Volume 1 - Quantitative Methods Flashcards
What are the 3 possible interpretations for a interest rate ?
An interest rate, r, can have three interpretations: (1) a required rate of return, (2) a discount rate, or (3) an opportunity cost. An interest rate reflects the relationship between differently dated cash flows.
An interest rate can be viewed as the sum of the real risk-free interest rate and a set of premiums that compensate lenders for bearing distinct types of risk: an inflation premium, a default risk premium, a liquidity premium, and a maturity premium.
r = Real risk-free interest rate + Inflation premium + Default risk premium +
Liquidity premium + Maturity premium.
What is A money-weighted return ?
A money-weighted return reflects the actual return earned on an investment after accounting for the value and timing of cash flows relating to the investment.
What is A time-weighted return ?
A time-weighted return measures the compound rate of growth of one unit of currency invested in a portfolio during a stated measurement period. Unlike a money-weighted return, a time-weighted return is not sensitive to the timing and amount of cashflows and is the preferred
performance measure for evaluating portfolio managers because cash
withdrawals or additions to the portfolio are generally outside of the control of the portfolio manager.
Gross return, return prior to deduction of managerial and administrative expenses (those expenses not directly related to return generation), is an appropriate measure to evaluate the comparative performance of an asset manager.
Net return, which is equal to the gross return less managerial and administrative expenses, is a better return measure of what an investor actually earned.
Annualizing periodic returns allows investors to compare differnt investments across different holding periods to better evaluate and compare their relative performance. With the number of compounding periods per year approaching infinity, the interest is compound continuously.
Real returns are particularly useful in comparing returns across time periods because inflation rates may vary over time and are particularly useful for comparing investments across time periods and performance between different asset classes with different taxation.
If USD 9,500 today and USD 10,000 in one year are equivalent in value, then USD 10,000 – USD 9,500 = USD 500 is the required compensation for receiving USD 10,000 in one year rather than now. The interest rate (i.e., the required compensation stated as a rate of return) is USD 500/USD 9,500 = 0.0526 or 5.26 percent.
An opportunity cost is the value that investors forgo by choosing a course of action. In the example, if the party who supplied USD 9,500 had instead decided to spend it today, he would have forgone earning 5.26 percent by consuming rather
than saving. So, we can view 5.26 percent as the opportunity cost of current
consumption.
What is The real risk-free interest rate ?
The real risk-free interest rate is the single-period interest rate for a completely risk-free security IF NO inflation were expected. In economic theory, the real risk-free rate reflects the time preferences of individuals for current versus future real consumption.
The sum of the real risk-free interest rate and the inflation premium is the nominal
risk-free interest rate
The nominal risk-free interest rate reflects the combination of a real risk-free rate plus an inflation premium:
(1 + nominal risk-free rate) = (1 + real risk-free rate)(1 + inflation premium)
In practice, however, the nominal rate is often approximated as the sum of the
real risk-free rate plus an inflation premium:
Nominal risk-free rate = Real risk-free rate + inflation premium.
Typically, interest rates are quoted in annual terms, so the interest rate on a 90-day government debt security quoted at 3 percent is the annualized rate and not the actual interest rate earned over the 90-day period.
Whether the interest rate we use is a required rate of return, or a discount rate,
or an opportunity cost, the rate encompasses the real risk-free rate and a set of risk premia that depend on the characteristics of the cash flows. All these premia vary over time and continuously change, as does the real risk-free rate. Consequently, all interest rates fluctuate, but how much they change depends on various economic fundamentals—and on the expectation of how these various economic fundamentals can change in the future.
The arithmetic mean return assumes that the amount invested at the beginning of each period is the same. In an investment portfolio, however, even if there are no cash flows into or out of the portfolio the base amount changes each year. The previous year’s earnings must be added to the beginning value of the subsequent year’s investment— these earnings will be “compounded” by the returns earned in that subsequent year. We can use the geometric mean return to account for the compounding of returns.
In general, the arithmetic return is biased upward unless each of the underlying
holding period returns are equal. The bias in arithmetic mean returns is particularly
severe if holding period returns are a mix of both positive and negative returns.
For reporting historical returns, the geometric mean has considerable appeal
because it is the rate of growth or return we would have to earn each year to match
the actual, cumulative investment performance.
The arithmetic mean is always greater than or equal to the geometric mean.
If we want to estimate the average return over a one-period horizon = we should use arithmetic, because it is the average of one-period returns.
If we want to estimate the average returns over more than one period = geometric, because the geometric mean captures how the total returns are linked over time.
The harmonic mean : sample of observations of 1, 2, 3, 4, 5, 6, and 1,000, the harmonic mean is 2.8560.
Compared to the arithmetic mean of 145.8571, we see the influence of the outlier (the 1,000) to be much less than in the case of the arithmetic mean. So, the harmonic mean is quite useful as a measure of central tendency in the presence of outliers.
Unless all the observations in a dataset are the same value, the harmonic mean is always less than the geometric mean, which, in turn, is always less than the arithmetic mean.
What is the trimmed mean ?
Both the trimmed and the winsorized means seek to minimize the impact of outliers in a dataset. Specifically, the trimmed mean removes a small defined percentage of the largest and smallest values from a dataset containing our observation before calculating the mean by averaging the remaining observations.
What is the winsorized mean ?
A winsorized mean replaces the extreme observations in a dataset to limit the
effect of the outliers on the calculations. The winsorized mean is calculated after
replacing extreme values at both ends with the values of their nearest observations,
and then calculating the mean by averaging the remaining observations.
