REP Mortgage Calculation

8/2/2019 REP Mortgage Calculation

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Mortgage calculations anddecisions

Real Estate Principles: A Value Approach

Ling and Archer


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Outline

Fixed-payment calculations with noprepayment

Fixed-payment calculations withprepayment

ARM calculations

Refinancing


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TVM

The calculations in this chapter isbased 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.


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Loan amount = PV

The maximum amount a lender will bewilling to loan is the PV of the futurepayments that it expect to receive.

A 30-year, fixed-rate, LPM mortgage. Thequoted interest rate is 6%. The monthlypayment is $1,000. What is the loanamount?

360 N; 0.5 I/Y; 1000 PMT; CPT PV.

The answer is: PV = -166,791,6144.


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Monthly loan payments

A 15 fixed-rate mortgage. The loan amountis $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.


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Quoted rate

A 20-year mortgage. The monthlypayment 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%.


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Loan balance

The remaining balance on a fixed-payment loan isthe 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. Theanswer 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. Theanswer is -141,492.0117.


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Discount points

The actual interest payments of a loan to thelender are usually higher than the quotedrate would suggest.

Discount points: advance interest the lendercharge at the beginning of the loan contract.

For example, in the previous example, if the

lender charges discount points in theamount of $5,307.5676. Then the actualpayout to the borrower is $145,000($150,307.5676 $5,307.5676 = $145,000).


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Lenders yield (LY)

Because of discount points, the lenderlearns a higher yield, called lenders

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%.


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Effective borrowing cost (EBC)

In addition to quoted rate and discountpoints, the borrower needs to incur othercosts 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 theborrower 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%.


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Usual up-front financing costs

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 up-

front). Recording fee ($40-$200). Survey costs ($200-$300). Etc.


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Annual percentage rate (APR)

The Trust-in-Lending Act: the lender needsto 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 fewrelevant ones.

The calculation of APR is based on theassumption of no prepayment.


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Prepayment

Prepayment is the norm for residentialmortgages; households sell their homes

frequently. The calculations of LY and EBC are

sensitive to when a prepayment mayhappen.

Note that the previous LY and EBCcalculations are based on the assumption ofno prepayment.


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LY with prepayment

Prepayment is a major risk that introduces re-investment risk.

However, a prepayment would increase the lenders

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).


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EBC with prepayment

Similarly, a prepayment would increase theEBC.

The loan balance is $137,006.1412. The actual proceed received by the

borrowers after discount points and closingcosts 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).


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Question

The difference between the LY (EBC)with prepayment and the LY (EBC)

without prepayment is larger when theprepayment occurs earlier. Why?


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ARMs

One of the most popular is 1-year ARMbased on a 30-year amortization; that is, theinitial 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 thenext.

Overall cap: the cap that limits interest ratechange over the life of the loan. Teaser rate: many ARM loans are marketed

with a temporarily reduced interest rate.


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ARM example, I

A 1-year $100,000 ARM with a 30-yearamortization. The index rate is 1-year T-bill

rate, which is 3.25% now. The margin is2.75%. The teaser rate is 4.5%, though.

The monthly interest rate for the 1st year: 4.5/ 12 = 0.375%.

The monthly payment for the 1st year: 360N; 0.375 I/Y; 100,000 PV; CPT PMT -506.6853.


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ARM example, II

The balance after 1 year is: 348 N; 0.375I/Y; 506.6853 PMT; CPT PV -98,386.7714.

Suppose that the index rate remains at3.25% 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: 348N; 0.5 I/Y; 98386.7714 PV; CPT PMT -597.2122 (vs. 506.6853 for 1st year).


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ARM example, III

The balance after 2 years: 336 N; 0.5I/Y; 597.2122 PMT; CPT PV -

97,088.0967.


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Refinancing

The borrower may refinance after interestrate falls.

Whether to refinance is a very complex

investment decision because refinancing isnot 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 todayseems to be a good deal compared with theexisting loan.

Timing option.


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Refinancing example, I

Suppose that Alan has an existing loan witha remaining term of 15 years, a remainingbalance 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-frontrefinancing costs (fees) are 5% (usually 4-9%) of the loan amount, i.e., $5,000. Thus,the actual loan amount is $105,000 if therefinancing costs are amortized.


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Refinancing example, II

If refinancing, the monthly rate is 5 / 12 = 0.4167%.

The monthly payment is: 180 N; 0.4167 I/Y; 100000PV; CPT PMT -790.81.

The reduction in monthly payment: 898.83 790.81= $108.02.

Suppose that Alan can earn 6% on the $108.02saving.

If Alan expects to sell his house in 8 years, the PVof the expected benefits of refinancing is: 96 N; 0.5I/Y; 108.02 PMT; CPT PV -8,219.8055.


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Refinancing example, III

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 taxis: (8219.8055 (1

20%)) 5000 = $1,575.884.

NPV > 0, so refinancing is not a bad idea.

We focus on NPV after tax because mortgageinterest payments are tax deductible; the existingloan has higher interest expense and higher taxbenefits than the new (refinancing) one.


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Refinancing example, IV

Suppose that Alan also expects that theinterest rate will drop from 5% to 4% in 1month.

In 1 month, the existing loan has aremaining 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.


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Refinancing example, V

The new monthly payment is: 179 N;0.3333 I/Y; 99,687.1661 PV; CPT PMT -740.36.

The reduction in monthly payment: 898.83740.36 = $158.47.

Alan can earn 6% on the saving and expectto sell his house in 7 years and 11 months.

The PV of the expected benefits ofrefinancing is: 95 N; 0.5 I/Y; 158.47 PMT;CPT PV -11,960.63.


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Refinancing example, VI

The NPV of refinancing after taxis:(11960.63 (1 20%)) 5000 = $4,568.50.

This NPV is higher than that of financingnow ($1,575.884).

Thus, Alan will prefer to wait even thoughthe NPV for acting today is positive.

Is it optimal for Alan to refinance twice: nowand 1 month later?

REP Mortgage Calculation

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