You are going to invest $5000 for 5 years. Choose the 3 scenarios which will yield the highest values.
Responses
A interest rate 4.1%, compounded monthlyinterest rate 4.1%, compounded monthly
B interest rate 4.2%, compounded quarterlyinterest rate 4.2%, compounded quarterly
C interest rate 3.95%, compounded continuouslyinterest rate 3.95%, compounded continuously
D interest rate 4%, compounded dailyinterest rate 4%, compounded daily
E interest rate 3.99%, compounded semiannuallyinterest rate 3.99%, compounded semiannually
F interest rate 4.3%, compounded every 2 months
To determine the scenarios that will yield the highest values, we need to calculate the future value (FV) of the $5000 investment for each scenario using the formula:
FV = P * (1 + r/n)^(n*t)
Where:
P = Principal amount ($5000)
r = Annual interest rate
n = Number of compounding periods per year
t = Number of years
Let's calculate the future value for each scenario:
A: interest rate 4.1%, compounded monthly
r = 4.1% / 100 = 0.041
n = 12
t = 5
FV(A) = 5000 * (1 + 0.041/12)^(12*5)
B: interest rate 4.2%, compounded quarterly
r = 4.2% / 100 = 0.042
n = 4
t = 5
FV(B) = 5000 * (1 + 0.042/4)^(4*5)
C: interest rate 3.95%, compounded continuously
r = 3.95% / 100 = 0.0395
n = Continuous compounding
t = 5
FV(C) = 5000 * e^(0.0395*5)
D: interest rate 4%, compounded daily
r = 4% / 100 = 0.04
n = 365
t = 5
FV(D) = 5000 * (1 + 0.04/365)^(365*5)
E: interest rate 3.99%, compounded semiannually
r = 3.99% / 100 = 0.0399
n = 2
t = 5
FV(E) = 5000 * (1 + 0.0399/2)^(2*5)
F: interest rate 4.3%, compounded every 2 months
r = 4.3% / 100 = 0.043
n = 6
t = 5
FV(F) = 5000 * (1 + 0.043/6)^(6*5)
Now let's calculate the future values for each scenario:
FV(A) ≈ $6,513.65
FV(B) ≈ $6,537.12
FV(C) ≈ $6,511.41
FV(D) ≈ $6,494.17
FV(E) ≈ $6,512.30
FV(F) ≈ $6,559.86
The three scenarios that will yield the highest values are B, C, and F.
To determine the three scenarios that will yield the highest values, we need to calculate the future value of each scenario using the compound interest formula:
Future Value = Principal * (1 + (Interest Rate / Compounding Frequency))^(Compounding Frequency * Time)
Let's calculate the future value for each scenario:
A. Interest rate 4.1%, compounded monthly:
Future Value (A) = $5000 * (1 + (0.041 / 12))^(12 * 5)
B. Interest rate 4.2%, compounded quarterly:
Future Value (B) = $5000 * (1 + (0.042 / 4))^(4 * 5)
C. Interest rate 3.95%, compounded continuously:
Future Value (C) = $5000 * e^(0.0395 * 5)
D. Interest rate 4%, compounded daily:
Future Value (D) = $5000 * (1 + (0.04 / 365))^(365 * 5)
E. Interest rate 3.99%, compounded semiannually:
Future Value (E) = $5000 * (1 + (0.0399 / 2))^(2 * 5)
F. Interest rate 4.3%, compounded every 2 months:
Future Value (F) = $5000 * (1 + (0.043 / 6))^(6 * 5)
Now, we can calculate the future values for each scenario:
A. Future Value (A) = $5000 * (1 + (0.041 / 12))^(12 * 5)
= $5000 * (1 + 0.00341)^(60)
= $5000 * (1.00341)^(60)
≈ $6093.75
B. Future Value (B) = $5000 * (1 + (0.042 / 4))^(4 * 5)
= $5000 * (1 + 0.0105)^(20)
= $5000 * (1.0105)^(20)
≈ $6109.12
C. Future Value (C) = $5000 * e^(0.0395 * 5)
≈ $5000 * 1.2077
≈ $6038.50
D. Future Value (D) = $5000 * (1 + (0.04 / 365))^(365 * 5)
= $5000 * (1 + 0.000109)^(1825)
= $5000 * (1.000109)^(1825)
≈ $6104.49
E. Future Value (E) = $5000 * (1 + (0.0399 / 2))^(2 * 5)
= $5000 * (1 + 0.01995)^(10)
= $5000 * (1.01995)^(10)
≈ $6117.30
F. Future Value (F) = $5000 * (1 + (0.043 / 6))^(6 * 5)
= $5000 * (1 + 0.00717)^(30)
= $5000 * (1.00717)^(30)
≈ $6125.00
Based on the calculations, the three scenarios with the highest values are:
1. Scenario F: Interest rate 4.3%, compounded every 2 months - Future Value ≈ $6125.00
2. Scenario E: Interest rate 3.99%, compounded semiannually - Future Value ≈ $6117.30
3. Scenario B: Interest rate 4.2%, compounded quarterly - Future Value ≈ $6109.12