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Title: CFA Level 1 - Quantitative Methods
Description: I create this summary of knowledge related to CFA level 1 for my 2017 December exam. I got into the top 10% with this. Hope this can help you. Please note that this does not guarantee for your pass, which requires dedication, hardwork and consistency. In case having trouble with any part, please refer to CFA notebook/Schwesser.
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Concepts
Nominal risk-free rate /
Real risk-free rate

Description
Time Value of Money
Nominal risk-free rate = Real risk-free rate + expected inflation rate

Required interest rate on a security Required interest rate on a securities = Nominal risk-free rate
+ default risk premium
+ liquidity premium
+ maturity risk premium
Effective annual rate (EAR)

𝐸𝐴𝑅 = 1 + π‘π‘’π‘Ÿπ‘–π‘œπ‘‘π‘–π‘ π‘Ÿπ‘Žπ‘‘π‘’

βˆ’1

In which :
π‘ƒπ‘’π‘Ÿπ‘–π‘œπ‘‘π‘–π‘ π‘Ÿπ‘Žπ‘‘π‘’ = π‘ π‘‘π‘Žπ‘‘π‘’π‘‘ π‘Žπ‘›π‘›π‘’π‘Žπ‘™ π‘Ÿπ‘Žπ‘‘π‘’ π‘š
π‘š = π‘›π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘π‘œπ‘šπ‘π‘œπ‘’π‘›π‘‘π‘–π‘›π‘” π‘π‘’π‘Ÿπ‘–π‘œπ‘‘π‘  π‘π‘’π‘Ÿ π‘¦π‘’π‘Žπ‘Ÿ
FV / PV formula

𝐹𝑉 = 𝑃𝑉 Γ— 1 + 𝐼 β„π‘Œ
𝑃𝑉 = 𝐹𝑉 Γ— 1 + 𝐼 β„π‘Œ

PV of a perpetuity

𝑃𝑉

Concepts
NPV decision rule

=

=

𝐹𝑉
1 + πΌβ„π‘Œ

𝑃𝑀𝑇
πΌβ„π‘Œ

Description
Discounted Cash Flow Applications
- (+) NPV β†’ ↑ shareholder wealth β†’ Accept
- (-) NPV β†’ ↓ shareholder wealth β†’ Reject
- 2 mutually exclusive projects β†’ Accept project with higher (+) NPV

IRR decision rule

- IRR > Firm's required rate of return β†’ Accept
- IRR < Firm's required rate of return β†’ Reject

Problems with NPV and IRR

Mutually exclusive projects β†’ might have conflict result between NPV and IRR, due to :
- Different size of initial costs
- Different timing of CF

Holding period return (HPR)
(or Holding period yield - HPY)

Holding period return : % change in investment value over the holding period

𝐻𝑃𝑅 =

Money-weighted return /
Time-weighted rate of return

𝐸𝑛𝑑𝑖𝑛𝑔 π‘£π‘Žπ‘™π‘’π‘’ βˆ’ 𝐡𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 π‘£π‘Žπ‘™π‘’π‘’ + 𝐢𝐹 π‘Ÿπ‘’π‘π‘’π‘–π‘£π‘’π‘‘ 𝐸𝑛𝑑𝑖𝑛𝑔 π‘£π‘Žπ‘™π‘’π‘’ + 𝐢𝐹 π‘Ÿπ‘’π‘π‘’π‘–π‘£π‘’π‘‘
=
βˆ’1
𝐡𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 π‘£π‘Žπ‘™π‘’π‘’
𝐡𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 π‘£π‘Žπ‘™π‘’π‘’

Money weighted return : IRR on a portfolio, taking into account all cash inflows and outflows
Time-weighted rate of return : compound growth

(1 + π‘‘π‘–π‘šπ‘’ π‘€π‘’π‘–π‘”β„Žπ‘‘π‘’π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ π‘œπ‘“ π‘Ÿπ‘’π‘‘π‘’π‘Ÿπ‘›) = 1 + 𝐻𝑃𝑅
Bank discount yield (BDY)

Γ— 1 + 𝐻𝑃𝑅

Γ— β‹― Γ— (1 + 𝐻𝑃𝑅 )

Bank discount yield : express the dollar discount from the face (par) value as a fraction of the face value
𝐷 360
π‘Ÿ = Γ—
𝐹
𝑑

