Chapter 15: Mortgage
calculations and decisions
Real Estate Principles: A Value Approach
Ling and Archer
Outline
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Fixed-payment calculations with no
prepayment
Fixed-payment calculations with
prepayment
ARM calculations
Refinancing
TVM
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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.
Loan amount = PV
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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.
Monthly loan payments
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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.
Quoted rate
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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%.
Loan balance
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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.
Discount points
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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).
Lender’s yield (LY)
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Because of discount points, the lender
learns a higher yield, called “lender’s
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%.
Effective borrowing cost (EBC)
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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%.
Usual up-front financing costs
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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 upfront).
Recording fee ($40-$200).
Survey costs ($200-$300).
Etc.
Annual percentage rate (APR)
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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.
Prepayment
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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.
LY with prepayment
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Prepayment is a major risk that introduces reinvestment 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 = 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% > 7.36% (LY w/o prepay).
EBC with prepayment
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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% > 7.55% (EBC w/o
prepay).
ARMs
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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.
ARM example, I
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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% for the 1st
year.
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.
ARM example, II
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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).
ARM example, III
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The balance after 2 years: 336 N; 0.5
I/Y; 597.2122 PMT; CPT PV  97,088.0967.
Refinancing
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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.
Refinancing example, I
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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 39%) of the loan amount, i.e., $5,000.
Refinancing example, II
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If refinancing, the monthly rate is 5 / 12 = 0.4167%.
Suppose the $5000 fee is not amortized. 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.
Refinancing example, III
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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 > 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.
Refinancing example, IV
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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.
Refinancing example, V
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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.
Refinancing example, VI
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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?
Rule of thumb
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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.
Fee paying options
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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.