Percentage Arithmetic – When Numbers Don’t Add Up

Tuesday, October 27, 2009

1117095_basic_math_57We are in a period in which percentages fill the media reports because they are so big. At the same time, many of those numbers and the way they are used can be misleading. Now is a good time to refresh ourselves on percentage arithmetic so that we draw the right conclusions.

First, a math primer:

Percent is a ratio of one number to another, expressed in 1/100s. A good example is the interest on a savings account. If the bank pays us $4 a year on a $100 deposit, our annual interest rate is 4/100 = .04 = 4 percent, or 4%.

Next, a complexity: compounding

Annual rates and returns are the accepted norm, so multiple periods need extra math to arrive at a total or an annual average. For example, we leave our deposit and interest earned at the bank for a second year, and the interest rate rises to 5 percent. This produces interest of $5.20 (.05 times $104.00), giving us a total of $109.20.

So, what have we earned on average? We know it isn’t 4.6% (one-half of the $9.20 total interest divided by the original $100 deposit) because the first year’s interest increased the money on deposit for the second year. Through math, we can get the average compound interest rate that our money earned over the two years: 4.5%. (At this rate for the entire two years, our interest earned would have been $4.50 in year one and $4.70 in year two, giving us the total of $9.20.)

Now, the problem:

Negative returns spoil the calculation, particularly when they are large like the ones we have just experienced. For example, both Harvard and Yale reported that their endowment funds decreased 30% for the year ended June 30, 2009. If they were now to earn a 30% positive return, they would still be underwater by over 9%. To offset the 30% decline, they need a positive return of almost 43%.

The reason for this apparent mismatch is that percents are always calculated using the beginning value. So, 100 to 70 is -30/100 = -30%. And 70 to 100 is +30/70 = +43%.

During the period we are in, we need to aware of these mismatches. Then, when we hear the Dow Jones Industrial Average dropped 54% from its October 2007 peak to its March 2009 low and has since risen 54%, we won’t be mystified to know that it is still 4,000 points, or almost 30% below its high. We will also know that means the Dow has to rise 43% to get all the way back.

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