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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
...
5 Γ— 𝑛 Γ— (𝑛 βˆ’ 1)
Confident interval

Confident interval : range of values around an expected outcome within which we expect the actual outcome to be some specific % of the time
𝑋 βˆ’ 1𝑠 ≀ 𝑋 ≀ 𝑋 + 1𝑠 β†’ 68% π‘π‘œπ‘›π‘“π‘–π‘‘π‘’π‘›π‘π‘’ π‘–π‘›π‘‘π‘’π‘Ÿπ‘£π‘Žπ‘™
𝑋 βˆ’ 1
...
65𝑠 β†’ 90% π‘π‘œπ‘›π‘“π‘–π‘‘π‘’π‘›π‘π‘’ π‘–π‘›π‘‘π‘’π‘Ÿπ‘£π‘Žπ‘™
𝑋 βˆ’ 1
...
96𝑠 β†’ 95% π‘π‘œπ‘›π‘“π‘–π‘‘π‘’π‘›π‘π‘’ π‘–π‘›π‘‘π‘’π‘Ÿπ‘£π‘Žπ‘™
𝑋 βˆ’ 2
...
58𝑠 β†’ 99% π‘π‘œπ‘›π‘“π‘–π‘‘π‘’π‘›π‘π‘’ π‘–π‘›π‘‘π‘’π‘Ÿπ‘£π‘Žπ‘™

Standard normal distribution /
z-value

Standard normal distribution : has mean = 0 ; standard deviation = 1
z-value : number of standard deviations a given observationis from the population mean
...

↑ Safety first ra o β†’ more preferable

π‘†π‘Žπ‘“π‘’π‘‘π‘¦ πΉπ‘–π‘Ÿπ‘ π‘‘ π‘…π‘Žπ‘‘π‘–π‘œ =

𝐸 𝑅

βˆ’π‘…
𝜎

In which:
𝑅 = π‘‡π‘Žπ‘Ÿπ‘”π‘’π‘‘ π‘Ÿπ‘’π‘‘π‘’π‘Ÿπ‘›
Lognormal distribution

πΏπ‘œπ‘”π‘›π‘œπ‘Ÿπ‘šπ‘Žπ‘™ π‘‘π‘–π‘ π‘‘π‘Ÿπ‘–π‘π‘’π‘‘π‘–π‘œπ‘› = 𝑒 , π‘€β„Žπ‘’π‘Ÿπ‘’ π‘₯ 𝑖𝑠 π‘›π‘œπ‘Ÿπ‘šπ‘Žπ‘™π‘™π‘¦ π‘‘π‘–π‘ π‘‘π‘Ÿπ‘–π‘π‘’π‘‘π‘’π‘‘
Characteristic of lognormal distribution :
- Lognormal distribution is skewed to the right
- Lognormal distribution is bounded from below by zero
Lognormal distribution is often used to model asset prices (because it cannot be negative, and can take any positive value)

Discretely compounded rate of
Discretely compounded rate of return : normal compound rate of return
return /
As the compounding periods get very shorter β†’ con nuously compounded rate of return
Continuously compounded rate of
𝑆
𝐸𝑓𝑓𝑒𝑐𝑑𝑖𝑣𝑒 π‘Žπ‘›π‘›π‘’π‘Žπ‘™ π‘Ÿπ‘Žπ‘‘π‘’ = 𝑒 βˆ’ 1 β†’ 𝑅 = 𝑙𝑛 1 + 𝐻𝑃𝑅 = 𝑙𝑛
return
𝑆

Holding period return for T years :
𝐻𝑃𝑅 = 𝑒 Γ— βˆ’ 1
Monte Carlo simulation

Monte Carlo simulation : uses randomly generated values for risk factors, based on their assumed distributions, to produce possible securities values
Limitation :
- Fairly complex
- Provide answer no better than the assumptions about the distributions of the risk factors and the pricing/valuation model used
- Statistic method β†’ cannot provide insights like analy c methods

Historical simulation

Historical simulation : uses random selected past changes in risk factors to generate distribution of possible securities values
Limitation :
- Cannot consider effect of significant events in the past that do not occur in the sample period

Concepts

Description

Simple random sampling

Sampling and estimation
Simple random sampling : method of selecting a sample in such a way that each item / person in the population being studied has the same likelihood of being included in the sample

