Suppose The Returns On An Asset Are Normally Distributed

Question

Suppose The Returns On An Asset Are Normally Distributed Suppose The Suppose the returns on an asset are normally distributed. The historical average annual return for the asset was 6.4 percent, and the standard deviation was 12.4 percent. To answer the following questions, we will utilize the properties of the normal distribution and Excel's NORMDIST function. First, we will determine the probability that the return in a given year will be less than –5.3 percent. Then, we will identify the return ranges that cover 95 percent and 99 percent of the outcomes respectively. Using the given data: - Mean (μ) = 6.4% - Standard deviation (σ) = 12.4% Probability that the return is less than –5.3% Expressing the return in decimal form: - μ = 0.064 - σ = 0.124 - Value for probability calculation = –5.3% = –0.053 Calculating the z-score: $$z = \frac{X - \mu}{\sigma} = \frac{-0.053 - 0.064}{0.124} \approx \frac{-0.117}{0.124} \approx -0.9435$$ Using Excel's NORMDIST function: which yields approximately 0.1722 or 17.22%. Therefore, the probability that the return will be less than –5.3% in a given year is approximately 17.22%. --- Range of returns for a 95% confidence interval For a normal distribution, approximately 95% of the outcomes lie within ±1.96 standard deviations from the mean. Calculating the range: - Lower bound: $$ \mu - 1.96\sigma = 0.064 - 1.96 \times 0.124 \approx 0.064 - 0.243 \approx -0.179 $$ - Upper bound: $$ \mu + 1.96\sigma = 0.064 + 0.243 \approx 0.307 $$ Expressed as percentages: - Lower bound = –17.90% - Upper bound = +30.65% The expected range of returns 95% of the time is from approximately –17.90% to +30.65%. --- Range of returns for a 99% confidence interval Similarly, for 99% confidence, the critical z-value is approximately 2.576. Calculating the range: - Lower bound: $$ 0.064 - 2.576 \times 0.124 \approx 0.064 - 0.319 \approx -0.255 $$ - Upper bound: $$ 0.064 + 2.576 \times 0.124 \approx 0.064 + 0.319 \approx 0.383 $$ Expressed as percentages: - Lo

Dr. Jack HW Helper · Accepted Answer

The analysis of asset returns assuming a normal distribution provides valuable insights into the likelihood of various outcomes over a given period. With a mean return of 6.4% and a standard deviation of 12.4%, investors can gauge the probability of experiencing negative returns, as well as the range within which most returns are likely to fall. Calculating the probability that returns fall below –5.3% involves standardizing this value to a z-score, which is approximately –0.9435. Using Excel’s NORMDIST function, this corresponds to a probability of roughly 17.22%. This indicates a moderate likelihood of experiencing a return less than –5.3% in any given year, highlighting the inherent risk in asset investments. Understanding the returns within a confidence interval helps investors anticipate the range of outcomes. For a 95% confidence level, the return range approximately spans from –17.90% to +30.65%. This suggests that in 95 out of 100 years, returns are expected to fall within this range. For a more conservative estimate, the 99% confidence interval extends from about –25.50% to +38.30%, implying greater uncertainty but also accounting for more extreme outcomes. There are practical implications for portfolio management. Investors seeking stability tend to focus on the narrower 95% interval, while those comfortable with higher risk may consider the broader 99% range. These intervals also assist in setting realistic expectations and aligning portfolios with risk tolerance levels. References Fabozzi, F. J. (2014). Bond Markets, Analysis, and Strategies. Pearson Education. Hull, J. C. (2018). Options, Futures, and Other Derivatives. Pearson Education. Sharma, S., & Kannan, R. (2018). "Measuring financial risk and return: An application of the normal distribution." International Journal of Financial Studies, 6(3), 76. Mandelbrot, B. (1963). "The variation of certain speculative prices." Journal of Business, 36(4), 394-419. Taleb, N. N. (2007). The Black Swan: The

Suppose The Returns On An Asset Are Normally Distributed

Suppose The Returns On An Asset Are Normally Distributed Suppose The

Probability that the return is less than –5.3%

Range of returns for a 95% confidence interval

Range of returns for a 99% confidence interval

Summary

Paper For Above instruction

References