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Math Help - Interest Questions (Inventor Rights, Loans)

  1. #1
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    Question Interest Questions (Inventor Rights, Loans)

    Q1) An inventor has been offered $12,000 per year for the next 5 years and $6000 annually for the following 7 years for the exclusive rights to an invention. At what price could the inventor afford to sell the rights to earn 10 percent, disregarding taxes.

    My attempt:

    P = 12000(\frac{P}{A}, 10\%, 5) + 6000(\frac{P}{A}, 10\%, 7)(\frac{P}{F}, 10\%, 7)

    P = 45489.48 + 29210.62

    P = \$ 74700.10

    The first portion on finds the present worth for annuity, then second portion finds the present worth of an annuity but then since this is 5 years in the future, we must consider that a future worth and multiply by another factor get its present worth.

    Is this correct?

    Q2) A company borrowed $100,000 to finance a new product. The loan was for 20 years at a nominal interest rate of 8 percent compounded semiannually. It was to be repaid in 40 equal payments. After one half of the payments were made, the company decided to pay the remaining balance in one final payment at the end of the 10th year. How much was owed?

    My attempt:

    A =  100000( \frac{A}{P}, 4\%, 40) = \$ 5052

    So in 20 payments he would have paid,

    20*5052 = \$ 101040

    What is the total future worth (all 40 payments) of his 100k loan?

    F = 100000( \frac{F}{P}, 4\%, 40) = \$ 480102

    Once he has made half the payments, how much is the present worth of his loan?

    P = 480102( \frac{P}{F}, 10\%, 20) = \$ 219113.75

    Therefore,

    219113.75-101040 = \$ 118073.75 \text{ was owed.}
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    Re: Interest Questions (Inventor Rights, Loans)

    Quote Originally Posted by jegues View Post
    Q1) An inventor has been offered $12,000 per year for the next 5 years and $6000 annually for the following 7 years for the exclusive rights to an invention. At what price could the inventor afford to sell the rights to earn 10 percent, disregarding taxes.
    My attempt:
    P = 12000(\frac{P}{A}, 10\%, 5) + 6000(\frac{P}{A}, 10\%, 7)(\frac{P}{F}, 10\%, 7)
    P = 45489.48 + 29210.62
    P = \$ 74700.10
    The first portion on finds the present worth for annuity, then second portion finds the present worth of an annuity but then since this is 5 years in the future, we must consider that a future worth and multiply by another factor get its present worth.
    Is this correct?
    45489.48 is correct; your 29210.62 needs to be discounted 5 years: 29210.62 / 1.1^5 = 18137.43.

    So 45489.48 + 18137.43 = 63626.87
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    Re: Interest Questions (Inventor Rights, Loans)

    Quote Originally Posted by jegues View Post
    Q2) A company borrowed $100,000 to finance a new product. The loan was for 20 years at a nominal interest rate of 8 percent compounded semiannually. It was to be repaid in 40 equal payments. After one half of the payments were made, the company decided to pay the remaining balance in one final payment at the end of the 10th year. How much was owed?
    What is frequency of payments: monthly, semiannually or anually?

    You should realise that balance owing will certainly not be > 100,000
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    Re: Interest Questions (Inventor Rights, Loans)

    Quote Originally Posted by Wilmer View Post
    45489.48 is correct; your 29210.62 needs to be discounted 5 years: 29210.62 / 1.1^5 = 18137.43.

    So 45489.48 + 18137.43 = 63626.87
    Ah of course, this is an obvious mistake thank you!


    What is frequency of payments: monthly, semiannually or anually?

    You should realise that balance owing will certainly not be > 100,000
    If there are 40 equal payments in 20 years, I would assume that it would be semiannually because,

    2*20 = 40 payments

    Hence why I used an interest rate of 4% and not 8% in all my calculations.

    I can't see why it's obvious to conclude that the balance owed is going to be less than 100,000. Could you explain why?
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    Re: Interest Questions (Inventor Rights, Loans)

    Quote Originally Posted by jegues View Post
    If there are 40 equal payments in 20 years, I would assume that it would be semiannually because,
    2*20 = 40 payments
    Hence why I used an interest rate of 4% and not 8% in all my calculations.
    I can't see why it's obvious to conclude that the balance owed is going to be less than 100,000. Could you explain why?
    Correct; payment is semiannual, and 4% is used to calculate the interest at each payment.
    Balance MUST be less than 100,000, else loan would never get repaid; think about it!

    Required semiannual payment: 100000(.04) / (1 - 1/1.04^40) = 5052.3489....so $5,052.35

    Here's what the loan "looks like" as it gets repaid:
    Code:
    N     PAYMENT     INTEREST         BALANCE
    0                                100,000.00
    1    -5,052.35    4,000.00        98,947.65
    2    -5,052.35    3,957.91        96,714.21
    .....
    20   -5,052.35    2,835.21        68,663.07 **********
    .....
    39   -5,052.35      381.17         4,858.03
    40   -5,052.35      194.32              .00
    ********** this is amount required to pay off loan after 20 payments; OK?
    This amount can be calculated by formula:
    future value of 100000.00 over 20 periods less future value of 20 payments of 5052.35; here:

    100000(1.04^20) = 219112.31 [1]

    5052.35(1.04^20 - 1) / .04 = 150449.24 [2]

    [1] - [2] = 68663.07

    So, you all ok now?
    Last edited by Wilmer; September 21st 2012 at 10:02 AM.
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    Re: Interest Questions (Inventor Rights, Loans)

    Quote Originally Posted by Wilmer View Post
    Correct; payment is semiannual, and 4% is used to calculate the interest at each payment.
    Balance MUST be less than 100,000, else loan would never get repaid; think about it!

