a period.
Write down the net cash flow on every date from $t = 0$ to the end, with the signs in the table below. Enter 0 for a date on which no money moves in or out. The calculator treats the entries as equally spaced, so leaving out an empty date would move every later cash flow one period earlier.
Solve for the IRR, for example with the calculator's cash flow worksheet (keystrokes below).

Sign convention (from the investor's point of view; the account's point of view simply reverses every sign):

Key concept

Item	Sign
Beginning value / initial purchase	outflow (−)
Additional deposits or share purchases	outflow (−)
Dividends or income withdrawn (not reinvested), withdrawals	inflow (+)
Ending value / sale proceeds	inflow (+)

A dividend received on the same date as a new purchase is netted with it into one cash flow. For example, a dividend of 2 and a purchase of 50 give $C F = - 48$ . The IRR is unchanged if every sign is flipped, as long as money going in and money coming out carry opposite signs.

Calculator (TI BA II Plus): [CF] [2ND] [CLR WORK], enter CF0, C01, C02 … (each frequency = 1), then [IRR] [CPT]. The resulting IRR is a rate per period. If the periods are years, it is already an annual rate.

2. Time-weighted rate of return (TWR)

The time-weighted return measures compound growth: the rate at which one unit of money invested at the start would have grown. Deposits and withdrawals do not affect it. Steps:

Value the portfolio immediately before each significant deposit or withdrawal, and split the evaluation period into subperiods at those dates.
Calculate each subperiod's holding period return (HPR). When there is a cash flow, the beginning value of the next subperiod is the value after that deposit or withdrawal:

H P R_{i} = \frac{V _{e n d, i} - V _{b e g in, i} + D _{i}}{V _{b e g in, i}}

Geometrically link the subperiod returns. If the total period is longer than one year, take the geometric mean to annualize, using the number of years (not the number of subperiods):

T W R_{ann u a l} = [(1 + H P R_{1}) (1 + H P R_{2}) \dots (1 + H P R_{n})]^{1/ years} - 1

For a single share bought and later added to, the per-share HPRs can be used directly: $H P R_{1} = (P_{1} - P_{0} + D_{1}) / P_{0}$ , and so on.

3. Example

An investor opens an account with $1,000. One year later the account is worth $1,150 (+15%), and she deposits another $500, bringing it to $1,650. At the end of year 2 the account is worth $1,782 (+8% in year 2).

TWR: $[(1.15) (1.08)]^{1/2} - 1 = 11.45%$ per year.
MWR: solve $- 1, 000 - \frac{500}{1 + r} + \frac{1 , 782}{( 1 + r ) ^{2}} = 0$ , which gives $r = 10.81%$ . (CF0 = −1,000; C01 = −500; C02 = +1,782.)

The MWR is lower than the TWR because the extra $500 was invested just before the weaker year. The MWR gives more weight to periods in which more money is invested.

4. Comparing the two measures

Key concept

Feature	Money-weighted return	Time-weighted return
Method	IRR of all cash flows	Geometric linking of subperiod HPRs
Affected by the timing and size of external cash flows?	Yes	No
Deposit made just before strong performance	MWR higher than TWR	unaffected
Deposit made just before poor performance	MWR lower than TWR	unaffected
No external cash flows during the period (no deposits, withdrawals or paid-out income)	equal to TWR	equal to MWR
Best use	Manager (or investor) controls the cash flows	Manager does not control the cash flows

The investment management industry prefers the time-weighted return because managers usually do not decide when clients add or withdraw money. The TWR isolates the manager's skill in selecting investments.
The Global Investment Performance Standards (GIPS) are voluntary. A firm that claims compliance must present performance using time-weighted returns, except in specific situations where money-weighted returns are more appropriate. These require that the manager controls external cash flows and, in addition, that the portfolio is closed-end, has a fixed life, calls committed capital from investors, or holds mainly illiquid investments (typical of private equity).
Withdrawals work the opposite way to deposits. Taking money out just before a strong period lowers the MWR relative to the TWR.

