Chapter 16: Mortgage calculations and decisions

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 -166,791,6144.

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 0.4583 N 15 12 180.
• 300000 PV 180 N 0.4583 I/Y CPT PMT.
• The answer is -2,451.1867.

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 -2000 PMT 300000 PV CPT I/Y. The answer
  is I/Y 0.4268.
• Quoted rate 0.4268 12 5.1216.

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 0.5833).
• 360 N 0.5833 I/Y 1000 PMT CPT PV. The answer
  is -150,307.5679.
• What is the loan balance at the end of 5 years?
  The remaining months 360 60 300.
• 300 N 0.5833 I/Y 1000 PMT CPT PV. The answer
  is -141,492.0117.

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
  5,307.5676. Then the actual payout to the
  borrower is 145,000 (150,307.5676 5,307.5676
  145,000).

9
Lenders yield (LY)

• Because of discount points, the lender learns a
  higher yield, called lenders yield), than the
  quoted rate.
• 145000 PV -1000 PMT 360 N CPT I/Y. The answer
  is I/Y 0.6133.
• 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 2692. Then,
  the actual loan received by the borrower is
  145,000 2,692 142,308.
• 142308 PV -1000 PMT 360 N CPT I/Y. The answer
  is I/Y 0.6292.
• 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 (gt2 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 lenders
  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 0.5833).
• The loan balance is 276 N 0.5833 I/Y 1000 PMT
  CPT PV ? 137,006.1412.
• 84 N -145,000 PV 1000 PMT 137006.1412 FV CPT
  I/Y ? 0.6399.
• LY I/Y 12 7.68 gt 7.36 (LY w/o prepay).

15
EBC with prepayment

• Similarly, a prepayment would increase the EBC.
• The loan balance is 137,006.1412.
• The actual proceed received by the borrowers
  after discount points and closing costs is
  142,308.
• 84 N -142,308 PV 1000 PMT 137006.1412 FV CPT
  I/Y ? 0.6695.
• EBC I/Y 12 8.03 gt 7.55 (EBC w/o prepay).

16
Question

• The difference between the LY (EBC) with
  prepayment and the LY (EBC) without prepayment is
  larger when the prepayment occurs earlier. Why?

17
ARMs

• One of the most popular 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.

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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
  0.375 I/Y 100,000 PV CPT PMT ? -506.6853.

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ARM example, II

• The balance after 1 year is 348 N 0.375 I/Y
  506.6853 PMT CPT PV ? -98,386.7714.
• Suppose that the index rate remains at 3.25
  after 1 year.
• The interest rate for the 2nd year 3.25 2.75
  6. Monthly rate is 0.5.
• The monthly payment for the 2nd year 348 N 0.5
  I/Y 98386.7714 PV CPT PMT ? -597.2122 (vs.
  506.6853 for 1st year).

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ARM example, III

• The balance after 2 years 336 N 0.5 I/Y
  597.2122 PMT CPT PV ? -97,088.0967.

21
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.

22
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 898.83.
• Alan can refinance the loan for 100,000, the
  same 15 years, for 5. But the up-front
  refinancing costs (fees) are 5 (usually 4-9) of
  the loan amount, i.e., 5,000. Thus, the actual
  loan amount is 105,000 if the refinancing costs
  are amortized.

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Refinancing example, II

• If refinancing, the monthly rate is 5 / 12
  0.4167.
• The monthly payment is 180 N 0.4167 I/Y 100000
  PV CPT PMT ? -790.81.
• The reduction in monthly payment 898.83 790.81
  108.02.
• Suppose that Alan can earn 6 on the 108.02
  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 108.02 PMT CPT PV ? -8,219.8055.

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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 (8219.8055
  (1 20)) 5000 1,575.884.
• NPV gt 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.

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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 898.83.
• Thus, the remaining balance in 1 month is 179 N
  0.5833 I/Y 898.83 PMT CPT PV ? -99,687.1661.

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Refinancing example, V

• The new monthly payment is 179 N 0.3333 I/Y
  99,687.1661 PV CPT PMT ? -740.36.
• The reduction in monthly payment 898.83 740.36
  158.47.
• 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 158.47 PMT CPT PV ?
  -11,960.63.

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Refinancing example, VI

• The NPV of refinancing after tax is (11960.63
  (1 20)) 5000 4,568.50.
• This NPV is higher than that of financing now
  (1,575.884).
• 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?