Q&A: An analyst wants to calculate a 95% confidence interval for...

...the projected mean return of a portfolio he manages (assuming the return is normal). His latest calculations show that this portfolio has a mean return of 8% and a variance of 36. What is the 95% confidence interval?

A: The 95% confidence interval for a normal variable X is given by: X-bar +- 1.96*s = X-bar +- 1.96*var^0.5 = 8 +- 1.96 *6 = [-3.76%, 19.76%] . That means that there is a 95% probability that the mean return lies in the range of -3.76% to 19.76%.

  • Confidence interval = Point Estimate +/- Reliability Factor * Standard Error, where Reliability factor is a number based on the sampling distribution of the point estimate and the degree of confidence (1 - alpha)
  • The reliability factor and the standard error, however, may vary depending on three factors:
    • Distribution of population: normal or non-normal.
    • Population variance: known or unknown.
    • Sample size: large or small.
  • For a normal variable X, the Reliability Factors are 1.645 for 90% confidence interval, 1.96 95%, and 2.58 99%.

Bonus Points

  • We can use the sample mean to estimate the population mean, and the sample standard deviation to estimate the population standard deviation. The sample mean and sample standard deviation are point estimates.
  • Confidence intervals use point estimates to make probability statements about the dispersion of the outcomes of a normal distribution. A confidence interval specifies the percentage of all observations that fall in a particular interval. ie. a 90% confidence interval means that 10% of the observations fall outside the 90% confidence interval, with 5% on each side.

Category: Quantitative Analysis > Probability

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