Unweighted arithmetic average should not be used to calculate total return of a portfolio where different holdings (e.g. stocks) have different weights. This page explains why.
Three Stocks Portfolio Example
Let's say we have one million dollars and want to invest it in three stocks. We particularly like stock ABC and allocate 80% of the portfolio to that stock ($800,000). We invest the rest in two other stocks, DEF and GHI.
At the beginning, out portfolio looks like this:
- Stock ABC: 80% of portfolio = $800,000
- Stock DEF: 10% of portfolio = $100,000
- Stock GHI: 10% of portfolio = $100,000
The important factor for illustrating the point of this example is that the positions have different weights: 80%, 10%, and 10%.
Performance of Individual Stocks
One year later, we are looking at our portfolio's performance. The three stocks performed like this:
- Stock ABC -20%
- Stock DEF +10%
- Stock GHI +40%
Arithmetic Average of Stock Returns
We can try and calculate arithmetic average of the three stocks' returns. Arithmetic average is sum divided by count:
(- 20% + 10% + 40%) / 3 = +10%
This calculation is simple, but unfortunately wrong for our unequally-weighted portfolio. We have not made 10%.
Total Portfolio Profit or Loss
How much money has the portfolio actually made?
- In stock ABC it has lost 20% of 800,000 = USD 160,000
- In stock DEF it has made 10% of 100,000 = USD 10,000
- In stock GHI it has made 40% of 100,000 = USD 40,000
The whole portfolio profit or loss is:
- 160,000 + 10,000 + 40,000 = – 110,000
The portfolio has lost $110,000 = 11% of its initial value.
Why Arithmetic Average Doesn't Work
Though two out of the three stocks have risen and arithmetic average of the three stocks' returns is +10%, the portfolio has lost money as a whole, because the biggest (and losing) position in stock ABC has outbalanced the two smaller positions.
If the portfolio had been equally weighted (each stock 33.3%), its return would have been +10%. Unfortunately, allocating a greater share to the stock that ended up losing money made the portfolio lose money as whole too.
With Unequal Weights, Use Weighted Average
For calculating average return of a portfolio or basket of stocks, arithmetic average is only suitable when all stocks have equal weights in the portfolio, which is rarely the case. When the weights are different, we need to take them into account and use weighted average instead.
In our example:
80% * -20% + 10% * 10% + 10% * 40% = -0.16 + 0.01 + 0.04 = -0.11 = -11%
See when to use arithmetic average and when not and why it also can't be used to calculate average percentage return over time.