QUANT METHODS 1, 3 ...

Question 1

1.1 what is the Real risk-free rate?

Answer 1

Is a theoretical rate on a single-period loan that contains no expectation of inflation and zero probability of default.

Represents in economic terms the time preference, the degree to which current consumption is preferred to equal future consumption

Question 2

1.1 What is the equilibrium interest rate?

Answer 2

Required rate of return for a particular investment

Question 3

1.1 What is the Real interest rate?

Answer 3

An interest rate from which the inflation premium has been subtracted (nominal rate minus risk of inflation)

Question 4

1.1 What is the Real rate of return?

Answer 4

Referring to an investor’s increase in purchasing power (after adjusting for inflation)

• Because expected inflation in future periods is not zero, the rates we observe on U.S. Treasury bills (T-bills), for example, are essentially risk-free rates, but not real rates of return.
• T-bill rates are nominal risk-free rates because they contain an inflation premium

Question 5

1.1 what is the relationship between nominal risk-free rate, real risk-free rate, and expected inflation rate?

Answer 5

(1+nominal risk-free rate) = (1+real risk-free rate)(1+expected inflation rate)

OR MORE SIMPLY (and in the curriculum)

nominal risk-free rate =(squiggly) real risk-free rate + expected inflation rate

Question 6

1.1 What are the three types of risk that may increase the required rate of return?

Answer 6

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

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

Maturity risk = as we will see in the FI topic area, the prices of longer-term bonds are more volatile than those of short-term bonds. Longer-maturity bonds have more maturity risk than shorter-term bonds and require a maturity risk premium

SO nominal rate of interest = real risk-free rate + inflation premium + default risk premium + liquidity premium + maturity premium

Question 7

1.1 What is HPR?

Answer 7

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

Question 8

1.1 What is the arithmetic mean return?

Answer 8

The simple average of a series of period returns.

Question 9

1.1 What is the downside of arithmetic mean?

Answer 9

It does not reflect multi-period compounding.

Therefore it is most appropriate for single-period returns.

Question 10

1.1 What is the geometric mean return?

Answer 10

It is a compound rate. when period rates of return vary from period to period, the geometric mean return will have a value less than the arithmetic mean return:

Question 11

1.1 in the light of arithmetic mean’s shortcoming, why is geometric mean a good thing?

Answer 11

because it DOES reflect multi-period compounding

therefore it is most appropriate for average return over multiple periods

Question 12

1.1 Fill the gap: The geometric mean is always ______ than or equal to the arithmetic mean

Answer 12

less

Question 13

1.1 What is the harmonic mean?

Answer 13

Used for certain computations, such as the average cost of shares purchased over time.

But most typically used for calculating ratios. PE ratio etc.

Because of its formula, it gives a lower weight to extreme observations.

Question 14

1.1 What is a caveat with the Harmonic mean and how is it overcome?

Answer 14

we can ONLY calculate a harmonic mean of positive numbers. For a set of returns that includes negative numbers, we can treat them in the same way we did with geometric means, using (1+return) for each period, then subtracting 1 from the result)

Question 15

1.1 Describe an equation that links the arithmetic mean, harmonic mean, and geometric mean

Answer 15

Question 16

1.1 What is the trimmed or winsorized mean?

Answer 16

The trimmed mean removes the highest and lowest values from a dataset, while the winsorized mean replaces them with the next-closest values

Question 17

1.1 Summarise the appropriate uses for each type of mean calculation

Answer 17

• Arithmetic mean - include all values, including outliers
• Geometric mean - compound the rate of returns over multiple periods
• Harmonic mean - calculate the average share cost from periodic purchases in a fixed money amount
• Trimmed or winsorized mean - decrease the effect of outliers

Question 18

1.1 What is the most appropriate method when calculating mean annual return

Answer 18

Geometric mean, because returns are compounded

Question 19

1.2 What is the Money-weighted return?

Answer 19

The Money-weighted return applies the concept of the internal rate of return (IRR) to investment portfolios. An IRR is the interest rate at which a series of cash inflows and outflows sum to zero when discounted to their present value. That is, they have a net present value (NPV) of zero.

The money-weighted rate of return is defined as IRR on a portfolio, taking into account all cash inflows and outflows. The beginning value of the account is an inflow, as are all deposits into the account. All withdrawals from the account are outflows, as is the ending value. (the specific application of IRR to a portfolio)

Question 20

1.2 What is the formula for money-weighted return?

Answer 20

Is an IRR calculation. Set to NPV of 0. THIS DOES REFLECT THE SIZE OF THE FUND

Question 21

1.2 When solving for MWR (IRR) how can e answer this question in the exam by trial and error?

Answer 21

Use on of the answers they give you - normally the middle one. If it equals 0, it is right, if not, does it need to be higher or lower?

Question 22

1.2 What is the Time-weighted rate of return?

Answer 22

Measures compound growth and is the rate at which $1 compounds over a specified performance horizon. Time-weighting is the process of averaging a set of values over time. The annual time-weighted return for an investment may be computed by performing the following steps:

CRUCIALLY - IT DOES NOT REFFLECT THE SIZE OF THE FUND.

Question 23

1.2 How is Time-weighted rate of return calculated? (step by step)

Answer 23

Question 24

1.2 What is the formula we can use for TWR?

Answer 24

Essentially geometric mean int he way that it is calculated

Question 25

1.2 Which measure for return do PMs usually prefer?

Answer 25

NOTE - the time-weighted rate of return is not affected by the timing of cash inflows and outflows. in the IM industry, time-weighted return is the preferred method of performance measurement because PMs typically do not control the timing of deposits to and withdrawals from the accounts they manage If funds are contributed to an investment portfolio just before a period of relatively poor portfolio performance, the money-weighted rate of return will tend to be lower than the time-weighted rate of return. On the other hand, if funds are contributed to a portfolio at a favorable time (just before a period of relatively high returns), the money-weighted rate of return will be higher than the time-weighted rate of return. The use of the time-weighted return removes these distortions, and thus provides a better measure of a manager's ability to select investments over the period. If the manager has complete control over money flows into and out of an account, the money-weighted rate of return would be the more appropriate performance measure.

