Volume 1 - Quantitative Methods Flashcards
What are the 3 possible interpretations for a interest rate ?
An interest rate, r, can have three interpretations: (1) a required rate of return, (2) a discount rate, or (3) an opportunity cost. An interest rate reflects the relationship between differently dated cash flows.
An interest rate can be viewed as the sum of the real risk-free interest rate and a set of premiums that compensate lenders for bearing distinct types of risk: an inflation premium, a default risk premium, a liquidity premium, and a maturity premium.
r = Real risk-free interest rate + Inflation premium + Default risk premium +
Liquidity premium + Maturity premium.
What is A money-weighted return ?
A money-weighted return reflects the actual return earned on an investment after accounting for the value and timing of cash flows relating to the investment.
What is A time-weighted return ?
A time-weighted return measures the compound rate of growth of one unit of currency invested in a portfolio during a stated measurement period. Unlike a money-weighted return, a time-weighted return is not sensitive to the timing and amount of cashflows and is the preferred
performance measure for evaluating portfolio managers because cash
withdrawals or additions to the portfolio are generally outside of the control of the portfolio manager.
Gross return, return prior to deduction of managerial and administrative expenses (those expenses not directly related to return generation), is an appropriate measure to evaluate the comparative performance of an asset manager.
Net return, which is equal to the gross return less managerial and administrative expenses, is a better return measure of what an investor actually earned.
Annualizing periodic returns allows investors to compare differnt investments across different holding periods to better evaluate and compare their relative performance. With the number of compounding periods per year approaching infinity, the interest is compound continuously.
Real returns are particularly useful in comparing returns across time periods because inflation rates may vary over time and are particularly useful for comparing investments across time periods and performance between different asset classes with different taxation.
If USD 9,500 today and USD 10,000 in one year are equivalent in value, then USD 10,000 – USD 9,500 = USD 500 is the required compensation for receiving USD 10,000 in one year rather than now. The interest rate (i.e., the required compensation stated as a rate of return) is USD 500/USD 9,500 = 0.0526 or 5.26 percent.
An opportunity cost is the value that investors forgo by choosing a course of action. In the example, if the party who supplied USD 9,500 had instead decided to spend it today, he would have forgone earning 5.26 percent by consuming rather
than saving. So, we can view 5.26 percent as the opportunity cost of current
consumption.
What is The real risk-free interest rate ?
The real risk-free interest rate is the single-period interest rate for a completely risk-free security IF NO inflation were expected. In economic theory, the real risk-free rate reflects the time preferences of individuals for current versus future real consumption.
The sum of the real risk-free interest rate and the inflation premium is the nominal
risk-free interest rate
The nominal risk-free interest rate reflects the combination of a real risk-free rate plus an inflation premium:
(1 + nominal risk-free rate) = (1 + real risk-free rate)(1 + inflation premium)
In practice, however, the nominal rate is often approximated as the sum of the
real risk-free rate plus an inflation premium:
Nominal risk-free rate = Real risk-free rate + inflation premium.
Typically, interest rates are quoted in annual terms, so the interest rate on a 90-day government debt security quoted at 3 percent is the annualized rate and not the actual interest rate earned over the 90-day period.
Whether the interest rate we use is a required rate of return, or a discount rate,
or an opportunity cost, the rate encompasses the real risk-free rate and a set of risk premia that depend on the characteristics of the cash flows. All these premia vary over time and continuously change, as does the real risk-free rate. Consequently, all interest rates fluctuate, but how much they change depends on various economic fundamentals—and on the expectation of how these various economic fundamentals can change in the future.
The arithmetic mean return assumes that the amount invested at the beginning of each period is the same. In an investment portfolio, however, even if there are no cash flows into or out of the portfolio the base amount changes each year. The previous year’s earnings must be added to the beginning value of the subsequent year’s investment— these earnings will be “compounded” by the returns earned in that subsequent year. We can use the geometric mean return to account for the compounding of returns.
In general, the arithmetic return is biased upward unless each of the underlying
holding period returns are equal. The bias in arithmetic mean returns is particularly
severe if holding period returns are a mix of both positive and negative returns.
For reporting historical returns, the geometric mean has considerable appeal
because it is the rate of growth or return we would have to earn each year to match
the actual, cumulative investment performance.
The arithmetic mean is always greater than or equal to the geometric mean.
If we want to estimate the average return over a one-period horizon = we should use arithmetic, because it is the average of one-period returns.
If we want to estimate the average returns over more than one period = geometric, because the geometric mean captures how the total returns are linked over time.
