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Cfa > Book 1 - Quant Methods > Flashcards

Book 1 - Quant Methods Flashcards

1
Q

Numerical discrete data

A

Countable values such as months, days or hours in a year

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2
Q

Numerical continuous data

A

Can take any fractional value (eg. the annual percentage return on an investment)

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3
Q

Default risk

A

The risk that a borrower will not make the promised payments in a timely manner 

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4
Q

Liquidity risk

A

 The risk of receiving less than fair value for an investment if it must be sold for cash quickly

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5
Q

Maturity risk

A

The risk associated withholding a longer term debt security. Longer, maturity bonds, have more maturity risk than shorter term bonds and require a maturity risk premium.

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6
Q

Nominal rate of interest = _ rate + _ premium + _ premium + _ premium

A

= nominal risk free rate + liquidity premium + maturity risk premium + default risk premium

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7
Q

Future value factor

A

= (1+ I/Y) ^N

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8
Q

Present value factor

A

= 1 / (1+ I/Y) ^N

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9
Q

Ordinary annuity

A

Most common type of annuity where cash flows occur at the end of each compounding. This is the typical cash flow pattern for many investment and business finance applications.

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10
Q

Annuity due

A

 a type of annuity where payments occur at the beginning of each period example; the first payment is today at T = 0

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11
Q

Perpetuity

A

A financial instrument that pays a fixed amount of money at set intervals over an infinite period of time can also be called perpetual annuity, example; preferred stocks, because they promise, fixed interest, or dividend payments forever = PMT / (I/Y)

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12
Q

Cash flow, additivity principle

A

 The fact that present value of any stream of cash flows equals the sum of the present values of the cash flows example; you can divide up a series of cash flows in anyway you like, and the present value of the pieces will equal the present value of the original series 

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13
Q

EAR Effective annual rate

A

 represents the annual rate of return actually being earned after adjustments have been made for different compounding periods

EAR = (1+ periodic rate)^M - 1
Periodic rate = stated annual rate/M
M = the number of compounding periods per year

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14
Q

Real risk free rate

A

A theoretical rate on a single period loan, where there is no expectation of inflation

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15
Q

Nominal risk free rate

A

Equal to real risk free rate plus expected inflation rate

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16
Q

Required rate of return on a security =

A

= real risk free rate + expected inflation + default risk premium + liquidity premium + maturity risk premium

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17
Q

Future value formula

A

= PV (1+ I/Y)^N
= PV multiplied by future value factor

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18
Q

Present value formula

A

= FV / (1+ I/Y)^N
= FV / present value factor

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19
Q

Categorical or qualitative data

A

Consist of labels that can be used to classify a set of data into groups categorical data may be nominal or ordinal

Cannot be used to perform mathematical operations

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20
Q

Nominal data

A

Labels that cannot be placed in order logically

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21
Q

Ordinal data

A

Can be ranked in a logical order based on a specific characteristic, then these categories are ordered with respect to that characteristic.

Example; when ranking 1000 small cap, stocks by performance, you may categorize Number one to the first 100 best performing stocks, number two to the second 100 best performing stocks and so on through number 10 for the 100 worst performing stocks. Based on these categories, we can say a stock ranked in category. Three performed better than a stock ranked in category four.

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22
Q

Time series

A

A set of observations taken periodically, most often at equal intervals, overtime example; daily closing prices of a stock over the past year

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23
Q

Cross sectional data

A

A set of comparable observations, all taken at one specific point in time example; today is closing prices of the 30 stocks in the Dow Jones industrial average

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24
Q

Structured data

A

 data organized in a defined way example; time series, cross-sectional, panel, data, market, data, fundamental data, analytical data

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25
Q

Panel data

A

A combination of time series and cross-sectional data. Often presented in tables where each row might represent cross-sectional data, and each column might represent time series data.

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26
Q

Unstructured data

A

Information that is presented in a form with no defined structure, unstructured data typically must be transformed into structured data for analysis

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27
Q

One dimensional array

A

And a ray that represents a single variable, a key feature of a time series is that new data can be added without affecting the existing data sequentially ordered data are used to identify trend cycles and other patterns

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28
Q

Two dimensional array, a.k.a. a data table

A

 example organized data sequentially with a cross-section of observations for each measurement date

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29
Q

Frequency distribution

A

Presentation of statistical data for large data sets. Summarizes statistical data by assigning data to specific groups or intervals.

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30
Q

Modal interval

A

The interval with the greatest frequency, a.k.a. the most number of observations

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31
Q

Relative frequency

A

Calculated by dividing the absolute frequency of each return by the total number of observations

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32
Q

Absolute frequency of an interval

A

The number of data points assigned to an interval

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33
Q

Contingency table

A

A two-dimensional array, where two variables can be analyzed at the same time. Rose represent attributes of one of the variables, columns represent attributes of the other variable.

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34
Q

Joint frequencies

A

Data in each cell of a contingency table, showing the frequency with which we observe two attributes simultaneously

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35
Q

Marginal frequency

A

The total total frequencies for a row or a column

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36
Q

Confusion matrix

A

A type of contingency table. For each of two possible outcomes, a confusion, matrix displays the number of occurrences predicted, and the number actually observed.

