Convexity

Convexity, in mathematics and economics, refers to a geometric or functional property where a line segment joining any two points on a curve or surface lies entirely on or above the curve or surface itself. It essentially describes a "bulging outwards" or "bowing outwards" shape. In economics, it's often associated with diminishing returns; as more of a resource is utilized, the marginal benefit gained from each additional unit decreases. convexity ensures that a local optimum is also a global optimum, making it a crucial concept in optimization problems. The degree of convexity can be quantified using second derivatives (in calculus) or through the comparison of chord length and arc length.

Convexity meaning with examples

• In portfolio optimization, convexity is a desirable characteristic of bond pricing. As interest rates change, the price of a bond with positive convexity will fluctuate less dramatically than a bond with negative convexity. A bond's convexity reflects how sensitive its duration is to changes in yield, helping investors manage interest rate risk and anticipate potential gains or losses based on market fluctuations and their investment horizons. This also plays a role in the yield curve as convexity will reflect the volatility for the specific term of an investment.
• In the context of a farmer planting crops, a convex production function illustrates that the yield increases at a diminishing rate with each additional unit of fertilizer used. The initially high return from the first few units of fertilizer decreases as more fertilizer is applied, representing the law of diminishing returns. The convex shape ensures that the most efficient point to maximise output is easily calculated.
• Consider the shape of a lens. A convex lens curves outwards, causing light rays to converge at a focal point. The curvature's convexity allows the lens to magnify images or focus light for various applications like eyeglasses or magnifying glasses. The extent of the curve determines the lens's focusing power, dictating how strongly it bends the light.
• In machine learning, the loss function for certain algorithms (like logistic regression or support vector machines) is often designed to be convex. This means that there is a single global minimum, allowing the algorithm to efficiently find the optimal solution by minimizing the loss function. Non-convex functions can result in getting trapped at local minima, leading to sub-optimal results and a longer training period.