What is the money-weighted return ?
The money-weighted return accounts for the money invested and provides the
investor with information on the actual return she earns on her investment. y. Amounts invested are cash outflows from the investor’s perspective and amounts returned or withdrawn by the investor, or the money that remains at the end of an investment cycle, is a cash inflow for the investor.
What is The internal rate of return ?
The internal rate of return is the discount rate at which the sum of present values
of cash flows will equal zero.
What are the 2 categories of Data and the 4 sub-categories?
1- Categorical: values that describe a quality or characteristic (must be mutually exclusive)
(N) Numerical: No logical order (Ex: Sectors of economy)
(O) Ordinal: has a logical order (no info about the distance between groups)
2- Numerical : Measured or counted quantities
(I) Integer/Discrete : Limited to a finite number of values
(R) Ratio/Continous : Can take any value within a range
What is Cross-sectional , Time-series and Panel Data ?
1- Cross-sectional : Multiple observations of a particular variable (stock prices of 60 companies)
2- Time-series : Multiple observations of a particular variable for the same observational unit over time (GM stock price in the last 5 days)
3- Panel Data : Cross-sectional + Time-series
What is structured and unstructured Data ?
1- Structured : Highly organized in a pre-defined manner
2- Unstructured Data : No organized form (social media, news) –> Also called alternative Data
In order to analyse Data, it must be transformed into structured data
For numerical data, how can we determine the interval width ?
Range (max - min) / K
K: number of intervals –> too few or too much can bring problems ; loss of info or too much noise
What are the uses of arithmetic mean and what to do if there’s a problem ?
The arithmetic mean can be usefull to explain the return for 1 year of an Index.
Cross-sectional mean : Average sales of 50 companies
Time-series mean : Average sales for the last 10 yrs for GM
This mean is susceptible to outliers: We can do nothing if they are legitimate and contain meaningful information
Or:
Delete the outliers by doing a trimmed mean. Excluding a small % of the lowest and highest values (Ex: 5% –> 2.5% highest and 2.5% lowest)
Or:
Replace the 2.5% by the value at which all others lie above –> the 96th observation 88 so 2.5% also become 88.
What are the different types of mode list ?
unimodal: only 1 value that is most frequent
bi-modial: two values have the highest frequency
….
Or no mode –> Uniform distribution
What is the main use for the Geometric mean ?
It is used to interpret the growth rate. Ex: The rate that makes your investement grow form initial enter into now.
Also referred to as compounded returns
What is the use of the harmonic mean ?
It is appropriate for averaging ratios when the ratios are repeatedly applied to a fixed quantity to yield a variable number of units.
Ex: Dollar cost averaging
What is dispersion ?
The variability around the central tendency
Two investors in the same mutual fund could have different money-weighted returns depending on the amount and timing of their contributions. A fund manager’s performance should only be judged on the basis of his or her decisions and actions. The money-weighted return can be skewed by the timing and amount of cash flows into and out of a fund, making it an inappropriate metric for assessing the performance of a manager who has no control over these.
Because TWR is unaffected by the timing and amount of cash flows, it is appropriate for assessing managers who do not control external cash flows, such as a mutual fund that is regularly receiving new contributions and making payouts to meet redemptions.
For an interest rate that compounds
times annually, the formulas for the future value and present value of an investment are:
FVn = PV(1+rs/m)** mn
PV = FVn (1+rs/m)** -mn
If we assume a 365-day year, the annualized return for an investment that generates a 0.6% return over 8 days is:
(1 + 0.006)** 365/8 - 1 = 31.4%
This example illustrates one of the limitations of annualizing returns, which is that the calculations are based on the assumption that short-term performance could be repeated over a longer period.
It is also possible to annualize returns that have been generated over holding periods of longer than one year. In these case, the number of periods per year, c, becomes a fraction. For example, if an investment earns a 17.8% return over two years, it’s annualized return is:
( 1 + 0.178) ** 1/2 - 1 = 8.53%
Gross returns are calculated on a pre-tax basis; trading expenses are accounted for in the computation of gross returns as they contribute directly to the returns earned by the manager.
The Time Value of Money in Finance
calculate and interpret the present value (PV) of fixed-income and equity instruments based on expected future cash flows
calculate and interpret the implied return of fixed-income instruments and required return and implied growth of equity instruments given the present value (PV) and cash flows
explain the cash flow additivity principle, its importance for the no-arbitrage condition, and its use in calculating implied forward interest rates, forward exchange rates, and option values
There are three basic categories of fixed-income instruments depending on how their cash flows are structured — discount instruments, coupon instruments, and annuity instruments.
1- Discount instruments (zero-coupons) have a very simple structure. One amount PV is borrowed today and a larger amount FV is repaid when the loan matures.
2- With coupon instruments, a principal amount PV is borrowed today and the same amount FV is repaid at maturity, but the borrower compensates that lender with periodic interest payments PMT at regular intervals during the term of the loan.
3- An annuity instrument is structured as a specified number of level cash flows. A common example of an annuity is a fixed-rate mortgage. Like a coupon instrument, the borrower makes payments at regular intervals to retire the debt.
A perpetual bond, also known as a perpetuity, is a special type of coupon instrument that makes fixed payments at regular intervals but never matures. The present value of an instrument that provides a perpetual stream of level payments is calculated as follows:
PV = PMT/r
Statistical Measures of Asset Returns
calculate, interpret, and evaluate measures of central tendency and location to address an investment problem
calculate, interpret, and evaluate measures of dispersion to address an investment problem
interpret and evaluate measures of skewness and kurtosis to address an investment problem
interpret correlation between two variables to address an investment problem
The mode is the most frequently occurring value in a distribution. Some distributions have more than one mode, while others have none. A distribution with just one mode is unimodal, with two modes is bimodal, and so on.
