if you wanted to save $ 50,000 to have it available in 20years provided the investment would return an APR of 12% compounded monthly.
a. how much would you need to have at the beggining (lump sum) in order to achieve the mark of $50,000.
b. how much would you have to save every year for 20 years?
c. how much would you have to save for the first 10years and nothing more?
For (a), solve the compound growth formula
X*(1.12)^20 = 50,000
X wil be the answer
For (b), use the calculation tool at
http://www.collegegold.com/calculatecost/savingsgrowthprojector
I get 619.59 per year
For (c), first calculate how much you need after 10 years to reach 50,000 with compound interest, without contributing for the last 10 years. Then use the computation tool of (b) to determine how much you must save per year for the first 10 years
a) amount = princ(1+i)^n
50000 = princ(1.01)^240
princ = 4590.29
b) amount = payment((1 + i)^n -1)/i
50000 = payment(1.01^240 - 1)/.01
payment = 50.54 per month.
c) so you are depositing every month for 10 years, then letting it "ride" for the next 10 years ?
50000 = [payment(1.01^120 - 1)/.01](1.01^120
50000 = payment(230.0386895)(3.300386895)
payment = 65.56
I didn't read the problem carefully, and assumed interest compounded annually.
Kudos to Reiny for doing it right, and knowing the formula for (b).
To find the solutions to these questions, we will use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
a. To find how much you need to have at the beginning (lump sum) in order to achieve $50,000 in 20 years with an APR of 12% compounded monthly, we need to solve for P in the compound interest formula.
A = P(1 + r/n)^(nt)
Plugging in the given values:
A = $50,000
r = 12% (0.12 as a decimal)
n = 12 (compounded monthly)
t = 20 years
$50,000 = P(1 + 0.12/12)^(12*20)
Simplifying further:
$50,000 = P(1.01)^(240)
To find P, divide both sides of the equation by (1.01)^240:
P = $50,000 / (1.01)^240
Using a calculator, the approximate value of P is $7,376.41.
Therefore, you would need to have approximately $7,376.41 at the beginning to achieve a final value of $50,000 in 20 years.
b. To calculate how much you would have to save every year for 20 years, we can rearrange the compound interest formula to solve for P:
A = P(1 + r/n)^(nt)
P = A / (1 + r/n)^(nt)
Since A is the target value of $50,000, and we now know P is $7,376.41, we can find how much you have to save (S) every year.
$7,376.41 = S * ((1 + 0.12/12)^(12*20) - 1) / (0.12/12)
Simplifying:
$7,376.41 = S * ((1.01)^240 - 1) / (0.01)
Rearranging and solving for S:
S = $7,376.41 * (0.12 / (1.01)^240 - 1)
Using a calculator, the approximate value of S is $387.40.
Therefore, you would need to save approximately $387.40 every year for 20 years in order to reach a total of $50,000.
c. To calculate how much you need to save for the first 10 years and nothing more, we will use the formula for the sum of a geometric series:
S = a * (1 - r^n) / (1 - r)
Where:
S = sum of the series (amount saved in the first 10 years)
a = first term (amount saved in the first year)
r = common ratio (equal to (1 + r/n))
Using the formula, we can calculate the amount saved in the first 10 years:
S = a * (1 - r^n) / (1 - r)
S = $387.40 * (1 - (1 + 0.12/12)^(12 * 10)) / (1 - (1 + 0.12/12))
Simplifying:
S = $387.40 * (1 - 1.01^120) / (1 - 1.01)
Using a calculator, the approximate value of S is $2,583.05.
Therefore, you would need to save approximately $2,583.05 for the first 10 years and nothing more to achieve your goal of $50,000 in 20 years.