Math Matters: Why Do I Need To Know This?
Bruce Kessler, Department of Mathematics
Western Kentucky University
Episode Fourteen
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Annuities – Investment strategies
Objective: To illustrate how knowing the mathematics behind annuities can
help us to understand and calculate the future value of investments. We also
illustrate how small changes in our contributions now can cause a large difference
in the future value of our investments. We develop the formula for the sum of a
geometric series, and the formula for the future value of an annuity.
Hello, and welcome to “Math Matters: Why Do I Need To Know This?” This is our last
episode of Math Matters, and we, we . . . it’s been a lot of fun. I enjoy getting the opportunity
to talk about the applications of mathematics, and hopefully it’s been an educational process
for all of us. I want to finish today with just a couple ideas from consumer mathematics,
things that don’t require heavy duty mathematics in order to understand, but it’s good for
you to know, so that you can kind of plot your course through life, planning your money and
so forth. And then I’ll kind of wrap up at the end. So let’s get going.
I’d like to talk to you about annuities, and first I should probably start and tell you what
an annuity is. An annuity is any type of account, a savings account kind of thing, where you
make a regular payment into the account, and then you get interest on the . . . not just the,
you’re doing more than just getting money out of it, you’re accruing compound interest on
this money that you put into the account. Examples would be IRA’s, individual retirement
accounts, or other types of retirement plans, where you’re making a regular contribution.
Things like whole-life insurance policies, not term, but whole-life, where you pay on this, and
then it has some value when you’re done paying on it. And even little things, like Christmas
Club accounts where you, you put money in monthly and then in December you get the money
out and you take that and you go shopping with it, or something like that.
Lets take a look at the timeline of how things work, I want to describe to you how to
calculate the value of these things, and to do that, lets talk about how these things would
work. You would typically think about in the timeline, and these are the are months, this
is segmented in months, so somewhere here you set this thing up, and then at the end of
the month, typically out of your paycheck, you’ll make a contribution. So let’s assume we’re
making a hundred dollar regular payment into the annuity, which would be made at the end
of the month, okay? Now, how much is that going to be worth when we’re done? Makes
sense it would be about $400, right? I’m making four $100 payments, so my answer should
be somewhere close to or slightly above $400. The way to kind of get to the future value of
this after a period of time is to take that monthly rate, we’ve done this before, this is the
compound interest formula, it’s 100% of what you have, plus the monthly rate, and you’re
going to compound that interest, but, in this first month, you only draw one, two, three months
of interest on it, so thats raised to the third power. The second payment only receives two
months of interest. So thats a two right there, the third payment only receives one month of
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interest, and the fourth receives no interest at all, it’s just the one hundred dollars that you
started with, and I went ahead and put the zero there, to kind of illustrate what’s happening.
To get to the future value of this we have to add these amounts up. (Figure 1)
Figure 1, Segment 1
So you’re adding up things raised to powers. Now, that can, you can do that the hard
way, I mean you get your calculator out, and just sit there and hunt-and-peck all the time,
but I’d like to look at things that happen over a long period of time, so i need a formula, I
need a short cut to the answer. So what I’m going to do is, I’m going to show you a little
trick, this is a cool little trick, where we add up things. What we’re talking about, when we
add up things raised to an increasing power it, we call that a geometric series. And we have
a nice way of generating the value of a geometric series. I’m going to show you the trick.
This is really slick.
Let’s say I want to determine what this would add up to, you know, r is some number,
and I’m raising to the zero power, and the first power, you know, adding it up, and adding it
up, and take this to the nth power. I add all these things up. Well let me call this S, okay?
I don’t know what S is, thats what I want to find out, and I’m going to pull a little stunt
here, I’m going to multiply S by r. Now what that does, is it raises the power by one. So I’m
lining things up here in powers of r. And then I will subtract this, from that. So subtract,
subtract, I’ll even put the little bar there. When I subtract these things, I get S − rS, and
I can factor out that S. Over here the r1 , the r2 , the r to the whatever, even the rn cancel
each other out. So, I’m just left with r0 , which is 1, minus rn+1 . And so to solve for S, I
just divide through by the junk, standard little trick. It actually, the way I set it up, it ends
up being one minus this, and one minus that, but if you multiply by − 1 in the top and the
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bottom, it would look like this, where you subtract the one in each case. And that looks a
little neater in the formula I’m getting ready to develop. (Figure 2)
Figure 2, Segment 1
I’m going to use this idea now to calculate the future value of that account. So we go back
to this, where I’ve got all the different accounts, all the different payments put in, and the
compound interest that they, that they draw, I’m going to add those things up. So each one of
those has a one hundred there that I can factor out, k, and then here is my geometric series.
