Q&A: The rates of return on the assets in a portfolio are normally distributed...

...with a mean of 20% and a standard deviation of 12%. What's the probability that the return on an asset falls between -3.52% and 50.96%?

CppQuant Answer

  • the z-value of -3.52%: z = (-3.52% - 20%)/12% = -1.96
  • the z-value of 50.96%: z = (50.96% - 20%)/12% = 2.58
  • The 95% confidence interval is -1.96 to 1.96. 99% is -2.58 to 2.58
  • Since a normal distribution is symmetrical, P(-1.96<=Z<=2.58) = 95%/2 + 99%/2 = 97%
    • Or with Excel: NORMSDIST(2.58) - NORMSDIST(-1.96) = 97%.

Bonus Points

  • z-score / z-value / z-statistic: There is unlimited number of normal distributions, each with a different mean or standard deviation. Therefore, it's impractical to provide a table of probabilities for each combination of mean and standard deviation. However, we can standardize the actual distribution for a normal random variable to a standard normal distribution with the formula: z = (X - m)/s, where x, m, s are a score, mean and standard deviation from the original distribution.
    • z distribution will only be a normal distribution if the original distribution (X) is normal.
  • The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

Category: Quantitative Analysis > Probability > Distribution

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