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} / t_{A}, the transformation from geometric to arithmetic moments is
$$1+{\text{m}}_{{G}_{i}}=\mathrm{exp}(t{\text{m}}_{{A}_{i}}+\frac{1}{2}t{\text{C}}_{{A}_{ii}})$$
$${\text{C}}_{{G}_{ij}}=(1+{\text{m}}_{{G}_{i}})(1+{\text{m}}_{{G}_{\text{j}}})(\mathrm{exp}(t{\text{C}}_{A}{}_{ij})-1)$$
For i,j = 1,..., n.
Note: 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);