Quantitative methods - Level I - CFA Program

THE TIME VALUE OF MONEY Quantitative Methods R-5 (SS-2)

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Interest rates: Interpretation o Nature:

Interest rates can be thought of as Discount Rates, or as Required Rate of Return (the minimum rate of return an investor must receive to accept the investment) or as Opportunity Cost of current consumption.

o Calculation: real risk free rate plus a set of premiums to compensate lenders for risk; determined through the forces of demand and supply as follows: Real risk-free interest rate + inflation premium + default risk premium + liquidity premium + maturity premium

The Future and Present Value of a single cash flow: o Discrete compounding (Discounting):

( )

( )

Where, rs (stated annual rate) and n (number of years) are compatible; and (rs/m) is the periodic rate. o Continuous compounding (Discounting):

Effective Annual Rate (EAR):

( )

The Future and Present Value of a series of cash flows: o Equal cash flows – ordinary annuity:

(( )

) (

( )

)

In the previous equations we have four variables, given three of them we can solve for the fourth. The PV of an annuity due = PV of ordinary annuity + ( ) The FV of an annuity due = FV of ordinary annuity + ( ) The Present Value of an infinite series of equal cash flows – Perpetuity:

(

)

The PV of an ordinary annuity = the difference between PV of receivable and payable perpetuities.o Unequal cash flows and :

Sum the (or ) for each cash flow individually.

Compounded growth rate as the single measure of growth over multiple-time periods (geometric mean).

Cash Flows additivity and equivalence principles: Present Values, Future Value, and Annuities can all be considered equivalent, thus can be added, as long as they are indexed at the same point in time. (or in a hypothetical condition where nominal interest rate equals, and expected to remain at, zero).

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DISCOUNTED CASH FLOW APPLICATIONS Quantitative Methods R-6 (SS-2)



Chief areas of financial decision making: o Capital Budgeting: the allocation of funds to relatively long range projects or investments. o Capital Structure: the choice of long-term financing vehicles for the investments. o Working Capital Management: the management of the company’s short-term assets and liabilities.

Net Present Value (NPV): o Definition:

For an investment; the PV of its cash inflows minus the PV of its cash outflows. o Calculation:

∑

( )

o Decision Rule: If NPV > 0 undertake the project, if NPV < 0 reject it. Positive NPV increases shareholders’ wealth, and vice versa. With zero NPV the company becomes larger, but shareholders’ wealth does not change. For mutually exclusive projects, choose the one with the highest NPV.

Internal Rate of Return (IRR): o Definition:

For an investment; the discount rate that makes NPV equal to zero. (or equates the inflows to outflows) o Calculation:

o Decision Rule:

If IRR > opportunity cost of capital accept the project, if IRR < opportunity cost of capital reject it. For mutually exclusive projects, choose the one with the highest IRR.

o Relation to NPV: For IRR > r NPV > 0; For IRR < r NPV < 0; For IRR = r NPV = 0.

o Problems with IRR: Assumes that cash flows are reinvested at the same IRR. However, NPV uses the market-determined

opportunity cost of capital, which is more realistic and economically relevant, as a discount rate. While gives the same accept/reject decision as NPV when projects are independent, it might conflict

with the profitability ranking of NPV for mutually exclusive projects, especially when the size or scale of projects differs and the timing of projects’ cash flows differs. In such cases, NPV ranking is superior over IRR ranking, as it directly reflects the effect on the shareholders’ wealth.

Portfolio return measurement: o Money Weighted Rate of Return (MWRR):

The same as IRR which equates (initial market value and additions) to (withdrawals, receipts, and ending market value).

To calculate TWRR, determine the timing and size of cash flows, insert the input on and get the answer from the calculator.

Has a serious drawback as it’s sensitive to factors out of the investment manager’s control. Namely, the timing and size of cash flows.

o Holding Period Return (total return):

o Time Weighted Rate of Return (TWRR): To calculate the annual TWRR, divide the holding period into sub-periods (separated by significant

additions or withdrawals), calculate HPR for each one, then geometrically link the returns using the appropriate equation of:

If sub-periods adds up to one year: ( ) ( ) ( )

If each sub-period is one year: [( ) ( ) ( )]

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DISCOUNTED CASH FLOW APPLICATIONS Quantitative Methods R-6 (SS-2)



Money Market Yields: For a T-Bill with a face value (par value) of 100,000 and 150 days until maturity, selling for 98,000:

o Bank Discount Yield ( )

Not a meaningful measure of investors return as the yield is based on the face value not the purchase price, the yield is annualized based on a 360 days year rather than 365, and based on a simple interest calculation (ignores compounding).

o Holding Period Yield ( )

o Effective Annual Yield ( ) ( )

( )

o CD equivalent or Money Market Yield ( )

( )

( ) ( )

( )( )

o The appropriate Yield to be used in discounting the short-term cash flow is HPY. If given any of the other three

yields we will need to convert it to HPY first.

o Annualizing a 4% semi-annual Yield to Maturity (YTM) can take two forms:

Considering compounding effect ( ) Ignoring compounding effect this is called Bond Equivalent Yield, annualizing

a semiannual yield by doubling is putting the yield on a bond-equivalent basis.

