Every day, millions of people check to see "how the stock market did". And, the percentage change number indubitably reigns king of all daily market indicators. However, this number can be notoriously misleading.
For example, pose the following scenario to anybody on the street (or even people in
the financial Industry):
Over the past 6 months, the Dow (DJIA) has dropped 30%. By what percentage will it need to rise over the next 6 months in order to recoup our losses?
Most people will answer that the market must simply rise another 30%. However, the market would actually need to rise 43% for it to be at its previous level.
(Math: Drop of .3 means market is at .7 of its previous level. Simple algebra shows that DJIA must be multiplied by the reciprocal of .7 to be back at it's original level. [Reciprocal of .7]=1.43, so 43%)
So, what we need is a new market change indicator that meets the following conditions:
Looks like a regular old percentage, reasonable to people accustomed to the old system
Acts like a percentage; varying as a function of the change in value of the stock/market, but is smaller when the market is larger (accounts for the size of the market)
A drop by a certain amount of this indicator would be offset by a gain of the same amount of this indicator.
Here is a rather obvious solution:
The new percentage is calculated by taking the change in points, divided by the average of the market's open and closing values. So, for a drop of 200 points from a market that opened at 8000 points, the percentages would be calculated as:
200/8000 = old percentage = .025 = 2.50% |
200/([8000+7800]/2) = new percentage = .0253 = 2.53%
Using the new system, the market dropped 2.53% that day. The next day, if the market rose by 2.53% in the new system, the market would be right back at 8000 points.
We would need some new simple equations to use with this new system. Simple algebra shows the following to be true using this new system if P=continuous percentage, S=Market's opening value, C=Market's closing value:
P = 2(C-S)/(C+S) |
S = (P+2)C/(2-P) |
C = (2-P)S/(P+2)
This system would clearly be much more intuitive and lead to more accurate judgment when people make financial decisions. However, I think there is an even better solution.
NOTE: Variables followed by a "#d' mean that value will be in the decimal form. So, a 2.5% drop in value will be represented by 0.975 in these variables. You can easily find the number the other way by subtracting 1 from these decimals.
This would be:
This leads to very similar numbers as the earlier formula. For the earlier scenario, it also results in a P of 2.53%, but varies only by 0.0000014%. Although it is less obvious, the calculations are actually simpler.
Also, using this system, one can easily convert between original percentages and "continuous percentages" The "obvious" system requires a more complex formula to do this. Here is the conversion from new percentage to old using the P=ln(c/s) system:
Old Percentage#d = e^(New Percentage)
New Percentage=ln(Old Percentage#d)
Therefore, I now favor this system for calculation "continuous percentages".