Sampling distribution

Sampling distribution : probability distribution of all possible sample statistics compounded from a set of equal equal-size samples that were randomly drawn from the sample population

Sampling error

Sampling error : difference between sample statistic (mean, variance, standard deviation of the sample) and its corresponding population parameter (mean, variance, standard deviation of
the population)
E
...
: Sampling error of mean = sample mean - population mean

Stratified random sampling

Stratified random sampling : use a classification system to separate the population β†’ smaller groups, based on 1 or more dis nguishing characteris cs

Time-series data /
Cross-sectional data /
Longtitudinal data /
Panel data

Time-series data : consist observations taken over a period of time at specific and equally spaced time intervals (e
...
: monthly returns of Stock A from 2014 to 2017)
Cross-sectional data : sample of observation taken at a single point in time (e
...
: EPS of all Stock as at 31/12/2017)
Longtitudinal data : observations over time of multiple characteristics of same entity (e
...
: GDP, inflation, unemployment rate of Vietnam from 2014 to 2017)
Panel data : observation over time of same characteristic for multipled entities (e
...
: EPS of all companies for the recent 3 years)

Central limit theorem

Central limit theorem : for simple random samples size n (from a population with mean ΞΌ, finite variance 𝜎 ), sampling distribution of the sample mean (π‘₯Μ… ) approaches a normal
probability distribution with mean ΞΌ and variance =

as the sample size becomes larger

Sample size (n ) is sufficiently large (nβ‰₯ 30) β†’ distribu on of sample means will be approximately normal
Mean of the population (ΞΌ) = mean of the distribution of all possible sample means
Standard error of the sample mean Standard error of the sample mean : standard deviation of the distribution of the sample means
𝜎
πœŽΜ…=
𝑛

in which :
𝜎 Μ… = π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘’π‘Ÿπ‘Ÿπ‘œπ‘Ÿ π‘œπ‘“ π‘‘β„Žπ‘’ π‘ π‘Žπ‘šπ‘π‘™π‘’ π‘šπ‘’π‘Žπ‘›
𝜎 = π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘‘π‘’π‘£π‘–π‘Žπ‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘‘β„Žπ‘’ π‘π‘œπ‘π‘’π‘™π‘Žπ‘‘π‘–π‘œπ‘›
𝑛 = π‘ π‘Žπ‘šπ‘π‘™π‘’ 𝑠𝑖𝑧𝑒
In case the standard deviation of the population is unknown, could use the standard deviaton of the sample
𝑠
𝑠̅=
𝑛

in which :
𝑠 Μ… = π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘’π‘Ÿπ‘Ÿπ‘œπ‘Ÿ π‘œπ‘“ π‘‘β„Žπ‘’ π‘ π‘Žπ‘šπ‘π‘™π‘’ π‘šπ‘’π‘Žπ‘›
𝑠 = π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘‘π‘’π‘£π‘–π‘Žπ‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘‘β„Žπ‘’ π‘ π‘Žπ‘šπ‘π‘™π‘’
𝑛 = π‘ π‘Žπ‘šπ‘π‘™π‘’ 𝑠𝑖𝑧𝑒
Desirable properties of an
estimator

Desirable properties of an estimator :
- Unbiasedness : sign of estimation error is random
- Efficiency : lower sampling error than any other unbiased estimator
- Consistency : variance of sampling error decreases with sample size

Point estimates /
Confidential internal

Point estimates : sample values used to estimate population parameter (e
...
: sample mean is an estimator of the population mean)
Confidential interval : range of values in which the population parameter is expected to lie

Student's t-distribution

Student's t-distribution is :
- bell-shaped probability distribution
- symmetrical about its mean (mean = 0)
- to construct confidence intervals based on small samples (n < 30) from populations with unknown variance and a normal distribution
- defined by a single parameter : degree of fredom (df) = number of sample observations (n) - 1 (for sample mean)
- more probability in the tails than normal distribution
- ↑ df β†’more observa ons near the center of the distribu on + ↓ % of observa ons in tails β†’ shape of t-distribu on more closely approaches a standard normal distribu on

Confidence interval

Confidence interval : range of values within which the actual value of parameter will lie, given the probability of 1 - Ξ±
In which:
Ξ± = level of significant
1 - Ξ± = degree of confidence
Formula for confidence interval for the population mean - normal distribution with a known variance

