What would be the consequences of your parents moving in with you NOW? How about 10 years from now? 20 years from now? This is an exercise in imagination which may usefully anticipate the planning exercise in this module.
Have a short discussion about this with the students. Some questions for them:
According to Inside Elder Care
3.6 million elderly parents are living with their adult children. These are only the ones needing care. According to this 2010 research from Duke University, over 16% of adults in the U.S. live in households containing two adult generations. This figure is up from 12% in 1980 and is attributed to the economic downturn. Of course, this figure includes adult children who move in with parents for economic reasons as well as adult parents moving in with children. The paper further points out that in 1900 70% of elderly mothers lived with their adult children, and attributes the drop since the 1920s to be due to increased wealth of the elderly due to Social Security.
If the class has previously completed the paycheck module, it would be good to recall the Social Security withholding and perhaps have a second look at that.
Have the students guess, then suggest how they might use mathematics to estimate this. Here are some leading questions.
It would make sense for the students to do their personal calculation in parallel with this one. For this they need to know how old their parents are and what they currently make (jointly). Retiring at 70 is an assumption they can use or they can ask their parents their plans. If it is unpleasant for students to ask their parents these questions then they can use an alternative scenario: themselves.
For the purposes of teaching the relevant mathematics it might be better not to use this spreadsheet until later. However if the purpose of the class is primarily financial, then use it early.
The main assumption in this slide is that the couple's salary will keep up with the cost of living. An exploration will illuminate this assumption. Have some students look up the increase in cost of living in each of the last five years. Have others look up median pay raises in various income categories for the last five years.
As usual, there is quite a bit of variation around this basic assumption. For example, one study projects (as of this writing) that median pay raises in 2012 will be 3%. Another article uses the Consumer Price Index to project an increase in cost of living of 3.6%.
What does it mean for wages not to “keep up with” the cost of living?
If the cost of living goes up 2% and this year's salary is $80,000, then what should next year's salary be? And how do you get that? Ask the students.
So here is a chance to remind students of the distributive property and the use of percents—adding 2% of 80,000 to 80,000 is the same as multiplying that 80,000 by 1.02.
The calculation also uses an iteration, which means repeating an operation using the output of the previous operation as input. In this case the operation is multiplying a given year's salary by 1.02 to get the salary for the next year.
The answer is expressed in exponential notation, which might require a reminder for some students.
The subscripts allow us to imagine any number instead of the $80,000 in the example.
Each student should do this calculation for his or her particular example.
This example rounds off the answer to $130,000. Now would be a good moment to remind students of rules for rounding off.
Here is an interesting way to look at the roundoff error. What happens if the cost of living increase is 2.1% or 1.99%? This is a rate that varies quite a bit and we are just estimating it anyway. How much does this change the projected salary and how does this compare to the rounding we just did?
Here is a chance to discuss the next assumption: that this salary will suffice until they die. Do students believe it will? How about for their own example?
The paper cited earlier from Duke University estimates that an elderly person requiring skilled nursing facilities currently pays an average of $75,000 per year for these. What does the cost of living increase project that these will cost in 25 years? And remember, our example is for a couple. What if they both require this care at the same time?
Of course, nursing care on this level usually is only required for a short amount of time—an average of about 3 months. But there is considerable variation in this estimate.
On the other hand, the parents may have paid off their house by the time they retire. If so, the reduction in expenses may result in their requiring less income than this estimate. What do mortgage payments run for your students' parents? This could be factored into the estimate.
So what does the slide mean by “access to”? Discuss this with the students. You want them to see that the couple only needs to obtain $130,000 in the first year. They don't need the last $130,000 until year 25.
What principles of money can we use to take advantage of this fact? Students should come up with ideas like compound interest, investment of the money, etc. So how much do they think the couple actually has to have on hand at the start of their retirement?
At this point you could do an easy experiment. Using the first withdrawals tab on the spreadsheet, students can put in an amount to start with (such as a million dollars), enter an average rate of return, and stipulate an annual withdrawal (such as $130,000). The spreadsheet will then add in the interest and subtract the withdrawal year after year until the money runs out. If you do this exercise, please point out that the formulas incorporate randomness in the APR.
So instead of programming a spreadsheet and using trial and error to figure out what will work (which is a completely legitimate way to proceed!), we could also do some algebra to find a formula that will tell us the answer. If your class has an algebra prerequisite (or is an algebra class) please do this. If not, you may want to stick to the spreadsheet method to see the same result.
At this point it would be good to put error bars around that 7%. How realistic is it on average? And what happens if it is more like 6% or 8%?
