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Have you ever pondered how mathematicians think about inequality and optimization? How do we apply theoretical concepts to solve real-world problems? Let me walk you through one of my favorite mathematical tools — Jensen’s inequality.
Jensen’s inequality is a simple yet powerful concept. In short, it states that for a convex function, the function’s value at the average of some points is less than or equal to the average of the function’s values at those points. At first glance, this may seem rather abstract. But its implications are profound. Jensen’s inequality allows us to derive bounds and build intuition about complex systems.
say you want to optimize an investment portfolio. The returns of each asset are uncertain and volatile. But by diversifying across assets, you can reduce risk. Jensen’s inequality gives us a mathematical justification for why diversification works — the volatility of a portfolio’s returns will be lower than the average volatility of its assets. This insight allows us to construct optimal portfolios that maximize return for a given level of risk.
still confusing? let’s work through it together…
Imagine you’re an investor with a variety of assets to choose from, each with its own expected return and risk level. Your goal is to maximize your expected return while minimizing your risk.
Let’s say you have two assets, A and B. Asset A is a low-risk, low-return asset, like a government bond. Asset B is a high-risk, high, asset-return like a tech startup’s stock.
If you put all your money in asset A, your portfolio’s return would be low but stable. If you put all your money in asset B, your portfolio’s return could be high, but it could also be very low or even negative.
Now, what if you split your investment between A and B? This is where Jensen’s inequality comes into play.
Let:
If you invest a fraction x of your money in asset A, you will get x*r_A return from it. Similarly, you invest the remaining (1-x) fraction of your money in asset B, so you get (1-x)*r_B return from it. The total return of your portfolio, r_P, is the sum of these two, hence the equation.
The risk of your portfolio, measured by the variance of its return, would be:
where Var denotes variance and Cov denotes the covariance.
The risk of a portfolio is typically measured by the variance of its returns. The variance measures how spread out or dispersed the returns are. If the returns are all close to the mean, the variance is small. If the returns are spread out over a large range, the variance is large.
The formula for the variance of the portfolio’s return, Var(r_P), is derived from the definition of variance and the properties of covariance. It’s a bit complex, but it essentially takes into account not just the variances of the returns of the individual assets (Var(r_A) and Var(r_B)), but also how these returns move together, which is measured by their covariance (Cov(r_A, r_B)).
The formula for Var(r_P) tells us that the risk of the portfolio depends on three things: the risks of the individual assets (as measured by their variances), the proportion of your money invested in each asset (x and 1-x), and the relationship between the assets (as measured by their covariance).
Here’s the key insight from Jensen’s inequality: the variance of your portfolio’s return is a convex function of your investment fraction x. This means that the risk of a diversified portfolio is less than or equal to the weighted average of the risks of the individual assets.
Jensen’s inequality tells us that for a convex function (like the function for variance), the function of the average is less than or equal to the average of the function. In the context of our portfolio, it means that the risk of the portfolio (which is the variance of the portfolio’s return, a convex function) is less than or equal to the average risk of the individual assets.
In other words, by diversifying your portfolio, you can achieve a higher expected return for a given level of risk, or a lower risk for a given expected return. This is the mathematical justification for the old adage, “Don’t put all your eggs in one basket.”
according to Jensen’s inequality, for a convex function, f(E[X]) ≤ E[f(X)]. This means:
- f(E[X]) represents the risk (y-value) of a diversified portfolio with average return E[X]
- E[f(X)] represents the average risk (y-value) of individual assets with returns X
So Jensen’s inequality shows that the risk of a diversified portfolio (f(E[X])) is less than or equal to the average risk of individual assets (E[f(X)]).
This proves that diversification reduces risk. Here’s why:
- If you invest in a single risky asset, your risk is simply f(X) — the risk of that single asset.
- If you diversify across multiple assets, your risk is f(E[X]) — the risk of the diversified portfolio.
- According to Jensen’s inequality, f(E[X]) ≤ E[f(X)] — the average risk of the individual assets.
- Therefore, the risk from diversifying, f(E[X]), is less than the risk from a single asset, f(X).
Jensen’s inequality also provides the foundation for many decision-making frameworks. In decision theory, we often have to choose between risky or uncertain options. How do we evaluate choices when we don’t know the exact outcomes? We can use Jensen’s inequality to determine bounds on the expected utility of each option. This helps us make more informed decisions, even with imperfect information.
At its core, mathematics is about developing logical reasoning skills and applying them to gain insights into complex systems. Jensen’s inequality is a prime example of how a simple mathematical concept can be a powerful tool for optimization, decision-making, and understanding of uncertainty. I hope this quick introduction to Jensen’s inequality has sparked your curiosity about mathematics and its applications!