Please could you help me solve this
Suppose you have $100 in an interest bearing account earning 10% interest. How much would you have after the first year under the following scenarios? .
annual compounding (every 12 months)- semi-annual compounding (every 6 months
quarterly compounding (every 3 months)
P1 = Po(1=r)^n.
Po = $100.
r = 10%/100% = 0.10.
n = 1Comp./yr. * 1yr. = 1 Compounding
period.
P1 = ?.
P2 = Po(1+r)^n.
Po = $.100
r = (10%/2)/100% = 0.05.
n = 2Comp./yr. * 1yr. = 2 Compounding periods.
P2 = ?.
P3 = Po(1+r)^n.
r = (10%/4)/100% = 0.025.
n = 4Comp./yr. * 1yr. = 4 Compounding periods.
P3 = ?.
Correction: P1 = Po(1+r)^n.
Sure, I can help you with that. To calculate the amount you would have after the first year under different compounding scenarios, we use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the amount you would have after the given time period
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
Let's calculate the amount under each of the given scenarios:
1. Annual Compounding (every 12 months):
In this scenario, interest is compounded only once at the end of the year.
Using the formula:
A = 100(1 + 0.10/1)^(1*1)
A = 100(1.10)
A = $110
Therefore, you would have $110 after the first year with annual compounding.
2. Semi-Annual Compounding (every 6 months):
In this scenario, interest is compounded twice a year.
Using the formula:
A = 100(1 + 0.10/2)^(2*1)
A = 100(1.05)^2
A = $110.25
Therefore, you would have $110.25 after the first year with semi-annual compounding.
3. Quarterly Compounding (every 3 months):
In this scenario, interest is compounded four times a year.
Using the formula:
A = 100(1 + 0.10/4)^(4*1)
A = 100(1.025)^4
A ≈ $110.38
Therefore, you would have approximately $110.38 after the first year with quarterly compounding.
So, the final amounts after one year are:
Annual compounding: $110
Semi-annual compounding: $110.25
Quarterly compounding: $110.38