Be sure to point out that this is almost the same calculation as in slide 7. It just includes a withdrawal.
If you want to avoid the algebra calculation and just use the spreadsheet at this point, notice that this formula is the same as the one used in the spreadsheet to get from one year's balance to the next. If you do this you can skip to slide 17
By all means have them do this calculation. Have them also do year 4 and then (by noticing the pattern) year 10.
This is a perfectly good formula for the relationship between the balance at the start and the balance at the end of year 25. But what might we not like about it?
You could actually have the students use this to do a calculation, but perhaps stop them before they finish the calculation. Ask them, why is it taking so long?
The main point here is that it is absolutely no fun to raise 1.07 to every single power from 1 to 24. How can we avoid this? By making use of our algebra.
If the students have not already seen these algebraic relationships, have them compute a few and then derive the general one.
In the formula we are using for the retirement account, what corresponds to x? They need to be able to identify this. (x = 1+i in the financial example.)
The point is supposed to be that (1-x25)/(1-x) is easier to compute than
(1 +x + x2 +x3 + . . . + x24). Is it?
How does your calculator calculate x25. What is a fast way to compute x24? Well, having computed x2, you can just square this to get x4, then square again to get x8 and again to get x16. That is just 4 calculations. A fifth multiplication of x8 times x16 gives the result. So this calculation is no harder than five multiplications. Plus there is very little adding.
This slide is using algebra to determine a formula for M25 that is easy to compute.
The spreadsheet simulations never give an exact balance of zero. Why not? Because the balance will go negative before year 25. Interest on negative money makes no sense, so you will get out garbage. Our formula is a very clean way of expressing what you have in year 25 if there is something left. However, the spreadsheet will tell you when you have run out of money.
But our formula is useful for working backwards to figure out the exact M0 that will make M25 equal zero.
Have them set M25 to zero and solve for M0.
At this point it would be good to remember the errors put on that 7%. Students can repeat the calculation at various return rates and see how the answer varies.
Of course, the average return is not the annual return—we know that varies a lot. The next few slides will show us how to simulate this randomness. These are the same formulas that were used to program the spreadsheet.
If you skipped the algebra above and just used the spreadsheet to estimate how much is needed, you can still compare the answers the class obtained for this number.
Note that, at this point, we have not taken into account any other sources of income, such as social security. It might be nice for students to get an idea of the contribution social security would make to their parents' income. They can ask directly or use an online calculator such as this one into which they can enter their parent's SS#.
Have the students look up some data on this. Each one could get the data on a different mutual fund, for example.
Students should try to identify what here is going to vary. (It's the rate of inflation or rise in cost of living, the 1.02)
You now have two choices with the spreadsheet.
Now here is the second part of the problem: figuring out how much your parents must have on hand to meet the projected annual income of $130,000.
The formulas on the slide are the same ones as earlier in slides 11 and 12. You might want to look at those again.
Again, students should be able to identify what is the randomly changing quantity. It is the 1.07, or expected annual return. If you wish you can also vary the withdrawal amount by using the "withdrawals (2)" tab.
It is really important to do this part of the problem. Whereas the last exercise results in a range of salaries at the 25 year mark, this investigation has dire implications.
At this point ask your students to create a spreadsheet or use the one provided.
With random variation thrown in, the couple often runs out of money before the 25 year mark. If each student ran the spreadsheet 5 times, you could get a count of how often this happens. This exercise is called a “numerical experiment” and it is done frequently in finance. For example,if the couple runs out of money before the 25 years are up on 5% of the simulations, you would say that there is a 95% chance of the money lasting 25 years. This numerical experiment is basically a “Monte Carlo” simulation, just like the professionals use, and which you often read about in business and economic literature. How accurate it is depends on what? Students should be able to figure this out—it depends on how well you know the statistics of the two varying rates (the rate of inflation and the rate of return on investment).
Ask the students why the authors of this module cannot give you those statistics. They should eventually figure out that it depends on HOW you invest the money. If students have completed the three investment modules in this series, they should understand that completely. Randomness is a factor for all investments except savings accounts and bonds that are held to maturity.
How can you make a projection? One way is to study the data for a class of investments that closely resembles what your parents plan to do in retirement. Students can make spreadsheets tracking some investments over the last 25 years and calculate mean return and variance. This will give a better estimate of the mean return and variation for a particular set of investments. It is important to discuss this if you plan to have them do the case study.
So in the next module, we are going to figure out how to save that million and a half. What should your parents be putting away now in order to be ready for retirement? It might be nice to get a sense of what students really think their parents will need upon retirement before starting that module.