πΆπ‘œπ‘›π‘“π‘–π‘‘π‘’π‘›π‘π‘’ π‘–π‘›π‘‘π‘’π‘Ÿπ‘£π‘Žπ‘™ = π‘ƒπ‘œπ‘–π‘›π‘‘ π‘’π‘ π‘‘π‘–π‘šπ‘Žπ‘‘π‘’ Β± π‘…π‘’π‘™π‘–π‘Žπ‘π‘–π‘™π‘–π‘‘π‘¦ π‘“π‘Žπ‘π‘‘π‘œπ‘Ÿ Γ— π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘’π‘Ÿπ‘Ÿπ‘œπ‘Ÿ = π‘₯Μ… Β± 𝑧

/

Γ—

𝜎
𝑛

Formula for confidence inerval for the population mean - normal distribution with unknown variance

πΆπ‘œπ‘›π‘“π‘–π‘‘π‘’π‘›π‘π‘’ π‘–π‘›π‘‘π‘’π‘Ÿπ‘£π‘Žπ‘™ = π‘₯Μ… Β± 𝑑

/

Γ—

𝜎
𝑛

Examples of interpretation:
- Probabilistic interpretation : 99% of resulting confidence intervals will include the population mean
- Practical interpretation : 99% confident that the population mean score is between 73
...
45
Criteria for selecting the
appropriate test statistic

Normal distribution - known variance
Normal distribution - unknown variance
Nonnormal distribution - known variance
Nonnormal distribution - unknown variance
(Note : samples are drawn randomly from the population)

Small sample (n < 30)
z - statistic
t - statistic
N/A
N/A

Large sample (n > 30)
z - statistic
t - statistic
z - statistic
t - statistic

Issues regarding of the appropriate Limitations of "larger is better" :
sample size
- May contain observations from a different population β†’ may reduce the precision of the popula on parameter es mates
- Increase costs
Data mining bias

Data mining bias : significant relationships occurr by chance
Warning signs :
- Evidence that many different variables were tested, most are unreported, until significant relationships were found
- Lack of economic theory that is consistent with the results
How to avoid : test by using out-of-sample data

Sample selection bias

Sample selection bias : selection is non-random (e
...
: due to lack of availability)

Survivorship bias

Survivorship bias : only using data from surviving mutual funds, hedge funds to study the performance, do not include funds that have ceased to exist due to closure or merger
- Would not be a problem if the characterisitcs of surviving funds and missing funds were the same
...
g
...
: π‘‘π‘€π‘œ βˆ’ π‘‘π‘Žπ‘–π‘™π‘’π‘‘ 𝑑𝑒𝑠𝑑 π‘“π‘œπ‘Ÿ π‘‘β„Žπ‘’ π‘π‘œπ‘π‘’π‘™π‘Žπ‘‘π‘–π‘œπ‘› π‘šπ‘’π‘Žπ‘› βˆ’ 𝐻 ∢ πœ‡ = πœ‡ π‘£π‘’π‘Ÿπ‘ π‘’π‘  𝐻 ∢ πœ‡ β‰  πœ‡
General decision for a one-tailed test :

π‘ˆπ‘π‘π‘’π‘Ÿ π‘‘π‘Žπ‘–π‘™ ∢
πΏπ‘œπ‘€π‘’π‘Ÿ π‘‘π‘Žπ‘–π‘™ ∢
Test statistic

𝐻 ∢ πœ‡ ≀ πœ‡ π‘£π‘’π‘Ÿπ‘ π‘’π‘ 
𝐻 ∢ πœ‡ β‰₯ πœ‡ π‘£π‘’π‘Ÿπ‘ π‘’π‘ 

𝐻 ∢ πœ‡ > πœ‡ , π‘œπ‘Ÿ
𝐻 ∢ πœ‡<πœ‡

Hypothesis testing includes :
- Test statistic : calculated from the sample data, by comparing the point estimate of the population parameter with the hypothesized value of the parameter (specified in the null
hypothesis)

𝑇𝑒𝑠𝑑 π‘ π‘‘π‘Žπ‘‘π‘–π‘ π‘‘π‘–π‘ =

π‘ π‘Žπ‘šπ‘π‘™π‘’ π‘ π‘‘π‘Žπ‘‘π‘–π‘ π‘‘π‘–π‘ βˆ’ β„Žπ‘¦π‘π‘œπ‘‘β„Žπ‘’π‘ π‘–π‘§π‘’π‘‘ π‘£π‘Žπ‘™π‘’π‘’
π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘’π‘Ÿπ‘Ÿπ‘œπ‘Ÿ π‘œπ‘“ π‘‘β„Žπ‘’ π‘ π‘Žπ‘šπ‘π‘™π‘’ π‘ π‘‘π‘Žπ‘‘π‘–π‘ π‘‘π‘–π‘

