An investment of $5,000 is made with three options:6% coumpounded annually, quarterly, and continuously.
1.) Create three functions
2.)What will be the value of each investment after 40 years?
3.)Which of the three investments will grow to a value of $30,000?
4.)What will be the value of each investment in 2024?
1.
a) amount = 5000(1.06)^n , n in years
b) amount = 5000(1.015)^(4n) , n in years
c) amount = 5000(e^(.06n) ) , n in years
2. number crunch, if n = 40
3. look at results of #2
4. depends on your "reference year"
If your "now" is 2012, then n = 2024-2012 = 12
For part 2, A(t)= $51,429
Q(t)=$54,142
C(t)= $55,116
For part 3, I meant when will each of the 3 investments grow to a value of $30,000. So, if I graphed the functions and look at the tables, would it help me determine the years?
To solve this problem, we need to use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = The final amount (value of the investment)
P = The principal amount (initial investment)
r = Annual interest rate (as a decimal)
n = Number of times the interest is compounded per year
t = Number of years
Let's break down the problem step by step:
1.) Creating Three Functions:
To calculate the final amount for each investment option, we can create three functions in a programming language, like Python. Here's an example:
```python
def compound_annually(principal, rate, time):
return principal * (1 + rate/100) ** time
def compound_quarterly(principal, rate, time):
return principal * (1 + rate/100/4) ** (4 * time)
def compound_continuously(principal, rate, time):
return principal * math.exp(rate/100 * time)
```
Here, `principal` refers to $5,000 (initial investment), `rate` refers to 6%, and `time` refers to 40 years.
2.) Value after 40 Years:
Using the created functions, we can calculate the value of each investment after 40 years:
```python
investment_annually = compound_annually(5000, 6, 40)
investment_quarterly = compound_quarterly(5000, 6, 40)
investment_continuously = compound_continuously(5000, 6, 40)
```
The value after 40 years for each investment option will be stored in the respective variables: `investment_annually`, `investment_quarterly`, `investment_continuously`.
3.) Investment Growing to $30,000:
We can use the compound interest formula in reverse to find how much time it will take for an investment to grow to $30,000. Again, using the created functions, we can solve for the time:
```python
time_to_reach_30000_annually = math.log(30000/5000, 1 + 6/100)
time_to_reach_30000_quarterly = math.log(30000/5000, 1 + 6/100/4)
time_to_reach_30000_continuously = math.log(30000/5000, math.exp(6/100))
```
The time it takes for each investment option to reach a value of $30,000 will be stored in the respective variables: `time_to_reach_30000_annually`, `time_to_reach_30000_quarterly`, `time_to_reach_30000_continuously`.
4.) Value in 2024:
To calculate the value of each investment in 2024, you need to find out how many years have elapsed from 2024 to the current year and use the created functions.
For example, if 2024 is 4 years from now:
```python
investment_in_2024_annually = compound_annually(5000, 6, 4)
investment_in_2024_quarterly = compound_quarterly(5000, 6, 4)
investment_in_2024_continuously = compound_continuously(5000, 6, 4)
```
The value of each investment option in 2024 will be stored in the respective variables: `investment_in_2024_annually`, `investment_in_2024_quarterly`, `investment_in_2024_continuously`.
Please note that the calculations provided are just examples. You can adjust the variables according to your specific requirements.