Mortgage Math

7/25/2019 Mortgage Math

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Mortgage Math

What is PV of $1000 per month for 15 monthsplus $10,000 paid 15 months from now at

10% nominal annual interest?

= (1!05"1000 # (0!0"10000

= $1,05 # $,0

= (PV&'!00$%%,15")PM* # (PV&'!00$%%,15")'V

( ) ( )1515 1210.1

000,10$

1210.1

000,1$

1210.1

000,1$875,22$

+

+

+

++

+

=


2/40

(With +al+ulator set to pmts at -./ of periods,and P23=14"

Mortgage Math Keys: DCF Keys:

1566667 . 8e9 1066667 &23 8e9

1066667 &23 8e9 0 66667 :'; 8e91000 66667 PM* 8e9 100066667 :'; 8e9

1000066667 'V 8e9 1 66667 .; 8e9

PV 66667 644,


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How the Calculator "Mortgage Math"Keys Work. . .

*he fie >mortgage math> 8e9s on 9our+al+ulator (.,&,PV,PM*,'V" sole

( ) ( ) ( ) NN rFV

r

PMT

r

PMTr

PMTPV

+

+

+

++

+

+

+

+=

11110

2


4/40

or0 = 6PV # (PV&'r,.")PM* #

(PV&'r,.")'V

where r = i m,

where i = .ominal annual interest rate

m = .um@er of pa9mentperiods per 9ear (mP23"!


5/40

-Aample

10%, 4069r full96amortiBing mortgage with pa9ments

of $1000month!*he +al+ulator soles the following eCuation for PV

*he result is PV = 10D45!

( ) ( ) ( )2402402 00833.1

0

00833.1

1000

00833.1

1000

00833.1

10000 +++++= PV


6/40

THE BASIC !ES #F CAC!ATI$% #A$&A'ME$TS ( BAA$CES

EetP = &nitial :ontra+t Prin+ipal (Eoan Falan+e at time Bero,

when mone9 is @orrowed"

rt= :ontra+t &nterest rate (per pa9ment period, e!g!,=im" appli+a@le for pa9ment in Period >t

&-t = &nterest portion of pa9ment in Period >t

PPt= Prin+ipal paid down (>amortiBed>" in the Period >t>

pa9mentGEFt= Gutstanding loan @alan+e after the Period >t>

pa9ment has @een made

PM*t= mount of the loan pa9ment in Period >t


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THE F#! BASIC !ES:

1" &-t = rt(GEFt61"

4" PPt = PM*tH &-t

" GEFt = GEFt61 6 PPtEquivalent to PV of remaining loan payments

" GEF0 = P

Know how to use these rules so that you cancalculate payment schedule, interest, principal,

and outstanding balance after each payment, for

any type of loan that can be dreamed up!


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A&&ICATI#$ #F THE F#! !ES T#S&ECIFIC #A$ T'&ES

1"'iAed63ate loans ('3Ms"

*he +ontra+t interest rate is +onstant

throughout the life of the loanrt=r, all t!

4" :onstant6Pa9ment loans (:PMs"

*he pa9ment is +onstant throughout the lifeof the loan

PM*t=PM*, all t!


9/40

" :onstant6mortiBation loans (:Ms"*he prin+ipal amortiBation is +onstant throughout the life

of the loan

PPt=PP, all t!

" 'ull96mortiBing loans&nitial +ontra+t prin+ipal is full9 paid off @9 maturit9 of

loan

PPt=P oer all t=1,,.!5" Partiall96mortiBing loans

Eoan prin+ipal not full9 paid down @9 due date of loanPPtIP, so GEF.must @e paid as @alloon at maturit9!


10/40

D" &nterest6Gnl9 loans

*he prin+ipal is not paid down until the end

PM*t=&-t, all t(eCuialentl9 GEFt=P, all t, and in +al+ulator eCuation 'V =

6PV"!


11/40

" d;usta@le63ate loans (3Ms"

*he +ontra+t interest rate aries oer time (rt

not +onstant, not 8nown for +ertain inadan+e, loan pa9ment s+hedules KeApe+ted 9ields must @e @ased onassumptions a@out future interest rates"!


