by Lemuel W. Davis, CFP®
Executive Summary
Financial calculators are programmed to solve present value and future value annuity problems based on two assumptions: (1) The series of cash flows or deposits occur at equal intervals and (2) the cash flows and deposits are nominal. Solutions to problems with inflation-adjusted cash flows and deposits are computed by entering an inflation-adjusted interest rate. In most cases, the calculator solution must be "adjusted" to yield the correct answer. The discrepancy between the calculator output and the true value may be significant and is easily quantifiable by using the appropriate equation in the section in this paper that covers required corrections to the calculator computed values.
Inflation-adjusted solutions are often required when evaluating college or retirement savings and other spending options. These problems are taught and tested in most financial courses where calculators are used to solve problems.
The purpose of this analysis is to derive the correct annuity equations for begin/end modes, nominal cash flows and deposits, and inflation-adjusted cash flows and deposits. These equations are then compared to the calculator equations when inflation-adjusted interest is used. The differences between the two sets of equations are used to compute adjustments to the calculator solutions to yield the correct answers. Calculators with a built-in equation solver can be programmed to solve these problems directly by eliminating the need for corrections or restrictions on input parameters.
Derivation of the Annuity Factors
Notational convention. To enhance readability, I will depart slightly from standard notation. I will use the present value annuity factor (PVAF ) to explain the notational differences. Two basic equations for PVAF are given: One is the fundamental equation (PVAFx,y,z) that directly computes the correct value, and the other is derived by substituting the inflation-adjusted interest into the calculator equation (PVAFCx,z). The latter equation yields the calculator equivalent solution which, in most cases, differs slightly from the PVAFx,y,z solution but is easily correctable.
Subscripts x, y, and z have the following meanings:
Subscripts are ordered as shown but less than three subscripts may suffice for some notations.
For example, PVAFCw,b is the calculator version of the present value of an annuity that applies inflation-adjusted interest to cash flows occurring at the beginning of n periods, and PVAFi,j,b is the present value derived using the time-value-of-money (TVM) equations.
An algebraic expression used to derive annuity factors. In general, the annuity factors can be derived using the following familiar algebraic expression:
Calculator Equations
Calculator equations for TVM problems. Most calculators including the HP 17bll+ use an iterative process to solve TVM problems. The HP 17bll+ is a powerful programmable calculator that solves TVM problems using equation (10).1 The correct solutions are obtained when solving annuity problems with nominal cash flows or deposits. Inflation-adjusted cash flows or deposits must be "adjusted" to yield the correct answers. The HP 17bll+ can be programmed to solve equation (11), which eliminates the need to "adjust" the answers.
An equation that allows one to solve problems with both nominal and inflation-adjusted cash flows and deposits is given by
Equivalent HP calculator equations. The basic calculator annuity equation (equation (10)) can be written as separate equations when the parameters and modes are set as indicated.
The calculator solutions. The equivalent calculator inflation-adjusted equations are determined by substituting the inflation-adjusted interest rate into equations (12), (13), (14), and (15) respectively, which yields equations (16), (17), (18), and (19).
The inverse of the above corrections must be applied to PMT when the inflation-adjusted present or future value is known and the calculator computes PMT. For example, when PVAFCw,e is given and one uses the calculator to solve for PMTC, the correction to the calculator value is PMT=PMTC(1+j)
Required adjustment to the inflation-adjusted interest rate when compounding intervals are less than a year. The inflation-adjusted interest is given as
Where i, j, and wp are annual interest rates.
Monthly compounding rates are determined by dividing wp by 12. This occurs when wp is entered into the HP 12C (wp, g, i) and other calculators causing the inflation-adjusted interest to change as follows:
The resulting calculator interest (equation (25)) is incorrect and the inflation-adjusted interest entered must be modified (equation (26)) to yield the correct solution. In the denominator of equation (26), j is divided by the number of compounding intervals in a year.
Conclusion
Financial calculators are programmed to solve time-value-of-money problems using an equation similar to equation (10), which yields the correct solution for nominal cash flows and deposits. Most solutions to inflation-adjusted problems must be corrected. Some may require several adjustments to yield the correct solution.
