Quant Flashcards
the difference between real and nominal GDP?
Real GDP = GDP with the inflation effect stripped out
Explain the concept of compound interest?
Compound interest: The process of the value of money growing over time due to the effect of interest accumulating on previously earned interest
What are the 3 things interest rates represent in terms of cash flow modelling?
The required rate of return
Discount rate
Opportunity cost
What is the formula for the nominal risk free rate?
Nominal Rf = Real Rf + Expected inflation rate
What are the 3 risks associated with investing in securities?
Default risk
Liquidity risk
Maturity risk
What is the formula for the required rate of return for a security incorporating maturity risk, default risk and liquidity risk?
Required rate of return on a security = Nominal Rf + Default risk + Liquidity risk + Maturity risk
What is the formula for the effective annual rate?
( EAR = (1 + \frac{I}{m})^m - 1 )
What are the formulas for quarterly, monthly and daily compounding frequencies?
( Quarterly = (1 + \frac{I}{4})^{4} -1 )
[ Monthly = (1 + \frac{I}{12})^{12} -1 ]
[ Daily = (1 + \frac{I}{365})^{365} -1 ]
What is the FV of a £100 investment after 2 years with a interest rate of 10% compounded annually?
N = 2, I/Y = 10, PV = -100, PMT = 0 ——> CPT FV = $121
What is the PV of a £100 FV with a interest rate of 10% compounded annually?
N = 2, I/Y = 10, PMT = 0, FV = 100 ——> CPT PV = $83
What is the difference between an ordinary annuity and an annuity due?
Ordinary Annuity: The most typical type of annuity, cash flows occur at the end of each period for a finite number of periods
Annuity Due: Annuities where cash flow occurs at the beginning of each period
What is the future value of an ordinary annuity that pays $1,000 per year at the end of the next 3 years, with a 10% interest rate?
N = 3, I/Y = 10, PV = 0, PMT = -1,000 —- > FV = $3,310
What is the present value of an ordinary annuity that pays $1,000 per year at the end of the next 3 years, with a 10% interest rate?
N = 3, I/Y = 10, PMT = -1,000, FV = 0 —- > PV = $2,487
What is the future value of an annuity due that pays $1,000 per year at the beginning of the next 3 years, with a 10% interest rate?
Set calculator to BGN mode — > N = 3, I/Y = 10, PV = 0, PMT = -1,000 —- > FV = $3,641
What is the present value of an annuity due that pays $1,000 per year at the beginning of the next 3 years, with a 10% interest rate?
Set calculator to BGN mode — > N = 3, I/Y = 10, PMT = -1,000, FV = 0 —- > PV = $2,735
What is the formula for the present value of a perpetual cash flow?
( Present\ value\ of\ a\ perpetuity = \frac{Cash\ flow}{(I/m)} )
What is the present value of the following uneven set of cash flows over a three year period at a 10% interest rate: Year 1=100, Year 2=200, Year 3=300?
Using the calculator:
Cash flow 1: N = 2, I/Y = 10, PV = -100, PMT = 0 —- > FV = $121
Cash flow 2: N = 1, I/Y = 10, PV = -200, PMT = 0 —- > FV = $220
Cash flow 3: N = 0, I/Y = 10, PV = -300, PMT = 0 —- > FV = $300
Therefore total future value = $641
What is the future value of the following uneven set of cash flows over a three year period at a 10% interest rate: Year 1=100, Year 2=200, Year 3=300?
Using the calculator:
Cash flow 1: N = 1, I/Y = 10, PMT = 0, FV = $100 —- > PV = $91
Cash flow 2: N = 2, I/Y = 10, PMT = 0, FV = $200 —- > PV = $165
Cash flow 3: N = 3, I/Y = 10, PMT = 0, FV = $300 —- > PV = $225
Therefore total present value = $481
What is the cash flow additivity principle?
Cash Flow Additivity Principle: PV of a set of cash flows = the present value of each cash flow individually… added together
What are some examples of measures of central tendency?
Arithmetic mean Geometric mean Weighted mean Median Mode
What are some examples of measures of dispersion?
Range
Variance
Standard deviation
What do measures of central tendency and measures of dispersion equate to in finance terms?
Measures of central tendency equate to expected return, measures of dispersion measures risk
What is the difference between descriptive and inferential statistics?
Descriptive statistics is the term given to the analysis of data that helps describe, show or summarize used to summarize the important characteristics of large data sets (populations) in a meaningful way such that, for example, patterns might emerge from the data.
