A company will need $3,000,000 to replace its equipment 5 years from now, so it decides to make equal payments at the end of each month into an account paying 5.6% compounded monthly.
(a) How much should each monthly payment be?
(b) How much interest is earned during the third year?
college level interest problem?
2009-05-20 9:41 pm
回答 (3)
2009-05-20 9:59 pm
✔ 最佳答案
It helps to have interest theory notation down, but I will assume you do not. Let i be the annualized interest rate and let R be the payment. there will be 60 payments. The monthly interest rate i= 5.6%/12.R * (1+i)^59 + R * (1+i)^58 ... + R = 3,000,000
R [(1+i)^60 - 1]/i = 3,000,000
If you change the 60 in the above formula to M, you will get the value of the account after M months. You will need this formula for the second part of this problem.
The interest factor, also called S-sub-60, or the future value of the annuity, is equal numerically to 69.06, (It should be greater than the # of months).
Thus, R = 43,442.06
The interest earned during the 3d year is determined as
Increase in value - value of your payments = Value Bank added
Balance(36) - Balance(24) - Sum of payments during the year.
This is 598,000 - 521,000 or about 77,000. I'll let you do the actual math.
Once you do this by hand, you can do the rest by a calculator. Example.
Set FV = 3,000,000
Set PV = 0
Set i = 0.056 / 12 (this part is the easiest to forget)
Make sure the annuity functions on your calculator are set for END of the period (this is usually the default).
Calculate PMT.
Using the TI-35, you can also get the total Principal and Total Interest between any two periods.
參考: Kellison, Interest Theory
2009-05-21 6:29 am
a)
Michael T is 100% correct here.
b)
I think Michael confused the formulas for interest PAID to a lender vs. interest EARNED from an investment:
Interest paid = total payments - total principal
= total payments - decrease in loan balance
Interest earned = future value of deposits - amount of total deposits
The problem is about an investment so we need to calculate interest EARNED from an investment.
Therefore, Interest earned during 3rd year
=future value of deposits during 3rd year - total deposits during third year
=43,442.0599 * [(1 + .056 / 12)^12 - 1] / (.056 / 12) - 12 * 43,442.0599
= 13,590.49
The term 43,442.0599 * [(1 + .056 / 12)^12 - 1] / (.056/12) uses the formula for the future value of an annuity-immediate (series of equal cash flows).
Michael T is 100% correct here.
b)
I think Michael confused the formulas for interest PAID to a lender vs. interest EARNED from an investment:
Interest paid = total payments - total principal
= total payments - decrease in loan balance
Interest earned = future value of deposits - amount of total deposits
The problem is about an investment so we need to calculate interest EARNED from an investment.
Therefore, Interest earned during 3rd year
=future value of deposits during 3rd year - total deposits during third year
=43,442.0599 * [(1 + .056 / 12)^12 - 1] / (.056 / 12) - 12 * 43,442.0599
= 13,590.49
The term 43,442.0599 * [(1 + .056 / 12)^12 - 1] / (.056/12) uses the formula for the future value of an annuity-immediate (series of equal cash flows).
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