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