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37
Q

Defining intervals of data

A

1.Range of values for each interval must have a lower and upper limit.
2. Intervals must be all inclusive.
3. intervals must not overlap
4. intervals must be mutually exclusive, so that each observation can be placed in only one interval.
Note: the number of intervals used is important. If two few intervals are used, the data may be two broadly summarized, or if too many intervals are used, the data may not be summarized enough 

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38
Q

Time preference

A

Economic term for real risk free rate

The degree to which current consumption is preferred to equal future consumption

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39
Q

HPR holding period return

A

The percentage increase in the value of an investment over a given period

HPR = ((end of period balance) / (beginning of period balance)) - 1

Ex. For a stock end of period balance includes stock price and dividends paid

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40
Q

Arithmetic mean

A

AM - Simple average of a series of periodic returns
AM = (R1 + R2 + R3) / N
Where R = return

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41
Q

Harmonic mean

A

Used to calculate the average share cost from periodic purchases

HM = N / sum of (1/Xi)
Where N = number of periods
And 1/Xi = 1 over each periodic return over
*denominator is summed

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42
Q

Geometric means

A

Compounded rate of returns over multiple periods

(Nth root of (1+ R1)(1+R2)(1+RN)) -1

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43
Q

Interest rates

A
  • required rate of return
  • discount rate
  • opportunity cost of consumption
  • can include a risk premium component but NOT always
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44
Q

IRR internal rate of return

A

The interest rate at which a series of cash inflows and outflows sum to zero when discounted to their present value

NPV = 0

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45
Q

Money weighted return

A

The IRR (internal rate of return) on a portfolio taking into account all cash inflows and outflows

1) determine the timing of each cash flow and whether the cash flow is an inflow + into the account or an outflow - to the investor
2) net the cash flows for each period and set the PV of cash inflows equal to the PV of cash outflows
3) solve for R using the calculator CF functions

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46
Q

Time weighted rate of return

A

Measures the compound growth and is the rate at which $1 compounds over a specified time horizon

  • preferred method of performance measurement for portfolio managers

1) break the evaluation into sub periods based on timing of cash flows (typically sub periods are years)
2) calculate the HPR for each holding period (end value/beinning value) - 1
3) solve for the geometric mean

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47
Q

Price relative

A

End of period value / beginning of period value

*part of the HPR formula not including -1

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48
Q

Continuous compounding

A

Mathematical limit or shortening the compounding period

Rcc = ln(1+ HPR) = ln (end value/beginning value)

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49
Q

Gross return

A

Total return on a security portfolio before deducting fees for the management and admin fees

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50
Q

Net returns

A

Return after management and admin fees have been deducted

*note commissions and other costs to generate the investment returns are deducted from both gross and net returns

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51
Q

Pretax nominal return

A

Return before paying taxes

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52
Q

After tax nominal return

A

Return after taxes are deducted

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53
Q

Real return

A

Nominal return adjusted for inflation

Ex. Nominal return is 7% and inflation is 2%
Real return = 7-2 or 5%

(1 + real return) = ((1+real risk free rate)(1+risk premium)) / (1+ inflation premium)

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54
Q

Leveraged return

A

Return to an investor that is a multiple of the return on the underlying asset - calculated as the gain or loss on the investment as a percentage of an investors cash investment

  • investment in a derivative produces a leveraged return because the cash deposited is only a fraction of the value of the assets underlying the derivative

LR = (r(V0 + VB) - rBVB) / V0

V0 = investment amount without leverage
VB = the amount of borrowed funds
r = rate of return
rB = borrowing rate for leveraged funds

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55
Q

Trimmed mean

A

Used to reduce the effect of outliers

Example; the trimmed mean excludes a stated percentage of the most extreme observations. A one percent trimmed mean for example, would discard the lowest 0.5%, and the highest 0.5% of the observations.

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56
Q

Relationship between money weighted and time weighted return calculation

A

If funds added before a period of poor performance, MWR < TWR

if funds added before a period of high returns, MWR > TWR

TWR preferred measure of the managers ability to pick investments

If the manager controls the money flows into and out of an account then MWR is preferred

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57
Q

Amortizing bond

A

One that pays a level amount each period including its maturity period - different from fixed coupon because each payment includes some portion of the principal

Annuity payment = (r * PV) / (1- (1+r)^-t)

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58
Q

DDMs - assume constant future dividend

A

Value CS same as PS
dividend payment / required rate of return

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59
Q

DDMs - assume constant growth rate of dividends

A

Apply the constant growth DDM aka the Gordon growth model

  • V0 represents the PV of ALL dividends in future period beginning with D1

V0 = D1 / (ke - gc)

V0 = value of a share this period
D1 = dividend expected to be paid NEXT period
Ke = required return on common equity
Gc = constant growth rate of dividends

  • ke must be greater than gc
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60
Q

DDMs - assume a changing growth rate of dividends

A

Can be done in many ways - testable example is multistage DDM

*assumes a pattern of dividends in the short term such as a period of high growth followed by a constant growth rate of dividends in the long term

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61
Q

Required rate of return formula

A

Ke = (D1/V0) + gc

  • rearrangement of the constant growth DDM where the required rate is the ratio of the expected dividend to the current price (dividend yield) plus the assumed constant growth rate
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62
Q

Implied growth rate formula

A

Gc = ke - (D1 / V0)

  • rearranged DDM constant growth formula where the implied growth rate is the required rate of return minus the dividend yield
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63
Q

No arbitrage principle

A

AKA law of one principle

If 2 sets of future cash flows are identical under all conditions, they will have the same price today (or if they don’t investors will quickly buy the cheaper security and sell the more expensive security driving their prices together)

Valuation based on no arbitrage principle are - forward interest rates, forward exchange rates, and option pricing using a binomial model

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64
Q

Forward interest rate

A

The interest rate for a loan to be made at some future date

Notation must identify the length of the loan and when in the future the money will be borrowed
Ex. 2y1y the rate for a 1 year loan to be made 2 years from now

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65
Q

Spot interest rate

A

Interest rate for a loan to be made today

S1 - a one year rate today
S2 - a two year rate today

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66
Q

No arbitrage - forward interest rates

A

Cash flow additive principle: borrowing for 3 years at the 3 year spot rate or borrowing for 1 year periods in 3 successive years should have the same cost today

(1+ S3)^3 = (1+ S1)(1+ 1y1y)(1+ 2y1y)

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67
Q

No arbitrage - forward currency rates

A

Forward / spot = (1+ IR of price currency) / (1+ IR of base currency)