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STATISTICAL CONCEPTS AND MARKET RETURNS Quantitative Methods R-7 (SS-2)



The nature of statistics: o Descriptive statistics:

The study of how data can be summarized effectively to describe the important aspects of large data sets. o Inferential statistics:

Founded on probability theory, and involves making forecasts, estimates, or judgments (hypothesis testing) about a population from a sample:

A population is all members of a specified group. Its descriptive measures are parameters. A sample is a subset of a population. Its descriptive measures are statistics.

Measurement scales: o Nominal scales:

The weakest level of measurement, as they categorize data without ranking. (e.g., mutual funds classifications) o Ordinal scales:

Rank data according to some characteristic. (e.g., risk ratings) o Interval scales:

The differences between ranks are equalscale values can be added or subtracted meaningfully. Absence of a true zero pointratios are meaningless. (e.g., risk aversion questionnaire)

o Ratio scales: The strongest level of measurement, as it has a true zero point. (e.g., rates of return, money)

Frequency distributions: o Summarize ascending ordered data into intervals. Where is the desired number of intervals; interval size =

(maximum value – minimum) / k, rounded up. For each range define the following items: Absolute Frequency: number of observations within the range. Relative Frequency: Frequency / Total number of observations Cumulative Absolute Frequency: Total frequency up to and including the specified range. Cumulative Relative Frequency: Cumulative frequency / Total number of observations

o Frequency distribution could be presented as a table or graph. Graphic presentation includes: Histogram: a bar chart of absolute frequencies. Shows how symmetric (bell shaped) the data are. Frequency polygon: a line graph of absolute frequencies, with intervals’ midpoints on the x-axis. Cumulative frequency distribution: plots absolute or relative cumulative frequency against upper

interval limit.

Measures of location: o Central tendency:

Mean:

Arithmetic: . Population mean is denoted μ while sample’s is . The sum of the deviations around the mean = 0. An advantage is that it uses all the information about the size and magnitude of observations. Its drawback is sensitivity to outliers.

Weighted: ∑

, where sum of .

e.g., Portfolio average return, market indexes, expected value.

Geometric:

[∏ ( ) ]

Mostly used to compute growth rate of a variable over time.

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Harmonic:

∑ ( ⁄ )

A special type of weighted mean in which an observation’s weight is inversely proportional to its magnitude. e.g., Cost averaging.

Harmonic mean < Geometric < Arithmetic (except if all observations are equal).

Median:

The middle item in a set of observations. For an odd observations =

, for even = average

(

)

Advantage: not affected by outliers.

Drawbacks: doesn’t use all information, and less mathematically tractable than the mean. Mode:

The most frequent occurring value in the distribution.

Uni-bi-tri or more modal or no mode.

Modal interval in a frequency distribution is the interval with the highest frequency.

The mode is the only measure of central tendency that can be used with nominal data.

o Other measures of location: Quantiles

( )

, if the location is a whole number use the corresponding value, otherwise, use

linear interpolation to get the needed value. Median divides the observations into half, quartiles into quarters, quintiles into fifths, deciles into

tenths, percentiles into hundredths. The th

percentile is the value at or below which percent of observations lie.

Quantiles are used in portfolio performance evaluation (ranking), as well as in investment strategy development and research (especially, quartiles).

Measures of dispersion: o Range:

Advantages: ease of computation Drawbacks: uses only two observations from the distribution, and sensitive to outliers.

o Mean Absolute Deviation:

∑ | |

Advantages: uses all observations. Drawbacks: difficult to manipulate mathematically.

o Variance:

∑ ( )

∑ ( )

(in units squared)

Sample variance is calculated using in the denominator as there are only independent deviations in the case of sample. Also, this correction converts sample variance from a biased estimator of population variance to unbiased estimator.

Standard Deviation (σ for a population, s for a sample) is the positive √ . (in units) SD ≥ MAD, as squaring gives more weight to large deviations than small ones.

o Semivariance, Semideviation:

∑( )

√

Focus on downside risk.

∑( )

√

Focus on values below some level other than the mean (B) Both semivariance and target semivariance are harder to work with mathematically than variance.

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Chebyshev’s inequality:

o The proportion of the observations within k standard deviations of the arithmetic mean ≥

o Its importance stems from its generality, as it’s applicable on samples and populations of discrete or continuous data regardless of the shape of the distribution.

Coefficient of Variation (CV):

o

o A scale-free (unitless) measure of relative dispersion, which is the amount of dispersion relative to a reference value or benchmark (risk per unit of return –mean in this case–).

o Useful for comparison between data sets that have different means or different units of measurement.

Sharpe Ratio:

o

o A measure of excess return (portfolio weighted mean) per unit of risk (standard deviation) o Drawbacks:

Negative Sharpe ratio is hard to interpret, as increasing the risk will result in higher ratios. One solution is to increase the evaluation period until it becomes positive, or use a different performance evaluation metric.

Standard deviation as a measure of risk assumes symmetric distribution, consequently, it’s understates risk inherent in options, which have asymmetric returns (e.g., left skewed distributions).

Symmetry and skewness in return distributions: o Symmetric distribution: (skewness = 0)

Normal distribution has the following characteristics:

Mean = median = mode

Completely described by two parameters–mean and variance.

