How to calculate Volatility using expected returns. Explanation and Overview.

In this article, I want to talk about how you can calculate volatility by using the expected return. So let's say that we have two firms, we've got firm A we've got firm B, and let's say that firm A they flip a coin and if it's heads they're gonna get an 11% return if its tail's they're gonna end up with a 9% return. So they've got a 50-50 shot of each of these different outcomes 11% or 9%


If we were to calculate the expected return of firm A we just end up having 0.5 which is the 1 out of 2 chance for a 50% chance multiplied by 0.11 which is that return. Then 0.5 is multiplied by 0.09 and this is just going to equal 0.10 or 10%. So the expected return for firm A is 10%.

Then firm B is going to have a 50% chance of an 80% return and a 50% chance of getting a return of negative 60%. So now that is actually going to also yield an expected return of 10%. So you'd have 0.5 multiplied by 0.8 and then you add in 0.5 multiplied by negative 0.6. I don't want to dwell too much on the expected return here but in each case, the firms have the same exact expected return but do you think that these firms have the same level of risk? 

Think about it, firm A their return it's pretty safe it's either gonna be 11 or it's gonna be 9 and so it's tightly bunched around the expected return. The two return here 11 and 9 they're both very close to 10. So there's not a lot of fluctuation there but with firm B on the other hand you're either gonna get a return of 80% or you're gonna lose 60%. Now ultimately that comes out to an expected return of 10% which is the same as firm A but you can see where you could say "Hey, this is much more of a gamble,  there's a lot more risk here in terms of whether we're gonna realize a return."


So people take that into consideration, I said look there's more volatility for firm B and what do we mean when we say volatility we basically talking about the standard deviation of the returns. So we can just calculate the standard deviation of the returns for firm A and for B and compare those and we'd expect that firm B because of that gamble there we could just kind of see that it's gonna have a higher standard deviation, it's gonna have more volatility. 

So basically what we would go ahead and calculate the volatility first. To get the standard deviation of course first we need to calculate the variance. For firm A we would just take and remember we've got two outcomes for firm A. So we would say the variance is going to be equal to 0.5 multiplied by 0.1  that's at 11% return minus the expected return of 0.1. Then we're gonna add in here plus 0.5  multiplied by 0.09 minus the expected return of 0.1. Then we'll square that. So when we add all that together that's going to give us 0.0001 for firm A in terms of the variance but remember the standard deviation is the square root of the variance. So to get the actual volatility or the standard deviation we're going to take the square root of 0.0001 and that is going to be equal to 0.01 which is equivalent to this is the same as saying we have the volatility of 1% for firm A.


So now let's go ahead and let's calculate it for a firm B so for firm B we're gonna have 0.5 multiplied by that 80% return but that's an 0.8 there 0.8 minus 0.1 the expected return and then we'll square that and when I'm talking about square I should make it clear we're squaring this (0.8 - 0.1) and then plus 0.5 multiplied by negative 0.6 minus 0.1 and then we're square that. Now that's going to give us 0.49 it's going to be our variance but again we have to take the standard deviation. So the square root of the variance, square root of 0.49 is going to be equal to 0.7. This is the same as basically saying we have the volatility of 70%. 


Now you see the important thing to note here is if you are just kind of naively looking at these two firms and you say "Yeah, I want to look at what kind of return I'm going to get from these two firms, I'm debating between two investing in firm A and firm B." If you just look at the expected return in each case it's going to be 10%, so that's why investors don't just look at the expected return, they also look at the risk and one very important component or measure of risk is volatility. We see here firm A has volatility of just 1% was firm B is actually 70%.

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