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68
Q

No arbitrage - Binomial model - options pricing

A

BM - based on the idea that over the next period some value will change to 1 of 2 possible values

Need
- value of the underlying asset at the beginning of the period
- exercise price for the option
- returns that result from an up move and a down move in the value of the underlying over one period
- risk free rate over the period

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69
Q

Measures of central tendency

A

Identify the center or average of a data set

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70
Q

Modal, interval

A

For continuous data such as investment returns, we typically do not identify a single outcome as the mode. Instead, we divide the relevant range of outcomes into intervals, and we identify the modal interval as the one into which the largest number of observations fall

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71
Q

Winsorized mean

A

Another technique for dealing with outliers instead of discarding, the highest and lowest observations, we substitute a value for them

Example; to calculate a 90% winsorized mean we would determine the fifth and 95th percentile of the observations substitute the 5th percentile for any values lower than that substitute the 95th percentile for any values higher than that and then calculate the mean of the revised data set

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72
Q

Dispersion

A

Defined as the variability around the central tendency

Common theme in finance investments, is the trade-off between reward, and variability, where the central tendency is the measure of the reward, and dispersion is the measure of risk

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73
Q

Mean, absolute deviation MAD

A

The average of the absolute returns of the deviations of individual observations from the arithmetic mean

= (SUM | Xi - arithmetic mean |) / N

Note: the computation of the MAD uses the absolute values of each deviation from the mean because the sum of the actual deviations from the arithmetic mean is zero

The resulting MAD can be interpreted to mean that on average an individual return will deviate plus or minus the MAD from the arithmetic mean return

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74
Q

Sample variance

A

The measure of dispersion that applies when a evaluating a sample of an observations from a population

S^2 = (SUM (Xi - arithmetic mean)^2) / (n - 1)

The denominator n - 1 is used because mathematical theory behind statistical procedures states that the use of the entire number of sample observations will systematically underestimate the population variance particularly for small sample sizes. This systematic underestimation causes the sample variance to be a biased estimator of the population variance

A major problem with using variance is the difficulty of interpreting it computed variance unlike the mean is in terms of squared units of measurement, typically percent squared, dollars squared etc.

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75
Q

Sample standard deviation

A

The square root of the sample variance

S = square root of ((SUM (Xi - arithmetic mean)^2) / (n - 1))

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76
Q

Relative dispersion / coefficient of variation

A

The amount of variability in a distribution around a reference point where benchmark relative dispersion is commonly measured with the coefficient of variation

CV = (standard deviation of X) / (average value of X)

CV is used to measure the risk per unit of expected return a lower CV is better CV enables the comparison of dispersion across different sets of data

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77
Q

Downside risk

A

In some situations it may be more appropriate to consider only outcomes less than the mean in calculating a risk measure

78
Q

Target downside deviation / target semi deviation

A

Calculating target downside deviation is similar to calculating standard deviation, but in this case, we choose a target value against which to measure each outcome, and only include deviations from the target value in our calculation, if the outcomes are below that target

S target = (SUM of all observations < B (Xi - B)^2) / (n - 1)

Where B = the target
Note: the denominator remains the same sample size N minus one even though we are not using all of the observations in the numerator

79
Q

Positively skewed distribution

A

Characterized by outliers greater than the mean in the upper region or right tail a positively skewed distribution is said to be skewed right because if it’s relatively longer upper right tail 

Mode < median < mean
* Mean is affected by outliers; in a positively skewed distribution, there are large positive outliers, which tend to pull the mean upward or more positive 

80
Q

negatively skewed distribution

A

Has a disproportionately large amount of outliers less than the mean that fall within its lower left tail a negatively skewed distribution is said to be skewed left because of its long lower tail

Mean < median < mode
* large negative outliers tend to pull the mean downward to the left

81
Q

Normal distribution

A

Also known as a symmetrical distribution. Distributional symmetry implies that intervals of losses and gains will exhibit the same frequency for a symmetrical distribution the mean median and mode are equal.

82
Q

Sample Skewness

A

Equal to the sum of the cubed, deviations from the mean divided by the cubed, standard deviation, and by the number of observations

The denominator is always positive, but the numerator can be positive or negative, depending on whether observations above the mean or observations below the mean tend to be farther from the mean on average

When a distribution is right skewed sample skewness is positive because the deviations above the main are larger on average

A skewed distribution has a negative sample skewness

If relative sus is equal to zero, the data are not skewed positive levels of relative skewness imply, a positively skewed distribution, whereas negative values of relative skewness imply, a negatively skewed distribution

Values of sample skew in excess of 0.5 absolute value are considered significant

In general greater excess ketosis and more negative skew in investment return, distributions indicate increased risk

83
Q

Kurtosis

A

A measure of the degree to which a distribution is more or less peaked than a normal distribution

84
Q

Platykurtic

A

A distribution that is less peaked, or flatter than a normal one

Has excess kurtosis less than zero

85
Q

Mesokurtic

A

A distribution that has the same kurtosis as a normal distribution

Has excess kurtosis equal to zero

86
Q

Leptokurtic

A

A distribution that is more peaked than a normal distribution

A Leptokurtic return distribution will have more returns, clustered around the mean, and more returns with large deviations from the mean a.k.a. fatter tails

Relative to a normal distribution, a Leptokurtic distribution will have a greater percentage of small deviation from the mean, and a greater percentage of extremely large deviations from the mean

This means that there is a relatively greater probability of an observed value, being either close to the mean, or far from the mean, in terms of investment returns. There is a greater likelihood of a large deviation from the main return, which is often perceived as an increase in risk.

