Part One
Our current loan balance=$ 130794.68, while monthly payments including premiums from escrew account=$1083.967
- Strategy for solving the problem
Looking at the loan duration, obviously the duration of twenty years will incur a higher monthly installment payment as opposed to the loan term of twenty five years. Assuming the escrow payment remains flat for the successive periods, we are going to calculate the monthly payments possibilities using financial annuity formulas. The interest rate 5.75% is assumed to be annual, so we will have to calculate a p-thly rate, in this case we compute a monthly interest rate.
Use annuity formulas for p-thly (in this case 12 months a year makes our p=12). Apply a discounting factor v in each annuity formula as will be done below. You can easily calculate the discounting factor by getting the reciprocal factor of the interest ratecharged by the bank.
- Step by step calculation
130794.68=X1a20¬p=X11-v20ipat i=5.75%, remember our p=12months.
v20=11+i20=11.057520=0.3269 and i12=121+i112-1
i12=1.0575112-1=0.05604
Therefore equating the above results to the annuity formula
130794.68=X11-v20i12=X11-0.32690.05604=12.0114X1
Getting the annual installment for a 20 year loan schedule
X1=130794.6812.0114=$10889.21 annualy
Hence the monthly payment excluding the escrow payment is got by dividing the above value by twelve.
=10889.2112=$907.43
Loan Schedule for 25 years
130794.68=X2a25¬p=X11-v25ipat i=5.75%, remember our p=12months.
v20=11+i25=11.057525=0.2472 and i12=121+i112-1
i12=1.0575112-1=0.05604
Therefore equating the above results to the annuity formula
130794.68=X21-v25i12=X21-0.24720.05604=13.4338X2
Getting the annual installment for a 25 year loan schedule
X2=130794.6813.4338=$9736.20 annualy
Hence the monthly payment excluding the escrow payment is got by dividing the above value by twelve.
=9736.2012=$811.35
We get the difference between the monthly payments for both loan schedules
=907.43-811.35=$96.08
Clearly $96<$100 of monthly expense.
- Clearly it would be a bad decision to take a loan schedule of 20 years since he will need an additional 96 dollars to meet the obligation of repayment. Considering he gets to uses less than 100 dollars on expenses left over he will need the 96 dollars to meet his expenses.
Part two
- Explaining the strategy
The strategy to use here is to use an extrapolation formula to predict the highest possible interest rate. At 5.75%the annuity factor is equal to 12.0114. Then you will get the annuity factor at a much higher interest level say 6%. Assuming for the remaining term of 20 years you still pay a level monthly of $1083.97, you get the annuity factor.
Using the three annuity factors and two known interest rates, you predict the third interest rate level by using an extrapolation method.
- Assuming level payment for the loan at $1083.97
Refinancing cost will increase the loan amount to a new level
=130794.97+2000=$132794.68
If the level monthly payment=$1083.97 then the annual payment is=1083.97*12=$13007.64
Hence 13007.64a20¬12=132794.68=>a20¬12=132794.6813007.64=10.2089
at an interest rate we want to get by using extrapolation formula.
We project the annuity factor at 6% and use it in the extrapolation formula
a20¬(12)=(1-v20)/i(12)
i12=121.06112-1=0.05841
v20=11.0620=0.3118
Therefore at 6%,a20¬12=1-0.31180.05841=11.782
Applying the extrapolation formula to our interest rate
i-5.756-5.75=10.2089-12.011411.782-12.0114
=>i-5.75=0.257.857=1.964
=>i=5.75+1.964=7.714%
This converted to the nearest quarter point =7.75% interest rate.
- Refinancing to repay the loan within the 20 years will not be favorable because of the increased interest payment plus you will not able to meet the monthly expenses that less than 100 dollars. An increase of 2% in the interest rate can greatly influence the interest payment but the capital payment may not be greatly affected.
References
Tietze, W., & Fallon, K. (2004). Financial maths. Sandgate, Qld: Knowledge Books and Software.
Financial Publishing Company. (1955). Direct reduction loan amortization schedules for loans with quarterly, semiannual & annual payments. Boston.
The amortization handbook: Covering monthly payment schedules, loan amounts, real estate. (1986). Stamford, Conn: Longmeadow Press.
