Statistics Stuff

We'll refer to variables x, y and z, each selected at random from some distribution.
(The distributions for x, y and z may or may not be the same.)
We'll refer to the Mean (or average) and Standard Deviation (or volatility) of the variables as (for example) M[x] and S[x].
(M[x], M[y], M[z] and S[x], S[y], S[z] may or may not be the same.)
1:     M[x + y] = M[x] + M[y] ... the Mean of a Sum is the Sum of the Means
    and M[x+C] = M[x] + C   if C is a constant
    and M[Cx] = CM[x]   if C is a constant

By definition, the Variance of the x-variable is VAR[x] = S2[x].
Further, by definition:
2:     VAR[x] = S2[x] = M[(x-M[x])2] = the average of (the deviation of x from its Mean)2
    and VAR[x+C] = VAR[x]   if C is a constant
    and VAR[Cx] = C2VAR[x]   if C is a constant
Increasing (or decreasing) M[x] shifts the distribution to the right (or left).

S[x] is a measure of how far the x-values vary from the mean, M[x]. (See Figure 1.)


Figure 1

By definition, the CoVariance of the x and y variables is given by:
3:     COVAR[x,y] = M[(x-M[x])(y-M[y])] = the average of (the deviation of x from its Mean)*(the deviation of y from its Mean)
    so VAR[x] = COVAR[x,x]

4:     COVAR[x,y] = M[xy] - M[x]M[y] = COVAR[x+C,y]
    so that adding a constant to either x or y (or both) doesn't change COVAR[x,y]
    Further: M[xy] = M[x]M[y] + COVAR[x,y]

    Also
    COVAR[x1+x2+...+xn,y] = COVAR[x1,y]+COVAR[x2,y]+...+COVAR[xn,y]

    If the variables have zero covariance (are "uncorrelated"), then
        M[xy] = M[x]M[y]

COVAR[x,y] = M[(x-M[x])(y-M[y])]     using 3
= M[xy - xM[y] - yM[x] + M[x]M[y]]
= M[xy] - M[x]M[y] - M[y]M[x] + M[x]M[y]     using 1
= M[xy] - M[x]M[y]   **
Further
COVAR[x+C,y] = M[(x+C)y] - M[x+C]M[y]     using **
= M[xy+Cy] - (M[x]+C)M[y]     using 1, where M[x+C] = M[x]
= M[xy]+M[Cy] - (M[x]+C)M[y]     again using 1
= M[xy]+CM[y] - M[x]M[y] - CM[y] = M[xy] - M[x]M[y]
Further
COVAR[x1+x2,y] = M[(x1+x2)y] - M[x1+x2]M[y]
= M[x1y+x2y] - (M[x1]+M[x2])M[y]
= M[x1y]+M[x2y] - M[x1]M[y]-M[x2]M[y]
= M[x1y]-M[x1]M[y] + M[x2y]-M[x2]M[y] = COVAR[x1,y]+COVAR[x2,y]
Further
COVAR[x1+x2+...+xn,y] = COVAR[x1,y]+COVAR[x2,y]+...+COVAR[xn,y]

By definition, the Pearson Correlation between the x and y variables is given by:
5:     PEARSON[x,y] = r(x,y) = COVAR[x,y] / (S[x] S[y] )

6:     VAR[x] = M[x2] - M2[x]
VAR[x] = COVAR[x,x] = M[x2] - M2[x]     from 3 and 4

7:     VAR[x+y] = VAR[x] + VAR[y] + 2 COVAR[x,y] = VAR[x] + VAR[y] + 2 r(x,y) S[x]S[y]
VAR[x+y] = M[(x+y-M[x]-M[y])2]     using 1 and 2
= M[(u + v)2]     where u = x-M[x] and v = y-M[y]
= M[u2] + M[v2] + 2 M[uv] = M[(x-M[x])2] + M[(y-M[y])2 + 2 M[(x-M[x])(y-M[y])]]
= VAR[x] + VAR[y] + 2 COVAR[x,y]     using 2 and 3
VAR[x1 + x2 +...] = M[(u1 + u2 + ...)2]     where, as above, u1 = x1-M[x1] etc.
= M[Σuk2] + 2 M[Σukuj]     k < j in the latter sum
= ΣVAR[uk] + 2 ΣCOVAR[uk, uj]
= ΣVAR[xk] + 2 ΣCOVAR[xk, xj]     using 2: VAR[x-C] = VAR[x] with C = M[x] ... and using 4

8:    if r(x,y) = 0 (that is, x and y have zero correlation) then
      VAR[x+y] = VAR[x] + VAR[y]
      and M[xy] = M[x]M[y]     the Mean of a Product = the Product of the Means
Put r(x,y) = 0 in 7   and COVAR[x,y] = 0 in 4.