Has excess kurtosis greater than zero

87
Q

Excess kurtosis

A

A distribution is said to exhibit excess kurtosis if it has either more or less kurtosis than the normal distribution, the computed kurtosis for all normal distributions is three

Excess kurtosis = kurtosis - 3

In general greater excess ketosis and more negative skew in returns, distributions indicates increased risk

88
Q

Scatterplot

A

A method for displaying the relationship between two variables, with one variable on the vertical access, and the other on the horizontal access, the paired observation can be plotted as a single point

Advantage of creating scatterplot: can reveal non-linear relationships, which are not described by the correlation coefficient, although the correlation coefficient for two variables may be close to zero a scatterplot can show clearly that they are related in a predictable way, or not related at all 

89
Q

Covariance and sample covariance

A

Covariance is a measure of how to variables move together

Sample covariance
Sxy = (SUM ([Xi - mean X][Yi - mean Y])) / (n - 1)

The value of covariance is hard to interpret because it depends on the units of the variables like the variance. The units of covariance are the square of the units used for the data

Covariance cannot be used to interpret the relative strength of the relationship between two variables, a positive covariance can only tell us that two variables tend to move together, and a negative covariance can only tell us that two variables tend to move an opposite directions

90
Q

Correlation coefficient

A

A standardized measure of the linear relationship between two variables

= covarianceXY / ((standard Dev X)(standard dev Y))

Cxy = (Sxy) / (SxSy)

Which implies Sxy = CxySxSy

-correlation measures the strength of the linear relationship between two random variables
-Correlation does not imply that changes in one variable cause changes in the other
-Correlation has no units
-Correlation ranges from negative one to positive one
-If correlation = -1 the random variables have perfect negative correlation this means that a movement in one random variable results in an exact opposite proportional movement in the other relative to its mean
-If correlation = +1 the random variables have perfect positive correlation. This means that a movement in one random variable result in a proportional positive movement in the other relative to its mean.
-If correlation = 0 there is no linear relationship between the variables, indicating that prediction of why cannot be made on the basis of X using linear methods
-which variable is causing change in the other is not revealed by correlation
-The role of outliers can be investigated using the correlation of two variables. If removing the outliers significantly reduces, the calculated correlation, further inquiries necessary into whether the outliers provide information or are caused by noise in the data used 

91
Q

Spurious correlation

A

Refers to correlation that is either the result of chance, or present due to changes in both variables overtime that is caused by their association, with a third variable

Example: we can find instances where two variables that are both related to the inflation rate exhibit significant correlation, but for which causation in either direction is not present 

92
Q

Compound annual growth rate

A

CAGR = [((end amount/ start amount) ^1/# of years) - 1]

Also equal to the geometric mean when interpreting returns over multiple years

93
Q

Expected value

A

The expected value of a random variable is the weighted average of the possible outcomes for this variable

E(X) = SUM P(Xi)Xi

Where P(Xi) = probability of outcome Xi

94
Q

Expected value variance, and standard deviation from probability model

A

P = probability
Ra = return
SUM (P * Ra) = E(Ra) or expected return
Variance = SUM (prob * [Ra - E(Ra)]^2 )
Square root of the variance is standard deviation

Note: for the sample standard deviation, we divide the sum of the square deviation from the mean by N - 1. Where and was the size of the sample. Here we have no N because we have no observations a probability model is forward looking. We use the probability weights instead as they described the entire distribution of outcomes.

95
Q

Probability tree

A

Used to show the probabilities of various outcomes

96
Q

Conditional expected values

A

Expected, values or expected returns can be calculated using conditional probabilities. As the name implies conditional expected, values are contingent on the outcome of some other event, an analyst would use conditional expected value to revise his expectations when new information arrives.

Example: consider the effect a tariff on steel imports, might have on the returns of a domestic steel producer stock. The stocks, conditional expected return given that the government imposes, the tariff will be higher than the conditional expected return if the tariff is not imposed.

97
Q

Bayes formula

A

Formula used to update a given set of prior probabilities for a given event in response to the arrival of new information

Updated probability = ((probability of new information for given event ) / (unconditional probability of new information)) * (prior probability of event)

P(BA) / P(B) = the joint probability of A and B divided by the unconditional probability of B

98
Q

Joint vs. conditional probability

A

Joint probability is the probability of two events occurring simultaneously. While conditional probability is the probability of one event occurring in the presence of the second event.

99
Q

Expected return of a portfolio

A

E(Rp) = SUM (WiE(Ri) = W1E(R1) + W2E(R2) +…+ WnE(Rn)

Wi = weight
Ri = expected returns

To establish the portfolio weight
Wi = (market value of investment in asset i / market value of the portfolio)

100
Q

Covariance

A

A measure of how to assets move together. The expected value of the product of the deviations of the two random variables from their respective expected values. Because we are concerned with the covariance of asset returns. The following formula has been written, in terms of the covariance of the return of asset I and the return of asset J

Cov(Ri, Rj) = E {[Ri - E(Ri)][Rj - E(Rj)]}

101
Q

Properties of covariance

A

The covariance of a random variable with itself is its variance that is COV(Ra, Ra) = Var(Ra)

Covariance may range from negative infinity to positive infinity

A positive covariance indicates that 11 random variable is above its mean the other random variable also tends to be above its mean

A negative covariance indicates that when one random variable is above its mean, and the other random variable tends to be below, it’s mean 

102
Q

Sample covariance

A

sx,y = [SUM {(R1i - mean R1)(R2i - mean R2)} ] / n-1

Where
R1i = an observation of returns on asset 1
R2i = an observation of returns on asset 2
n = number of observations in the sample 

103
Q

Covariance matrix

A

Shows the cove variances between returns on a group of assets

Diagonal terms from top left are the variances of each assets returns

The covariance between returns on two assets does not depend on order so in a covariance matrix, not all covariance terms are unique

In general for an assets, there are in variance terms on the diagonal, and n(n-1) / 2 unique covariance terms 

104
Q

Portfolio variance

A

For two asset portfolio
Var(Rp) = Wa^2(Var(Ra)) + Wb^2(Var(Rb)) + 2*(WaWb(Cov(Ra, Rb))

For a three asset portfolio
Var(Rp) = Wa^2(Var(Ra)) + Wb^2(Var(Rb)) + Wc^2(Var(Rc)) + 2(WaWb(Cov(Ra, Rb)) + 2(WaWc(Cov(Ra, Rc)) + 2*(WbWc(Cov(Rb, Rc))

105
Q

Shortfall risk

A

The probability that a portfolio value or return, will fall below a particular target value or return over a given period

106
Q

Roy safety first criterion

A

States that the optimal portfolio minimizes the probability that the return of the portfolio falls below some minimum acceptable level. This minimum acceptable level is called the threshold level symbolically Roy safety first criterion can be stated as follows.

