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[Number 21]

Make Three solved problems each using Mutual Funds in Mathematics in Modern World

 

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Problem 1: Calculating Returns on Mutual Funds
Suppose you invest $5,000 in a mutual fund that has an average annual return of 7%. After 10 years, how much will your investment be worth?

Solution:
To solve this problem, we can use the compound interest formula:

A = P(1 + r/n)^(nt)

Where:
A = the final amount
P = the initial investment
r = the annual interest rate (expressed as a decimal)
n = the number of times that interest is compounded per year
t = the number of years

In this case, the initial investment is $5,000, the annual interest rate is 7% (or 0.07), the compounding is done annually (n = 1), and the investment period is 10 years (t = 10).

Plugging in the values, we get:

A = 5000(1 + 0.07/1)^(1*10)
A = 5000(1.07)^10
A ≈ 5000 * 1.96715
A ≈ $9,835.75

Therefore, after 10 years, your investment in the mutual fund will be worth approximately $9,835.75.

Problem 2: Evaluating Risk and Return of Mutual Funds
You are comparing two mutual funds, Fund A and Fund B. Fund A has an average annual return of 10% with a standard deviation of 5%, while Fund B has an average annual return of 8% with a standard deviation of 3%. Which fund would you consider a better investment based on the risk and return trade-off?

Solution:
To evaluate the risk and return trade-off, we need to consider both the average annual return and the standard deviation of each mutual fund.

Fund A:
Average Annual Return: 10%
Standard Deviation: 5%

Fund B:
Average Annual Return: 8%
Standard Deviation: 3%

Fund A has a higher average annual return (10% compared to 8%), indicating a potentially higher return on investment. However, Fund A also has a higher standard deviation (5% compared to 3%), indicating a higher level of risk or volatility.

To determine which fund is a better investment, you need to consider your risk tolerance. If you are comfortable with higher risk and volatility in exchange for potentially higher returns, Fund A may be a better choice. However, if you prefer lower risk and are willing to accept slightly lower returns, Fund B may be a more suitable option.

Problem 3: Assessing Diversification in Mutual Funds
You are building an investment portfolio and considering adding two mutual funds to diversify your holdings. Fund X has an average annual return of 12% and a standard deviation of 8%, while Fund Y has an average annual return of 10% and a standard deviation of 4%. How would including both funds in your portfolio affect your overall risk and return?

Solution:
To evaluate the impact of including both funds in your portfolio, you need to consider the risk and return characteristics of each fund as well as their correlations.

Fund X:
Average Annual Return: 12%
Standard Deviation: 8%

Fund Y:
Average Annual Return: 10%
Standard Deviation: 4%

By including both funds in your portfolio, you are introducing diversification. Diversification can help reduce the overall risk of your portfolio if the funds have a low correlation. The correlation coefficient ranges from -1 to +1, where -1 indicates a perfect negative correlation, 0 indicates no correlation, and +1 indicates a perfect positive correlation.

If Fund X and Fund Y have a low correlation (close to 0), adding both funds to your portfolio would help reduce the overall risk. However, if they have a high positive correlation, the risk reduction benefit may be limited.

To fully assess the impact on risk and return, you would need to consider the weightage or proportion of each fund in your portfolio and calculate the overall risk and return using portfolio theory or techniques like the mean-variance analysis.