Inflation-Adjusted Retirement Planning

Model Development

Pre-retirement planning model

Model
Development

 

Pre-retirement planning model Post-retirement planning model Other useful equations & formulas

Symbolic notation is used throughout the model development, so it would be wise to review the meanings of the symbols before perusing the modeling.

Symbolic notations

A time-tested, finite mathematical series, well-known in college-level engineering economics courses as the geometric gradient series, is used as the base for model development.  Two variations of this series are used to calculate the inflation-adjusted savings relations, and the inflation-adjusted retirement income relations.  Other useful interest formulas and equations are also presented

Item Cash-flow diagram Equations
Modeling the savings years

The future-sum form of the geometric gradient series is used for inflation-adjusted retirement savings / investment plan.  The cash flow diagram is shown in blue to the left.  The equation is as follows:

The entire inflation-adjusted savings series is calculated from A1 as follows:

 

Modeling the retirement years The present-value form of the geometric gradient series is used for inflation-adjusted retirement income planning.  The cash flow diagram is shown in red to the right.  The equation is as follows:

The entire inflation-adjusted retirement series is calculated from R1 as follows:

 

Including an initial lump sum amount

If, in pre-retirement planning, we have an initial lump sum, BO, at time j=0, we will need to convert it to an equivalent amount Bm, at time j=m, as shown in the cash flow diagram to the left, as follows:

 Retirement income
in today's $

In pre-retirement planning, we want to work with current-year estimates of our annual retirement need, RO, in today's dollars (at time j=1).
We then calculate
R1 from RO as follows:

Linking the equations

For pre-retirement planning, we must connect the above three equation sets at the point j=m, to be coincident with the point k=0, and set the equality relation as follows: 

The combined equation model

The Consolidated Pre-retirement Planning Equation

 

 

Note that the the consolidated equation is in terms of  RO, and A1 , which are terms stated in current dollars at time j=1 , and BO stated in dollars which are still essentially current dollars from j=0.  This gives you solutions in terms you can easily relate to.

Three pre-retirement planning solutions can be generated form the consolidated equation:

  • Solution 1: Solving for A1, given RO, i, r, and both m and n (which means we must also know DO, DR , and DD)

  • Solution 2:  Solving for RO, given A1, i, r, and both m and n (which means we must also know DO, DR , and DD)
     

 

  • Solution 3: Solving for DR, given A1, ROi, r, and (m+n) (which means we must also know DO and DD, but not m or n)

(See example to see how this is done)

Auxiliary Equations

From DO, DR , and DD we calculate m and n:

m = DR - DO
 n
=
DD  - DR

A complete example of each variation on a theme is provided.  These examples also allow you to substitute in your own data to personalize your own solution(s).  

Example solutions

©2008, Simon Revere Mouer III, PhD, PE
all rights reserved