## Sunday, January 24, 2021

### 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.

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