Inferential statistics: Stats made to make predictions, forecasts and judgements about a wider population using only a smaller subset or sample
What is the difference between a population and a sample?
Population: Set of all possible members of a stated group. e.g. returns of all the stocks on the NYSE
Sample: Subset of the population
What are the 4 levels of measurement scales?
NOIR
- Nominal
- Ordinal
- Interval
- Ratio
What is the difference between a parameter and a sample statistic?
Sample statistic: A measure used to describe a characteristic of a sample
Parameter: A measure used to describe a characteristic of a population
What are the three steps to building a frequency distribution?
Define the intervals
Tally the observations / assign to relevant interval
Count the observations
What is a relative frequency chart?
Relative frequency: A histogram with entries in the bar chart out of described out of 100%
What is a histogram?
Histogram: Bar chart of absolute frequencies of a distribution (same as a frequency distribution)
What is a frequency polygon?
Frequency polygon: Absolute frequency line chart with midpoints of the intervals plotted
What is the sum of mean deviations formula?
( Sum\ of\ mean\ deviations: \Sigma(X_i - \mu) = 0 )
What is the formula for the median?
( Median = \frac{(n+1)}{2}th\ entry )
What is the formula for the geometric mean?
( Geometric\ mean = (X_1X_2….X_n)^{1/n} )
What is the formula for the geometric mean for percentage returns?
( Geometric\ mean = ((1+X_1)…(1+X_n))^{1/n} -1 )
When is the geometric mean more appropriate than the arithmetic mean?
When looking at multi year returns
What is the harmonic mean used to calculate?
Average cost of shares purchased over time when you buy same dollar amount in shares
What is the formula for the harmonic mean?
( Harmonic\ mean = \frac{n}{\Sigma 1/X_i} )
When should you use arithmetic mean over harmonic mean?
You use the arithmetic mean when you purchase the same number of shares, use harmonic mean when you purchase same dollar amount of shares
What is the formula for the measure of location in a dataset?
( Measure\ of\ location\ in\ a\ dataset: L_y = (n+1)\frac{y}{100} )
What is a decile?
Decile: A distribution divided into tenths
What is a quintile?
Quintile: A distribution divided into fifths
What is dispersion?
Dispersion: Variability around the central tendency
What is the formula for the mean average deviation?
( MAD = \frac{\Sigma |X_i - \mu |}{n} )
What is the mean average deviation?
The average that an individual return will deviate from the mean return
What is the formula for the population variance?
( Variance = \frac{\Sigma (X_i - \mu)^2}{n} )
What is the formula for the sample variance?
( Variance = \frac{\Sigma (X_i - \mu)^2}{n-1} )
What is a biased estimator?
An estimator where the estimator’s expected value and the true value of the parameter are unequal
What is Chebyshev’s inequality?
Chebyshevs inequality: 1- 1/k^2 The % of observations that lie within k standard observations of the mean is at least 1-1/k^2 for all k > 1
How do we calculate k in Chebyshev’s inequality?
k = range we are looking around the mean divided by the standard deviation
What is the formula for the coefficient of variation?
( c_v = \frac{\sigma}{\mu} )
What is the formula for the sharpe ratio?
( Sharpe\ Ratio = \frac{R_p - R_f}{\sigma} )
For a positvely skewed distribution what is the greater the mean or the median?
The mean
What is more prominently impacted by outliers, the mean or the median?
Mean
What is kurtosis?
Kurtosis is a measure of how more or less peaked a distribution is and also how fat the tails are compared to a normal distribution
What are the three different states of kurtosis?
- Leptokurtic
- Mesokurtic
- Platykurtic
What is the formula for excess kurtosis and what does it measure?
Excess kurtosis = Kurtosis – 3. Measures whether the distribution has more or less kurtosis than a normal distribution
What is the kurtosis of a normal distribution?
3
What are the 2 criteria that need to be met for a probability to exist?
- 0 =< p =< 1
2. If a set of events is mutually exclusive and exhaustive the sum of all these outcomes = 1
What is an empirical probability?
Empirical probability: A probability established by analysing past data
What is a priori probability?
Priori probability: A probability determined using a formal reasoning and inspection process
What is a subjective probability?
A probability based on a personal feeling or hunch
Which 2 types of probability are completely objective in nature?
Empirical and priori probabilities
What is the formula for odds?
( Odds = \frac{p}{1-p} )
What is the formula for probability? (Hint: includes odds)
( p = \frac{1}{1+odds} )
What is the multiplication rule of probability formula?
