Bill Is Considering Investing 120,000 Among Two Investments

Bill Is Considering Investing 120000 Among Two Investments He Will

Bill is contemplating an investment of $120,000 allocated between two distinct investment options. The first investment involves placing $50,000 into a security whose annual rate of return follows a uniform distribution ranging from -10% to 25%. The second investment involves investing $70,000 into a financial instrument whose annual return follows a normal distribution with an average of 10% and a standard deviation of 3%. This assignment requires simulating the annual return rates over three years for both investments using given random numbers, and then calculating the cumulative balances over the period. The goal is to analyze how different probabilistic models influence the investment outcomes over time, thus providing insight into risk and return profiles.

Paper For Above instruction

Investment decisions are inherently probabilistic due to the uncertainty of financial markets. When managing a portfolio, investors must consider various stochastic factors that influence returns, which in turn affect the overall profitability and risk levels of their investments. In the case of Bill's scenario, understanding the distribution characteristics of each investment type is crucial, as it guides the simulation of future outcomes and informs risk management strategies.

Simulation of Investment Returns Using Probabilistic Models

To begin, it is essential to comprehend the distribution types governing each investment. The first investment operates under a uniform distribution, which implies that each return within the specified range is equally likely. Specifically, the rate of return varies from -10% (a loss) to 25% (a gain). This uniform model simplifies the simulation process as it allows the use of straightforward random sampling within the specified bounds. Conversely, the second investment follows a normal distribution, characterized by a bell-shaped curve centered around an average return of 10% with a standard deviation of 3%. This reflects more realistic financial return behaviors, where most outcomes cluster around the mean with fewer extreme results.

Using the provided random numbers, the simulation for three years involves generating return rates for each investment and updating the balances cumulatively. For Investment 1, the uniform distribution-based return rate can be calculated by scaling the random number to the range from -10% to 25%. Specifically, a random number (RN) between 0 and 1 maps linearly to this range, computed as:

Rate of Return = -10% + RN (25% - (-10%)) = -10% + RN 35%

For example, if RN = 0.05, the corresponding return is:

-10% + 0.05 * 35% = -10% + 1.75% = -8.25%

This process repeats for each year. For Investment 2, with a normal distribution, the return rate is simulated by applying the inverse transform sampling method, which might involve applying the z-score corresponding to the random number and then scaling by the standard deviation to compute the return:

Return Rate = Mean + Z * Standard Deviation

where Z is the z-score associated with the random number, derived from the inverse cumulative distribution function (CDF) of the standard normal distribution. For simplicity, standard software functions or tables can be used to obtain Z for the provided random numbers.

Simulation Procedure and Calculations

Assuming the random numbers provided are 0.05 for both investments in the first year, the simulation proceeds as follows:

Year 1

• Investment 1: Using RN = 0.05, Rate = -10% + 0.05 35% = -8.25%. Balance = $50,000 (1 - 0.0825) = $45,875.
• Investment 2: Using RN = 0.05, determine Z-score from the inverse normal CDF. For example, Z ≈ -1.64, then Return = 10% + (-1.64)3% ≈ 10% - 4.92% ≈ 5.08%. Balance = $70,000 (1 + 0.0508) ≈ $73,555.60.

And so on for subsequent years, updating the balances cumulatively, considering the simulated return rates each year. The cumulative balances are then recorded after each year, illustrating investment performance over time.

Implications and Analysis of Results

Simulating the investment returns with probabilistic models demonstrates the potential variability in outcomes. The uniform distribution’s symmetric nature around the mean indicates a relatively higher probability of encountering extreme negative or positive returns, highlighting the risk inherent in the first investment. Conversely, the normal distribution model suggests that most returns cluster near the mean, with fewer extreme deviations, reflecting a more probabilistic stability in the second investment.

Over the three-year period, the cumulative effect of these return rates will show the growth or decline of each investment, illustrating how risk and return profiles influence overall portfolio performance. Such simulations are vital tools for financial planning, risk assessment, and strategic diversification.

Conclusion

In conclusion, the simulation exercise underscores the importance of understanding the statistical distribution of investment returns in decision-making. The use of uniform and normal models provides insights into the range and likelihood of possible outcomes, helping investors like Bill to evaluate risk and optimize their investments accordingly. Incorporating probabilistic models into investment analysis is essential for realistic planning and managing uncertainty in financial portfolios.

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