HP 12C: Solving Two Actuarial Problems

 HP 12C: Solving Two Actuarial Problems

Solve for Interest and Number of Periods Given Present Value Annuity Factor and Future Value Annuity Factor

Given the following:

* Present Value Annuity Factor (PVAF)

* Future Value Annuity Factor (FVAF)

Solve for both:

*  the interest rate

*  the number of periods of the annuity

Keep in mind that present value and future value annuity factors are not the same of present value and future value, respectively.

Recall that:

PVAF = (1 - (1 + i)^-n) / i

FVAF = ((1 + i)^n - 1) / i

Note that:

FVAF = ( (1 + i)^n - 1 ) / i

FVAF = (1 + i)^n * (1 - (1+i)^-n) / i

FVAF= (1 + i)^n * PVAF

Then:

FVAF / PVAF = (1 + i)^n

ln (FVAF / PVAF) = ln (1 + i)^n

ln (FVAF / PVAF) = n * ln(1 + i)

ln (FVAF / PVAF) / ln(1 + i) = n

Also:

1 / FVAF + i 

= i / ((1 + i)^n - 1) + i

= i / ((1 + i)^n - 1) + (i * (1 + i)^n - 1)) / ((1+ i)^n - 1)

= (i + i * (1 + i)^n - i) / ((1 + i)^n - 1)

= (i * (1+i)^n) / ((1 + i)^n - 1)

= (1 + i)^n / (1 + i)^n * ( i / (1 - (1 + i)^-n)

= 1 * ( i / (1 - (1 + i)^-n)

= 1 / PVAF

Then:

1 / FVAF + i = 1 / PVAF

i = 1 / PVAF - 1 / FAVF

To summarize: 

i = 1 / PVAF - 1 / FAVF

n = ln (FVAF / PVAF) / ln(1 + i)

The following program solves for interest and number of payments. 

HP 12C (Classic) Program

Stack set up:

Y: PVAF

X:  FVAF

Step: Key: Code

01:  STO 1 :  44, 1

02:  x<>y :  34

03:  STO 2 :   44, 2

04:  1/x :   22

05:  x<>y :   34

06:  1/x :   22

07:  - :   30

08:  STO 3 :  44, 3

09:  ENTER :  36

10:  ENTER :  36

11:  1 :   1

12:  + :   40

13:   LN : 43, 23

14:  RCL 1 : 45, 1

15:  RCL 2 : 45, 2

16:  ÷  :  10

17:  LN :  43, 23

18:  x<>y : 34

19:  ÷  : 10

20:  STO 4 :  44, 4

21:  GTO 00 :  43, 33, 00

Example:

Input:

Y:  PVAF = 22.3965

X:  FVAF = 40.5681

Results:

Y:  i = 0.02   (2%)

X:  n = 30  (30 periods)

Present Value of an Annuity Due with an Effective Discount Rate

The problem determines the present value of an annuity due with an effective discount rate.  The effective discount rate is different from the interest rate (i).  Convert the effective discount rate to interest rate by:

i = d / (1 - d)   [i, d are in decimal form]

HP 12C (Classic) Program

Stack set up:

Clear TVM values

Store n:  [ n ]

Store payment: [ PMT ]

X:  discount rate (as a percentage.  Example: for 10%, enter 10)

Step: Key: Code

01:  BEG  : 43, 7

02:  ENTER  : 36

03:  CLx  :  35

04:  R↓   :  33

05:  1   :  1

06:  %  :  25

07:  1   :  1

08:  x<>y   :  34

09:  -  :  30

10:  LSTx  : 43, 36

11:  x<>y   :  34

12:  ÷  : 10

13:  1  :  1

14:  EEX  : 26

15:  2  :  2

16:  ×  :  20

17:  [ i ] :  12

18:  [ PV ]  :  13

19:  GTO 00  :  43,33,00

Example:

Input:

n = 36, PMT = -250.00

X:  d = 5%;  (5  [R/S])

Results:

X:  4211.10

Present Value:  $4,211.10

Source:

Finan, Marcel B.  A Basic Course in the Theory of Interest and Derivatives Markets:  A Preparation for the Actuarial Exam FM/2   Arkansas Tech University, 2017.  

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Time Value of Money: Using Present Value Factors

Time Value of Money:  Using Present Value Factors

Introduction

The time value of money equation is:

0 = PV + PMT * PVAF + FV * PVSF

where

PVAF = present value of annuity factor = Σ( (1 + i%)^(-k), k, 1, n )
PVSF = present value of single factor = (1 + i%)^-n
n = number of payments
i% = periodic interest rate (as a decimal)

The cash flow convention is maintained with this model.  Cash outflows (payments) are negative, cash inflows (receipts) are positive.

