Jensens inequality

What Is Jensen's Inequality?

Jensen's inequality is a fundamental concept in financial mathematics that establishes a relationship between the expected value of a function and the function of an expected value. Specifically, for a convex function, Jensen's inequality states that the function of the expected value of a random variable is less than or equal to the expected value of the function of that random variable. Conversely, for a concave function, the inequality is reversed. This principle is crucial for understanding how averages behave under non-linear transformations, particularly in contexts involving uncertainty and decision making.

History and Origin

Jensen's inequality is named after the Danish mathematician Johan Jensen, who formally proved it in 1906. His work built upon earlier findings by other mathematicians, notably Otto Hölder in 1889, who demonstrated a version of the inequality for twice-differentiable functions. The inequality's generalization by Jensen established its broad applicability across various mathematical fields.⁴ Jensen's contribution solidified the understanding of how convexity and concavity interact with statistical averages, laying groundwork for its later use in economics and finance.

Key Takeaways

Jensen's inequality relates the expected value of a function to the function of an expected value.
For a convex function, the function of the mean is less than or equal to the mean of the function.
For a concave function, the function of the mean is greater than or equal to the mean of the function.
It is a foundational tool in probability theory, mathematical analysis, and financial modeling, particularly in areas like utility theory and option pricing.
The difference between the two sides of the inequality is often referred to as the "Jensen gap."

Formula and Calculation

Jensen's inequality can be stated formally as follows:

For a convex function (\phi) and a random variable (X),
$\phi(E[X]) \le E[\phi(X)]$

For a concave function (\phi) and a random variable (X),
$\phi(E[X]) \ge E[\phi(X)]$

Where:

(E[X]) represents the expected value of the random variable (X).
(\phi(X)) represents the function applied to the random variable (X).
(E[\phi(X)]) represents the expected value of the function applied to the random variable (X).

The inequality holds provided that (E[X]) and (E[\phi(X)]) are finite.

Interpreting Jensen's Inequality

Interpreting Jensen's inequality hinges on the nature of the function involved—whether it is convex or concave. If a function is convex, its graph "bows upward," meaning the line segment connecting any two points on the graph lies above or on the graph. In this case, the value of the function at the average of inputs will always be less than or equal to the average of the function's values for those inputs. This concept is vital in understanding phenomena like the value of derivatives, where the payoff function is often convex.

Conversely, for a concave function, which "bows downward" (the line segment connecting any two points on the graph lies below or on the graph), the value of the function at the average of inputs will be greater than or equal to the average of the function's values for those inputs. This is particularly relevant in risk aversion, where utility functions are typically assumed to be concave.

Hypothetical Example

Consider an investor evaluating a risky asset whose price, (X), can be either $50 or $150, each with a 50% probability. The expected price of the asset is (E[X] = (0.5 \times $50) + (0.5 \times $150) = $100).

Now, imagine a financial instrument whose payoff function is (f(X) = X^2). This is a convex function.

Function of the Expected Value: The payoff if the asset price was its expected value would be (f(E[X]) = f($100) = ($100)^2 = $10,000).
Expected Value of the Function: The expected payoff from the instrument is (E[f(X)] = (0.5 \times f($50)) + (0.5 \times f($150))).
(E[f(X)] = (0.5 \times ($50)^{2) + (0.5 \times ($150)}2))
(E[f(X)] = (0.5 \times $2,500) + (0.5 \times $22,500))
(E[f(X)] = $1,250 + $11,250 = $12,500)

In this example, (f(E[X]) = $10,000) and (E[f(X)] = $12,500). As predicted by Jensen's inequality for a convex function, (f(E[X]) \le E[f(X])), or $10,000 \le $12,500. This demonstrates how the non-linearity of the payoff (convexity) means that the expected payoff is higher than the payoff calculated from the expected underlying value. This principle is fundamental in understanding the value inherent in certain derivatives.