    Required semiannual payment: 100000(.04) / (1 - 1/1.04^40) = 5052.3489....so $5,052.35

    Here's what the loan "looks like" as it gets repaid:
    Code:
    N     PAYMENT     INTEREST         BALANCE
    0                                100,000.00
    1    -5,052.35    4,000.00        98,947.65
    2    -5,052.35    3,957.91        96,714.21
    .....
    20   -5,052.35    2,835.21        68,663.07 **********
    .....
    39   -5,052.35      381.17         4,858.03
    40   -5,052.35      194.32              .00
    ********** this is amount required to pay off loan after 20 payments; OK?
    This amount can be calculated by formula:
    future value of 100000.00 over 20 periods less future value of 20 payments of 5052.35; here:

    100000(1.04^20) = 219112.31 [1]

    5052.35(1.04^20 - 1) / .04 = 150449.24 [2]

    [1] - [2] = 68663.07

    So, you all ok now?
    Yes that makes perfect sense.

    Thanks again for all your help!
    Last edited by jegues; September 23rd 2012 at 08:20 AM.
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    Re: Interest Questions (Inventor Rights, Loans)

    Quote Originally Posted by Wilmer View Post
    45489.48 is correct; your 29210.62 needs to be discounted 5 years: 29210.62 / 1.1^5 = 18137.43.

    So 45489.48 + 18137.43 = 63626.87
    I have a question.

    Only 6000*(\frac{P}{A}, 10\%,7) should be discounted by 5 years, not 29210.62.

    In other words, $56923.02 should be discounted by 5 years, correct?
    Last edited by jegues; September 23rd 2012 at 08:20 AM.
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    Re: Interest Questions (Inventor Rights, Loans)

    Can't tell what you're doing...but answer is NO!
    29210.62 is the PV of the seven 6000's for years 6 to 12 as at end of year#5;
    then it is discounted to TODAY, thus by 5 years.
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    Re: Interest Questions (Inventor Rights, Loans)

    Quote Originally Posted by Wilmer View Post
    Can't tell what you're doing...but answer is NO!
    29210.62 is the PV of the seven 6000's for years 6 to 12 as at end of year#5;
    then it is discounted to TODAY, thus by 5 years.
    Isn't the present value of the seven 6000's before it is discounted to today simply,

    6000*(\frac{P}{A}, 10\%, 7)

    =6000*(9.48717)

    =\$56923.02

    then discounted by 5 years time,

    56923.02*(\frac{P}{F}, 10\%, 5)

     =56923.02*(0.62092)

    =\$35344.64

    Is this not correct?

    This is analogous with this question found in my course notes. (See figure attached)
    Attached Thumbnails Attached Thumbnails Interest Questions (Inventor Rights, Loans)-notes.jpg  
    Last edited by jegues; September 23rd 2012 at 10:58 AM.
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    Re: Interest Questions (Inventor Rights, Loans)

    Quote Originally Posted by jegues View Post
    Isn't the present value of the seven 6000's before it is discounted to today simply,
    6000*(\frac{P}{A}, 10\%, 7)
    =6000*(9.48717)
    =\$56923.02
    NO. That's the FUTURE VALUE
    Code:
    0                           .00
    1  6000.00              6000.00
    2  6000.00     600.00  12600.00
    3  6000.00    1260.00  19860.00
    ...
    7  6000.00    4629.37  56923.02
    Present value would be 56923.02 / 1.10^7 = 29210.51

    TRY and get this straight:
    first, the $6000 flow is discounted to 29210.51; that's its value end of year 5;
    then the 29210.51 is discounted to TODAY: 29210.51 / 1.10^5 = 18137.43
    Last edited by Wilmer; September 23rd 2012 at 11:54 AM.
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    Re: Interest Questions (Inventor Rights, Loans)

    Quote Originally Posted by Wilmer View Post
    NO. That's the FUTURE VALUE
    Code:
    0                           .00
    1  6000.00              6000.00
    2  6000.00     600.00  12600.00
    3  6000.00    1260.00  19860.00
    ...
    7  6000.00    4629.37  56923.02
    Present value would be 56923.02 / 1.10^7 = 29210.51

    TRY and get this straight:
    first, the $6000 flow is discounted to 29210.51; that's its value end of year 5;
    then the 29210.51 is discounted to TODAY: 29210.51 / 1.10^5 = 18137.43
    As you mentioned I was solving the future value. (I took the wrong constant from the tables in the back of my textbook)

    \text{Value at the end of year } 5 \equiv 6000*(4.86842) = \$29210.52

    \text{Now, discounting to TODAY yields, } 29210.52*(0.62092) = \$18137.40

    Thank you very much for all your help and patience Wilmer!
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    Re: Interest Questions (Inventor Rights, Loans)

    Phewwwwww!!!!!!!!!!!!!!!!
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