CFA 2018 level 1 fintree quantitative methods

The Time Value of Money LOS aLess liquidity = High LRP Shorter maturity = Low MRP Return on Non-investment grade bond > Return on Investment grade bond, because risk of Non-investment g

The Time Value of Money LOS a

Less liquidity

= High LRP

Shorter maturity

= Low MRP

Return on Non-investment grade bond > Return on Investment grade bond, because

risk of Non-investment grade bond > risk of Investment grade bond

International Fischer Relationship (approx.)

Consumption cost -

107

True saving - 3

Treasury bonds = Real RFR + Expected inflation

Corporate bonds = Real RFR + Expected inflation + Risk premium

Nominal risk-free rate = Real risk-free rate + Expected inflation

@ 10% p.a.

Default risk

Liquidity risk

Maturity risk

return

Risk that borrower will not make promised payments

in a timely manner

Risk of receiving less than FV for an investment if it must be sold for cash quickly

Risk of volatility

of price of a bond because of its longer maturity

Types of risks

Default risk premium

Liquidity risk premium

Maturity risk premium

Interest rate can be interpreted as

-2

3 1

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LOS c

LOS d

LOS e

EAR on TI BA II Plus Professional

-Effective Annual Rate =

It is a stream of equal cash flows occurring at equal intervals.

Calculation and interpretation of effective annual rate

TVM with different compounding frequencies

PV = -100

FV = 113.68

N = 1 X 4 = 4 I/Y = 13.25/4 = 3.3125

PV = -100

FV = 113.92

N = 1 X 12 = 12 I/Y = 13.25/12 = 1.104

PV = -100

FV = 114.08

The rate of interest that an investor actually earns as a result of

compounding is known as EAR

CF Disc rate

Principal repayment (Inst - Int.)

Closing loan (Op loan - Princ

2 Using amort function in TI BA II plus professional

Using CF function in TI BA II plus professional

Discounted Cash Flow Applications LOS a

LOS b

LOS c

LOS d

NPV IRR

PV of inflows − PV of outflows

Rate at which PV of inflows = PV of outflows

At IRR, NPV = 0

Decision rule

Holding period return (HPR)

Money-weighted rate of return

(MWRR)

Time-weighted rate of return

(TWRR) Total return

For a single project NPV and IRR rules lead to same accept/reject decision

If IRR > WACC, NPV =+ve

If IRR < WACC, NPV =−ve

If IRR > WACC = Accept

If IRR < WACC = Reject

+ve = Accept

−ve = Reject Mutually exclusive projects - Accept project with highest NPV

Ending value - beginning value + CF received

− 1

− 1

Or Or

! TWRR is not affected by timing of the cash flows, therefore it is more preferred method of

performance measurement

! If funds are contributed to a portfolio just prior to a period of relatively poor performance,

MWRR < TWRR

! If funds are contributed to a portfolio just prior to a period of relatively high returns,

Provides better measure of manager’s ability to select investments Appropriate if manager has complete

T - bill

Or

3 mistakes analogy to remember the formulas (indicated in red)

j No Compounding

k 360 days

l Face value as base

365/90 (1+3.09%) - 1

12.36%

3.09%

360 90

Measure used to describe a characteristic of a population

It is used to measure

a characteristic of a sample

Statistical Concepts and Market Returns

LOS a

LOS b

Descriptive statistics

Sample statistic Frequency distribution

-Tabular presentation of statistical data

Data employed with a frequency distribution may be measured using any type of measurement scale

Parameter

Types of measurement scales

Inferential statistics

Used to summarize important characteristics of large data

Eg Average of weekly tests Eg Forecast on pass or not

Used to make forecasts of

Higher level of measurement than nominal scales Observation is assigned to a category

Eg MF’s star rating

Most refined level

of measurement Provides ranking and equal differences between scale values Has a true zero point as origin

Provides relative ranking and assurance that differences between scale values are equal Weakness - Zero doesn’t mean total absence

