Learning Module 3 of Quantitative Methods 11–14% of the exam

Benchmarking Returns

5 lessons. About 32 minutes in cram mode, 144 minutes in full.

Start with lesson 1 Learn with Sensei Read the module summary Practice this module

Lessons

1
Money-weighted vs time-weighted rate of return
calculate 6 min cram, 30 min full
2
Index value, price return and total return
calculate 5 min cram, 22 min full
3
Price-weighted and equal-weighted indexes
calculate 8 min cram, 32 min full
4
Market-cap, float-adjusted and fundamental weighting
calculate 8 min cram, 34 min full
5
Rebalancing, reconstitution and what indexes are used for
describe 5 min cram, 26 min full

What the exam expects you to do

Calculate and compare money-weighted and time-weighted rates of return.Money-weighted vs time-weighted rate of return
Describe the choices and the implications of the different weighting methods used in index construction and management, and calculate, interpret, and explain the value and the returns of an index.Index value, price return and total return

Module summary

The whole module on one sheet, organised by lesson: what to memorize, how the ideas fit together, the formulas and the key terms. No paragraphs, just what you need to revise.

Hide what I know

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

Money-weighted vs time-weighted rate of return

Known

MemorizeMoney-weighted vs time-weighted rate of return

	Money-weighted	Time-weighted
What it is	internal rate of return of the net flows	sub-period returns chained
Size and timing of flows matter?	Yes	No
Use to judge	the investor's own result; a manager who controls the flows (private equity)	a manager who does not control the flows
Global Investment Performance Standards	allowed only if the manager controls flows AND the fund is closed-end, fixed-life, committed-capital or illiquid	required otherwise
Sub-annual	annualize: $(1 + r_{MW})^k - 1$	chain all sub-periods in the year

Memory hook Money-weighted = my timing. Time-weighted = the manager.

Know it cold Money added before a strong period → money-weighted above time-weighted. Before a weak period → below. No flows → equal.

Key terms

money-weighted rate of return Memorize
dollar-weighted rate of return Know the idea
time-weighted rate of return Memorize
internal rate of return Use it
external cash flows Use it
Global Investment Performance Standards (GIPS) Use it
daily valuation Know the idea

Which rate of return under the Global Investment Performance Standards?

Does the manager control external cash flows?the first test

No Memorize time-weighted rate of return
Yes Memorize and the fund has one of four features
- Closed-end money-weighted allowed
- Fixed life money-weighted allowed
- Committed capital called on demand money-weighted allowed
- Significant illiquid holdings money-weighted allowed

Formulas

Money-weighted rate of return Use it

0 = \sum_{t=0}^{T} \frac{CF_t}{(1 + r_{MW})^t}

Quick example

Flows of −80 at t = 0, −100 at t = 1 and +216 at t = 2. The internal rate of return of those flows, and so the money-weighted rate of return, is 13.30% a year.

On your calculator

CF2NDCLR WORK open the cash flow worksheet and clear it
80+|-ENTER↓ CF0 = −80
100+|-ENTER↓ C01 = −100, and F01 stays at 1
↓216ENTER C02 = 216
IRRCPT 13.30

CF cash flow worksheet · 2ND second function (the shifted label above a key) · CLR WORK clear the worksheet · +|- change sign · ENTER enter (on the HP, puts the number on the stack) · IRR internal rate of return · CPT compute

fCLEAR REG clear the cash flow registers
80CHSgCF0 the first flow
100CHSgCFj t = 1
216gCFj t = 2
fIRR 13.30

f gold shift key · CLEAR REG clear all registers · CHS change sign · g blue shift key · CF0 cash flow at time zero · CFj cash flow j · IRR internal rate of return

Time-weighted rate of return Memorize

r_{TW} = (1 + r_1)(1 + r_2)\cdots(1 + r_T) - 1

Quick example

Quarterly returns of 5%, −5%, 5% and 4% between cash flows. 1.05 × 0.95 × 1.05 × 1.04 − 1 = 8.93% for the year.

