Equity Return Attribution: Arithmetic and Geometric
Suppose you had a portfolio benchmark return of 4.8%. You chose a tilted allocation that earned you a 0.4% added return, and some stock selection that earned an additional 0.8%. Nice work!
What would your excess return be?
Right.
No; that's the total return, but your excess return is just the sum of those two pieces: 1.2%.
That's not it. Your excess return is just the sum of those two pieces: 1.2%.
Does this sound like something that could be easily explained to someone who doesn't have a finance background?
Sure.
Oh, it could be. It's not that hard:
"I was up 0.4% in one effort and 0.8% in another, so I gained 1.2% overall." Simple enough. This is __arithmetic attribution__, and it is popular because of its simplicity.
But here's a different question: if instead of the benchmark earning 4.8% for the year and you adding 1.2% on top of that, suppose the portfolio gained 0.4% per month, and you outperformed by 0.1% per month. These monthly values are 1/12 of the total in the first question. Is that the same thing?
Incorrect. Please don't forget about the time value of money.
No, getting 1/12 of the return per month won't hurt your overall returns at all.
Exactly. This is essentially changing a couple APRs to monthly compounding, which results in a larger yield overall. But when you're dealing with multiple periods, that's the math involved for accuracy. __Geometric attribution__ links these returns together:
>$$ G = \frac{1+R}{1+B} - 1$$.
$$R$$ is your periodic return, $$B$$ is the benchmark, and $$G$$ then is the geometric excess return.
What would each monthly $$G$$ factor be in this case, in percentage terms?
You got it!
Not quite.
>$$\displaystyle G = \frac{1+R}{1+B} - 1 = \frac{1+0.005}{1+0.004} - 1 \approx 0.0996 \% $$
Does this seem like a fair estimate of what you earned each month?
You're right, in a way; opinions are divided. On one hand, you did earn that 0.1% differential each month, but on the other hand, the total return was 0.5% for the month, and that was obtained by combining two returns. The monthly 0.4% compounds to 4.9070% for the year, and your 0.1% monthly performance differential compounds to 1.2066% for the year.
Combining these returns suggests a portfolio return of
>$$ (1 + 4.9070 \%)(1 + 1.2066 \%) - 1 = 6.1728 \% $$
for the year. Does this match the portfolio's compounded return?
It's not, actually. Compound 0.5% per month to see that you get something different.
That's right.
The total monthly return of 0.5% compounds to just 6.1678% for the year. This is why the $$G$$ calculation necessarily reduces the compounded differential effects so that the numbers in the end aren't inflated above that of the total.
Good job, though... 0.0996% is still a good monthly effort!
No, it isn't. Compound 0.5% per month to see what you get.
To summarize:
[[summary]]
1.2
Other responses
6
Yes
No
Yes
No, it's less
No, it's more
Other responses
0.0996
Yes
No! It should be the 0.1% difference!
Yes
No, it's too much
No, it's too little
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