Data grouped in intervals have modal intervals. This is the highest bar in a histogram.
There are three ways to deal with outliers:
1- No adjustments : This is appropriate if all values are equally important and meaningful.
2- Remove all outliers : A trimmed mean is calculated by discarding a certain percentage of the highest and lowest values. For example, with a sample of 100 observations, a 2% trimmed mean would be the arithmetic mean without the highest value (top 1%) and the lowest value (bottom 1%).
3- Replace outliers with another value : A winsorized mean adjusts any outliers’ values to either an upper or lower limit. No observations are excluded from the calculation.
If we arrange the observations in ascending order, then the quantile is a value at or below which a stated fraction of the data is found.
There are many common quantiles used in practice. Distributions are often divided into four quartiles, five quintiles, ten deciles, or one hundred percentiles.
For example, the 90th percentile score (P90) on an exam is the number that separates the top 10% scores from the bottom 90%.
The interquartile range (IQR) is the difference between the third quartile and the first quartile.
The most common measures of absolute dispersion are range, mean absolute deviation, variance, and standard deviation.
The range is the difference between the maximum and minimum values
The mean of the deviations around the mean will always be zero. Therefore, it would not be a useful measure of dispersion. The mean absolute deviation (MAD) adjusts for this.
Variance is the average of the squared deviations around the mean, while standard deviation is the positive square root of the variance.
Downside Deviation :
Variance and standard deviation take into account returns above and below the mean. But investors care for downside risk.
Target semideviation, or target downside deviation, captures dispersion of observations below a specified target value (e.g., 10%).
Coefficient of Variation :
The coefficient of variation (CV) is a relative dispersion measure. It allows comparisons between data sets with very different means. CV is the ratio of the standard deviation, S, to the mean, X.
Mean has to be positive.
No units of measurement
Skewness :
A positively skewed distribution (long right tail) has frequent small losses and a few extreme gains. The mode is less than the median, which is less than the mean.
A negatively skewed distribution (long left tail) has frequent small gains and a few extreme losses. The mean is less than the median, which is less than the mode.
!! Investors should be concerned if returns have this distribution.
Kurtosis :
Kurtosis measures if a return is more or less peaked than a normal distribution. A normal distribution has a kurtosis value of 3.
(L) : Leptokurtic ( greater than 3, Fat Tails) –> Meaning that extreme returns are more common
(M) : Mesokurtic (normal distribution)
(P) : Platykurtic ( less than 3, Thin Tails)
Excess kurtosis equals kurtosis minus 3. This measures kurtosis relative to the kurtosis of a normal distribution.
The excess kurtosis (0.15) is positive, indicating that the distribution is “fat-tailed”; therefore, there is more probability in the tails of the distribution relative to the normal distribution.
The correlation metric quantifies the linear relationship between two variables.
Properties of Correlation :
1- The correlation coefficient is bounded by -1 and +1.
2- A correlation of 0 indicates that there is no linear relationship between the two variables.
3- A positive correlation coefficient (i.e., rxy > 0) indicates a positive linear relationship between the variables. In other words, an increase in X is associated with an increase in Y. When rx = 1, the variables have a perfect positive linear relationship.
4- A negative correlation coefficient (i.e., rxy < 0) indicates a negative linear relationship between the variables. In other words, an increase in X is associated with an decrase in Y. When rx = -1, the variables have a perfect inverse linear relationship or perfect negative linear relationship.
Limitations of Correlation Analysis :
1- The correlation coefficient is not a reliable measure when the variables have a nonlinear relationship.
2- The correlation coefficient is very sensitive to outliers.
3- Correlation does not imply causation.
4- The conclusions may not be valid. A spurious correlation refers to:
- The correlation between two variables that reflects chance relationships (i.e., just a coincidence)
- The correlation produced by a calculation that mixes the two variables with a common third variable
- The correlation between two variables that arises due to their relation to a third variable (although the two variables are not correlated)
5- Correlation may not produce a full picture of the data. (Same correlation but different relationship)
The correlation coefficient only measures the degree of linear association between two variables. It does not explain the amount each variable changes.
A correlation coefficient can only be between –1 and +1, but covariance is not subject to the same constraint.
Probability Trees and Conditional Expectations
calculate expected values, variances, and standard deviations and demonstrate their application to investment problems
formulate an investment problem as a probability tree and explain the use of conditional expectations in investment application
calculate and interpret an updated probability in an investment setting using Bayes’ formula
Forecasts of a random variable are often based on an expected value, which is the probability-weighted average of its possible outcomes.
The variance of a random variable is the probability-weighted average of the squared deviations from its expected value.
A random variable’s variance must be greater than zero because, if there is no dispersion of outcomes, the expected value is known with certainty and the variable is not random.
The probabilities that are used as the basis for forecasts are rarely static. Analysts are continually updating their unconditional (marginal) probabilities to reflect the latest information. This type of forecasting produces conditional expected values.
The conditional expected value of X given that scenario S occurs is expressed as
E( X|S ).
The total probability rule for expected value : states that the unconditional expected value is equal to the probability-weighted average of the conditional expected values.
Bayes’ formula is a rational way to adjust viewpoints based on new information. It is based on the total probability rule.
Prior probabilities represent the probabilities before the arrival of any new information. The posterior probability reflects the new information.
