E Flashcards
calculate and interpret measures of central tendency, including the population mean, sample mean, arithmetic mean, weighted average or mean, geometric mean, harmonic mean, median, and mode;
Central Tendency
A quantitative measure that specifies where the data are centered
Common Measures of Central Tendency
“the arithmetic mean, the median, the mode, the weighted mean, and the geometric mean.”
Population Mean (an example of a ‘parameter’: any descriptive measure of a population characteristic) ‘Mu’
Aritmetic mean (average) for a finite population
**A pop. could be a portfolio managers investment universe
Property: A given population has only 1 mean
Sample Mean ‘ Xbar’ (A ‘statistic’: A descriptive measure of a sample)
Arithmetic mean (average) for a sample
“We should not expect any of the actual observations to equal the mean, because sample means provide only a summary of the data being analyzed”
“The mean is generally the statistic that you will use as a measure of the typical outcome for a distribution”
Arithmetic Mean (focus: Average single period performance)(“all observations are equally weighted by the factor 1/n (or 1/N).”) (Mean addresses’ reward)
The most used measure of central or middle data that describes a representative outcome of an investment decision. USE: AS a measure for the typical outcome of an asset. Mathematically, the sum of deviations around the mean =’s 0. Pay attention to outcomes above the mean and below in analysis, describe them, etc
Definition: Sum of the observations / #of observations
Adv: Uses all the information about the size and magnitude of observations
Disadv: Can be skewed by large values, so use the median in addition to the mean
Weighted Average Mean (allows different weights on different observations) (“X-bar sub-w”)(“value is price multiplied by quantity”)( Also time series, but considers portfolio weights and rebalances compared to geometric means which consider compound-returns)
Weight (return), but express weight as decimal - expresses the fact that a portfolio return is a weighted sum
A weighted average of forward looking date = expected value
Different weights = portfolio weights = Stocks .N and bonds .N = 1 (the sum of the weights should equal 1)
“value is price multiplied by quantity, thus, price fluctuations cause portfolios weights to change.
“must reflect the fact that stocks have a 70 percent weight in the portfolio and bonds have a 30 percent weight”
Cal: “reflect this weighting is to multiply the return on the stock investment by 0.70 and the return on the bond investment by 0.30, then sum the two results.”
Calc: Multiply the respective Return to their Respective Weight, then sum the results
Interps: +weights = long positions
- weights = short positions
Geometric Mean (aka, compound returns)(“With its focus on the profitability of an investment over a multiperiod hori- zon, the geometric mean is of key interest to investors.”)
less sensitive to extreme values than regular mean. Geometric mean returns use time-series
Calc: Multiply the values together and take the ‘nth’ root: 1/n
Calc: Ln G = 1/n Ln (product of all observations), then G= e^LnG
If a return is (-n), then add 1 to each return, then subtract 1 at the end
Rules: Product under the radical sign is a non-neg, all observations Xi are greater than or equal to 0
Geometric avg’s should be used with investments
If returns vary from year to year the geometric return will always be less than that arithmetic return
“In general, the difference between the arithmetic and geometric means increases with the variability in the period-by-period observations”
INterpretation: “The arithmetic and geometric mean also rank the two funds differently. Although SLASX has the higher arithmetic mean return, PRFDX has the higher geometric mean return. How should the analyst interpret this result?”
-“The geometric mean return represents the growth rate or compound rate of return on an investment”
Harmonic Mean
Limited uses: “appropriate when averaging ratios (“amount per unit”) when the ratios are repeatedly applied to a fixed quantity to yield a variable number of units.” aka ‘cost averaging’ - the periodic investment of a fixed amount of money ( Ratio - 40$/share, per month, yields a variable # of units aka G or L) “The average price paid is in fact the harmonic mean of the asset’s prices at the purchase dates, assuming we invest FIXED amounts of MONEY)
‘ratios’ - amount per unit
Median (divides a distribution in half)
the value of the middle item of a set of items that has been sorted into ascending or descending order.
A distribution has only 1 median, and values lie above or below the median
Adv: Unaffected by extreme value
Disadv: Doesnt use all info about size and magnitude of observations, as it focus’ only on the relative positions of the ranked observations
Mode
The most frequently occurring value in a distribution
Unique: A dist. can have more than 1 mode or ‘no mode’: (Stock return data and other data from continuous distributions may not have a modal outcome.)
1 mode: Unimodal
2 modes: Bimodal
3 modes: Trimodal
Modal Interval: An interval with the most values = the highest bar in the histogram.
Deviations around the arithmetic mean are important information because…
..they indicate risk. The concept of deviations around the mean forms the foundation for the more complex concepts of variance, skewness, and kurtosis
“A mathematical fact concerning the harmonic, geometric, and arithmetic means is that…
…unless all the observations in a data set have the same value, the harmonic mean is less than the geometric mean, which in turn is less than the arithmetic mean.”