The harmonic mean : sample of observations of 1, 2, 3, 4, 5, 6, and 1,000, the harmonic mean is 2.8560.
Compared to the arithmetic mean of 145.8571, we see the influence of the outlier (the 1,000) to be much less than in the case of the arithmetic mean. So, the harmonic mean is quite useful as a measure of central tendency in the presence of outliers.
Unless all the observations in a dataset are the same value, the harmonic mean is always less than the geometric mean, which, in turn, is always less than the arithmetic mean.
What is the trimmed mean ?
Both the trimmed and the winsorized means seek to minimize the impact of outliers in a dataset. Specifically, the trimmed mean removes a small defined percentage of the largest and smallest values from a dataset containing our observation before calculating the mean by averaging the remaining observations.
What is the winsorized mean ?
A winsorized mean replaces the extreme observations in a dataset to limit the
effect of the outliers on the calculations. The winsorized mean is calculated after
replacing extreme values at both ends with the values of their nearest observations,
and then calculating the mean by averaging the remaining observations.
What is the money-weighted return ?
The money-weighted return accounts for the money invested and provides the
investor with information on the actual return she earns on her investment. y. Amounts invested are cash outflows from the investor’s perspective and amounts returned or withdrawn by the investor, or the money that remains at the end of an investment cycle, is a cash inflow for the investor.
What is The internal rate of return ?
The internal rate of return is the discount rate at which the sum of present values
of cash flows will equal zero.
What are the 2 categories of Data and the 4 sub-categories?
1- Categorical: values that describe a quality or characteristic (must be mutually exclusive)
(N) Numerical: No logical order (Ex: Sectors of economy)
(O) Ordinal: has a logical order (no info about the distance between groups)
2- Numerical : Measured or counted quantities
(I) Integer/Discrete : Limited to a finite number of values
(R) Ratio/Continous : Can take any value within a range
What is Cross-sectional , Time-series and Panel Data ?
1- Cross-sectional : Multiple observations of a particular variable (stock prices of 60 companies)
2- Time-series : Multiple observations of a particular variable for the same observational unit over time (GM stock price in the last 5 days)
3- Panel Data : Cross-sectional + Time-series
What is structured and unstructured Data ?
1- Structured : Highly organized in a pre-defined manner
2- Unstructured Data : No organized form (social media, news) –> Also called alternative Data
In order to analyse Data, it must be transformed into structured data
For numerical data, how can we determine the interval width ?
Range (max - min) / K
K: number of intervals –> too few or too much can bring problems ; loss of info or too much noise
What are the uses of arithmetic mean and what to do if there’s a problem ?
The arithmetic mean can be usefull to explain the return for 1 year of an Index.
Cross-sectional mean : Average sales of 50 companies
Time-series mean : Average sales for the last 10 yrs for GM
This mean is susceptible to outliers: We can do nothing if they are legitimate and contain meaningful information
Or:
Delete the outliers by doing a trimmed mean. Excluding a small % of the lowest and highest values (Ex: 5% –> 2.5% highest and 2.5% lowest)
Or:
Replace the 2.5% by the value at which all others lie above –> the 96th observation 88 so 2.5% also become 88.
What are the different types of mode list ?
unimodal: only 1 value that is most frequent
bi-modial: two values have the highest frequency
….
Or no mode –> Uniform distribution
What is the main use for the Geometric mean ?
It is used to interpret the growth rate. Ex: The rate that makes your investement grow form initial enter into now.
Also referred to as compounded returns
What is the use of the harmonic mean ?
It is appropriate for averaging ratios when the ratios are repeatedly applied to a fixed quantity to yield a variable number of units.
Ex: Dollar cost averaging
What is dispersion ?
The variability around the central tendency
Two investors in the same mutual fund could have different money-weighted returns depending on the amount and timing of their contributions. A fund manager’s performance should only be judged on the basis of his or her decisions and actions. The money-weighted return can be skewed by the timing and amount of cash flows into and out of a fund, making it an inappropriate metric for assessing the performance of a manager who has no control over these.
Because TWR is unaffected by the timing and amount of cash flows, it is appropriate for assessing managers who do not control external cash flows, such as a mutual fund that is regularly receiving new contributions and making payouts to meet redemptions.
For an interest rate that compounds
times annually, the formulas for the future value and present value of an investment are:
FVn = PV(1+rs/m)** mn
PV = FVn (1+rs/m)** -mn
If we assume a 365-day year, the annualized return for an investment that generates a 0.6% return over 8 days is:
(1 + 0.006)** 365/8 - 1 = 31.4%
This example illustrates one of the limitations of annualizing returns, which is that the calculations are based on the assumption that short-term performance could be repeated over a longer period.
