True Annual Growth Rate of an Investment Portfolio

This program is Copyright © 1978 by Hewlett-Packard and is used here by permission. This program was originally published in "HP-38E/38C Personal Finance Applications" book. This program was transcribed by Kris Collins.

This program is supplied without representation or warranty of any kind. Hewlett-Packard Company and The Museum of HP Calculators therefore assume no responsibility and shall have no liability, consequential or otherwise, of any kind arising from the use of this program material or any part thereof.

Overview

Finding the true rate of return on an investment portfolio where variable amounts are invested or withdrawn at indefinite or irregular intervals is a complicated calculation. However, it is a highly useful tool in money management.

This program uses "Fisher's algorithm", a powerful technique to solve for the true realized rate of return.

Notes:

The program can accommodate a maximum of nine transactions.
All transactions must be input in succession, oldest to newest.
Enter only whole dollars (no cents) for P_j.
For problems of this type, more than one, or no solutions may exist. The program is written to allow you to watch the solution converge, that is, program execution pauses to display the differences between successive approximations to the true interest rate. If this quantity does not get smaller as execution continues, no solution may exist. You may continue searching for a solution or try another solution by using a different initial guess for i (program instructions, step 5).
The financial registers are used by this program to store miscellaneous data. Do not use them for other purposes when running the program.

Reference:

1) Lawrence Fisher, "An Algorithm for finding Exact Rates of Return", The JOURNAL OF BUSINESS, University of Chicago Press, Volume XXXIX No. 1 part 2, January 1966.

KEY ENTRY	DISPLAY	KEY ENTRY	DISPLAY
g P/R g CLP	00-	g LASTX	25 31
RCL PMT	01-24 14	g FRAC	30-25 61
f Change DAYS	02-24 41	2	31-2
3	03-3	g y^x	32-25 21
6	04-6	f √ x	33-25 21
5	05-5	g EEX	34-25 32
0	06-0	2	35-2
0	07-0	×	36-61
÷	08-71	ENTER↑	37-31
RCL PV	09-22 13	ENTER↑	38-31
g x⇔y	10-25 5	RCL i	39-22 12
g GTO 15	11-25 7 15	×	40-61
÷	12-51	g e^x	41-25 22
g CF_i	13-25 14	RCL PV	42-22 13
g GTO 00	14-25 7 00	×	43-61
x⇔y	15-33	STO / 0	44-21 51 0
-	16-41	×	45-61
g GTO 13	17-25 7 13	STO / 1	46-21 51 1
1	18-1	RCL n	47-22 11
0	19-0	3	48-3
n	20-11	g x⇔y	49-25 5
0	21-0	g GTO 24	50-25 7 24
STO 0	22-21 0	RCL i	51-22 12
STO 1	23-21 1	RCL FV	52-22 15
RCL g CF_i	24-22 25 14	RCL 0	53-22 0
g x=0	25-25 6	-	54-41
g GTO 24	26-25 7 24	RCL 1	55-22 1
f INTGR	27-24 61	÷	56-71
PV	28-13	÷	57-51

KEY ENTRY	DISPLAY	KEY ENTRY	DISPLAY
i	58-12	CHS	66-32
g LASTX	59-25 31	g x⇔y	67-25 5
g PSE	60-25 4	g GTO 18	68-25 7 18
2	61-2	1	69-1
g y^x	62-25 21	RCL i	70-22 12
f √ x	63-24 21	f %T	71-24 22
g EEX	64-25 32	g P/R
7	65-7

REGISTERS
R₀ Sum numerator	R₁ Sum denominator	R₂ CF₁	R₃ CF₂
R₄ CF₃	R₅ CF₄	R₆ CF₅	R₇ CF6
R₈ CF₇	R₉ CF₈	R_.0 CF₉	R_n n
R_i i_k	R_PV T	R_PMT p_j	R_FV V_T

Instructions:

Key in the program.
Set switch to M.DY and press f CLEAR ALL.
Key in the date of the final fund evaluation and press PMT.
Key in the final fund value and press FV.
Key in an initial guess for the interest rate, .1, and press i.
Key in 2, press n.
Key in the transaction (whole dollars only) and press PV.
Key in the date of that transaction and press R/S.
Repeat steps 7 and 8 until all transactions are entered. Then press g GTO 18 R/S to compute the continuously compounded interest rate.
To compute the annual rate, use the following keystroke sequence: RCL i 1 / g LN 100 x.

Example:

You invest in a pension fund and after anumber of year you are told that the fund value was 9,050 dollars on 7/01/1977. Determine the effective (continuously compounded) and nominal rate of return from the record of your transactions which are as follows:

    an investment of $4,150 on 1/01/1972;
    an investment of $2,005 on 1/01/1973;
    a withdrawal of $950 on 1/01/1974;
    a withdrawal of $3,111 on 1/01/1975;
    an investment of $2,000 on 10/01/1975;
    an investment of $2,005 on 1/01/1976; and
    a final investment of $1,000 on 7/01/1976.

Keystrokes	Display
D.MY/M.DY = M.DY
f CLEAR ALL	0.00
7.011977 PMT	7.01
9050 FV	9,050.00
.1 I	0.10
2 N	2.00
4150 PV	4,150.00
1.011972 R/S	4,150.06
2005 PV	2,005.00
1.011973 R/S	2,005.04
950 CHS PV	-950.00
1.011974 R/S	-950.03
3111 CHS PV	-3,111.00
1.011975 R/S	-3,111.02
2000 PV	2,000.00
10.011975 R/S	2,000.02
2005 PV	2,005.00
1.011976 R/S	2,005.01
1000 PV	1,000.00
7.011976 R/S	1,000.01
g GTO 18
R/S	(-.04) Differences between
	(-0.004) successive approximations
	(-0.00005) of the correct interest rate.
	(-0.00000001)
	5.90 effective continuous rate

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