You are reading the 2026 curriculum (exams up to November 2026). Sitting in February 2027 or later? Go to the 2027 Quantitative Methods guide.

Learning Module 1 of Quantitative Methods 6–9% of the exam

Rates and Returns

6 lessons. About 30 minutes in cram mode, 140 minutes in full.

Start with lesson 1 Learn with Sensei Read the module summary

Lessons

1
What an interest rate really is
interpret 4 min cram, 20 min full
2
Holding period return
calculate 4 min cram, 16 min full
3
Arithmetic, geometric and harmonic mean returns
calculate 6 min cram, 28 min full
4
Money-weighted vs time-weighted return
calculate 6 min cram, 28 min full
5
Annualized and continuously compounded returns
calculate 6 min cram, 24 min full
6
Gross, net, after-tax, real and leveraged returns
calculate 4 min cram, 24 min full

What the exam expects you to do

Interpret interest rates as required rates of return, discount rates, or opportunity costs and explain an interest rate as the sum of a real risk-free rate and premiums that compensate investors for bearing distinct types of risk.What an interest rate really is
Calculate and interpret different approaches to return measurement over time and describe their appropriate uses.Holding period return
Compare the money-weighted and time-weighted rates of return and evaluate the performance of portfolios based on these measures.Money-weighted vs time-weighted return
Calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses.Annualized and continuously compounded returns
Calculate and interpret major return measures and describe their appropriate uses.Gross, net, after-tax, real and leveraged returns

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.

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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.

What an interest rate really is

Known

MemorizeThe five building blocks of a rate

Component	Pays you for
Real risk-free rate	Waiting
Inflation premium	Lost purchasing power
Default risk premium	The borrower might not pay
Liquidity premium	Selling quickly at a discount
Maturity premium	Long bonds swing more when rates move

Memory hook Real + inflation = what a T-bill pays. Then the three risks, D, L, M: "Don't Lend Long".

Know it cold $1 + r_{nominal} = (1 + r_{real})(1 + \pi)$ . A short government bill has no D, L or M premium.

Key terms

opportunity cost Know the idea
real risk-free interest rate Memorize
inflation premium Memorize
default risk premium Memorize
liquidity premium Memorize
maturity premium Memorize
nominal risk-free interest rate Memorize

What a quoted interest rate is made of

Quoted interest ratethe required return, discount rate and opportunity cost, all at once

Nominal risk-free rate Memorize what a short government bill pays
- Real risk-free rate the reward for waiting, with no inflation and no risk
- Inflation premium expected loss of purchasing power
Risk premiums Memorize all zero for a short government bill
- Default risk premium the borrower may not pay in full and on time
- Liquidity premium selling quickly costs you a discount
- Maturity premium long debt loses more value when rates rise

Formulas

Interest rate as a sum of components Memorize

r = r_{real} + \text{IP} + \text{DRP} + \text{LP} + \text{MRP}

Quick example

Real risk-free 1.5%, inflation 3.0%, default 1.2%, liquidity 0.4%, maturity 0.6%. Yield ≈ 1.5 + 3.0 + 1.2 + 0.4 + 0.6 = 6.7%.

Nominal risk-free rate (exact) Memorize

1 + r_{nominal} = (1 + r_{real})(1 + \text{inflation})

Quick example

Real 2.0%, inflation 3.0%. 1.02 × 1.03 − 1 = 5.06%, against 5.00% from the shortcut.

On your calculator

1.02×1.03−1= 0.0506, so 5.06%

1.02ENTER1.03×1− 0.0506

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

Holding period return

Known

MemorizeOne period and several

Case	Formula
One period	$(P_1 - P_0 + I_1) / P_0$
Several periods	$(1+R_1)(1+R_2)\cdots - 1$

Memory hook "Gain plus income, over what you paid." Then: returns multiply, they never add.

Know it cold The base is always $P_0$ , and income counts.

Key terms

capital gain Know the idea
income yield Use it
total return Know the idea
holding period return Memorize

Formulas

Holding period return Memorize

R = \frac{P_1 - P_0 + I_1}{P_0}

Quick example

Buy at 40.00, receive 0.80 of dividends, sell at 43.00. (43.00 − 40.00 + 0.80) ÷ 40.00 = 9.5%.

On your calculator

43−40+0.8=÷40= 0.095

43ENTER40−0.8+40÷ 0.095

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

Multi-period holding period return Memorize

R = (1 + R_1)(1 + R_2)\cdots(1 + R_n) - 1

Quick example

Returns of 8%, −5% and 12%. 1.08 × 0.95 × 1.12 − 1 = 14.91%.

