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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