( Multiplication\ rule\ of\ probability = P(ANB) = P(A/B) * P(B) )
What is the conditional probability formula?
( Conditional\ probability: P(A/B) = \frac{P(ANB)}{P(B)} )
What is the addition rule of probability?
( Addition\ rule\ of\ probability: P(ANB) = P(A) + P(B) - P(ANB) )
What is the total probability rule?
(Total\ probability\ rule: P(A) = P(A/B_1) * P(B_1) +…+ P(A/B_n) * P(B_n))
What is the impact on a joint probability if 2 events are mutually exclusive
P(ANB) = 0
What is the impact on a joint probability if 2 events are independent?
P(ANB) = P(A)P(B)
What is the formula for the law of expected total probability?
( E[X] = \Sigma E[X|A_i] * P(A_i) )
What is covariance?
Covariance measures how one random variable moves in relation to another
What is the formula for covariance?
( Cov(X,Y) = E[XY] - E[X]E[Y] )
What is the range of covariance?
Covariance ranges from minus infinity to plus infinity
What is the formula for the correlation coefficient?
( Correlation\ coefficient: p_{ij} = \frac{Cov_{ij}}{\sigma_i \sigma_j} )
What is the range for the correlation coefficient?
It ranges from -1 to +1
What is a spurious correlation?
Spurious correlation: 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
How do you calculate the expected return of a 2 stock portfolio?
( E[R_p] = w_AE[R_A] + w_BE[R_B] )
How do you calculate the variance for a 2 stock portfolio?
( \sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 +2w_Aw_B\sigma_A \sigma_B p_{AB} )
What is the purpose of bayes theorem?
To update a probability based on new information available
What is Bayes formula?
( P(A/B) = \frac{P(B/A)P(A)}{P(B)} )
When should you use the permutation and combination formulas?
When ordering matters in the question, use the permutation formula. If ordering does not matter in the question, use the combination formula
What are the properties of a discrete random variable?
A probability distribution where the set of all possible outcomes can be counted
What are the properties of a continuous random variable?
A probability distribution where the set of all possible outcomes cannot be counted
What is a continuous uniform distribution (CDF)?
It represents the cumulative probability up until a random variable
What are the properties of a discrete uniform distribution?
Discrete Uniform Distribution: All outcomes are equally likely to occur. 2 parameters “a” and “b”
What are the properties of a bernoulli random variable?
Bernoulli Random variable: the discrete probability distribution of a random variable which takes the value 1 with probability p and the value 0 with probability q=1-p, that is, the probability distribution of any single experiment that asks a yes–no question. It is binary
What are the properties of a binomial distribution?
Binomial distribution: binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n Bernoulli trials
What are the mean and variance of a binomial distribution?
Mean = np Variance = np(1-p)
What is the probability of one outcome of a continuous distribution
zero
What % of observations lie within one standard deviation either side of the mean?
68%
What % of observations lie within two standard deviations either side of the mean
95%
What is a confidence interval?
Confidence Interval: The range around the expected value that the outcome us expected to be found in
What is the general confidence interval formula?
( Confidence\ interval = Point\ estimate \pm reliability\ factor * standard\ error )
What is the formula for a standardised normal distribution?
( Z = \frac{(x-\mu)}{\sigma} )
Define shortfall risk?
Shortfall Risk: The risk that the return Rp falls below a threshold level Rl over a given time period
What are the properties of a lognormal distribution?
Bounded below at 0
Positively skewed
What is the formula for continuously compounded returns?
LN(S1/S0)
What are the properties of monte carlo simulation?
Fairly complex
Provides answers that are no better than the assumptions / inputs around the distribution
Cannot provide the insight that an analytic solution can
What are the properties of historical simulation?
Uses historical information so no estimates required
Past changes in risk factors are not necessarily a good indicator of future changes
Cannot address “what if” questions like monte carlo simulation can (sensitivity analysis)
What is the difference between simple random sampling, systematic sampling and stratified random sampling?
Simple random sampling: A sampling method where every member of a population has equal chance of being chosen for the sample
Systematic sampling: Selecting the nth member from a population for a sample
Stratified random sampling: Uses a classification system to separate the population into smaller groups based on one or more distinguishing characteristics into stratum
What are the benefits of using stratified random sampling?
Guarantees we have a certain number of samples from each stratum in our pool
What is sampling error?
Sampling error: The difference between the population metric and the equivalent sample metric
What is the difference between time series and cross sectional data and what is longitudinal and panel data?
