Two partners agree to invest equal amounts in their business. One will contribute $10,000 immediately. The other plans to contribute an equivalent amount in 8 years. How much should she contribute at that time to match her partner's investment now, assuming an interest rate of 4% compounded quarterly?
To solve this problem, we need to use the formula for future value of a lump sum:
FV = PV x (1 + r/n)^(n*t)
Where:
FV = future value
PV = present value
r = interest rate
n = number of compounding periods per year
t = number of years
First, we need to find out how much the $10,000 investment will be worth in 8 years at a 4% quarterly interest rate. We can plug in the values:
FV = 10,000 x (1 + 0.04/4)^(4*8) = 10,000 x 1.3659 = $13,659
So the partner who plans to invest in 8 years needs to contribute $13,659 to match her partner's investment now.
To find out how much the second partner should contribute in 8 years to match the first partner's investment now, we can use the concept of compound interest.
First, let's calculate the future value of the first partner's investment of $10,000 over 8 years using quarterly compounding at an interest rate of 4%.
To calculate the future value, we can use the formula:
FV = PV * (1 + r/n)^(n*t)
Where:
FV = future value
PV = present value
r = interest rate
n = number of compounding periods per year
t = number of years
Let's plug in the values:
PV = $10,000
r = 4% (or 0.04 as a decimal)
n = 4 (quarterly compounding)
t = 8 years
FV = $10,000 * (1 + 0.04/4)^(4*8)
FV = $10,000 * (1 + 0.01)^(32)
FV = $10,000 * (1.01)^(32)
Using a calculator, we find that the future value is approximately $14,625.89.
Now, since the second partner wants to match this amount with their investment in 8 years, they need to contribute the same future value. They can use the present value formula to calculate the equivalent contribution:
PV = FV / (1 + r/n)^(n*t)
Using the same values as before:
FV = $14,625.89
r = 4% (or 0.04 as a decimal)
n = 4 (quarterly compounding)
t = 8 years
PV = $14,625.89 / (1 + 0.04/4)^(4*8)
PV = $14,625.89 / (1 + 0.01)^(32)
PV = $14,625.89 / (1.01)^(32)
Using a calculator, we find that the second partner should contribute approximately $10,000 to match the first partner's investment now.
To find out how much the second partner should contribute in 8 years to match the investment of $10,000 made by the first partner, we can use the concept of compound interest.
The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
A = the future value (amount accumulated after time t)
P = the principal amount (initial investment)
r = annual interest rate (written as a decimal)
n = number of times that interest is compounded per year
t = number of years
In this case, the first partner invests $10,000 immediately, so P = $10,000. The interest rate is 4% per year and compounded quarterly, so r = 0.04 and n = 4.
Let's calculate the future value of the $10,000 investment after 8 years:
A = $10,000(1 + 0.04/4)^(4*8)
A = $10,000(1 + 0.01)^32
A = $10,000(1.01)^32
Using a calculator or spreadsheet, we find that A ≈ $10,998.87. This is the future value of the first partner's investment in 8 years.
Now, we need to find out how much the second partner should contribute in 8 years to match this amount. Let's call this amount X.
X(1 + 0.04/4)^(4*8) = $10,998.87
X(1.01)^32 = $10,998.87
Dividing both sides by (1.01)^32, we get:
X = $10,998.87 / (1.01)^32
Using a calculator or spreadsheet, we find that X ≈ $8,664.62. Therefore, the second partner should contribute approximately $8,664.62 in 8 years to match the first partner's investment of $10,000 made now.