Quantitative Methods Flashcards

Question

**Statistics** _Measurement_ Definition of Nominal Scale

Answer 1

Assigning items to groups or categories, such as race or sex, qualitative rather than quantitative. No ordering or ranking implied, but can allocate numbers to the groups (eg, 1 - value funds, 2 - growth funds).

Answer 2

Allocated to each item a ranking for a certain characteristic (eg scale of 1 to 10 between worst and best performing manager). There is an ordering implied but the scale is arbitrary and distances between the ranks not necessarily consistent.

Answer 3

Items in population given a ranking for a certain characteristic, where an order is implied and the distance between each rank is standardised. Such that the difference between 0 and 10 is the same as 20 to 30. However no zero point is defined therefore doubling from 10 to 20 is not the same as doubling from 20 to 40.

Answer 4

Ranking of items within a population on a given parameter, which has an ordering, where difference between each ranking is standardised and there is a defined zero point. So the effect of doubling from 10 to 20 is the same as from 20 to 40. e.g. temperature on the Kelvin scale (NOT farenheit since the zero point in farenheit is arbitrary)

Answer 5

N - Nominal O - Ordinal I - Interval R - Ratio

Answer 6

Number of actual observations in a given interval.

Answer 7

The result from dividing the absolute frequency of a return interval by the total number of observations.

Answer 8

Result of cumulating the results form absolute and relative frequency as you move from one interval to the next.

Answer 9

See formula. * Used when calculating returns over multiple periods * Exists only if all values are greater than zero * Always less than arithmetic mean unless numbers are all the same (in which case they're the same)

Answer 10

Special cases also given: H(x₁,x₂) = 2x₁x₂ / (x₁ + x₂) H(x₁,x₂,x₃) = 3x₁x₂x₃ / (x₁x₂ + x₂x₃ + x₁x₃)

Answer 11

* There are 3 quartiles, 4 quintiles, 9 deciles and 99 percentiles in a data set * They split the population into 4, 5, 10 and 100 groups * Q₁ is the first or lowest quartile, Q₃ the highest * Distance from Q₁ to Q₃ is the _inter-quartile range_

Answer 12

Difference between the lowest and higest values in a population of numbers.

Answer 13

Take the absolute difference between each value and the mean, then take the average of those results.

Answer 14

This is the average squared deviation of each value from the mean.

Answer 15

Same as population, average of squares of difference between values and the _sample_ mean. Dividing by n results in a _baised_ estimator of population variance, using n - 1 gives an _unbaised_ estimator.

Answer 16

Simply the square root of the variance. Normal distn - 68% within 1 sd, 95% within 2 sd. Standard Deviation not directly comparable between different data sets since means are different sizes (not a relative measure).

Answer 17

A relative dispersion measure, so allows comparison between different data sets. Simply divide standard deviation by the mean. a.k.a. Relative Standard Deviation

Answer 18

For any sample/population, the proportion of observations within c standard deviations of the mean is _at least_ 1 - 1/c². Works on sample and population, discrete or continuous data. Allows us to measure minimum amount of dispersion from the standard deviation.

Answer 19

r_p= mean return of portfolio r_f = risk free mean return σ_p = standard deviation of portfolio a.k.a. reward to variability ratio Measures the reward to volatility trade-off and recognises the existence of a risk-free return.

Answer 20

The _degree of asymmetry_ of a data set. Normal distribution has zero skewness. Positively Skewed Distribution means a long tail to the right (a few big wins, lots of small losses). Also say skewed to the right.

Answer 21

Mode: The peak of the distribution. Mean: Pulled towards the long tail (extreme values). Median: In-between.

Answer 22

Valid when N is large. Zero for normal distribution.

Answer 23

Measure of how much a distribution is stretched out to the tails or peaked around the mean, regardless of skew. Normal distribution has a neutral kurtosis.

Answer 24

* Platykurtic: Low peak, more returns with large deviations from the mean. * Leptokurtic: Tall peak, few returns with large deviations from the mean. * Mesokurtic: Same kurtosis as normal distribution.

Answer 25

Formula is for _excess kurtosis_. Normal distribution has kurtosis of 3 and excess kurtosis of 0.

Answer 26

A probability based on frequency of occurence in a set of historical data.

Answer 27

A probability based on logical analysis rather than historical observation (eg 50% chance of heads on coin toss).

Answer 28

A probability based on subjective judgement (eg John thinks there is a 60% chance of a merger occuring). Priori and Empirical probabilities by contrast are considered to be objective.

Answer 29

Another name for unconditional probability, i.e. just the probability that an event will occur, NOT conditional on any other events having occured.

Answer 30

Means P(A) = P(A|B) or vice versa. In this case P(AB) = P(A) \* P(B) which is the multiplication rule.

