RPN With DM32: Interest Conversions
Today’s edition of RPN will focus on the Swiss Micros DM32 and HP 32SII calculator family (32SII, 33s 35s). My plan is to expand the range of calculators used in the RPN series to include the HP 41C/DM41X, HP 42S/DM42(n), HP 11C in addition to the HP 32SII and HP 15C families.
Nominal Interest Rate (EFF/NOM) and Effective Interest Rate (EFF)
When loans, annuities, mortgage, and other time-valued financial instruments are executed, the interest rate typically given is known as the nominal interest rate (NOM), also known as the APR. In mortgages and auto loans where the payments typically take place every month (12 per year), the periodic interest rate is divided by 12 to come up with the periodic rate. The periodic rate is compounded every month.
For example, if a loan has an APR (nominal rate, NOM) of 6% and payments take place monthly, the periodic interest rate is:
6% / 12 = 0.5%
The effective interest (EFF) rate is the nominal interest rate calculated as if was compounded annually.
Paid in Arrears
Typically, payments are paid in arrears, meaning payments are made after the service or the use of associated goods were provided. An example is the mortgage payment for the month of June is paid in the beginning of July.
The effective rate for payments paid in arrears is:
EFF% = (1 + NOM% / PY) ^ PY - 1
EFF%: effective interest rate
NOM%: nominal interest rate, APR
PY: payments per year
The effective rate for a APR of 6%, compounded monthly, paid in arrears is:
EFF% = (1 + 6% / 12) ^ 12 – 1 = (1 + 0.06 / 12)^12 – 1 ≈ 0.06168 (6.168%)
Paid in Advance
Sometimes, payments are paid in advance, where payments are made before the service or use of associated goods are provided. For example, the mortgage payment for the month of June is paid at the beginning of June or the end of May. The mortgage is paid in advance.
The effective rate for payments paid in arrears is:
EFF% = (1 - NOM% / PY) ^ (-PY) - 1
EFF%: effective interest rate
NOM%: nominal interest rate, APR
PY: payments per year
The effective rate for a APR of 6%, compounded monthly, paid in advance is:
EFF% = (1 - 6% / 12) ^ (-12) – 1 = (1 - 0.06 / 12)^(-12) – 1 ≈ 0.06200 (6.2%)
Conversion Formulas
The conversion formulas are (see source):
Payments in Arrears
EFF% = (1 + NOM% / PY) ^ PY – 1
NOM% = PY * ((1 + EFF%) ^ (1 / PY) – 1)
Payments in Advance
EFF% = (1 – NOM% / PY) ^ (-PY) – 1
NOM% = PY * (1 – (1 + EFF%)^(-PY))
DM32: Two Ways to Solve
There are two approaches to converting interest rates: directly through a program and using the solver.
Direct Programs
LBL A: NOM to EFF, Payments in Arrears
LBL
A
STO Z
÷
1
x<>y
%
+
RCL
Z
y^x
1
-
2
10^x
×
RTN
Syntax:
Y: NOM
X: PY
XEQ A
(Y: 5, X: 12, XEQ A: 5.11619)
LBL B: NOM to EFF, Payment in Advance
LBL B
STO Z
÷
1
x<>y
%
-
RCL
Z
+/-
y^x
1
-
2
10^x
×
RTN
Syntax:
Y: NOM
X: PY
XEQ B
(Y: 5, X: 12, XEQ B: 5.13809)
LBL C: EFF to NOM, Payments in Arrears
LBL C
STO
Z
1/x
x<>y
1
x<>y
%
+
x<>y
y^x
1
-
RCL×
Z
2
10^x
×
RTN
Syntax:
Y: EFF
X: PY
XEQ C
(Y: 5, X: 12, XEQ C: 4.88895)
LBL D: EFF to NOM, Payment in Advance
LBL D
STO
Z
1/x
+/-
x<>y
1
x<>y
%
+
x<>y
y^x
1
x<>y
-
RCL×
Z
2
10^x
×
RTN
Syntax:
Y: EFF
X: PY
XEQ D
(Y: 5, X: 12, XEQ D: 4.86911)
Using The Solver
The DM32, and by extension, the HP 32SII, HP 33s, and HP 35s, has a solver that can solve for any variable. The solver uses a program in the format:
LBL α
INPUT (var)
INPUT (var)
…
function(var) = 0
RTN
To solve the equation (DM32, HP 32SII):
1. Press [ blue shift/right shift ] [ XEQ ] (FN=). An FN= prompt appears asking for a label.
2. Press [ blue shift/right shift ] [ 7 ] (SOLVE). You will be prompted to enter the variable to be solved.
3. The calculator prompts for values for any of the values. Press [ R/S ] to accept.
4. The solution is shown. I think the values and solutions are stored in the solutions. It is the case for the DM32, HP 32SII, and HP 32S.
Notes:
* HP 32S has the FN= and SOLVE in the SOLVE/∫ menu ([orange shift ] [ 1 ]).
* The right shift turned lavender in later HP 32SII calculators (late 1990s/early 2000s).
LBL S: Solver
LBL S
INPUT E
INPUT
M
INPUT N
1
RCL M
RCL÷ N
+
RCL
N
y^x
1
-
RCL- E
RTN
The equation is:
E = (1 + M/N) ^ N – 1
E = EFF%
M = NOM% (or APR%)
N = payments per year
The EFF% and NOM% must be entered in decimal. For payments in arrears, enter N as positive. For payments in advance, enter N as negative.
Example
Set function as:
FN= S |
NOM to EFF, pmts in arrears
Solve E
M = 5% = 0.05 N = 12
E = 0.05116 (5.116%) |
NOM to EFF, pmts in advance
Solve E
M = 5% = 0.05 N = -12
E = 0.05138 (5.138%) |
EFF to NOM, pmts in arrears
Solve M
E = 5% = 0.05 N = 12
M = 0.04889 (4.889%) |
EFF to NOM, pmts in advance
Solve M
E = 5% = 0.05 N = -12
M = 0.04869 (4.869%) |
Source
J.J. Rose and E.M. Reeves VALPAC: A Discounted Cash Flow Approach To Property Valuation. (user manual) The Incorporated Society of Valuers and Auctioneers. An HP 41C Pac.
Eddie
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