Jimmy opens a savings account with a $200 deposit at the beginning of the month. The account earns 4.6% annual interest compounded monthly. At the beginning of each subsequent month, Jimmy deposits an additional $200. How much will the account be worth at the end of 12 years?
The standard formula
amount = paym( (1+i)^n - 1)/i
assumes that payments are made at the end of a payment periods, yours are made at the beginning of each period.
I will pretend we have 145 payments, but then subtract the last one
i = .046/12 = .003833... (I stored in my calculator's memory)
amount = 200( 1.003833...^145 - 1)/.0038333 - 200
= 38,489.12
or , take the present value
PV = 200 + 200(1 - 1.003833..^-143)/.0038333
= 22,185.33
now we have to "move this forward" 144 periods
= 22185.33(1.003833..)^144
= 38,489.13 just like above
To calculate the future value of Jimmy's savings account after 12 years, we can use the formula for compound interest:
Future Value (FV) = P(1 + r/n)^(nt)
Where:
- P is the principal amount (initial deposit)
- r is the annual interest rate (expressed as a decimal)
- n is the number of times the interest is compounded per year
- t is the number of years
Given:
- Initial deposit (P) = $200
- Annual interest rate (r) = 4.6% = 0.046 (expressed as a decimal)
- Interest compounded monthly, so n = 12 (number of times per year)
- Number of years (t) = 12
First, let's calculate the future value of Jimmy's initial deposit:
FV_initial_deposit = $200(1 + 0.046/12)^(12*12)
FV_initial_deposit ≈ $200(1 + 0.003833)^144
FV_initial_deposit ≈ $200(1.003833)^144
FV_initial_deposit ≈ $200(1.734224)
FV_initial_deposit ≈ $346.84 (rounded to the nearest cent)
Now, let's calculate the future value of the monthly deposits:
FV_monthly_deposits = $200[(1 + 0.046/12)^(12*12) - 1]/(0.046/12)
FV_monthly_deposits ≈ $200[(1.003833)^144 - 1]/(0.046/12)
FV_monthly_deposits ≈ $200[1.734224 - 1]/(0.046/12)
FV_monthly_deposits ≈ $200[0.734224]/(0.046/12)
FV_monthly_deposits ≈ $200[0.734224]/(0.003833)
FV_monthly_deposits ≈ $200[191.135155]
FV_monthly_deposits ≈ $38,227.03 (rounded to the nearest cent)
Finally, we can calculate the total future value by adding the future values of the initial deposit and the monthly deposits:
Total Future Value = FV_initial_deposit + FV_monthly_deposits
Total Future Value ≈ $346.84 + $38,227.03
Total Future Value ≈ $38,573.87
Therefore, at the end of 12 years, Jimmy's savings account will be worth approximately $38,573.87.
To find out how much the account will be worth at the end of 12 years, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value (the amount the account will be worth at the end)
P = the principal amount (the initial deposit)
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
In this case:
P = $200
r = 4.6% annual interest rate = 0.046 (in decimal form)
n = 12 (since interest is compounded monthly)
t = 12 years
First, let's calculate the future value at the end of each month for the first year:
For the first month:
P = $200
r = 0.046 / 12 (since interest is compounded monthly)
n = 12 (since interest is compounded monthly)
t = 1 month
Using the formula, we get:
A1 = 200(1 + 0.046/12)^(12*1)
A1 = 200(1 + 0.00383)^12
A1 ≈ $202.57
For the second month:
P = $200 + $200 (since Jimmy deposits an additional $200)
r = 0.046 / 12
n = 12
t = 2 months
Using the formula, we get:
A2 = (A1 + 200)(1 + 0.046/12)^(12*2)
A2 ≈ ($202.57 + $200)(1 + 0.00383)^24
A2 ≈ $406.05
By repeating this calculation for each subsequent month for the entire 12-year period and summing up the results, we can find the total future value of the account at the end of 12 years.