“The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others.”^[1]

This kind of average is called the weighted mean, and is given by the following formula:

\[x_w =\frac{\Sigma x\cdot w}{\Sigma w}\]

Where the x’s are the data you are looking to average and the w’s are the weights assigned to each of those data points.

Illustration

Suppose you have $3,000, and make 3 different investments, some invested at 5%, some at 6%, and some at 7%.

Question: Is your average rate of return equal to [latex]\frac{(5 + 6 + 7)\%} {3} = 6%[/latex]?

Answer: The answer ONLY if you invested equal amounts in each investment, and NO otherwise. In the case where you invest different amounts, a weighted average is required to determine the average rate of return.

Example: Investing Different Amounts in Three Investments

Example 4.1.1

Let us suppose you invested $30,000 at 5%, $6,000 at 6%, and $4,500 at 7%.

Question: Is the overall (average) rate of return for your investment 6%?

Answer: No, it is given by the following calculation:

[latex]\begin{align*} x_w &=\frac{5\%\cdot$30,000 +6\%\cdot$6,000 + 7\%\cdot$4,500}{$30,000+$6,000+$4,500}\\ &=\frac{$1,500 +$360 + $315}{$30,000+$6,000+$4,500}\\ &= \frac{$2,175}{$40,500}\\ &= 0.0537 = 5.37\% \end{align*}[/latex]

Example: Tables for Weighted Mean Calculations

Example 4.1.2

Let us, instead, use a table to organize the calculations from the previous example. Note: the x · w column is the product of the x and w values.

x	w	x · w
5%	$30,000	0.05 × $30,000 = $1,100
6%	$6,000	0.06 × $6,000 = $360
7%	$4,500	0.07 × $4,500 = $315
Total	$40,500	$2,175

This gives the following results:

[latex]\begin{align*} x_w &=\frac{\Sigma x\cdot w}{\Sigma w} =\frac{$40,500}{$2,175} =0.0537=5.37\% \end{align*}[/latex]

Example: Excel for Weighted Mean (Video)

Example 4.1.3

We can also use Excel’s =sum() and =sumproduct formulas to calculate the weighted means:

Click here for the Excel solutions shown in the above video.

Example: Solving for Missing x-value (X)

Example 4.2.1

Problem Setup: You purchased shares several years ago and sold them today:

• You purchased 100 shares of Moderna for $200 each
• You purchased 40 shares of Microsoft for $40 each
• You earned 12% on your investments today (when you sold all of the shares).
• But, you LOST 15% on the sale of your Moderna shares.

Question: What rate of return did you earn on the sale of your Microsoft shares?

Company	x	Quantity	Price	Amount Invested (w)	Interest Earned (x · w)
Moderna	−15%	100	$200	100×$200 = $20,000	−15% × $2,000 = −$3,000
Microsoft	x	40	$400	40×$40 = $16,000	x×$16,000 = 16,000x
Totals =		140	–	Σw = $36,000	Σx·w = −$3,000+16,000x

This gives the following results:

[latex]\begin{align*} 12\% &=\frac{-$3,000 +$16,000 x}{$36,000} \end{align*}[/latex]

Steps: We need to use algebra to solve for this equation:

1. Cross multiply: [latex]12\%\cdot$36,000 =-$3,000 +$16,000 x[/latex]
2. Collect like terms: [latex]$4,320+$3,000 =$16,000 x[/latex]
3. Divide both sides by $16,000: [latex]\frac{$7,320}{$16,000}= \frac{$16,000 x}{$16,000}[/latex]
4. Simplify: [latex]0.4575=x=45.75\%[/latex]

Answer: You earned 45.75% on your Microsoft Shares.

Example: Solving for Missing Weight (Exercise)

Example 4.2.2

You also had an opportunity to invest in some Amazon stock for $150/share. The shares earned 20% over the same time period. If you bought these shares, you would end up earning 18% overall over the investment period.

Question: How much money would you have had to invest in Amazon stock to achieve an overall average rate of 15% on your investments?

Company	x	Quantity	Price	Amount Invested (w)	Interest Earned (x · w)
Moderna	−15%	100	$200	100×$200 = $20,000	−15%×$20,000 = −$3,000
Microsoft	45.75%	40	$400	40×$400 = $16,000	45.75%×$16,000 = $7,320
Amazon	20%	w	$150	w×$150 = 150w	20%×150w = 30w
Totals =		140+w	–	Σw = $36,000+150w	Σx·w = −$3,000+$7,320+30w

Instructions: Drag and drop the values below into the correct boxes:

Key Takeaways (Exercise)

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