Bayes Formula :
P (Event | Info) = ( P (Info| Event) /
(Event | Info) ) * P (Event)
Portfolio Mathematics
calculate and interpret the expected value, variance, standard deviation, covariances, and correlations of portfolio returns
calculate and interpret the covariance and correlation of portfolio returns using a joint probability function for returns
define shortfall risk, calculate the safety-first ratio, and identify an optimal portfolio using Roy’s safety-first criterion
Investment opportunities should be evaluated in the context of how they impact the tradeoff between a portfolio’s expected return and the level of portfolio risk, as measured by variance.
Covariance measures the tendency for two variables to move in sync.
Covariance is positive if, when one asset is generating above-average returns, the other asset is as well. Both assets will also tend to generate returns below their respective averages in the same periods.
Covariance is negative if one asset is generating above-average returns while the other’s returns are below its average (or vice versa).
The covariance of an asset’s returns with itself (own covariance) is equal to its variance.
If there are n securities in a portfolio, there are n (n-1) / 2 covariances to estimate
A zero correlation only tells us that there is no linear relationship, but it doesn’t tell us anything about the possibility of a non-linear relationship.
Random variables are independent if, and only if, P (XY) = P(X) * P(Y). If two random variables are independent, they must also be uncorrelated. By contrast, knowing that variables are uncorrelated does not allow us to conclude that they are independent.
If two random variables are uncorrelated, then the multiplication rule for the expected value of the product of uncorrelated random variables tells us that E(XY) = E(X) * E(Y). If we cannot assume that the variables are uncorrelated, we must calculate a conditional expected value.
Safety-first rules are designed to address shortfall risk, which is the possibility that portfolio returns will fail to meet a specified threshold.
The safety-first ratio can be used in a portfolio context to account for correlations between returns on individual assets.
Mean-variance analysis considers risk symmetrically by relying on a measure of dispersion (standard deviation) that captures the volatility of returns both above and below the mean. By contrast, the safety-first ratio is only concerned about the risk that returns will fall below a minimum acceptable level.
Simulation Methods
explain the relationship between normal and lognormal distributions and why the lognormal distribution is used to model asset prices when using continuously compounded asset returns
describe Monte Carlo simulation and explain how it can be used in investment applications
describe the use of bootstrap resampling in conducting a simulation based on observed data in investment applications
Unlike a symmetric normal distribution, a lognormal distribution is bounded by zero on the left, which creates a long right tail (i.e., positive skew). This feature make it a useful distributional assumption for asset prices, which are often not normally distributed and cannot fall below zero.
By definition, lognormal random variables cannot have negative values.
If a random variable Y follows a lognormal distribution, then ln(Y) is normally distributed. Conversely, if we know that ln(Y) is normally distributed, it must be true that the distribution of Y is lognormal.
Continuously Compounded Rates of Return :
An asset’s continuously compounded return can be calculated :
r0,T = ln(ST/S0)
Difference between inital price and end of investment horizon
As noted, if an asset’s continuously compounded return is normally distributed, then its price is lognormally distributed. In fact, the future price may be lognormally distributed even if the returns are not normally distributed.
Monte Carlo simulation is particularly useful for valuing European-style options and it is often used to value complex financial instruments, such as mortgage-backed securities. This method can also be used in cases when there are no applicable analytical models.
For example, there is no option pricing model for Asian-style options, which make payoffs based on the difference between the strike price and the average price of the underlying asset over a specific period.
The following steps provide a general overview of the Monte Carlo simulation:
1- Specify the quantities of interest in terms of underlying variables.
2- Split the time horizon into subperiods.
3- Specify the method for generating the data used in the simulation.
4- Use a computer program to draw K
random values of each risk factor and produce a value for the variable of interest.
5- Calculate the average value of the variable of interest.
6- Repeat Steps 4 and 5 for N
trials. The Monte Carlo estimate is the mean quantity of interest over the N
trials.
Bootstrapping :
Resampling is the process of repeatedly drawing samples from a larger pool of sample data in order to make statistical inferences about the population parameters. One of the most commonly-used methods, known as bootstrap resampling relies on computer simulation rather than analytical tests, such as the t-test or z-test.
The main drawback of this technique is that it produces only statistical estimates, while analytical methods provide exact results and insights into cause-and-effect relationships. In practice, bootstrap resampling is often used as a complement to analytical methods.
Monte Carlo simulation can only provide statistical estimates, not precise valuations. And, unlike analytical methods, Monte Carlo simulation does not provide insight into the causal relationships between variables.
However, Monte Carlo simulation can be used to value many types of options that cannot be priced with analytical methods (e.g., Asian call options).
Bootstrapping through random sampling generates the observed variable from a random sampling with unknown population parameters. The analyst does not know the true population distribution, but through sampling can infer the population parameters from the randomly generated sample.
Analysts performing bootstrap: seek to create statistical inferences of population parameters from a single sample.
Estimation and Inference
compare and contrast simple random, stratified random, cluster, convenience, and judgmental sampling and their implications for sampling error in an investment problem
explain the central limit theorem and its importance for the distribution and standard error of the sample mean
describe the use of resampling (bootstrap, jackknife) to estimate the sampling distribution of a statistic
Sampling is used to get information about a parameter of a population. A statistic from a sample is used because measuring the parameter from the population is either not possible or cost-prohibitive.
The two types of sampling methods :
1- Probability sampling: Every member of the population has the same chance of being selected. The sample created is usually representative of the population.
