Chebyshev's Theorem: Analyzing Investment Portfolio Risk

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In summary, the conversation discusses the comparison of two investment portfolios, one consisting of all stocks and the other consisting of 60% stocks and 40% bonds. The data for the past several years shows the annual returns for both portfolios. The conversation also mentions the use of Ex, Ex2, Ey, and Ey2 to compute the sample mean, variance, and standard deviation for both portfolios. Finally, the concept of a 75% Chebyshev interval is brought up to compare the two portfolios.
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cyod22
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1. do bonds reduce the overall risk of an investment portfolio? let x be a random variable representing annual % return for Vangaurd Total Stock Index (all stocks). Let y be a random variable representing annual return for Vangaurd Balanced Index(60% stock and 40% bond). For the past several years we have the following data.

x: 11 0 36 21 31 23 24 -11 -11 -21
y: 10 -2 29 14 22 18 14 -2 -3 -10

a.) Compare Ex, Ex2, Ey and Ey2 (2 = squared)

b.) use results in part (a) to compute the sample mean, variance, and standard deviation for x and for y.

c.) Compute a 75 % Chebyshev interval about the mean for x values and also for y values. Use interval to compare funds.

I was able to do part a & b but have no idea what they want for c. i do have the answer but i am not sure what they used to get the answer.



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What is a Chebyshev interval?
 
  • #3
There are two possibilities - I'm not sure what level of work you're at.

First (and simplest): how many standard deviations around the mean does chebyshev's theorem say you must go to include 75% of the data values? (remember chebychev's theorem says the percentage of values between [tex] \bar x \pm ks [/tex] is at least
[tex] 1 - {1}/{k^2}[/tex].

Second (and more complicated) is the idea discussed at the following link:
http://www.quantdec.com/envstats/notes/class_12/ucl.htm

I'm guessing it is option 1 you need to use.
 

1. What is Chebyshev's Theorem and how is it used in analyzing investment portfolio risk?

Chebyshev's Theorem, also known as the Chebyshev Inequality, is a mathematical concept that allows us to make general statements about the spread of data around the mean. In the context of investment portfolio risk, it is used to understand the probability of a certain percentage of the portfolio's returns falling within a certain number of standard deviations from the mean. This helps investors evaluate the potential risk of their portfolio.

2. How does Chebyshev's Theorem differ from the Empirical Rule in analyzing investment risk?

Chebyshev's Theorem is a more general concept than the Empirical Rule, which is a specific case of the theorem that only applies to data sets that follow a normal distribution. In contrast, Chebyshev's Theorem can be applied to any data set, regardless of its distribution, making it a more versatile tool for analyzing investment risk.

3. Can Chebyshev's Theorem be used to make accurate predictions about investment returns?

No, Chebyshev's Theorem does not make predictions about specific investment returns. It only provides a general understanding of the spread and probability of returns, based on the mean and standard deviation of the data set.

4. How can investors use Chebyshev's Theorem to make informed decisions about their investment portfolios?

Investors can use Chebyshev's Theorem to set realistic expectations for their portfolio returns and understand the potential risk involved. By knowing the percentage of returns that are likely to fall within a certain range, investors can make more informed decisions about diversifying their portfolio and managing risk.

5. Are there any limitations to using Chebyshev's Theorem in investment analysis?

Yes, Chebyshev's Theorem assumes that the data set is complete and accurate, and that the mean and standard deviation are representative of the entire population. In reality, these assumptions may not always hold true, which can affect the accuracy of the results. Additionally, the theorem does not take into account any external factors that may impact investment returns, such as economic conditions or company-specific events.

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