1. Chapter 16: Mortgage calculations and decisions
Real Estate Principles: A Value Approach
Ling and Archer

2. Outline Fixed-payment calculations with no prepayment
Fixed-payment calculations with prepayment
ARM calculations
Refinancing

3. TVM
The calculations in this chapter is based on time-value-of-money (TVM), which you learned in BSAD 180, 181, etc.
The financial calculator used in this course is Texas Instruments BAII Plus.

4. Loan amount = PV
The maximum amount a lender will be willing to loan is the PV of the future payments that it expect to receive.
A 30-year, fixed-rate, LPM mortgage. The quoted interest rate is 6%. The monthly payment is $1,000. What is the loan amount?
360 N; 0.5 I/Y; 1000 PMT; CPT PV.
The answer is: PV = -166,

5. Monthly loan payments
A 15 fixed-rate mortgage. The loan amount is $300,000. The quoted interest rate is 5.5%. What is the monthly payment?
I/Y = 5.5 / 12 = ; N = 15 × 12 = 180.
PV; 180 N; I/Y; CPT PMT.
The answer is: -2,

6. Quoted rate
A 20-year mortgage. The monthly payment is $2,000. The loan amount is $300,000. What is the quoted rate?
240 N; PMT; PV; CPT I/Y. The answer is: I/Y =
Quoted rate = × 12 = %.

7. Loan balance
The remaining balance on a fixed-payment loan is the PV of the remaining payments.
A 30-year mortgage. The monthly payment is $1,000. The quoted rate is 7% (monthly rate = 7% / 12 = %).
360 N; I/Y; 1000 PMT; CPT PV. The answer is -150,
What is the loan balance at the end of 5 years? The remaining months = 360 – 60 = 300.
300 N; I/Y; 1000 PMT; CPT PV. The answer is -141,

8. Discount points
The actual interest payments of a loan to the lender are usually higher than the quoted rate would suggest.
Discount points: advance interest the lender charge at the beginning of the loan contract.
For example, in the previous example, if the lender charges discount points in the amount of $5, Then the actual payout to the borrower is $145,000 ($150, – $5, = $145,000).

9. Lender’s yield (LY)
Because of discount points, the lender learns a higher yield, called “lender’s yield), than the quoted rate.
PV; PMT; 360 N; CPT I/Y. The answer is: I/Y =
The LY = I/Y × 12 = 7.36%.

10. Effective borrowing cost (EBC)
In addition to quoted rate and discount points, the borrower needs to incur other costs at the closing, called closing costs, such as title insurance, appraisal fee, etc.
Suppose that the closing costs are $ Then, the actual loan received by the borrower is $145,000 – $2,692 = $142,308.
PV; PMT; 360 N; CPT I/Y. The answer is: I/Y =
EBC = I/Y × 12 = 7.55%.

11. Usual up-front financing costs
Discount points.
Loan origination fee (e.g., 1% of the loan amount).
Loan application and document fees ($200-$700).
Appraisal ($250-$400).
Credit check ($35-$75).
Title insurance (0.5-1% of the loan).
Mortgage insurance (>2% of the loan if pay up-front).
Recording fee ($40-$200).
Survey costs ($200-$300).
Etc.

12. Annual percentage rate (APR)
The Trust-in-Lending Act: the lender needs to disclose APR of the loan to the borrower.
APR can be thought as a proxy for EBC.
The expense (closing costs) items to be included in calculating APR may omit a few relevant ones.
The calculation of APR is based on the assumption of no prepayment.

13. Prepayment
Prepayment is the norm for residential mortgages; households sell their homes frequently.
The calculations of LY and EBC are sensitive to when a prepayment may happen.
Note that the previous LY and EBC calculations are based on the assumption of no prepayment.

14. LY with prepayment
Prepayment is a major risk that introduces re-investment risk.
However, a prepayment would increase the lender’s return, i.e., LY, as well.
Suppose that the loan is expected to be paid off at the end of 7 years (84 months). Quoted rate is 7% (monthly rate = 7% / 12 = %).
The loan balance is: 276 N; I/Y; 1000 PMT; CPT PV  $137,
84 N; -145,000 PV; 1000 PMT; FV; CPT I/Y 
LY = I/Y × 12 = 7.68% > 7.36% (LY w/o prepay).

15. EBC with prepayment Similarly, a prepayment would increase the EBC.
The loan balance is $137,
The actual proceed received by the borrowers after discount points and closing costs is $142,308.
84 N; -142,308 PV; 1000 PMT; FV; CPT I/Y 
EBC = I/Y × 12 = 8.03% > 7.55% (EBC w/o prepay).

