Chapter 18
Real Estate Finance Tools:
Present Value and Mortgage
Mathematics
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
Major Topics
 Time value of money calculations
 Present value of a single sum or annuity
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payment
Future value of a single sum or annuity
Mortgage loan constants
Mortgage balance calculations
Point charges and their effects on borrowing
costs or yields
Annual Percentage Rate
Effective Cost of Borrowing
Net present value and IRR calculations
Refinancing decisions
Adjustable Rate Mortgage or ARM Calculations
Price Level Adjusted Mortgage
Reverse Annuity Mortgages (Future Value of
Annuity)
Supportable mortgage calculations
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
Introduction to the Time
Value of Money
 A dollar today is worth more than a dollar
received in future
 In most economies we expect a return on
money or capital related to the productivity
of things capital can buy
 This is the fundamental source of the real
returns (not just inflationary increases)
 The required returns are cumulatively
known as the opportunity cost of capital
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
Present & Future Value of a Single Sum
 PV = FV / (1+r)
 FV = PV (1+r)
 PV is the present value
 FV is future value
 r is the total expected rate of return

r includes the risk free and risk premium
rates

r is called “discount rate” when solving
for PV

r is called “rate of return” when solving
for FV
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
PV & FV over Multiple Periods of
Time (Contd.)
 General formula for PV and FV across
multiple periods:
 PV = FV / (1+r)
 FV = PV (1+r)
N
N
 N is the number of periods between FV and
PV
 If FV and PV are known the rate of return
can be found by the formula:
r = (FV/PV)
1/N
–1
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
PV of an Annuity
 Annuity: stream of regular payments of
equal amounts

E.g.: monthly rental payments,
mortgage payments
N
1 – 1/(1+r)
PV = PMT ----------------r
 ‘PMT’ is the equal amount of payments
occurring at end the of each consecutive
equal length period of time
 ‘N’ is the number of payments
 ‘r’ is the interest rate per period to time,
compounded at the end of each period
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
PV of Annuity (Contd.)
 For payments in advance the PV formula
changes to:
1 – 1/(1+r)
PV = PMT (1+r) --------------r
N
 Expressed in simple interest annual rate
terms, the annuity formula assumes the
forms:
(Tm)
1 – 1/(1 + i/m)
PV = PMT ---------------------------i/m
i/m
PMT = PV ---------------------------1 – 1/(1 + i/m)
(Tm)
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
Mortgage Constant
‘MMC’ is the monthly mortgage constant
It is the monthly payment per dollar of loan
and it includes both interest and principal
amortization
r
MMC = -----------------N
1 – 1/(1+r)
Here N & r are in months
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
Calculating a Loan Balance
 Outstanding Loan Balance (OLB) equals the
present value of the remaining loan
payments
 Original mortgage was for ‘T’ years at a rate
of ‘i’
 If ‘q’ payments have been made, the formula
will be:
(mT-q)
1 – 1/(1 + i/m)
OLB = PMT ---------------------------i/m
(12T-q)
1 – 1/(1 + i/12)
OLB = PMT ---------------------------i/12
(with m=12)
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
Calculating the Principal and Interest
Separation of a Mortgage (Contd.)
 Example: A $150,000 30yr mortgage at 9%
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
Future Value of an Annuity
 The FV of an annuity is the result of equal
payments compounding over time at a given
interest rate
 Used in RAM (Reverse Annuity Mortgage)
 Formula:
N
(1+r) – 1
FV = PMT ----------------r
 ‘PMT’ is the annuity paid every month
 ‘r’ is the interest per period (month)
 ‘n’ is the number of months
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
Calculating Yields or
Borrowing Costs
Recap of terms:
 Contract interest rate
 Index
 Spread
 Prime
 Prime Rate of Interest
 Discount Rate
 Carry cost
 Effective or true cost of borrowing
 Effective yield
 Contract rate
 Points
 Yield
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
More Mortgage Calcs on a
Financial Calculator
Inputs:
 PV =

