arith2geom

Arithmetic to geometric moments of asset returns

Syntax

[mg,Cg = arith2geom(ma,Ca)

[mg,Cg = arith2geom(___,t)

Description

[mg,Cg = arith2geom(ma,Ca) transforms moments associated with a simple Brownian motion into equivalent continuously compounded moments associated with a geometric Brownian motion with a possible change in periodicity.

[mg,Cg = arith2geom(___,t) adds an optional argument t.

Examples

Obtain Arithmetic to Geometric Moments of Asset Returns

This example shows several variations of using arith2geom.

Given arithmetic mean m and covariance C of monthly total returns, obtain annual geometric mean mg and covariance Cg. 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 ];
[mg, Cg] = arith2geom(m, C, 12)

mg = 4×1

    0.8934
    2.9488
    4.9632
   17.0835

Cg = 4×4
103 ×

    0.0003    0.0004    0.0003         0
    0.0004    0.0065    0.0065    0.0110
    0.0003    0.0065    0.0354    0.0536
         0    0.0110    0.0536    1.0952

Given annual arithmetic mean m and covariance C of asset returns, obtain monthly geometric mean mg and covariance Cg. 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.

[mg, Cg] = arith2geom(m, C, 1/12)

mg = 4×1

    0.0044
    0.0096
    0.0125
    0.0203

Cg = 4×4

    0.0005    0.0003    0.0002         0
    0.0003    0.0025    0.0017    0.0010
    0.0002    0.0017    0.0049    0.0029
         0    0.0010    0.0029    0.0107

Given arithmetic mean m and covariance C of monthly total returns, obtain quarterly continuously compounded return moments. In this case, the output is 3 of the input periods so that the optional input t = 3.

[mg, Cg] = arith2geom(m, C, 3)

mg = 4×1

    0.1730
    0.4097
    0.5627
    1.0622

Cg = 4×4

    0.0267    0.0204    0.0106         0
    0.0204    0.1800    0.1390    0.1057
    0.0106    0.1390    0.4606    0.3418
         0    0.1057    0.3418    1.8886

Input Arguments

ma — Arithmetic mean of asset-return data

vector

Arithmetic mean of asset-return data, specified as an n-vector.

Data Types: double

Ca — Arithmetic covariance of asset-return data

matrix

Arithmetic covariance of asset-return data, specified as an n-by-n symmetric, positive semidefinite matrix. If Ca 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 geometric moments in terms of periodicity of arithmetic 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

mg — Continuously compounded or "geometric" mean of asset returns over the target period

vector

Continuously compounded or "geometric" mean of asset returns over the target period (t), returned as an n-vector.

Cg — Continuously compounded or "geometric" covariance of asset returns over the target period

matrix

Continuously compounded or "geometric" covariance of asset returns over the target period (t), returned as an n-by-n matrix.

Algorithms

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

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

and

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

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

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

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

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

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

$${\text{C}}_{G_{ij}}=(1+{\text{m}}_{G_i})(1+{\text{m}}_{G_{\text{j}}})(\exp (t{\text{C}}_A{}_{ij})-1)$$

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

Note

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

The arith2geom function has no restriction on the input mean ma but requires the input covariance Ca to be a symmetric positive-semidefinite matrix.

The functions arith2geom and geom2arith are complementary so that, given m, C, and t, the sequence

[mg,Cg] = arith2geom(m,C,t);   	
[ma,Ca] = geom2arith(mg,Cg,1/t); 

yields ma = m and Ca = C.

Version History

Introduced before R2006a