12/40

:lassi+al 'iAed63ate

Mortgage*he +lassi+al mortgage is @oth '3M K

:PM

PM* = P(PV&'r,." = P L(1 H 1(1#r". "r


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)*+,+++, -/, 0+1year C&M...

MONTH BEG.

BAL.

INTEREST PMT PRIN END BAL.

1 $60,000.

00

$600.00 $617.17 $17.17 $59,982.

83

2 $59,982.

83

$599.83 $617.17 $17.34 $59,965.

49

3 $59,965.

49

$599.65 $617.17 $17.51 $59,947.

98


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Using Your Calculator

-2 Calculate oa3 &ay4e3ts:

E5a46le:$100,000 069ear 10% mortgage

with monthl9 pa9mentsD066667 .

1066667 &23

100000 66667 PV

0 66667 'V

PM*66667 6


15/40

2 Calculate oa3 A4ou3t7A88or9a;l;ty2:

E5a46le:2ou +an afford $500monthpa9ments on 069ear, 10% mortgage

D066667 .

1066667 &23

50066667 PM*066667 'V

PV66667 6 5D,N


16/40

If you can borrower 80% of house value,

how much can you affor !o "urchase#

urchase r&ce ' $5(,)75 * 0.80

urchase r&ce ' $71,218


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02 Calculate #utsta39;3g oa3 Bala3ce:E5a46le: What is the remaining @alan+e on $100,000,

10%, 069ear, monthl96pa9ment loan after 5 9ears

(after D0 pa9ments hae @een made"?'irst get loan terms in the registers

D066667 .

1066667 &23

10000066667 PV

066667 'VPM*66667 6 *hen +al+ulate remaining @alan+e either wa9 @elow. 66667 D0 .66667 00

'V 66667 6 ND,5


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19/40

*2 Meet a88or9a;l;ty co3stra;3t y tra9;3g o886ay4e3t a4ou3t w;th a4ort;=at;o3 rate:

E5a46le:Jo @a+8 to eAample 4 on the preiouspage! *he afforda@ilit9 +onstraint was a $500mopa9ment limit! Ouppose the $5D,N


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-nter

&23 = 10, PV = 65000, PM* = 500, 'V = 0,

:ompute . = 10!

*hus, the amortiBation rate would hae to @e10 months, or 9ears!

$ote:*his does not mean loan would hae to

hae a 69ear maturit9, it +ould still @e a 069ear 6art;ally1a4ort;=;3gloan, with @alloonof $40,45 due after 0 9ears!


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?2 Deter4;3;3g 6r;3c;6al ( ;3terest co46o3e3tso8 6ay4e3ts:E5a46le:'or the $100,000, 069ear, 10% mortgage in

pro@lem 1 on the preious page, @rea8 out the+omponents of the 14 pa9ments num@ering 50 throughD1!

&n the QP610F, after entering the loan as in pro@lem 1, enter50, &.PR*, D1, MG3*, = $N,DND!0D int, = $!0 prin, =$ND,501

GEFD1!*o get the +orresponding alues for the su@seCuent +alendar

9ear, press MG3* again, to get = $N,D0!D5 int, = $N44!41prin, =$N5,5


22/40

Eoan 2ields and Mortgage

ValuationEoan 2ield = -ffe+tie &nterest 3ate

2ield = &33 of loan

3e+all &33 @ased on +ash flows!


23/40

Using calculator equation:

( ) ( ) ( ) NN rFV

r

PMT

r

PMTr

PMTPV

+

+

+

++

+

+

+

+=

11110

2


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et:

"# $%&

'# $%t, t#,*,...,+

' - %" # $% +

+# olding eriod

where :';represents a+tual +ash flow at

end of period >;>!


25/40

*hen, @9 the definition of >r> in theeCuation a@oe, we hae

( ) ( ) NN

r

CF

r

CF

r

CFCFNPV

+

++

+

+

+

+==

1110

2

21

0


26/40

(@earing in mind that

-Apressed in nominal per annum terms (i=mr,where m=P23", we +an thus find the 9ield @9

+omputing the &23, proided the alues in the., PV, PM*, and 'V registers eCual theappropriate a+tual +ash flow and holding periodalues!

