Steve wants to have $25000 in 25 years, he can only get 3.2% interest compounded quarterly. his bank will guarantee the rate for either 5 or 8 years
in 5 years he can get 4% compounded quarterly for the remainder of the term
in 8 years he can get 5% compounded quarterly for the remainder of the term
which option should steve take the 5 or 8 year?
how much more does he need to invest?
case 1:
3.2% for 5 years, then 4% for 8 years
P(1.008)^20 (1.01)^80 = 25000
P = 9616.55
case 2:
P(1.008)^32 (1.0125)^68 = 25000
P = 8324.16
clearly since he needs less money to invest using the second option to obtain the same goal, he should clearly take the second option.
Well, Steve clearly has two options: a 5-year plan with 4% interest or an 8-year plan with 5% interest. Now, let's put on our clown noses and do some math!
For the 5-year plan, Steve will have his money invested at 3.2% interest for the first five years and then switched to 4% interest for the remaining term. So, the first step is figuring out what the amount will be after five years at 3.2% interest.
To do that, we'll use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (in this case, $25,000)
P = the principal amount (unknown for now)
r = annual interest rate as a decimal (3.2% becomes 0.032)
n = number of times interest is compounded per year (quarterly, so n = 4)
t = number of years (5)
25,000 = P(1 + 0.032/4)^(4 * 5)
Solving this equation gives us P ≈ $21,942.56.
Now, for the 8-year plan, Steve would have his money invested at 3.2% interest for the first eight years and then switched to 5% interest for the remaining term. We can use the same formula with the appropriate changes:
25,000 = P(1 + 0.032/4)^(4 * 8)
Solving this equation gives us P ≈ $19,087.38.
So, Steve needs to invest $21,942.56 - $19,087.38 = $2855.18 more for the 8-year plan compared to the 5-year plan.
In short, Steve should go for the 5-year plan to avoid having to invest those extra clams.
To determine which option Steve should take, we can calculate the future value of his investment using both options and see which one yields a higher amount.
Option 1: 5 years at 4% compounded quarterly, and 20 years at 3.2% compounded quarterly.
Option 2: 8 years at 5% compounded quarterly, and 17 years at 3.2% compounded quarterly.
Let's calculate the future value for each option.
Option 1:
Initial investment = $25,000
Interest rate = 4% / 4 (quarterly compounding) = 1% per quarter
Number of quarters in 5 years = 5 * 4 = 20 quarters
Future value (after 5 years) = $25,000 * (1 + 1%)^(20) = $29,900.27
Interest rate for the remaining 20 years = 3.2% / 4 (quarterly compounding) = 0.8% per quarter
Future value (after 20 years) = $29,900.27 * (1 + 0.8%)^(80) = $52,626.66
Option 2:
Initial investment = $25,000
Interest rate = 5% / 4 (quarterly compounding) = 1.25% per quarter
Number of quarters in 8 years = 8 * 4 = 32 quarters
Future value (after 8 years) = $25,000 * (1 + 1.25%)^(32) = $35,709.76
Interest rate for the remaining 17 years = 3.2% / 4 (quarterly compounding) = 0.8% per quarter
Future value (after 17 years) = $35,709.76 * (1 + 0.8%)^(68) = $50,231.88
Therefore, Steve should choose the 8-year option as it yields a higher future value.
To calculate how much more he needs to invest, we subtract the future value of the 8-year option from the desired $25,000:
Additional investment = $25,000 - $50,231.88 = -$25,231.88
Since the result is negative, it means that Steve does not need to invest any additional funds beyond the initial $25,000 for the 8-year option to reach his goal.
To determine which option Steve should take (5 years or 8 years), we need to calculate the future value of his investment for both scenarios and compare them.
Let's break down the calculations step by step:
1. Option 1: 5 years at 4% compounded quarterly
If Steve chooses this option, he will have 5 years of compound interest at 4% and 20 years of compound interest at 3.2%. Let's calculate the future value for this option.
The formula to calculate the future value (FV) of an investment with compound interest is:
FV = P(1 + r/n)^(nt)
Where:
FV = Future Value
P = Principal amount (initial investment)
r = Annual interest rate (in decimal form)
n = Number of times the interest is compounded per year
t = Number of years
For Steve's investment, the principal amount (P) is $25,000, the annual interest rate (r) is 4% (or 0.04 in decimal form), the interest is compounded quarterly (n = 4), and the time period (t) is 25 years.
Calculating FV for the first 5 years:
FV1 = 25000(1 + 0.04/4)^(4*5)
FV1 = 25000(1 + 0.01)^20
FV1 ≈ 25000(1.01)^20
FV1 ≈ 25000(1.2214)
FV1 ≈ $30,535.09
Calculating FV for the remaining 20 years (compounded at 3.2%):
FV2 = 30535.09(1 + 0.032/4)^(4*20)
FV2 = 30535.09(1 + 0.008)^80
FV2 ≈ 30535.09(1.008)^80
FV2 ≈ $56,875.51
Total Future Value (FV) for Option 1:
FV_Total = FV1 + FV2
FV_Total ≈ $30,535.09 + $56,875.51
FV_Total ≈ $87,410.60
2. Option 2: 8 years at 5% compounded quarterly
If Steve chooses this option, he will have 8 years of compound interest at 5% and 17 years of compound interest at 3.2%. Let's calculate the future value for this option.
Calculating FV for the first 8 years:
FV1 = 25000(1 + 0.05/4)^(4*8)
FV1 = 25000(1 + 0.0125)^32
FV1 ≈ 25000(1.0125)^32
FV1 ≈ $40,316.94
Calculating FV for the remaining 17 years (compounded at 3.2%):
FV2 = 40316.94(1 + 0.032/4)^(4*17)
FV2 = 40316.94(1 + 0.008)^68
FV2 ≈ 40316.94(1.008)^68
FV2 ≈ $78,045.52
Total Future Value (FV) for Option 2:
FV_Total = FV1 + FV2
FV_Total ≈ $40,316.94 + $78,045.52
FV_Total ≈ $118,362.46
Comparing the two options:
Option 1: Future Value ≈ $87,410.60
Option 2: Future Value ≈ $118,362.46
Based on these calculations, Steve should choose Option 2 (8 years at 5% compounded quarterly). This option will yield a higher future value of approximately $118,362.46.
To calculate how much more Steve needs to invest to reach $25,000, we need to subtract the initial investment ($25,000) from the Future Value of Option 2.
Additional investment needed:
Additional Investment = FV_Total - P
Additional Investment ≈ $118,362.46 - $25,000
Additional Investment ≈ $93,362.46
Steve needs to invest an additional amount of approximately $93,362.46 to reach his goal of $25,000 in 25 years by choosing Option 2.