LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
Portfolio theory, the CAPM, value at risk, and the drift in a Black-Scholes stock all ask for an expected return. A die has one. The average of its faces is 3.5, and a long sequence of rolls converges to that number. A stock return series does not, because the distribution is not the same one being sampled forever. The lecture's Nvidia charts, split around 2008, 2016, and 2024, do not level off. The expected return behaves like a random process. Models that need a single number are using a proxy, and the lecture's point is to admit the proxy.
Why it matters
Weak stationarity, the assumption that opens many time-series courses, freezes the mean so the estimator has a target. Markets are a poor fit for that freeze. Nvidia early in its life and Nvidia later are not the same firm, and the lecture treats that as the reason a cumulative average should not be asked to converge. Higher moments depend on the mean, so if the mean wanders, variance, skewness, and kurtosis wander with it. There is no single window, daily through yearly, that the lecture will crown as correct.
The idea in plain words
Expectation is a probability-weighted average. For a fair die it is (1+2+3+4+5+6)/6 = 3.5. The law of large numbers says the sample average approaches 3.5 if every roll comes from that same die. A weak version and a strong version differ in the mode of convergence. The lecture does not need the distinction. It needs the hypothesis both versions share: the distribution stays put. A running plot starts wherever the first roll landed, jitters, and settles on the line at 3.5. The fraction of each face approaches one sixth. That is a frequency reading, not a prior updated by belief. Add time and the average can split. The lecture's ergodic game pays 1 dollar on a six and loses 20 cents otherwise. The expected gain per roll is zero. Start at 100. Some paths reach about 130 and some about 80, and at 1,000 plays the time average and the cross-sectional average agree at 100. A non-ergodic game starts at 100 and multiplies wealth by 1.5 on heads and by 0.6 on tails. The expected multiplier is 1.05, so the cross-section of players grows. Most individual paths decay toward zero, and a few paths hold an extreme share of the wealth. The time average and the ensemble average are not the same object. The lecture simulates both and treats the pictures as the definitions. A mean can also fail to exist. The St.
How to use it
Plot the cumulative mean of the return series you intend to plug into a model, on more than one window. The lecture's full Nvidia history looks calmer than it is, because early volatility dominates the scale. Isolated stretches in 2008, 2016, and 2024 do not converge. Assuming the mean was 0.003 would already be wrong by the end of the 2008 window. Assuming 0.00185 would be wrong by the end of the 2016 window. The 2024 window fails the same way. A 2025 slice of the long cumulative mean bounced near 0.002, from about 0.00216 down toward 0.002 and back near 0.00213, which is not a settled level. Restricting the average to 2025 alone shows a negative stretch the lecture ties to the tariff selloff. Both pictures cannot be "the" expected return.
The lecture's constructive suggestions are a Kalman filter, a regime-switching model, or a modified GARCH. One Kalman illustration fits on 2024 and then rolls forward. It is aimed at the expected-return level, not at tomorrow's return. The lecture does not report an error for that filter. It offers the filter as an example of a proxy, useful or not depending on the job. "All models are wrong, some are useful" is the closing slogan, and the useful part is knowing which assumption, here stationarity, you violated.
What usually goes wrong
Quoting a historical average as if the die theorem applied. The theorem needed a fixed distribution. Using the all-time Nvidia mean and the 2025-only mean as if a ritual could pick between them. And treating a non-ergodic wealth game's expected multiplier, 1.05, as the growth a typical player experiences. The typical path in that simulation decayed. The cross-section was carried by a few outliers.
Questions people ask
What does the die establish?
That a fixed distribution has a mean, 3.5, which the running average approaches. Stocks are being denied that premise, not the arithmetic of the die.
What did the Nvidia windows show?
No leveling in the cumulative mean for the stretches around 2008, 2016, and 2024. A 2025 slice moved, including a negative stretch, instead of locking onto one rate.
What can replace a single expected return?
A model of a moving level, such as the Kalman filter the lecture sketches, a regime model, or a GARCH-style variance model. Each is a proxy. None restores a constant that the charts do not have.