Common exam traps

Leaving out dividends that are paid out and not reinvested. For the MWR they are inflows on their payment date. For the TWR they belong in the numerator of that subperiod's HPR.
Computing a TWR subperiod return without first adding the deposit just made to (or removing the withdrawal from) the beginning value, which mixes new money with investment results. In the example, measuring year 2 from $1,150 instead of the $1,650 after the deposit gives 54.96% instead of 8%.
Annualizing over the number of subperiods instead of the number of years. Three quarterly subperiods inside one year need no root.
Averaging subperiod returns arithmetically. The TWR links them geometrically. In the example, $(15% + 8%) /2 = 11.50%$ instead of the TWR of 11.45%.
The "required return" of an investor is irrelevant to both calculations. It is a hurdle rate for judging the result.
If the subperiods are of equal length and each earns the same rate, both measures equal that rate whatever the cash flows. Differences come from cash flows combined with uneven subperiod returns.

Exam shortcuts

With no deposits, withdrawals or paid-out income during the period, the MWR equals the TWR, so no IRR has to be computed.
A deposit made just before the stronger subperiod puts the MWR above the TWR, and one made just before the weaker subperiod puts it below; withdrawals work the opposite way, so comparison questions need no IRR.
If the subperiods are of equal length and each earns the same rate, the MWR and the TWR both equal that rate whatever the cash flows.

Bottom line

The money-weighted return is the IRR of the portfolio's cash flows, the rate that gives them an NPV of zero, with money going in and money coming out entered with opposite signs.
The time-weighted return geometrically links the holding period returns of subperiods formed at each significant deposit or withdrawal, and is annualized with a root equal to the number of years, not the number of subperiods.
The MWR depends on the timing and size of external cash flows: a deposit just before strong performance makes it higher than the TWR and a deposit just before poor performance makes it lower, while the TWR is unaffected.
The MWR and the TWR are equal when there are no external cash flows during the period.
The investment management industry prefers the TWR because managers usually do not control when clients add or withdraw money; the MWR suits a manager or investor who controls the cash flows.
Firms claiming compliance with the voluntary GIPS must present time-weighted returns, except where the manager controls external cash flows and the portfolio is also closed-end, has a fixed life, calls committed capital or holds mainly illiquid investments.

Quick check

Question 1Core

Hannah Kowalczyk makes the following trades in Silverleaf Foods shares, with no taxes or transaction costs:

Today: buys one share at $30.00.
One year from today: buys a second share at $36.00.
Two years from today: sells both shares at $41.00 each.

Silverleaf pays a dividend of $1.50 per share at the end of year 1 and $2.00 per share at the end of year 2. Dividends are not reinvested. Kowalczyk's required return is 10%. Her money-weighted rate of return is closest to:

Show answer and explanation

Correct answer: B

The money-weighted return is the internal rate of return of the cash flows the investor actually pays and receives. At the end of year 1 the $1.50 dividend on the first share is netted against the $36.00 purchase. At the end of year 2 she receives the sale proceeds and the dividends on two shares. The 10% required return is not an input.

t	Cash flow
0	−30.00
1	+1.50 − 36.00 = −34.50
2	2 × 41.00 + 2 × 2.00 = +86.00

- 30.00 - \frac{34.50}{1 + r} + \frac{86.00}{( 1 + r ) ^{2}} = 0 \Rightarrow r = 0.2131 \approx 21.3%

TI BA II Plus: [CF] [2ND] [CLR WORK]; CF0 = −30; C01 = −34.5, F01 = 1; C02 = 86, F02 = 1; [IRR] [CPT] = 21.31.

Why the other options are wrong

A. 15.9% is the IRR after ignoring both dividends (CF1 = −36, CF2 = +82). Dividends that are paid out are cash received by the investor.
C. 22.2% is the time-weighted return: $H P R_{1} = (36 - 30 + 1.5) /30 = 25.0%$ and $H P R_{2} = (41 - 36 + 2) /36 = 19.4%$ , so $1.250 \times 1.194 - 1 = 22.2%$ . This measure ignores the fact that more money was invested in year 2.