Eg Temperature measurement

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LOS c

LOS d

LOS e

Relative frequency and cumulative relative frequency

Histogram and Frequency polygon

Measures of central tendency

Interval / class

WM = 8.2

HM =

4 1/10 + 1/14 + 1/4 +1/8

ª Sum of deviations from arithmetic mean is always zero

ª To calculate portfolio return, weighted mean is used

ª Geometric mean is used for calculating investment returns over multiple periods

ª Harmonic mean is used to calculate average of ratios

ª Arithmetic mean > Geometric mean > Harmonic mean

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Standard deviation Variance

It is the midpoint of a data set

Value that occurs most frequently in a data set

A data set can have more than one mode or even no mode

If a data set has one/two/three modes it is said to be unimodal/bimodal/trimodal

th

Median = [(n+1) X 50%] observation Data needs to be arranged in ascending order

to calculate median using above formula

SD can be calculated directly on TI BA II plus professional.

Ÿ Use DATA (2nd 7) to enter data then,

Ÿ Use STAT (2nd 8) to see SD

Distribution is divided into quarters

Distribution is divided into fifths

Distribution is divided into tenths

Distribution is divided into hundreds

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CV = SDx X Lower the better

It is used to measure excess return per unit of risk aka reward-to-variability

ratio

SR = Portfolio return RFR −

SD of portfolio Higher the better

Motorcycle takes 3 ltrs of petrol

to cover the entire distance

Motorcycle takes 2.2 ltrs of petrol

to cover the entire distance

20 − 10 2.2

Locations of mean, median and mode

right skew

Mesokurtic distribution

Leptokurtic distribution

Platykurtic distribution

Mean = Median = Mode Mean > Median > Mode

Skewness: Extent to which data is not symmetrical Negative skew in returns distributions indicates increased risk

Mean < Median < Mode

Asymmetrical distribution

Kurtosis: Measures the peakedness of a distribution Positive kurtosis in returns distributions indicates increased risk

Mean, median,

1

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S > 0.5 indicates significant level of skewness k

Excess kurtosis > 1 is considered a large value

LOS m

Arithmetic mean return is appropriate for forecasting single period returns in future periods

Geometric mean return is appropriate for forecasting future compound returns over multiple periods

Use of arithmetic mean and geometric mean when analyzing investment returns

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Probability Concepts LOS a

LOS b

LOS c

Random variable Outcome

Two defining properties of probability

Probability of an event in terms of odds

Mutually exclusive events

Uncertain quantity/

number

Observed value

of a random variable

An outcome or

a set of outcomes

Events that can not happen together

All possible events

è Probability is always between 0 & 1

è If we have mutually exclusive and exhaustive events then sum of probabilities of those events will always be 1

If probability of an event is 20%

Then for 10 experiments, success = 2 failure = 8

data

Eg Historical pass rates

Determined using formal reasoning

Eg Throwing a die

= 1/6

Least formal method of developing probabilities Involves personal judgement

LOS d

LOS e

LOS f

Unconditional and conditional probabilities

Multiplication, addition and total probability rules

Conditional Unconditional

µ Refers to probability of an event

regardless of occurrence of other

µ A conditional probability of an occurrence is also called its likelihood

Used to determine unconditional probability of an event,

given conditional probabilities

Apply this rule when a question says ‘or’

P(AB) = P(A|B) × P(B) P(A|B) = P(AB)

P(B)

P(A or B) = P(A) + P(B) − P(AB)

P(A) = P(A|B ) × P(B ) + P (A|B ) x P(B ) + P(A|B ) × P(B ) 1 1 2 2 n n

For mutually exclusive events the joint probability is zero

Joint probability - Probability that all the events will occur at the same time

For events that are not mutually exclusive, joint probability must be subtracted from the total of unconditional probabilities to avoid double counting

P(B)

P(AB) P(A)

Eg.

