Splits and Price Weighted Averages
Suppose XYZ were to split two for one so that its share price fell to $50. We would not want the average to fall, as that would incorrectly indicate a fall in the general level of market prices. Following a split, the divisor must be reduced to a value that leaves the average unaffected. Table 2.4 illustrates this point. The initial share price of XYZ, which was $100 in Table 2.3, falls to $50 if the stock splits at the beginning of the period. Notice that the number of shares outstanding doubles, leaving the market value of the total shares unaffected. The divisor, d, which originally was 2.0 when the two stock average was initiated, must be reset to a value that leaves the “average” unchanged. Because the sum of the post split stock prices is 75, while the presplit average price was 62.5, we calculate the new value of d by solving 75/d _ 62.5. The value of d, therefore, falls from its original value
of 2.0 to 75/62.5 _ 1.20, and the initial value of the average is unaffected by the split:
75/1.20=62.5.
At period end, ABC will sell for $30, while XYZ will sell for $45, representing the same negative 10% return it was assumed to earn in Table 2.3. The new value of the price weighted average is (30 _ 45)/1.20 _ 62.5. The index is unchanged, so the rate of return is zero, rather than the _4% return that would be calculated in the absence of a split.
This return is greater than that calculated in the absence of a split. The relative weight of XYZ, which is the poorer performing stock, is reduced by a split because its initial price is lower; hence the performance of the average is higher. This example illustrates that the implicit weighting scheme of a price weighted average is somewhat arbitrary, being determined by the prices rather than by the outstanding market values (price per share times number of shares) of the shares in the average.