Question

(use equations, not computer)
Suppose that you are given the option to borrow a fixed rate US mortgage of $80,000 at 12% for 25 years with monthly payments. Alternatively, you may borrow another fixed rate US mortgage of $90,000 for 25 years with monthly payments at a contract interest rate to be determined. The lender would like to have an effective annual yield of 25% on the incremental cost of borrowing (i.e., on the $10,0000), reflecting the borrower’s increased default risk. Formulate how you would compute the contract interest rate on the entire $90,000 loan.

Answer 1

Effective monthly interest rate of the 12% p.a. mortgage=

(1+mthly rate)^12-1=12%

(1+mthly rate)=((0.12+1)^(1/12))

Mthly rate=((0.12+1)^(1/12))-1

0.009488793

0.9489%

As the lender would like to have an effective annual yield of 25% on the incremental cost of borrowing

Effective monthly interest rate of the incremental $ 10000 at 25% p.a. mortgage=

(1+mthly rate)^12-1=25%

(1+mthly rate)=((0.25+1)^(1/12))

mthly rate=((0.25+1)^(1/12))-1

0.018769

1.877%

Now, we can find the weighted average monthly interest as under:

Mortgage amt.	Wt. to total	Mthy.int.	Wt.*mthly int.
1	2=1/$ 90000	3(as above)	4=2*3
80000	88.89%	0.009489	0.00843
10000	11.11%	0.018769	0.00209
90000	1		0.01052
Converting to annual rate
(1+0.01052)^12-1=		0.13381
	ie.	13.38%
		p.a.

So, the ANSWER:

The contract interest rate on the entire $90,000 loan

will be the weighted average mthly .interest ratesso as to fetch effective annual interest rates of

12% p.a.on the first $ 80000 &

25% p.a.on the incremental $ 10000

ie. 13.38%