Question 26

1.3 What is the formula for annualised return?

Answer 26

Interest rates and market returns are typically stated as annualised returns, regardless of the actual length of the time period over which they occur To annualise an HPR that is realised over a specific number of days, use the following formula:

Question 27

1.3. what is non-annual compounding?

Answer 27

DOES NOT REFLECT COMPOUNDING. if we wanted to compound - (1 + semi-annual rate) ^2 then -1.

Question 28

1.3 What is the EAR (Effective Annual Rate) if given a stated/quoted interest rate?

Answer 28

If an interest rate is stated as 8% per year, compounded semi-annually, for example: You compound by 4% twice. so do 1 (x1.04), then ans (x1.04) = 1.0816 -1 = 8.16%. (look at formula) So EAR = 8.16%

Question 29

1.3 What is the formula for Continuously Compounded rates?

Answer 29

Showcases how EAR increases in marginally smaller increments. If we tend toward infinity - we can find a ceiling to EAR. Make sure to decimalise Rcc when calculating.

Question 30

1.3 how do we solve for the quoted continuously compounded rates given actual HPRs?

Answer 30

taking logs of each side of the previous equation

Question 31

1.3 Why may it be useful to use continuously compounded returns?

Answer 31

1. They are additive over multiple periods 2. e^Rcc can be used as a multiplying and a discounting factor - ease

Question 32

1.3 What is the formula for the present value of a future cash flow:

Answer 32

Question 33

1.3 What formula can you use to calculate continuously compounded returns

Answer 33

and note: the periods are additive. therefore CC return from t=0 to t=1, and t=1 to t=2, is the same as t=0 to t=2 when added.

Question 34

1.3 What is the gross return?

Answer 34

Gross return refers to the total return on a security portfolio before deducting fees for the management and administration of the investment account. Net return refers to the return after these fees have been deducted. Commissions on trades and other costs that are necessary to generate the investment returns are deducted in both gross and net return measures.

Question 35

1.3 What is Pretax nominal return?

Answer 35

Pretax nominal return refers to the return before paying taxes. Dividend income, interest income, short-term capital gains, and long-term capital gains may all be taxed at different rates.

Question 36

1.3 What is After-tax nominal return?

Answer 36

After-tax nominal return refers to the return after the tax liability is deducted.

Question 37

1.3 What is real return?

Answer 37

Real return is nominal return adjusted for inflation. Consider an investor who earns a nominal return of 7% over a year when inflation is 2%. The investor's approximate real return is simply 7 − 2 = 5%. The investor's exact real return is slightly lower: 1.07 / 1.02 − 1 = 0.049 = 4.9%.

Question 38

1.3 How can we exactly calculate real return?

Answer 38

Question 39

1.3 What does Real return measure?

Answer 39

Real return measures the increase in an investor's purchasing power—how much more goods she can purchase at the end of one year due to the increase in the value of her investments.

Question 40

1.3 What is a leveraged return?

Answer 40

A leveraged return refers to a return to an investor that is a multiple of the return on the underlying asset. The leveraged return is calculated as the gain or loss on the investment as a percentage of an investor's cash investment. An investment in a derivative security, such as a futures contract, produces a leveraged return because the cash deposited is only a fraction of the value of the assets underlying the futures contract. Leveraged investments in real estate are common: investors pay only a portion of a property's cost in cash and borrow the rest.

Question 41

1.3 What is a formula for calculating leveraged return?

Answer 41

Vo = the amount the fund can invest without leverage Vb = the amount the fund can borrow rb = interest rate of the borrowed money r = amount earns by investing the proceeds of the leveraged money

Question 42

1.3 Different formula for leveraged returns?:

Answer 42

OR do can do Rl = (Rp x Va) - (Rd x Vb) then /2

Question 43

3.1 What are measures of central tendency? And a prominent example.

Answer 43

Measures of central tendency identify the center, or average, of a dataset. This central point can then be used to represent the typical, or expected, value in the dataset. The arithmetic mean is the sum of the observation values divided by the number of observations. It is the most widely used measure of central tendency.

Question 44

3.1 What is a sample mean?

Answer 44

An example of an arithmetic mean is a sample mean, which is the sum of all the values in a sample of a population, ΣX, divided by the number of observations in the sample, n. It is used to make inferences about the population mean. The sample mean is expressed as follows:

Question 45

3.1 What is the median?

Answer 45

The median is the midpoint of a dataset, where the data are arranged in ascending or descending order. Half of the observations lie above the median, and half are below. To determine the median, arrange the data from the highest to lowest value, or lowest to highest value, and find the middle observation. The median is important as the mean can be much affected by outliers, take the middle number, if even, take the mean of the middle two numbers.

Question 46

3.1 What is the mode? And what does it mean to be unimodal, bimodal, or trimodal?

Answer 46

The mode is the value that occurs most frequently in a dataset. A dataset may have more than one mode, or even no mode. When a distribution has one value that appears most frequently, it is said to be unimodal. When a dataset has two or three values that occur most frequently, it is said to be bimodal or trimodal, respectively.

Question 47

3.1 What are two methods for dealing with outliers?

Answer 47

Trimmed mean, winsorized mean

Question 48

3.1 What is the trimmed mean?

Answer 48

A trimmed mean excludes a stated percentage of the most extreme observations. A 1% trimmed mean, for example, would discard the lowest 0.5% and the highest 0.5% of the observations.

Question 49

3.1 What is a winsorized mean?

Answer 49

Instead of discarding the highest and lowest observations, we substitute a value for them. To calculate a 90% winsorized mean, for example, we would determine the 5th 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 dataset.