On your calculator

1.05×0.95×1.05×1.04−1= 0.0893

1.05ENTER0.95×1.05×1.04×1− 0.0893

ENTER enter (on the HP, puts the number on the stack)

Time-weighted rate of return over several years (annualized) Memorize

r_{TW} = \left[(1 + r_1)(1 + r_2)\cdots(1 + r_T)\right]^{1/T} - 1

Quick example

Year 1: 80 to 100 (+25%). Year 2: 100 to 108 (+8%). (1.25 × 1.08)^(1/2) − 1 = 16.19% a year.

On your calculator

1.25×1.08= year 1 × year 2 growth, 1.35
√x−1= two years, so the square root: 0.1619

√x square root

1.25ENTER1.08× 1.35
g√x1− 0.1619

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Annualizing a sub-period money-weighted rate of return Use it

r_{MW,\,annual} = (1 + r_{MW})^{k} - 1

Quick example

A quarterly internal rate of return of 2.0371% annualizes to 1.020371⁴ − 1 = 8.40%, not 4 × 2.0371% = 8.15%.

On your calculator

1.020371y^x4−1= 0.0840

y^x power

1.020371ENTER4y^x1− 0.0840

ENTER enter (on the HP, puts the number on the stack) · y^x power

Index value, price return and total return

Known

MemorizeIndex value and returns

Measure	What it is	Formula
Index value	aggregate ÷ divisor	$V = \text{aggregate} \div D$
Price return	price changes only	$(V_1 - V_0) \div V_0$
Total return	price changes + reinvested income	$(V_1 - V_0 + Inc) \div V_0$
From constituents	start weights × returns	$\sum w_i r_i$

Memory hook Total = price + income, always over the START value. Weights are start-of-period weights.

Know it cold Divisor: set for a base of 100 or 1,000, changed only for non-investment events. Total return index ≥ price index whenever income is paid.

Key terms

security market index Use it
constituent securities Know the idea
index construction steps Know the idea
divisor Use it
aggregate reference value Know the idea
price return Memorize
price return index Use it
price index Know the idea
total return Memorize
total return index Use it

Five steps to an index

Building an indexeach step changes the return

Objective Know the idea the market or sector it represents
Selection of constituents Know the idea by predefined criteria
Weighting method Memorize decides how each price change moves the index
- Price
- Equal
- Market cap (float-adjusted)
- Fundamental factor
Calculation Know the idea a start date and a start value
Maintenance Know the idea rebalancing, reconstitution, corporate actions and periodic review of the rules

Two returns from the same constituents

Return on an indexone set of securities, two published versions

Price return Memorize price changes only, shown by the price return index
- Also called price index
Total return Memorize price changes plus income, shown by the total return index
- Price return change in level ÷ start level
- Income return dividends and interest ÷ start level

Formulas

Index value Use it

V_t = \frac{\text{aggregate reference value}_t}{D}

Quick example

Constituents worth 10,800 in total and a divisor of 10.8 give an index of 1,000. When they are worth 11,360 the index is 11,360 ÷ 10.8 = 1,051.85.

On your calculator

11360÷10.8= 1051.85

11360ENTER10.8÷ 1051.85

ENTER enter (on the HP, puts the number on the stack)

Index price return Memorize

r_{price} = \frac{V_1 - V_0}{V_0}

Quick example

A price return index moves from 2,000 to 2,060. (2,060 − 2,000) ÷ 2,000 = 3.00%.

On your calculator

2060−2000=÷2000= 0.03

2060ENTER2000−2000÷ 0.03

ENTER enter (on the HP, puts the number on the stack)

Index total return Memorize

r_{total} = \frac{V_1 - V_0 + Inc}{V_0}

Quick example

Index 2,000 to 2,060 with 24 points of income (capital distributions). (2,060 − 2,000 + 24) ÷ 2,000 = 4.20%.

