2.4 - Statistical Foundations

Question 1

Variance

Answer 1

• 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

Question 2

Sample Variance

Answer 2

σ^2 = Σ(R-μ)^2 / n-1

*same as variance but divide by n-1

Question 3

3rd Central Moment

Answer 3

• skewness

Skewness = Σ(R-μ)^3 / σ^3

Positive skewness = right skew, mean>median>mode

Negative skewness = left skew, mean

Question 4

4th Central Moment

Answer 4

• 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)

Question 5

Covariance

Answer 5

• 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)

Question 6

Sample Covariance

Answer 6

• Covariance divided by number of observations minus one

Cov(Ri, Rj) = σi,j = E(Ri-μi)(Rj-μj) / t - 1

Question 7

Correlation Coefficient

Answer 7

• 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

Question 8

Beta

Answer 8

• 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)

Question 9

Autocorrelation

Answer 9

• 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)

Question 10

Properties of Standard Deviation - scaling volatility

Answer 10

• 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%

Question 11

Properties of Standard Deviation - levered vs unlevered portfolio

Answer 11

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%

Question 12

Portfolio STD. Dev.

Answer 12

√(wi^2σi^2) + (wj^2σj^2) + 2(wi)(wj)(corr i,j)(σi)(σj)