Question 50

3.1 What is a quantile + examples?

Answer 50

Quantile is the general term for a value at or below which a stated proportion of the data in a distribution lies. Examples of quantiles include the following: - Quartile. The distribution is divided into quarters. - Quintile. The distribution is divided into fifths. - Decile. The distribution is divided into tenths. - Percentile. The distribution is divided into hundredths (percentages).

Question 51

3.1 what is the Interquartile range?

Answer 51

The difference between the third quartile and the first quartile (25th percentile) is known as the interquartile range.

Question 52

3.1 What is a box and whisker plot?

Answer 52

To visualize a dataset based on quantiles, we can create a box and whisker plot. In a box and whisker plot, the box represents the central portion of the data, such as the interquartile range. The vertical line represents the entire range.

Question 53

3.1 How can the Box and whisker Plot be adjusted to show for outliers?

Answer 53

Can add 'fences' that are positioned at 1.5x the IQR of the data

Question 54

3.1 What is 'dispersion'?

Answer 54

Dispersion is defined as the variability around the central tendency. The common theme in finance and investments is the tradeoff between reward and variability, where the central tendency is the measure of the reward and dispersion is a measure of risk.

Question 55

3.1 What is the range?

Answer 55

range = maximum value − minimum value

Question 56

3.1 What is the mean absolute deviation?

Answer 56

The mean absolute deviation (MAD) is the average of the absolute values of the deviations of individual observations from the arithmetic mean: 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. Absolute is SIZE (rather than sign) HELP TO REMEMBER

Question 57

3.1 What is the sample variance?

Answer 57

The sample variance, s2, is the measure of dispersion that applies when we are evaluating a sample of n observations from a population. The sample variance is calculated using the following formula: REMEMBER THE (N - 1) Normally have to square route to get back to sample standard deviation.

Question 58

3.1 What is the sample standard deviation?

Answer 58

The sample standard deviation is the square root of the sample variance. The sample standard deviation, s, is calculated as follows:

Question 59

3.1 What is relative dispersion?

Answer 59

Relative dispersion is the amount of variability in a distribution around a reference point or benchmark.

Question 60

3.1 What is the coefficient of variation?

Answer 60

Relative dispersion is the amount of variability in a distribution around a reference point or benchmark. Relative dispersion is commonly measured with the coefficient of variation (CV), which is computed as follows: CV measures the amount of dispersion in a distribution relative to the distribution's mean. This is useful because it enables us to compare dispersion across different sets of data. In an investments setting, the CV is used to measure the risk (variability) per unit of expected return (mean). A lower CV is better. ULTIMATELY - CV IS VARIATION PER UNIT OF RETURN YOU WOULD PREFER THE LOWER CV

Question 61

3.1 What is downside risk?

Answer 61

When we use variance or standard deviation as risk measures, we calculate risk based on outcomes both above and below the mean. In some situations, it may be more appropriate to consider only outcomes less than the mean (or some other specific value) in calculating a risk measure. In this case, we are measuring downside risk.

Question 62

3.1 What is target downside deviation? (Also known as target semideviation).

Answer 62

One measure of downside risk is target downside deviation, which is also known as target semideviation. 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.

Question 63

3.1 What is the formula for calculating target downside standard deviation?

Answer 63

In the data, just ignore all returns that are above the target (eg all data points above 3% return) With the remainder, take the data point and subtract the target, use this summation in the equation.

Question 64

3.2 What is Skewness, or skew?

Answer 64

Skewness or skew, refers to the extent to which a distribution is not symmetrical. Nonsymmetrical distributions may be either positively or negatively skewed and result from the occurrence of outliers in the dataset.

Question 65

3.2 What is an outlier?

Answer 65

Outliers are observations extraordinarily far from the mean, either above or below:

Question 66

3.2 What is a positively skewed distribution?

Answer 66

A positively skewed distribution is 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 of its relatively long upper (right) tail.

Question 67

3.2 What is a negatively skewed distribution?

Answer 67

A negatively skewed distribution 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.

Question 68

3.2 How are the mean, median, and mode, related to each other in a symmetrical distribution?

Answer 68

For a symmetrical distribution, the mean, median, and mode are equal.

Question 69

3.2 How are the mean, median, and mode, related to each other in a positively skewed distribution?

Answer 69

For a positively skewed, unimodal distribution, the mode is less than the median, which is less than the mean. The mean is affected by outliers; in a positively skewed distribution, there are large, positive outliers, which will tend to pull the mean upward, or more positive.

Question 70

3.2 How are the mean, median, and mode, related to each other in a negatively skewed distribution?

Answer 70

For a negatively skewed, unimodal distribution, the mean is less than the median, which is less than the mode. In this case, there are large, negative outliers that tend to pull the mean downward (to the left).

Question 71

3.2 How do we calculate the sample skewness?

Answer 71

Sample skewness is equal to the sum of the cubed deviations from the mean divided by the cubed standard deviation and by the number of observations. Sample skewness for large samples is approximated as follows:

Question 72

3.2 Once calculated, what does the number delivered form the sample skewness formula tell us?

Answer 72

Note that the denominator is always positive, but that 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 mean are larger, on average. A left-skewed distribution has a negative sample skewness. If 0 - it is not skewed

Question 73

3.2 What is kurtosis?

Answer 73

Kurtosis is a measure of the degree to which a distribution is more or less peaked than a normal distribution. It is critical in a risk management setting. Most research about the distribution of securities returns has shown that returns are not normally distributed. Actual securities returns tend to exhibit both skewness and kurtosis. Skewness and kurtosis are critical concepts for risk management because when securities returns are modeled using an assumed normal distribution, the predictions from the models will not take into account the potential for extremely large, negative outcomes.