And if you’ll notice, that power of r is always one more than the highest power in the series.
So this would be a four, which is actually equal to the number of payments I made, okay,
and then it’s just a matter of deciding what the monthly rate is. I do this with, uh, oh, there
is a little simplification, one plus M minus one is just M. If I assume a 6% interest rate,
that’s 6% annual, divide that by twelve, then thats 12 % per month. So my monthly interest
rate would be 0.005. And then, when I do the little calculation, it works out to be $403.01.
So that means I gained $3.01 from this process, other than just what I paid in. Not a big
wad of money, I know. (Figure 3)
But now lets think about, I’d like to think about what happens if I try to do this over a
long period of time. So, first let me say that the general formula, let me just go back, and
I’ll take the one-hundred, make that a P, n is the number of payments, okay? So here’s the
formula that I would use to calculate the value of the annuity after so many payments. Let’s
say for example that I pay a hundred dollars from each paycheck for thirty years. And I’ll
say still I’m getting 6%, okay? So, taking those amounts, here’s the 100, there’s the 21 %
interest, there’s the 12 % interest. When I do that calculation, I end up with over $100,000.
Now, when you think about how much I paid in, I paid in $100, 360 times. So thats $36,000
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Figure 3, Segment 1
that I paid in. I’ve almost tripled my money by doing that over a long period of time, okay?
You know, I’m trying to show you the value of this kind of account. (Figure 4)
The additional thing to kind of point out about this, is how it reacts to small changes.
Now, lets say instead of $100, I put in $101. That means I’m going to contribute an additional
$360, okay? So, how does that affect the calculation? Well, it raises it by a little over $1,000,
so again, I put in an extra $360, I get out an additional $1,000. I could also look at changes
to the percentage rate. One percent of $36,000 is $360. However, compound interest works
so that I draw interest on my interest, and that snowball effect, if I raise this up to 7%, as
opposed to 6%, the end value, the future value of this account, is $142,330, basically, thats a
41, or $42,000 increase. That – that’s dramatic, okay? So it shows how affected that is by
the interest rate. (Figure 5)
I’ve actually got a couple of summary pages that have those formulas on them, that will
kind of summarize what we just talked about, and I’ll get us set-up for the next segment.
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Figure 4, Segment 1
Figure 5, Segment 1
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Summary page 1, Segment 1
Summary page 2, Segment 1
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Fair division – Estate settlement
Objective: To illustrate how some very simple arithmetic can be used to divide
an estate with several non-dividable objects in a fair manner among several heirs.
The idea I’d like to talk to you about today involves the fair division of estates. Perhaps
you’ve inherited some money, or you’ve inherited a house, or a boat, or things like this, and
you have siblings, so you have to share it. Well, some things share very well. Money, you
can take a wad of money, if there’s three of us, we divide by three, we each take that amount
of the money, and we go with it. It’s not so easy when I talk about things like houses, and
boats, and so forth. You can cut them into thirds, but then you’ve destroyed the value of
them. You can sell things, but, thats not always a good option either. If it’s the old family
home, you don’t want to sell the old family home, you know, you want to, someone needs
to hold on to the possession of that in the family. And so the question becomes how do you
divide things up so that everyone feels like they are getting their fair share of the estate. And
it turns out there’s a nice little mathematical algorithm for doing this that I’m going to, I’m
going to take you through. It’s very simple; it really just involves addition and subtraction,
and division and so forth, but it’s pretty slick.