It is also possible to annualize returns that have been generated over holding periods of longer than one year. In these case, the number of periods per year, c, becomes a fraction. For example, if an investment earns a 17.8% return over two years, it’s annualized return is:
( 1 + 0.178) ** 1/2 - 1 = 8.53%
Gross returns are calculated on a pre-tax basis; trading expenses are accounted for in the computation of gross returns as they contribute directly to the returns earned by the manager.
The Time Value of Money in Finance
calculate and interpret the present value (PV) of fixed-income and equity instruments based on expected future cash flows
calculate and interpret the implied return of fixed-income instruments and required return and implied growth of equity instruments given the present value (PV) and cash flows
explain the cash flow additivity principle, its importance for the no-arbitrage condition, and its use in calculating implied forward interest rates, forward exchange rates, and option values
There are three basic categories of fixed-income instruments depending on how their cash flows are structured — discount instruments, coupon instruments, and annuity instruments.
1- Discount instruments (zero-coupons) have a very simple structure. One amount PV is borrowed today and a larger amount FV is repaid when the loan matures.
2- With coupon instruments, a principal amount PV is borrowed today and the same amount FV is repaid at maturity, but the borrower compensates that lender with periodic interest payments PMT at regular intervals during the term of the loan.
3- An annuity instrument is structured as a specified number of level cash flows. A common example of an annuity is a fixed-rate mortgage. Like a coupon instrument, the borrower makes payments at regular intervals to retire the debt.
A perpetual bond, also known as a perpetuity, is a special type of coupon instrument that makes fixed payments at regular intervals but never matures. The present value of an instrument that provides a perpetual stream of level payments is calculated as follows:
PV = PMT/r
Statistical Measures of Asset Returns
calculate, interpret, and evaluate measures of central tendency and location to address an investment problem
calculate, interpret, and evaluate measures of dispersion to address an investment problem
interpret and evaluate measures of skewness and kurtosis to address an investment problem
interpret correlation between two variables to address an investment problem
The mode is the most frequently occurring value in a distribution. Some distributions have more than one mode, while others have none. A distribution with just one mode is unimodal, with two modes is bimodal, and so on.
Data grouped in intervals have modal intervals. This is the highest bar in a histogram.
There are three ways to deal with outliers:
1- No adjustments : This is appropriate if all values are equally important and meaningful.
2- Remove all outliers : A trimmed mean is calculated by discarding a certain percentage of the highest and lowest values. For example, with a sample of 100 observations, a 2% trimmed mean would be the arithmetic mean without the highest value (top 1%) and the lowest value (bottom 1%).
3- Replace outliers with another value : A winsorized mean adjusts any outliers’ values to either an upper or lower limit. No observations are excluded from the calculation.
If we arrange the observations in ascending order, then the quantile is a value at or below which a stated fraction of the data is found.
There are many common quantiles used in practice. Distributions are often divided into four quartiles, five quintiles, ten deciles, or one hundred percentiles.
For example, the 90th percentile score (P90) on an exam is the number that separates the top 10% scores from the bottom 90%.
The interquartile range (IQR) is the difference between the third quartile and the first quartile.
The most common measures of absolute dispersion are range, mean absolute deviation, variance, and standard deviation.
The range is the difference between the maximum and minimum values
The mean of the deviations around the mean will always be zero. Therefore, it would not be a useful measure of dispersion. The mean absolute deviation (MAD) adjusts for this.
Variance is the average of the squared deviations around the mean, while standard deviation is the positive square root of the variance.
Downside Deviation :
Variance and standard deviation take into account returns above and below the mean. But investors care for downside risk.
Target semideviation, or target downside deviation, captures dispersion of observations below a specified target value (e.g., 10%).
Coefficient of Variation :
The coefficient of variation (CV) is a relative dispersion measure. It allows comparisons between data sets with very different means. CV is the ratio of the standard deviation, S, to the mean, X.
Mean has to be positive.
No units of measurement
Skewness :
A positively skewed distribution (long right tail) has frequent small losses and a few extreme gains. The mode is less than the median, which is less than the mean.
A negatively skewed distribution (long left tail) has frequent small gains and a few extreme losses. The mean is less than the median, which is less than the mode.
!! Investors should be concerned if returns have this distribution.