Minimize P(Rp<Rl)
Where
Rp = portfolio return
Rl = threshold level return

If portfolio returns are normally distributed, then Roy’s safety first criterion can be stated as follows

Maximize safety first ratio = [E(Rp) - Rl] / standard deviation of the portfolio

Assuming that returns are normally distributed the portfolio with the larger safety first ratio using 0% as a threshold return will be the one with the lower probability of negative returns

Note: the safety first ratio is the number of standard deviations below the mean thus the portfolio with the larger safety first ratio has the lower probability of returns below the threshold return

107
Q

Lognormal distribution

A

Generated by the function e^x where x is normally distributed

Because the natural logarithm, ln, of e^x is x, the logarithms of lognormal distributions are normally distributed hence the name

Helpful for modeling asset prices if we think of an assets future price as the result of a continuously compounded return on its current price

Pt = P0e^r0,T

Pt = asset price at time T
P0 = asset price at time 0 (today)
r0,T = continuously compounded return on the asset from time 0 to time T

Positively skewed and bounded by zero

108
Q

Independently and identically distributed returns

A

Independently distributed - past returns not useful for predicting future returns

Identically distributed - mean and variance do not change over time (a property known as stationarity)

109
Q

Monte Carlo simulation

A

A technique based on the repeated generation of one or more risk factors that affect security values to generate a distribution of security values

For each of the risk factors, the analyst must specify the parameters of the probability distribution that the risk factor is assumed to follow a computer, then used to generate random values for each risk factor based on its assumed probability distributions each set of randomly generated risk factors is used with a pricing model to value the security procedures repeated many times and the distribution of simulated asset values is used to draw inferences about the expected value of the security, and possibly the variance of security values about the mean as well

Monte Carlo simulation is used to do the following
Value complex securities
Simulate the profits and losses from trading activity
Calculate estimates of value at risk
Simulate the pension fund assets and liabilities overtime
Value portfolios of assets that have non-normal return distributions

Advantage: inputs not limited to historical data, not based on historical data and can answer what if if scenarios
Disadvantage: complexity, garbage in; garbage out 

  1. Specified distributions of random variables, such as interest rates and underlying stock prices.
  2. Use random generation of variables.
  3. Value the derivative or portfolio, using those values
  4. Repeat steps, two and 3000s of times.
  5. Calculate mean and variance of distribution of outcomes 
110
Q

Re-sampling

A

Another method for generating data inputs to use in a simulation to conduct re-sampling start with the observed sample and repeatedly draw sub samples from each with the same number of observations from these samples, we can infer parameters for the population such as its mean and variance

111
Q

Bootstrap re-sampling

A

In bootstrap Rampling, we draw repeated samples of size N from the full data set, replacing the sampled observations each time so that they may be re-drawn in another sample can directly calculate the standard deviation of the sample means as our estimate of the standard error of the sample mean

** one of the strengths of boot dropping is that it offers a good representation of the statistical features of a population
** the method does not provide exact results, and the inputs can be limited by the distribution of actual outcomes

112
Q

Probability sampling

A

Selecting a sample when we know the probability of each sample member in the overall population

113
Q

Random sampling

A

Each item is assumed to have the same probability of being selected if we have a population of data and select our sample by using a computer to randomly select a number of observations from the population each date of point has an equal probability of being selected we call the simple random sampling

114
Q

Non-probability sampling

A

Based on either low-cost or easy access to some data items or on using the judgment of the researcher and selecting specific data items

Less randomness in selection may lead to greater sampling error

115
Q

Systematic sampling

A

Sampling every Nth member from a population

116
Q

Stratified random sampling

A

Uses a classification system to separate the population into smaller groups based on one or more distinguishing characteristics from each subgroup or stratum a random sample is taken, and the results are pulled. The size of the samples from each stratum is based on the size of the stratum relative to the population.

Stratified sampling is often used in bond indexing because of the difficulty and cost of completely replicating. The entire population of bonds in this case bonds in a population are categorized, according to major bond risk factors, including, but not limited to duration, maturity and coupon rate then samples are drawn from each separate category , and combined to form a final sample

117
Q

Cluster sampling

A

Based on subsets of a population, but in this case, we assume that each subset or cluster is representative of the overall population with respect the item we are sampling for example, we may have incomes for states residence by county. The data for each county is a cluster.

To the extent that the subgroups do not have the same distribution as the entire population of the characteristic of interest cluster sampling will have greater sampling error than simple random sampling

The advantages of cluster sampling are that it is low cost, and requires less time to assemble the sample. It also may be appropriate for a smaller pilot study.

118
Q

One stage cluster sampling

A

A random sample of clusters is selected, and all the data in those clusters comprise the sample

119
Q

 two-stage cluster sampling

A

Random samples from each of the selected clusters comprise the sample

Two stage cluster sampling can be expected to have greater sampling error than one stage cluster sampling

120
Q

Convenience sampling

A

Refers to selecting sample data based on ease of access, using data that are readily available. The sample is not random so sampling error will be greater.