Roughly 68% of observations lie within ± 1σ, 95% within ± 2 σ, 99% within ± 3 σ. o Skewed distribution:

Negatively (left) skewed:

Mode > median > mean (very risky)

Limited, though frequent, upside compared with unlimited, but less frequent, downside. Positively (right) skewed:

Mode < median < mean (attractive)

Limited, though frequent, downside compared with unlimited, but less frequent, upside.

Kurtosis in return distributions: o o Mesokurtic distribution:

Normal distribution, Excess kurtosis = 0 o Leptokurtic distribution:

Excess kurtosis > 0 More peaked with fatter tails higher large negative and positive frequencies very risky More small surprises and more big surprises.

o Platykurtic distribution: Excess kurtosis < 0 Less peaked with thinner tails lower large negative and positive frequencies less risky Less small surprises and less big surprises.

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Using Geometric and Arithmetic means: o Geometric mean:

The appropriate measure of past performance and multi-period returns, as it represents the compounded growth rate.

Accordingly, semilogarithmic scale should be used to graph the historical value of investment, where slope is analogous to growth rate.

o Arithmetic mean: The appropriate measure of forward-looking returns as it captures volatility (uncertainty), and the

appropriate measure of average one-period return.

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PROBABILITY CONCEPTS Quantitative Methods R-8 (SS-2)



Probability general rules: o A random variable is a variable whose outcomes (possible values) are uncertain, a specified set of outcomes is

called event. o The properties of a probability:

The probability of any event E is a number between 0 and 1. ( ) . The sum of probabilities of any set of mutually exclusive and exhaustive events = 1. Exclusive events

don’t overlap, and exhaustive events cover all possible outcomes. o The types of probability:

Empirical probability: the relative frequency of occurrence based on historical data. A priori probability: based on logical analysis rather than on observation or personal judgment.

Grouped together, empirical and a priori types are called objective probabilities. Subjective: based on personal judgment without reference to any particular data.

o Probability stated as odds:

Given odds against E of “a to b” ( )

o Dutch Book Theorem states that inconsistent probabilities create profit opportunities. Investors by their buy and sell decisions to exploit the inconsistent probabilities, should eliminate the profit opportunity and inconsistency.

o Conditional probability: ( | ) ( )

( )

o Multiplication rule for probability (joint probability): ( ) ( | ) ( ) ( ) ( | ) ( ) o Addition rule for probability: ( ) ( ) ( ) ( ) the minus term is to correct for double

counting. o Independent events: ( | ) ( ) ( | ) ( ) o Multiplication rule for independent events: ( ) ( ) ( ) ( ) ( ) ( ) ( ) o Total probability rule: to get the unconditional probability from conditional probabilities, based on # of

scenarios (S): ( ) ( ) ( ) ( | ) ( ) ( | ) ( ) SC

: S complement ( ) ( ) ( ) ( ) ( | ) ( ) ( | ) ( ) ( | ) ( ) The total probability rule is a weighted average, probabilities of scenarios are used as weights, given

that: P(S) +P( SC

) = 1, also P(S1) + P( S2) +…+ P(Sn) = 1.

For a random variable: o Expected value (E):

The probability weighted average of its possible outcomes. ( ) ( ) ( ) ( ) Conditional EV: ( | ) ( | ) ( | ) ( | ) Unconditional EV, based on # of scenarios (S):

( ) ( | ) ( ) ( | ) ( ) ( ) ( | ) ( ) ( | ) ( ) ( | ) ( )

o Variance: ( ) ( )[ ( )]

( )[ ( )] ( )[ ( )]

o Standard deviation: ( ) √

For a portfolio: o Expected return (E):

( ) ( ) ( ) ( ) o Variance:

For a portfolio consisting of items, the overall variance is determined by the weights and relationships between all of its items . The breakdown of this relations is relations between each item and itself (variances) and ( ) unique relations between each item and other items (covariances).

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PROBABILITY CONCEPTS Quantitative Methods R-8 (SS-2)



Accordingly, for a 2-items portfolio, the variance is calculated as: ( )

And for a 3-items portfolio: ( )

Standard deviation: ( ) √ Covariance:

∑ ( )( )( ) ∑ ( )

Measures the linear relationship (co-movement) between two variables.

Its value is not very meaningful as it ranges from positive to negative infinity and presented in terms of squared units (i.e. %

2, $

2)

Covariance matrix: shows all inter-relations between items (which are ), with variances on the diagonal.

Correlation:

Standardized measure of the linear relationship between two variables.

Its value has no measurement unit and ranges from -1 (perfectly negatively correlated) to +1 (perfectly positively correlated).

Correlation matrix: shows all inter-relations between items (which are ), with +1 on the diagonal.

As long as security returns are not perfectly positively correlated, risk reduction (variance reduction) through diversification benefits is possible. The smaller the correlation the greater the diversification benefits, which is a key insight of MPT mean-variance optimization.

