2.4 - Statistical Foundations Flashcards

1
Q

Variance

A
  • 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
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2
Q

Sample Variance

A

σ^2 = Σ(R-μ)^2 / n-1

*same as variance but divide by n-1

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3
Q

3rd Central Moment

A
  • skewness

Skewness = Σ(R-μ)^3 / σ^3

Positive skewness = right skew, mean>median>mode

Negative skewness = left skew, mean

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4
Q

4th Central Moment

A
  • 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)

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5
Q

Covariance

A
  • 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)

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6
Q

Sample Covariance

A
  • Covariance divided by number of observations minus one

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

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7
Q

Correlation Coefficient

A
  • 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
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8
Q

Beta

A
  • 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)

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9
Q

Autocorrelation

A
  • 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)

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10
Q

Properties of Standard Deviation - scaling volatility

A
  • 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%

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11
Q

Properties of Standard Deviation - levered vs unlevered portfolio

A

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%

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12
Q

Portfolio STD. Dev.

A

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

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