Chapter 18 Real Estate Finance Tools: Present Value and Mortgage Mathematics : Chapter 18 Real Estate Finance Tools: Present Value and Mortgage Mathematics
Major Topics : 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
Introduction to the Time Value of Money : 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
Present & Future Value of a Single Sum : 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
PV & FV over Multiple Periods of Time (Contd.) : PV & FV over Multiple Periods of Time (Contd.) General formula for PV and FV across multiple periods:
PV = FV / (1+r)N
FV = PV (1+r)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
PV of an Annuity : 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/(1+r)N
r
PV of Annuity (Contd.) : PV of Annuity (Contd.) For payments in advance the PV formula changes to:
PV = PMT (1+r) ---------------
Expressed in simple interest annual rate terms, the annuity formula assumes the forms:
1 – 1/(1+r)N
r PV = PMT ---------------------------- 1 – 1/(1 + i/m)(Tm)
i/m PMT = PV ---------------------------- i/m
1 – 1/(1 + i/m)(Tm)
Slide8 : 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/(1+r)N
Calculating a Loan Balance : 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 m=12) 1 – 1/(1 + i/m)(mT-q)
i/m 1 – 1/(1 + i/12)(12T-q)
i/12
Calculating the Principal and Interest Separation of a Mortgage (Contd.) : Calculating the Principal and Interest Separation of a Mortgage (Contd.) Example: A $150,000 30yr mortgage at 9%
Future Value of an Annuity : 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
(1+r)N – 1
r
Calculating Yields or Borrowing Costs : 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
More Mortgage Calcs on a Financial Calculator : 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
Effective Yield Calculation : 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)
Effective Yield Calculation : 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
Annual Percentage Rate (APR) : 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
Points – A tool to increase Yield : 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
Mortgage Pricing (Contd.) : 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
ARM and FRM : 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
ARM and FRM : 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
Choosing b/w FRMs and ARMs : 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
Refinancing : 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
END : END