Practical Applications

Jensen's inequality has widespread practical applications across financial markets and economic theory:

Risk Aversion and Utility Theory: In utility theory, individuals typically exhibit risk aversion, meaning they prefer a certain outcome to a risky gamble with the same expected value. This behavior is modeled using concave utility functions. Jensen's inequality, applied to a concave utility function, shows that the expected utility of a risky prospect is less than the utility derived from the expected value of that prospect, (E[U(X)] \le U(E[X])). This formalizes the concept of a risk premium.
Option Pricing: The payoff functions of many derivatives, such as call options, are convex. Jensen's inequality explains why the expected payoff of an option is greater than or equal to the payoff calculated using the expected price of the underlying asset. This "convexity bias" contributes to the inherent value of options and is crucial in option pricing models.
*³ Diversification and Portfolio Management: Jensen's inequality helps illustrate the benefits of diversification in portfolio management. For instance, if volatility (a measure of risk) is a convex function of an asset's returns, then the risk of a diversified portfolio (the function of the average return) is less than or equal to the average risk of individual assets. T²his provides a mathematical justification for why combining assets can reduce overall portfolio volatility.
Arbitrage Theory: While not directly proving arbitrage opportunities, Jensen's inequality underpins theoretical models that demonstrate how expected values behave under different market conditions and functions, influencing the understanding of equilibrium and potential mispricings.

Limitations and Criticisms

While profoundly influential, Jensen's inequality is a mathematical statement of an inequality, not an equality. This means it provides a lower or upper bound, but not necessarily the exact value, for the relationship between the function of an expectation and the expectation of a function. The difference between the two sides of the inequality is known as the "Jensen gap".

The inequality is considered "sharp" only in specific cases, such as when the function is linear (in which case equality holds, (\phi(E[X]) = E[\phi(X)])), or when the random variable (X) is constant (i.e., has zero volatility). In most real-world applications where functions are strictly convex or concave and random variables are not constant, a significant "gap" exists. Therefore, while Jensen's inequality confirms the direction of the relationship (e.g., that convexity adds value to the expected function), it does not quantify how much value is added without further analysis. Research continues into "sharpening" the inequality to provide tighter bounds or more precise estimations in various contexts.

¹## Jensen's Inequality vs. Concave Function

Jensen's inequality and the concept of a concave function are intrinsically linked, but they are not the same thing. A concave function is a type of mathematical function characterized by its "downward-bowing" shape, where the line segment connecting any two points on its graph lies below or on the graph. Mathematically, a twice-differentiable function (f(x)) is concave if its second derivative (f''(x)) is less than or equal to zero for all (x) in its domain.

Jensen's inequality is a theorem or statement that applies to concave (and convex) functions. For a concave function (\phi), Jensen's inequality states that the function of the expected value of a random variable (X) is greater than or equal to the expected value of the function of (X): (\phi(E[X]) \ge E[\phi(X)]). This relationship is critical in fields like economics, where diminishing marginal utility is modeled by concave utility functions, directly implying risk aversion. Without the definition of a concave function, Jensen's inequality as applied to utility theory would not hold its significant implications for financial behavior.

FAQs

What does Jensen's inequality tell us about risk?

For a risk-averse investor, whose utility function is typically concave, Jensen's inequality shows that the expected utility from a risky investment is lower than the utility they would receive from the expected value of that investment if it were certain. This mathematically explains why investors demand a "risk premium" to undertake uncertainty.

Is Jensen's inequality always true?

Yes, Jensen's inequality is a mathematical theorem, and it is always true under the conditions that the function is either convex or concave and the relevant expected values exist and are finite. The direction of the inequality depends on whether the function is convex or concave.

How is Jensen's inequality used in financial modeling?

In financial modeling, Jensen's inequality is used to analyze non-linear payoffs, such as those found in derivatives and option pricing. It helps to understand why the expected value of a non-linear financial instrument (like an option, which has a convex payoff) can exceed the value derived simply from the expected underlying price. It also underpins models in portfolio management and mean-variance analysis by formalizing the benefits of diversification.