# 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, a `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), and`Ca`, 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);`