Question

This problem is a complex financial problem that requires several skills, perhaps some from previous sections. During four years of college, Nolan MacGregor's student loans are $4,000, $3,500, $4,400, and $5,000 for freshman year through senior year, respectively. Each loan amount gathers interest of 2%, compounded quarterly, while Nolan is in school and 3%, compounded quarterly, during a 6-month grace period after graduation. (a) What is the loan balance (in dollars) after the grace period? Assume the freshman year loan earns 2% interest for 3/4 year during the first year, then for 3 full years until graduation. Make similar assumptions for the loans for the other years. (Round your answer to the nearest cent.)

(b) After the grace period, the loan is amortized over the next 10 years at 3%, compounded quarterly. Find the quarterly payment (in dollars). (Round your answer to the nearest cent.)

(c) If Nolan decides to pay an additional $90 per payment, how many payments will amortize the debt? (Round your answer to two decimal places.)
payments

What amount (in dollars) should be added to the last payment to pay the loan in full? (Round your answer to the nearest cent.)
$  

(d) How much will Nolan save (in dollars) by paying the extra $90 with each payment? (Round your answer to the nearest cent.)
$

Answer 1

FV of first year loan of $ 4000 at 2%/4= 0.5% or 0.005 per quarter ,at end of 9+4+4+4+1.5 =22.5 quarters---using FV of a single sum formula,

4000*1.005^22.5=

4475.03

FV of second year loan of $ 3500 at 2%/4= 0.5% or 0.005 per quarter ,at end of 9+4+4+1.5 =18.5 quarters---using FV of a single sum formula,

3500*1.005^18.5=

3838.31

FV of third year loan of $ 4400 at 2%/4= 0.5% or 0.005 per quarter ,at end of 9+4+1.5 =14.5 quarters---using FV of a single sum formula,

4400*1.005^14.5=

4729.99

FV of fourth year loan of $ 5000 at 2%/4= 0.5% or 0.005 per quarter ,at end of 9+1.5 =10.5 quarters---using FV of a single sum formula,

5000*1.005^10.5=

5268.82

(a)Loan balance (in dollars) after the grace period

Summation of the above 4 FVs at end of grace period,

4475.03+3838.31+4729.99+5268.82=

18312.15

18312

ANSWER:(a)Loan balance (in dollars) after the grace period = 18312

b. Loan amortisation--quarterly payment (in dollars)

The present value of the total loan = 18312

interest rate= 3% /4= 0.75% or 0.0075 per quarter

no.of quarterly periods=10*4=40

Qrtly payment --needs to be found out

using the formula for PV of ordinary annuity,

PV=Periodic Pmt.*(1-(1+r)^-n)/r

ie. 18312=pmt.*(1-1.0075^-40)/0.0075

Qtrly pmt.=18312/((1-1.0075^-40)/0.0075)=

531.60

or

532

ANSWER: b.quarterly payment (in dollars)= 532

c.If an additional $ 90 per payment is made , no.of payments that will amortize the debt:

Using the above same formula for PV of ordinary annuity,

& filling up all the known values , except n for no.of quarters, as under,

ie. 18312=(532+90)*(1-1.0075^-n)/0.0075

& solving for n, we get the no.of qrtrly. Payments as:

33 .39 payments

ANSWER:

c.If an additional $ 90 per payment is made , no.of payments that will amortize the debt= 33

Amount (in dollars) that should be added to the last payment to pay the loan in full:

can be found out by knowing the principal balance at end of 33rd pmt.

By using the foll.formula,

FV of loan bal.=FV of original loan value-FV of the annuity

ie. FV =(PV*(1+r)^n)-(Pmt.*((1+r)^n-1)/r)

where,

FV= Future value (principal) remaining of the loan

PV=Present value /original loan balance= 18312

Pmt.=qrtly pmt.= 532+90=622

r= rate /period, ie. 0.0075/qtr.

n=no.of qrtrly. Pmts.= 33

So, plugging in the values.

ie. FV =(18312*(1+0.0075)^33)-(622*((1+0.0075)^33-1)/0.0075)=

241.48

ANSWER:

Amount (in dollars) that should be added to the last payment to pay the loan in full= 241

(d) Amt. Nolan will save (in dollars) by paying the extra $90 with each payment:

Difference between the total pmts.

(532*40)-((622*33)+241)=

513