Or

P(A) = 60%

P(Both) = P(At least one) = P(None) =

LOS g

LOS h, i, j

Dependent and independent events

Unconditional probability using total probability rule

Ÿ Occurrence of one event has no

influence on occurrence of other events

Ÿ P (A|B) = P(A)

Ÿ Getting 5 on a second roll of die is

independent of getting 5 on the first roll

of die

Ÿ Occurrence of one event is dependent

on occurrence of other events

Good economy next year

Interest rates increase

Prob of good economy and rate increase

Prob of good economy and rate decrease

Interest rates decrease

Conditional probability

Joint probability 1

EPS = 3

c

P(Good ) = 60%

Economy

0.4 × 0.6 × 10 = 2.4

0.4 × 0.4 × 8 = 1.28 0.6 × 0.7 × 6 = 2.52

0.6 x 0.3 x 3 = 0.54

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LOS k

LOS l

LOS n

Covariance and correlation

Expected value, variance and standard deviation of portfolio

µ Only +ve and −ve sign matters for

determining relationship b/w the variables

µ Standardized measure of covariance

µ Measures strength of linear relationship between two random variables

µ Does not have a unit

It is used to calculate updates probability

W E(R ) + W E(R ) +W E(R ) + +W E(R ) 1 1 2 2 3 3 n n

P(A) = 30%

c

P(B ) = 70%

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LOS o Counting problems

n!

n ! x n ! x x n ! 1 2 k

n

P r

Eg A person has 8 cars He uses 3 cars

for work, 3 other for long distance trips

and 2 other for commute other than

work Calculate the no of different

ways to label them.

Eg How many different ways are there to select

3 players from 5, if the order of selection is important ?

Eg How many different ways are there to select

3 players from 5, if the order of selection is not important ?

Permutation is used when order of selection is important Combination is used when order of selection is not important

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Common Probability Distribution LOS a

Discrete random variable

-Discrete random variable

Eg.

Continuous random variable

-Continuous random variable

Describes the probabilities of all possible outcomes

of a random variable Probabilities of all outcomes should equal to 1

There is a finite number of possible outcomes Eg

number of stocks in portfolio There is an infinite number of possible outcomes Eg Return earned in portfolio

Number of people present in class Probability distribution of a roll of a die

Continuous non - uniform

Cumulative distribution function

+1

Binomial random variable

Outcome can be either

Ÿ Mean of binomial distribution = np

Ÿ Variance of binomial distribution = npq

Ÿ q = 1 − P

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708 x 1/1.2

= 590

850 x 1/1.2

= 708 Stock price (S) = 850 Uptick (u) = 1.2 Downtick (d) = 1/U = 1/1.2

Return on portfolio − Return on benchmark

Sud Sdu

Sdd

15 6

Sd S

ª For all a < x1 < x2 < b (i.e for all x1 and x2 between the boundaries a and b)

ª P(X < a or X > b) = 0 (i.e probability of X outside the boundaries is zero)

ª P(x < X < x ) = (x - x )/b - a 1 2 2 1

(This defines the probability of outcomes between x and x ) 1 2

Properties of continuous uniform distribution

ª Continuous uniform distribution will always have lower and upper bound (a,b)

ª Probability of X taking any value below ‘a’ or above ‘b’ will be zero

X is uniformly distributed between 2 & 20 Calculate the probability that X will be between 6 & 15.

Because it is a continuous distribution

not go to zero (i.e the tails get very thin but extend infinitely

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LOS k

LOS l

LOS m

Univeriate distribution

Multiveriate distribution

Confidence interval

Standard normal distribution

Single variable

Standard normal distribution - Z-value =

It is a normal distribution that is standardized so that its

mean = 0 and standard deviation = 1

Observation − Population mean standard deviation

More than one variables

A multivariate distribution with 10 variables has - 10 means 10 Variances 45 Correlations

10 x 9

2 = 45

=

3σ 2σ

1σ -1σ

-2σ -3σ

-At 1.5 Z-value, Probability = 93.32%

Therefore probability of value less than 400 = 1 − 0.9332 = 6.68%

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LOS n

LOS o

LOS p

Shortfall risk and Safety first ratio

Normal and lognormal distribution

Discrete and continuous compounding

Shortfall risk

Probability that portfolio value

or return will fall below a particular value or target over a given period of time

Lower the better

Excess return per unit of risk over minimum acceptable return/threshold level.