Question 74

3.2 Normal distribution has a kurtosis (when you calculate it) of...

Answer 74

3

Question 75

3.2 What is Leptokurtic?

Answer 75

Leptokurtic describes a distribution that is more peaked than a normal distribution. We don't like this as there are bigger outliers on both sides. Less chance of a mean sized deviation from the mean. Kurtosis > 3

Question 76

3.2 What is Platykurtic?

Answer 76

platykurtic refers to a distribution that is less peaked, more domed, or flatter than a normal one. (with thinner tails - so prefer) LOOKS LIKE A PLATYPUS BILL. Kurtosis < 3

Question 77

3.2 What is Mesokurtic?

Answer 77

A distribution is mesokurtic if it has the same kurtosis as a normal distribution.

Question 78

3.2 What does it mean for a distribution to exhibit excess kurtosis?

Answer 78

A distribution is said to exhibit excess kurtosis if it has either more or less kurtosis than the normal distribution. Excess kurtosis = kurtosis -3

Question 79

3.2 How much is the kurtosis of all normal distributions when computed? And how does this affect how it is reported?

Answer 79

The computed kurtosis for all normal distributions is three. Statisticians, however, sometimes report excess kurtosis, which is defined as kurtosis minus three. Thus, a normal distribution has excess kurtosis equal to zero, a leptokurtic distribution has excess kurtosis greater than zero, and platykurtic distributions will have excess kurtosis less than zero.

Question 80

3.2 How is sample kurtosis for large samples calculated?

Answer 80

Sample kurtosis for large samples is approximated using deviations raised to the fourth power:

Question 81

3.2 What is a scatter plot?

Answer 81

Scatter plots are a method for displaying the relationship between two variables. With one variable on the vertical axis and the other on the horizontal axis, their paired observations can each be plotted as a single point.

Question 82

3.2 What is a key advantage of scatter graphs?

Answer 82

A key advantage of creating scatter plots is that they can reveal nonlinear relationships, which are not described by the correlation coefficient. Panel C illustrates such a relationship. Although the correlation coefficient for these two variables is close to zero, their scatter plot shows clearly that they are related in a predictable way.

Question 83

3.2 What is Covariance?

Answer 83

Covariance is a measure of how two variables move together. BUT KEEP IN MIND In practice, the covariance is difficult to interpret. The value of covariance depends on the units of the variables. The covariance of daily price changes of two securities priced in yen will be much greater than their covariance if the securities are priced in dollars. Like the variance, the units of covariance are the square of the units used for the data. AND we cannot interpret the relative strength of the relationship between two variables.

Question 84

3.2 What is the sample covariance formula?

Answer 84

Question 85

3.2 What is the problem with calculating covariance?

Answer 85

The answer does not tell us much, can be any number. We just need to see if positive or negative to see if they mope with each other positively or negatively. This is why we use correlation.

Question 86

3.2 What is correlation?

Answer 86

A standardized measure of the linear relationship between two variables is called the correlation coefficient, or simply correlation.

Question 87

3.2 What is the correlation formula?

Answer 87

in other words: correlation = covariance/SDX*SDY

Question 88

3.2 Please describe some properties of the correlation of two variables:

Answer 88

1. Correlation measures the strength of the linear relationship between two random variables. 2. Correlation has no units. 3. The correlation ranges from –1 to +1. That is, –1 ≤ ρXY ≤ +1. 4. If ρXY = 1.0, the random variables have perfect positive correlation. This means that a movement in one random variable results in a proportional positive movement in the other relative to its mean. 5. If ρXY = –1.0, 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. 6. If ρXY = 0, there is no linear relationship between the variables, indicating that prediction of Y cannot be made on the basis of X using linear methods.

Question 89

3.2 What is spurious correlation?

Answer 89

Spurious correlation refers to correlation that is either the result of chance or present due to changes in both variables over time that is caused by their association with a third variable. For 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. EG: In his book Spurious Correlation,1 Tyler Vigen presents the following examples. The correlation between the age of each year's Miss America and the number of films Nicolas Cage appeared in that year is 87%.

Question 90

4.1 Wha is a discrete probability distribution?

Answer 90

There are a finite number of outcomes

Question 91

4.1 What is the expected value? (of a random variable)

Answer 91

The expected value of a random variable is the probability weighted average of the possible outcomes for the variable. The mathematical representation for the expected value of random variable X, that can take on any of the values from x1 to xn, is:

Question 92

4.1 What is a probability tree?

Answer 92

A general framework, called a probability tree, is used to show the probabilities of various outcomes.

Question 93

4.1 What are conditional expected values?

Answer 93

As the name implies, conditional expected values are contingent on the outcome of some other event. An analyst would use a conditional expected value to revise his expectations when new information arrives.

Question 94

4.1 What is Bayes' formula used for?

Answer 94

Bayes' formula is used to update a given set of prior probabilities for a given event in response to the arrival of new information.

Question 95

4.1 What is Bayes' formula? (words)

Answer 95

Question 96

4.1 When simplified, what is Bayes' formula reduced to?

Answer 96

Question 97

5.1 Formula for the expected return of a portfolio:

Answer 97

Question 98

5.1 What is the weight of an asset within the portfolio? Formula:

Answer 98

Question 99

5.1 What is the formula for Expected Return of a portfolio of two assets?

Answer 99

Just weighted return

Question 100

5.1 What is the formula for portfolio variance (and a way of remembering it)?

Answer 100

and then imagine the maths of just squaring added numbers. variance is an averaged squared distance. (weightA*sigmaA + weightB*sigmaB) if this is squared, remember the way two brackets are multiplied. so then just add the 'Rho' (correlation coefficient) to the doubled part in the middle. Then remember the relationship between correlation and covariance.

Question 101

5.1 How does this work for 3 assets? (portfolio variance)

Answer 101

Question 102

5.1 What is the formula for covariance between the return of two assets? (expected)

Answer 102

Question 103

5.1 What is the formula for a sample covariance for a sample of returns data?

Answer 103

Question 104

5.1 What does a covariance matrix look like, and what is the value of the covariance down the diagonal?

Answer 104

the variance. and note - covariance does not depend on order.

