Squared error of regression line (video) | Khan Academy
To analyse the relationship between portfolio and index as well as its volatility, statistical tools like R-squared, Beta and Standard deviation are. In this article we will cover significance of Beta and Standard Deviation. Beta Variance = (Sum of squared difference between each monthly return and its. Read 13 answers by scientists with 7 recommendations from their colleagues to the question asked by Saraswati Bhandari on May 29,
The correlation with the market will be zero, but it is certainly not a risk-free endeavor. On the other hand, if a stock has a moderately low but positive correlation with the market, but a high volatility, then its beta may still be high. A negative beta simply means that the stock is inversely correlated with the market. A negative beta might occur even when both the benchmark index and the stock under consideration have positive returns.
It is possible that lower positive returns of the index coincide with higher positive returns of the stock, or vice versa. The slope of the regression line in such a case will be negative. Using beta as a measure of relative risk has its own limitations.
Most analyses consider only the magnitude of beta. Beta is a statistical variable and should be considered with its statistical significance R square value of the regression line. Closer to 1 R square value implies higher correlation and a stronger relationship between returns of the asset and benchmark index. If beta is a result of regression of one stock against the market where it is quoted, betas from different countries are not comparable.
Utility stocks commonly show up as examples of low beta. These have some similarity to bonds, in that they tend to pay consistent dividends, and their prospects are not strongly dependent on economic cycles. They are still stocks, so the market price will be affected by overall stock market trends, even if this does not make sense. Staple stocks are thought to be less affected by cycles and usually have lower beta.
This is based on experience of the dot-com bubble around year Although tech did very well in the late s, it also fell sharply in the early s, much worse than the decline of the overall market. More recently, this is not a good example. During the market fall, finance stocks did very poorly, much worse than the overall market. Then in the following years they gained the most, although not to make up for their losses.
Foreign stocks may provide some diversification. However, this effect is not as good as it used to be; the various markets are now fairly correlated, especially the US and Western Europe.
Beta relies on a linear model. An out of the money option may have a distinctly non-linear payoff. The change in price of an option relative to the change in the price of the underlying asset for example a stock is not constant. Beta views risk solely from the perspective of market prices, failing to take into consideration specific business fundamentals or economic developments. The price level is also ignored, as if IBM selling at 50 dollars per share would not be a lower-risk investment than the same IBM at dollars per share.
Beta fails to allow for the influence that investors themselves can exert on the riskiness of their holdings through such efforts as proxy contestsshareholder resolutions, communications with management, or the ultimate purchase of sufficient stock to gain corporate control and with it direct access to underlying value. Beta also assumes that the upside potential and downside risk of any investment are essentially equal, being simply a function of that investment's volatility compared with that of the market as a whole.
This too is inconsistent with the world as we know it. I don't even know what it looks like right now. And what we want to do is minimize this squared error from each of these points to the line.
Beta (finance) - Wikipedia
So let's think about what that means. So if the equation of this line right here is y is equal to mx plus b. And this just comes straight out of Algebra 1. This is the slope on the line, and this is the y-intercept. This is actually the point 0, b. What I want to do, and that's what the the topic of the next few videos are going to be, I want to find an m and a b.
So I want to find these two things that define this line. So that it minimizes the squared error. So let me define what the error even is. So for each of these points, the error between it and the line is the vertical distance. So this right here we can call error one. And then this right here would be error two. It would be the vertical distance between that point and the line.
Or you can think of it as the y value of this point and the y value of the line. And you just keep going all the way to the endpoint between the y value of this point and the y value of the line. So this error right here, error one, if you think about it, it is this value right here, this y value. It's equal to y1 minus this y value. Well what's this y value going to be?
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Well over here we have x is equal to x1. And this point is the point m x1 plus b. You take x1 into this equation of the line and you're going to get this point right over here.
So that's literally going to be equal to m x1 plus b.
Squared error of regression line
That's that first error. And we can keep doing it with all the points. This error right over here is going to be y2 minus m x2 plus b. And then this point right here is m x2 plus b.
The value when you take x2 into this line. And we keep going all the way to our nth point. This error right here is going to be yn minus m xn plus b.
Now, so if we wanted to just take the straight up sum of the errors, we could just some these things up. But what we want to do is a minimize the square of the error between each of these points, each of these n points on the line. So let me define the squared error against this line as being equal to the sum of these squared errors. So this error right here, or error one we could call it, is y1 minus m x1 plus b. And we're going to square it.
So this is the error one squared. And we're going to go to error two squared. Error two squared is y2 minus m x2 plus b. And then we're going to square that error. And then we keep going, we're going to go n spaces, or n points I should say. We keep going all the way to this nth error.
The nth error is going to be yn minus m xn plus b.