log return to simple return

The above are pages from CAIA level 1, chapter 3 quantitative foundation. I know it is from CAIA, but the quant stuff is basically similar to CFA quant stuff.

using the example on page 47, an asset earns return of 10% in period 1 and 20% in period 2.

So the arithmetic mean log return using equation 3.4 is (10%+20%)/2 = 15%

Now, using equation 3.5 to calculate the geometric return, sqrt{1.1 x 1.2} - 1 = 14.89%

on page 47, it says, when the arithmetic mean log return is converted into an equivalent simple rate, that rate is referred to as the geometric mean return.

But according to equation 3.1

0.15 should equal to ln(1.1489)

But it doesn’t.

So assuming “Continuous/Log” returns

1)“arithmetic mean log return using equation 3.4 is (10%+20%)/2 = 15%” -> agreed

2)“geometric return, sqrt{1.1 x 1.2} - 1 = 14.89%” -> ??

I am not sure you can do this to get Geometric Mean if the given periodic returns are log (continuous) returns since “total return” (Ro,T) is simply (r1+r2 ) the sum of periodic returns, instead of (1+R1)(1+R2) -1

To convert the “arithmetic mean log (continuous) return” into an equivalent simple (discrete) rate:

We know r = ln (1+R) where r is log/continuous return and R is simple/discrete return

=> R = e ^r - 1

Alternatively, if you started out assuming the periodic returns were discrete 1) above would be incorrect and 2) would be fine. We need to use a consistent set of assumptions.

I’m not making any assumptions, I’m simply following the book. Using formula 3.4, I got 15%, using forumla 3.5, I got 14.89%. The book says when the result from 3.4 is converted to equavalent simple rate, you should get the result found in 3.5.

The book made it clear that formula 3.4 is the means of log return and 3.5 is using periodic return (since formula 3.5 has 1+R_{0,T} inside the root, and 1+R_{0,T} is defined in formula 3.2 as (1+R1)(1+R2).

No, it isn’t.

That’s the arithmetic mean effective return.

The arithmetic mean log return is:

(ln(1.1) + ln(1.2)) / 2

= (0.0953 + 0.1823) / 2

= 0.1388

And ln(1.1489) = 0.1388.

Um . . . voilà!