May Wattson has 50000 to invest in a 4-year certificate of deposit that earns an interest rate of 3.75% compounded monthly or 3.50% compounded daily. Based on annual yield which is the better investment?
The first part of the question is irrelevant.
all we need is to compare 3.76% compounded monthly with 3.5% compounded daily
let the first rate be i
1+i = (1 + .0376/12)^12 = 1.038254..
i = appr 3.825%
let the second rate be j
1+j = (1 + .035/365)^365 = 1.0356..
= appr 3.56%
take over
To compare the two investment options, we can calculate the future value of the investments using the compound interest formula.
For the first investment option, the interest rate is 3.75% compounded monthly. The formula to calculate the future value is:
FV = P(1 + r/n)^(nt)
Where:
FV = future value
P = principal amount
r = interest rate per compounding period
n = number of compounding periods per year
t = number of years
For the second investment option, the interest rate is 3.50% compounded daily. The formula to calculate the future value is:
FV = P(1 + r/n)^(nt)
Where:
FV = future value
P = principal amount
r = interest rate per compounding period
n = number of compounding periods per year
t = number of years
Let's calculate the future value for each investment option:
1. For the 4-year certificate of deposit with an interest rate of 3.75% compounded monthly:
FV1 = 50000(1 + 0.0375/12)^(12*4)
FV1 ≈ 50000(1.003125)^(48)
FV1 ≈ 50000(1.160654018)
FV1 ≈ 58032.7009
2. For the 4-year certificate of deposit with an interest rate of 3.50% compounded daily:
FV2 = 50000(1 + 0.0350/365)^(365*4)
FV2 ≈ 50000(1.0000958904)^(1460)
FV2 ≈ 50000(1.1689767319)
FV2 ≈ 58448.8366
Therefore, the better investment option, based on the annual yield, is the second option with an interest rate of 3.50% compounded daily. It will result in a higher future value of approximately $58,448.84, compared to the first option which has a future value of approximately $58,032.70.
To compare which investment is better based on annual yield, we need to calculate the future value of the investment for each option.
Let's start with the 4-year certificate of deposit that earns an interest rate of 3.75% compounded monthly. The formula to calculate the future value of a compound interest investment is:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (initial investment)
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years
For this case, P = $50,000, r = 3.75% or 0.0375 (as a decimal), n = 12 (monthly compounding), and t = 4 (4 years).
Using the formula, we can calculate the future value of the investment:
A1 = $50,000 * (1 + 0.0375/12)^(12*4)
Now, let's calculate the future value of the investment for the second option, which is a 4-year certificate of deposit with an interest rate of 3.50% compounded daily. We'll use the same formula, but with the new interest rate.
Using the formula A = P(1 + r/n)^(nt), and the values P = $50,000, r = 3.50% or 0.035 (as a decimal), n = 365 (daily compounding), and t = 4 (4 years):
A2 = $50,000 * (1 + 0.035/365)^(365*4)
Now that we have the future values for both investment options, we can compare them to determine which is better based on annual yield. The higher future value represents the better investment option.
Compare A1 and A2. If A1 is greater than A2, then the investment with a 3.75% interest rate compounded monthly is the better option. If A2 is greater, then the investment with a 3.50% interest rate compounded daily is the better option.
By calculating the values, you can determine which investment provides a higher annual yield.