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Markov, Chebyshev and Jensen: bounds without a full distribution
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Start with the idea
- You may be able to limit the chance of a large loss without knowing its exact probability. The limit can be much larger than the real chance.
Symbols, units & horizon
- Y: nonnegative loss in dollars
- a: positive dollar loss threshold
- X: one-trade P&L in dollars
- μ,σ: finite mean and positive standard deviation in dollars
- k: positive dimensionless number of standard deviations
- P: probability
- φ: convex function on the support of X with expectations defined
- no multi-period extrapolation is implied
When and why to use this
Use inequalities as conservative checks on risk outputs or to explain why nonlinear losses and utility require more than a mean forecast.
Start here · the same idea in everyday words
| Technical term | Everyday meaning | Concrete example |
|---|---|---|
| Upper bound | A value the actual quantity cannot exceed under the stated conditions. | At most 20% does not mean exactly 20%. |
| Convex | A function whose graph lies below the line joining any two points on it. | Squaring is convex: larger deviations receive increasingly large squared values. |
| Support of a distribution | The set of values an outcome can take. | For a two-outcome example of −1 and 3 dollars, those two values are its support. |
Unpack a technical sentence
Technical wording: A distribution-light bound need not identify the tail probability.
Read it as: You may be able to limit the chance of a large loss without knowing its exact probability. The limit can be much larger than the real chance.
A slower worked example · one step at a time
- Suppose a nonnegative loss is $10 with probability .10 and $0 with probability .90.
- The expected loss is .10 times $10 plus .90 times $0, or $1.
- Markov’s inequality bounds the probability of a loss of at least $5 by expected loss divided by $5: 1/5=.20.
- The actual probability of a loss of at least $5 is .10 because only the $10 outcome qualifies. Both .10 and its upper bound .20 are correct, but they answer different questions.
- The loss is nonnegative, as the theorem requires. Applying the same reasoning to signed net profit without a justified transformation would not meet that condition.
Self-assessed: Why would a conservative bound differ from the true probability?
It uses limited information, here only nonnegativity and the mean. Many different loss distributions can share those facts.
Study loop: predict the result, work through the arithmetic, run the Python, then explain which assumption would make the result unreliable.
- Markov bounds the upper tail of a nonnegative random variable using its mean. Chebyshev applies Markov to a squared deviation and bounds either tail using variance.
- Jensen says a convex function of an average is no greater than the average function value. Reverse the inequality for a concave function, when the expectations exist.
- The square is convex: squaring expected P&L understates expected squared P&L. Log wealth is concave, which is why average growth depends on the distribution rather than just the arithmetic mean.
Before reading the working, evaluate 1/2^2. Give the result in probability upper bound.
Chebyshev gives a 25% upper bound at two standard deviations; it does not assert the tail probability is 25% or that the distribution is Gaussian.
Markov, Chebyshev and Jensen: bounds without a full distribution
- Markov: if expected nonnegative loss is $2 and threshold a=$10, the probability of loss at least $10 is at most 2/10=.2.
- Chebyshev: with k=2 the probability of being at least two standard deviations from the mean is at most 1/4=.25.
- Jensen: for equally likely dollar outcomes −1 and 3, mean is 1. Its square is 1, while mean squared outcome is (1+9)/2=5, confirming 1≤5.
Chebyshev gives a 25% upper bound at two standard deviations; it does not assert the tail probability is 25% or that the distribution is Gaussian.
An analogy to remember
A speed limit bounds what is allowed without predicting the actual speed. A probability bound likewise limits a tail without identifying it.
How this becomes a building block
Distribution-light bounds can act as sanity checks alongside a richer risk model, with their conservatism made explicit.
Python · run and reuse
Architecture pseudocode
# Python 3.10+; standard library only unless NumPy is imported.
# Synthetic teaching inputs; no historical performance claim.
def markov_bound(expected_nonnegative_loss, threshold):
if expected_nonnegative_loss < 0 or threshold <= 0: raise ValueError("Invalid loss or threshold")
return min(1.0, expected_nonnegative_loss/threshold)
def chebyshev_bound(k):
if k <= 0: raise ValueError("Positive k required")
return min(1.0,1/k**2)
def jensen_square(values):
values = list(values)
if not values: raise ValueError("Nonempty sample required")
return (sum(values)/len(values))**2, sum(x*x for x in values)/len(values)
print(markov_bound(2,10), chebyshev_bound(2), jensen_square([-1,3]))
# Replay the slower, synthetic example above. Units are stated in its steps.
def plain_language_example():
from math import exp, e # Standard library; no extra dependency.
return {
"expected_loss": .1*10+.9*0,
"bound": 1/5,
"actual_probability": .1,
}
print(plain_language_example())Self-assessed: explain the calculation, its output units, and one condition under which it fails.
Use inequalities as conservative checks on risk outputs or to explain why nonlinear losses and utility require more than a mean forecast. Markov requires a nonnegative quantity, Chebyshev requires finite variance, and Jensen requires the stated convexity and expectations. Weak bounds may be too loose for position sizing.
Python implementation
Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.
# Python 3.10+; standard library only unless NumPy is imported.
# Synthetic teaching inputs; no historical performance claim.
def markov_bound(expected_nonnegative_loss, threshold):
if expected_nonnegative_loss < 0 or threshold <= 0: raise ValueError("Invalid loss or threshold")
return min(1.0, expected_nonnegative_loss/threshold)
def chebyshev_bound(k):
if k <= 0: raise ValueError("Positive k required")
return min(1.0,1/k**2)
def jensen_square(values):
values = list(values)
if not values: raise ValueError("Nonempty sample required")
return (sum(values)/len(values))**2, sum(x*x for x in values)/len(values)
print(markov_bound(2,10), chebyshev_bound(2), jensen_square([-1,3]))
# Replay the slower, synthetic example above. Units are stated in its steps.
def plain_language_example():
from math import exp, e # Standard library; no extra dependency.
return {
"expected_loss": .1*10+.9*0,
"bound": 1/5,
"actual_probability": .1,
}
print(plain_language_example())Continue learning
Statistics & Probability — all lessons- Start with counts, probabilities and averages
- Random outcomes, sample averages and the limits of the bell curve
- Conditional probability: update a belief with evidence
- Prediction markets: probability, price and net expected value
- Expected value: measure the payoff before choosing the risk
- Correlation: how many strategies do you really have?
- Conditional probability and Bayes: where the edge actually lives
- The central limit theorem: sampling means under explicit assumptions
- Law of large numbers: what averaging can and cannot establish
- Central limit theorem, standard errors and confidence intervals
- Total expectation and total variance: within-regime and between-regime risk
- Markov, Chebyshev and Jensen: bounds without a full distribution
- Gauss–Markov theorem: what a regression hedge ratio guarantees
- Delta method: propagate estimator uncertainty through a smooth function
- Multiple testing and the union bound: account for the whole search
- Estimation uncertainty and Bayesian updating
Quantitative finance and development glossary · Python resources and libraries · Research sources and limitations