Question 105

5.1 What is the formula for the portfolio variance?

Answer 105

All variations of the two

Question 106

5.1 What is joint probability?

Answer 106

The probability of two things happening at once

Question 107

5.1 What is a joint probability function for returns? Show me what it looks like

Answer 107

Question 108

5.1 Once in the joint probability function of returns - what do we need to do to calculate the covariance of asset A and asset B?

Answer 108

1. calculate expected values 2. use the attached table to calculate each of the covariances under each probability 3. add the together

Question 109

5.1 What are the key distances in a. normal distribution?

Answer 109

it is 1.96 standard deviations - rounded to 2 here Area under the graph is a confidence level NOTE ALSO - if 5% tail, then from middle is 1.65 sd

Question 110

5.1 List of standard deviation distances for tail sizes in a normal distribution

Answer 110

LEARN

Question 111

5.1 If given the question 'calculate a 95% confidence interval for the next year's return' given a mean annual return and standard deviation, how would you do it?

Answer 111

Question 112

5.1 what is a z-value?

Answer 112

number of standard deviations we need to go to capture the confidence level %. this distance is referred to as a z value

Question 113

5.1 What is a standard normal distribution?

Answer 113

a normal distribution that has been standardised so that mean = 0 and standard deviation = 1

Question 114

5.1 What is the formula for calculating z value?

Answer 114

NOTE - this is with the backdrop of being a standard normal distribution. So our calculation is translated into this model when calculating the z value.

Question 115

5.1 What is shortfall risk?

Answer 115

Shortfall risk is the probability that a portfolio value or return will fall below a particular target value or return over a given period.

Question 116

5.1 What is Roy's safety-first criterion?

Answer 116

Roy's safety-first criterion 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.

Question 117

5.1 Symbolically, how can Roy's safety-first criterion be stated?

Answer 117

Also think about it like the calculation of a z value

Question 118

5.1 If portfolio returns are normally distributed, how can Roy's safety-first criterion be stated? (clue: SFR)

Answer 118

Question 119

5.1 What are the two steps for choosing among portfolios with normally distributed returns using Roy's safety-first criterion?

Answer 119

But then after the largest ratio has been chosen, this needs to be used in the z-table of negative values to find the percentage value that represents the shortfall risk of a portfolio.

Question 120

6.1 What is a lognormal distribution?

Answer 120

replace this with 'log is normal' in your head. So random variable Y is lognormal is ln(Y) is normal. It is used to model price 'relatives': (Pt/Po)

Question 121

6.1 Please explain the relationship between HPR and the modelling of price relatives on a lognormal distribution.

Answer 121

because Pt/Po is the price in future over the price today, this is equal to 1 + HPR (as covered previously). therefore, ln(Y) = ln (Pt/Po) = ln(1 + HPR) = Rcc (continuously compounding returns = NORMAL DSTRIBTUION so LOGARITHM OF YOUR PRICE RELATIVE IS YOUR CONTINUOUSLY COMPOUNDED RETURNS

Question 122

6.1 Describe the visual difference between a normal and lognormal distribution:

Answer 122

Lognormal is always positive and positively skewed

Question 123

6.1 What do we use the lognormal distribution for?

Answer 123

The lognormal distribution is useful for modeling asset prices if we think of an asset's future price as the result of a continuously compounded return on its current price.

Question 124

6.1 how can we model asset prices if we think of an asset's future price as the result of a continuously compounded return on its current price?

Answer 124

Question 125

6.1 How do we scale volatility?

Answer 125

Scale up by the square root of time.

Question 126

6.1 How would you do this: "daily volatility of FTSE100 index returns is estimated to be 0.86%. Calculate the annualised estimated volatility of FTSE 100 returns assuming 250 trading days in the year."

Answer 126

Question 127

6.1 What is a Monte Carlo simulation?

Answer 127

Monte Carlo simulation is 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 is 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. This procedure is repeated many times (100s, 1,000s, or 10,000s), and the distribution of simulated asset values is used to draw inferences about the expected (mean) value of the security—and possibly the variance of security values about the mean as well.

Question 128

6.1 What is the Monte Carlo simulation used for?

Answer 128

1. Value complex securities. 2. Simulate the profits/losses from a trading strategy. 3. Calculate estimates of value at risk (VaR) to determine the riskiness of a portfolio of assets and liabilities. 4. Simulate pension fund assets and liabilities over time to examine the variability of the difference between the two. 5. Value portfolios of assets that have nonnormal return distributions.

Question 129

6.1 What are the advantages and limitations of the Monte Carlo simulation?

Answer 129

An advantage of Monte Carlo simulation is that its inputs are not limited to the range of historical data. This allows an analyst to test scenarios that have not occurred in the past. The limitations of Monte Carlo simulation are that it is fairly complex and will provide answers that are no better than the assumptions about the distributions of the risk factors and the pricing/valuation model that is used. Also, simulation is not an analytic method, but a statistical one, and cannot offer the insights provided by an analytic method.

Question 130

6.1 What is resampling?

Answer 130

Resampling is another method for generating data inputs to use in a simulation. Often, we do not (or cannot) have data for a population, and can only approximate the population by sampling from it. (For example, we may think of the observed historical returns on an investment as a sample from the population of possible return outcomes.) To conduct resampling, we start with the observed sample and repeatedly draw subsamples from it, each with the same number of observations. From these samples, we can infer parameters for the population, such as its mean and variance.

Question 131

6.1 What is bootstrap resampling?

Answer 131

In bootstrap resampling, we draw repeated samples of size n from the full dataset, replacing the sampled observations each time so that they might be redrawn in another sample. We can then directly calculate the standard deviation of these sample means as our estimate of the standard error of the sample mean.