Answer 31

P(A) = P(A|S₁) + P(A|S₂) + ... + P(A|S_n) Where S₁ to S_n are mutually exclusive and exhaustive. P(A) = P(A|S) + P(A|S^c)

Answer 32

σ²(X) = E{ [X - E(X)]² }

Answer 33

The positive square root of variance.

Answer 34

Cov( R_i, R_j) = E[( R_i - E( R_i ) ) \* ( R_j - E( R_j ) )] Note, covariance of a random variable with itself is: Cov( R_i, R_i ) = E[( R_i - E( R_i ) )²] i.e. it's own variance

Answer 35

Divide covariance by the standard deviation of each random variable.

Answer 36

Expected return is weighted average of individual returns. Expected variance via formula below: σ²(R_p) = w₁²σ²(R₁) + w₂²σ²(R₂) + 2w₁w₂Cov(R₁,R₂)

Answer 37

Answer 38

In Combinations order does NOT matter: _nC_r = n! / (n-r)r!

Answer 39

For Permutations order DOES matter: _nP_r = n! / (n-r)!

Answer 40

Equal probability (straight line) within a given range. f(x) = 1/(b-a) for a\<=x\<=b F(x) = (x-a) / (b-a) for a\<=x\<=b µ(x) = (a+b)/2 σ²(x) = (b-a)²/ 12

Answer 41

An experiment with two possible outcomes (i.e. single experiment in a binomial distribution)

Answer 42

Binomial Random Variable B(n,p) is the number of successes in n bernoulli trials where probability of success in an individual trial is p. Probability of r succeses in n trials is: p(r) = _nC_r \* p^r \* (1-p)^n-r µ(B(n,p)) = np σ²(B(n,p)) = np\*(1-p)

Answer 43

90% x-bar - 1.645σ to x-bar + 1.645σ 95% x-bar - 1.96σ to x-bar + 1.96σ 99% x-bar - 2.58σ to x-bar + 2.58σ

Answer 44

Z table is based on a normal distribution with mean 0 and standard deviation 1. Z-value is therefore the number of standard deviations from the mean (z = (x - µ)/σ). For a given potential outcome convert using the mean and SD of that test to a z value to find the probability of outcome being above/below that potential outcome.

Answer 45

Distribution of a group of random variables, in this case several normally distributed variables e.g. the distribution of a portfolio of normally distributed stock returns. Dependent on means and standard deviations of individual stocks plus the correlation matrix.

Answer 46

Shortfall is risk that portfolio falls below an acceptable value. Saftey First criterion is to choose a portfolio with the lowest possible probability of falling below this value. For normal distribution this equates to maximising the SF-Ratio: SF Ratio = (E(R_P) - R_L) / σ_P Where R_P is portfolio return, R_L is minimum return.

Answer 47

A random variable Y follows a lognormal distribution if it's natural logarithm lnY is normally distributed. So it's lognormal if its log is normal.

Answer 48

Denoted by r_t,t+1 (cont. comp. rate of return between t and t+1): r_t,t+1 = ln(S_t+1/S_t) = ln(1+R_t,t+1) Gives a more reasonable average return as: [(1.2/1 - 1) + (1/1.2 - 1)] / 2 = 2.5% [ln(1.2/1) + ln(1/1.2)] / 2 = 0%

Answer 49

A sample that has been taken using a biased method, resulting in a sample with different characteristics than the population. Note that a sample with different characteristics than the population as a result of randomness in a fair sample, is NOT a biased sample.

Answer 50

For a given sample size, and sample statistic (e.g. the sample mean), the sample distribution is the distribution of outcomes of that statistic across an infinite number of samples. Alternatively it's the relative frequency of outcomes of the statistic for every possible sample of the population, of the given sample size.

Answer 51

Split the population up into sub-populations based on given characteristics, then select a sub-sample from each with size based on relative size of the sub-population. Put the sub-samples together to get a sample of the population. This ensures proportional representation across the characteristics used to split the population.

Answer 52

As opposed to time-series data, cross-sectional data consists of observations at a single point in time, such as the closing prices of 20 different stocks at close on a given date.

Answer 53

Given a distribution with mean µ and variance σ², the sampling distribution of the mean x-bar approaches a normal distribution with mean µ and variance σ²/N as the sample size (N) increases. With N \> 29 it is normal EVEN IF underlying distribution is not normal. This allows us to construct confidence intervals on the population mean from sample data using normal distn., regardless of whether the population is normally distributed!

Answer 54

The standard error of a statistic is the standard deviation of the sampling distribution of that statistic. σ_m = σ / sqrt(N) Where σ_m is the standard error of the mean, σ is the standard deviation of the population and N is the sample size. This DOES NOT require the pop distn to be normal. Estimate using sample standard deviation (s) if population standard deviation (σ) is not known.