Time series data: Observations taken over a period of time at specific and equally spaced intervals e.g. McDonalds P/E ratio from 2004 to 2019
Longitudinal data: Observations over time of multiple characteristics of the same entity e.g. McDonalds P/E ratio, share price and dividend yield from 2004 to 2019
Panel data: Observations over time of multiple characteristics of multiple entities. e.g. McDonalds and Apples, P/E ratio, share price and dividend yield from 2004 to 2019
Cross sectional data: Observations taken at a single point in time e.g. EPS of all S&P 500 companies at 31/12/04
What is the central limit theorem?
Central limit theorem: In a population of size n with mean µ and variance σ^2, the sampling distribution of the sample mean x approaches a normal distribution N-( µ, σ^2/n) as the sample size becomes large (n> 30)
What are the 3 properties of a reliable estimator?
- Unbiased
- Efficient
- Consistent
Why is a normal distribution more appropriate than a t-distribution for larger sample sizes?
As n gets larger the distribution closer approaches the normal distribution - therefore a normal distribution is more appropriate
What are the properties of a student t distribution?
Symmetric
Appropriate to use when constructing confidence intervals on small samples (n<30) on a distribution with an unknown variance
Defined by a single parameter: Degrees of freedom = n-1
Fatter tails than normal distributions
What are the different components of a confidence interval?
- Point estimate
- Reliability factor
- Confidence intervals
If the sample is normal with an unknown variance from a sample size, N<30. What is the appopriate test statistic?
t-test
If the sample is normal with an unknown variance from a sample size, N>30. What is the appopriate test statistic?
t or Z statistic
What is survivorship bias?
Survivorship bias: Only include funds in a sample that over the years have survived. They do not include funds that have ceased to trade / failed. Very common bias in mutual funds
What is sample selection bias?
Sample selection bias occurs when the sample selected does not represent the overall population (does not have the same characteristics as the population).
What is look ahead bias?
Look ahead bias occurs when test is conducted using data that was not available at the time of the test (ex predicting earnings for the year whereas fourth quarter earnings are not yet available.)
What is time period bias?
Time period bias: Using sample series which is too short or not seasonally adjusted or even too long sample time series can cause biased estimators
What are the 7 steps to perform a hypothesis test?
- State the hypothesis test
- Select the test statistic
- Specify the significance level (alpha)
- Set a decision rule
- Collect data and calculate sample statistics
- Make decision on the hypothesis test
- Make decision as to the validity of the overall test
What is the null hypothesis?
Null hypothesis: The hypothesis the researcher wants to reject
What is the alternative hypothesis?
Alternative hypothesis: The conclusion made if there is significant evidence to reject the null hypothesis
What must a null hypothesis always contain?
Null hypothesis always contains an equals sign
What is the formula for the Z-test test statistic?
( Z = \frac{\bar{x}-\mu_0}{\sigma/\sqrt{n}} )
What is a type I error?
Type 1 error: Probability of rejecting null hypothesis when it is actually true
What is a type II error?
Type 2 error: Probability of not rejecting null hypothesis when it is actually false
What is the power of a test?
1- P(Type II error)
What is the relationship between a type I error and a type 2 error?
If a P(type 1 error) goes up the P(type II error) goes down, they have an inverse relationship
What is the p-value?
p-value: the lowest level of significance at which we can reject the null hypothesis
What are the factors that determine whether we use a Z test or a T test for a hypothesis test?
Sample size
Distribution (normal or non-normal)
Variance (known or non- known)
How many degrees of freedom are used for a difference in mean test?
n1+n2 - 2
What test statistic do we use to test mean differences for 2 DEPENDENT samples of data
t-test paired comparisons test
What is the chi squared test to test?
Chi-square test is used for hypothesis tests concerning the value of the variance of a normally distributed population.
What is a unique property of a chi squared test compared to a normal distribution?
It is asymmetric
What is the formula for the test statistic for a chi squared distribution?
( X_{n-1}^2 = \frac{(n-1)s^2}{\sigma_0^2} )
When do we use the F test?
When we test whether the variance of one population is equal to the variance of another population
What is the test statistic for the F test?
( F = \frac{S_1^2}{S_2^2} )
What is the difference between a parametric and non-parametric test?
Parametric Tests: E.g. t test, F test, chi squared test - rely on the test statistic to test statistical significance
Non-Parametric Tests: distribution-free tests by some researchers – are tests that do not make any assumption regarding the distribution of the parameter under study