121
Q

Judgmental sampling

A

Refers to samples for which each observation is selected from a larger data set by the researcher based on one’s experience and judgment

122
Q

Central limit theorem

A

For simple, random samples of size N from a population with a mean, and a finite variance, the sampling distribution of the sample mean approaches a normal probability distribution with mean and variance equal to variance/n as the sample size becomes large

The sample size must be significantly large, which usually means N is greater than or equal to 30

If the sample size N is sufficiently large, the sampling distribution of the sample means will be approximately normal

The meaning of the population and the mean of the distribution of all possible sample means are equal

The variance of the distribution of sample means is variance/n

The standard error of the sample mean is the standard deviation of the distribution of sample means

Standard error of the sample mean = standard deviation of the population / square root of N

Standard error of the sample mean = Sample standard deviation / square root of N

As the sample size increases, the sample mean gets closer on average to the true mean of the population in other words, the distribution of the sample means around the population mean gets smaller and smaller, so the standard error of the sample mean decreases

123
Q

Jackknife sampling

A

Calculates multiple sample means each with one of the observations removed from the sample. The standard deviation of the sample means can then be used as an estimate of the standard error of sample means.

This is a simple tool and can be used when the number of observations available is relatively small. This method can remove bias from statistical estimates.

124
Q

Hypothesis testing procedure

A
  1. State the hypothesis.
  2. Select the appropriate test statistic.
  3. Specify the level of significance.
  4. State the decision rule regarding the hypothesis.
  5. Collect the sample and calculate the sample statistics.
  6. Make a decision regarding the hypothesis.
  7. Make a decision based on the result of the test.
125
Q

Null hypothesis

A

H0
The hypothesis that the researcher wants to reject. It is the hypothesis that is actually tested and is the basis for the selection of the test statistics
The hypothesis is always an equal to condition

-if the null is true, do not reject the null
If the all is true and is rejected, this is a type one error (alpha = probability of a type one error)

126
Q

Alternative hypothesis

A

Ha
The alternative hypothesis is what is concluded if there is sufficient evidence to reject the null hypothesis and is usually what you are really trying to assess. When the no hypothesis is discredited, the implication is that the alternative hypothesis is valid.

-if the null is false reject the null (power of the test = 1-P(type 2 error)
-if the Nolas false and you do not reject the no this is a type 2 error

127
Q

Test statistic

A

Test statistic = sample statistic - hypothesized value /(standard error)

Standard error when population standard deviation is known
= popSD / square root of n

Standard air one population standard deviation is unknown
= sampleSD / square root of n

128
Q

Type one error

A

Rejection of the null hypothesis when it is actually true

129
Q

Type two error

A

Failure to reject the Null hypothesis when it is actually false

130
Q

Significance level

A

The probability of making a type one error and is designated by the Greek letter alpha for example a significance level of 5% alpha = .05 means there’s a 5% chance of rejecting a true null hypothesis when conducting a hypothesis test, a significance level must be specified to identify the critical values needed to evaluate the test statistic 

131
Q

Power of a test

A

The probability of correctly rejecting the Null hypothesis when it is false

1-p(type 2 error)
The probability of rejecting the null when it is false, a.k.a. the power of the test equals one minus the probability of not rejecting the null when it is false a.k.a. a type two error

132
Q

P value

A

Probability of obtaining a test statistic that would lead to a rejection of the null hypothesis, assuming the null hypothesis is true. It is the smallest level of significance for which the null hypothesis can be rejected.

P value > significance then test statistic is NOT in the tail - fail to reject

P value < significance then test statistic is in the tail - reject

133
Q

Confidence level

A

Confidence level = 1 - significance level

Or 1 - the chance of error (rejecting the null hypothesis when it is actually true

134
Q

Types of hypothesis test for population means and population variances

A

T test - hypothesis concerning value of the population mean
Z test - hypothesis concerning value of the population when the sample sizes
T test - Hypothesis concerning the equality of two population means the nature of that test depends on whether the samples are independent (a difference in means test) or dependent (appeared comparison test)
Chi square - hypothesis concerning the value of a population variance
F test - hypothesis concerning the equality of two population variances

135
Q

Parametric tests

A

Rely on assumptions regarding the distribution of the population and are specific to population parameters for example, the Z test relies upon a mean and a standard deviation to define the normal distribution. The Z test also requires that either the sample is large, relying on the central limit theorem to assure a normal sampling distribution, or the population is normally distributed

136
Q

Nonparametric tests

A

Either do not consider a particular population parameter or have few assumptions about the population that is sampled. Non-parametric tests are used when there is concern about quantities other than the parameters of a distribution or when the assumptions of parametric tests can’t be supported they are also used when the data are not suitable for parametric test example ranked observations.

Situations where non-parametric test is used
-if the parametric test assumptions are not met, an example would be a hypothesis test of the mean value for a variable that comes from a distribution that is not normal and is small
-A non-parametric is ranked
-The hypothesis does not involve the parameters of the distribution, such as testing whether a variable is normally distributed we can use a non-parametric test called a run test to determine whether data are random. This test provides an estimate of the probability that series of changes are random.

137
Q

Z test

A

HO: sample mean = population mean
HA: sample mean does not equal pop mean

Compare sample mean to population
Normal distribution
Known population variance
If sample > or equal to 30 sample is considered large and Z test can be used without knowing pop variance

Test statistic = sample mean - hypothesized population mean / (pop standard deviation/ square root of N)

Standard error = standard deviation / square root of n

Denominator is the standard error of the sample mean

138
Q

Independent - difference in means
T test

A

2 types of equality of pop means (independent and dependent)
1) independent - difference in means test - assumes sample variances are equal and samples are taken from 2 normally distributed populations

H0: mean sample1 - mean sample2 = 0
HA: mean sample1 - mean sample2 does not equal 0

T statistic = sample mean - hypothesized sample mean /(sample standard deviation/square root of N)

Denominator is the estimated sample error

Degrees of freedom = n-1

139
Q

Dependent - paired comparisons test
T test

A

2 types of equality of pop means (independent and dependent)
2) dependent - paired comparisons test - normally distributed samples