In which :
π‘Ÿ = π‘Žπ‘›π‘›π‘’π‘Žπ‘™π‘–π‘ π‘’π‘‘ 𝑦𝑖𝑒𝑙𝑑 π‘œπ‘› π‘Ž π‘π‘Žπ‘›π‘˜ π‘‘π‘–π‘ π‘π‘œπ‘’π‘›π‘‘ π‘π‘Žπ‘ π‘–π‘ 
𝐷 = π‘‘π‘œπ‘™π‘™π‘Žπ‘Ÿ π‘‘π‘–π‘ π‘π‘œπ‘’π‘›π‘‘ = πΉπ‘Žπ‘π‘’ π‘£π‘Žπ‘™π‘’π‘’ βˆ’ π‘π‘’π‘Ÿπ‘β„Žπ‘Žπ‘ π‘’ π‘π‘Ÿπ‘–π‘π‘’
𝐹 = π‘“π‘Žπ‘π‘’ π‘£π‘Žπ‘™π‘’π‘’
𝑑 = π‘›π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘Ÿπ‘’π‘šπ‘Žπ‘–π‘›π‘–π‘›π‘” π‘‘π‘Žπ‘¦π‘  𝑑𝑖𝑙 π‘šπ‘Žπ‘‘π‘’π‘Ÿπ‘–π‘‘π‘¦
BDY is not representative of the return earned by an investor, due to :
- BDY annualises using simple interest β†’ ignore effects of compound interest
- Based on bond's Face value instead of purchase price
- BDY annualises on a 360-day year
Effective annual yield (EAY)

Effective annual yield (EAY) : annualised value, based on a 365-day year, that accounts for compound interest rate

πΈπ΄π‘Œ = 1 + π»π‘ƒπ‘Œ
Money market yield (or CD
equivalent yield)

Bond equivalent yield

/

βˆ’1

Money market yield (or CD equivalent yield) : annualised holding period yield, assuming a 360-day year

π‘Ÿ

= π»π‘ƒπ‘Œ Γ— 360⁄𝑑

π‘Ÿ

=

360 Γ— π‘Ÿ
360 βˆ’ 𝑑 Γ— π‘Ÿ

Bond equivalent yield : 2 Γ— semiannual discount rate (because the coupon interest is paid in 2 semiannual payments)

π΅π‘œπ‘›π‘‘ π‘’π‘žπ‘’π‘–π‘£π‘Žπ‘™π‘’π‘›π‘‘ 𝑦𝑖𝑒𝑙𝑑 = 2 Γ—

1 + 𝐻𝑃𝑅

π΅π‘œπ‘›π‘‘ π‘’π‘žπ‘’π‘–π‘£π‘Žπ‘™π‘’π‘›π‘‘ 𝑦𝑖𝑒𝑙𝑑 = 2 Γ— 1 + πΈπ΄π‘Œ

βˆ’1

...
Nominal scale - data is put into categories that have no particular order
2
...
Interval scale - Differences in data values are meaningful, but ratios are not meaningful
4
...
Procedures to construct a frequency distribution :
- Step 1 : Define the intervals - Too few intervals β†’ data might be too broadly summarised ; Too many intervals β†’ data might not be summarised enough
- Step 2 : Tally (assign) the observations
- Step 3 : Count the observations
Relative frequency = Absolute frequency Γ· Total number of observations
Cumulative frequency for an interval = sum of all absolute / relative frequencies for all values ≀ that interval's max value

Histogram /
Frequency polygon

Histogram : bar chart of data that has been grouped into a frequency distribution

Frequency polygon :
- Horizontal axis : midpoint of each interval
- Vertical axis : absolute frequency
- Each point is connected with a straight line

Measurement of central tendency : Measurement of central tendency : to identify the center, or average, of a data set β†’ used to represent the typical, or expected, value in the data set
Population mean / Sample mean / Arimethic mean : sum of all observation value divided by the number of observations
arimethic mean / weighted mean / Population mean : mean of all observed values in the population
geometric mean / harmonic mean /
βˆ‘ 𝑋
πœ‡ π‘π‘œπ‘π‘’π‘™π‘Žπ‘‘π‘–π‘œπ‘› π‘šπ‘’π‘Žπ‘› =
median / mode
𝑁
Sample mean : mean of all sample values
βˆ‘ 𝑋
𝑋 π‘ π‘Žπ‘šπ‘π‘™π‘’ π‘šπ‘’π‘Žπ‘› =
𝑛
Weighted mean :

𝑋 =

𝑀 𝑋 = 𝑀 𝑋 +𝑀 𝑋 +β‹―+ 𝑀 𝑋

Geometrical mean :

𝐺 = 𝑋 Γ— 𝑋 Γ— β‹―Γ— 𝑋 = 𝑋 Γ— 𝑋 Γ— β‹―Γ— 𝑋
Harmonic mean : used to find the average purchasing price