Let me run this situation by you, lets say there are three heirs to an estate, that includes
a house, a boat, a car, and then there’s $75,000 in cash. Okay, now the cash division is easy,
they’ll each, you know we can say you’ll each get twenty-five thousand of that. But let me
hold on to that cash for just a moment, and let’s talk about how we can divide the estate up
fairly, okay? The first thing I would do if I’m trying to work this problem is I would say,
okay, I would like each of the heirs to bid on the items. And this will take place in complete
secrecy, and I’ll show you later, that it actually works to their advantage to bid on it’s value
to them, not some kind of inflated value. So, we go through, and all the heirs bid on the
items, and I’m showing you some examples of what those bids could be. (Figure 1)
Then what we need to do is go through, and, based on these bids, determine what they
would consider to be a fair share of the estate. And the way we do that is we add up the
value of everything according to what they called the value. So for Heir 1, we take the values
that they gave here, add those up, and then the $75,000 is worth $75,000 to everybody, so we
add that up, and that makes the estate worth $168,000 – to this heir. We’re going to share
it equally, so we divide by three, and that gives us what we call their “fair share,” according
to them. We do this to each of the three heirs, so we do this across, we add everything up,
divide it by 3, add everything up, divide by three, so they’re being fairly consistent here, but
they’re not identical, and that’s kind of important. But this is what every one of them should
consider a fair share of the estate, okay? (Figure 2)
Then, we go in and we say, okay, you bid the highest for this item, you consider it to be
the most valuable item of the three of you, so you get that item. Now in our case here, that
would mean that Heir 1 gets the house, Heir 2 bid the highest with $7,000 for the boat, and
Heir 3 bid highest on the car, so they get the car, okay? You get those items. Now here’s
the problem: this house is valued at, actually by all of them, at much more than their fair
shares. These are valued much less than their fair shares. (Figure 3)
So what we do is we then use the cash, that’s in the account, to kind of adjust everybody
down to the fair share, okay? So the first person, the house is worth more than their fair
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Figure 1, Segment 2
Figure 2, Segment 2
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Figure 3, Segment 2
share, that means they need to pay some back, and they end up paying the difference, which is
$29,000. That’s paid into the estate, okay? Now the folks paid, they received less than their
fair share, so we’re going to make up that difference in cash, okay? So, Heir 2, when we
subtract, they would receive $48,000, paid from the estate. That’s removed from the amount
of cash we have on hand. And Heir 3 received the car I think. That difference in their fair
share, what they consider their fair share, and the cars value, what they consider the cars
value, is $47,000, so we take that out of the estate, pay that to them. And when we’re all
done with this, we still have a little bit of money left over. There’s nine thousand dollars left
out of the cash. (Figure 4)
And so the next step is easy. Divide 9,000 by three, give each of the people left, give each
of them what’s left, $3,000 to each of the heirs. So what does that really give, what do they
end up with, that’s kind of the point of this. The first heir gets the house, which they value
at $85,000. They had to pay some cash, but they get $3,000 of it back, for a grand total of,
in their mind, $59,000. Okay, now that’s $3,000 more than what they considered to be their
fair share. The second heir got the, I think it was the boat, received some cash, and received
the additional cash. That additional cash is over and above what they considered to be their
fair share. And then the third heir, the same thing happens. This is the system, because
everybody ended up getting more than what they considered to be fair. Everyone should be
happy with this arrangement. Which is nice, that’s a good system, that’s a good way to kind
of handle things. (Figure 5)
Even beyond that, it’s a good system in that it discourages folks from bidding artificially,
you know, to make sure they get it. Everyone has to play honest, or they’re going to get
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Figure 4, Segment 2
Figure 5, Segment 2
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caught, and I’ll show you how that works. Let’s say that the first heir really, really, really
wanted the house. So they gave this exorbitant price, this exorbitant bid on the house. Even
though they really thought it was valued at $85,000, they bid $200,000 on it, okay? Now
watch how this effects their totals. Their fair share now is going to go up, right when I
change this to $200,000, the value goes up here, and then when you divide by three, it goes
up. So, their fair share is inflated as well, okay, the rest of them stay the same. (Figure 6)
Figure 6, Segment 2
They were going to get the house anyway, and so they still get the house. What changes
now though, is how much is paid to the estate. If this is two-hundred thousand, this is now
a pretty big sum, right? This is way over and above what they considered their fair share.
So they have to pay quite a bit back to the estate, and what that does is that puts a lot more
cash on hand for the estate, okay? (Figure 7 and 8) Everything else is as it was, but what
changes is, instead of taking the amount $9,000, dividing by three, and getting $3,000, now
when you divide by three you get $28,556. (Figure 9)
So that’s paid out, and watch what happens to the value here with each of the heirs, okay?
And what I’ve done is, instead of the $200,000, I used the actual value, what they thought
it was actually valued at. They end up with a lot less than the others. And what they’ve
done is they cheated themselves out of, they did get the item they wanted, but they cheated
themselves out of actual value of the estate. (Figure 10)
This is a remarkable little system for doing this. Chances are at some point in your life,
you might encounter a situation like this. Now you kind of have an idea how to handle the
mathmatics of it. I’ve summarized the algorithm for this in a couple of summary pages, and
I’ll get set up for the last little segment. (Figure 10)
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Figure 7, Segment 2
Figure 8, Segment 2
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Figure 9, Segment 2
Figure 10, Segment 2
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Summary page 1, Segment 2
Summary page 2, Segment 2
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Wrap-up
The last thing I’d like to do with you today, since it’s our last episode, is I would like to kind
of wrap things up, and end with some encouraging words. Now I realize, at this stage of the
game, you’re probably getting ready for finals, you may be getting ready for finals, and you
may even be a little sick of mathmatics at this point if you’re finishing up the semester. But
I would like to encourage you to follow up, and take some more mathmatics. And I have a
couple of reasons for telling you this. We’ve already demonstrated that we could handle some
very tough situations with the mathmatics that we find in the general math courses. We’ve
looked, we spent a semester looking at things like that. It’s also true though, that you can
get into some fairly routine situations that have tough mathmatical answers, and that’s the
kind of thing that we handle in the upper level courses.
