Teacher Pension Crisis in your State?

Pension Crisis? A=Pert Linda Blanco Math Department Joliet Junior College Joliet, IL The oldest public community college! lblanco@jjc.edu 630-818-5967 (cell) 2

just to see what their understanding was? Did you ever ask your students a simple question, just to see what their understanding was? Ever wonder if your high school grads had a basic understanding of the difference between simple interest and compound interest?

And, no they don’t know the difference. I do it regularly. And, no they don’t know the difference. Many have never heard of simple interest. And regarding compound interest, others immediately try to “research” on Google for a “formula.”

Here is one of my attempts at assessing them prior to studying compound interest in Business Calculus:

I got two answers to each question, almost simultaneously.

How did they arrive at the answers? To get $1120 the student showed: to justify the answer $1123.20

Remember – We are still figuring simple interest. Looks good. But another student showed: to justify the answer $1123.20 Remember – We are still figuring simple interest.

Again, two answers were offered. How about the question regarding compounding semiannually? Again, two answers were offered.

Here are the justifications for those answers: Notice that one student just dittoed the previous work done to get simple interest (no edits).

I contend that they really don’t understand the principle of compounding of interest. To test that out, I have them complete a chart as if it were a bank record.

Now explore what’s going on; slow rise in interest amount, Now change the word annually to semiannually and see what they don’t understand: Now explore what’s going on; slow rise in interest amount, Slight increase in earnings over annual compounding Interest Period (6 months) Principal Investment ($) Interest Earned this Period ($) using APR of 6% New Balance of Account ($) 1 1000 30 1030 2 30.9 1060.9 3 31.83 1092.73 4 32.78 1125.51

Now explore what’s going on; slow rise in interest amount, Slight increase in earnings over annual compounding And $1125.51 is about $1000(1.03)4 Emphasize significance of 3% and 103% and 4. Interest Period (6 months) Principal Investment ($) Interest Earned this Period ($) using APR of 6% New Balance of Account ($) 1 1000 30 1030 2 30.9 1060.9 3 31.83 1092.73 4 32.78 1125.51

whether your book has them or not. When the time is right, introduce the formula: But don’t let them forget the basics! Ask good conceptual questions, whether your book has them or not.

For instance: If $1000 is invested at an APR of 6%, compounded semiannually, what will be the total amount of money at the end of 45 years? If $1000 is invested at an APR of 6% compounded semiannually, how much interest will be earned the first half of the 46th year? If $1000 is invested at an APR of 6% compounded semiannually, how fast is the money growing when the amount reaches $50K? (Calculus question)

. Interest Period ( ) Principal Investment ($) Interest Earned this Period ($) using APR of ______ New Balance of Account ($)   .

. Interest Period ( ) Principal Investment ($) Interest Earned this Period ($) using APR of ______ New Balance of Account ($)   .

and ? ? So what about the confusion between If they understand that Is it a stretch to assert that ?

and t as independent variable, In the formula, how about we regard as the “base” and t as independent variable,

then gets really close to when the value of n is really large.

And what if we don’t like the “base?” Can we only use base e when growth is continuous? We teach the “Change of Base” property for logs, but we usually don’t change the base of an exponential function. Why not? We generally cover the inverse function relation: So why not capitalize on this?

In Business Calculus we discuss the relationship: If then or Since the rate at which the function is changing is directly proportional to the amount present, the larger the amount, the faster the growth.

So, it might be an advantage to convert from an original base b to base e. P = C(1.03)t becomes P = C(eln1.03)t or P = Cet(ln1.03) So P’ = C(eln1.03)t (ln1.03) P’ = (ln1.03)C If the initial investment is $1000, the initial rate of growth is 1000(ln1.03) or about $29.56 per unit of time But, by the time the investment grows to $100000, the rate of growth is about $2956 per same unit of time.

Worthy of note is that ln1.03 is not quite 3%. So an investment following P = Ce.03t is not precisely the same as one following P=Celn1.03t. The first has a “growth rate” of 3% per unit of time yielding over 3%. But the second yields 3% during each unit of time. So be wary of formulas that round early and don’t fit the data one is trying to achieve.

So what happens when we continue to contribute to our investment regularly over a long period of time?  Consider an investment of $1000 invested at an annual rate of 10%, compounded annually. If that amount is left to sit for 45 years, it grows to 1000(1.1)45. If we invest another $1000 a year later and let it sit for the next 44 years, it grows to 11000(1.1)44. 000(1.1)44. And if we do it again the following year, that $1000 will grow to 1000(1.1)43.

If we continue up until the last year before retirement, the last contribution will grow for just that last year to 1000(1.1)1. At the end of 45 years, our contributions and earned interest will have grown to: 1000(1.1)1+1000(1.1)2+1000(1.1)3+ …+1000(1.1)45 a geometric series of 45 terms written from most recent to farthest back.

Since this is a geometric series, we can evaluate as follows: The total amount comes to $790,795.32 Not a bad return on $45,000 investment.

So what happens if I am not diligent with my contributions? Or what happens if someone else responsible for those contributions fails to contribute? Let’s say that contributions are not made for 5 consecutive years. Does it matter which years?

For this scenario, let’s go back to the original order of the terms representing the compounded contributions: 1000(1.1)45+1000(1.1)44+1000(1.1)43+ …+1000(1.1)1 How much less would be available at retirement if we skip the last five years? 1000(1.1)5+1000(1.1)4+1000(1.1)3+1000(1.1)2+1000(1.1)1 So, by not contributing the last $5000, we would be about $6715 short of the previously computed total.

Again, let’s go back to the original order of the terms representing the compounded contributions: 1000(1.1)45+1000(1.1)44+1000(1.1)43+ …+1000(1.1)1 How much less would be available at retirement if we skip some in between years? Let’s say years 11-15. 1000(1.1)35 +1000(1.1)34+1000(1.1)33+1000(1.1)32+1000(1.1)31 So, by not contributing this particular $5000, we would be about $117,183 short of the previously computed total.

Percentagewise, that’s a huge chuck of change! $117183/$790795 is about 15% Skipping the last 5 years by comparison would only lose us about 0.8%.

Questions: If we are going to “pay back” the missing contributions, how much is owed? How much would have to be paid in over a period of, say, five years in order to regain the loss?  If we take five more years to pay back into the fund, shouldn’t we also be taking into account the amount that would have accrued during those five additional years? Aren’t there elements of this concept that could be introduced as early as in developmental courses, or high school courses?

Seize the teachable moment when you realize they don’t have a clue bring it back to basics don’t just give the formula link concepts taught within the course or curriculum

Linda Blanco Joliet Junior College lblanco@jjc.edu 630-818-5967 (cell)