Expected returns vs annualized returns

Question

Im doing an excercise where it is assumed that there is 50% chance of market gaining 30% and 50% chance market losing 10%, every year. I have to calculated annualized return.

To compute annualized return, i thought i can simply take expected return ie 0.5*(1+0.3) + 0.5* (1-0.1) = 1.1

but if i model it as 1st year 30% gain and 2nd year loss of 10%, then return over 2 years is 1.3*0.9 = 1.17, therefore annualized return is sqrt(1.17)-1 ie 8.17%

which is the correct approach and why? Thanks a lot

Answer 1

I think you are confusing the expected return of a combination of two assets with the expected return of one asset over two periods.

You have 4 possible combinations with (in this case) equal probability:

+30%, +30% = 1.69
+30%, -10% = 1.17
-10%, +30% = 1.17
-10%, -10% = 0.81

So your expected return is the sum of the probability and return of each case:

  (0.5 * 0.5 * 1.69) 
+ (0.5 * 0.5 * 1.17) 
+ (0.5 * 0.5 * 1.17) 
+ (0.5 * 0.5 * 0.81) 
= 1.21

So +21% would be the expected 2-year return. Annualizing that would give you an expected return of

SQRT(1.21) = 1.10 = +10% (Annualized)

In general, one would build a tree with each possibility and use the probability of each node to construct the result and probability of each path. In this case, since both paths are equally likely, each combination is also equally likely.

Answer 2

This question highlights the difference between using arithmetic mean (for expected return) and geometric mean (for annualised return).

The expected return is covered here, e.g. (figures updated)

if an investment has a 50% chance of gaining 30% and a 50% chance of losing 10%, the expected return would be 5% = (50% x 30% + 50% x -10% = 10%).

In other words, arithmetic mean: (0.3 - 0.1)/2 = 0.1

Use of the geometric mean is described here:-

Why Use the Geometric Mean Instead of the Arithmetic Mean for Returns?

It’s used because it includes the effect of compounding growth from different periods of return. Therefore, it’s considered a more accurate way to measure investment performance.

((1 + 0.3)*(1 - 0.1))^(1/2) - 1 = 8.16654 %

Quoting Investopedia with updated figures:-

As shown, at 8.16654%, the geometric mean provides a return that’s worse than the 10% arithmetic mean. But it is the result that represents reality in this case.

How does the geometric mean work?

The geometric mean effectively "averages" the growth factors to finds the equivalent single growth factor that, when applied consistently over multiple periods, yields the same final value as the original, varying growth rates. It is designed to handle multiplicative relationships, which is exactly what we have with growth factors. By finding the equivalent single growth factor, it provides a meaningful average that accurately reflects the overall growth over multiple periods.

Confirming results with simulation

The check below runs three 10m simulations all resulting quite close to the expected 0.0816654. 10m were needed to smooth out the noise.