Answer 55

_Unbaisedness_: The estimators expected value (mean of its sampling distn) equals the parameter it is meant to estimate. _Efficiency_: Estimator is efficient if no other unbaised estimator of the parameter value has a lower standard error. _Consistency_: Means the standard error of the unbaised estimator approaches zero as sample size increases.

Answer 56

The single estimate of an unknown population parameter calculated as a sample mean is called the point estimate of the mean. The formula used to calculate it is called the estimator.

Answer 57

Degree of confidence is 95% Level of significance is 5% 20 and 40 are the lower and higher confidence limits There is a 95% probability that the population mean lies between 20 and 40

Answer 58

If the variance is not known use t-distribution instead of z-distribution for confidence intervals (with t table based on N-1 degrees of freedom, where N is the sample size). If the population is not normal make sure you have sample size of 30+. If the sample size is very high using the z distn is usually ok.

Answer 59

Bias as a result of using somebody else's results of empirical (historic data) analysis to guide your own analysis over essentially the same historical data. Can be avoided by carrying out analysis on new data, although this is difficult in investment analysis because the set of historical data is limited.

Answer 60

Bias created as a result of finding forecasting models through extensive searches of databases to find patterns. Highly likely when there is no economic justification for the pattern. Can be avoided by testing models on a different set of data.

Answer 61

Caused when analysis is carried out on a data set that has excluded (e.g.) stocks that have gone bankrupt. The data has an upward bias since it excludes a section of the population which performed very poorly.

Answer 62

Bias caused based on the assumption that fundamental information was available at a point in time when it isn't. For example assuming that people know their earnings data in January for January, when they might not find out till March. Usually results in upwards bias.

Answer 63

Where a conclusion drawn relates only to a particular time period which may make that conclusion time-specific. Usually a problem if the time period is too short (eg covers only an economic upswing). Also a problem if the time period is too long since fundamental economic structure may have changed.

Answer 64

Designated H₀ this is the hypothesis regarding the population (eg mean monthly income is $5000) which you desire to test. It is either rejected or failed to be rejected, never accepted.

Answer 65

Designated H₁, this is the statement which is accepted if the sample data provides sufficient evidence that H₀ is false.

Answer 66

A test statistic is a number calculated from the sample whose value (relative to its probability distn) provides statistical evidence against the null hypothesis. Typically of the form: test statistic = (sample statistic - H₀ parameter value) / Standard error of sample statistic

Answer 67

Chosen in testing, this is the probability that a null hypothesis which is true is rejected. Designated by greek letter alpha.

Answer 68

The decision rule is the statement of the conditions under which H₀ is rejected or not. The critical value (2 of them if two-sided) is the dividing point past which H₀ will be rejected. The critical region is the area where if the test stat falls in it, we REJECT H₀.

Answer 69

Note: With n=1 the sample distribution of the mean is just the distribution of the population, so the below simplifies to the z value. This usually follows the normal distn (central limit theorem) therefore:

Answer 70

Type one error is the when H₀ is true but is rejected. Type two error is when H₀ is false but is not rejected. Type one is more serious and is controlled via the level of significance (alpha), which is the probability of a type one error. Reducing alpha to reduce the risk of a type one error increases the risk of a type two error (probability designated as ß). Can only reduce both alpha and ß by increasing sample size.

Answer 71

The probability of correctly rejecting a false null hypothesis (1-ß). If the power of an experiment is low there is a high chance of an inconclusive result. Assumes that H₀ is false.

Answer 72

Instead of stating a decision rule, calculate the test statistic and the calculate the probability of getting a value more extreme than that (one or two-sided) assuming H₀ is correct. This probability can be compared to alpha value to decide whether to reject H₀, but also provides extra information about how strong the rejection is.

Answer 73

Below formula is the test statisticfor the hypothesis that the difference between two population means is a given number (or zero, ie that they are the same). Degrees of freedom are n₁ + n₂ - 2 if population variances are assumed to be the same. Requires normal distribution

Answer 74

Used on differences between related data (eg before and after data, results for twins). H₀ : µ_d = µ_d0 Where µ_d is the difference between the means and µ_d0 is some fixed value (typically zero). Requires Normal distribution

Answer 75

d-bar is the sample mean of differences Standard error for the sample mean of differences: s_d-bar is s_d/sqrt(n) Standard deviation of the differences: s_d = sqrt( Σ(d_i - d-bar)² / (n-1) ) n-1 degrees of freedom Note: This is basically just normal distn.

Answer 76

Subject is the testing of the differences between two means. Treatment depends on whether the two populations are related. If the population is the same (eg before or after) or is some way dependent then it is paired testing. If populations are different (typically also sample sizes will be different) it is independent.

