➡️ Quant Finance from scratch part 5: Measure return Two questions guide the video: why does a fall hurt more than the same rise helps, and how do quants measure returns so that they add up? Step 1, the base. A return is always measured from where you start. From $100, minus 50 % takes away $50. From $50, plus 50 % adds back only $25, so you are still $25 short. Step 2, percent moves multiply. $100 × ½ = $50, and only × 2 brings you back to $100. Times 2 is +100 %. Step 3, percent returns don't add up. −50 % + 100 % makes +50 % on paper, yet the money is right back at $100. Step 4, the log return: the natural log of the new price divided by the old one. The fall becomes ln(0.5) = −0.69, the rise ln(2) = +0.69, and together they make exactly 0. Step 5, the idea behind it. On a log scale every halving is the same step down and every doubling the same step up. The log return is the length of that step, so a fall and the rise that undoes it have the same size. Step 6, a whole year. A simulated stock moves randomly, about 2 % a day. Adding up its 252 daily percent moves gives −24.0 %, yet it really lost 25.0 %. Adding up the daily log returns and turning the sum back into percent gives exactly the real −25.0 %. Why quants love log returns: they add up over time, a price built from them can never fall below zero, and in the classic model of a stock the log returns follow the bell curve from part 4. The honest part: for one small day both are almost the same, since +1 % is 0.00995 as a log return. The difference grows with big moves and over time. This video is for education only, and nothing in it is financial advice. #quantfinance #logreturns #returns #compounding #python - @quantfinancetogo