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 $\frac{(5 + 6 + 7)\%} {3} = 6%$?

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 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:

$\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}{$40,500}\\ &= 0.0537 = 5.37\% \end{align*}$

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.

This gives the following results:

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

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 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?

This gives the following results:

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

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

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

Answer: You earned 45.75% on your Microsoft Shares.

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?

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