2- Non-probability sampling: Non-probability considerations (such as the convenience to access data or the sampler’s judgment) are used in sample selection. The sample created may not be representative of the population.
Probability sampling :
1- Simple random sampling: each population element has an equal probability of being selected. It is also often called just a random sample.
2- Systematic sampling: This method chooses every Kth member until the desired sample size is reached. It is useful if the analyst cannot identify all members of a population.
3- Stratified random sampling: the population is first divided into subgroups (strata) based on some criteria. Simple random samples are drawn from each subgroup in proportion to the subgroup’s relative size to the entire population. This method results in less variance than estimates derived from simple random sampling.
4- Cluster sampling : divides the population into subgroups known as clusters. However, unlike stratified random sampling which defines subgroups based on certain criteria, cluster sampling divides the entire sample into mini-representations of the population. In other words, each cluster will consist of samples with different characteristics.
The sampling error is the difference between the sample statistic and the population parameter (e.g., the sample mean and the population mean).
The sampling distribution of a statistic is the distribution of all statistic values calculated from the same sample size from the same population.
Stratified sampling is commonly used to create portfolios that are meant to track a bond index. First, the entire population of bonds in the index is divided into subgroups based on factors such as maturity, sector, credit quality, etc. The manager then selects a sampling of bonds from within each subgroup.
Non-probability sampling:
1- Convenience sampling: selects samples based on how accessible they are for the researcher. This method allows samples to be collected quickly at a low cost.
2- Judgmental sampling: the researcher handpicks samples based on their knowledge and professional judgment. This sampling method is beneficial when there is a time constraint because the researcher can quickly select a more representative sample using their expertise.
Example of cluster :
A research analyst may want to study the average annual return of institutional investors in Europe. Since there are too many institutional investors to be included in the study, the analyst can use a two-stage cluster sampling as follows:
Stage 1: Group the institutional investors by countries in Europe where each country represents a cluster of investors. Then, randomly select a few clusters (i.e., a few countries).
Stage 2: Randomly choose a selection of institutional investors from the clusters that were identified in Stage 1 and calculate an average of their returns.
Systematic sampling can be used when not all members of a population can be coded or even identified to be placed in groups.
The sample mean is a random variable with a probability distribution called the statistic’s sampling distribution.
Central Limit Theorem :
The sampling distribution of a sample mean X will be approximately normal with a mean of u (the population mean) and variance of o**2 / n, provided the sample size n is large (usually greater than 30). This is true for a population with any probability distribution provided it has a finite variance o **2.
According to the central limit theorem, when the sample size increases, the distribution of the sample mean will converge to a normal distribution. This is true regardless of the actual distribution of the population.
Note that the standard error describes the accuracy of an estimate (from sampled data) relative to its true value.
A population has a non-normal distribution with mean and variance. The sampling distribution of the sample mean computed from samples of large size from that population will most likely have:
A
the same distribution as the population distribution.
B
its mean approximately equal to the population mean.
C
its variance approximately equal to the population variance.
B) Given a population described by any probability distribution (normal or non-normal) with finite variance, the central limit theorem states that the sampling distribution of the sample mean will be approximately normal, with the mean approximately equal to the population mean, when the sample size is large.
According to the central limit theory, the variance of the distribution of the sample mean is equal to the variance of the population divided by sample size.
Resampling is a process that allows analysts to repeatedly draw samples from the original data set. This is important when the sample size is too small to accurately estimate the population parameter.
Bootstrap :
Usually requires computer simulation. Using this method, each sample drawn is being replaced with an identical element for the next draw, so the sample size stays the same after each draw. The size of each resample is also same as the size of the original sample.
The greater the number of resamples, the smaller the estimated standard error of the sample mean.
Boostrap is able to determine the standard error and confidence intervals for statistics such as the median. In addition, it produces accurate estimates without relying on any analytical formula.
Jackknife :
While bootstrap repeatedly draws samples with replacement, jackknife draws samples by leaving out one observation at a time (without replacement).
Hypothesis Testing
explain hypothesis testing and its components, including statistical significance, Type I and Type II errors, and the power of a test.
construct hypothesis tests and determine their statistical significance, the associated Type I and Type II errors, and power of the test given a significance level
compare and contrast parametric and nonparametric tests, and describe situations where each is the more appropriate type of test
Hypothesis testing is used to determine whether a sample statistic is likely from a population with the hypothesized value of the population parameter.
Provide an insight to this question by examining how a sample statistic describes a population parameter.
The steps in the hypothesis testing process are:
1- State the hypotheses.
2- Identify the appropriate test statistic.
3- Specify the level of significance.
4- State the decision rule.
5- Collect data and calculate the test statistic.
6- Make a decision.
Stating the Hypotheses
Two hypotheses are always stated:
1- Null hypothesis: H0 -> This is assumed true until the test proves otherwise.
2- Alternative hypothesis: H1 -> This is only accepted if there is sufficient evidence to reject the null hypothesis.
Together, the null and alternative hypotheses must be collectively exhaustive, which means that they account for every possible outcome. They must also be mutually exclusive, meaning that any outcome must either confirm the null hypothesis or provide sufficient evidence to indicate that the null hypothesis should not be accepted.
For example, if an analyst is attempting to show that the mean annual return of a stock index has exceeded 10%, the null hypothesis should be that the mean return is less than or equal to 10%. The alternative hypothesis should only be accepted if statistical tests provide sufficient evidence that the mean return is not less than or equal to 10%.
The null hypothesis can be rejected or not rejected after the test statistic has been calculated. The decision is based on a comparison that assumes a specific significance level, which establishes how much evidence is required to reject the null hypothesis.