⁄

𝑁

𝐻=

1
𝑋
In which :
𝑁 = π‘›π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘£π‘Žπ‘™π‘’π‘’π‘  π‘œπ‘“ 𝑋 (π‘‘π‘–π‘šπ‘’π‘  π‘œπ‘“ π‘π‘’π‘Ÿπ‘β„Žπ‘Žπ‘ π‘’π‘ )
Median : midpoint of a data set when the data is arranged from smallest to largest
Mode : Value that occurs ost frequently in a data set
- Unimodal : 1 value that occurs most frequently
- Bimodal : 2 values that occur most frequently
- Trimodal : 3 values that occur most frequently
βˆ‘

Quartiles /
Quintiles /
Deciles /
Percentiles

Quartiles - distribution is divided into quarters
Quintiles - distribution is divided into fifth
Deciles - distribution is divided into tenth
Percentiles - distribution is divided into hundredth (percents)
Formula for the position of the observation at given percentiles
𝑦
𝐿 = (𝑛 + 1) Γ—
100

In which :
𝑦 = π‘π‘œπ‘ π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘‘β„Žπ‘’ π‘œπ‘π‘ π‘’π‘Ÿπ‘£π‘Žπ‘‘π‘–π‘œπ‘›
𝑛 = π‘›π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘‘π‘Žπ‘‘π‘Ž π‘π‘œπ‘–π‘›π‘‘π‘ 
Dispersion

Dispersion : variability around the central tendency

Range

Range : relative simple measure of variability
Range = Maximum value - Minimum value

Mean absolute deviation

Mean absolute deviation : average of the absolute vlues of the deviations of individual observations from the arithmetic mean

𝑀𝐴𝐷 =

Population variance

βˆ‘

𝑋 βˆ’π‘‹
𝑛

Population variance : average of the squared deviations from the mean

𝜎 =

βˆ‘

𝑋 βˆ’πœ‡
𝑁

βˆ‘

𝑋 βˆ’π‘‹
𝑁

βˆ‘

𝑋 βˆ’π‘‹
π‘›βˆ’1

βˆ‘

𝑋 βˆ’π‘‹
π‘›βˆ’1

Population standard devation

𝜎=

Sample variance

𝑠 =

Sample standard deviation

𝑠=
Chebyshev's inequality

Chebyshev's inequality : For any set of observations, whether sample or population data, regardless of the shape of the distribution, % of observations that lie within k standard deviations
(k > 1) of the mean is at least :

π‘œπ‘π‘ π‘’π‘Ÿπ‘£π‘Žπ‘‘π‘–π‘œπ‘› π‘‘β„Žπ‘Žπ‘‘ 𝑙𝑖𝑒 π‘€π‘–π‘‘β„Žπ‘–π‘› π‘˜ π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘‘π‘’π‘£π‘–π‘Žπ‘‘π‘–π‘œπ‘› β‰₯ 1 βˆ’

Coefficient variation

Relatie dispersion : amount of variability in a distribution relative to a reference point or benchmark, commonly measured with the coefficient variation

𝐢𝑉 =

Sharpe ratio

𝑠
π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘‘π‘’π‘£π‘–π‘Žπ‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘₯
=
π‘Žπ‘£π‘’π‘Ÿπ‘Žπ‘”π‘’ π‘£π‘Žπ‘™π‘’π‘’ π‘œπ‘“ π‘₯
𝑋

Sharpe ratio : measures the excess return per unit of risk

π‘†β„Žπ‘Žπ‘Ÿπ‘π‘’ π‘Ÿπ‘Žπ‘‘π‘–π‘œ =

π‘Ÿ βˆ’π‘Ÿ
𝜎

in which :
π‘Ÿ = π‘π‘œπ‘Ÿπ‘‘π‘“π‘œπ‘™π‘–π‘œ π‘Ÿπ‘’π‘‘π‘’π‘Ÿπ‘›
π‘Ÿ = π‘Ÿπ‘–π‘ π‘˜ βˆ’ π‘“π‘Ÿπ‘’π‘’ π‘Ÿπ‘’π‘‘π‘’π‘Ÿπ‘›
𝜎 = π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘‘π‘’π‘£π‘–π‘Žπ‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘π‘œπ‘Ÿπ‘‘π‘“π‘œπ‘™π‘–π‘œ π‘Ÿπ‘’π‘‘π‘’π‘Ÿπ‘›π‘ 
Skewness

Distribution in symmetrical if it is shaped identically on both sides of its mean
Skewness : describe the extent to which a distribution is not symmetrical
- Positively skewness : many outliers in the upper region / right tail (said to be skewed right)
- Negatively skewness: many outliers in the lower regin / left tail (said to be skewed left)
Sample skewness :

