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arith2geom

Arithmetic to geometric moments of asset returns

Syntax

[mg, Cg] = arith2geom(ma,Ca)
[mg, Cg] = arith2geom(ma,Ca,t)

Arguments

ma

Arithmetic mean of asset-return data (n-vector).

Ca

Arithmetic covariance of asset-return data, an n-by-n symmetric, positive-semidefinite matrix.

t

(Optional) Target period of geometric moments in terms of periodicity of arithmetic moments with default value 1 (scalar).

Description

arith2geom 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(ma,Ca,t) returns mg, continuously compounded or "geometric" mean of asset returns over the target period (n-vector), and Cg, which is a continuously compounded or "geometric" covariance of asset returns over the target period (n-by-n matrix).

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

E[X]=mA

and

cov(X)=CA

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

E[Y]=1+mG

cov(Y)=CG

Given t = tG / tA, the transformation from geometric to arithmetic moments is

1+mGi=exp(tmAi+12tCAii)

CGij=(1+mGi)(1+mGj)(exp(tCAij)1)

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

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

This 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.

Examples

Example 1. 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 t = 12 with

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

Example 2. 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 t = 1/12 with

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

Example 3. Given arithmetic means m and standard deviations s of daily total returns (derived from 260 business days per year), obtain annualized continuously compounded mean mg and standard deviations sg with

[mg, Cg] = arith2geom(m, diag(s .^2), 260);
sg = sqrt(diag(Cg));

Example 4. 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 t = 3 with

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

Example 5. Given arithmetic mean m and covariance C of 1254 observations of daily total returns over a 5-year period, obtain annualized continuously compounded return moments. Since the periodicity of the arithmetic data is based on 1254 observations for a 5-year period, a 1-year period for geometric returns implies a target period of t = 1254/5 so that

[mg, Cg] = arith2geom(m, C, 1254/5); 

Introduced before R2006a

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