Key takeaway For an MWR share problem, net the dividend received against any purchase on the same date, and include dividends on all shares held at the end.

Module 3.2

Security Market Indexes

LOS 3.b — Security market indexes: construction, weighting and returns

1. What an index is and how its return is measured

A security market index represents the performance of an asset class, a security market or a segment of a market. It is built as a portfolio of constituent securities. Its value at any time is calculated from the market prices of those securities. The percentage change in the index value over a period is the index return.

Key concept

Version	What it uses	Return it produces
Price index (price return index)	Constituent prices only	Price return
Return index (total return index)	Prices plus income (dividends, interest, other distributions), assumed reinvested	Total return

When constituents pay cash distributions, the total return is greater than the price return, and a total return index cannot grow more slowly (or fall faster) than the matching price index. With no distributions, the two are identical.

Returns for successive periods are geometrically linked:

Key concept

R_{period} = (1 + R_{1}) (1 + R_{2}) \dots (1 + R_{n}) - 1 V_{n} = V_{0} (1 + R_{1}) (1 + R_{2}) \dots (1 + R_{n})

Example. Monthly index returns of +1.8%, −0.6% and +2.1% link to $(1.018) (0.994) (1.021) - 1 = 3.31%$ . Their sum, 3.3%, is only an approximation. Starting from 500, the index ends at $500 \times 1.0331 = 516.6$ .

2. Building an index: five decisions, in order

Target market / objective: which market or segment the index should measure. This is always the first decision, and it drives everything else.
Constituent securities: the selection criteria and the securities that meet them.
Weighting method: price, equal, market-capitalization (possibly float-adjusted) or fundamental weighting.
Calculation: the starting index value and how the index is computed on each later date (divisor and base value).
Maintenance: rebalancing, reconstitution and periodic review of the method.

Common index types by target market (preview of Reading 41, LOS 41.d):

Type	What it covers
Broad-based equity index	A whole market: its largest listed companies, usually float-adjusted and market-cap-weighted. A composite market index measures the overall market and usually holds more than 90% of its total value.
Multi-market index	Several national markets (e.g. a regional, emerging-market or global index)
Sector index	One industry or economic sector (e.g. financials, health care), in one country or globally
Investment theme index	Stocks tied to an investor trend beyond industry or company size (e.g. an ESG-screened index)

3. Weighting methods

Price-weighted index

A price-weighted index is the sum of the constituent prices divided by a divisor:

Index = \frac{\sum P _{i}}{D} w_{i} = \frac{P _{i}}{\sum P _{j}}

At launch, $D = \sum P_{i} / base value$ .
The divisor must be changed whenever a stock split, reverse stock split or stock dividend occurs, or when the constituents change, so that the event alone does not move the index. A stock dividend works like a split stated as a percentage: after a 10% stock dividend, one share becomes 1.1 shares. Unlike a cash dividend, it pays out nothing and leaves each holder's ownership unchanged. For a split, recompute the sum with the post-split price and solve $D_{n e w} = \sum P_{n e w} / index value$ .
Higher-priced stocks have more weight. A 10% move in the highest-priced stock moves the index most, whatever the company's size.
A portfolio holding an equal number of shares of each constituent replicates the index.
It is simple to compute, but a share price says nothing about a company's value, because share counts differ widely. It is therefore a suitable benchmark mainly for a portfolio whose results depend disproportionately on the price changes of particular securities. Examples: the DJIA (30 US stocks) and the Nikkei 225.

Example. Prices of $14, $38 and $73 (sum $125) with a base value of 100 give $D = 1.25$ . If the $73 stock splits 2-for-1, the sum becomes $14 + 38 + 36.50 = 88.50$ , and the new divisor is $88.50/100 = 0.885$ .

