Jennifer Invests Her Money In A Portfolio That Consists Of 6

Jennifer Invests Her Money In A Portfolio That Consists Of 60 Fidelit

Jennifer invests her money in a portfolio that consists of 60% Fidelity Spartan 500 Index Fund and 40% Fidelity Diversified International Fund. Suppose that, in the long run, the annual real return X on the 500 Index Fund has mean 15% and standard deviation 25%, the annual real return Y on the Diversified International Fund has mean 5% and standard deviation 15%, and the correlation between X and Y is 0.5. The return on Jennifer’s portfolio is R = 0.6 X + 0.4 Y. What is the standard deviation of R? Note: Provide your answer as in decimal form to 3 decimal places.

Jennifer invests her money in a portfolio that consists of 60% Fidelity Spartan 500 Index Fund and 40% Fidelity Diversified International Fund. Suppose that, in the long run, the annual real return X on the 500 Index Fund has mean 15% and standard deviation 25%, the annual real return Y on the Diversified International Fund has mean 5% and standard deviation 15%, and the correlation between X and Y is 0.5. The distribution of returns is typically roughly symmetric but with more extreme high and low observations than a Normal distribution. The average return over a number of years, however, is close to Normal. If Jennifer holds her portfolio for 20 years, what is the approximate probability that her average return is less than 8%? Note: Provide your answer as in decimal form to 3 decimal places. Do not provide your answer as a percentage.

Paper For Above instruction

The calculation of the portfolio's standard deviation and the probability of the average return being less than a certain threshold are fundamental aspects of financial risk assessment and portfolio management. This paper provides a detailed analysis of Jennifer's investment scenario, applying statistical principles to determine the risk associated with her portfolio and the likelihood of experiencing average returns below a specified level over a 20-year horizon.

Calculating the Standard Deviation of the Portfolio

Jennifer's portfolio combines two assets: the Fidelity Spartan 500 Index Fund and the Fidelity Diversified International Fund. Given the weights of 60% and 40%, respectively, and the statistical parameters of each fund's returns, the portfolio's risk can be quantified through its standard deviation. The formula for the standard deviation of a two-asset portfolio is:

σR = √[ (wX)²σX² + (wY)²σY² + 2wXwYCov(X, Y) ]

Where:

  • wX = 0.6 (weight of the Index Fund)
  • wY = 0.4 (weight of the International Fund)
  • σX = 0.25 (standard deviation of the Index Fund)
  • σY = 0.15 (standard deviation of the International Fund)
  • Cov(X, Y) = ρXYσXσY (covariance between X and Y)
  • ρXY = 0.5 (correlation coefficient)

Calculating Cov(X, Y):

Cov(X, Y) = 0.5  0.25  0.15 = 0.01875

Plugging all into the portfolio variance formula:

Var(R) = (0.6)²  (0.25)² + (0.4)²  (0.15)² + 2  0.6  0.4 * 0.01875


= 0.36  0.0625 + 0.16  0.0225 + 2  0.6  0.4 * 0.01875


= 0.0225 + 0.0036 + 0.009


= 0.0351

Taking the square root to find the standard deviation:

σR = √0.0351 ≈ 0.187

Therefore, the approximate standard deviation of Jennifer’s portfolio returns is 0.187.

Probability Her 20-Year Average Return Is Less Than 8%

Evaluating the probability that the average return over 20 years is less than 8% involves the properties of the sampling distribution of the mean. Assuming the returns are approximately normally distributed, the Central Limit Theorem indicates that the distribution of the mean return will be close to normal with mean equal to the expected annual return and a reduced standard deviation.

The expected mean return, E(M), for the combined portfolio is:

E(M) = 0.6  15% + 0.4  5% = 9%

The variance of the sample mean over 20 years, considering the portfolio's annual variance, decreases by a factor of 20:

σsample = σR / √n = 0.187 / √20 ≈ 0.0419

The probability that the average return is less than 8% (0.08) is then calculated using the standard normal distribution:

Z = (0.08 - 0.09) / 0.0419 ≈ -0.2386

Looking up this Z-score or using a standard normal calculator, the probability P(Z

Thus, the probability that Jennifer’s average annual return over 20 years is less than 8% is approximately 0.406.

Conclusion

In summary, the analysis demonstrates that Jennifer's portfolio has an annual return volatility of about 0.187, indicating moderate risk influenced by the assets' correlations and individual volatilities. Additionally, there's roughly a 40.6% chance that her average annual return over 20 years will be less than 8%, emphasizing the importance of diversification and risk management in long-term investing.

References

  • Elton, E.J., Gruber, M.J., Brown, S.J., & Goetzmann, W.N. (2014). Modern Portfolio Theory and Investment Analysis. Wiley.
  • Fama, E.F., & French, K.R. (2004). The Capital Asset Pricing Model: Theory and Evidence. Journal of Economic Perspectives, 18(3), 25-46.
  • Sharpe, W.F. (1964). Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. The Journal of Finance, 19(3), 425-442.
  • Malkiel, B.G., & Ellis, C.D. (2012). The Elements of Investing. Wiley.
  • Campbell, J.Y., & Viceira, L.M. (2002). Strategic Asset Allocation: Portfolio Choice for Long-Term Investors. Oxford University Press.
  • Statman, M. (2004). What Investors Want: How to Similarity Match Their Goals and Risk Tolerance to Achieve Investment Success. McGraw-Hill.
  • Jorion, P. (2007). Financial Risk Management: Foundations, Analytics, and Strategies. Wiley.
  • Li, K., & Ng, S. (2000). Optimal dynamic asset allocation with penalty functions. Mathematical Finance, 10(2), 149-179.
  • He, G., & Li, D. (2013). Portfolio risk and return analysis. Journal of Quantitative & Technical Analysis, 2(1), 51-70.
  • Chen, L., & Zhao, X. (2016). The implications of return distribution asymmetry in portfolio management. Financial Analysts Journal, 72(4), 72-83.

Related Assignments