Blogs, Calculus, Modeling


Happy Labor Day everyone!  If you have the day off, I hope you enjoy the day and do something fun.  Since it is labor day and I have the day off, I decided to do a post that is labor related.  I love my job, as I believe most educators do, but the question of when you are going to be able to retire is still one that everyone in the work force has to think about.

Before I get into my retirement planning, I should state that the following can provide a general outline; however, I am not an accountant and I don’t know the current tax laws, so you should put more research in before finalizing a course of action.  On the other hand, I have run through this problem in many of my classes before and it really seems to grab my student’s attention.  In particular, I’ll look at the problem of planning for retirement in my college algebra or general education requirement type class that covers math that should appear in algebra 1 or 2 in high school.  Therefore, it is an exercise you can run through with high school or college students to get them thinking about planning for their futures at a young age, thus providing them plenty of time to get ahead on saving.

Now, let’s look at the our planning and lay out our assumptions.  While it makes most sense to put money away every paycheck, I am going to work with the plan that money will be put away annually.  In this savings process, we will plan on putting money away in some account that earns interest.  Also, I want to account for inflation and raises.  Since it is impossible to exactly predict what interest rates, inflation rates or raises you will receive over the years, I am going to choose constant rates of 5% interest earned with 2% inflation per year and any raises earned are in order to keep up with inflation.  Furthermore, we will count interest, inflation, and raises compounded annually to go along with our annual payments.  While the assumptions are not likely to be exact they should at least give an idea of what your retirement will look like over the years.  Furthermore, I am going to work under the assumption that I begin my career making a salary of $S per year.  I wanted to leave the salary as a parameter so that you can choose what you started at, or, if showing this to your class, you can make them look up average salaries for the career they want to have when they finish school.

For the equations, I will let S=salary, R=amount of money at retirement, d=deposit made annually, w=withdrawal made annually, n=time in years before retirement, m=time in years after retirement, r=interest rate per year (since I am using annual compounding, the apy and apr are equal), and i=inflation rate per year.  Now, if you are going to retire today, you will be able to make one payment toward your retirement account, so you will have


in your account.  If you have one year until retirement, you will then haveretirement

in your account since your first deposit will have earned one years worth of interest and your second deposit will be based on your new salary.  In general, if you have n years until retirement, you will have


in your account.  We can then simplify this so that we get the equation


Upon retirement, I assume you will take out withdrawals that are equal in value.  That is, you will actually have to withdraw an additional 2% per year to account for inflation.  Starting with a withdrawal the day you retire and assuming your last withdrawal the beginning of the final planned year, we can follow a similar process to what we did for deposits.  If you plan to live off your retirement for m years after retirement, your retirement amount will have to be


Now, as a general rule, it is suggested that you are able to withdraw 80% of your salary after you retire.  Therefore, I will use w=.8*S.  From this, you can then find the amount required for deposit in a two step process.  First, you find the amount you need to retire based on how long you have before you retire and how long you plan on living off your retirement.  You then can find the required deposit amount based on how long you have left before you retire.  If you combine these, you end up with


Then, if you plan on starting to save at 25, retiring at 65 and living to 95, then you should deposit .2168*S per year.  On the other hand, if you plan on beginning to save at 35, retiring at 65 and living to 95, then you need to put deposit .3398*S per year.  If doing this in a class, I like to take this time to point out that if they start saving for retirement early, it will significantly lower the percentage of their salary they will have to put away each year.  Applying this to the salary they found for their chosen career will often help instill the importance of starting retirement savings.

Note that small changes in the interest or inflation rate can cause large changes in the required savings amount, so I tried to leave the equations so that they wouldn’t be hard to change if you want to use different rates.  In case it’s wanted, I’ve also given the equation for deposit amount below.


Again, I hope you enjoy your labor day.  If you happen to use this for a lecture idea, I hope your students enjoy it and learn something helpful.  If you use this to go over your own planning, I hope things work out so that you get to retire when you want to.


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