π‘†π‘Žπ‘šπ‘π‘™π‘’ π‘ π‘˜π‘’π‘€π‘›π‘’π‘ π‘  𝑆

=

1 βˆ‘
Γ—
𝑛

𝑋 βˆ’π‘‹
𝑠

in which : s = sample standard deviation
Sample skewness > 0 β†’ right skewed
Sample skewness < 0 β†’ le skewed
|Sample skewness|β‰₯ 0
...
Multiplication rule of probability : used to determined th joint probability of 2 events
P(AB) = P(A and B) = P(A|B) Γ— P(B) = P(B|A) Γ— P(A)
2
...
Total probability rule : used to determine the unconditional probability of an event, given conditional probabilities
P(A) = P (A|B1) Γ— P(B1) + P(A|B2) Γ— P(B2) +
...
, Bn is a mutually exclusive and exhaustive set of outcomes

Dependent event /
Independent event

Independent events : the occurrence of one event has no influence on the occurrence of the others
...
Probability of one events is affected by the occurence of other events

Expected value

𝐸 𝑋 =

Variance / Standard deviation

𝑃 𝑋 ×𝑋 =𝑃 𝑋

×𝑋 +𝑃 𝑋

×𝑋 +β‹―+ 𝑃 𝑋

×𝑋

Variance

𝜎 =𝑀 Γ— 𝑋 βˆ’πΈ 𝑋

+𝑀 Γ— 𝑋 βˆ’πΈ 𝑋

+ β‹―+𝑀 Γ— 𝑋 βˆ’πΈ 𝑋

Standard deviation

π‘†π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘‘π‘’π‘£π‘–π‘Žπ‘‘π‘–π‘œπ‘› = 𝜎 = π‘‰π‘Žπ‘Ÿπ‘–π‘Žπ‘›π‘π‘’
Covariance

Covariance : measure of how 2 assets move together

𝑃 Γ— 𝐴 βˆ’ 𝐴̅ Γ— 𝐡 βˆ’ 𝐡

πΆπ‘œπ‘£ 𝐴, 𝐡 =

Correlation coefficient

πΆπ‘œπ‘Ÿπ‘Ÿ 𝑅 , 𝑅
Portfolio variance

Type
equation
here
...
Total number of ways that the labels can be assigned :

𝑛!
𝑛 ! Γ— 𝑛 ! Γ— β‹― Γ— (𝑛 !)
Factorial : n! = n Γ— (n - 1) Γ— (n - 2) Γ— (n - 3) Γ— … Γ— 1
Combination : Choose r items (2 labels - chosen and not chosen) with no specific ordering

π‘›πΆπ‘Ÿ =

𝑛!
𝑛 βˆ’ π‘Ÿ ! Γ— π‘Ÿ!

Permutation : Choose r items (2 labels - chosen and not chosen) with specific ordering

π‘›π‘ƒπ‘Ÿ =

𝑛!
π‘›βˆ’π‘Ÿ !

Concepts
Probability distribution
Probability function

Discrete random variable vs
...

Sum of all probabilities of all possible outcomes = 1
Probability function : probability that a random variable = a specific value
p(x) = P(X=x)
Discrete randome variable

Continuous random variable

- Limited number of possible outcomes
- A measurable and positive probabilities for each outcome

- Unlimited number of possible outcomes
- Only measurable probabilities for a range of outcome
...
g
...
g
...
g
...
g
...
g
...
g
...
55 and 86
...
g
...
However, funds that are dropped from the sample have lower return relative to
surviving funds β†’ biased toward be er funds β†’ yield overes mated results

Look-ahead bias

Look-ahead bias : using sample data that was not available on the test date (e
...
: Test Price-to-book ratio @ year end β†’ year-end BV is not available un l 30-60 days a er year end)
Solution : estimate BV

Time-period bias

Time over which data is gathered is too short / too long
Too short β†’ resutl may reflect phenomena specific to that me period
Too long β†’ underlying fundamental economic rela onships might have changed

Concepts

Description

Hypothesis

Hypothesis testing
Hypothesis : statement about the value of a population parameter developed
Title: CFA Level 1 - Quantitative Methods
Description: I create this summary of knowledge related to CFA level 1 for my 2017 December exam. I got into the top 10% with this. Hope this can help you. Please note that this does not guarantee for your pass, which requires dedication, hardwork and consistency. In case having trouble with any part, please refer to CFA notebook/Schwesser.
Buy These NotesPreview