Question 132

6.1 Why is bootstrap resampling different from the Monte Carlo simulation?

Answer 132

Simulation using data from bootstrap resampling follows the same procedure as Monte Carlo simulation. The difference is the source and scope of the data. For example, if a simulation uses bootstrap resampling of historical returns data, its inputs are limited by the distribution of actual outcomes.

Question 133

7.1 What is probability sampling?

Answer 133

Probability sampling refers to selecting a sample when we know the probability of each sample member in the overall population.

Question 134

7.1 What is random sampling?

Answer 134

With random sampling, each item is assumed to have the same probability of being selected.

Question 135

7.1 What is simple random sampling?

Answer 135

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 data point has an equal probability of being selected—we call this simple random sampling. If we want to estimate the mean profitability for a population of firms, this may be an appropriate method.

Question 136

7.1 What is nonprobablity sampling?

Answer 136

Nonprobability sampling is based on either low cost and easy access to some data items, or on using the judgment of the researcher in selecting specific data items. Less randomness in selection may lead to greater sampling error.

Question 137

7.1 What is systematic sampling?

Answer 137

Another way to form an approximately random sample is systematic sampling—selecting every nth member from a population.

Question 138

7.1 What is stratified random sampling?

Answer 138

Stratified random sampling 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 pooled. 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 (stratified) according to major bond risk factors such as duration, maturity, coupon rate, and the like. Then, samples are drawn from each separate category and combined to form a final sample.

Question 139

7.1 what is cluster sampling?

Answer 139

Cluster sampling is also based on subsets of a population, but in this case, we are assuming that each subset (cluster) is representative of the overall population with respect to the item we are sampling. For example, we may have data on personal incomes for a state's residents by county. The data for each county is a cluster.

Question 140

7.1 What is one-stage cluster sampling?

Answer 140

In one-stage cluster sampling, a random sample of clusters is selected, and all the data in those clusters comprise the sample.

Question 141

7.1 what is a two-stage cluster sampling?

Answer 141

In two-stage cluster sampling, random samples from each of the selected clusters comprise the sample. Contrast this with stratified random sampling, in which random samples are selected from every subgroup.

Question 142

7.1 what is convenience sampling?

Answer 142

Convenience sampling refers to selecting sample data based on ease of access, using data that are readily available. Because such a sample is typically not random, sampling error will be greater. An analyst should initially look at the data before adopting a sampling method with less sampling error.

Question 143

7.1 What is judgemental sampling?

Answer 143

Judgmental sampling refers to samples for which each observation is selected from a larger dataset by the researcher, based on one's experience and judgment. As an example, a researcher interested in assessing company compliance with accounting standards may have experience suggesting that evidence of noncompliance is typically found in certain ratios derived from the financial statements. The researcher may select only data on these items. Researcher bias (or simply poor judgment) may lead to samples that have excessive sampling error. In the absence of bias or poor judgment, judgmental sampling may produce a more representative sample or allow the researcher to focus on a sample that offers good data on the characteristic or statistic of interest.

Question 144

7.1 What is the central limit theorem?

Answer 144

AND - The bigger the sample size, the lower the dispersion of the sample mean around the true population mean. it is a direct relationship. NORMAL DISTRIBUTION. Dispersion of this is the SIGMAsquared / n (variance/n). So this is the variance of Xbar.

Question 145

7.1 What is the difference between the sample statistic and the true population parameter? eg:

Answer 145

This is the sampling error. The sample statistic (eg sample mean) will bounce around the population mean, and will have a certain volatility as well. remember the video example of the dice averaging. The sample means generated will jump around a distribution with the true population mean in the centre.

Question 146

7.1 What is the Standard error of the Sample mean?

Answer 146

The standard deviation of the distribution of sample means (Xbar) is called the standard error of the sample mean. Often, the true population standard deviation is not known. In these cases, we use the sample standard deviation.

Question 147

7.1 What is the relationship between n and the SE of the sample mean?

Answer 147

As n increases, SExbar goes down. If you have more data, more accuracy.

Question 148

7.1 Standard deviation vs. Standard error

Answer 148

- SD is the dispersion of a single observation, X, from a distribution. - SE, is the dispersion of the sample mean around the distribution's true population mean, mui

Question 149

7.1 How can our knowledge of the characteristics of normal distributions translate to the Xbar dispersion?

Answer 149

CONFIDENCE INTERVALS work in the same way here.

Question 150

7.1 example of previous card

Answer 150

Question 151

7.1 Describe how and why we use the Student's t-distribution in calculating confidence intervals?

Answer 151

When standard deviation of the true population is not known, we use Sx in the calculation of SE. Sample mean follows the t-distribution instead of a normal distribution (with degrees of freedom n-1) Therefore, instead of 1.96, we use the t-distribution critical value instead of z-distribution to calculate the confidence interval.

Question 152

7.1 Properties of Student's t-distribution (4 points)

Answer 152

- symmetrical (bell shaped) - fatter tails than a normal distribution - defined by a single parameter, degrees of freedom (df) , where df = n - 1 - as df increases, t-distribution approaches standard normal distribution Notice - how on the t-stat table, as the df increase, it falls towards the z-value of normal distribution.

Question 153

7.1 When can you use z-values when calculating confidence intervals of sample mean?

Answer 153

IF n is larger or equal to 30. BUT AS AN APPROXIMATION.

Question 154

7.1 Confidence intervals for mean - table to remember.

Answer 154

Question 155

7. What is resampling? And provide to examples:

Answer 155

Computational methods to estimate the standard error of the sample mean include the following: - Bootstrapping - Jackknife

Question 156

7. What is Bootstrapping resampling?

Answer 156

resampling from original with replacement of items when drawn, calculating the sample mean each time; calculate the sample standard decision of these sample means (which would be the standard error).

Question 157

7. What is Jackknife resampling?

Answer 157

Calculate multiple sample means, each with one observation removed; calculate standard deviation of these means.

