Thread: 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:







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:



So in 20 payments he would have paid,



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



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



Therefore,

Originally Posted by jegues

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:



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

Originally Posted by jegues

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

Originally Posted by Wilmer

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?

Originally Posted by jegues

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?

Originally Posted by Wilmer

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!

Originally Posted by Wilmer

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 should be discounted by 5 years, not 29210.62.

In other words, $56923.02 should be discounted by 5 years, correct?

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.

Originally Posted by Wilmer

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,







then discounted by 5 years time,







Is this not correct?

This is analogous with this question found in my course notes. (See figure attached)

Originally Posted by jegues

Isn't the present value of the seven 6000's before it is discounted to today simply,

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

Originally Posted by Wilmer

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)





Thank you very much for all your help and patience Wilmer!

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