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At 9:23 AM -0700 1/12/01, cwest wrote:
>When someone says the following what is the "math" behind the statement?
>
>"our annualized gains represent geometric mean returns. It's the amount of
>return you'd have to achieve each and every year"
The ratio of final value (vn) to initial value (v0) is:
vn/v0 = v1/v0 * v2/v1 * v3/v2 * v4/v3 * ... * vn/v(n-1)
where
v0 is the value of your account in the beginning
v1 is the value of your account at the end of period 1, etc.
Defining the periodic returns:
1+R = (1+r1) * (1+r2) * (1+r3) * (1+r4) * ....
where
r1 = (v1-v0)/v0, etc. (the return in period 1)
so that:
1+r1 = (v0+v1-v0)/v0 = v1/v0
These terms must be multiplied to include the effects of compounding.
Addition is usually preferable to multiplication so we can convert
this equation to additions by using logarithms:
LN(1+R) = LN(v1/v0) + LN(v2/v1) + LN(v3/v2) + LN(v4/v3) +
and
R = EXP(LN(1+R) - 1
This is an exact form and these are "geometric returns". Assuming
compounding - profits are allowed to accumulate in the account.
The equation above:
1+R = (1+r1) * (1+r2) * (1+r3) * (1+r4) + ....
can be expanded to:
1+R = 1+r1+r2+r3+r4... + 2*r1*r2 + 2*r2*r3 + ...
so that:
R = r1+r2+r3+r4...
is an approximation that leaves out a lot of higher order terms.
These are usually called "arithmetic returns".
Bob Fulks
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