The Haystack: What Do You Mean?
Schools may be closed, but it is finals week here at Monument. And here’s a computational nugget for the mathletes out there.
You are probably familiar with averages. Three children. Ages 3, 5, and 9. Average age = 5.7 years.
*Yawn*
And you are probably familiar with financial ratios – say, price-to-earnings (or P/E). A stock trades at a price of $100 with earnings per share of $10. P/E = 10x.
*Double yawn*
But how do you average P/E ratios? Say, the average P/E for a basket of stocks?
You own an equal amount of 3 stocks. One trading at 5x, another a 10x, and the last at 15x. The average of your basket is 10x, right?
The answer is 8.2x.
The arithmetic mean (from our first example above) gives undue weighting to extreme measures and will produce an overestimation. It is not equipped to deal with rates, or in this case, earnings yields.
The harmonic mean – and more specifically for a basket of stocks, the weighted harmonic mean – yields the accurate result.
But do not take my word for it. Let us use another real-life example.
Say you take a 5-mile trip or 10-mile round trip. On the way there, you travel 30mph. On the way back, you travel 10mph. Your intuition might be to say your average speed is 20mph. But alas…
• Leg 1 of your trip takes 10 minutes. Leg 2 of your trip takes 30 minutes. Forty total minutes.
• Leg 1 accounts for 25% of your trip, and leg 2 accounts for 75%.
• 0.25 x 30mph + 0.75 x 10mph = 15mph
• [1/10 + 1/30] / 2 = 1/15, and the reciprocal of 1/15 = 15…this is your harmonic mean!
The more you know!
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Erin M. Hay, CFA, CMT
Private Wealth Advisor, Portfolio Manager
A graduate of the University of Oklahoma, Erin began his investment management career with J.P. Morgan Private Bank. There, he worked with portfolio managers, traders and asset class specialists to bring tailored investment solutions to his team’s high net worth clients. However, intellectual curiosity would dictate both a change in geography and job description.
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