Answer:
- second account makes more for the first 20 years; after 20.6 years, the first account makes more.
- first account: 10 yrs: $9743; 20 yrs: $21096; 50 yrs: $214,137; 85 yrs: $3,198,369
- second account: 10 yrs: $10802; 20 yrs: $21216; 50 yrs: $160,734; 85 yrs: $1,706,582
Explanation:
You want to compare the future values of two accounts after 10, 20, 50, and 85 years:
- $4500 at 7.75%, compounded monthly
- $5500 at 6.75%, compounded continuously
Compounded monthly
The formula for account value with compounding monthly is ...
A = P(1 +r/12)^(12t) . . . . . . principal P invested at rate r for t years
A = 4500(1 +0.075/12)^(12·t) . . . . . using given values for P and r
The first line of calculator output shown in the attachment gives the value after 10, 20, 50, and 85 years.
Compounded continuously
The formula for account value with continuous compounding is ...
A = P·e^(rt)
A = 5500·e^(0.0675t) . . . . . using given values for P and r
The second line of calculator output shown in the attachment gives the value after 10, 20, 50, and 85 years.
You will notice this (second) account has higher value up to 20 years, then lower value after that.
The account with the higher interest rate will make more money after the "break-even" period of about 20.6 years.
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Additional comment
We have rounded the account values to the nearest dollar. For the purpose of value comparison, three significant figures would be sufficient.
The second attachment shows the different account values over the long term. You can see the higher interest rate really makes a difference.
The third attachment shows the shorter-term account values, and the "break-even" time period of about 20.6 years, when the values in both accounts are about $22,063.