It is important to note the above corrections to calculator solutions for present values (PV) are valid only if the future value is zero, and when computing future values (FV) the present value must be zero. One can derive corrections to the calculator solutions when these parameters are not zero, but the corrections are complicated and unwieldy. Some solutions are easily adjusted, but many problems must be solved using equations (2) through (9) and other basic TVM equations. The two problems given in appendix A are examples of the complexity involved in using the HP 12C and other calculators to solve inflation-adjusted problems.
Clearly, trying to remember which corrections to apply to the various parameters can be confusing, but one's options may be limited when using older calculators; either one makes the corrections or use TVM equations to solve the problems. Fortunately, the latest calculators are more flexible. The HP 17 bll+ calculator can be programmed to solve equation (11) using its built-in equation solver. The program and operating instructions given in appendix B will yield true values for inflation-adjusted annuity problems and other TVM problems without requiring corrections or constraints on input parameters.
Endnote
- Owner's Handbook and Problem-Solving Guide, HP 17bll+ [VERIFY] financial calculator, p. 247.
Appendix A
Problem 1
What is the future value of a series of inflation-adjusted deposits made at the end of each month for six months if the first month's deposit is $500, the investments earn 8 percent annually, and inflation is 3 percent annually?
The future value of an ordinary annuity compounded monthly requires two adjustments: First, the inflation-adjusted interest is determined by equation (26); second, the calculated future value is adjusted using equation (22).
Problem 2
John is retiring with $2,537,735.29 in saving. He is planning for 25 years in retirement and would like to leave the Red Cross $2 million. Assume his investments earn 8 percent and inflation is 3 percent over the entire period. What payment will he receive at the beginning of the first year of retirement?
The apparent calculator solution is obtained by setting begin mode, inputting the inflation-adjusted interest, the given parameters, and solving for the first year's payment (PMT) using the following HP 12C key strokes:
[Clear Reg.]; [g, Begin Mode]; [Compute I = 4.854369 (Equation 24)]; [I, i]; [–2,537,735.24, PV]; [25, n]; [2,000,000, FV]; [PMT = ?]
The apparent solution is PMT = 128,450.57, which is incorrect. The correct solution is obtained as follows: When the calculator is set to begin mode and inflation-adjusted interest is used, the calculator equation (equation (10)) becomes
This equation is almost identical to equation (11) except the last term is slightly different. Since equation (11) solutions don't require adjusting, the objective is to make the above equation look like equation (11) in begin mode.
Setting
forces the calculator to use the correct future value.
The correct answer is obtained by entering FV1 for the future value when solving for PMT. The S 12C key strokes are
[Clear Reg.]; [g, Begin Mode]; [Compute I = 4.854369]; [Compute 955,211.14]; [25, n]; [I, i]; [-2,537,735.29, PV]; [ ]; [PMT = ?] PMT = 149,750.71, which is correct.
Appendix B
An HP 17bll+ program that directly solves inflation-adjusted annuity problems:
TVM: (IF(P/YR=0:L(P:1):L(P:P/YR))+L(K:(Ix.01)÷G(P))+L(L:(Jx.01)÷G(P)))x0=PV+(1+MODExG(K))x(PMT÷(G(K)-G(L)))x(1-((1+G(L))÷(1+G(K)))^(N/YRxG(P)))+FV÷(((1+G(K)))^(N/YRxG(P)))
Operating Instructions
The operation of this program is similar to the operation of the HP installed program. The first step before inputting data is to press the Shift Key and then the CLR DATA Key. This sets all parameters to zero, which initializes the program in end mode and initially sets the number of payments per year (P/YR) equal to zero, which is changed to one at execution time. Yearly payments or deposits are entered by setting P/YR to zero or one. Setting P/YR to any other number sets the program for that number of payments or deposits per year. The program is capable of solving for all unknown parameters when the other inputs are known except P/YR cannot be determined. The number of years (N/YR) can be a fraction if the period of interest is less than a year. For example, monthly payments for six months would be set as follows: P/YR = 12 and N/YR = 0.5. Since j is initially set to zero, equation (11) is identical to equation (10) before inputting any parameters. Inflation adjusted problems are solved by setting j to the assumed annual inflation rate. Interest i is also an annual rate. When entering data, one must observe all the "sign" rules the HP program currently requires. End mode is entered by setting MODE = 0 and begin mode is entered by setting MODE = 1.
Lemuel W. Davis, CFP®, is an adjunct instructor at Jacksonville State University, College of Commerce and Business Administration.
