**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**

EXPORT GRADMORT()

BEGIN

// Graduated Mortgage

// Winkler, Jud

// 2018-02-11 EWS

LOCAL A,I,N,C,T,mat;

LOCAL P,X,Y;

INPUT({A,I,N,C,T},"Graduated

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

mat:=ADDROW(mat,

[k+1,mat[k,2]*(1+C)],k+1);

END;

RETURN mat;

END;

**TI-84 Plus CE Program GRADMORT**

"GRADUATED
MORTGAGE"

"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.