A Type I error occurs if a true null hypothesis is mistakenly rejected. A Type II error occurs if a false null hypothesis is mistakenly accepted.
The probability of a Type I error is the level of significance of the test, which is denoted as ‘a’. The complement of this probability,
1- ‘a’, is the confidence level. For example, a level of significance of 5% corresponds to a confidence level of 95%. There is a 5% chance of incorrectly rejecting a true null hypothesis.
Reducing the probability of a Type I error by decreasing ‘a’ will increase the probability of a Type II error.
The complement of the probability of a Type II error is the power of a test. This is the probability of correctly rejecting the false null hypothesis. The power equals
1 -‘B’ .
The decision rule must be stated when comparing the test statistic’s calculated value to a given value based on the significance level of the test. The critical value of the test statistic is the rejection point of the null hypothesis. It is set based on the level of significance and the probability distribution associated with the test statistic.
The null hypothesis is rejected when the test statistic is calculated to be more extreme than the critical value(s). In this case, the result is known to be statistically significant.
The smallest level of significance at which a null hypothesis can be rejected is called the p-value.
For a two-tailed test, or two-sided test, there are two ways to reject the null hypothesis:
1- The sample/estimator is significantly smaller than the hypothesized value of the population parameter.
2- The sample/estimator is significantly larger than the hypothesized value of the population parameter.
When performing a two-sided test, the 5% level of significance is split evenly into two regions of rejection (i.e., one region in each tail of the distribution). By contrast, a one-tailed test has only one region of rejection.
A right-side test because the null hypothesis of u =< u0 will be rejected if the estimator is significantly greater than the hypothesized value. For a left-side test, the null hypothesis would be u >= u0 , which will be rejected if the estimator is significantly less than the hypothesized value.
A t-test is commonly used to test the value of an underlying population mean. The t-distribution is similar to the standard normal distribution, but it is impacted by the degrees of freedom. Smaller degrees of freedom produce fatter tails.
Can be used for a population with unknown variance and large sample (>= 30)
Test Concerning Differences Between Means With Independent Samples : Often, one desires to compare the means between groups and find if the populations are approximately normally distributed and the samples are independent.
Test Concerning Differences Between Means With Dependent Samples : The prior tests were only valid if the samples were unrelated to each other (i.e., independent).
A paired comparison test is a statistical test for differences in dependent items.
The difference between two random variables taken from dependent samples, denoted di, is calculated. Then, the list of differences is statistically analyzed.
A pooled estimator is most likely used when testing a hypothesis concerning the:
difference between the means of two at least approximately normally distributed populations with unknown but assumed equal variances.
The assumption that the variances are equal allows for the combining of both samples to obtain a pooled estimate of the common variance.
A paired comparisons test is appropriate to test the mean differences of two samples believed to be dependent.
The chi-square test can be used to test the relationship between an observed sample variance and its hypothesized value. The chi-square distribution is an asymmetrical family of distributions defined with degrees of freedom. It does not take on negative values.
Test Concerning a Single Variance
Test Concerning the Equality of Two Variances :
We can use the F-test to examine the equality/inequality of two population variances.
The F-test is used based on the ratio of the sample variances. The F-distribution is bounded below by 0 and defined by two values of degrees of freedom – one for the numerator, and one for the denominator.
The hypothesis testing procedures discussed thus far deal with parameters and are dependent on assumptions (e.g., normality of distribution). A test that has either characteristic is known as a parametric test.
On the other hand, a nonparametric test is not concerned with parameters or makes minimal assumptions on the underlying population.
Nonparametric procedures are used when:
1- The hypotheses do not concern a parameter.
2- The data are given in ranks or use an ordinal scale : Nonparametric tests are more appropriate for ranked data.
3- The data are subject to outliers: Extreme values may influence parametric statistics, but they have no effect on nonparametric statistics (e.g., median).
4- The data do not meet distributional assumptions : The sample may be too small (i.e., less than 30 observations), or not normally distributed / its variance may be unknown.
When data is ranked, such as when it has been sorted according to an ordinal scale, the assumptions of parametric tests (e.g.,
z-test, t-test) do not hold and a nonparametric test should be used. Parametric tests require a stronger measurement scale.
Parametric and Non-Parametric Tests of Independence
explain parametric and nonparametric tests of the hypothesis that the population correlation coefficient equals zero, and determine whether the hypothesis is rejected at a given level of significance
explain tests of independence based on contingency table data
The sample correlation between variables X and Y is denoted as rXY. This sample statistic allows us to test hypotheses about the population correlation coefficient, denoted as p.
If the hypothesis being tested is that the two variables have a statistically significant relationship in the population, a two-sided test is used and the null hypothesis is p = 0
. The alternative hypothesis, Ha, that p has a non-zero value is only accepted if there is sufficient evidence to reject the null hypothesis.
If an analyst hypothesizes a positive relationship between two variables in a population, a one-sided test is used and the null hypothesis is p =< 0 with a single region of rejection in the right tail.
The parametric correlation coefficient (a.k.a. the Pearson correlation or bivariate correlation) between two variables can be tested using their sample correlation statistic.
The Spearman rank correlation coefficient, denoted rS, is similar to the correlation coefficient but does not rely on the same underlying assumptions. Notably, this statistic is appropriate to use when it is believed that the underlying population is not normally distributed.
To test the independence between the sectors and analyst ratings for this sample, calculate the chi-square test statistic ;
The null hypothesis is rejected if the calculated chi-square test statistic exceeds a critical value, indicating that the variables are not independent.