On your calculator

2060−2000+24=÷2000= 0.042

2060ENTER2000−24+2000÷ 0.042

ENTER enter (on the HP, puts the number on the stack)

Index return from constituent returns Memorize

r_{index} = \sum_i w_i r_i

Quick example

Weights 50%, 30% and 20%. Price returns of 6%, 2% and −4% give 2.8%. Adding dividend yields of 2%, 1% and 3% makes total returns of 8%, 3% and −1%, so the index total return is 0.50 × 8 + 0.30 × 3 + 0.20 × (−1) = 4.7%.

On your calculator

(0.5×8)+(0.3×3)−(0.2×1)= 4.7

0.5ENTER8×0.3ENTER3×+0.2ENTER1×− 4.7

ENTER enter (on the HP, puts the number on the stack)

Price-weighted and equal-weighted indexes

Known

MemorizePrice weighting versus equal weighting

	Price-weighted	Equal-weighted
Weight of i	$P_i \div \sum P$	$1 \div N$
Index value	$\sum P \div D$ (base 100 or 1,000)	$\sum n_i P_i \div D$
Return	change in $\sum P$ ÷ start $\sum P$	average of returns since the last rebalancing
Bias	high-priced shares; splits shift weights	small companies overweighted
Upkeep	new divisor after a split	frequent rebalancing

Memory hook Price: one share of each, so the dearest share leads. Equal: the same money in each, so a small firm counts like a giant.

Know it cold After a split, new divisor = new sum of prices ÷ old index value, so the index does not move. A split matters to a price-weighted index, not to an equal-weighted one.

Key terms

price weighting Memorize
divisor Memorize
stock split Use it
equal weighting Memorize

Price and equal weighting at a glance

Two simple weightingsvalue = aggregate ÷ divisor

Price-weighted Memorize aggregate = sum of one share of each
- Weight price ÷ sum of prices
- Return change in the sum of prices
- Upkeep new divisor after splits and swaps
Equal-weighted Memorize aggregate = equal holdings n_i × P_i
- Weight 1 ÷ N at each rebalancing
- Return average of returns since the last rebalancing
- Upkeep frequent rebalancing

Formulas

Price-weighted index value Memorize

V_{PRI} = \frac{\sum_i P_i}{D}

Quick example

Prices 30, 60 and 90 with a divisor of 0.18 (set for a base of 1,000). 180 ÷ 0.18 = 1,000.

On your calculator

30+60+90=÷0.18= 1000

30ENTER60+90+0.18÷ 1000

ENTER enter (on the HP, puts the number on the stack)

Price weight and price-weighted return Memorize

w_i = \frac{P_i}{\sum_j P_j}, \qquad r = \frac{\sum P_1 - \sum P_0}{\sum P_0}

Quick example

Prices go from 30, 60, 90 to 33, 57, 99. The sum rises from 180 to 189, so the return is 9 ÷ 180 = 5.00%. The 90 share carries 90 ÷ 180 = 50% of the weight.

On your calculator

189÷180−1= 0.05

189ENTER180÷1− 0.05

ENTER enter (on the HP, puts the number on the stack)

Divisor after a split Memorize

D_{new} = \frac{\sum P_{after}}{V_{before}}

Quick example

Prices 40, 60 and 100 with a divisor of 2.00 give an index of 100. The 100 share splits 2-for-1, so the sum falls to 150. New divisor = 150 ÷ 100 = 1.50.

On your calculator

40+60+50=÷100= 1.5

40ENTER60+50+100÷ 1.5

ENTER enter (on the HP, puts the number on the stack)

Equal-weighted index value Use it

V_{EQI} = \frac{\sum_i n_i P_i}{D}, \quad n_1 P_1 = n_2 P_2 = \cdots = n_N P_N

Quick example

Hold 90 X at 80, 288 Y at 25 and 160 Z at 45 (7,200 in each). 21,600 ÷ 21.6 = 1,000.

