Question

a) A borrower is purchasing a property for $180,000 and can choose between two possible loan alternatives. The first is a 90% loan for 25 years at 9% interest and 1 point and the second is a 95% loan for 25 years at 9.35% interest and 1 point. Assume the loan will be held to maturity. What is the incremental cost of borrowing the extra money?

b) Use the information in the above problem, except assume that the loan will be repaid in 5 years. What is the incremental cost of borrowing the extra money?

Answer 1

a. The first is a 90% loan for 25 years at 9% interest and 1 point

so ,Loan amount=180000*90%= 162000

Less: 1 point upfront(162000*1%)= 1620

So, net loan amt. = 162000-1620=160380

Calculating the monthly pmt.on this loan

using the PV of ordinary annuity formula,

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

with monthly r= 9%/12=0.0075 p.m. for 25*12=300 mths.

160380=Pmt.*(1-1.0075^-300)/0.0075

1345.90

Same way,

calculating the second loan alternative

so ,Loan amount=180000*95%= 171000

Less: 1 point upfront(162000*1%)= 1710

So, net loan amt. =171000-1710=169290

Calculating the monthly pmt.on this loan

using the PV of ordinary annuity formula,

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

with monthly r= 9.5%/12=0.0078 p.m.

169290=Pmt.*(1-1.0078^-300)/0.0078

1462.64

Difference (increased) in mthly pmts. For 25*12=300 mths.=

1462.64-1345.90=

116.74

Here, the extra cost towards points=

1710-1620=

90

Extra money got by increased interest rate=

171000-162000=

9000

Forming an equation of the PV of the annuity of additional amts. Borrowed less difference in points, to the increased annuity of mthly. Pmts.

(9000-90)=116.74*(1-(1+r)^-300)/r

solving for r, we get the increased cost as

1.28%

the incremental cost of borrowing the extra money =1.28%

b. Assuming that the loan will be repaid in 5 years

With the same net loan amt. after points,

Calculating the monthly pmt.on this loan

using the PV of ordinary annuity formula,

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

with monthly r= 9%/12=0.0075 p.m. for 5*12=60 mths.

160380=Pmt.*(1-1.0075^-60)/0.0075

3329.23

Calculating the monthly pmt.for the 2nd loan

using the PV of ordinary annuity formula,

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

with monthly r= 9.5%/12=0.0078 p.m. for 60 mnths.

169290=Pmt.*(1-1.0078^-60)/0.0078

3543.83

Now the

Difference (increased) in mthly pmts. For 5*12=60 mths.=

3543.83-3329.23=

214.6

Again, forming an equation of the PV of the annuity of additional amts. Borrowed less difference in points, to the increased annuity of mthly. Pmts.

(9000-90)=214.6*(1-(1+r)^-60)/r

solving for r, we get the increased cost as

1.30%

the incremental cost of borrowing the extra money =1.30%