find the accumulated value of an investment of 25,000 for 4 years at an interest rate of 5%, if the money is a compounded semiannually; b. coumpounded quaterly; c. compounded monthly d. coumpounded continuously.
I'll try the first one. The remaining cases are similar and serve as exercises for you (as they should).
Principal, P = 25000
Period, t = 6 months.
Time, T = 4 years.
Number of periods, n = 4/0.5=8
Annual interest rate, = 5%
Interest rate per period, i = 5%*0.5=2.5%
Future value (compound interest)
= P(1+i)^n
= 25000*(1+2.5%)^8
= 25000*1.2184
= $30,460.07
Thank you but I need answer for 4.5% interest rate
To find the accumulated value of an investment with different compounding frequencies, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A is the accumulated value
P is the principal amount (initial investment)
r is the annual interest rate
n is the number of times interest is compounded per year
t is the number of years
Let's calculate the accumulated value for each compounding frequency:
a. Compounded semiannually:
n = 2 (since it is compounded twice a year)
r = 0.05 (5% expressed as a decimal)
t = 4
Substituting the values in the formula, we get:
A = 25000(1 + 0.05/2)^(2 * 4)
A = 25000 * (1 + 0.025)^8
A = 25000 * (1.025)^8
A ≈ 25000 * 1.22020
A ≈ 30,505
b. Compounded quarterly:
n = 4 (compounded four times a year)
r = 0.05 (5% expressed as a decimal)
t = 4
Substituting the values in the formula, we get:
A = 25000(1 + 0.05/4)^(4 * 4)
A = 25000 * (1 + 0.0125)^16
A = 25000 * (1.0125)^16
A ≈ 25000 * 1.2190
A ≈ 30,475
c. Compounded monthly:
n = 12 (compounded twelve times a year)
r = 0.05 (5% expressed as a decimal)
t = 4
Substituting the values in the formula, we get:
A = 25000(1 + 0.05/12)^(12 * 4)
A = 25000 * (1 + 0.00417)^48
A = 25000 * (1.00417)^48
A ≈ 25000 * 1.2184
A ≈ 30,460
d. Compounded continuously:
For continuous compounding, we need to use the formula:
A = Pe^(rt)
Where:
e is Euler's number (approximately 2.71828)
r = 0.05 (5% expressed as a decimal)
t = 4
Substituting the values in the formula, we get:
A = 25000 * e^(0.05 * 4)
A = 25000 * e^(0.2)
A ≈ 25000 * 1.2214
A ≈ 30,535
Therefore, the accumulated values are:
a. Compounded semiannually: $30,505
b. Compounded quarterly: $30,475
c. Compounded monthly: $30,460
d. Compounded continuously: $30,535