Kurtosis :
Kurtosis measures if a return is more or less peaked than a normal distribution. A normal distribution has a kurtosis value of 3.
(L) : Leptokurtic ( greater than 3, Fat Tails) –> Meaning that extreme returns are more common
(M) : Mesokurtic (normal distribution)
(P) : Platykurtic ( less than 3, Thin Tails)
Excess kurtosis equals kurtosis minus 3. This measures kurtosis relative to the kurtosis of a normal distribution.
The excess kurtosis (0.15) is positive, indicating that the distribution is “fat-tailed”; therefore, there is more probability in the tails of the distribution relative to the normal distribution.
The correlation metric quantifies the linear relationship between two variables.
Properties of Correlation :
1- The correlation coefficient is bounded by -1 and +1.
2- A correlation of 0 indicates that there is no linear relationship between the two variables.
3- A positive correlation coefficient (i.e., rxy > 0) indicates a positive linear relationship between the variables. In other words, an increase in X is associated with an increase in Y. When rx = 1, the variables have a perfect positive linear relationship.
4- A negative correlation coefficient (i.e., rxy < 0) indicates a negative linear relationship between the variables. In other words, an increase in X is associated with an decrase in Y. When rx = -1, the variables have a perfect inverse linear relationship or perfect negative linear relationship.
Limitations of Correlation Analysis :
1- The correlation coefficient is not a reliable measure when the variables have a nonlinear relationship.
2- The correlation coefficient is very sensitive to outliers.
3- Correlation does not imply causation.
4- The conclusions may not be valid. A spurious correlation refers to:
- The correlation between two variables that reflects chance relationships (i.e., just a coincidence)
- The correlation produced by a calculation that mixes the two variables with a common third variable
- The correlation between two variables that arises due to their relation to a third variable (although the two variables are not correlated)
5- Correlation may not produce a full picture of the data. (Same correlation but different relationship)
The correlation coefficient only measures the degree of linear association between two variables. It does not explain the amount each variable changes.
A correlation coefficient can only be between –1 and +1, but covariance is not subject to the same constraint.
Probability Trees and Conditional Expectations
calculate expected values, variances, and standard deviations and demonstrate their application to investment problems
formulate an investment problem as a probability tree and explain the use of conditional expectations in investment application
calculate and interpret an updated probability in an investment setting using Bayes’ formula
Forecasts of a random variable are often based on an expected value, which is the probability-weighted average of its possible outcomes.
The variance of a random variable is the probability-weighted average of the squared deviations from its expected value.
A random variable’s variance must be greater than zero because, if there is no dispersion of outcomes, the expected value is known with certainty and the variable is not random.
The probabilities that are used as the basis for forecasts are rarely static. Analysts are continually updating their unconditional (marginal) probabilities to reflect the latest information. This type of forecasting produces conditional expected values.
The conditional expected value of X given that scenario S occurs is expressed as
E( X|S ).
The total probability rule for expected value : states that the unconditional expected value is equal to the probability-weighted average of the conditional expected values.
Bayes’ formula is a rational way to adjust viewpoints based on new information. It is based on the total probability rule.
Prior probabilities represent the probabilities before the arrival of any new information. The posterior probability reflects the new information.
Bayes Formula :
P (Event | Info) = ( P (Info| Event) /
(Event | Info) ) * P (Event)
Portfolio Mathematics
calculate and interpret the expected value, variance, standard deviation, covariances, and correlations of portfolio returns
calculate and interpret the covariance and correlation of portfolio returns using a joint probability function for returns
define shortfall risk, calculate the safety-first ratio, and identify an optimal portfolio using Roy’s safety-first criterion
Investment opportunities should be evaluated in the context of how they impact the tradeoff between a portfolio’s expected return and the level of portfolio risk, as measured by variance.
Covariance measures the tendency for two variables to move in sync.
Covariance is positive if, when one asset is generating above-average returns, the other asset is as well. Both assets will also tend to generate returns below their respective averages in the same periods.
Covariance is negative if one asset is generating above-average returns while the other’s returns are below its average (or vice versa).
The covariance of an asset’s returns with itself (own covariance) is equal to its variance.
If there are n securities in a portfolio, there are n (n-1) / 2 covariances to estimate
A zero correlation only tells us that there is no linear relationship, but it doesn’t tell us anything about the possibility of a non-linear relationship.
Random variables are independent if, and only if, P (XY) = P(X) * P(Y). If two random variables are independent, they must also be uncorrelated. By contrast, knowing that variables are uncorrelated does not allow us to conclude that they are independent.
