# 8 – M4 L3b 08 Winners And Losers Approximating Curves With Polynomials V4

Now that we have a sense for how accelerated or decelerated gains and losses look like visually, how do we represent their relative convexity or concavity in numbers? One way to approximate a curve, is with a formula that looks like y equals x squared. Or more accurately, y equals ax plus bx squared. The authors use t as the independent variable name representing time, such as the number of days from the start of the stocks trajectory. They also use beta as the coefficient for t, and gamma as the coefficient for t squared. To make the discussion easier to follow, let’s just give descriptive names to the coefficients. The coefficient for t will be called gain, and the coefficient for t squared will be called accelerate. You’ll see why we chose these names in a minute. Let’s see what a positive or negative direction coefficient might look like. If we just ignore the t squared term, a positive gain coefficient, will make a line that slopes upward which could represent a stock price that is gaining. Conversely, a negative gain coefficient, makes a line that slopes downward which could represent a stock price that is losing. Next, we can look at the accelerate coefficient and how it affects the shape of the curve. If the gain coefficient is positive, and the accelerate coefficient is also positive, both the t term and the t squared term are pointing up and to the right, so the curve is convex and looks like an accelerated gain. If the gain coefficient is still positive but the accelerate coefficient is negative, the t term is being counteracted by the t squared term, so the curve looks like a decelerated gain. Let’s look at two more cases when the gain coefficient is negative. If the accelerate coefficient is also negative, then both terms are pushing the trajectory downward, so the curve looks like an accelerated loss. Lastly, if the gain coefficient is negative but the accelerate coefficient is positive, the two terms counteract each other and the curve looks like a decelerated loss. Okay, now for the fun part. Based on whether the gain coefficient and the accelerated coefficient are positive or negative, can you decide which scenarios you would rather go long, or short?