( ) ( ) ( ) ( ) NN

NNN

r

CF

r

FVPMT

r

FV

r

PMT

+=

+

+

=+

++ 1111


27/40

/n *ndary m0t, loans are priced so their yields

equal the 1m0t2s required yield3 (li0e e4pected

total return, 5(r)#rf-6, from before).

7t the time when a loan is originated (primary

mar0et), the loan yield is usually

appro4imately equal to its contract interest

rate. (8ut not e4actly9)


28/40

*he tr;cky 6artin loan 9ield +al+ulation(a" *he holding period oer whi+h we wish to +al+ulate

the 9ield ma9 not eCual the maturit9 of the loan (e!g!,

if the loan will @e paid off earl9, so . ma9 not @e theoriginal maturit9 of the loan" + maturity S

(@"*he a+tual time6Bero present +ash flow of the loanma9 not eCual the initial +ontra+t prin+ipal on theloan (e!g!, if there are >points> or other +losing +osts

that +ause the +ash flow dis@ursed @9 the lenderandor the +ash flow re+eied @9 the @orrower to noteCual the +ontra+t prin+ipal on the loan, P" $%&S


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(+"*he a+tual liCuidating pa9ment that pa9s off theloan at the end of the presumed holding periodma9 not eAa+tl9 eCual the outstanding loan

@alan+e at that time (e!g!, if there is a >prepa9mentpenalt9> for pa9ing off the loan earl9, then the@orrower must pa9 more than the loan @alan+e, so'V is then different from GEF" $%+'-O8+

Oo we must ma8e sure that the amounts in the., PV, and 'V registers refle+t the a+tual+ash flows


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Example

$200,000 mortgage, 30-year

maturity, monthly payments

10% annual interest

The loan has 2 points

(discount points or prepaid

interest) Also a 3 point prepayment

penalty through end of 5thyear.


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T What is 9ield (effe+tie interest rate" assumingholding period of 9ears (i!e!, @orrower will pa9loan off after months"?

T Frea8 this pro@lem into steps(1":ompute the loan +ash flows using the +ontra+t alues

of the parameters(.=D0, &=10%, PV=400000, 'V=0, :ompute PM*=$1


32/40

;tep ) +

&> /?6

*&&&&& > "& > %"

'> @AA.B

;tep *)BC> +

%"> DBC&B E .&< # *&&,=BD > %"D=&&& > "

;tep .**F


33/40

54pected yield (li0e 5(r) or 1goingin /663)

is .**F, eGen though 1contractual

interest rate3on the loan is only &F.(Hhen closing costs and prepayment

penalties are quoted in IpointsI, you do not

need to 0now the amount of the loan to find

its yield.)


34/40

General rule to calculate yield:

Change the amount in the PV Register

last,

(just prior to computing the yield!


35/40

5quiGalent solution to preGious problem:

Use $% 0eys instead of mortgage math 0eys9

D=&&& > $%J 0ey @AA.B > $%J 0ey

B@ > +J 0ey

*&*B&B > $%J 0ey/66 > .**F


36/40

Using Market Yields to

Value Mortgages(+ote: his is performing a $% +"

analysis of the loan as an inGestment,

finding what price can be paid for theloan so the deal is +"#&. 'ar0et2s

required yield is 1r3, the opportunity cost

of capital for the loan.)


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Example

T $100,000 mortgage, 069ear, 10%, points prepa9ment penalt9 @efore 5

9ears!T -Ape+ted time until @orrower prepa9s

loan = 9ears!

T Qow mu+h is the loan worth toda9 if themar8et 9ield is 11!00%?


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Otep 1"

D06667.,

106667&23,1000006667PV,

06667'V,

:ompute PM*6667 6


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Otep 4"6667.,

'V6667 6N


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Deter4;3;3g re@u;re9 9;scou3t6o;3ts 7or or;g;3at;o3 8ee2:

*o aoid lender doing .PV I 0 deal inma8ing loan, we need

(100,000 6 N,DN

Mortgage Math

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