Question

2. Irene and Frank Adam plan to purchase an 80 acre tract of land valued at $2,200 per acre. The lender charges a $500 loan application fee and $250 for a real estate appraisal. A stock requirement of 5% of the loan amount (after addition of fees) is required. The fees and stock requirement can be added to the original loan amount. The original stock value will be returned upon retiring the loan. The contractual rate is 8%. The fixed annual payments are based on a 20 year amortization period. The interest is calculated using the remaining balance method.

a. What is the effective annual interest rate for the loan?

b. What is the effective annual interest rate for the loan if there is no stock requirement?

c. What is the effective annual interest rate if the amortization period is lengthened to 30 years using the conditions of question a? Explain the difference from the answer in a.

Answer 1

The loan amount required = $2200 per acre * 80 acres = $176,000

Fees = $500+$250 = $750

Total Amount for stock requirement =$176,750

Stock requirement =$176750*5%=$8837.50 which will be released upon last payment

Total amount on which interest is charged = $176750+$8837.50 = $185587.50

So, A/0.08*(1-1/1.08^20) = 185587.50 where A is annual payment

=> A = $18902.50

a) Annual Effective rate (r) is given by

18902.50/r*(1-1/(1+r)^20)-8837.50/(1+r)^20 = 176000

Using hit and trial method

If r=0.09, Left hand side of above equation =170975.42

r=0.085, Left hand side of above equation =177151.93

r=0.086, Left hand side of above equation =175888.41

So, r lies between 8.5% and 8.6%, Using linear approximation method to get approximate value of r

r = 0.085+ (177151.93-176000)/(177151.93-175888.41)*(0.086-0.085) =0.085912

So,effective annual interest rate for the loan is 8.59% (Correct to three decimal places)

b) If there is no stock requirement

So, A/0.08*(1-1/1.08^20) = 176750 where A is annual payment

=> A = $18002.38

Annual Effective rate (r) is given by

18002.38/r*(1-1/(1+r)^20) = 176000

Using hit and trial method

If r=0.09, Left hand side of above equation =164335.53

r=0.081, Left hand side of above equation =175442.41

r=0.0805, Left hand side of above equation =176094.28

So, r lies between 8.05% and 8.1%, Using linear approximation method to get approximate value of r

r = 0.0805+ (176094.28-176000)/(176094.28-175442.41)*(0.081-0.0805) =0.080572

So,effective annual interest rate for the loan is 8.057% (Correct to three decimal places)

c)If amortization period is extended to 30 years

So, A/0.08*(1-1/1.08^30) = 185587.50 where A is annual payment

=> A = $16485.26

a) Annual Effective rate (r) is given by

16485.26/r*(1-1/(1+r)^30)-8837.50/(1+r)^30 = 176000

Using hit and trial method

If r=0.09, Left hand side of above equation =168697.78

r=0.085, Left hand side of above equation =176399.92

r=0.086, Left hand side of above equation =174812.79

So, r lies between 8.5% and 8.6%, Using linear approximation method to get approximate value of r

r = 0.085+ (176399.92-176000)/(176399.92-174812)*(0.086-0.085) =0.085252

So,effective annual interest rate for the loan is 8.525% (Correct to three decimal places)

The effective rate is slightly lesser than in a) as the extra cost incurred is spread over 30 years rather than 20 years. Hence the effective rate was 8.525% as compared to 8.591% in a)