Answer 77

Chi-squared statistic. s² is the sample variance σ₀² is the hypothesised value for σ² Population MUST be random AND normally distributed n-1 degrees of freedom (similar to t-distn)

Answer 78

Fischer distribution (F-distribution) F = S₁² / S₂² Convention states larger sample variance goes on top. Requires both populations to be normally distributed. Note that degrees of freedom are excluded since n₁-1 in both denominator and numerator.

Answer 79

F-distribution tables denoted by F(n₁ - 1, n₂ - 2) where n₁ and n₂ are sample sizes of numerator and denominator. F-distribution is one-directional, H₀ is that σ₁² \<= σ₂² because we chose 1 & 2 such that s₁² \> s₂². We then test whether the ratio is high enough to reject H₀. For two-sided tests (H₀: σ₁² = σ₂²) need to halve alpha. As with chi-squared, distn MUST be random AND normal.

Answer 80

This is testing of something other than a parameter of the population (such as mean or variance). Parametric tests are preferred where relevant, but may not be due to the distribution of the data, nature of the data (eg ordinal/ranked data) or where no parameter is involved (eg testing whether data is indeed random).

Answer 81

Both agree that supply and demand control prices, however technical analysts believe this information feeds into prices gradually over a period of time and trends can be taken advantage of.

Answer 82

Xs represent upwards price trends and Os represent donwards trends. X axis not linear in time, dependent on price movements. Chart switches between X's and O's when trend deemed to have shifted.

Answer 83

For upwards trends draw an upwards pointing support line below the low points. For a downwards trend draw a downwards trending resistance line joining the peaks.

Answer 84

When a price breaks through a resistance line that same price level then becomes a support (or vice versa), referred to as a change in polarity.

Answer 85

Head and shoulders is a reversal pattern (as is inverse HaS). Look for higher volume on the left shoulder rally than the head rally, increases in volume on the sell-offs. Target price = neckline - (head peak - neckline)

Answer 86

Double (or triple) top/bottom is a reversal pattern characterised by the price hitting a support/resistance level twice and failing to break it.

Answer 87

Traingles are continuation patterns, either ascending (supported by ascending trend line, top is a flat resistance line), descending (descending resistance line with stock bouncing off a flat support line) and symmetrical (both support ascending and resistance descending). The original pattern is expected to assert itself (ie the horizontal line for asc/desc is only temporary).

Answer 88

Horizontal support and resistance lines, ie rangebound. Can stay in the range for a while, but usually expect the trend before the pattern was established to eventually continue.

Answer 89

Short-term continuation patterns representing a consolidation for a short time. A flag has parallel support and resistance lines in a different direction to the larger trend. A pennant is a symmetrical triangle (ascending support, descending resistance) with a typically neutral overall slope.

Answer 90

When a short-term moving average breaks out above a longer-term moving average this is considered a bullish signal. The opposite is a dead cross.

Answer 91

Measure the percentage price change over a given period (eg charts the % price change over the last 20 days).

Answer 92

Compares the average price change during advancing periods to average price change during declining periods. RSI = 100 - 100/(1 + RS) Where RS = average gain / average loss RSI is a range of zero to 100 where above 70 considered overbought and below 30 considered oversold.

Answer 93

This is simply the level of the close expressed as a percentage between the lowest low and highest high in its current range (so at 50% it's in the middle of its established range).

Answer 94

Simply the difference between a short and long term moving average, so when they converge the value is zero, and as the moving averages diverge the MACD becomes negative or positive.

Answer 95

Indicator that measures the extent to which money is moving into or out of rising or declining stocks. Ratio of 1 means market is in balance. Ratio \> 1 means more money is moving into declining stocks. Ratio \< 1 means more money moving into advancing stocks.

Answer 96

Sinusoidal like waves which the world economy follows on a periodicity of 40 to 60 years.

Answer 97

States that the market isn't random but moves in cycles. Says the market moves in five waves of the prevaling trend (impulse waves) and three corrective waves which counter the trend. The waves count is a Fibonnaci sequence.

Answer 98

Simply the practice of looking at several related markets at the same time to determine patterns (eg US bond market and US equity market).

Answer 99

HPR = 1 + HPY

Answer 100

Leptokurtic distributions have FAT tails. Platykurtic distributions have more observations far away (so tails stretch further away) but observations in the tails less bunched together, so they look LONG and THIN.

Answer 101

Use Chebyshev's equation to get confidence intervals based on # standard deviations for any distribution.

Answer 102

Amount of time since price change (along with % of income spent on the product and closeness of substitutes) impacts elasticity of demand. Changes in price expectations shifts the aggregate demand curve to the left or right, but doesn't impact elasticity of demand.

Answer 103

Backfill bias is when a new item is added to an index and performance from before the date of its addition is included in the index.