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Assume stock prices follow a random walk. If my investments go up by 1,000 dollars on the stock market today and I keep that money invested, in expectation, how much are my investments worth 1 year from now?

I see that the answer is 1,000 dollar, i.e., I should think about my new level of wealth as permanent.

But can someone provide me with the math and the intuition for this?

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  • $\begingroup$ It is just a consequence of the assumption of a "random walk" $\endgroup$ – Alex C Nov 14 '19 at 21:04
  • $\begingroup$ But can you explain why? Intuitively and also with some math? $\endgroup$ – Aaaaad Nov 14 '19 at 21:15
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I guess the concept you're looking for are martingales. These are stochastic processes which remain on their current level (in expectation!).

Ignoring some technical conditions, a stochastic process $(X_t)$ is called a martingale if for all time points $t\geq s$, $$\mathbb{E}[X_t|\mathcal{F}_s]=X_s.$$ Here, $(\mathcal{F}_s)$ refers to a filtration, the information set available at time $s$. So, given the knowledge (information) at time $s$, your best prediction for the future value $X_t$ is the current value $X_s$.

For instance, the process $(X_t)$ could model the wealth of a portfolio. This would align with your question. However, the question is whether such a portfolio really satisfies the above property. Probably it will not. Real life stock prices are not really martingales: is the best guess (expected value) for Apple's price in one year's time really today's price? Perhaps not. If you however simplify reality and assume that stock prices are simple random walks, then stock prices are indeed martingales, see here.

Derivatives pricing is obviously built upon the concept of martingales and indeed, given the absence of arbitrage strategies, there exist artifical probability measures under which discounted stock prices are indeed martingales. But the expectation in this risk-neutral world is vastly different from the real world expectations (because real world investors are risk averse).

A related concept would be the Markov property. Here, you require that the filtration $(\mathcal{F}_s)$ is generated by the random variable $X_s$, i.e. $\mathbb{E}[X_t|\mathcal{F}_s]=\mathbb{E}[X_t|\sigma(X_s)]$ for all $t\geq s$. This means, past information does not matter at all for predicting stock prices. Again, in the real world, you will find plenty of violations of the Markov property.

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  • $\begingroup$ Very helpful, but I am not sure about your comment that "stock prices are not really martingales". I thought the evidence suggests that stock prices generally evolve according to a random walk with drift and violations of this are minor? So I might not expect Apple's price to be the same in year - it will likely be just a bit (the drift term) higher? $\endgroup$ – Aaaaad Nov 14 '19 at 22:43
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    $\begingroup$ @Aaaaad violations of this are massive. KeSchn is correct; prices are not really martingales. There has never been a successful validation study of mean-variance finance. Indeed, academic conferences would be empty, devoid of anything to talk about if there were. Probably fully half of any conference is built around an anomaly in the data. You should accept KeSchn's answer. It is a good answer to your question as to the intuition of what is going on. $\endgroup$ – Dave Harris Nov 14 '19 at 23:55
  • $\begingroup$ @DaveHarris - I have accepted the answer - It really was incredibly helpful. However, I still feel like I want the intuition for how people should think about winnings and losses on their portfolio in real life - should they consider the change in their wealth level a permanent change, in expectation? And if so, why? $\endgroup$ – Aaaaad Nov 15 '19 at 13:46

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