Equal-weighted index

In an equal-weighted index, every constituent gets the same weight $1/ N$ at each rebalancing.

At inception, each stock is given the same reference value. The number of shares held is reference value ÷ price. The aggregate reference value is $N \times$ reference value, and $D = aggregate reference value / base value$ .
For a period that starts at equal weights, the index return is the arithmetic mean of the constituent returns. The geometric mean does not apply, because the stocks are held side by side.
Weights drift as prices move. Winners grow above $1/ N$ and losers fall below it. To restore equal weights, the index must be rebalanced frequently, which means selling winners and buying losers. This is costly for a tracking portfolio.
Small companies are overweighted relative to their size. The index is a suitable benchmark for a strategy that limits exposure to the largest companies, or for a fund that invests equal amounts in each holding. Examples: Value Line Composite Average, FT Ordinary Share Index.

Market-capitalization-weighted (value-weighted) index

In a market-capitalization-weighted index, each constituent's weight is its share of the total market value of the constituents:

w_{i} = \frac{P _{i} N _{i}}{\sum P _{j} N _{j}} Index = \frac{\sum P _{i} N _{i}}{D}, D = \frac{base-period market cap}{base value}

Stock splits and stock dividends do not change a company's market value, so an ordinary stock split or stock dividend by itself leaves the stock's weight and the divisor unchanged. The divisor is adjusted when constituents are added or removed (reconstitution). The new divisor is the new total market cap divided by the unchanged index value.
It tracks changes in aggregate investor wealth. A tracking portfolio rebalances itself automatically, and turnover is low.
The main drawback is that a stock's weight rises as its price rises. Possibly overvalued stocks get more weight and possibly undervalued stocks get less.
It is the usual benchmark for a passive strategy that aims to earn the market return (e.g. S&P 500).

Float adjustment

The market float is the value of shares actually available to the investing public. It excludes shares held by controlling (strategic) shareholders, and often those held by other corporations or governments. Free float is narrower still, because it also excludes shares that foreign investors cannot buy. A float-adjusted market-capitalization-weighted index weights each firm by float-adjusted capitalization = market cap × float %. Firms with large controlling holdings get less weight than in an unadjusted index. Float adjustment belongs to market-cap weighting only.

Fundamental-weighted index

In a fundamental-weighted index, weights come from company fundamentals like earnings, dividends, cash flow, sales or book value. These weights are not driven by the share price, although over long periods they are related to it. Such an index has a value tilt: compared with market-cap weighting, it overweights stocks with low P/E or high book-to-market ratios. It suits value-oriented and other fundamentals-driven active strategies. Its advantage is that it looks beyond prices and market values. Its drawback is that it may leave out firms without positive earnings or without dividends, which can bias it toward or away from some sectors. Fundamental weights also drift, so the index needs rebalancing.

Key concept

Method	Weight driven by	Adjust divisor for splits?	Rebalancing need	Typical bias
Price	Share price	Yes	Automatic	Toward high-priced shares
Equal	$1/ N$	No	High	Toward small caps
Market cap / float-adjusted	Market value	No	Automatic	Toward large and recently risen stocks
Fundamental	Earnings, book value, etc.	No	Periodic	Value tilt

4. Example: price return vs. total return

A market-cap-weighted index has a base value of 1,000 when constituent capitalization is $500 million, so $D = 500, 000$ . One period later the capitalization is $515 million, and constituents paid $6 million of dividends.

Price index value $= 515, 000, 000/500, 000 = 1, 030$ , so the price return is 3.0%.
Total return index value $= (515 + 6) million /500, 000 = 1, 042$ , so the total return is 4.2%.
The capital distribution return is $6/500 = 1.2%$ .

From per-share data, compute each constituent's market value as price $\times$ shares at the start and at the end, and its dividends as dividend per share $\times$ shares. The market-cap total return is then (ending value $+$ dividends $-$ beginning value) $\div$ beginning value.