Multiplication rule for Expected Value of the product of uncorrelated random variables: ( ) ( ) ( ) e.g., ( ) ( ) ( )

Bayes’ Formula: o Given a set of prior probabilities for an event of interest, if you receive new information, the rule for updating

your probability of the event (to get the posterior probability) is:

( )

( | ) ( | )

( ) ( )

o When the decision maker has no prior beliefs or views, the prior probabilities are equal (called diffuse priors). In this case, prior probability of event will be equal to unconditional probability of information

( ) ( ( )

Principles of counting: o Multiplication rule of counting:

For sequential tasks with ways of doing each task, the number of ways the tasks can be done o The number of ways we can assign objects to tasks ( )( ) (by convention 0! = 1) o Multinomial formula (general formula for labeling problems):

The number of ways that objects can be labeled with different labels, with of the first type, of the second type, and so on, with , is given by: ( )

o Combination formula (binomial formula): A special case of multinomial formula is when , and is called combination. It answers the question, in how many ways we can choose objects from a total of objects when order doesn’t matter?

nCr ( )

( )

o Permutation formula:

Answers the same question as combination formula but when order does matter. nPr

( )

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COMMON PROBABILITY DISTRIBUTIONS Quantitative Methods R-9 (SS-3)



Types of random variables: o Discrete random variable: has a countable number of possible outcomes. o Continuous random variable: has unlimited number of possible outcomes (e.g., rate of return)

Probability distribution: a function that describes the random variable, has two forms:

o Probability Function: For a discrete variable ( ) ( ), For a continuous variable ( )which is called probability density function (PDF).

o Cumulative distribution function (CDF): For both discrete and continuous variables;

( ) ( ) ( ) . Parallel to cumulative relative frequency.

The discrete uniform distribution: o The simplest of all probability distributions, it has a finite number of outcomes, and each outcome is equally

likely. With ( ) and ( ) ( ) ( ) o The base for generating random numbers.

The binomial distribution: o A Bernoulli trial is a trial that produces one of two outcomes. A binomial random variable is defined as the

number of successes in Bernoulli trials. The binomial distribution assumes: The probability, , of success is constant for all trials. The trials are independent.

o Based on the above assumptions, a binomial random variable is described by two parameters, : probability of success, and : number of trials. ( ), accordingly, A Bernoulli random variable is ( )

o The probability of successes in trials = number of possible ways * probability of successes

( ) ( )

( ) ( )

o The binomial distribution is symmetric when , positively skewed when , and negatively skewed when .

o Mean and variance: Bernoulli, ( ): Mean , Variance ( ) Binomial, ( ): Mean , Variance ( )

o The binomial pricing model: Using a binomial tree, which consists of nodes showing the potential stock price at a specified time. The binomial random variable in this case is the number of up moves (not the stock price), defined as

( ) where, is the number of periods, is the up transition probability. ( ) is the down transition probability.

Final stock price distribution is a function of the initial stock price, the number of up moves, and the size of up and down moves.

Assumes constant volatility, but flexible to change this assumption by changing up/down probabilities at different nodes.

Mostly used for pricing options, where the terminal value is discounted recursively into today value.

Continuous random variables: o Continuous uniform distribution: (with lower limit and upper limit ): ( ) ( )

( ) {

( ) {

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o The normal distribution: The central limit theorem states that the sum (and mean) of a large number of independent random

variables is approximately normally distributed. Characteristics:

The distribution of one normal random variable (univariate distribution) is described by its mean and variance ( ), while, a multivariate distribution of random variables as

a group (e.g., portfolio of securities) is described by ( ) ( ) ( )

( ).

Has a skewness and excess kurtosis of 0.

A linear combination of two or more normal random variables is also normally distributed. Standard normal distribution (unit normal distribution) has a and as ( ). To

standardize a normal random variable with any mean or standard deviation use ( ) , so that, ( ) ( ) ( ) ( ).

Drawbacks:

A normal random variable has the range of . Having no lower limit, makes

normal distribution inappropriate to model asset prices (≥ 0) and less appropriate to model

asset returns (≥ -100%). Despite that, normal distribution is a good approximation of returns

as the normal probability of outcomes below 100% is very small.

Assuming normality is problematic for equity and options returns. Equity returns mostly has

an excess kurtosis > 0 (fat-tails problem), and options are skewed. Accordingly, normal

distribution underestimates the probability of extreme events for equity and options

returns.

Applications:

Roy’s safety-first criterion states that, if returns are normally distributed, the safety-first optimal portfolio maximizes the safety-first ratio (minimizes short fall risk). ( ( ) ) . The probability that will be less is ( ).

o The lognormal distribution: A random variable follows a lognormal distribution if its natural logarithm, , is normally

distributed. In other words, , is lognormally distributed if is normally distributed. The lognormal distribution is described by and , which are a function of the associated variable and .

This distribution is bounded below by 0 accurately describe the distribution of asset prices, and right skewed confidence intervals around the mean cannot be constructed in the same way as in the symmetric normal distribution.

( ) if continuously compounded returns are normally

distributed (or approximately normally distributed according to central limit theorem), then asset prices are lognormally distributed.

An estimate of volatility is crucial for using option pricing models such as the Black-Scholes-Merton model. Volatility measures the standard deviation of the continuously compounded returns on the underlying assets.

To estimate annual volatility, we start with standard deviation of the continuously compounded daily

returns then annualize it using √

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Monte Carlo simulation: o Involves the use of computer to generate a large number of random samples (using a uniform random variable

between 0 and 1) from a specified probability distribution(s) to produce a frequency distribution for changes in simulated subject (e.g., portfolio value).

o Uses: Financial planning. Estimate VAR (Value at Risk) Value complex securities for which no analytic pricing formula is available (e.g., European-style

options). o Limitations:

Fairly complex Sensitive to assumptions about the distribution of the risk factors and the pricing model used. Provides only statistical estimates, not exact results. Analytical methods, where available, provide

more insight into cause-and-effect relationships. o Historical simulation (back simulation):

An alternative to Monte Carlo simulation. It samples from an actual historical record of returns (or other underlying variables) to simulate a process.

