## Documentation Center |

Geometric to arithmetic moments of asset returns

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

| Continuously-compounded or "geometric" mean of asset returns (positive n-vector). |

| Continuously-compounded or "geometric" covariance of asset returns (n-by-n symmetric, positive-semidefinite matrix). |

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

`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 *t** _{G}* are modeled as multivariate
lognormal random variables with moments

and

Arithmetic returns over period *t** _{A}* are modeled as multivariate
normal random variables with moments

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

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

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`.

**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);

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