On your calculator

1.08×0.95×1.12−1= 0.1491

1.08ENTER0.95×1.12×1− 0.1491

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

Arithmetic, geometric and harmonic mean returns

Known

MemorizeThree means, three jobs

Mean	How	Use for
Arithmetic	Sum ÷ n	Next period's expected return
Geometric	(Product of 1 + R)^(1/n) − 1	Past compound growth
Harmonic	n ÷ Σ(1/x)	Average cost, fixed amount each period

Memory hook Smallest to largest: H ≤ G ≤ A ("HGA"). All equal only when every return is the same.

Know it cold $R_G \approx \bar R - \sigma^2/2$ : the more returns swing, the bigger the gap.

Key terms

arithmetic mean return Memorize
geometric mean return Memorize
cost averaging Know the idea
harmonic mean Use it

Three means, three jobs

Mean returnharmonic ≤ geometric ≤ arithmetic, equal only when every value is the same

Arithmetic Memorize the simple average of the returns
- Use for next period's expected return, statistics
- Question words expected, sample mean, single period
Geometric Memorize the constant compound growth rate
- Use for past performance over several periods
- Question words compound, annualized, over the period
Harmonic Use it the average when a fixed amount buys varying units
- Use for average cost per share, averaging ratios such as P/E
- Question words equal amount each month, average cost

Formulas

Arithmetic mean return Memorize

\bar{R} = \frac{R_1 + R_2 + \cdots + R_T}{T}

Quick example

+60%, −40% and +10%. (60 − 40 + 10) ÷ 3 = 10% a year.

Geometric mean return Memorize

R_G = \left[(1 + R_1)(1 + R_2)\cdots(1 + R_T)\right]^{1/T} - 1

Quick example

+60%, −40% and +10%. (1.60 × 0.60 × 1.10)^(1/3) − 1 = 1.056^(1/3) − 1 = 1.83% a year.

On your calculator

1.6×0.6×1.1= growth of 1 over three years, 1.056
y^x31/x= the cube root, 1.0183
−1= 0.0183

y^x power · 1/x reciprocal

1.6ENTER0.6×1.1× 1.056
31/xy^x the cube root, 1.0183
1− 0.0183

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal · y^x power

Harmonic mean Memorize

\bar{X}_H = \frac{n}{\sum_{i=1}^{n} 1/X_i}

Quick example

EUR 1,000 a month at unit prices of 20, 25 and 40. 3 ÷ (1/20 + 1/25 + 1/40) = EUR 26.09 a unit.

On your calculator

201/x+251/x+401/x= the sum of reciprocals, 0.115
1/x×3= 26.09

1/x reciprocal

201/x251/x+401/x+ 0.115
1/x3× 26.09

1/x reciprocal

Money-weighted vs time-weighted return

Known

MemorizeMoney-weighted vs time-weighted

	Money-weighted	Time-weighted
What it is	IRR of the investor's flows	Chained sub-period returns
Timing of flows matters?	Yes	No
Use to judge	The investor's own result	The manager (GIPS)

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

Know it cold Money added before a strong period → MWR > TWR. Before a weak period → MWR < TWR.

Key terms

money-weighted return Memorize
time-weighted return Memorize
internal rate of return Use it

Formulas

Money-weighted return (net present value of flows = 0) Use it

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

Quick example

Flows of −100 at t = 0, −120 at t = 1 and +260 at t = 2. The IRR, and so the money-weighted return, is 12.05% a year.

On your calculator

CF2NDCLR WORK open the cash flow worksheet and clear it
100+|-ENTER↓ CF0 = −100
120+|-ENTER↓ C01 = −120, and F01 stays at 1
↓260ENTER C02 = 260
IRRCPT 12.05

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
100CHSgCF0 the first flow
120CHSgCFj t = 1
260gCFj t = 2
fIRR 12.05

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 return (annualized) Memorize

R_{TW} = \left[(1 + R_1)(1 + R_2)\cdots(1 + R_n)\right]^{1/T} - 1

Quick example

Year 1: 100 to 120 (+20%). Year 2: 120 to 130 (+8.33%). (1.20 × 1.0833)^(1/2) − 1 = 14.02% a year.

