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

Chapter 18

Real Estate Finance Tools: Present Value and Mortgage

Mathematics


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

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

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.)

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 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.) 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)


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 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.)

Example: A $150,000 30yr mortgage at 9%


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

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

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 CalculationLoan 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 10yearsPMT = ($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

Step 3: Calculate the lender's yield on the amount disbursed, considering earlyrepaymentEnter 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)

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 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.)

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 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 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 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 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

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

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