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Math Help - Amortization of a Loan (Time Value of Money)

  1. #1
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    Amortization of a Loan (Time Value of Money)

    I'm having trouble with these 3 questions. I can't figure out how much the interest is applied to, because I don't know how much the loan is amortized up until any specific point. This is for a beginner level finance class.

    1.) Your firm had received a $2,000,000 loan from its bank. The loan carries an annual interest rate of 6% and is being amortized in equal quarterly payments (payable at the end of each quarter). The loan has a maturity of 10 years. You now want to repay the remaining balance of the loan in full. At this point of time, there are 10 payments left. What is the loan balance you need to pay now?

    2.) When Ms. Jones retired, she received a lump sum of $1,200,000 from her pension plan. She then invested it in an annuity account that would pay her an equal amount at the end of each year for the next 25 year, so that the initial capital of $1,200,000 will be totally depleted. The interest rate for this annuity account is 4% per year. Ms. Jones passed away after receiving the 15th annual payment. Her estate lawyer must now determine the amount of money left that will go to Ms. Jones's heirs. What is this amount?

    3.) You have just renegotiated the interest rate of your home mortgage loan. The original loan of $500,000 carries an interest rate of 7%, has an original maturity of 15 years, and requires monthly payments (at the end of the month). There are 80 months remaining to repay the loan. The new mortgage has a lower rate of 6% and will still have 80 months left to maturity. The lender requires a $1500 rate modification fee. First estimate the monthly savings if you switch to the new lower interest rate and then the new gain or loss in present value dollars today if you go through with this rate modification deal.

    Any help would be greatly appreciated. Thanks in advance.
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  2. #2
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    (1)
    L=2000000
    n=10*4=40
    r=.06/4=.015 (assuming that 6% is nominal annual interest rate, compounded quarterly)..
    If there are 10 payments left, you have made p=30 payments. Then the value on the outstanding loan, assuming that you have just made the 30th payment, is;
    B=L\cdot \frac{(1+r)^n-(1+r)^p}{(1+r)^n-1}=2000000\cdot \frac{(1.015)^{40}-(1.015)^{30}}{(1.015)^{40}-1}=\$616541.8

    (2)
    Similiar to (1). L=1200000, n=25, r=.04.
    So she has received p=15 payments. Then the amount outstanding in the annuity is calculated as (1) was..

    (3)
    Assuming that the 100th payment was just made (there is a full month until the next payment is due), after 100 payments on the mortage, the outstanding balance is;
    B=500000\cdot \frac{(1.005833)^{180}-(1.005833)^{100}}{(1.005833)^{180}-1}=\$286644.1

    The monthly repayment on the mortgage is calculated using the formula
     <br />
P=\frac{L}{ \frac{r}{1- \left( \frac{1}{1+r}\right)^n}}<br />
    So under the original contract, the monthly payment was;
    L=$500000, r=.07/12=.005833, n=180
     <br />
P=\frac{500000}{ \frac{.005833}{1- \left( \frac{1}{1.005833}\right)^{180}}}=\$4494.1<br />

    If they refinance the outsanding loan balance of $286644.1 at 6% for the remaining 80 periods, the repayment will be;
    P=\frac{286644.1}{ \frac{.005}{1- \left( \frac{1}{1.005}\right)^{80}}}=\$4219.284

    so they will save $274.857 a month.
    The PV of this saving is, at 6% discount rate (Im not sure if you should use 6%?), less the rate modification fee;
     <br />
PV=274.857\cdot \left[ \frac{1-(1.005)^{-80}}{.005}\right]-1500=18086.24-1500=\$16586.24<br />


    I hope this helps/is correct.
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  3. #3
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    Quote Originally Posted by GB89 View Post
    1.) Your firm had received a $2,000,000 loan from its bank. The loan carries an annual interest rate of 6% and is being amortized in equal quarterly payments (payable at the end of each quarter). The loan has a maturity of 10 years. You now want to repay the remaining balance of the loan in full. At this point of time, there are 10 payments left. What is the loan balance you need to pay now?
    This is what happens "on the way to the Forum!":
    Code:
    0                                2,000,000.00
    1    -66,854.20   30,000.00      1,963,145.80
    2    -66,854.20   29,447.18      1,925,738.78
    ....
    30   -66,854.20   10,099.45        616,541.80
    Notice the ending 616,541.80 is what Robb calculated.

    Can you follow that?
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  4. #4
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    Quote Originally Posted by Robb View Post

    (3)


    The monthly repayment on the mortgage is calculated using the formula
     <br />
P=\frac{L}{ \frac{r}{1- \left( \frac{1}{1+r}\right)^n}}<br />
    So under the original contract, the monthly payment was;
    L=$500000, r=.07/12=.005833, n=180
     <br />
P=\frac{500000}{ \frac{.005833}{1- \left( \frac{1}{1.005833}\right)^{180}}}=\$4494.1<br />

    If they refinance the outsanding loan balance of $286644.1 at 6% for the remaining 80 periods, the repayment will be;
    P=\frac{286644.1}{ \frac{.005}{1- \left( \frac{1}{1.005}\right)^{80}}}=\$4219.284

    [/tex]


    I hope this helps/is correct.
    Thanks so much. I'm just having trouble following those 2 equations - I typed the numbers into my calculator and I'm not getting that result.
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  5. #5
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    Quote Originally Posted by Robb View Post
    The monthly repayment on the mortgage is calculated using the formula
     <br />
P=\frac{L}{ \frac{r}{1- \left( \frac{1}{1+r}\right)^n}}<br />
    So under the original contract, the monthly payment was;
    L=$500000, r=.07/12=.005833, n=180
     <br />
P=\frac{500000}{ \frac{.005833}{1- \left( \frac{1}{1.005833}\right)^{180}}}=\$4494.1<br />

    If they refinance the outsanding loan balance of $286644.1 at 6% for the remaining 80 periods, the repayment will be;
    P=\frac{286644.1}{ \frac{.005}{1- \left( \frac{1}{1.005}\right)^{80}}}=\$4219.284
    Formula for payment calculation is:
    P = Payment
    A = Amount of loan
    i = periodic interest rate
    n = number of payments.

    P = Ai / (1 - f) where f = 1 / (1 + i)^n

    Robb's 1st calculation is correct: 4,494.03
    His 2nd one not correct should be: 4,356.14

    Difference of 137.89 in payments

    Calculation of 4356.14:
    f = 1 / 1.005^80
    P = 286644.10 * .005 / (1 - f) = 4356.14
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  6. #6
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    GB, one way to "see" what goes on with loans being repaid is assume
    an account is used to receive the payments, while the loan receives
    no payments; like this, using a loan of $1000 repaid at $100 monthly,
    with interest at 1% per month (12% annual):
    Code:
           Loan Amount           Payments account      Net owing
    0             1000.00                        .00     1000.00
    1    10.00    1010.00     100.00    .00   100.00      910.00
    2    10.10    1020.10     100.00   1.00   201.00      819.10
    3    10.20    1030.30     100.00   2.01   303.01      727.29
    4    10.30    1040.60     100.00   3.03   406.04      634.56
    This clearly shows that the loan balance at any month-end
    is the future value of the loan amount less the future value
    of the payments to the end of that month.
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