I=
 N=
$240,000
8%
360
(Amount of Loan)
(divide by 12)
(30 year loan x 12
months/year)
 Solve for PMT
 Result
 PMT = ($1,761.03)
 The payment is based on the annuity that
equates to a present value of the
mortgage loan when discounted at the
contract rate of interest
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
Effective Yield Calculation
Loan Amount is $240,000 with 1.5 points and
prepayment expected in 10 years without
penalty
Step 1: Calculate actual loan amount
Loan Amount Disbursed
= $240,000 – 1.5%(240,000)
= $236,400 net dollars
Step 2: Calculate loan balance due at end of
10
years
PMT = ($1,761.03)
I = 8% (convert to monthly)
N = 240 (Months Remaining on the loan)
Compute
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
Effective Yield Calculation
Step 3: Calculate the lender's yield on the
amount disbursed, considering early
repayment
Enter PV = $ 236,400
Enter PMT = $(1,761.03)
Enter
N = 120 (The expected time until
prepayment)
Enter FV = $ (210,539)
Compute I = 8.23%
This is the effective cost of borrowing
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
Annual Percentage Rate (APR)
 When loans are held over full amortization
term the effective borrowing costs are
based on APR for annual percentage rate
 Truth in lending Act
 If there are no point charges, APR is equal
to effective borrowing costs
 APR is the yield which brings the future
payment stream back to present value such
that it exactly equals the net cash
disbursed by the lender
PV = Mortgage – Points = [1-{1/(1+APR12)N}/APR/12]* PMT
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
Points – A tool to increase Yield
 Lender’s perspective: Decrease contract
rate (looks attractive to borrower) and
increase points to compensate for it
 Question: How many points are needed to
bring a mortgage yield up given the contract
rate is lower than required yield?
 Steps (using business calculator)
 Find monthly payment and input as PMT
 Find mortgage balance (considering
payout) input as FV
 Input monthly interest rate (Required
yield/12)
 Input the number of periods
 Compute for PV
 Loan amount – PV will give the points
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
Mortgage Pricing (Contd.)
 Which loan is best for a borrower depends
on the expected tenure or time they expect
to hold the loan
 The 7.5% loan with 7 points is better if the
borrower is fairly certain they will hold the
loan for more then 10 years and if they don’t
believe rates will come down allowing them
to refinance before 10 years
 If the borrower is uncertain about holding
periods or future rates, the 8.6% loan is the
best choice with the lowest cost for
anything under a 10 year hold
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
ARM and FRM
 Fixed Rate Mortgage (FRM), where the rate
of interest charged remains constant
throughout the term
 Adjustable Rate Mortgage (ARM), where the
rate of interest and hence the mortgage
payment is variable due to the link with an
index
 Spread is the amount above the index that
is added to determine the new contract rate
of interest
 Typically ARMs are priced at significantly
lower interest rates as much of the future
interest rate risk is borne by the borrower
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
ARM and FRM
 Annual rate caps is the maximum increase
in the rate that is possible per year
 Life time caps is the maximum total
increase in the rate that is possible during
the loan term
 A 1.0% to 2.0% annual rate cap is common
 Typical life caps are 5% or 6% over the
course of the loan, so a loan that starts at
6% can never be higher then 11% if the life
cap is 5%
 To calculate the new payment we first need
the balance of the loan and then we use this
balance over the remaining term or N to
calculate payments at the new rate
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
Choosing b/w FRMs and ARMs
 FRM interest rate risk is borne by lender
 With ARMs much of the interest rate risk is
borne by the borrower
 Borrowers who are just able to qualify for
the mortgage with little excess in their
budget for the risk of higher payments will
often opt for the FRM, while wealthier
borrowers with few liquidity concerns will
often opt for the ARMS
 Rather than lower aspirations many
households will start to consider taking on
the risk of an ARM as rate rise and the
spread in the market between FRMs and
ARMs increases
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
Refinancing
 Refinancing can save borrower money if
there is a drop in mortgage interest rates
 Situations when refinancing is not
advisable:
 Remaining term of the loan is short or
expected tenure with new loan is short
 Mortgage rates are expected to further
drop
 Prepayment penalties are higher than
benefits
 Deciding whether refinancing is profitable
or not:
 NPV of expected savings exceeds the
cost of refinancing then it is advisable
and vice-versa
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner
END
“Real Estate Principles for the New Economy”: Norman G. Miller and David M. Geltner