You invest $5,500 into a 4.5% one-year CD. Find the balance of the account after five
years if it's compounded:
a. Quarterly Equation:______________________ Balance: $_________________
b. Annually Equation:______________________ Balance: $_________________
c. Continuously Equation :___________________ Balance: $_______________
The only one you should have any difficulty with is the last one, the others are very
straightforward. You MUST know how to do these if you are studying this topic.
c. amount = 5500(e^(5*.045) = 5500(e^.225) = .... use your calculator
a. P = Po(1+r)^n.
Po = $5,500.
r = 0.045/4 = 0.01125 = Quarterly rate.
n = 4comp./yr. * 5yrs. = 20 Compounding periods.
b. P = Po(1+r)^n.
r = 0.045/yr.
n = 1Comp./yr * 5yrs. = 5 Compounding periods.
To find the balance of the account after five years with different compounding intervals, we'll use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the balance after t years
P = the principal amount (initial investment)
r = the annual interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years
For all calculations, the principal amount (P) is $5,500, the annual interest rate (r) is 4.5% (or 0.045 as a decimal), and the number of years (t) is 5.
a. Quarterly Compounding:
In this case, the interest is compounded four times per year (n = 4).
The equation becomes:
A = 5500(1 + 0.045/4)^(4*5)
Simplifying:
A = 5500(1.01125)^(20)
Calculating the power:
A ≈ 5500(1.24969)
A ≈ $6,873.30
Therefore, the balance of the account after five years with quarterly compounding is $6,873.30.
b. Annually Compounding:
In this case, the interest is compounded once per year (n = 1).
The equation becomes:
A = 5500(1 + 0.045/1)^(1*5)
Simplifying:
A = 5500(1.045)^(5)
Calculating the power:
A ≈ 5500(1.2214)
A ≈ $6,717.70
Therefore, the balance of the account after five years with annual compounding is $6,717.70.
c. Continuously Compounding:
The equation for continuous compounding is slightly different:
A = Pe^(rt)
Where:
A = the balance after t years
P = the principal amount (initial investment)
r = the annual interest rate (as a decimal)
t = the number of years
e = Euler's number, approximately 2.71828
The equation becomes:
A = 5500e^(0.045*5)
Calculating:
A ≈ 5500(1.63117)
A ≈ $8,971.44
Therefore, the balance of the account after five years with continuous compounding is $8,971.44.