On your calculator

1.2×130÷120= year 1 × year 2 growth, 1.30
√x−1= two years, so the square root: 0.1402

√x square root

1.2ENTER130×120÷ 1.30
g√x1− 0.1402

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

Annualized and continuously compounded returns

Known

MemorizeFour conversions

Convert	Do
Period return → annual	$(1+R)^{c} - 1$ , c = periods a year
Stated → effective	$(1 + r_s/m)^m - 1$
Continuous rate → effective	$e^{r} - 1$
Prices → continuous return	$\ln(P_1 / P_0)$

Memory hook "Annualize" = a power. "Continuous" = e or ln. Never multiply.

Know it cold Continuous returns add across periods. Holding period returns multiply.

Key terms

annualized return Memorize
stated annual rate Use it
effective annual rate Memorize
continuously compounded return Memorize

One year of growth, four ways to state it

A rate for one yearalways convert to the same basis before comparing

Stated annual rate Use it the quoted rate, before compounding within the year
Periodic rate Use it stated rate ÷ m, credited each period
Effective annual rate Memorize what you actually earn in a year: (1 + r_s ÷ m)^m − 1
Continuously compounded rate Memorize ln(1 + HPR). Adds across periods

Formulas

Annualized return Memorize

R_{annual} = (1 + R_{period})^{c} - 1

Quick example

2.5% over 3 months, so c = 4. 1.025⁴ − 1 = 10.38%.

On your calculator

1.025y^x4=−1= 0.1038

y^x power

1.025ENTER4y^x1− 0.1038

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

Effective annual rate Memorize

\text{EAR} = \left(1 + \frac{r_s}{m}\right)^{m} - 1

Quick example

8% stated, compounded quarterly. (1 + 0.08 ÷ 4)⁴ − 1 = 8.24%.

On your calculator

2NDICONV opens on NOM
8ENTER↓↓ NOM = 8, skip to C/Y
4ENTER↑CPT EFF = 8.24

2ND second function (the shifted label above a key) · ICONV interest conversion worksheet · ENTER enter (on the HP, puts the number on the stack) · CPT compute

0.08ENTER4÷1+ 1.02
4y^x1− 0.0824

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

Effective annual rate, continuous compounding Memorize

\text{EAR} = e^{r} - 1

Quick example

6% compounded continuously. e^0.06 − 1 = 6.18%.

On your calculator

0.062NDe^x−1= 0.0618

2ND second function (the shifted label above a key) · e^x e to the power x

0.06ge^x1− 0.0618

g blue shift key · e^x e to the power x

Continuously compounded return Memorize

r_{cc} = \ln(1 + \text{HPR}) = \ln\left(\frac{P_1}{P_0}\right)

Quick example

A price rises from 45 to 52. ln(52 ÷ 45) = ln 1.1556 = 14.46%.

On your calculator

52÷45=LN 0.1446

LN natural logarithm

52ENTER45÷gLN 0.1446

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · LN natural logarithm

Gross, net, after-tax, real and leveraged returns

Known

MemorizeThe return ladder

Step	Take away	Use for
Gross	Nothing (trading costs already in)	Comparing managers
Net	Fees and expenses	What the investor got
After-tax	Taxes	What the investor kept
Real	Inflation (divide)	Comparing across inflation

Memory hook Gross for skill, net for the client, real for comparing countries or decades.

Know it cold $R_L = R + \frac{V_B}{V_E}(R - r_D)$ and real $= \frac{1 + R}{1 + \pi} - 1$ .

Key terms

gross return Use it
net return Use it
after-tax nominal return Use it
real return Memorize
leveraged return Memorize

Formulas

Real return (exact) Memorize

R_{real} = \frac{1 + R_{nominal}}{1 + \text{inflation}} - 1

Quick example

Nominal 6.5%, inflation 2.5%. 1.065 ÷ 1.025 − 1 = 3.90%, against 4.00% from subtraction.

On your calculator

1.065÷1.025−1= 0.0390

1.065ENTER1.025÷1− 0.0390

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

Leveraged return Memorize

R_L = R + \frac{V_B}{V_E}\left(R - r_D\right)

Quick example

Equity 600,000, borrowed 400,000 at 5%, portfolio return 9%. 9% + (400 ÷ 600)(9% − 5%) = 11.67%.

On your calculator

9−5=×400÷600+9= in percent: 11.67

9ENTER5−400×600÷9+ 11.67

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

Cheat sheet · PDF · 3 A4 pages

Rates and Returns on 3 pages

What to memorize, the concept maps, every formula with a worked example and keystrokes for the TI BA II Plus and HP 12C, how each idea is asked, and the traps. Print it, or keep it on your phone.

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