16. ARMs
One of the most popular ARMs is 1-year ARM based on a 30-year amortization; that is, the initial contract rate remains in effect for 1 year and adjusts annually thereafter.
Periodic cap: the cap that limits change in the interest rate from one change date to the next.
Overall cap: the cap that limits interest rate change over the life of the loan.
Teaser rate: many ARM loans are marketed with a temporarily reduced interest rate.

17. ARM example, I
A 1-year $100,000 ARM with a 30-year amortization. The index rate is 1-year T-bill rate, which is 3.25% now. The margin is 2.75%. The teaser rate is 4.5%, though.
The monthly interest rate for the 1st year: 4.5 / 12 = 0.375%.
The monthly payment for the 1st year: 360 N; I/Y; 100,000 PV; CPT PMT 

18. ARM example, II
The balance after 1 year is: 348 N; I/Y; PMT; CPT PV  -98,
Suppose that the index rate remains at 3.25% after 1 year.
The interest rate for the 2nd year: = 6%. Monthly rate is 0.5%.
The monthly payment for the 2nd year: 348 N; 0.5 I/Y; PV; CPT PMT  (vs for 1st year).

19. ARM example, III
The balance after 2 years: 336 N; 0.5 I/Y; PMT; CPT PV  -97,

20. Refinancing The borrower may refinance after interest rate falls.
Whether to refinance is a very complex investment decision because refinancing is not a one-time decision.
You can refinance later (say, 1 year later) when interest rate could be lower, instead of doing it today even though doing it today seems to be a good deal compared with the existing loan.
Timing option.

21. Refinancing example, I
Suppose that Alan has an existing loan with a remaining term of 15 years, a remaining balance of $100,000, and an interest rate of 7%. The existing monthly payment is $
Alan can refinance the loan for $100,000, the same 15 years, for 5%. But the up-front refinancing costs (fees) are 5% (usually 3-9%) of the loan amount, i.e., $5,000.

22. Refinancing example, II
If refinancing, the monthly rate is 5 / 12 = %.
Suppose the $5000 fee is not amortized. The monthly payment is: 180 N; I/Y; PV; CPT PMT 
The reduction in monthly payment: – = $
Suppose that Alan can earn 6% on the $ saving.
If Alan expects to sell his house in 8 years, the PV of the expected benefits of refinancing is: 96 N; 0.5 I/Y; PMT; CPT PV  -8,

23. Refinancing example, III
The up-front refinancing costs (fees) are 5% (usually 4-9%) of the loan amount, i.e., $5,000.
Suppose that Alan has a 20% marginal income tax rate.
The NPV of refinancing after tax is: ( × (1 – 20%)) – 5000 = $1,
NPV > 0, so refinancing is not a bad idea.
We focus on NPV after tax because mortgage interest payments are tax deductible; the existing loan has higher interest expense and higher tax benefits than the new (refinancing) one.

24. Refinancing example, IV
Suppose that Alan also expects that the interest rate will drop from 5% to 4% in 1 month.
In 1 month, the existing loan has a remaining term of 14 years and 11 months. The interest rate on the existing loan is 7%. The existing monthly payment is $
Thus, the remaining balance in 1 month is: 179 N; I/Y; PMT; CPT PV  -99,

25. Refinancing example, V
The new monthly payment is: 179 N; I/Y; 99, PV; CPT PMT 
The reduction in monthly payment: – = $
Alan can earn 6% on the saving and expect to sell his house in 7 years and 11 months.
The PV of the expected benefits of refinancing is: 95 N; 0.5 I/Y; PMT; CPT PV  -11,

26. Refinancing example, VI
The NPV of refinancing after tax is: ( × (1 – 20%)) – 5000 = $4,
This NPV is higher than that of financing now ($1, ).
Thus, Alan will prefer to wait even though the NPV for acting today is positive.
Is it optimal for Alan to refinance twice: now and 1 month later?

27. Rule of thumb
A widely used rule of thumb by practitioners and news media is that: refinance when the interest rate spread between existing loan and a new loan reaches about 2%.
Of course, this rule of thumb is very rough.

28. Fee paying options
3 main options for paying refinancing fees: (1) pay the fees up front, (2) opt for a higher mortgage rate instead of paying the fees, and (3) have the fees tacked on to the principal of the mortgage.