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April 4th, 2014, 12:29 AM   #1
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finance: monthly, balance, and interest

tim takes out a 15-year loan of the amount $300,000. The annual interest rate is 4% , compounded monthly. Answer the following questions.

i What will Bob's monthly payment be?
ii: Ten years into his repayment program, what will tim's balance owed be?
iii: After 15 years, how much will Bob have spent on interest?

please help me. DX

Last edited by looni; April 4th, 2014 at 12:31 AM.
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April 4th, 2014, 12:46 AM   #2
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If you have not been given a formula for the monthly payment amount, this can be derived as follows:

Let $P$ = monthly payment, $A$ = amount borrowed, $i$ = monthly interest rate, and $n$ = the number of payments.

Also, let $D_n$ be the debt amount after payment $n$.

Consider the recursion:

(1) $\displaystyle D_{n}=(1+i)D_{n-1}-P$

(2) $\displaystyle D_{n+1}=(1+i)D_{n}-P$

Subtracting (1) from (2) yields the homogeneous recursion:

$\displaystyle D_{n+1}=(2+i)D_{n}-(1+i)D_{n-1}$

whose associated auxiliary equation is:

$\displaystyle r^2-(2+i)r+(1+i)=0$

$\displaystyle (r-(1+i))(r-1)=0$

Thus, the closed-form for our recursion is:

$\displaystyle D_n=k_1(1+i)^n+k_2$

Using initial values, we may determine the coefficients $k_i$:

$\displaystyle D_0=k_1+k_2=A$

$\displaystyle D_1=k_1(1+i)+k_2=(1+i)A-P$

Solving this system, we find:

$\displaystyle k_1=\frac{Ai-P}{i},\,k_2=\frac{P}{i}$ and so we have:

$\displaystyle D_n=\left(\frac{Ai-P}{i} \right)(1+i)^n+\left(\frac{P}{i} \right)=\frac{(Ai-P)(1+i)^n+P}{i}$

Now, equating this to zero, we can solve for $P$:

$\displaystyle \frac{(Ai-P)(1+i)^n+P}{i}=0$

$\displaystyle (Ai-P)(1+i)^n+P=0$

$\displaystyle (P-Ai)(1+i)^n=P$

$\displaystyle P\left((1+i)^n-1 \right)=Ai(1+i)^n$

$\displaystyle P=\frac{Ai(1+i)^n}{(1+i)^n-1}$

$\displaystyle P=\frac{Ai}{1-(1+i)^{-n}}$

Now, you can plug in the given values to get $P$ and use other formulas in the above to answer the remaining questions as well.
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April 6th, 2014, 09:20 AM   #3
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Quote:
Originally Posted by looni View Post
tim takes out a 15-year loan of the amount $300,000. The annual interest rate is 4% , compounded monthly. Answer the following questions.

i What will Bob's monthly payment be?
ii: Ten years into his repayment program, what will tim's balance owed be?
iii: After 15 years, how much will Bob have spent on interest?

please help me. DX
1: Mark gave you formula
2: That'll the PV of the payment at month 120 (n = 60)
3: payment * 180 - 300000
OK?
Thanks from MarkFL
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April 6th, 2014, 12:20 PM   #4
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Quote:
Originally Posted by Denis View Post
2: That'll the PV of the payment at month 120 (n = 60)
Clarification:
That'll the PV of the payments left following month 120 (n = 60)

Can't edit no more?!
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April 6th, 2014, 10:08 PM   #5
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Quote:
Originally Posted by Denis View Post
Clarification:
That'll the PV of the payments left following month 120 (n = 60)

Can't edit no more?!

i got it thanks! ^^
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May 22nd, 2015, 04:22 AM   #6
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Hello,
This Loan Payment Calculator computes an estimate of the measure of your monthly credit instalments and the yearly income needed to manage them without too much financial trouble. This loan calculator expects that the interest rate stays consistent for the duration of the life of the credit.
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May 22nd, 2015, 10:40 PM   #7
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Quote:
Originally Posted by looni View Post
tim takes out a 15-year loan of the amount $300,000. The annual interest rate is 4% , compounded monthly. Answer the following questions.

i What will Bob's monthly payment be?
ii: Ten years into his repayment program, what will tim's balance owed be?
iii: After 15 years, how much will Bob have spent on interest?

please help me. DX
Bob's payment will be zero as it was Tim who took out the loan and not Bob unless Tim stole Bob's identity in which case now Bob has a dilemma
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