Its concept is that the historical record provides the most direct evidence on distributions (and that the past applies to the future).

Drawbacks:

Any risk factor not represented in the time period selected will not be simulated.

Unlike Monte Carlo simulation, doesn’t provide “what if” analyses.

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SAMPLING AND ESTIMATION Quantitative Methods R-10 (SS-3)



Sampling: o Simple random sample: sampling with each element of the population has an equal probability of being

selected. Systematic sampling is one such sampling plan where every kth

member is selected until the desired size is completed.

o Stratified random sampling: a simple random sample drawn from subsets of the population (strata) based on a classification criteria. Advantages of stratified over simple random sampling:

Guarantees that population subdivisions of interest are represented. Estimates of parameters have greater precision (smaller variance).

o Sampling error: the difference between sample statistic and population parameter. o Sampling distribution of a statistic: the distribution of all the distinct possible values that the statistic can

assume when computed from samples of the same size randomly drawn from the same population. o Time-series and cross-sectional data:

Time-series: data concerning one subject over many periods of time. Caution should be taken not to select data that have witnessed a structural (regime) change. In such case, conclusions will be sourced from two different populations (with different distributions) and thus flawed.

Cross-sectional: data concerning many subjects at a single point in time. Subjects selected should be homogeneous to ensure homogenous distributions.

The Central Limit Theorem: o Given a population with any probability distribution having and finite , the sampling distribution of the

sample mean computed from samples of size will be approximately normal with and , when is large (≥ 30).

o Standard error of : √ , when we know the population , otherwise, √ .

Point and interval estimates of population mean ( ): o Properties of sound estimators:

Unbiasedness: expected value of the estimator (the mean of its sampling distribution) = parameter. Efficiency: has the least variance among other unbiased estimators of the same parameter. Consistency: the probability of estimates close to the parameter increases as sample size increases.

o Confidence interval: A range for which one can assert with a given probability , that it will contain the parameter it’s

intended to estimate. Note that is the probability size in each tail. Constructed as . Reliability factor is a

number based on the distribution of the point estimate and the degree of confidence. As the degree of confidence increases, reliability factor will increase leading to a wider interval.

-distribtion:

A family of distributions defined by a single parameter which is degrees of freedom ( ).

Symmetric with a mean of 0, and less peaked with fatter tails ( ) compared to standard normal -distribution, which causes reliability factor to be higher than value for the same degree of confidence, thus, more conservative.

As the number of increases, -distribution approaches -distribution and reliability factors becomes closer (lower).

Confidence interval for :

Normally distributed population with known variance: √

Large sample, population variance unknown -Alternative: √

Population variance is unknown–large sample regardless of population distribution–small

sample from normal or approximately normal population: √ , with

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SAMPLING AND ESTIMATION Quantitative Methods R-10 (SS-3)



Basis for computing reliability factor:

Sampling from: Small sample size Large sample size

Normal distribution with known variance z z

Normal distribution with unknown variance t t or z (t is preferred)

Non-normal distribution with known variance NA z

Non-normal distribution with unknown variance NA t or z (t is preferred)

Selection of sample size:

All else equal, the larger the sample size, the narrower is the confidence interval (more precision).

To increase the sample size; the analyst should weigh the need for precision against the risk of sampling from more than one population and additional expenses.

Issues in sampling plans: o Data-mining bias:

The practice of determining a model by extensive searching (drilling) through a dataset for statistically significant patterns. Intergenerational data mining involves using information developed by previous researchers to guide current research using the same or related dataset.

Detection:

Conduct an out-of-sample test, if the proposed variable or investment strategy remains statistically and economically significant then it’s valid (have a story), otherwise, data-mining might be present. The most crucial out-of-sample test is future investment success (have a future), however, when become public; prices may adjust to reflect the new strategy.

Be alert to phrases such as “we (or someone) noted that,…” o Sample selection bias:

Appears when data availability leads to certain assets being excluded from the analysis. Such as:

Survivorship bias: sampling from a database that tracks only companies currently in existence.

Removal (or delisting) of a company’s stock from an exchange.

Hedge funds with poor performance will not wish to make their records public, creating a self-selection bias in hedge fund databases.

o Look-ahead bias: If the test uses information that was not available on the test date. e.g., using P/B multiple as of end

of year in equity valuation at the end of the year, as Book Value was not available back then. o Time-period bias:

Conclusions drawn from a time-series test are sensitive to the time-period selected, as follows:

If it’s too-short, the results are likely time-period specific. e.g., market anomalies, like value stocks (with low P/B) outperform growth stocks, should be tested over several business cycles.

If too long, dataset might have witnessed a structural change.

