Modern Portfolio Theory     ... Frontier Math

  • We start with N portfolio components and devote fractions x1, x2, ... xN to each, so
    x1+x2+ ... +xN = 1.
  • Their Mean returns over the past M months are: r1, r2, ... rN
    and we assume the Mean return on our portfolio - over the past M months - is
    R = x1r1+x2r2+ ... +xNrN.
  • The N components have Standard Deviations: S1, S2, ... SN.
Suppose the monthly returns of component "1", over each of the M months, are g11, g12, g13, ... g1M
(where g13, for example, is the return for the component #1 during month #3) and
the returns of component "2" are g21, g22, ... g2M and so on and
the deviations from the Mean (over all M months) can be displayed as an N x M matrix:
G =
g11 - r1 g12 - r1 g13 - r1   ...   g1M - r1
g21 - r2 g22 - r2 g23 - r2   ...   g2M - r2
  ...     ...     ...     ...  
gN1 - rN gN2 - rN gN3 - rN   ...   gNM - rN
Note that column#1 of G is devoted to the deviations for month#1 and column#2 is devoted to the deviations for month#2, etc.

Further, we can display the fractions devoted to each asset via the 1xN matrix vector:

XT =
x1 x2 x3   ...   xN
where the superscript T denotes the transpose of the corresponding Nx1 vector.

The M monthly deviations from the mean Portfolio Return, R = x1r1+x2r2+ ... +xNrN (as described in Part I) are the components of the 1xM matrix:

XT G =
x1(g11-r1)+x2(g21-r2)+ ...+xN(gN1-rN) x1(g12-r1)+x2(g22-r2)+ ...+xN(gN2-rN)     ...     x1(g1M-r1)+x2(g2M-r2)+ ...+xN(gNM-rN)

In order to compute the sum of the squares of this 1xM matrix, we multiply by its transpose. That is, if UT is a 1xM matrix, the sum of the squares of its components is the 1x1 scalar: UTU

Hence, the sum of squares for our matrix (namely XT G) is the quadratic:

Q = XT G { XT G)T = XT GGT X = XTWX
where W is the covariance matrix. Its diagonal elements, when divided by N, give the square of the Standard Deviation - that is, the Variance - of each asset class. The off-diagonal elements give the co-variances between classes.

to return to Part I