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Can you explain what a weighted average is concerning the following?
Five bonds and their percents are laddered in the following manner:
Year 5 ---- 4.40%
Year 4 ---- 4.05%
Year 3 ---- 3.63%
Year 2 ---- 3.08%
Year 1 ---- 2.50%
This is an investment in bonds with $100,000 invested equally for five years yielding a weighted average of 3.53% or $3,532.00 annually.
(17.66%/5 years = 3.532%/1 year)
My questions are as follows:
1. Why is the percent of 3.53% regarded as a weighted average in this example?
2. What would need to change in the above example so that the percent would be a simple average?
I thank you for your answers.
Answer Let me just clarify what is meant by a weighted average (WA). A weighted average over N items is given by (1/N)* SUM [w(i)*f(i)], where
w(i) represents the weight (value or significance) of a single occurence of type i.
f(i) represents the frequency of i (i.e., number of times it appears).
eg., to calculate the weighted average of 9 scores {1,5,9,5,1,1,1,9,9}, take a tally.
Here, the total count is N=9. We have three different types i={1,2,3}. The weights are given by w(1)=1, w(2)=5, w(3)=9.
Frequency f(i) refer to the number of times type i appears, we have f(1)=4, f(2)=2, f(3)=3.
Applying the formula, weighted average = (1/9)*[4*1+2*5+3*9]=41/9=4.555...
In the example that you referred to, there is evidently no weighted averaging whatsoever (if you just look at the plain figures). It is simply an average. You can think of it as a special case, where all types are valued (weighted) equally. I am not sure how weighted percentages are used in bonds by the professionals. I can only presume that the quoted figures refer to current yield (annual payout as a percentage of the current market price). Be careful though, I have seen people over simplify things and make claims which are untrue. Simplifying things does not always provide true insight. As this article [www.moneychimp.com/articles/finworks/fmbondytm.htm] shows, common problem (as innocent as finding the annual rate of return) may not have a closed form solution. Sometimes, the equations are not so straight-forward and need to be solved using numerical approximation techniques.
On second thought, I think what you've outlined makes sense in terms of diversified investment. There are five bonds in your portfolio, you invest an equal amount in each of them at the beginning, but these will change unless their performance are identical. In which case, there will be no need for weighted averages.
I believe there are two levels of averaging alluded to here.
I think the way the term "weighted average" is used is a little misleading. The 3.532% figure is merely a simple average of the consolidated figures across the five year period. But each of the annual percentage figure derives from the combined performance of five different bonds. This is the source of weighted averaging. Because captial gain/loss, fluctuation in market price will affect the annual payout on each bond differently. So, the figures you have available conceals much information. It concerns with not only the growth percentage, but how much it is growing on, term by term. The adjustment it makes takes into account what happens when all things are equal.
