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{SECT 0 {PARA 4 "" 0 "" {TEXT -1 29 "Module 3 : Finite Mathematics" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 23 "303 : Amo
rtizing A Loan" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 17 "O B J E C T I V E" }}{PARA 0 "" 0 "" {TEXT -1 269 "We wil
l continue with defining financial functions and show how they can be \+
used to analyze amortization situations numerically and graphically. W
e will also look at graphs showing how much principal and interest are
 being at any given time as a loan is being paid off." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 9 "S E T U P" }}{PARA 0 
"" 0 "" {TEXT -1 252 "In this project we will use the following comman
d packages. Type and execute this line before begining the project bel
ow. If you re-enter the worksheet for this project, be sure to re-exec
ute this statement before jumping to any point in the worksheet." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "restart; with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the
 name changecoords has been redefined\n" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "__________________________________
_________________________________________________" }}{PARA 4 "" 0 "" 
{TEXT -1 26 "A. Amortized Loan Payments" }}{PARA 0 "" 0 "" {TEXT -1 
83 "__________________________________________________________________
_________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 21 "COMPUTING THE PAYMENT" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "First w
e define a financial function to give us the monthly payment for an am
ortized loan." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 138 " Monthly_Payment := (Principal,annual_rate, yea
rs) -> \n                    Principal*annual_rate/ (12*(1-(1+annual_r
ate/12)^(-12*years)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0Monthly_P
aymentGR6%%*PrincipalG%,annual_rateG%&yearsG6\"6$%)operatorG%&arrowGF*
*&*&9$\"\"\"9%F1F1,&\"#7F1*&F4F1),&F1F1*&#F1F4F1F2F1F1,$9&!#7F1!\"\"F=
F*F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
198 "The inputs to this function are the principal of the loan, the an
nual percentage interest rate in decimal form and the number of years \+
to pay it off. The result is the amount of the monthly payment." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Here is a
n example of a loan of $165,000 @ 7.375% for 30 years." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " Monthl
y_Payment(165000, .07375 , 30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
+5ShR6!\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 
"_____________________________________________________________________
______________" }}{PARA 4 "" 0 "" {TEXT -1 42 "B. How Does Interest Ra
te Affect Payments?" }}{PARA 0 "" 0 "" {TEXT -1 83 "__________________
_________________________________________________________________" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 178 "How does the interest rate affect the payment?
 If we fix a loan amount of $165,000 and repayment period of 30 years,
 we can create a graph which will illustrate the relationship." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
121 " plot( [0, Monthly_Payment( 165000,  x, 30)], \n         x = .06.
.(.09),  labels = [`interest rate`, `monthly payment` ]);" }}{PARA 13 
"" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"3y****
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$\"3o***\\Peui='F*F+7$$\"3[++D'3&o]iF*F+7$$\"3:+](oX*y9jF*F+7$$\"3#)**
\\P9CAujF*F+7$$\"3\\+]P*zhdV'F*F+7$$\"3]**\\P>fS*\\'F*F+7$$\"3))**\\(=
$f%Gc'F*F+7$$\"3l****\\#y,\"GmF*F+7$$\"37++Dr\"zbo'F*F+7$$\"31++](4&G]
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P%y!eP(F*F+7$$\"3K+](=WU[V(F*F+7$$\"3U++DJ#>&)\\(F*F+7$$\"3E+]P>:mkvF*
F+7$$\"3o**\\iv&QAi(F*F+7$$\"3O++vtLU%o(F*F+7$$\"37+++bjm[xF*F+7$$\"38
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F+7$$\"3J+]PfL'zV)F*F+7$$\"3u*******)>=+&)F*F+7$$\"39***\\i_4Qc)F*F+7$
$\"3d**\\P%>5pi)F*F+7$$\"3#3++]:$*[o)F*F+7$$\"3w++Dr\"[8v)F*F+7$$\"3c+
++Ijy5))F*F+7$$\"3Y**\\P/)fT())F*F+7$$\"3!***\\i0j\"[$*)F*F+7$$\"3o***
************)F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-F$6$7S7$F($\"3!*40-l
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$\"3%)=%)=7:&o0\"F[v7$FL$\"3mk,L338j5F[v7$FO$\"3s-eyG\"=-2\"F[v7$FR$\"
3EBoLPRNx5F[v7$FU$\"3Ys=&)e)[U3\"F[v7$FX$\"3gkH4;b_!4\"F[v7$Fen$\"3aN!