One region of rejection, on the right side of the distribution. All else equal, the critical chi-square value will be higher as the number of degrees of freedom increases and the region of rejection narrows.
It would be valuable for the analyst to know which cells have observations that deviate significantly from their expectations, assuming the variables are independent. This deviation is captured by the standardized residual (a.k.a. Pearson residual).
Simple Linear Regression
describe a simple linear regression model, how the least squares criterion is used to estimate regression coefficients, and the interpretation of these coefficients
explain the assumptions underlying the simple linear regression model, and describe how residuals and residual plots indicate if these assumptions may have been violated
calculate and interpret measures of fit and formulate and evaluate tests of fit and of regression coefficients in a simple linear regression
describe the use of analysis of variance (ANOVA) in regression analysis, interpret ANOVA results, and calculate and interpret the standard error of estimate in a simple linear regression
calculate and interpret the predicted value for the dependent variable, and a prediction interval for it, given an estimated linear regression model and a value for the independent variable
describe different functional forms of simple linear regressions
Linear Regression :
- One variable is the dependent variable or the explained variable. This refers to the variable whose variation is being explained. It is typically denoted by Y.
- The other variable is the independent variable or the explanatory variable. This refers to the variable whose variation is being used to explain the variation of the dependent variable. It is typically denoted by X.
The variation of Y is also referred to as the sum of squares total (SST), or the total sum of squares.
Note the difference between the error term and the residual:
- The error term corresponds to the true underlying population relationship.
- The residual corresponds to the estimated linear relationship based on the sample.
Interpreting the Regression Coefficients :
The intercept is the value of the dependent variable if the value of the independent variable is set to zero. Note that the intercept may not have meaning if it is not realistic for the independent variable to be zero.
Cross-sectional data has many observations for X and Y for the same time period. Typical cross-sectional data are observations from different companies or different investment funds. For example, you could test if the predicted earnings-per-share growth impacted the price-to-earnings ratio in a given time period.
Typical time-series data has many observations from different time periods for the same company or asset class. For example, you could collect monthly data on inflation rates to determine if they impact short-term interest rates.
The four assumptions underlying the simple linear regression model are:
1- Linearity: The relationship between the dependent variable and the independent variable is linear.
2- Homoscedasticity: The variance of the residuals is constant for all observations.
3- Independence: The pairs (X,Y) are independent of each other. This implies the residuals are uncorrelated across observations.
4- Normality: The residuals are normally distributed.
If the relationship is not linear, the model will underestimate or overestimate the dependent variable at certain points, and thus produce biased results.
Another implication of the linearity assumption is that the independent variable must not be random (i.e., it must be non-stochastic). This is because the linear relationship between the dependent variable and the independent variable would not exist if the independent variable is random.
Consequently, the residuals are random.
The residuals are assumed to be homoscedastic, which means the variance of residuals is constant for all observations.
A violation of this assumption indicates that the data series may come from two different regimes.
Independence :
The pairs (X,Y) are assumed to be uncorrelated with one another, which means they are independent. This implies residuals will be uncorrelated across observations. This assumption is also needed to correctly estimate the variances of ^b0 and ^b1.
Correlation of residual errors indicates that this assumption of independence has been violated.
The normality assumption does not require the dependent and independent variables to be normally distributed; only residuals must be normally distributed. A violation of this assumption may lead to invalid test statistics of the regression coefficients.
For large sample sizes, the central limit theorem applies, and consequently we may drop the normality assumption.
The test statistics of the regression coefficients are still valid even if residuals are not normally distributed.
The homoskedasticity assumption states that the variance of the error term is the same for all observations. It does not imply that this measure has an expected value of zero.
The assumption of linearity states that the dependent and independent variables are linearly related. This implies that the independent variable is not random. If it were, there would be no linear relationship with the dependent variable.
The sum of squares total (SST), which is the total variation in Y, can be broken into two components:
1- Sum of squares error (SSE), which is the unexplained variation in Y
2- Sum of squares regression (SSR), which is the explained variation in Y
Measures to evaluate how well the regression model fits the data include:
- The coefficient of determination
- The F-statistic for the test of fit
- The standard error of the regression
Coefficient of Determination :
The coefficient of determination (a.k.a.
R-squared or R^2) measures the fraction of the total variation in the dependent variable that is explained by the independent variable
If there is only one independent variable in the regression, then the coefficient of determination is equal to the square of the correlation between the dependent variable and the independent variable :
R^2 = r^2
F-satistic :
While the coefficient of determination is descriptive, it is not a statistical test. To evaluate if our regression model is statistically meaningful, an F-test statistic is needed.
The standard error of the estimate is also known as the standard error of the regression or the root mean square error. The smaller the se, the more accurate the regression.
In a simple linear regression, the T test statistic used to test whether the slope coefficient is zero is equal to the T test statistic used to test whether the pairwise correlation is 0
An indicator variable (a.k.a. dummy variable) takes on only the values 0 or 1 as the independent variable. To illustrate, consider a company’s monthly stock returns over a 5-year period. A simple linear regression with monthly returns as the dependent variable and the indicator variable as the independent variable can be used. The indicator variable takes on a value of 0 if there is no earnings announcement in that month and 1 if there is an earnings announcement
This regression allows testing whether there are different returns for months with an earnings announcement compared to months without an earnings announcement.
The common choice for the level of significance is 0.05, which indicates a 5% probability of rejecting a true null hypothesis (a Type I error).