H0: meand = meandz (0)
HA: meand does not equal meandz (0)

Where dz = hypothesized mean of paired differences, which is commonly zero

DOF = n-1

140
Q

Chi squared test

A

Used for the variance of a normally distributed population

Sigma squared = pop variance
Sigma squared0 = hypothesized variance

2 tailed test
H0: pop variance = hypothesized variance
HA: pop variance does not equal hypothesized variance

1 tailed test also possible

141
Q

Simple linear regression

A

The purpose of a simple linear regression is to explain the variation in a dependent variable in terms of the variation in a single independent variable

Variation is interpreted as the degree to which a variable differs from its mean value

Variation in Y = SUM (Yi - meanY) ^2

142
Q

Dependent variable

A

The variable who is variation is explained by the independent variable

143
Q

Independent variable

A

The variable used to explain the variation of the dependent variable

144
Q

 simple linear regression model

A

Y = mx + b

Yi = b0 + b1xib + E1

Yi = ith observation of the dependent variable Y
Xi = ith observation of the independent variable X
B0 = regression intercept term
B1 = regression slope coefficient
Ei = residual for the ith observation also referred to as to disturbance term or error term

Regression line = The estimated equation for a line through a scatterplot of the data that best explains the observed value for Y in terms of the observed values for X

The equation is the same, but all variables, except the observed value of X are predicted

145
Q

Some of squared errors SSE

A

The sum of the vertical distances between the estimated and actual Y values 

146
Q

Ordinary least squares OLS

A

The simple linear regression is frequently referred to as OLS because the regression line is the line that minimizes the sum of squared errors SSE

The values determined by the estimated regression equation are called lease squares estimates

147
Q

Slope coefficient

A

Estimated slope coefficient for the regression line describes the change in why for a one unit change in X it can be positive negative or zero depending on the relationship

B1 hat = covXY / variance X

148
Q

The estimated intercept

A

Lines intersection with the X axis X equals zero it can be positive negative or zero

B0 hat = mean of Y - (B1 hat * mean of X)

The intercept is an estimate of the dependent variable when the independent variable is zero

149
Q

Beta

A

The Slowe coefficient in a regression of the excess returns of an individual security variable on the return on the market. X variable is called the stocks beta, and is an estimate of systematic risk of a stock.

A stock with a beta or regression slope, coefficient of one has an average level of systematic risk in a stock with a beta of greater than one has a more than average level of systematic risk if beta is less than one, the stock is less risky than the average

150
Q

Assumptions of linear regression

A
  1. Linear relationship exist between the dependent and the independent variables.
  2. The variance of the residual term is constant for all observations.
  3. the residual term is independently distributed a.k.a. the residual for one observation is not correlated with that of another observation (and observations are independent of each other)
  4. The residual term is normally distributed.
151
Q

Homoskedasticity

A

Refers to the case where prediction errors all have the same variance

152
Q

Heteroskedasticity

A

Refers to the situation when the assumption of homoskedasticity is violated

Occurs when a plot of observations around a fitted regression line has prediction areas or residuals that increase in magnitude with larger values of the independent variable X. It is likely that variance is not constant for all observations.

Residuals can be measured in magnitude with larger values of the independent variable, or they can be measured if variance of the error changes over time

153
Q

Independence

A

An observation that some prediction errors are noticeably larger than others

Can be investigated by plotting the residuals overtime

If the plot illustrates that there are large predictive errors at a regular cadence, this suggest that there is seasonality in the data

If the relationship is not independent, the residuals are not independent, and our estimates of the models parameters variances will not be correct

154
Q

Normality

A

When the residuals aka prediction errors are normally distributed we can conduct hypothesis testing for a evaluating the goodness outfit of the model

With a large sample size based on the central limit theorem, our parameter estimates may be valid, even when the residuals are not normally distributed

155
Q

Outliers

A

Observations that are far from our regression line those that have large prediction errors or X values that are far from the others

Outliers will influence our perimeter estimates so that the OLS model will not fit the other observations well

156
Q

ANOVA Analysis of variance

A


 a statistical procedure for analyzing the total variability of the dependent variable

The output of the ANOVA procedure is in ANOVA table which is a sum of the variation in the dependent variable

Can also be thought of as the source of data for the computation of many regression concepts

For a simple linear regression with one independent variable, the coefficient of determination may be computed by simply squaring the correlation coefficient

R^2 = r^2

157
Q

The total sum of squares SST

A

Measures the total variation in the dependent variable STT is equal to the sum of the squared differences between the actual values and the mean of Y

STT = SUM (Yi - mean Y)^2

158
Q

SSR Some of squares regression

A

Measures the variation in the dependent variable that is explained by the independent variable SSR is the sum of the squared distances between the predicted Y values, and the mean of Y

SSR = SUM (Yi hat - mean Y)^2

159
Q

MSR Mean square regression

A

SSR / number of independent variables

A simple linear regression only has one independent variable so MSR = SSR

CFA LEVEL ONE DOES NOT ADDRESS MULTIPLE REGRESSIONS 

160
Q

SSE Some of squared error

A

Measures the unexplained variation in the dependent variable SSE is the sum of the squared vertical distances between the actual values and the predicted values on the regression line

SSE = SUM (Yi - Y hat)

161
Q

MSE Mean squared error

A

SSE / DOF

N -1 minus number of independent variables

A simple linear regression has only one independent variable so degrees of freedom are N - 2

162
Q

SST = SSR + SSE

A

Total variation = explained variation + unexplained variation

SST = Yi - mean Y

163
Q

SEE standard error of estimate

A

The standard deviation of the residuals of a regression, the lower, the SEE the better the model fit

SEE = square root MSE

MSE = SEE / n-2

164
Q

R^2 coefficient of determination

A

The percentage of the total variation in the dependent variable explained by the independent variable

