Geometric to arithmetic moments of asset returns
[ma, Ca] = geom2arith(mg, Cg) [ma, Ca] = geom2arith(mg, Cg, t)
Continuously-compounded or "geometric" mean of asset returns (positive n-vector).
Continuously-compounded or "geometric" covariance of asset returns (n-by-n symmetric, positive-semidefinite matrix).
(Optional) Target period of arithmetic moments in terms of periodicity of geometric moments with default value 1 (positive scalar).
geom2arith transforms moments associated with a continuously-compounded geometric Brownian motion into equivalent moments associated with a simple Brownian motion with a possible change in periodicity.
[ma, Ca] = geom2arith(mg, Cg, t) returns ma, arithmetic mean of asset returns over the target period (n-vector), andCa, which is an arithmetric covariance of asset returns over the target period (n-by-n matrix).
Geometric returns over period tG are modeled as multivariate lognormal random variables with moments
Arithmetic returns over period tA are modeled as multivariate normal random variables with moments
Given t = tA / tG, the transformation from geometric to arithmetic moments is
For i,j = 1,..., n.
This function requires that the input mean must satisfy 1 + mg > 0 and that the input covariance Cg must be a symmetric, positive, semidefinite matrix.
The functions geom2arith and arith2geom are complementary so that, given m, C, and t, the sequence
[ma, Ca] = geom2arith(m, C, t); [mg, Cg] = arith2geom(ma, Ca, 1/t);
yields mg = m and Cg = C.
Example 1. Given geometric mean m and covariance C of monthly total returns, obtain annual arithmetic mean ma and covariance Ca. In this case, the output period (1 year) is 12 times the input period (1 month) so that t = 12 with
[ma, Ca] = geom2arith(m, C, 12);
Example 2. Given annual geometric mean m and covariance C of asset returns, obtain monthly arithmetic mean ma and covariance Ca. In this case, the output period (1 month) is 1/12 times the input period (1 year) so that t = 1/12 with
[ma, Ca] = geom2arith(m, C, 1/12);
Example 3. Given geometric means m and standard deviations s of daily total returns (derived from 260 business days per year), obtain annualized arithmetic mean ma and standard deviations sa with
[ma, Ca] = geom2arith(m, diag(s .^2), 260); sa = sqrt(diag(Ca));
Example 4. Given geometric mean m and covariance C of monthly total returns, obtain quarterly arithmetic return moments. In this case, the output is 3 of the input periods so that t = 3 with
[ma, Ca] = geom2arith(m, C, 3);
Example 5. Given geometric mean m and covariance C of 1254 observations of daily total returns over a 5-year period, obtain annualized arithmetic return moments. Since the periodicity of the geometric data is based on 1254 observations for a 5-year period, a 1-year period for arithmetic returns implies a target period of t = 1254/5 so that
[ma, Ca] = geom2arith(m, C, 1254/5);