## Wednesday, February 14, 2018

### HP Prime and TI-84 Plus CE: Graduated Mortgage Payments

HP Prime and TI-84 Plus CE:  Graduated Mortgage Payments

Introduction

The program GRADMORT calculates the payment schedule of a graduated mortgage.  A graduated payment mortgage (GPM) is a mortgage option that allows home buyers, particularly young, first-time home buyers, to purchase a home.  The payments increase by a set percentage of at annual rate for several years until it reaches a plateau.

According to Daniel T. Winkler’s and G. Donald Jud’s article, “The Graduated-Payment Mortgage: Solving the Initial Payment Enigma” (see Source at the end of the blog entry), the HUD (United States Department of Housing and Urban Development), offers five graduated payment schedule plans, known as Section 245 loans.

In the article Winkler and Jud derive a formula for solving for the first tier payment.  The article states the solution for a two-tier case and a general case.  The program GRADMORT covers the general case.

Variables:

A = amount of the mortgage (PV).

I = periodic monthly interest of the mortgage, I%YR/1200.

T = Number of years the payment increases.  For example, if T = 5, then the payment would increase 5 times, once a year, until it reaches the final constant payment on year 6

C = The payment’s annual percent increase, C%/100.

N = The term of the mortgage, in years.

P = The initial payment (PMT).  To calculate the payments for subsequent years, multiply the last result by (1 + C).  (hence, add C% every year for each tier)

The general formula:

P = A / ( (1 + Σ((1 + C)^(k)/(1 + I)^(12*k), k, 1, N-1) * ( 1/I * (1 + I)^(-12) )
+  (1 + C)^N/(1 + I)^(12*N) * (1/I * (1 – (1 + I)^-(12*T – 12*N) )

The program breaks the formula into smaller parts for calculation purposes.

GRADMORT returns a two column matrix, the first column is the year and the second column is the payment.  The last row shows the payment when it stabilizes (stops increasing).

Monthly payments and end of period payments (end of month) are assumed.  Property taxes and interest are not calculated in this program.

HP Prime Program GRADMORT

BEGIN
// Winkler, Jud
// 2018-02-11 EWS

LOCAL A,I,N,C,T,mat;
LOCAL P,X,Y;

Mortgage",{"Loan Amount:",
"Loan Rate: ","# Tiers: ",
"% Increase: ","Term (yrs): "});

I:=I/1200;
C:=C/100;

X:=Σ((1+C)^K/(1+I)^(12*K),K,0,N-1);
X:=X/I*(1-(1+I)^(−12));
Y:=(1+C)^N/(1+I)^(12*N);
Y:=Y/I*(1-(1+I)^−(12*T-12*N));
P:=A/(X+Y);

mat:=[[1,P]];
LOCAL k;
FOR k FROM 1 TO N DO
[k+1,mat[k,2]*(1+C)],k+1);
END;

RETURN mat;

END;

TI-84 Plus CE Program GRADMORT

"WINKLER/JUD"
"EWS 2018-02-11"
Input "LOAN AMT: ",A
Input "LOAN RATE: ",I
I/1200→I
Input "NO. TIERS: ",N
Input "INCREASE: ",C
C/100→C
Input "LOAN TERM (YRS): ",T

Σ((1+C)^K/(1+I)^(12*K),K,0,N-1)→X
X/I*(1-(1+I)^(­12))→X

(1+C)^N/(1+I)^(12*N)→Y
Y/I*(1-(1+I)^­(12T-12N))→Y

A/(X+Y)→P
{1}→L1
{P}→L2
For(K,1,N)
augment(L1,{K+1})→L1
augment(L2,{L2(K)*(1+C)})→L2
End
List→matr(L1,L2,[A])
Disp [A]

Example

A young couple, who just graduated from college and starting on their careers, have qualified to participate in a GPM.  They will finance a \$200,000 mortgage at a fixed annual interest of 4.4%.  Payments increase 2.1% for the first five years of the 35 year mortgage.

Variables (the program will prompt you for the amounts):

Loan Amount:  200000
Loan Rate:  4.4
# Tiers:  5
% Increase: 2.1
Term (yrs): 35

Results:

Matrix:
[ [1, 855.231019434]
[2, 873.190870842]
[3, 891.52787913]
[4, 910.249964592]
[5, 929.365213848]
[6, 948.881883339] ]

Year 1:  \$855.23
Year 2:  \$873.19
Year 3:  \$891.53
Year 4:  \$910.25
Year 5:  \$929.37
Year 6 (years 6 – 35):  \$948.88

Source

Winkler, D.T. an G.D. Jud.  “The Graduated Payment Mortgage:  Sovling the Initial Payment Enigma”  Journal of Real Estate Practice and Education, vol. 1, 1998, pp 67-79.  Retrieved February 10, 2018.  Link:  https://libres.uncg.edu/ir/uncg/f/D_Winkler_Graduated_1998(MULTI%20UNCG%20AUTHORS).pdf

Eddie

This blog is property of Edward Shore, 2018.