SF ratio = R − Threshold return p

20%

because they can not take negative values

-Annual, semi-annual, quarterly, monthly etc.

No of compounding periods within a given time period

Lognormal distribution

SF ratio =

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Continuous compounding calculations

Technique based on repeated generation of one or more risk factors that

affect security values, to generate a distribution

It is based on actual change in value or actual change in risk factor for some prior period

Each iteration of simulation involves randomly selecting one of these past changes for

each risk factor and calculating the value of the asset or portfolio in question, based

on those changes in risk factor Its advantage is that it uses actual distribution of risk factors, which need not be estimated.

Its limitations are : Past changes in risk factor may not be a good indication of future changes

It can not address the sort of ‘whatif’ questions that Monte Carlo simulation can

It is used to

Ÿ Value complex securities

Ÿ Simulate profits/losses from a trading strategy

Ÿ Calculate estimates of VaR to determine the

riskiness of a portfolio

Ÿ Simulate pension fund assets and liabilities to

examine the variability of the differences

between the two

Ÿ Value portfolios of assets that do not have

normal returns distribution

Its limitations are

Ÿ It is complex

Ÿ It is subject to model risk and input risk

Ÿ Simulation is not an analytic method, but a statistic one.

Ÿ Increased complexity does not necessarily ensure accuracy

Sampling and Estimation LOS a

LOS b

LOS c

Simple random sampling and sampling distribution

It is a probability distribution of all possible sample statistic computed from samples drawn from the population

Method of selecting a sample in such a way that each item in the population has same likelihood

of being included in the sample.

Another way to form an approximately random sample.

Eg Drawing a sample of 5 apples from 50 to calculate average weight.

th

Eg Selecting every n item from the population

Sampling error = Sample statistic − Population parameter

Stratified random sampling -

Sampling distribution

-Sampling distribution does not have to be normal distribution

Mean, Variance, Standard Deviation of

sample

Avg calorie intake of a nation Results of these samples

are then pooled to form a combined sample

Mean, Variance, Standard Deviation of

ª Sample mean(x) approaches population mean(μ) as sample size becomes large

2

ª Variance equals ‘σ /n’ as sample size becomes large

ª If sample size n, is sufficiently large (n ≥ 30), the sampling distribution of the sample

means will be approximately normal

ª If central limit theorem works, population mean(μ) = mean of sampling distribution

ª Standard deviation of sampling distribution = σ/√n (standard error)

LOS d

LOS e

LOS f

LOS g

Time-series and Cross-sectional data

Central limit theorem

Standard error of sample mean

Describe properties of an estimator

Time-series data

Cross-sectional data

It consists of observations taken

over a period of time

ª Unbiasedness - It is one for which the expected value of the estimator is equal to the

parameter you are trying to estimate

ª Efficiency - Unbiased estimator is also efficient if the variance of its sampling distribution

is smaller than other unbiased estimators of parameter you are trying to estimate

ª Consistency - An estimator for which the accuracy of the parameter estimate increases as the sample size increases

It consists of observations taken

at a single point in time

Time-series and cross-sectional data can be pooled in the same data set.

Panel and longitudinal data are typically presented in table or spreadsheat form.

Observations over time of multiple characteristics of the

same entity Eg Unemployment, GDP growth rates, inflation of a country over 10 years.

Observations over time of same characteristic of the

multiple entities Eg analysis of D/E ratio of 20 companies over 8 quarters.