I really need some help with this one.
Envision that you have served as business manager of Media World for over 2 years. You have noticed that for the last 12 months the business has regularly had cash assets of $20,000 or more at the end of each month. You have found a 6-month certificate of deposit that pays 6% compounded monthly. To obtain this rate of interest, you must invest a minimum of $2,000. You have also found a high interest savings account that pays 3% compounded daily. Based on the cash position of the business at this time, assume that you decide to invest $4,000
1. Assume that you will invest the full amount in a certificate of deposit.
a. What would be the future value of the CD at the end of the investment term?
b. How much interest would the investment earn for the period?
c. What would be the effective rate of the investment?
Six-month certificate of deposit at 6% p.a. compounded monthly, equivalent to 6/12=0.5% per period (month).
Principal = 4000
Future value, FV = 4000*1.005^6 = 4121.51
Interest earned = FV-principal
Effective rate = (interest/principal)*12months/6 months
Thanks MathMate. The second part of the question is about the high interest savings account that pays 3% compounded daily. How do you do it daily?
Assume that you decide to invest the $4,000 in the high-interest savings account.
a. What future value would you expect to receive at the end of 6 months?
b. How much interest would the investment earn for the period?
c. What would be the effective rate of the investment?
I would rather stay with the 6%.
Daily interest sounds good, but not at half the interest rate of 3%. You basically follow the same procedure as in the 6% compounded monthly, but divide the interest rate by 365, and compound every day.
Here's how it works:
High interest at 3% p.a. compounded daily, equivalent to 3/365% per period (day). Six-months = 183 days, approx.
Principal = 4000
Future value, FV = 4000*(1+3/365)^183 = 4060.62
Interest earned = FV-principal
Effective rate = (interest/principal)*12months/6 months
Note that the effective rate is for six-months prorated to a year. It would be slightly higher if the money was deposited for a whole year.
Also, if the six-month certificate had been compounded daily, the future value would have added $0.63 to the pot, even though daily interest sounds soooo good!
Thanks sgain for the help, I really appreciate it.
You're welcome!
To determine the future value of the certificate of deposit (CD), you can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = future value
P = principal amount (initial investment)
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
Based on the information provided:
P = $4,000
r = 6% (0.06)
n = 12 (compounded monthly)
t = 6 months (since it's a 6-month CD)
a. What would be the future value of the CD at the end of the investment term?
Using the formula, we can calculate:
A = $4,000(1 + 0.06/12)^(12*0.5)
A = $4,000(1 + 0.005)^(6)
A = $4,000(1.005)^6
A ≈ $4,000(1.0304)
A ≈ $4,121.60
Therefore, the future value of the CD at the end of the 6-month investment term would be approximately $4,121.60.
b. How much interest would the investment earn for the period?
To calculate the interest earned, you can subtract the initial investment from the future value:
Interest = Future Value - Principal
Interest = $4,121.60 - $4,000
Interest ≈ $121.60
Therefore, the investment would earn approximately $121.60 in interest for the 6-month period.
c. What would be the effective rate of the investment?
The effective rate takes into account the compounding frequency and provides an annualized rate for comparison purposes. It can be calculated using the following formula:
Effective Rate = (1 + r/n)^n - 1
For the CD investment:
r = 6% (0.06)
n = 12 (compounded monthly)
Using the formula, we can calculate:
Effective Rate = (1 + 0.06/12)^12 - 1
Effective Rate ≈ (1 + 0.005)^12 - 1
Effective Rate ≈ (1.005)^12 - 1
Effective Rate ≈ 0.061678 - 1
Effective Rate ≈ 0.061678
Therefore, the effective rate of the CD investment would be approximately 6.17%.