Question 158

7. Example of bootstrapping

Answer 158

Question 159

8.1 What is a hypothesis?

Answer 159

A hypothesis is a statement about the value of a population parameter developed for the purpose of testing a theory or belief. Hypotheses are stated in terms of the population parameter to be tested, like the population mean, µ.

Question 160

8.1 What is the hypothesis testing procedure?

Answer 160

Question 161

8.1 what is the null hypothesis?

Answer 161

The null hypothesis, designated H0, is 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 null is generally stated as a simple statement about a population parameter. It always includes the 'equal to' condition

Question 162

8.1 What is the alternative hypothesis?

Answer 162

The alternative hypothesis, designated Ha, is what is concluded if there is sufficient evidence to reject the null hypothesis and is usually what you are really trying to assess. Why? You can never really prove anything with statistics—when the null hypothesis is discredited, the implication is that the alternative hypothesis is valid.

Question 163

8.1 What is the decision rule?

Answer 163

If the test statistic falls outside the critical z-values, then the null hypothesis is rejected. if it falls within, we fail to reject the null hypothesis. Hypothesis testing involved two statistics: 10 the test statistic calculated from the sample data, and 2) the critical value of the test statistic.

Question 164

8.1 What are the critical values I need to remember for hypothesis testing?

Answer 164

Question 165

8.1 What is a test statistic?

Answer 165

A test statistic is calculated by comparing the point estimate of the population parameter with the hypothesized value of the parameter (i.e., the value specified in the null hypothesis). As indicated in the following expression, the test statistic is the difference between the sample statistic and the hypothesized value, scaled by the standard error of the sample statistic:

Question 166

8.1 What are the four distributions for test statistics?

Answer 166

- t-distribution - z-distribution (standard normal distribution) - chi-square distribution - f-distribtuion

Question 167

8.1 what are type I and type II hypothesis testing errors?

Answer 167

Question 168

8.1 What is the power of a test?

Answer 168

The power of a test is the probability of correctly rejecting the null hypothesis when it is false. 1 - P(Type II error)

Question 169

8.1 Table of type I and II Errors in Hypothesis Testing.

Answer 169

Question 170

8.1 What is a p-value?

Answer 170

Is the 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.

Question 171

8.1 Two-tailed test characteristics

Answer 171

Question 172

8.1 What is a p-value?

Answer 172

The p-value of a test is the probability of getting the test statistic (or a result more extreme) if the null were true. p-value < significance level -> REJECT (MOST IMPORTANT IDEA) A p-value is the smallest level of significance at which the null can be rejected. (I think this makes things confusing but is common sense - look at pic)

Question 173

8.2 When hypothesis testing for the difference between means, what is the equation for the test-statistic? (plus explain it)

Answer 173

Remember the s2p is calculated with the formula that s1squared = sum of deviations squared / n-1 So the numerator of s2p is sum of squared deviation of both samples, over the denominator of both degrees of freedom added !

Question 174

8.2 What null and alternative hypothesis should you apply to the below question?

Answer 174

Ho: mean1 - mean2 = 0 Ha: mean1 - mean2 =(with cross) 0

Question 175

8.2 FOR CLARITY ON TESTS: what are the two approaches to hypothesis testing

Answer 175

A. p-value vs significance level In this approach, we compare the p-value computed from the data, and the significance level in the question This approach emphasises the strength of evidence B. Test statistic vs critical value (e.g. z = 1.96) In this approach, you compare the test-statistic with the critical value form the sampling distribution. If the test statistic falls in the rejection region (e.g. |z| > 1.96) we reject null hypothesis. The critical value defines a boundary beyond which results are considered too extreme under Ho. SUMMARY: Comparing to 0.05 and comparing to 1.96 are two ways of expressing the same decision—one in terms of probability (p-value), the other in terms of distance from the null (test statistic).

Question 176

8.3 What is the t-stat formula for the test statistic when we Test for the Mean Difference between Dependent samples from related distributions?

Answer 176

(eg. same people taking same test but under different conditions) d(bar) is the mean of all the differences between the two samples. SE of differences (the denominator) is calculated in the exact same way as the sample mean SE)

Question 177

8.4 Testing a Single Variance Formula (hypothesis testing). What is the t-stat formula?

Answer 177

Testing how much squared deviation we have in our distribution. Hypothesis would relate to a hypothesised level of a true blue variance within the distribution. Numerator = sum of squared deviations Chi-square distribution, does not go below zero (because variance won't)

Question 178

REMEMBER: critical values come from a table, test statistics come from a calculation!!!!

Question 179

8.5 Testing Two Variances (hypothesis testing) (more straightforward one)

Answer 179

Hypothesis relates to the ratio of variances Follows F-distribution (2 degrees of freedom) n1 - 1 degrees of freedom on numerator and n2 - 1 on the denominator

Question 180

8. What is a Parametric / Nonparametric test?

Answer 180

Everything so far has been parametric

Question 181

9.1 What is the t-stat formula for the parametric test of correlation?

Answer 181

Basically testing whether the population correlation coefficient is different from zero (so testing for any correlation)

Question 182

9.1 Why in the parametric test of correlation do we have n-2 degrees of freedom?

Answer 182

because when calculating correlation, we base this on covariance. and when calculating covariance, we fix xbar and ybar. Each loses a degree of freedom.

Question 183

9.2 What is the Spearman rank correlation test?

Answer 183

Whether two sets of ranks are correlated. This is used in the Non-parametric test: rank correlation This finds the correlation number r that we can then use in our test statistic formula that we used before.

Question 184

9.2 When r is found using the spearman rank correlation test, how do we find a solution (performing the test of independence on the data?)