Example: and our squared value of 0.63 indicates that the variation of the independent variable explains 63% of the variation in the dependent variable

R^2 = SSR / SST

165
Q

F statistic

A

F test assesses how well a set of independent variables as a group explains the variation in the dependent variable

F = MSR / MSE = (SSR/k) / (SSE/ n - k - 1)

MSR = mean regression some of squares
MSE = mean squared error

For simple linear regression there’s only one independent variable, so the F test is equivalent to a T test of the statistical significance of the slope coefficient

H0: b1 = 0
HA: b1 does not equal 0

DOF numerator = k = 1
DOF Denominator = n - k - 1 = n - 2

Reject H0 if F > Fe

Rejecting the null hypothesis that the value of the slope coefficient equals zero at a stated level of significance indicates that the independent variable in the dependent variable have a significant linear relationship 

166
Q

Hypothesis test of a regression coefficient

A

A t test may also be used to test the hypothesis that the true slope coefficient b1 is equal to a hypothesized value letting b1 hat, be the point of estimate for b1 the appropriate test statistic with N -2° of freedom is

Tb1 = (b1 hat - b1) / sb1 hat

Sb1 hat = standard error

Reject NULL if T > +T critical
OR
T < -T critical

Rejecting, then, no hypothesis supports the alternative hypothesis that the slope coefficient is different from the hypothesize value of b1

H0: b1 = 0 HA: b1 does not equal 0

To test whether an independent variable explains the variation in the dependent variable (that it is statistically significant) the null hypothesis is that the true slope is zero

Note: the t test for a simple linear regression is equal to a t test for the correlation coefficient between X and Y

167
Q

Predicted values

A

Values of the dependent variable based on the estimated regression coefficient in a prediction about the value of the independent variable. They are the values that are predicted by the regression equation given an estimate of the independent variable.

Y hat = b0 hat + (b1 hat)(Xp)

168
Q

Confidence intervals for predictive values

A

Y hat - (tc * Sf) < Y < Y hat + (tc * Sf)

Te = two tailed critical T value at the desired level of significance with degrees of freedom equal to N -2
Sf = standard of the forecast

169
Q

Log Lin model

A

If the dependent variable is logarithmic while the independent variable is linear

Ln Yi = b0 + b1Xi + Ei

In this model, the slope coefficient is interpreted as the relative change independent variable for an absolute change in the independent variable

170
Q

Lin log model

A

If the dependent variable is linear while the independent variable is logarithmic

Yi = b0 + b1ln(Xi) + Ei

The slope coefficient is interpreted as the absolute change independent variable for a relative change in the independent variable

171
Q

Log log model

A

Both the dependent and independent variable are logarithmic

Ln Yi = b0 + b1ln(Xi) + Ei

In this model, the slope coefficient is interpreted as the relative change independent variable for a relative change in the independent variable

172
Q

Characteristics of big data

A

Volume, velocity, variety

Volume - amount of data, growing
Velocity - how quickly data is communicated
Variety - varying degrees of structure in which data may exist (structured, semi structured and unstructured)

Low latency - real time data
High latency - data communicated periodically or with a lag

173
Q

Data science

A

Methods for processing and visualizing data

  1. Capture - collecting data and transforming it into usable forms
  2. Curation - assuring data quality by adjusting for bad or missing data
  3. Storage - archiving and accessing data
  4. Search - examining stored data to find needed information
  5. Moving data from their source or a storage medium to where they are needed
174
Q

Neural networks

A

Example of artificial intelligence in that they are programmed to process information in a way similar to a human brain

175
Q

Machine learning

A

Computer algorithm is given inputs of data and may be given outputs of target data

designed to learn, without human assistance, how to model the output data based on the input data or how to detect and recognize patterns in the input data

Process starts with training dataset in which the algorithm looks for relationships

A validation data set is used to refine these relationship models which can be applied to a test dataset to analyze their predictive ability

176
Q

Supervised learning

A

The input and output data are labeled the machine learns to model the outputs from the inputs, and then the machine is given new data on which to use the model

177
Q

Unsupervised learning

A

The input data are not labeled and the machine learns to describe the structure of the data

178
Q

Deep learning

A

A technique that uses layers of neural networks to identify patterns beginning with simple patterns and advancing to more complex ones - may employ supervised or unsupervised learning

Ex. Image and speech recognition

179
Q

Overfitting

A

Occurs when the machine learns the input and output data too exactly, treats noise as true parameters and identifies spurious patterns and relationships - creates a model that is too complex

180
Q

Under fitting

A

Occurs when the machine fails to identify actual patterns and relationships, treating true parameters as noise - the model is not complex enough to describe the data

181
Q

Black box

A

Outcomes produced by machine learning that are based on relationships that are not readily explainable

182
Q

Text analytics

A

Analysis of unstructured data in text or voice forms

Finance example: partially automate specific tasks such as evaluating company regulatory filings

183
Q

Natural language processing

A

Use of computers and artificial intelligence to interpret human language

Ex. Speech recognition and language translation

Finance example: check for regulatory compliance to examine employee communications, evaluate large volumes of research reports

Machine learning and other techniques can be useful in modeling and testing risk

184
Q

Algorithmic trading

A

Computerized securities trading based on a predetermined set of rules

Ex. May be designed to enter the optimal execution instructions for any given trade based on real time price and volume data

185
Q

High frequency trading - algo trading

A

A type of algo trading that identifies and takes advantage of intraday securities mispricings

186
Q

Subjective probability

A

Based on personal judgement

187
Q

Joint probability

A

Probability that two or more events happen concurrently

188
Q

Priori probability

A

One based on logical analysis rather than on observation or personal judgement

189
Q

Empirical probability

A

Calculated using historical data

190
Q

Conditional probability

A

Probability of one event happening on the condition that another event is certain to occur

Cfa (2 decks)

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