9:    if r(x,y) = 1 (that is, x and y have perfect correlation) then
      S[x+y] = S[x] + S[y]     the Volatility of a Sum = the Sum of the Volatilities
VAR[x+y] = S2[x+y]= S2[x] + S2[y]+ 2 S[x]S[y]     using 7 with r(x,y) = 1
= (S[x]+S[y])2
so S[x+y]= S[x] + S[y]   if r(x,y) = 1
and, similarly, S[x+y]= |S[x] - S[y]|   (the absolute value)   if r(x,y) = -1

10:    if r(x,y) = 0 (that is, x and y have zero correlation) then
        M[xy] = M[x]M[y]
        VAR[xy] = M2[x]VAR[y] + M2[y]VAR[x] + VAR[x]VAR[y]
M[xy] = M[(x-M[x]+M[x])(y-M[y]+M[y])]     adding & subtracting the Means
= M[(x-M[x])(y-M[y])+M[x](y-M[y])+M[y](x-M[x])+M[x]M[y]]     multiplying
= COV[x,y]+M[x]0+M[y]0+M[x]M[y]]     using 1 and the fact that M[x-M[x]] = 0
= COV[x,y] + M[x]M[y]]
= M[x]M[y]]     if COVAR[x,y] = 0
VAR[xy] = M[x2y2] - M2[xy]     using 6
= (M[x2]M[y2] + COVAR[x2,y2]) - (M[x]M[y] + COVAR[x,y])2     using 4 (for each term)
= M[x2]M[y2] - (M[x]M[y])2     setting correlations (or CoVariances) to 0
= (M2[x] + VAR[x])(M2[y] + VAR[y]) - M2[x]M2[y]     using 6, again!
= M2[x]VAR[y] + M2[y]VAR[x] + VAR[x]VAR[y]

11:     if r(x,y) = 0 (that is, x and y have zero correlation) then
      COVAR[x,xy] = VAR[x]M[y]
COVAR[x,xy] = M[x2y] - M[x]M[xy]     using 4
= M[x2]M[y] - M[x](M[x]M[y])     using 8 (Mean of Product = Product of Means) when correlations are zero
= (M2[x] + VAR[x])M[y] - M2[x]M[y]     using 6
= VAR[x]M[y]

Summary Stuff

If
      r(x,y) = Pearson Correlation between variables x and y
      Σ x stands for x1 + x2 + ... + xn
      Σ xy stands for x1y1 + x2y2 + ... + xnyn
      M[x] = (1/n) Σ x = the Mean of the xs
      SD2[x] = (1/n) Σ (x - M[x])2 = (1/n) Σ x2 - M2[x] their Variance or (Standard Devation)2
      Beta[x,y] = slope of the regression line, plotting (xk,yk)
      Error = the mean square deviation of the yk from the regression line
then:
      r = {M[xy]-M[x] M[y]}/{SD[x]SD[y]} = (1/n) Σ (x-M[x]) (y-M[y]) / {SD[x]SD[y]}
      Beta[x,y] = COVAR[x,y] / SD2[x] = r SD[y] / SD[x]
      Error2 = SD2[y] (1-r2)

If
      X is the vector with components (xk - M[x]) / SD[x]√n        
      Y is the vector with components (yk - M[y]) / SD[y]√n
then
      X and Y are of unit length. That is: ||X|| = ||Y|| = 1
      r = XY = ||X|| ||Y|| cos(θ) = cos(θ)
      Error = SD[y] |sin(θ)|
and
      Y = ( cos(θ) + isin(θ) )X = exp(iθ) X
where i rotates a vector by 90 degrees (in the plane of X and Y   ... so i2 = -1).

If the weights of our portfolio are described by the n-vector W
and the covariance is described by the n x n matrix Θ,
then the Standard Deviation is the positive scalar σ, where:
σ2 = XT Θ X

See: R-squared
Correlation Stuff
Linear Regression