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-Stock 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.