2.4 - Statistical Foundations Flashcards
Variance
- measures dispersion of data
- sum of squared deviations divided by the number of observations
σ^2 = Σ(R-μ)^2 / n
- standard deviation is the sq. root of the variance = σ
Sample Variance
σ^2 = Σ(R-μ)^2 / n-1
*same as variance but divide by n-1
3rd Central Moment
- skewness
Skewness = Σ(R-μ)^3 / σ^3
Positive skewness = right skew, mean>median>mode
Negative skewness = left skew, mean
4th Central Moment
- kurtosis
Clustering of data around mean
Kurtosis = Σ(R-μ)^4 / σ^4
Normal distribution Kurtosis is 3 — excess kurtosis is over 3, leptokurtic (high peak, fat tails, higher chance of large losses)
Low kurtosis is platykurtic (flat peak, smaller % of small deviations from the mean)
Covariance
- unscaled measure of how two assets move together
Cov(Ri, Rj) = σi,j = Σ(Ri-μi)(Rj-μj) / n-1
Or… (return a1 minus mean a) * (return b1 minus mean b)
Sample Covariance
- Covariance divided by number of observations minus one
Cov(Ri, Rj) = σi,j = E(Ri-μi)(Rj-μj) / t - 1
Correlation Coefficient
- Scaled version of the covariance
Correlation of returns between assets i and j = ρi,j = Covariance(Ri,Rj) / σiσj
- ranges from +1 to -1
- +1 is perfect positive correlation
- -1 is perfect negative correlation
Beta
- related to correlation
- measures the slope of the linear relationship between the two assets
- sensitivity of asset returns compared to changes in the broader market
β = Correlation of i,j * (σi / σj)
Autocorrelation
- correlation over time
- returns that are correlated over time (autocorrelated) have predictability in the returns — autocorrelated returns are said to follow an autoregressive process
K-order autocorrelation = E(Rt- μ)(Rt-k - μ)
Properties of Standard Deviation - scaling volatility
- Scaling σ up from single period to multi-period —> multiplyσT = √T * σ1
Ie: monthly return std dev of 10%, calc the annualized std dev?
σ12 = √12 * .10 = .3464 = 34.64%
- Scaling down σ from multi-period to single —> divideσT = σ1 / √T/n
Ie: annual return std dev of 30%, calc the 6 month std dev?
σ.5 = .30 / √(12/6) = .2121 = 21.21%
Properties of Standard Deviation - levered vs unlevered portfolio
σL = (assets / equity) * σu
Ie: assume an index w/ annual volatility of 30%, calc the vol of a portfolio that leverages an index 1.5 to 1
σL = 1.5 * .30 = 45%
Portfolio STD. Dev.
√(wi^2σi^2) + (wj^2σj^2) + 2(wi)(wj)(corr i,j)(σi)(σj)