In which : Standard error of the statistic is calculated as :
𝜎
𝜎 =
π‘€β„Žπ‘’π‘› π‘‘β„Žπ‘’ π‘π‘œπ‘π‘’π‘™π‘Žπ‘‘π‘–π‘œπ‘› π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘‘π‘’π‘£π‘–π‘Žπ‘‘π‘–π‘œπ‘› 𝜎 𝑖𝑠 π‘˜π‘›π‘œπ‘€π‘›
𝑛
𝑠
𝑠 =
π‘€β„Žπ‘’π‘› π‘‘β„Žπ‘’ π‘π‘œπ‘π‘’π‘™π‘Žπ‘‘π‘–π‘œπ‘› π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘‘π‘’π‘£π‘–π‘Žπ‘‘π‘–π‘œπ‘› 𝜎 𝑖𝑠 π‘’π‘›π‘˜π‘›π‘œπ‘€π‘›
𝑛
Type I errors / Type II errors

Type I error : reject the null hypothesis when it is actually true
Type II error : failure to reject the null hypothesis when it is actually false
Significant level (Ξ±) : probability of makign Type I error

Decision rule /
Power if a test /
Relation between confidence
intervals and hypothesis tests

Decision rule for rejecting / failing to reject the null hypothesis : based on the distribution of the test statistic, by comparing the computed test statistic to a critical value at a stated level of
significant
Power of a test : probability of correctly rejecting the null hypothesis when it is false
...
economic
significance

Statistic significance does nto necessary imply economic significance, due to :
- Transactions costs
- Taxes
- Risk

p-value

p-value : probability of obtaining a test statistic that would lead to a rejection of the null hypothesis, assuming the null hypothesis is true (smallest level of significant for which the null
hypothesis can be rejected

Criteria for selecting the
appropriate test statistic when the Normal distribution - known variance
population is normally distributed, Normal distribution - unknown variance
and the variance is known /
unknown

Small sample (n < 30)
z - statistic
t - statistic

Large sample (n > 30)
z - statistic
t - statistic

Hypothesis test concerning the
t-statistic for 2 normally distributed populations with variances of the population is unknown, t-statistic for 2 normally distributed population with variance of the population is unknown,
equality of the population means of but assumed to be equal:
but assumed to be unequal:
π‘₯ βˆ’π‘₯
2 independent, normally
π‘₯ βˆ’π‘₯
𝑑=
/
𝑑=
distributed populations with equal
/
𝑠
𝑠
𝑠
𝑠
+
/ unequal assumed variances
𝑛
𝑛
𝑛 +𝑛

πœ‡ π‘£π‘’π‘Ÿπ‘ π‘’π‘  πœ‡
in which L πœ‡ and πœ‡ are
independent

Where :
𝑛 βˆ’1 ×𝑠 + 𝑛 βˆ’1 ×𝑠
𝑠 =
𝑛 +𝑛 βˆ’2
π·π‘’π‘”π‘Ÿπ‘’π‘’ π‘œπ‘“ π‘“π‘Ÿπ‘’π‘’π‘‘π‘œπ‘š = 𝑛 + 𝑛 βˆ’ 2

Where :

