Chapter 18 Real Estate Finance Tools: Present Value and Mortgage Mathematics

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Chapter 18Real Estate Finance Tools Present
Value and Mortgage Mathematics
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Major Topics

• Time value of money calculations
• Present value of a single sum or annuity 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

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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

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Present Future Value of a Single Sum

• PV FV / (1r)
• FV PV (1r)
• 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

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PV FV over Multiple Periods of Time (Contd.)

• General formula for PV and FV across multiple
  periods
• PV FV / (1r)N
• FV PV (1r)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

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PV of an Annuity

• Annuity stream of regular payments of equal
  amounts
• E.g. monthly rental payments, mortgage payments
• PV PMT -----------------
• 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

1 1/(1r)N r

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PV of Annuity (Contd.)

• For payments in advance the PV formula changes
  to
• PV PMT (1r) ---------------
• Expressed in simple interest annual rate terms,
  the annuity formula assumes the forms

1 1/(1r)N r
1 1/(1 i/m)(Tm) i/m
PV PMT ----------------------------
i/m 1 1/(1 i/m)(Tm)
PMT PV ----------------------------

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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
• MMC ------------------
• Here N r are in months

r 1 1/(1r)N

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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
• OLB PMT ----------------------------
• OLB PMT ----------------------------
• (with m12)

1 1/(1 i/m)(mT-q) i/m
1 1/(1 i/12)(12T-q) i/12

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Calculating the Principal and Interest Separation
of a Mortgage (Contd.)

• Example A 150,000 30yr mortgage at 9

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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
• FV PMT -----------------
• PMT is the annuity paid every month
• r is the interest per period (month)
• n is the number of months

(1r)N 1 r

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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

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More Mortgage Calcs on a Financial Calculator

• Inputs
• PV 240,000 (Amount of Loan)
• I 8 (divide by 12)
• N 360 (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

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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 PV (210,539) (Use as FV
input)
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Effective Yield Calculation
Step 3 Calculate the lender's yield on the
amount disbursed, considering early repayment En
ter 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
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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/(1APR12)N/APR/12
  PMT

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Points A tool to increase Yield

• Lenders 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

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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 dont
  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

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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

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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

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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

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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

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