## Documentation Center |

Arithmetic to geometric moments of asset returns

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

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

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

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

`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 *t** _{A}* are modeled as multivariate
normal random variables with moments

and

Geometric returns over period *t** _{G}* are modeled as multivariate
lognormal random variables with moments

Given *t* = *t** _{G}* /

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

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`.

**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);

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