# How to Calculate the Expected Move of a Stock

I've been trying to determine how to calculate the expected move of a stock, and I've found there are several ways to do this but most of my findings usually lead me to this formula: Unfortunately nobody seems to provide an reasoning for why and how this formula works, I was wondering if someone could help walk me through the reason of why this formula works. Also, if you know of a more appropriate formula I should be using I'd appreciate knowing that too. Thank you!

• To the close voter: this is primarily about how statistics can be applied to finance; the expectation here is more along the lines of a mean than anything else. As maths applied to finance it is on topic and not opinion based. – MD-Tech Jul 6 '17 at 13:16
• That formula is incorrect. The expected value of a stock in the future should be nothing more than the future value (in time-value-of-money terms) of the current price of the stock at some risk-free interest rate, minus any dividends paid. The volatility tells you how big of a range you can expect the price to be within. – D Stanley Jul 6 '17 at 14:07
• Are you looking to calculate the expected move over a short period of time (e.g. to create a random walk) or to estimate what a particular stock will be say, 6 months from now? – D Stanley Jul 6 '17 at 16:08
• Yes I am trying to calculate the expected move over a short period of time such as a couple of days, but I would be just as interested to know how to estimate what a stock value could be within 6 months from now. If you could answer either or both, I'd be happy! – HappyTurtle Jul 6 '17 at 20:36

This formula omits some important parts.

1. This gives an absolute number which doesn't mention or care if the stock will go up or down by that number.
2. It's entirely useless for predicting the future value of the stock. It's used for the pricing of derivatives such as options, and for showing how risky/volatile the stock is.
3. It's missing the second part, which is about interest rate and about how much we expect the value of the stock to increase.

The formula assumes that the price of the stock moves a little bit each day. So the more days you have the more it moves. That's what the Number of Calendar Days is for. Some of these moves will cancel each other out, that's what the square root is for. What's remaining is really just a magic number.

The stock price and sqrt(365) are there because that's how Implied Volatility is defined. Implied Volatility is a number that measures how much/often a price changes, in this case over a year (that's where the 365 comes from), and relative to the value of the stock (that's why we multiply by the value of the stock).

Implied Volatility instead of just Volatility means we derive volatility by locking at past prices of the stock. One way to do that is to use the same formula backwards.

The missing part about interest rate is really just a prediction of how much we except the value of the stock to increase, on average, which is just a bog standard high school compound interest calculation.

It is not true except in a special case. The complete math is here https://stats.stackexchange.com/questions/31177/does-the-variance-of-a-sum-equal-the-sum-of-the-variances

Essentially, it is only true if the distribution has a variance and all events are independent. These are very strong assumptions. If it were true, then the variance each day would be added together. The standard deviation is the square root of the variance and so the number of days would be taken to its square root.

A far sounder rule of thumb would be to order historical observations from low to high. Find the observation at the 25th percentile and the 75th percentile. Subtract the 25th from the 75th. Take that difference and divide by two. That will give you the semi-interquartile range. It is NOT the "expected move" as you have put it, but it is the move you would expect to see twenty-five percent of the time. So the median plus or minus that number is a reasonable amount of movement to anticipate.

I am not providing the math because it is far more complex than the sum of the variances rule above and this forum does not permit the use of LaTeX making the work daunting to try, at best.

The formula I use is Stock Price x IV x Sq root of 28 / Sq root of 365 = Expected move. I was given this by a CBOE rep at a TDAmeritrade Market Drive Event in NY.

• That's just the OP's formula for a 4-week period. – TripeHound Apr 30 '18 at 7:33