SOLUTION: 5. A mutual fund is an investment company that distributes shares of its own stock continuously and is obligated to repurchase the stock from the shareholders upon request. A mutu

Question 1115943: 5. A mutual fund is an investment company that distributes shares of its own stock continuously and is obligated to repurchase the stock from the shareholders upon request. A mutual fund owns the securities of several corporations and receives dividends on the shares that it holds. These cash dividends are then paid to its shareholders, who in turn may reinvest the dividends automatically in additional shares. In mutual funds, it is hoped that a loss in one holding will be compensated for by a gain in another holding. One important advantage of these funds is that they usually choose stocks or bonds that are most likely to yield profits, reducing your chances of losing money.
A popular type of mutual fund is a money market fund, which invests only in high-yielding government and corporate short-term securities. These types of funds provide an alternative to a savings account in a bank because they usually have higher interest rates. A certificate of deposit, commonly known as a CD, is a type of money-market fund.
The mutual fund actually originated in Europe in the early part of the nineteenth century and did not become popular in the United States until the 1920s. Mutual funds experienced a period of tremendous growth after World War II.
Suppose you began making annual investments several years ago in a mutual fund in order to save for retirement. You initially invested $1000 and increased this amount by 3% per year as your salary rose. Shares of the fund initially cost $50 each and have been increasing in price by 12% each year.
a) Write a rule for the amount invested in year n.
b) Write a rule for the share price in year n at the time you invested.
c) Write a rule for the number of shares bought in year n.
d) Even though you have been investing larger amounts of money each year, your recent annual statements from your mutual fund company show that the number of shares you own has barely increased at all. Has the company made an error? Justify your answer mathematically.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
your investment in the shares in year 1 is equal to 1000.

the formula for your investment in the shares in year n is equal to 1000 * (1.03) ^ (n-1)

this is because you are increasing the amount you invest each year by 3% per year.

in year 1, it is 1000 * 1.03 ^ 0 = 1000

in year 2, it is 1000 * 1.03 ^ 1 = 1030

in year 3, it is 1000 * 1.03 ^ 2 = 1060.9

etc.....

this can be seen in column C of the excel spreadsheet printout.

the shares cost 50 dollars each in year 1.
they increase in value by 12% a year.

the formula for the price of each share in year n is 50 * (1.12) ^ (n-1)

in year 1, they cost 50 * 1.12 ^ 0 = 50

in year 2, they cost 50 * 1.12 ^ 1 = 56

in year 3, they cost 50 * 1.12 ^ 2 = 62.72

etc.....

this can be seen in column D of the excel spreadsheet printout.

the formula for the number of shares bought in year n is (1000 * (1.03) ^ (n-1)) / (50 * (1.12) ^ (n-1))

the number of shares bought in year 1 is (1000 * 1.03 ^ 0) / (50 * 1.12 ^ 0) = 20
the number of shares bought in year 2 is (1000 * 1.03 ^ 1) / (50 * 1.12 ^ 1) = 18.39285714.

the number of shares bought in year 3 is (1000 * 1.03 ^ 2) / (50 * 1.12 ^ 2) = 16.91485969.

etc.....

this can be seen in column E of the excel spreadsheet printout.

the number of shares bought each year is being reduced because your investment is increasing 3% per year and the price of each share has been increasing 12% per year.

the number of shares you buy each year is therefore equal to (3000 * 1.03 ^ (n-1)) / (50 * 1.12 ^ (n-1)

this formula can be shown as (3000 / 50) * (1.03 ^(n-1)) / 1.12 ^ (n-1)).

this can also be shown as (3000 / 50) * (1.03/1.12) ^ (n-1)

simplify this to get the number of shares that can be bought in year n is 20 * .9196428571 ^ (n-1)

for example, the number of shares that can be bought in year 9 is equal to 50 * .9196428571 ^ (8) which is equal to 10.23254379.

look at cell E11 in the excel spreadsheet printout and you will see that the number of shares bought in year 9 is equal to 10.23254379.

cell E11 is the cell that is in column E row 11.

as you can see in the excel spreadsheet printout, the number of shares bought in each succeeding year is less even though the amount of money invested in each year is more because the price of each share is increasing at a greater rate than the amount being invested.

here's the excel spreadsheet printout.

$$$

here's a reference that can help you understand how mutual shares work.

https://www.thebalance.com/buying-mutual-funds-is-different-than-buying-stocks-3140909

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