Quant Flashcards
coefficient of variation
CV = (σ/mean) –> standard deviation / mean
Express how much dispersion exists relative to the mean and allow for comparison of dedree of dispersion
Choose investment with lower CV
calc a confidence interval
step 1 - calc the standard error of the mean = standard deviation / sq rt of the sample size
step 2 - calc confidence interval
This is the confidence interval range.
- 645 for 90% confidence intervals (the significance level is 10%, 5% in each tail).
- 960 for 95% confidence intervals (the significance level is 5%, 2.5% in each tail).
- 58 for 99% confidence intervals (the significance level is 1%, 0.5% in each tail).
Step 3 calc the confidence interval. For example, mean of 25, w 95% conficence interval, standard error is 2
Sample Mean +/- (standard error * 1.96)
Lower tail: 25 - (2 * 1.96) = 21.08
Upper tail: 25 + (2 * 1.96) = 28.92
Money Weighted Return
Concept here is you’re looking for IRR based on cash into (+ buys) and out of (- divds and sells) of the account.
Use cash flow function on calculator.
Time Weighted Return
Concept here is this is the geometric mean of each holding period return. Time weighted is not affected by the timing of cashflows
step 1 - calc holding period return for each period.
([price + divd) / purchase price] - 1
step 2 - use geometric mean to get TWR
When calculating the geometric mean for a returns data set: (1+R1) * (1+R2) * (1+Rn)^1/n
The time-weighted rate of return is the preferred method of performance measurement, because it is not affected by the timing of cash inflows and outflows.
How to Calc Continuous Compound Returns - Given a $ amount or Rate
Use LN button on calculator
If given $ amount –> ln(price/purchase price)
e.g. LN(120/100) = LN(1.20) = .1823 = 18.23%
If given a rate –> ln(1+rate)
e.g. rate is 20% = ln(1+.20) = .1823 = 18.23%
Describe unbiased, consistent and efficient estimators.
An unbiased estimator has an expected value equal to the true value of the population parameter.
A consistent estimator is more accurate the greater the sample size.
An efficient estimator has the sampling distribution that is less than that of any other unbiased estimator.
Covariance of a 2 security portfolio
Sum of: [(return on Asset 1 in period t - mean return on Asset 1) * (return on Asset 2 in period t - mean return on Asset 2)] / # of periods
Covariance measures the extent to which two variables move together over time. Same rules as correlation
Correlation of 2 stock portfolio
Remember correlation symbol looks like a little p
= covariance of asset 1 & 2 / (standard deviation of asset 1 * standard deviation of asset 2)
Note covariance can then be expressed as: = correlation * standard deviation of asset 1 * standard deviation of asset 2)
Calc the variance and standard deviation of a portfolio
variance = (weigh of asset 1^2 * variance of asset 1) + (weigh of asset 2^2 * variance of asset 2) + 2 * weight of asset 1 * weight of asset 2 * the covariance of asset 1 & 2.
Note: the covariance of asset 1&2 is also = correlation of asset 1&2 * standard deviation of asset 1 * standard deviation of asset 2
Note: the variance of asset 1 is the standard deviation of asset 1^2. think of the symbols.
Standard Deviation of the Portfolio = square root of the portfolio variance.
steps for hypothesis testing
1 - Stating the hypotheses.
2 - Identifying the test statistic and its probability distribution.
3 - Specifying the significance level.
4 - Stating the decision rule.
5 - Collecting the data and performing the calculations.
6 - Making the statistical decision.
7 - Making the economic or investment decision.
covariance of 2 stock
first calc the mean return for both
covariance = sum of: {(return on X period 1 - Mean of X) * (return on y period 1 - mean of y)] + {(return on X period 2 - Mean of X) * (return on y period 2 - mean of y)] / (n - 1)
The effective annual rate, based on continuous compounding for a stated annual rate of %, can be calculated from the formula:
effective annual rate = e^x [calc button] - 1
Based on a stated rate of 10%, the effective rate with continuous compounding is .1 (2nd LN for e^x] - 1 = .10517 or 10.517%.
what does the “sampling error” represent?
The difference between the true population parameter (i.e. mean index return) you’re trying to estimate and the sample statistic calculated is called the sampling error. The arithmetic mean is the appropriate estimator of the next period’s return.
Geometric Mean
DON’T FORGET TO MINUS THE 1. The geometric mean is always less than or equal to the arithmetic mean. The only time the two means will be equal is when there is no variability in the observations.
frequently used to average rates of change over time or to compute the growth rate of a variable. we frequently use the geometric mean to average a time series of rates of return on an asset or a portfolio, or to compute the growth rate of a financial variable such as earnings or sales.
Rg = {[(1+R1) * (1+R2) * (1+Rn)] ^ 1/n} - 1
The greater the variability of returns, the greater the difference between this and arithmetic mean. Arithmetic mean will always be greater
F - test
Used to compare the variances of 2 populations
Define a Type I Error
Rejecting the a true null. The probability of this is the significance level. You’re 95% confident the bull is false, so your 5% significance is that the null is true
The II error
Failing to reject the null when it is false
Power of the test
The probability of correctly rejecting the null
P value
The smallest level of significance at which the null can be rejected. The smaller the p-value, the stronger the evidence against the null.
Bayes Formula
Used to update probability given new info.
P (Event | New Info) = {P (New Info | Event) / (New Info)} * P(Event)
Z score
(Observation - population mean) / standard deviation
Continuous compounding
Continuously compounded rate from stated rate:
= ln(S1/So) # ln (1+HPR)
Effective annual rate from continuously compounded rate:
= e^r - 1
OR
deposits 2,000 into an account that pays 6% per annum compounded continuously. Value after 4 years is
.06 * 4 2nd e^x * 2000 = 2,542.25
cross sectional data
Data on some characteristics of multiple companies at a single point in time are cross-sectional data.
Monte Carlo Simulation
Monte Carlo simulation is better suited to hypothetical “what if” analysis than historical simulation. Monte Carlo simulation is not an analytical method that can provide great insights into cause and effect relationships.
Monte Carlo simulation is used to:
- Value complex securities.
- Simulate the profits/losses from a trading strategy.
- Calculate estimates of value at risk (VaR) to determine the riskiness of a portfolio of assets and liabilities.
- Simulate pension fund assets and liabilities over time to examine the variability of the difference between the two.
- Value portfolios of assets that have non-normal returns distributions.
Statistically significant vs economically meaningful.
A statistically significant result might not be economically meaningful after an analyst accounts for the risk, transaction costs, and applicable taxes.
standardize a random vairable
X - mean of X / standard deviation of x
this is the z formula z = (x - u) / standard deviation
Test statistic
(sample mean - pop mean) / standard error
Standard error is standard deviation / square root of sample size