Solving for Payment (PMT)

0 = PV + PMT * PVAF + FV * PVSF
-PMT * PVAF = PV + FV * PVSF
-PMT = (PV + FV * PVSF) / PVAF
PMT = -(PV + FV * PVSF) / PVAF

Example

PV = $50,000.00
FV = -$10,000.00
n = 60  (5 years of monthly payments)
i% = 0.05/12

PVAF = 52.99071
PVSF = 0.77921

PMT = -(50000 - 10000 * 0.77921) / 52.99071 = -796.5150873
(cash payment of $796.52)

Solving for Future Value (FV)

0 = PV + PMT * PVAF + FV * PVSF
-FV * PVSF = PV + PMT * PVAF
FV = -( PV + PMT * PVAF ) / PVSF

Example

PV = -$1,000.00
PMT = -$250.00
n = 60  (5 years of monthly payments)
i% = 0.05/12

PVAF = 52.99071
PVSF = 0.77921

FV = -( -1000 - 250 * 52.99071) / 0.77921 = 18284.7724
(future value of deposit $18,284.77)

We can use this method with any calculator.  A partial present value factor table, one for single payment, and one for annuity, is presented below.

Source:

"HP 17BII Financial Calculator Owner's Manual"  Edition 1  Hewlett Packard.  Corvallis, OR.  December 1989.

Have fun,

Eddie

All original content copyright, © 2011-2019.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Fun with the TI-80

Fun with the TI-80

TI-80 Program D2DMS - Decimal to Degrees-Minutes-Seconds

Variables:

Decimal Format:

D = decimal

DMS Format:

H = degrees/hours, M = minute, S = seconds

INPUT “DEC:”,D

IPART D→H

IPART (60*FPART D)→M

60 * FPART (60 * FPART D)→S

DISP “H,M,S:”,H,M,S

TI-80 Program DMS2D - Degrees-Minutes-Seconds to Decimal

Variables:

Decimal Format:

D = decimal

DMS Format:

H = degrees/hours, M = minute, S = seconds

INPUT “H:”,H

INPUT “M:”,M

INPUT “S:”,S

H + M/60 + S/3600 → D

DISP “DEC:”,D

TI-80 Program QUADRAT - Quadratic Equation

Variables:

A, B, C are coefficients of the equation Ax^2 + Bx + C, where the discriminant D:

D = B^2 – 4*A*C

If D≥0, then the roots are real and stored in X and Y.

If D<0, then the roots are complex and are in the form of conjugates X ± Yi.  X is the real part, Y is the imaginary part.

DISP “AX^2+BX+C=0”

INPUT “A:”,A

INPUT “B:”,B

INPUT “C:”,C

B^2 – 4AC → D

DISP D 

-B / (2A) → E

IF D≥0

THEN

E + √D/(2A) → X

E - √D/(2A) → Y

DISP “R1:”,X

DIPS “R2:”,Y

ELSE

E → X

√-D / (2A) → Y

DISP “RE:”,X

DISP “IM :”,Y

END

Annuity Factors

Variables:

I = periodic interest rate

N = number of payments/periods/deposits

TI-80 Program USFV – Annuity Future Value Factor

INPUT “I:”,I

INPUT “N:”,N

( (1+.01)^N – 1)/(.01I) → F

DISP F

TI-80 Program USPV – Annuity Present Value Factor

INPUT “I:”,I

INPUT “N:”,N

(1 – (1 + .01I)^-N)/(.01I) → P

DISP P

Two Dimensional Vector Operations

Let two vectors be defined as V1 = [A, B] an V2 = [C, D].  The program calculates the dot product, stored in E, norm of V1, stored in F, norm of V2, stored in G, and the angle between V1 and V2 in degrees, stored in H.

TI-80 Program VECTOR2

DEGREE

DISP “V1:”
INPUT A

INPUT B

DISP “V2:”

INPUT C

INPUT D

AC + BD → E

√(A^2 + B^2) → F

√(C^2 + D^2) → G

COS^-1 (E /(F*G)) → H

DISP “NORM V1:”, F

DISP “NORM V2:”, G

PAUSE

DISP “DOT:”, E

DISP “ANGLE:”, H

Simplistic Logistic Regression

Fit data (x,y) to the equation:

Y = 1 / (A + B*e^(-X))

Refer to http://edspi31415.blogspot.com/2017/04/hp-prime-and-ti-84-plus-ce-simple.html

TI-80 Program SIMPLOG

INPUT “L1:”, L1

INPUT “L2:”, L2

e^-L1 → L1

1/L2 → L2

LINREG(aX+b) L1, L2

a→A: b→B

DISP “1/(B+Ae^X)”,A,B

PAUSE

DISP “CORR^2”,r^2

Eddie

This blog is property of Edward Shore, 2017