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HYPOTHESIS TESTING Quantitative Methods R-11 (SS-3)



General notes: o Test structure:

is the null hypothesis. Always have the equal sign. is the alternative hypothesis or the condition that we presume as true. Depending on the

relational operator of , the test is two-tailed for , one-tailed for or . o Test statistic:

o Significance level ( ): For a hypothesis test, there are four possibilities:

Reject a true null hypothesis: Type I error (probability of which = )

Don’t reject a false null hypothesis: Type II error (probability of which (= ) is inversely related to )

Don’t reject a true null hypothesis: this is a correct decision.

Reject a false null hypothesis: this is a correct decision. Probability of which is power of the test ( )

There is a trade-off between and . The only way to decrease both is by increasing sample size ( ). o Decision rule:

Rejection point (critical value) for the test statistic: A value, based on the test distribution and significance level, with which the computed test statistic is compared to decide whether to reject or not reject the null.

o An significance level in a two-tailed hypothesis test, is equivalent to a ( ) confidence interval. o -value (marginal significance level): is the smallest significance level at which the null can be rejected.

Meaning that if -value reject , otherwise don’t reject.

Hypothesis tests concerning the mean: o Tests concerning a single mean:

Population variance unknown, and is large, or small with normal or approximately normal population: ( -test with ). e.g., of a one-tailed test:

Test structure:

Test statistic:

√

Decision rule:

Interpretation: If null cannot be rejected, we conclude that is not statistically significant at the given significance level (i.e 5%).

Population variance unknown and is large: (alternative -test)

Test statistic:

√

Population is normal and variance is known (regardless of ): ( -test)

Test statistic:

√

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o Tests concerning differences between means (from independent samples): ( -test): Test Structure:

Normally distributed populations, population variances unknown but assumed to be equal: -test with ( )

Using a pooled estimator of the common variance to calculate the standard error in the test statistic. A pooled estimate is an estimate drawn from the combination of two different samples.

Normally distributed populations, population variances unknown but cannot be assumed equal: approximate -test with modified

Using each sample variance to calculate the standard error in the test statistic. The assumption of equal or unequal variances will be given, or can be deduced from the difference

between sample variances (or standard deviations).

o Tests concerning mean differences (from dependent samples): ( -test with ): Paired comparisons test for paired observations, for random variables and calculated as:

Test Structure:

Test statistic:

From the paired observations, calculate , and as usual, then use:

√

Hypothesis tests concerning variance: o Tests concerning a single variance from normal population: (chi-square test with )

Unlike distributions, is asymmetrical, bounded below by 0, and not robust sensitive to violation of its assumptions.

Test Structure:

Test statistic: ( )

o Tests concerning differences between the variances of normal populations: ( -test with ( ) for the numerator and ( ) for the denominator)

-test is the ratio of sample variances. Like distribution, -distribution is asymmetrical, bounded below by 0, and not robust.

Test Structure:

Test statistic:

following the convention reject if

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Nonparametric inference: o A test that is not concerned with a parameter (e.g., is the sample random or not?) or a test that makes

minimal assumptions about the population, or when the data are given in ranks. o It will frequently involves the conversion of observations into ranks according to magnitude, and sometimes it

will involve working with only greater than (denoted ) and less than (denoted ) relationships. o If the assumptions of the parametric test are met, the parametric is generally preferred to the nonparametric

test as it usually permits us to draw sharper conclusions. o The Spearman correlation coefficient:

Same as in , it’s a number between and . Based on the ranks of the two random variables under test. For , a -test utilizing Spearman correlation coefficient can be used to determine statistical

significance of whether , with . For Spearman rank correlation distribution for sample size.

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TECHNICAL ANALYSIS Quantitative Methods R-12 (SS-3)



Definition and scope: An analysis that uses price and volume data for freely traded securities, often graphically, in decision making. The logic is that supply and demand determine prices and their changes cause changes in prices which discount all information about the security, and can be projected with charts and other technical tools. The need for the security to be freely traded means that the application of technical analysis is limited in markets subject to large outside manipulation.

Principles and assumptions: o Unlike fundamental analysis, the major principle is that humans are emotional, thus, irrational and tend to

behave similarly in similar circumstances, accordingly, patterns (which represent a graphical depictions of the collective psychology of the market) tend to repeat.

o The greater the volume of a participant’s trade, the more impact that market participant will have on price.

Technical and fundamental analysis: o Fundamental analysis relies on wide sources of data, while technical analysis uses almost only price and

volume data which are less wide but more concrete and objective. o Fundamental analysis is the more theoretical approach (what prices should be), while technical analysis is

more practical (what prices will be). Noting that deviations from intrinsic values can persist for long periods, and that technicians are better in identifying market moves after the moves are already underway, as trends must be in place for some time before they are recognizable.

o Fundamental analysis is a younger field than technical analysis. o Technical analysis is used with all free traded security types, especially, commodities, currencies, and futures

for which there are no accompanying financial statements or income stream to get intrinsic value using fundamental analysis.

o Technical analysis is practically useful for short-term traders, as it requires less time than fundamental analysis.

o Technical analysis is superior to fundamental analysis in the case of fraud (e.g., fraudulent financial statements).

Technical analysis tools: o Charts:

Line chart: with time on -axis and price (usually closing) on the -axis. Bar chart:

Each entry has 4 prices; high, low, opening to the left, and closing to the right.

Useful in showing divergence (volatility) which is indicated by long bars. Candlestick chart:

Each entry has 4 prices, but in addition to bar charts the body of the candle is shadowed if the closing price was lower than the opening price, and clear vice versa.

