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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, an |

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

`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

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

and

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

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

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

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

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

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

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

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

If *t* = 1, then **Y** = exp(**X**).

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);