['>!3!)4\"F[v7$Fhn$\"3V0FAwEO/6F[v7$F[o$\"3>3batHx66F[v7$F^o$\"3I(pOSm
[$=6F[v7$Fao$\"3s)=\"))R2eD6F[v7$Fdo$\"3Pk\"*z\"G%[K6F[v7$Fgo$\"3>i<#*
R[qR6F[v7$Fjo$\"3gU)oM8^j9\"F[v7$F]p$\"3?7RRSm``6F[v7$F`p$\"3_*e**HW=5
;\"F[v7$Fcp$\"3#*H&3zDYv;\"F[v7$Ffp$\"3I$)fe<>hu6F[v7$Fip$\"3!**z$yX\"
G>=\"F[v7$F\\q$\"3;s5&3&>5*=\"F[v7$F_q$\"3q*f?A?eg>\"F[v7$Fbq$\"31Kn1N
$*z.7F[v7$Feq$\"3AM\\%)z0x57F[v7$Fhq$\"3Kd$[[#*H#=7F[v7$F[r$\"3;W%fPS.
]A\"F[v7$F^r$\"3,8\"o5$RUK7F[v7$Far$\"3q3%f#p0UR7F[v7$Fdr$\"3OK*=g(*[n
C\"F[v7$Fgr$\"33(eq'R%GRD\"F[v7$Fjr$\"3qbH_:+Yh7F[v7$F]s$\"3%**\\/$H&G
(o7F[v7$F`s$\"3i-zkMk<w7F[v7$Fcs$\"3'*)>06KxNG\"F[v7$Ffs$\"3Q6?zB2R!H
\"F[v7$Fis$\"3#)yA?[X@)H\"F[v7$F\\t$\"3F5.VMcA08F[v7$F_t$\"3b[x$Gp9FJ
\"F[v7$Fbt$\"3NZUS\")e*)>8F[v7$Fet$\"3&))))ezJFwK\"F[v-Fht6&FjtF+F[uF+
-%+AXESLABELSG6$%.interest~rateG%0monthly~paymentG-%%VIEWG6$;$\"\"'!\"
#$\"\"*Fc_l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 105 "We can see that as interest rates increa
se, payments increase also in an essentially linear relationship." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 83 "____________________________________________________
_______________________________" }}{PARA 4 "" 0 "" {TEXT -1 40 "C. How
 Big of A Mortgage Can You Afford?" }}{PARA 0 "" 0 "" {TEXT -1 83 "___
______________________________________________________________________
__________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 319 "The first question a home buye
r needs to ask is How big of a mortgage loan can I afford? We can answ
er this question knowing the prevailing interest rate, the length of t
he loan, and a comfortable payment for the buyer. Lets say that a buye
r is comfortable paying $1,000 per month and get a loan at 7.375% for \+
30 years." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 " fsolve( Monthly_Payment(x, .07375, 30) = 1000,x);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+['eyW\"!\"%" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 
"_____________________________________________________________________
______________" }}{PARA 4 "" 0 "" {TEXT -1 41 "D. How Do Interest Rate
s Affect Payments?" }}{PARA 0 "" 0 "" {TEXT -1 83 "___________________
________________________________________________________________" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 71 "For a given interest rate, plot the mortgage ve
rsus the payment amount." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 53 "For a given interest rate, plot mortgage vs. payme
nt." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 115 " plot( solve( Monthly_Payment(y, .07375, 30) = x, y)
, x = 900..1200, \n               labels = [payment, mortgage]);" }}
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3d**\\PC#)GA\"*F1$\"3!4*4_*Qy2K\"F-7$$\"3Y***\\Peui=*F1$\"3(R])*)4F/I8
F-7$$\"3]++D'3&o]#*F1$\"3.p^?S%o$R8F-7$$\"3f**\\(oX*y9$*F1$\"39pN%p%)
\\'[8F-7$$\"3?+]P9CAu$*F1$\"3eGG4**[Dd8F-7$$\"3,+]P*zhdV*F1$\"3a8$p@$
\\;m8F-7$$\"3;+]P>fS*\\*F1$\"3i[H7,(z`P\"F-7$$\"3O+](=$f%Gc*F1$\"3'Qo&