On your calculator

(90×80)+(288×25)+(160×45)=÷21.6= 1000

90ENTER80×288ENTER25×+160ENTER45×+21.6÷ 1000

ENTER enter (on the HP, puts the number on the stack)

Equal weight and equal-weighted return Memorize

w_i = \frac{1}{N}, \qquad r = \frac{1}{N}\sum_i r_i \ \text{(since the last rebalancing)}

Quick example

Four shares return 8%, −2%, 5% and 1%. (8 − 2 + 5 + 1) ÷ 4 = 3.00%.

On your calculator

8+2+|-+5+1=÷4= 3

+|- change sign

8ENTER2CHS+5+1+4÷ 3

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Market-cap, float-adjusted and fundamental weighting

Known

MemorizeWeighting methods: weight driver, bias, upkeep

Method	Weight driven by	Tilt	Rebalancing
Price	share price	high-priced shares	divisor after splits
Equal	$1 \div N$	small caps	high
Market cap (float-adjusted)	market value of the float	large caps, momentum	low (self-rebalancing)
Fundamental	earnings, sales, book value…	value, contrarian	periodic

Memory hook Cap weighting buys more of what has risen. Fundamental weighting buys more of what has become cheap.

Know it cold Cap-weighted return = change in the TOTAL market value, not an average of returns. Free float also drops shares foreigners cannot buy.

MemorizeOne table, five answers (our three-stock table)

Method	Price return	Total return
Price-weighted	10.67%	12.33%
Equal-weighted	8.67%	10.17%
Cap-weighted	5.19%	7.04%
Float-adjusted (Y float 60%)	7.27%	9.09%
Fundamental (earnings)	4.25%	6.19%

Memory hook The bigger the weight on Y (the one that fell), the lower the return.

Know it cold Exam options are the other methods' answers. Compute the one asked for, then match.

Key terms

market-capitalization weighting Memorize
value weighting Know the idea
market float Use it
float-adjusted market-capitalization weighting Memorize
free float-adjusted index Know the idea
fundamental factor weighting Memorize
contribution to index return Use it

Weighting methods and what each favors

Index weighting methodsthe same shares, a different rule

Price weighting Memorize weight follows share price
- Aggregate sum of one share of each
- Favors high-priced shares; splits shift weights
Equal weighting Memorize weight 1 ÷ N
- Aggregate number of constituents
- Favors small companies; needs frequent rebalancing
Market-cap weighting Memorize weight follows market value
- Aggregate total market value (of the float)
- Float-adjusted version counts only shares the public can buy
- Favors large, recently risen shares (momentum)
Fundamental factor weighting Memorize weight follows earnings, sales, book value, cash flow, dividends
- Aggregate total of the chosen fundamental
- Favors value; contrarian when rebalanced

Formulas

Market-cap weight Memorize

w_i = \frac{Q_i P_i}{\sum_j Q_j P_j}

Quick example

Share A: 20 million shares at 35. Share B: 15 million shares at 28. Values 700 and 420, total 1,120. A carries 700 ÷ 1,120 = 62.5%.

On your calculator

(20×35)+(15×28)= 1120
700÷1120= 0.625

20ENTER35×15ENTER28×+ 1120
700ENTER1120÷ 0.625

ENTER enter (on the HP, puts the number on the stack)

Market-cap-weighted index value Memorize

V_{MCI} = \frac{\sum_i Q_i P_i}{D}

Quick example

Total market value 10,800 at the start, divisor 10.8: index 1,000. Market value 11,360 at the end: 11,360 ÷ 10.8 = 1,051.85.

On your calculator

11360÷10.8= 1051.85

11360ENTER10.8÷ 1051.85

ENTER enter (on the HP, puts the number on the stack)

Market-cap-weighted index return Memorize

r = \frac{\sum_i Q_i P_{i,1} + Div}{\sum_i Q_i P_{i,0}} - 1

Quick example

Total market value of the constituents goes from 1,200 to 1,230, with 15 of dividends (Div, the total paid by the constituents). (1,230 + 15) ÷ 1,200 − 1 = 3.75%. Without dividends the price return is 2.50%.