5. Maintenance: rebalancing vs. reconstitution

Rebalancing means resetting the constituents' weights to target once price moves have shifted them, usually every quarter. It matters mainly for equal-weighted and fundamental-weighted indexes. When an equal-weighted index is rebalanced, the total value (and the index value) is left unchanged. Each position is simply reset to total ÷ N. For example, with two holdings worth 6,300 and 5,500, each is reset to 5,900: sell 400 of the first and buy 400 of the second. Selling recent winners and buying recent losers runs against a trend-following approach.
Reconstitution means changing which securities are in the index. Securities are deleted when they no longer meet the criteria (for example, they fall out of the size range), mature, are acquired, go bankrupt or are delisted. They are replaced by securities that do meet the criteria. The divisor is adjusted on that date so the index value does not jump.
Prices react. An added security tends to rise as index-tracking managers buy it, and a deleted security tends to fall as they sell it. The weights of the remaining constituents are also adjusted to fit the weighting scheme.
Changing the calculation formula or the rules is neither rebalancing nor reconstitution.

Common exam traps

Confusing the aggregate dollar value, the divisor and the index value.
After a split in a price-weighted index, dividing every price by the split ratio or keeping the old divisor. In the example, keeping $D = 1.25$ after the split gives an index of $88.50/1.25 = 70.80$ instead of 100.
Changing a stock's market-cap weight, or the divisor of a market-cap-weighted index, because of a split.
Compounding the constituent returns of an equal-weighted index within one period, or adding index returns across periods instead of linking them.
Leaving income out of a total return, or adding it to a price return.
Finding a weight after price moves by growing the old weight by the stock's own return. Divide each position's new value by the new total.
Weighting a float-adjusted index by floating share counts instead of float-adjusted market value (market cap × float %).

Exam shortcuts

For a period that starts at equal weights, an equal-weighted index return is the arithmetic mean of the constituent returns, so no share counts or divisor are needed.
Only a price-weighted index changes its divisor for a stock split or stock dividend; equal-weighted, market-cap-weighted and fundamental-weighted indexes leave it unchanged.

Bottom line

A price index uses constituent prices only and gives the price return; a total return index adds income, assumed reinvested, and its total return exceeds the price return when constituents pay distributions.
A price-weighted index is the sum of prices divided by a divisor that is changed for splits, reverse splits, stock dividends and constituent changes, and its higher-priced stocks carry more weight.
An equal-weighted index gives each constituent $1/ N$ at each rebalancing, overweights small companies relative to their size and needs frequent, costly rebalancing as weights drift.
A market-capitalization-weighted index weights each stock by its share of total market value, tracks aggregate investor wealth and rebalances automatically, but gives more weight to stocks whose prices have risen; a float-adjusted version uses market cap × float %.
A fundamental-weighted index weights constituents by fundamentals such as earnings, dividends, cash flow, sales or book value and has a value tilt.
Rebalancing resets weights to target after price moves and matters mainly for fundamental-weighted and equal-weighted indexes; reconstitution changes the constituents, with the divisor adjusted on that date.

Quick check

Question 2Core

Which of the following statements about a total return index is most accurate?

Show answer and explanation

Correct answer: B

The value of a total return index grows by compounding its periodic total returns: $V_{n} = V_{0} (1 + R_{1}) (1 + R_{2}) \dots (1 + R_{n})$ . Each periodic total return includes both price changes and income such as dividends and interest.

Why the other options are wrong

A. An index whose value reflects only price changes is a price return index. A total return index also includes the income (dividends, interest, other distributions) paid by the constituents.
C. Income paid by the constituents can never be negative. A total return index therefore always rises at least as fast as (or falls no faster than) the matching price return index.

Key takeaway Total return index ≥ price return index; values compound by geometric linking.