Hypothesis test concerning the
𝑑̅ βˆ’ πœ‡
𝑑̅
=
equality of the population means of 𝑑 =
𝑠
𝑠
2 dependent, normally distributed
populations
in which :
𝑑̅ = π‘Žπ‘£π‘’π‘Ÿπ‘Žπ‘”π‘’ π‘‘π‘–π‘“π‘“π‘’π‘Ÿπ‘’π‘›π‘π‘’ π‘œπ‘“ 𝑛 π‘π‘Žπ‘–π‘Ÿπ‘’π‘‘ π‘œπ‘π‘ π‘’π‘Ÿπ‘£π‘Žπ‘‘π‘–π‘œπ‘›π‘ 
πœ‡ = β„Žπ‘¦π‘π‘œπ‘‘β„Žπ‘’π‘ π‘–π‘§π‘’π‘‘ π‘šπ‘’π‘Žπ‘› π‘œπ‘“ π‘π‘Žπ‘–π‘Ÿπ‘’π‘‘ π‘‘π‘–π‘“π‘“π‘’π‘Ÿπ‘’π‘›π‘π‘’π‘  = 0
πœ‡ π‘£π‘’π‘Ÿπ‘ π‘’π‘  πœ‡
𝑠
𝑠 = π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘œπ‘“ π‘’π‘Ÿπ‘Ÿπ‘œπ‘Ÿ =
𝑛
/
in which : πœ‡ and πœ‡ are
βˆ‘
𝑑 βˆ’ 𝑑̅
paired comparisons
𝑠 = π‘ π‘Žπ‘šπ‘π‘™π‘’ π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘‘π‘’π‘£π‘–π‘Žπ‘‘π‘–π‘œπ‘› =
π‘›βˆ’1
(dependent)
𝑛 = π‘›π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘π‘Žπ‘–π‘Ÿπ‘’π‘‘ π‘œπ‘π‘ π‘’π‘Ÿπ‘£π‘Žπ‘‘π‘–π‘œπ‘›π‘ 
Hypothesis test concerning the
Hypothesis test concerning the variance of a normally distributed population β†’ using chi-square test
variance of a normally distributed
2 π‘‘π‘Žπ‘–π‘™π‘’π‘‘ 𝑑𝑒𝑠𝑑 ∢
𝐻 ∢ 𝜎 =𝜎
π‘£π‘’π‘Ÿπ‘ π‘’π‘  𝐻 ∢ 𝜎 β‰  𝜎
population
1 π‘‘π‘Žπ‘–π‘™π‘’π‘‘ 𝑑𝑒𝑠𝑑 ∢
𝐻 ∢ 𝜎 β‰€πœŽ
π‘£π‘’π‘Ÿπ‘ π‘’π‘  𝐻 ∢ 𝜎 > 𝜎
π‘œπ‘Ÿ
𝐻 ∢ 𝜎 β‰₯𝜎
π‘£π‘’π‘Ÿπ‘ π‘’π‘  𝐻 : 𝜎 < 𝜎
In which :
𝜎 = π‘‘π‘Ÿπ‘’π‘’ π‘£π‘Žπ‘Ÿπ‘–π‘Žπ‘›π‘π‘’ π‘œπ‘“ π‘‘β„Žπ‘’ π‘π‘œπ‘π‘’π‘™π‘Žπ‘‘π‘–π‘œπ‘›
𝜎 π‘£π‘’π‘Ÿπ‘ π‘’π‘  𝜎
𝜎 = β„Žπ‘¦π‘π‘œπ‘‘β„Žπ‘’π‘ π‘–π‘ π‘’π‘‘ π‘£π‘Žπ‘Ÿπ‘–π‘Žπ‘›π‘π‘’ π‘œπ‘“ π‘‘β„Žπ‘’ π‘π‘œπ‘π‘’π‘™π‘Žπ‘‘π‘–π‘œπ‘›

Chi-square test statistic is computed as:
π‘›βˆ’1 ×𝑠
Ο‡
=
𝜎

Hypothesis test concerning the
equality fo the variances of 2
normally distributed population
based on 2 independent ramdom
samples

𝜎 π‘£π‘’π‘Ÿπ‘ π‘’π‘  𝜎

Parametric tests / Nonparametric
tests

Hypothesis test concerning the equality of the variances of 2 populations β†’ F-distributed test sta s c
(assumption : normally distributed populations and independent samples)

