# arith2geom

Arithmetic to geometric moments of asset returns

## Syntax

``[mg,Cg = arith2geom(ma,Ca)``
``[mg,Cg = arith2geom(___,t)``

## Description

example

````[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.```

example

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

## Examples

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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

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

Data Types: `double`

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`

(Optional) Target period of geometric moments in terms of periodicity of arithmetic moments, specified as a scalar positive numeric.

Data Types: `double`

## Output Arguments

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

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 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

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

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

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`.