Point and figure chart:

Unlike other charts, time is not represented on the -axis, alternatively, number of changes in price.

Drawn on a grid consisting of columns of X’s (increasing prices) alternating with columns of O’s (decreasing price).

If we are in an X column, a new entry is made only when price changes by the box size or reversal size, either another X if price is increasing or a new column of Os if the price is decreasing.

The box size is the height of box in the same direction; the reversal size is a multiple of box size (3 for example) in the opposite direction as a rule to start a new different column.

Being a multiple; use of reversal rule eliminates noise in the price data.

Clearly illustrate the price levels that may signal the end of a decline or advance, and the levels at which a security may frequently trade.

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Major sustained price moves are represented by long columns of X’s (when prices are moving up), or O’s (when prices are moving down). Short columns of X’s and O’s signal congestion areas where security trades up and down in a narrow range.

Scale:

-axis: The appropriate time interval depends on the nature of the underlying data and the specific use of the chart; for short-term, medium or long-term decision. Generally, the greater the volatility of the data, the more frequent time interval is useful to uncover information.

-axis: Price is plotted on a linear (arithmetic) or algorithmic scale depending on the magnitude of moves. Arithmetic scale is more appropriate with small magnitudes, while algorithmic scale is more appropriate for large scales.

Volume:

Increase in volume confirms the current market trend (up or down), while decreasing volume might signal a reversal (e.g., when prices are increasing at a decreasing volume “divergence”).

Volume charts are usually displayed below price charts, with each period’s volume shown as a vertical line.

Relative strength analysis:

To show the relative out or underperformance of some security, a line chart of the ratio of two prices, with the asset under analysis as the numerator and with a benchmark or other security as the denominator is plotted. A rising line shows the asset is outperforming, and a declining line shows the opposite.

o Trend: Uptrend occurs when forces of demand are greater than supply, thus, leading to higher highs and,

after retracement, higher lows. Uptrend is drawn by connecting the lows of the price chart. Major breakthroughs (5-10%) indicate that uptrend is over, while minor breakthroughs call for the line to be moderately adjusted.

Downtrend is the opposite of uptrend. Non-trending (sideways) occurs when the price fluctuate in a narrow range. Little useful information

can be revealed using technical analysis tools in this case. Support is defined as a low price range in which buying activity is sufficient to stop the decline in

price, resistance is the opposite. Change in polarity principle states that once a support level is breached, it becomes a resistance level

and vice versa.

o Chart patterns: For a pattern to have a predictive value, there should be a clear trend in place prior to the pattern.

Reversal patterns:

Head and shoulders: o The prior trend must be an uptrend. For an inverse head and shoulders; it’s

preceded by a downtrend. o Volume is important in interpreting the pattern. Usually volume decreases in rallies

confirming that a reversal is expected. Strong volume in the left shoulder which decreases in the formation of head, and further decreases in the right shoulder.

o The neckline is the line connecting the lows of the pattern (highs of the inverse pattern) and acts as a support level (resistance in the inverse pattern) during the formation of the pattern, and a resistance level (support in the inverse pattern) when the pattern is completed.

o Price target: - For head and shoulders ( ) - For inverse head and shoulders ( )

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Double tops and bottoms: o Double tops pattern occurs when an uptrend reverses twice at roughly the same

high price level. Double bottom is the opposite. o The longer the time between the two tops (bottoms), and the deeper the sell-off

(demand) after the first top (bottom), the more significant the pattern is considered to be.

o Price target: - For double tops ( ) - For double bottoms ( )

Triple tops and bottoms: o Traders have no way to determine whether a double top or bottom will be followed

by a third one. o More significant than double tops and bottoms as the tops, bottoms, valley levels

are more tested, and as the time required to form the pattern is longer.

Continuation patterns: Indicates a change of ownership from one group of investments to another. called “a healthy correction”.

Triangles: o Formed by two intersecting trendlines connecting the highs and the lows. o Ascending triangles are formed in uptrends. With same highs and lower lows,

buyers are becoming more bullish and the pattern breaks in uptrend. o Descending triangles are formed in downtrends. With same lows and lower highs,

sellers are becoming more bearish and the pattern breaks in downtrend. o Symmetric triangles are formed in up or downtrends. With lower highs and higher

lows, buyers are becoming more bullish at the same time that sellers are becoming more bearish and the pattern breaks in the same direction as the preceding trend.

o Price target (according to the preceding pattern: ( )

Rectangle pattern: o Formed by two parallel trendlines connecting the highs and the lows. o Bullish rectangle in uptrend and bearish rectangle in downtrend.

Flags and pennants: o Form over short periods of time. o Flags are rectangular with the parallel trendlines slope in a direction opposite to the

preceding trend. o Pennants are triangular. The difference between a pennant and triangle is that a

pennant is a short-term formation whereas a triangle is a long-term formation. o Price target: Flags and pennants form almost at the half way through the trend ( )

o Technical indicators:

Measure how potential changes in supply and demand might affect prices. Price based indicators:

Moving average: o The average of the closing price of a security over a specified number of periods.

Moving average smooth out noise from the price data and give a clearer image of market trend. The longer the time period considered the smoother the data becomes.