The smaller the P-value, the smaller the probability of Type I error, the more likely the regression is valid.
In a simple regression with a single indicator variable, the intercept is the mean of the dependent variable when the indicator variable takes on a value of zero, which is before the shift in policy in this case.
The P-value is the smallest level of significance at which the null hypotheses concerning the slope coefficient can be rejected. In this case, the P-value is less than 0.05, and thus the regression of the ratio of cash flow from operations to sales on the ratio of net income to sales is significant at the 5% level.
The P-value corresponding to the slope is less than 0.01, so we reject the null hypothesis of a zero slope
If the relationship between the independent variable and the dependent variable is not linear, one or both of these variables can be transformed to convert this relationship to a linear form so that the simple linear regression can still be used.
3 forms :
1- Log-lin : The slope coefficient is the relative change in the dependent variable for an absolute change in the independent variable.
2- Lin-log : The slope coefficient is the absolute change in the dependent variable for a relative change in the independent variable. Useful for significant difference in variables scale’s.
3- Log-log : The slope coefficient is the relative change in the dependent variable for a relative change in the independent variable. Useful when calculating elasticities (relative change).
A direct comparison between a log-lin model and a lin-lin model should not be done because the dependent variables are not in the same form.
Selecting the Correct Functional Form :
The appropriate function form of the simple linear regression depends on goodness of fit:
The coefficient of determination, R^2
The F-statistic
The standard error of the estimate, se
Introduction to Big Data Techniques
describe aspects of “fintech” that are directly relevant for the gathering and analyzing of financial data
describe Big Data, artificial intelligence, and machine learning
describe applications of Big Data and Data Science to investment management
Key areas of fintech include:
- Analysis of large datasets, such as from social media and sensor networks
- Analytical tools, including artificial intelligence techniques
Big Data datasets have traditionally been defined by three key characteristics:
- Volume: Data volumes have grown from megabytes to petabytes.
- Velocity: High velocity is possible because more data are now available in real-time.
- Variety: Dataset were once limited to structured data (e.g., SQL table), but now include semi-structured data (e.g., HTML code) and even unstructured data from non-traditional sources such as social media, emails, and text messages.
- “Veracity” : Analysts must be able to trust the reliability and credibility of data sources.
Big Data sources include:
financial markets
businesses (e.g., corporate financials)
governments (e.g., trade and economic data)
individuals (e.g., credit card purchases, internet search logs, and social media posts)
sensors (e.g., satellite imagery and traffic patterns)
Internet of Things (e.g., data from “smart” buildings)
3 main sources of alternative data are generated by individuals, business processes, and sensors.
Alternative data are used to identify factors that affect security prices, which can then be used to improve asset selection and trading. But investment professionals should be cautious about collecting personal information that is protected by regulations.
Challenges with Big Data include quality, volume, and appropriateness. Dataset may contain outliers or reflect biases, so they must be sourced, cleansed, and organized before they can be used.
Big Data is collected from many different sources and is in a variety of formats, including structured data (e.g., SQL tables or CSV files), semi-structured data (e.g., HTML code), and unstructured data (e.g., video messages).
It involves formats with diverse structures.
Machine learning (ML) algorithms are computer programs that learn how to complete tasks, improving with time as more data have become available. ML models are trained to map relationships between inputs and outputs.
Potential errors include overfitting, which occurs when an ML model learns the inputs and target dataset too well. Overfitting can cause ML models to treat noise in the data as true parameters.
- Underfitting can cause the ML model to treat true parameters as noise.
- Supervised learning, computers learn from labeled training data. The inputs and outputs are identified for the algorithm ( identify the best variable to forecast future stock returns).
- Unsupervised learning, only the dataset is provided to the algorithm. The inputs and outputs are not labeled (group companies into peer groups).
Deep learning utilizes neural networks to identify patterns with a multistage approach.
Algorithms are not explicitly programmed, which can result in outcomes that are not easily understood.
Supervised learning involves explicitly labeling data as either inputs or outputs. However, it is not essential to label outputs.
An algorithm initially identifies relationships in a training dataset before further testing is performed using an evaluation or validation dataset.
Data Processing Methods :
- Data capture – Collecting data and transforming them to a usable format.
- Low-latency systems communicate high volumes of data with minimal delay, which is needed for automated trading.
- Data curation – Cleaning data to ensure high quality.
- Data storage – Recording, archiving, and accessing data.
- Search – Locating specific information in large datasets.
- Transfer – Moving data from their source or storage location to the analytical tool.
Tag clouds can be used to show the frequency of keywords in the data. Words that appear more often are in a larger font.
Text analytics use computer programs to analyze unstructured text- or voice-based datasets.
Natural language processing (NLP) is an application of text analytics that focuses on interpreting human language. It is commonly used for tasks such as translation, speech recognition, text mining, and sentiment analysis.
monitor communications among employees to ensure compliance with policies.
Text analytics is appropriate for application to:
A
economic trend analysis.
B
large, structured datasets.
C
public but not private information
A is correct. Through the text analytics application of NLP, models using NLP analysis might incorporate non-traditional information to evaluate what people are saying—via their preferences, opinions, likes, or dislikes—in the attempt to identify trends and short-term indicators about a company, a stock, or an economic event that might have a bearing on future performance.
Data-mining is the practice of determining a model by extensive searching through a dataset for statistically patterns.
- The absence of an explicit economic rationale for a variable of trading strategy is the “no story” warning sign of a data-mining porblem.
- The testing of many variables by the researcher is the “too much digging” warning sign of a data-mining porblem.
