Jane is having difficulty deciding whether to put her savings in the Mystic Bank or in the Four Rivers Bank. Mystic offers a 12% rate compounded quarterly, and Four Rivers offers 14% compounded semiannually. Jane has $40,000 to invest and expects to withdraw the money at the end of five years. Using the tables in the Business Math Handbook that accompanies the course textbook, determine which one of the following is the best deal.
A. Mystic for last two years
B. Mystic
C. Four Rivers
D. Four Rivers for first two years
Answer: A??
I don't have your handbook, but
all you have to do is to compare the equivalent annual rates of the two accounts.
No tables are needed
annual rate for 12% per annum compounded quarterly
= (1+.12/4)^4 = 1.1255% represents 12.55%
annual rate for 14% compounded semi-annually
= (1 + .14/2)^2 = 1.1449 represents 14.49%
so clearly C
choice A and D are mere distractors.
To determine the best deal, we need to calculate the future value of the investment at each bank and compare the results.
For Mystic Bank:
- Interest rate: 12% compounded quarterly
- Time: 5 years
Using the formula for compound interest, the future value (FV) can be calculated as:
FV = P(1 + r/n)^(nt)
Where:
P = principal amount (initial investment) = $40,000
r = annual interest rate (12%) = 0.12
n = number of times interest is compounded per year (quarterly compounded = 4)
t = number of years (5 years)
FV for Mystic Bank:
FV = $40,000(1 + 0.12/4)^(4*5)
FV = $40,000(1.03)^20
FV = $40,000(1.806111)
FV = $72,244.44
For Four Rivers Bank:
- Interest rate: 14% compounded semiannually
- Time: 5 years
Using the same formula as before, the future value (FV) for Four Rivers Bank can be calculated as:
FV = P(1 + r/n)^(nt)
Where:
P = principal amount (initial investment) = $40,000
r = annual interest rate (14%) = 0.14
n = number of times interest is compounded per year (semiannually compounded = 2)
t = number of years (5 years)
FV for Four Rivers Bank:
FV = $40,000(1 + 0.14/2)^(2*5)
FV = $40,000(1.07)^10
FV = $40,000(1.967151)
FV = $78,686.06
Now, let's consider the options mentioned:
A. Mystic for the last two years:
To calculate the future value for this option, we will need to calculate the interest earned from Mystic Bank in the first three years and then add it to the future value at the end of the five years.
Interest earned from Mystic Bank in the first three years:
FV = $40,000(1 + 0.12/4)^(4*3)
FV = $40,000(1.03)^12
FV = $40,000(1.425970)
FV = $57,038.82
Now, let's add this value to the future value of Mystic Bank from above (after five years):
FV = $57,038.82 + $72,244.44
FV = $129,283.26
B. Mystic:
We have already calculated the future value for Mystic Bank, which is $72,244.44.
C. Four Rivers:
We have already calculated the future value for Four Rivers Bank, which is $78,686.06.
D. Four Rivers for the first two years:
To calculate the future value for this option, we will need to calculate the interest earned from Four Rivers Bank in the last three years and then add it to the future value at the end of the five years.
Interest earned from Four Rivers Bank in the last three years:
FV = $40,000(1 + 0.14/2)^(2*3)
FV = $40,000(1.07)^6
FV = $40,000(1.501229)
FV = $60,049.16
Now, let's add this value to the future value of Four Rivers Bank from above (after five years):
FV = $60,049.16 + $78,686.06
FV = $138,735.22
So, the best deal among the given options is D. Four Rivers for the first two years, with a future value of $138,735.22.
To determine which bank offers the best deal, we need to compare the amount of money Jane would have at the end of five years if she were to invest in Mystic Bank, Four Rivers Bank, or a combination of the two.
Using the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the initial principal amount ($40,000)
r = annual interest rate (12% for Mystic, 14% for Four Rivers)
n = number of times interest is compounded per year (4 times for Mystic, 2 times for Four Rivers)
t = number of years (5 years)
For Mystic Bank:
A1 = $40,000(1 + 0.12/4)^(4 * 5)
A1 ≈ $40,000(1 + 0.03)^20
A1 ≈ $40,000(1.03)^20
A1 ≈ $40,000(1.806111)
A1 ≈ $72,244.44
For Four Rivers Bank:
A2 = $40,000(1 + 0.14/2)^(2 * 5)
A2 ≈ $40,000(1 + 0.07)^10
A2 ≈ $40,000(1.07)^10
A2 ≈ $40,000(1.967151)
A2 ≈ $78,686.04
Option A suggests investing in Mystic Bank for the last two years. To determine the value of this investment, we first need to find the future value of a $40,000 investment in Mystic Bank for the initial three years.
A3 = $40,000(1 + 0.12/4)^(4 * 3)
A3 ≈ $40,000(1 + 0.03)^12
A3 ≈ $40,000(1.03)^12
A3 ≈ $40,000(1.430467)
A3 ≈ $57,218.68
Then, we calculate the future value of this amount after two more years:
A = $57,218.68(1 + 0.12/4)^(4 * 2)
A ≈ $57,218.68(1 + 0.03)^8
A ≈ $57,218.68(1.03)^8
A ≈ $57,218.68(1.264728)
A ≈ $72,377.82
Therefore, the amount of money Jane would have at the end of five years if she chose option A is $72,377.82.
Comparing the amounts:
Option B (Investing in Mystic Bank): $72,244.44
Option C (Investing in Four Rivers Bank): $78,686.04
Option D (Investing in Four Rivers Bank for the first two years): Not calculated yet
Option A (Investing in Mystic Bank for the last two years): $72,377.82
From the calculations, we can see that option C (Investing in Four Rivers Bank) offers the highest amount of money at the end of five years, making it the best deal.