Documentation |
Arithmetic to geometric moments of asset returns
[mg, Cg] = arith2geom(ma, Ca) [mg, Cg] = arith2geom(ma, Ca, t)
ma | Arithmetic mean of asset-return data (n-vector). |
Ca | Arithmetic covariance of asset-return data (n-by-n symmetric, positive-semidefinite matrix. |
t | (Optional) Target period of geometric moments in terms of periodicity of arithmetic moments with default value 1 (scalar). |
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
$$1+{\text{m}}_{{G}_{i}}=\mathrm{exp}(t{\text{m}}_{{A}_{i}}+\frac{1}{2}t{\text{C}}_{{A}_{ii}})$$
$${\text{C}}_{{G}_{ij}}=(1+{\text{m}}_{{G}_{i}})(1+{\text{m}}_{{G}_{\text{j}}})(\mathrm{exp}(t{\text{C}}_{A})-1)$$
For i,j = 1,..., n.
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);