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o The average could be set on an equal weight basis, or giving exponentially higher weight to the most recent prices (exponentially smoothed moving average).

o Used in conjunction with price trend or other longer/shorter moving average(s). This serves as a way to set support/resistance levels, or selling and buying signals. When a short-term moving average crosses from underneath a longer-term one, this is a bullish sign (golden cross) and thereby a buying signal. When a short-term moving average crosses from above a longer-term one, this is a bearish sign (dead cross) and thereby a selling signal.

Bollinger bands: o Consist of a moving average plus a higher line representing the moving average plus

a set number of standard deviations from the average price, and a lower line that is the moving average minus same number of standard deviations. Consequently, the more volatile are the prices, the wider is the band.

o The upper band serves as a resistance level, and the lower band serves as a support level.

o Major breakout above/below the bands may trigger a continuation in the trend, according to a specific rule (i.e. % or time), buy at the upper band breakout, and sell at the lower band breakout.

Momentum oscillators: Momentum refers to out of ordinary market sentiment (that is hard to discern through price charts only), to measure it oscillations around a specific value (0 or 100) or between a specific range (0-100) are used. It can determine the strength of a trend (general trend must always be taken into account first when using oscillators) or signal a reversal through determining overbought/oversold areas and convergence/ divergence from trend, also, it can be used in sideways market.

Momentum (or Rate of Change (ROC)) oscillator: o To oscillate around 0: ( )

o To oscillate around 100: (

)

Where : oscillator value, : last closing price, and : closing price days ago.

Relative Strength Index (RSI): (default range 30-70)

o

∑( )

∑(| |)

Stochastic oscillator: (default range 20-80) o Composed of two lines; %K (act like short-term Moving Average) and the signal line

%D (act like long-term Moving Average)

o (

)

Where : last closing price, : lowest 14 days price, and : highest 14 days price

Moving Average Convergence/Divergence (MACD) oscillator: o Composed of two lines; MACD line (act like short-term Moving Average) and the

signal line (act like long-term Moving Average). o MACD line: difference between 2 (short and long) exponentially smoothed MAs.

While the signal line is an exponentially smoothed average of MACD line. o Oscillates around 0 with no default range, rather, the current level is compared to

historical levels. Sentiment indicators:

Opinion polls

Calculated statistical indices: o Put/call ratio:

A contrarian indicator of sentiment as it measures bearishness/bullishness. At extremely low levels (sentiment is overly positive) a decrease in prices is likely, and vice versa.

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o Volatility index (VIX): A contrarian indicator calculated by Chicago Board Options Exchange, with high reading refers to high bearishness.

o Margin debt: There is a strong positive correlation between margin debt and price levels. At extremely high levels, margin debt could signal an overbought market.

o Short interest ratio: Has mixed interpretation; one is that it’s a contrarian indicator, the other is that high readings suggests increase in future demand to cover short positions and thus

increase in prices.

(in days)

Flow-of-funds indicators:

Arms index (short-term trading index (TRIN):

o

o A reading of balance, more volume in declining stocks (bearish), more volume in rising stocks (bullish).

Margin debt: Besides being a sentiment indicator, it’s a Flow of Funds indicator as well, as increasing margin debt may signal potential increase in demand, and vice versa.

Mutual fund cash position: The percentage of mutual fund assets held in cash. Has mixed interpretation; one is that it’s a contrarian indicator, the other is that high readings suggests increase in future mutual funds purchases (demand) thus increase in prices.

New equity issuance (IPOs): A contrarian indicator, as IPOs are made at market tops and increase the supply of securities.

Secondary offerings: A contrarian indicator due to increase in the supply of securities.

o Cycles: The primary problems with cycles are the small sample size, and that data don’t clearly fit the cycle.

Kondratieff (K) wave: 54 years. 18-year cycle: 3 cycles make up a K wave. Decennial pattern: years ending with a 0 have the worst performance and years ending with a 5 have

the best performance. Presidential cycle: pre-election year (3

rd year) has the best performance, followed by the election

year(4th

year). Post-election year (1st

year) and midterm year (2nd

year) have the worst performance. One explanation is that politicians tends to inject stimulus into the economy in the pre-election year to improve their chances of being re-elected.

Elliott Wave Theory: o Market moves in 9-cycles (waves) from grand supercycle to subminuette. o In a bull market, the cycle follows a pattern of an impulsive wave (3 ups and 2 corrections), followed by a

corrective wave (2 downs and 1 correction). The other way around for the bear market. o Each wave in the same trend of the market consists of 5 subwaves, and each wave in contrary to the trend

consists of 3 subwaves. o Waves follow patterns that are ratios of the numbers in the Fibonacci sequence (0,1,1,2,3,5,…) where each

number representing the addition of the preceding two numbers in the sequence. The ratio of Fibonacci number to the next (1/2, 2/3, 3/5,…)(and its inverse 2, 3/2, 5/3,…) are of great importance to the waves formulation.

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Intermarket Analysis: o Based on the principle that all markets are interrelated. o Involves using relative strength analysis for different groups of securities (e.g., stocks vs. bonds, sectors in an

economy, and securities within the same sector). o Technicians often look for inflection points in one market as a warning sign to start looking for a change in

trend in a related market.

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Quantitative methods - Level I - CFA Program

Education

Transcript of Quantitative methods - Level I - CFA Program