A $250 position falls 12%, then the remainder rises 9%. What is the ending value, and by how much does adding the two moves misstate the net change?

A $250 position falls 12%, then the remainder rises 9%. What is the ending value, and by how much does adding the two moves misstate the net change?

Approach: Apply the two multipliers in order, then compare the exact net multiplier against the sum of the two moves and identify the size of the missing term.

$239.80. Apply the factors in order: 250 * 0.88 = 220, then 220 * 1.09 = 239.80. The net multiplier is 0.88 * 1.09 = 0.9592, a fall of 4.08%. Adding the moves gives -12 + 9 = -3%, which is 1.08 percentage points too generous. The gap is the cross term a*b = -0.12 * 0.09 = -0.0108, since the exact net is a + b + a*b. Percentage changes compound, so the second move applies to a changing base that the first move already shrank. The working rule is that the sum is right to first order and the correction is the product of the two moves, carrying a minus sign whenever the moves have opposite signs.

Follow-up: What single percentage move applied twice gives the same 4.08% loss, and how does that relate to the geometric mean?

Key concepts: compounding, cross term, changing base.