On your calculator

1230+15=÷1200−1= 0.0375

1230ENTER15+1200÷1− 0.0375

ENTER enter (on the HP, puts the number on the stack)

Float-adjusted market-cap weight Memorize

w_i = \frac{f_i Q_i P_i}{\sum_j f_j Q_j P_j}

Quick example

A holder owns 40% of a company with 100 million shares at 30. With float 60%, the index counts 0.60 × 100 × 30 = 1,800 of the 3,000 market value.

On your calculator

0.6×100×30= 1800

0.6ENTER100×30× 1800

ENTER enter (on the HP, puts the number on the stack)

Contribution to index return Use it

\text{contribution}_i = w_i \times r_i, \qquad r_{index} = \sum_i w_i r_i

Quick example

Weight 34.09%, return −4%: contribution 0.3409 × (−4%) = −1.36 points.

On your calculator

0.3409×4+|-= −1.36

+|- change sign

0.3409ENTER4CHS× −1.36

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Fundamental weight Memorize

w_i = \frac{F_i}{\sum_j F_j}

Quick example

Earnings of 60, 30 and 10 give weights of 60%, 30% and 10%, whatever the share prices are.

Rebalancing, reconstitution and what indexes are used for

Known

MemorizeRebalancing versus reconstitution

	Rebalancing	Reconstitution
Changes	weights	membership
Trigger	weights drift away from the rule	securities no longer meet the criteria, or new ones qualify
Frequency	usually quarterly	on set review dates (some indexes yearly)
Needs it most	equal-weighted, fundamental (often quarterly)	every index, cap-weighted included

Memory hook Rebalancing = same list, new weights. Reconstitution = new list.

Know it cold Price- and cap-weighted indexes rebalance automatically with prices. Equal weighting trades back to the same value in each holding: n = (value ÷ N) ÷ price.

MemorizeWhat indexes are for

Use	What it does
Benchmark	judge a portfolio against the market it invests in
Blueprint for passive funds	index funds and exchange-traded funds copy it
Custom index	factor, thematic, income, risk-weighted, active

Memory hook Measure it, copy it, customize it.

Know it cold The benchmark must match the manager's universe and weighting style.

Key terms

rebalancing Memorize
counter-trend trading Use it
reconstitution Memorize
banding Know the idea
benchmark Use it
passive investing Use it
active investing Know the idea
custom index Know the idea

Keeping an index up to date

Index managementboth create turnover and trading costs

Rebalancing Memorize reset the weights of the existing constituents
- Equal-weighted and fundamental on a schedule, often quarterly
- Price- and cap-weighted rebalance automatically with prices; divisor changes for corporate actions
Reconstitution Memorize change which securities are in the index
- Add securities that now meet the rules
- Remove no longer qualify, bankrupt, delisted, merged
- Adjust divisor, sometimes past values

Reconstitution, step by step

Reconstitutionthree steps

Rank and test Know the idea rank securities, usually by market value, and test them against the inclusion rules
Announce Know the idea preliminary changes a few weeks ahead, then the final list
Implement Know the idea at the close of the reconstitution date

Custom indexes

Custom indexesrules-based, built for one objective

Market factor Know the idea momentum, value
Thematic Know the idea for example clean energy
Income-oriented Know the idea high dividend yield shares, bonds
Risk-weighted Know the idea for example minimum variance
Active strategy Know the idea shifts with economic, credit and market cycles

Formulas

Shares after rebalancing an equal-weighted index Use it

n_i^{new} = \frac{V_{aggregate} \div N}{P_i}

Quick example

Holdings worth 23,472 across 3 shares; Y trades at 24. 23,472 ÷ 3 ÷ 24 = 326 shares.

On your calculator

23472÷3=÷24= 326

23472ENTER3÷24÷ 326

ENTER enter (on the HP, puts the number on the stack)