Answer 184

1. Find critical value in a t-table 2. Calculate t-stat

Question 185

9.3 What is a contingency table?

Answer 185

A way to store categorical data

Question 186

9.3 In a contingency table, what is the formula to find the expected value within the table IF the value is independent?

Answer 186

Question 187

9.3 What is the t stat formula for a contingency table (when calculating whether the observed value is statistically significant from the expected value)?

Answer 187

Sum of every cell and sum - for a chi squared statistic

Question 188

9.3 How many degrees of freedom would we have in a contingency table?

Answer 188

rows and columns

Question 189

9.3 How do we then calculate wether the values in the table are independent?

Answer 189

So, we have used expected values, then the t-stat equation to find the t-stat of the table. Then use the degrees of freedom eqn to find the critical value within the t-table. Use the significance level with this too. Chi is a ONE TAILED TEST. (it is a squared distance test so can only be +ve). SO WOULD BE THE 0.95 PART OF T-TABLE

Question 190

10.1 What are the assumptions of linear regression?

Answer 190

Question 191

10.1 What would be a breach of linearity?

Answer 191

NOTE: linearity - breach of linearity would be some kind of LOBF that is curved/quadratic where the residuals are therefore not independent.

Question 192

10.1 What would be a breach of homoskedasticity?

Answer 192

HETEROSKADASCITY: if we don't have homoskedasticity - then our data is not behaving in the same way across all of our regression range, and so the nature of the relationship is changing. We are trying to capture a static equation. B

Question 193

10.1 What would be a breach of independence?

Answer 193

That we seasonality in our data This is called AUTOCORRELATION

Question 194

10.1 What would be a breach in residuals being normally distributed?

Answer 194

note (we dont need to know its normality if it is a large sample size)

Question 195

10.2 What is SSE?

Answer 195

sum of squared errors

Question 196

10.2 What is SSR?

Answer 196

sum of squares of the regression

Question 197

10.2 What is SST?

Answer 197

sum of squares total (is actually the numerator of the sample variance) (so SSE and SSR is breaking SST up into explained and unexplained variation)

Question 198

10.2 Describe SSE SSR and SST on a scatter graph:

Answer 198

Question 199

10.3 What is an ANOVA table?

Answer 199

It is a table that provides and analysis of variance ANOVA compares: variability between groups and variability within groups. An analysis of how good your x variable is at explaining variation in y

Question 200

10.3 Steps for filling in the ANOVA table

Answer 200

1. add the SumSquares regression and residual, so the SSR and SSE. Add them for a total SST. 2. Establish the degrees of freedom. For the regression, as we have one x variable (normally), this will be 1. (in the example used there are 5 degrees of freedom so total has to be n-1, therefore 4) - and therefore those left for the residual are 3. 3. The Mean Square column (a "variance column". Do column 2 (SumSquares so SSR or SSE or SST), then divide by column 1 (degrees of freedom). This is finding the variance explained by each part of the Regression and residual. 4. Calculate the F-stat by doing MSR/MSE from column 3. 5. RSquared (coefficient of determination). This is the level of variation in Y that is explained by the model.

Question 201

10.3 How do you calculate the strength of relationship, as measured by the correlation coefficient?

Answer 201

Square route the coefficient of determination (Rsquared)

Question 202

10.3 Whta is the degrees of freedom of a regression line?

Answer 202

n-2

Question 203

10.3 How do we calculate the F-stat within the ANOVA table?

Answer 203

SO WE ONLY GET ONE F-STAT WITHIN THE TABLE

Question 204

10.3 What would be a 'global test' of regression?

Answer 204

Ho: all of our coefficients are zero H1: There is at least one of our coefficients is non-zero

Question 205

en you have the F-stat, how do we use it to test the hypothesis?

Answer 205

We find the critical F-value from statistical tables (95th percentile - if at 5% significance), and compare to F-stat as normal

Question 206

10.3 How do you calculate the standard error of estimate?

Answer 206

Better understood as volatility of the residual Route MSE

Question 207

10.4 What is a regression coefficient t-test?

Answer 207

Rather than testing the regression as a whole, this is testing a single coefficient. Example, if you know from the global regression test that at leats one of the coefficients is non-zero, then it is worthwhile interrogating individually which one it is

Question 208

10.4 What is the formula for calculating the test-stat for the regression coefficient t-test?

Answer 208

This is pretty much the same as when calculating the test-state for a mean, with the sample mean - observed mean / SE

Question 209

10.5 What are prediction intervals?

Answer 209

Intervals outside of the regression line, where, for example, we are 95% sure that we will observe our Y value given a value for X

Question 210

10.5 what is the standard error of the forecast? And formula (but not important to learn lol)

Answer 210

(notation Sf). The unit of distance you need to go on either side of the regression line to capture 95% chance that the true Y will be in that range.

Question 211

10.5 How do we use the standard error of the forecast to create the prediction intervals for a regression line?

Answer 211

Use the Sf and critical value product to create positive and negative boundaries with the predicted value.

Question 212

10.6 Change in a logarithm is...

Answer 212

is the same as the continuously compounded return

Question 213

10.6 Functional form table.

Answer 213

Question 214

11. What is Big Data?

Answer 214

Extremely large and complex datasets

Question 215

11. What are traditional and nontraditional sources of Big Data?

Answer 215

Question 216

11.1 What are the Big V's that are considerable factors that define Big Data?

Answer 216

veracity - accuracy and unbiased nature

Question 217

11.1 what is artificial intelligence?

Answer 217

Question 218

11.1 What is machine learning?

Answer 218

Question 219

11.1 What is supervised learning?

Answer 219

Human labels what goes in and out

Question 220

11.1 What is unsupervised learning?

Answer 220

Human does not label what goes in and out

Question 221

11.1 What is deep learning?

Answer 221

Called deep learning because you can have multiple layers

Question 222

11.1 What are the three stages of machine learning?

Answer 222

Question 223

11.1 What is Overfitting and Underfitting within the stages of machine learning?

Answer 223

Question 224

11.2 name some applications of data science

Answer 224