>=\\c%Q\"F-7$$\"3N++]#y,\"G'*F1$\"37%ohH/8SR\"F-7$$\"3=++Dr\"zbo*F1$\"
3n)H)Rc\\L-9F-7$$\"37++](4&G](*F1$\"3')*3)))fMq69F-7$$\"3f****\\7nD:)*
F1$\"3S<LNJ/6@9F-7$$\"3U****\\-*oy()*F1$\"3L*[H9zv,V\"F-7$$\"3b**\\Ppn
sM**F1$\"3npc&o+3%Q9F-7$$\"3/++DFOB+5!#9$\"37@VB2p>[9F-7$$\"3.+++R5'f+
\"Fgp$\"3O:s')*Q*[c9F-7$$\"3.+vV!QBE,\"Fgp$\"3cGM`=b8m9F-7$$\"3-++]\"o
?&=5Fgp$\"3))Gm(o(Rnu9F-7$$\"3\"**\\Pa&4*\\-\"Fgp$\"3!p'>&*=?/%[\"F-7$
$\"3\"**\\7j=_6.\"Fgp$\"3WgmN5E'H\\\"F-7$$\"3!***\\P%y!eP5Fgp$\"34nfl6
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if,'=P+_\"F-7$$\"3%**\\P>:mk0\"Fgp$\"3_\"*>SaOhH:F-7$$\"3#**\\iv&QAi5F
gp$\"3$ey$G)**\\z`\"F-7$$\"33+]PPBWo5Fgp$\"3kFx0yM&pa\"F-7$$\"3/++]Nm'
[2\"Fgp$\"3#*Q`l`\\Dc:F-7$$\"3)***\\(yb^63\"Fgp$\"3Y\"*z'Gja`c\"F-7$$
\"3++vVVDB(3\"Fgp$\"31?ISS!fTd\"F-7$$\"35+]7TW)R4\"Fgp$\"3Ky-'Q$[$Re\"
F-7$$\"34++]@80+6Fgp$\"3q.w2>)=Ff\"F-7$$\"3%****\\7,Hl5\"Fgp$\"3w,7?)f
(4-;F-7$$\"3!**\\P4w)R76Fgp$\"3(Q%y/mhf5;F-7$$\"33++vZf\")=6Fgp$\"3'R:
qaM())>;F-7$$\"3\"**\\P/-a[7\"Fgp$\"3C.!G7iH'G;F-7$$\"3-+v=Yb;J6Fgp$\"
3_:sO=ywP;F-7$$\"3,++D;iLP6Fgp$\"3)3;\"Qw?qY;F-7$$\"3')*\\PfL'zV6Fgp$
\"3YAy`;a0c;F-7$$\"35+++*>=+:\"Fgp$\"3QN&zZzj]m\"F-7$$\"3'***\\i_4Qc6F
gp$\"3_%eSE;wUn\"F-7$$\"3-+vV>5pi6Fgp$\"3w?XWZAT$o\"F-7$$\"3)*****\\:$
*[o6Fgp$\"3UvAagt!=p\"F-7$$\"3%***\\7<[8v6Fgp$\"3')H.a2\"H9q\"F-7$$\"3
3+++L'y5=\"Fgp$\"3_oqv7\\.5<F-7$$\"3#**\\P/)fT(=\"Fgp$\"3JL)\\g\\5#><F
-7$$\"3'**\\i0j\"[$>\"Fgp$\"3&p0R+q#*zs\"F-7$$\"%+7F*$\"32++gx.VP<F--%
'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fa[l-%+AXESLABELSG6$%(paymentG%)mortgag
eG-%%VIEWG6$;F(Ffz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 334 "Often when interest rates go down, home \+
prices go up. This occurs because lower interest rates mean more peopl
e can afford more expensive homes. There is an inverse relationship be
tween interest rate and mortgage amount if payment and repayment perio
d are fixed. For example, let's consider a payment of $1,200 per month
 for 30 years." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 96 "array( [ seq( [ evalf(6+k/4,4),fsolve(Monthly_Pa
yment( P, (6+k/4)/100,30)=1200 )], k = 0..12)]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7/7$$\"\"'\"\"!$\"+t$*\\,?!\"%7$$\"%]i!\"$$
\"+!pY*[>F-7$$\"%+lF1$\"+M)H&)*=F-7$$\"%]nF1$\"+!>W,&=F-7$$\"\"(F*$\"+
:3p.=F-7$$\"%]sF1$\"+9h2f<F-7$$\"%+vF1$\"+G:@;<F-7$$\"%]xF1$\"+XK,v;F-
7$$\"\")F*$\"+I>SN;F-7$$\"%]#)F1$\"+nCI(f\"F-7$$\"%+&)F1$\"+@Pkg:F-7$$
\"%]()F1$\"+3$e`_\"F-7$$\"\"*F*$\"+)Q#Q\"\\\"F-" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "This gives a table which
 shows interest rates from 6% to 9% in 1/4% increments and the corresp
onding mortgage amount." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 83 "____________________________________________________
_______________________________" }}{PARA 4 "" 0 "" {TEXT -1 19 "E. Pay
off of A Loan" }}{PARA 0 "" 0 "" {TEXT -1 83 "________________________
___________________________________________________________" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "When you pay o
ff a mortgage loan, part of your payment reduces the principal and par