Practice Questions

Question 3Core

The prices of the stocks in a price-weighted index add up to $270, and the divisor is 1.80. One constituent, trading at $90, then completes a 3-for-1 stock split. The divisor that should be used immediately after the split is closest to:

Show answer and explanation

Correct answer: B

A stock split must not change the value of a price-weighted index. The divisor is reset so that the new sum of prices, with the splitting stock at its post-split price, gives the same index value as before.

Index value before the split: $270/1.80 = 150.0$

Post-split price of the stock: $90/3 = $30$ , so the sum of prices falls by $60 to $210.

D_{n e w} = \frac{210}{150.0} = 1.40

Why the other options are wrong

A. 0.60 treats every constituent as splitting 3-for-1 ( $270/3 = 90$ , and $90/150 = 0.60$ ). Only the one stock's price is divided by 3.
C. 1.80 is the old divisor. If it were kept, the index would drop to $210/1.80 = 116.7$ purely because of the split.

Key takeaway Price-weighted index + split: recompute the sum with the new price, keep the index value, solve for the new (smaller) divisor.

Question 4Core

An equal-weighted index contains the three stocks shown in the table below. After the period's price changes, and before any rebalancing, Kestrel Aerospace's index weight is closest to:

Equal-weighted index: reference values and price returns for the period

Stock	Reference value at start	Price return for the period
Kestrel Aerospace	$25,000	+12%
Linden Packaging	$25,000	+4%
Oakmere Retail	$25,000	−6%

Show answer and explanation

Correct answer: C

An equal-weighted index starts with the same weight in each stock. After prices move, each stock's weight is its new value divided by the new total, until the index is rebalanced. Stocks that outperformed now carry more than one-third.

Stock	Start value	Return	End value	End weight
Kestrel	25,000	+12%	28,000	36.1%
Linden	25,000	+4%	26,000	33.5%
Oakmere	25,000	−6%	23,500	30.3%
Total	75,000		77,500	100%

w_{K es t r e l} = \frac{28 , 000}{77 , 500} = 36.1%

A tempting shortcut is to grow the starting one-third weight by Kestrel's own return: $33.33% \times 1.12 = 37.33%$ . That is wrong because it keeps the old 75,000 total in the denominator; the new weight must divide the new position value by the new total value (77,500).

Why the other options are wrong

A. 33.3% is the starting weight. It assumes the weights stay equal after price changes, but equal weights are only restored when the index is rebalanced.
B. 34.8% drops the minus sign on Oakmere's −6% return, so the new total becomes $25, 000 \times (1.12 + 1.04 + 1.06) = 80, 500$ and $28, 000/80, 500 = 34.8%$ . Oakmere lost value, so the total is only 77,500.

Key takeaway Between rebalancing dates, equal-weighted index weights drift toward the winners. New weight = new position value ÷ new total.

Question 5Core

The sponsor of the Harrowgate Midcap 200 Index removes Velden Tools, which was delisted after filing for bankruptcy, and adds Corvin Optics, which now meets the index's size criteria. This change is best described as:

Show answer and explanation

Correct answer: A

Reconstitution changes the index's membership. It is needed when a constituent ceases to exist (for example after a merger, a bankruptcy or delisting, or a bond maturing) or no longer meets the index criteria, and replacements that meet the criteria are added.

Why the other options are wrong

B. Rebalancing resets the weights of existing constituents to their target weights. It does not swap one security for another.
C. Redefinition is not one of the index maintenance steps. The index's objective and rules stay the same; only its membership changes.

Key takeaway Membership change = reconstitution; weight reset = rebalancing.

This reading has 61 questions in the full bank. Practice all of them.

Key Takeaways

For an MWR share problem, net the dividend received against any purchase on the same date, and include dividends on all shares held at the end.
Total return index ≥ price return index; values compound by geometric linking.
Price-weighted index + split: recompute the sum with the new price, keep the index value, solve for the new (smaller) divisor.
Between rebalancing dates, equal-weighted index weights drift toward the winners. New weight = new position value ÷ new total.
Membership change = reconstitution; weight reset = rebalancing.