geom2arith

Geometric to arithmetic moments of asset returns

Syntax

[ma,Ca] = geom2arith(mg,Cg)

[ma,Ca] = geom2arith(___,t)

Description

[ma,Ca] = geom2arith(mg,Cg) 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(___,t) adds an optional argument t.

Examples

Obtain Geometric to Arithmetic Moments of Asset Returns

This example shows several variations of using geom2arith.

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 the optional input t = 12.

m = [ 0.05; 0.1; 0.12; 0.18 ];
C = [ 0.0064 0.00408 0.00192 0; 
    0.00408 0.0289 0.0204 0.0119;
    0.00192 0.0204 0.0576 0.0336;
    0 0.0119 0.0336 0.1225 ];
[ma, Ca] = geom2arith(m, C, 12)

ma = 4×1

    0.5508
    1.0021
    1.0906
    1.4802

Ca = 4×4

    0.0695    0.0423    0.0196         0
    0.0423    0.2832    0.1971    0.1095
    0.0196    0.1971    0.5387    0.3013
         0    0.1095    0.3013    1.0118

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 the optional input t = 1/12.

[ma, Ca] = geom2arith(m, C, 1/12)

ma = 4×1

    0.0038
    0.0070
    0.0076
    0.0103

Ca = 4×4

    0.0005    0.0003    0.0001         0
    0.0003    0.0020    0.0014    0.0008
    0.0001    0.0014    0.0037    0.0021
         0    0.0008    0.0021    0.0070

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 the optional input t = 3.

[ma, Ca] = geom2arith(m, C, 3)

ma = 4×1

    0.1377
    0.2505
    0.2726
    0.3701

Ca = 4×4

    0.0174    0.0106    0.0049         0
    0.0106    0.0708    0.0493    0.0274
    0.0049    0.0493    0.1347    0.0753
         0    0.0274    0.0753    0.2530

Input Arguments

mg — Continuously compounded or geometric mean of asset returns

vector

Continuously compounded or geometric mean of asset returns, specified as an n-vector.

Data Types: double

Cg — Continuously compounded or geometric covariance of asset returns

matrix

Continuously compounded or geometric covariance of asset returns, specified as an n-by-n symmetric, positive semidefinite matrix. If Cg is not a symmetric positive semidefinite matrix, use nearcorr to create a positive semidefinite matrix for a correlation matrix.

Data Types: double

t — Target period of arithmetic moments in terms of periodicity of geometric moments

1 (default) | scalar positive numeric

(Optional) Target period of geometric moments in terms of periodicity of arithmetic moments, specified as a scalar positive numeric.

Data Types: double

Output Arguments

ma — Arithmetic mean of asset returns over the target period

vector

Arithmetic mean of asset returns over the target period (t), returned as an n-vector.

Ca — Arithmetic covariance of asset returns over the target period

matrix

Arithmetic covariance of asset returns over the target period (t), returned as an n-by-n matrix.

Algorithms

Geometric returns over period t_G are modeled as multivariate lognormal random variables with moments

$$E[\text{Y}]=1+{\text{m}}_G$$

and

$$\mathrm{cov}(\text{Y})={\text{C}}_G$$

Arithmetic returns over period t_A are modeled as multivariate normal random variables with moments

$$E[\text{X}]={\text{m}}_A$$

$$\mathrm{cov}(\text{X})={\text{C}}_A$$

Given t = t_A / t_G, the transformation from geometric to arithmetic moments is

$${\text{C}}_{A_{ij}}=t\log \left(1+\frac{{\text{C}}_{G_{ij}}}{(1+{\text{m}}_{G_i})(1+{\text{m}}_{G_j})}\right)$$

$${\text{m}}_{A_i}=t\log (1+{\text{m}}_{G_i})-\frac{1}{2}{\text{C}}_{A_{ii}}$$

For i,j = 1,..., n.

Note

If t = 1, then X = log(Y).

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.

Version History

Introduced before R2006a