Normal distribution probability

Question

The annual returns on stocks and treasury bonds over the next 12 months are uncertain. Suppose that these returns can be described by normal distributions with stocks having a mean of 15% and a standard deviation of 20%, and bonds having a mean of 6% and a standard deviation of 9%. Which asset is more likely to have a negative return?

Answer 1

The objective of this exercise is to help you develop your ability to reason with probability distributions. You would like to get to the point where your reflex is to think through such problems this way:

"A negative return is anything less than 0%.

"For stocks, 0% is three-quarters of a standard deviation (i.e., three quarters of 20%) less than the mean (15%). Therefore we can think of the chance of a negative return as the area to the left of -3/4 under a standard bell curve.

"For bonds, 0% is only two-thirds of a standard deviation less than its mean (0% - 6% = -2/3 of 9%). The chance of a negative return is represented by the area to the left of -2/3 under the same bell curve.

"Because negative two-thirds (bonds) is greater than negative three-quarters (stocks), negative returns occupy more of the left tail of the bond return distribution. That makes a negative bond return more likely."

The whole point is to translate information about means and standard deviations into areas under a distribution function.

Notice that the only calculations required in this case are simple ones; with textbook problems like this, you can do them in your head. In "real world" problems you can usually still think a problem through by approximating the calculations. This gives you the ability to think on your feet, which is one key to mastering any subject.

Answer 2

Here is the R code to quickly solve this:

   > pnorm(0, mean=15, sd=20)
        [1] 0.2266274

   > pnorm(0, mean=6, sd=9) 
        [1] 0.2524925

So bonds will be more likely to have a negative return

Answer 3

Stock: P( (X-15)/20 < (0-15)/20 ) = P(Z<-3/4) = .2266, Bond: P( (Y-6)/9 < (0-6)/9 ) = P(Z<-2/3) = .2525

More likely the bond.

Note that the price is log-normally distributed.