Tamra is having difficulty deciding whether to put her savings in the Albina Community Bank or in the Umpqua Bank. Albina offers a 12% rate compounded quarterly, and Umpqua offers 14% compounded semiannually. Tamra has $40,000 to invest and expects to withdraw the money at the end of five years. Using the tables found in your textbook, determine which one of the following is the best deal.
a. albina for last two years
b. umpqua for first two years
c. umpqua
d. albina
my answer is C.
so, did you consult your tables? The final balances will be
albina: (1 + .12/4)^(4*5)
umpqua: (1 + .14/2)^(2*5)
for the last two years, at rate r, compounded n times/year, the growth is
40000((1+r/n)^(5n) - (1 + r/n)^(3n))
That is, subtract the balance at the end of 3 years from the final balance after 5 years. Just plug in the rates etc. as needed.
[1 + (.12 / 4)]^20 = 1.806
[1 + (.14 / 2)]^10 = 1.967
To determine which option is the best deal, we need to calculate the future value of the investment for both banks.
For Albina Community Bank:
Rate = 12%
Compounding Period = Quarterly
Principal = $40,000
Time = 5 years
Using the formula for compound interest, we can calculate the future value for Albina:
Future Value = Principal * (1 + (Rate / Compounding Period))^(Compounding Period * Time)
Future Value = $40,000 * (1 + (0.12 / 4))^(4 * 5)
Future Value = $40,000 * (1 + 0.03)^20
Future Value = $40,000 * (1.03)^20
Future Value = $40,000 * 1.80611194
Future Value = $72,244.48
For Umpqua Bank:
Rate = 14%
Compounding Period = Semiannually
Principal = $40,000
Time = 5 years
Calculating the future value for Umpqua using the same formula:
Future Value = Principal * (1 + (Rate / Compounding Period))^(Compounding Period * Time)
Future Value = $40,000 * (1 + (0.14 / 2))^(2 * 5)
Future Value = $40,000 * (1 + 0.07)^10
Future Value = $40,000 * (1.07)^10
Future Value = $40,000 * 1.967151
Future Value = $78,686.04
Comparing the future values, we can see that Umpqua Bank offers a higher return on investment. Therefore, the best deal for Tamra would be to invest in Umpqua Bank, which is option C.
To determine which option is the best deal, we need to compare the future values of the savings at the end of five years for both Albina Community Bank and Umpqua Bank.
For Albina Community Bank:
Interest rate (r) = 12% = 0.12
Compounding period (n) = quarterly = 4 times a year
Principal amount (P) = $40,000
Using the compound interest formula:
Future value (FV) = P * (1 + r/n)^(n*t)
Where t is the number of years
Plugging in the values for Albina Community Bank:
FV = $40,000 * (1 + 0.12/4)^(4*5) = $40,000 * (1 + 0.03)^20 = $40,000 * (1.03)^20 ≈ $79,170.64
For Umpqua Bank:
Interest rate (r) = 14% = 0.14
Compounding period (n) = semiannually = 2 times a year
Principal amount (P) = $40,000
Plugging in the values for Umpqua Bank:
FV = $40,000 * (1 + 0.14/2)^(2*5) = $40,000 * (1 + 0.07)^10 = $40,000 * (1.07)^10 ≈ $83,987.98
Comparing the future values:
- The future value for Albina Community Bank is approximately $79,170.64.
- The future value for Umpqua Bank is approximately $83,987.98.
Therefore, the best deal for Tamra's savings is option c. Umpqua Bank, as it offers the highest future value at the end of five years.