t is interest. But how much the payment goes toward each?" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 261 "The answer will t
urn out to vary month by month. This is important because when you hav
e a home loan you can get a tax write-off only for the portion of your
 payment that is interest. The part of the payment which is paying off
 principal is not a tax deduction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 73 "To answer this question, we will create a
 special graph function called \"" }{TEXT 256 6 "payoff" }{TEXT -1 2 "
\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1621 "\011restart;\n\011payoff := proc( principal, annual
_interest_rate, years)    \n\011\011local monthly_rate,payment_no,new_
princ, payment,n,po,ip, x1,x2, y1,y2, y3,intpaid;\n\011\011new_princ :
= principal;\n\011\011monthly_rate  := evalf(annual_interest_rate/1200
); \n\011\011payment := principal*monthly_rate/(1-(1+monthly_rate)^(-1
2*years));\n\011\011n := 12*years; x2 := 0;\n\011\011for payment_no fr
om 1 to n do\n\011\011intpaid := new_princ*monthly_rate;  new_princ :=
 new_princ - payment + intpaid; \n\011\011x1 := x2; x2 := evalf(paymen
t_no/12,4);\n\011\011ip[payment_no] := plots[polygonplot]( [[x1,0],[x1
,intpaid],[x2, intpaid],[x2,0]], \n\011\011\011\011\011\011\011color =
 blue, style=patchnogrid );  \n\011\011if (payment_no = round(3*years)
) then y1:= intpaid;fi;\n\011\011if (payment_no = round(6*years)) then
 y2:= intpaid;fi;\n\011\011if (payment_no = round(9*years)) then y3:= \+
intpaid;fi;   \n\011od;    \n\011plots[display]( seq( ip[i], i = 1..n)
,\n\011plot( \{payment, [[n,0],[n,payment]]\},x = 0..years, thickness \+
= 2, color = red),\n\011plot( \{y1,y2,y3\},x = 0..years,thickness = 1,
 color = green),\n\011plot(\{\011[[years,0],[years,payment]],[[years/4
,0], [years/4,payment]],\n\011\011\011\011[[years/2,0],[years/2,paymen
t]],[[3*years/4,0],[3*years/4,payment]]\},\n\011\011\011\011x = 0..yea
rs, thickness = 1, linestyle = 2, color = coral),\n\011plots[textplot]
([evalf(years/4),y1, cat(convert(evalf(100*y1/payment,4) ,string),\"%
\")],\n\011\011\011\011align=\{ABOVE,RIGHT\},font=[HELVETICA,BOLD,12])
,\n\011plots[textplot]([evalf(years/2),y2,cat(convert(evalf(100*y2/pay
ment,4) ,string),\"%\")],\n\011\011\011align=\{ABOVE,RIGHT\},font=[HEL
VETICA,BOLD,12]),\n\011plots[textplot]([evalf(3*years/4),y3, cat(conve
rt( evalf(100*y3/payment,4) \011\011\n\011\011\011,string),\"%\")],ali
gn=\{ABOVE,RIGHT\},font=[HELVETICA,BOLD,12]) );\nend:" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "Lets see how it work
s. Lets take a mortgage amount of $177,00 for 30 years @ 7 5/8%." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 " payoff( 177000, 7 + 5/8, 30);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 
300 300 {PLOTDATA 2 "6bbl-%)POLYGONSG6%7&7$$\"\"!F)F(7$F($\"++voC6!\"'
7$$\"%L$)!\"&F+7$F/F(-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%&STYLEG6#%,
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