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# geom2arith

Geometric to arithmetic moments of asset returns

## Syntax

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

## Arguments

 mg Continuously-compounded or "geometric" mean of asset returns (positive n-vector). Cg Continuously-compounded or "geometric" covariance of asset returns (n-by-n symmetric, positive-semidefinite matrix). t (Optional) Target period of arithmetic moments in terms of periodicity of geometric moments with default value 1 (positive scalar).

## Description

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

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

and

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

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

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

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

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

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

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

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.

## Examples

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);`