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Homework answers / question archive /  A loan is amortized over 8 years with monthly payments at a nominal interest rate of 9% compounded monthly

 A loan is amortized over 8 years with monthly payments at a nominal interest rate of 9% compounded monthly

Finance

 A loan is amortized over 8 years with monthly payments at a nominal interest rate of 9% compounded monthly. The first payment is $1200 and is to be paid one month from the date of the loan. Each succeeding monthly payment will be 2% lower than the prior payment. Calculate the outstanding loan balance immediately after the 79th payment is made. ANSWER = $

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First, we will find the Present value, PV of the loan
Using the formula for PV of growing annuity & assigning -ve sign, for the growth , to denote decrease
PV(g)A=Pmt./(r-g)*(1-((1+g)/(1+r))^n)
where,
PV(g)A----- needs to be found out---??
Pmt.=the first end-of-month payments, ie. $ 1200
r= 9%/12= 0.75% or 0.0075 p.m
g= negative growth rate , ie. -2% or -0.02 p.m.
n= no.of compounding mths., ie. 8 yrs.*12= 96
Now, we will plug in the values , in the formula,
PV=(1200/(0.0075-(-0.02)))*(1-((1-0.02)/(1+0.0075))^96)
40574.21
So, PV of the loan= $ 40574.21
 
Now, the balance o/s on the loan after the 79 th payment is
FV of the single sum of $ 40574.21 at end of 79 th mthly .pmt.-FV of 79-mths , ordinary mth.end 2% decreasing annuity of $ 1200
ie. FV of single sum-FV of negatively growing mthly.annuity
Balance o/s at end mth.79=(PV*(1+r)^n)-(Pmt.*(((1+r)^n-(1+g)^n)/(r-g)))
We use the same values as above, for all the variables
ie.( 40574.21*(1+0.0075)^79)-(1200*((1+0.0075)^79-(1-0.02)^79)/(0.0075-(-0.02)))
3319.53
So, the answer--- o/s loan bal. after the 79th mthly. Pmt. Is
3319.53
(Answer)
 

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