2 π‘‘π‘Žπ‘–π‘™π‘’π‘‘ 𝑑𝑒𝑠𝑑 ∢
𝐻 ∢ 𝜎 = 𝜎 π‘£π‘’π‘Ÿπ‘ π‘’π‘  𝐻 ∢ 𝜎 β‰  𝜎
1 π‘‘π‘Žπ‘–π‘™π‘’π‘‘ 𝑑𝑒𝑠𝑑 ∢
𝐻 ∢ 𝜎 ≀ 𝜎 π‘£π‘’π‘Ÿπ‘ π‘’π‘  𝐻 ∢ 𝜎 > 𝜎
π‘œπ‘Ÿ
𝐻 ∢ 𝜎 β‰₯ 𝜎 π‘£π‘’π‘Ÿπ‘ π‘’π‘  𝜎 < 𝜎
In which : 𝜎 π‘Žπ‘›π‘‘ 𝜎 π‘Ÿπ‘’π‘π‘Ÿπ‘’π‘ π‘’π‘›π‘‘ π‘‘β„Žπ‘’ π‘£π‘Žπ‘Ÿπ‘–π‘Žπ‘›π‘π‘’π‘  π‘œπ‘“ π‘›π‘œπ‘Ÿπ‘šπ‘Žπ‘™ π‘ƒπ‘œπ‘π‘’π‘™π‘Žπ‘‘π‘–π‘œπ‘› 1 π‘Žπ‘›π‘‘ π‘ƒπ‘œπ‘π‘’π‘™π‘Žπ‘‘π‘–π‘œπ‘› 2
F-statistic is computed as :
𝑠
𝐹=
𝑠
Parametric tests
- Rely on assumptions regarding the distribution of the population
- Specific to population parameters
(e
...
: z-test, t-test, chi-square test, F-test)

Nonparametric tests
- Have few assumptions about the population that is sampled
- Do not consider a particular population parameter
Situations for using non-parametric tests:
- Assumptions about the distribution of the random variable that support a parametric test
are not met
- When data are ranks rather than values
- Hypothesis does not involve the paramenter of the distribution

Concepts
Technical analysis

Description
Technical Analysis
Technical analysis : an analysis methodology for forecasting the direction of prices through the study of past market data, primarily price and volume with assumption that a security’s price
already reflects all publicly-available information
...
Line charts : show closing prices for each as a continuous line
2
...

- Closing price : a point/dash on the right side of the line
- Opening price : a point/dash on the left side of the line
3
...

- Closing price < Opening price β†’ Filled box
- Closing price > Opening price β†’ Clear box
4
...
Horizontal axis represents the number of changes in direction
- X : increase of 1 box size
- O : decrease of 1 box size
- Price continue to move in the same direction in the next periods β†’ fill the same column
- Price reverses by at least the reversal size (usually 3Γ— box size) β†’ begin the next column
5
...
g
...
Trend :
- Uptrend : prices are consistently reaching higher highs, and retracing to higher lows
- Downtrend : prices are consistently reaching lower lows, and retracing to lower highs
2
...
Trendline represent a level of support/resistance
- Support level : buying is expected to emerge β†’ prevent further price decrease
- Resistance level : selling is expected to emerge β†’ prevent further price increase
4
...
Head and shoulder pattern : reversal patterns at the end of uptrends
...
Double top and Triple top : reversal patterns at the end of uptrends
...
Inverse head and shoulder pattern : inverse double bottom and triple bottom : reversal patterns at the end of downtrends
...
Continuation patterns : a pause in a trend rather than a reversal
5
...
Rectangles : Trading forms a range between a support level and a resistance level

Common technical analysis
indicators Price based indicators

1
...
Moving average = mean of the last n closing prices
- Short-term average crosses long-term average β†’ indicate changes in price trend
- ST average crosses above LT average β†’ emerging uptrend β†’ buy
- ST average crosses below LT average β†’ emerging downtrend β†’ sell
2
...

- ↑ bands β†’ ↑ vola lity
- ↓ bands β†’ ↓ vola lity
- Price above Bollinger bands β†’ overbought β†’ price is too high, and likely to decrease in the near term
- Price below Bollinger bands β†’ oversold β†’ price is too low, and likely to increase in the near term
3
...
g
...
g
...

- Rate of change oscillator = 100 x (latest closing price - closing price of n period earlier) β†’ oscillate around 0
...

+ Sustainable uptrend : close price is nearer to the recent high
+ Sustainable downtrend : close price is nearer to the recent low

Common technical analysis
indicators Non-price based indicators

1
...

- Opinion polls : directly measure investor sentiment
- Put / Call ratio = Put volume / Call volume
...
↑ VIX β†’ ↓ investors fear in the stock market
- Margin debt : ↑ margin debt β†’ agressive buying by bullish margin investors β†’ ↑ market price and future price decrease, and vice versa
- Short interest ratio : ↑ short interst ra o β†’ expect stock price to decrease and future increase in buying demand
2
...
Mutual fund cash position tend to increase when market is failing, and vice versa
...
g
...
After that, analyst could apply relative strength analysis to identify which assest within these classes are outperforming others
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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