This is a very basic article explaining why unweighted arithmetic average is not suitable when portfolio is not equally weighted. See more about When to Use Arithmetic Average and When Not.
3 stock portfolio example
Let’s say you have 1 million dollars and you want to invest it in stocks. You construct a fairly concentrated portfolio – you particularly like stock ABC. At the beginning of the year you have invested 1 million dollars in 3 stocks:
- Stock ABC: 80% of portfolio = USD 800,000
- Stock DEF: 10% of portfolio = USD 100,000
- Stock GHI: 10% of portfolio = USD 100,000
The important factor for illustrating the point of this article is that the positions have different weights in the portfolio: 80%, 10%, and 10%.
Performance of individual stocks
One year later, you are looking at your portfolio’s performance. The 3 stocks performed like this:
- Stock ABC -20%
- Stock DEF +10%
- Stock GHI +40%
Arithmetic average of the stocks’ returns
What is the arithmetic average? The sum of -20%, +10% and +40% equals +30%. Divided by 3, the arithmetic average of the stocks’ returns in the last year is +10%. (See How to Calculate Arithmetic Average: The Very Basics).
Percentage return and P/L of total portfolio
How much money have you made?
- In stock ABC you have lost 20% of 800,000 = USD 160,000
- In stock DEF you have made 10% of 100,000 = USD 10,000
- In stock GHI you have made 40% of 100,000 = USD 40,000
The whole portfolio P/L is negative 160,000 plus 10,000 plus 40,000, which equals a loss of USD 110,000. On percentage basis, you have lost 11% of your initial investment.
Why is arithmetic average not working?
Though 2 of the 3 stocks have risen and arithmetic average of the 3 stocks’ returns is +10%, your portfolio has lost money, as your biggest (and losing) position in stock ABC has outbalanced the two smaller positions. Had your portfolio been equally weighted (each stock 33.3%), its return would have been +10%. But you were unlucky to allocate more to ABC, the one stock that has lost in the end.
Weighted average is better for calculating total portfolio return
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, you need to take it into account and use weighted average instead.