This is an unusual Index in that it multiplies together all the N prices in the index then takes the Nth root of that product.
>Huh?
Suppose there are 1800 prices in the Index and the product is P. Then the Value Line Geometric Index is A1/1800.
We could, of course, just take the Nth root of each price and multiply that together.
>Huh?
That’s why it’s called geometric. It uses the Geometric Mean of all the prices.
There’s also an Arithmetic Index which uses the Arithmetic Mean by taking the average …
>Can you just give an example?
Okay, suppose there are just ten stocks in the Index whose prices change, from one day to the next, like so:
| First Day | 74.70 | 43.21 | 10.27 | 61.85 | 39.72 | 9.27 | 44.59 | 97.05 | 54.61 | 27.41 |
| Next Day | 76.97 | 44.53 | 10.18 | 57.21 | 41.21 | 10.02 | 45.00 | 95.85 | 53.37 | 26.15 |
Now we take the 10th root of each price, getting:
| First Day | 1.544 | 1.462 | 1.261 | 1.499 | 1.450 | 1.259 | 1.463 | 1.578 | 1.488 | 1.386 |
| Next Day | 1.54 | 1.462 | 1.261 | 1.499 | 1.450 | 1.259 | 1.463 | 1.578 | 1.488 | 1.386 |
Now we multiply them all together getting:
First Day ValueLine = 37.0623
Next Day ValueLine = 37.1160
>And that’s the Index value?
Yes, for ten (fictitious) stocks and two successive days.
The beauty of this index is that, if you divide today’s close by yesterday’s close you’ll get (approximately) the median return on a portfolio that has equal dollar amounts in each stock.
>It is?
Yes.
If you have $1.00 in each stock, and stock#1 goes from $74.70 to $76.97 (as above), then the gain is 76.97/74.70 = 1.0304
… from $1.00 to $1.0304 for that stock. That’s a “Gain Factor” of 1.0304.
The Gain Factors for all ten stocks are (76.97/74.70) and (44.53/43.21) … and (26.15/27.41).
The median Gain Factor is (approximately) the Geometric Mean of these ten ratios, namely:
[(76.97/74.70)(44.53/43.21)…(26.15/27.41)]1/10 = 1.00155 or a 0.155%% median return.
>Why does the geometric mean give the …?
The median?
The argument goes like this:
- Assume that the returns for each of the N stocks have a lognormal distribution.
- Assume that the Gain Factors are 1+R1, 1+R2, … 1+RN.
- Then (since we’re assuming a lognormal distribution) their logarithms have a normal distribution. That is:
x1 = log(1+R1), x2 = log(1+R2), … xN = log(1+RN) are normally distributed. - That means that half of these numbers lie below the mean (a property of normal distributions).
- That is: Half of the x’s lie below (1/N) [ x1+x2+…+xN ]
- But (1/N) [ x1+x2+…+xN ] = (1/N)[log(1+R1)+ log(1+R2)+ … +log(1+RN)] = log{[(1+R1)(1+R2)…(1+RN) ]1/N}
and (surprise!) {[(1+R1)(1+R2)…(1+RN) ]1/N} is the geometric mean.
>That does it?
That does it.