** 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

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