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, a |

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

`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 arithmetic 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

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

and

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

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

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

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

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

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

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

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

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`

.

**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);