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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 (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

$E\left[\text{X}\right]={\text{m}}_{A}$

and

$\mathrm{cov}\left(\text{X}\right)={\text{C}}_{A}$

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

$E\left[\text{Y}\right]=1+{\text{m}}_{G}$

$\mathrm{cov}\left(\text{Y}\right)={\text{C}}_{G}$

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

$1+{\text{m}}_{{G}_{i}}=\mathrm{exp}\left(t{\text{m}}_{{A}_{i}}+\frac{1}{2}t{\text{C}}_{{A}_{ii}}\right)$

${\text{C}}_{{G}_{ij}}=\left(1+{\text{m}}_{{G}_{i}}\right)\left(1+{\text{m}}_{{G}_{\text{j}}}\right)\left(\mathrm{exp}\left(t{\text{C}}_{A}\right)-1\right)$

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); `