70 a stock had annual returns of 36 percent 87 percent 56...

70. A stock had annual returns of 3.6 percent, -8.7 percent, 5.6 percent, and 12.5 percent over the past four years. Which one of the following best describes the probability that this stock will produce a return of 22 percent or more in a single year? A. less than 0.1 percent B. less than 0.5 percent but greater than 0.1 percent C. less than 1.0 percent but greater the 0.5 percent D. less than 2.5 percent but greater than 0.5 percent E. less than 5 percent but greater than 2.5 percent Average return = (0.036 - 0.087 + 0.056 + 0.125)/4 = 0.0325 ∑ = √[1/(4 - 1)] [(0.036 - 0.0325)2 + (-0.087 - 0.0325)2 + (0.056 - 0.0325)2 + (0.125 - 0.0325)2] = 0.0883 Upper end of 95 percent range = 0.0325 + (2 × 0.0883) = 20.91 percent Upper end of 99 percent range = 0.0325 + (3 × 0.0883) = 29.75 percent A return of 22 percent or more in a single year has between a 1 percent and a 2.5 percent probability of occurring in any one year.

Transcript

00:01 We have been told the annual returns for a stock over the past four years.

00:04 Which of these best describes the probability of it producing a return of 22 % or more in a single year? so you want to know how this has happened.

00:14 So what they've done is they've found the mean and standard deviation of this sample.

00:19 So the sample mean, x -bar, can be calculated by taking each value, adding them up, dividing by the total number.

00:27 And we're doing this in decimal form rather than percentage.

00:33 Percentages don't tend to play well with calculations, so this is a good idea.

00:37 So we add up our four numbers in decimal form and we divide by four, giving us a mean of 0 .0325.

00:52 So 3 .25 % is the average return for the past four years.

00:56 Is.

00:57 Now for the sample standard deviation, we take each value, subtract the mean, square the difference, get the sum of squared differences, divide by n minus one, and then square root it.

01:11 So it's important that i know this is a sample, otherwise i would be dividing by n and finding the population standard deviation.

01:20 But we're only looking at the past four years, i assume the stock's been around longer than that, so we're finding the sample standard deviation.

01:30 So take a value, subtract the mean, square the result, and repeat for the others, divide by 3, square root, we get a sample standard deviation 0 .0883.

01:50 When you're looking at any kind of gambling or investing, the standard deviation is a measure of the risk.

01:59 Looking at how high that standard deviation is compared to the mean, this is a very risky stock.

02:05 Now for what they've done next.

02:07 They have made a huge assumption.

02:09 They have assumed that the returns year on year follow a normal distribution and that they are independent of each other.

02:24 That's quite an assumption here, saying that okay whatever's happened in the last four years just to represent a normal distribution and these are independent of each other, they're not going to influence the next year, that's a huge assumption.

02:36 But that is what they have done.

02:37 And by doing that they have allowed themselves to use the empirical rule, also called the 68, 95, 99 .7 % rule.

02:52 This says that, actually i'll draw a curve to help us here, it says that approximately approximately 68 % of a normal distribution is within one standard deviation of the mean.

03:05 So if i draw a normal curve, 68 % of the time, it'll be between minus 1 and 1, if these are standard deviations away.

03:16 95 % of the time, it is within 2, so from minus 2 to 2...

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