In: Finance
(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.
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% |