For example, here’s a very easy problem. If I say okay, what are all the solutions to this
equation, well, you’ve probably realized by now that that’s a line. It’s a linear equation, so
the graph of that is a line, and the solutions are all the points on that line. That’s pretty
easy, a graphing calculator can do that for us. (Figure 1) However, if I ask the question
this way, I say, what are the integer solutions to that equation? That is a lot less exacting.
And, a lot of the solutions here will not work. They have to fall along an integer lattice here,
like these points do. This one actually occurs at (1, 1), this one occurs at (−2, 3). Those
are integer solutions, they’re not always, not everything on the line though will work for this
answer. And so we’re getting into a field of mathematics called discrete mathematics, and
that requires some further study, although that’s a very, that’s a very simple kind of question.
(Figure 2)
Figure 1, Segment 3
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Figure 2, Segment 3
Here’s another one. Let’s say I want exact solutions to this quadratic equation. Now
you folks have probably had, know the quadratic formula, some of you even, I had some
kids yesterday singing me, singing me the quadratic formula, which I thought was pretty
cool. We have a way to do that, and I’ll show you the graph. We use quadratic formula,
we can even factor this one if we need to to solve it exactly. (Figure 3) However, if I take
anyone of those powers up above 2, if I make that 2 a 3, that becomes a very difficult thing
to solve symbolically. Even this one, which has nice answers, it has nice integer answers
− 2 and 1. If you ask me to do that symbolically, that all of a sudden becomes a pretty
difficult problem, because I don’t have a cubic formula. I have a quadratic formula. I know
how to solve linear things, but something like that is actually kind of tough. We can talk
about numerical solutions, we can talk about some different ways of trying to factor that,
but as a rule, multiplying is much easier than factoring. And that’s fairly difficult to factor.
(Figure 4) That, by the way, is the idea behind public key encryption – that you have this big
long number that is the product of two primes. It’s easy to multiply the primes together. It’s
very difficult to divide that composite number into the two primes, and that’s, a lot of our
national security is based on that very idea right there.
Here’s something else, if we look at two lines, a system of equations, linear equations,
well, we’ve talked about actually in this program ways of solving that exactly, you know,
getting exact solutions to that. The solution will be this point where the two lines overlap.
(Figure 5) However, if I jump above the power of one on any one of these things, like I make
that x3 , make that y 4 , all of the sudden, this has become an extremely difficult problem to
solve exactly. And it really doesn’t get much better as you go up the higher level mathmatics,
this is just a hard problem, okay? Now, we can do it graphically. Obviously, I can graph
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Figure 3, Segment 3
Figure 4, Segment 3
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these things, and I can numerically isolate that point, and there are other techniques for
doing that. But exact solutions are very difficult for non-linear systems. So, these are the
kind of things that can pop up, and, and in day-to-day applications, I mean not everything
is linear, thats just the sad fact of things. (Figure 6)
Figure 5, Segment 3
Here’s another example of a very tough problem. It sounds very simple. We have a bakery
that makes five different types of fudge – you know it sounds like something you would ask
out of just being naive – how many different ways can we fill a package with twelve pieces
of fudge? I’ve got five, I can make them all one type, I can make them all the other types,
make half, . . . . I mean, you start thinking about all the ways to do this, and there are a lot
of ways to do this. That is a very difficult question to answer. That’s also something we’d
handle in the field of mathematics called discrete mathematics. We have counting techniques
for doing things like that. But it’s a little difficult. If I go back and say, well okay, let me put
a restriction on that. Let’s make sure we have one of each type in the package, then it gets
just that much tougher to kind of figure out, okay, well, how many ways can you do this?
You certainly don’t want to try and list them all out by process of elimination, because the
answer could be up in the thousands, ten-thousands, whatever it is, okay? So this is some
fairly difficult mathmatics to deal with, although it’s very simply stated, okay? (Figure 7)
The other kind of approach, the other kind of thing I would say as far as why I would
encourage you to continue to study math is, you know, I think we jump the gun sometimes
when we’re studying, and we make this assumption that because we can’t immediately find
some applications for things, that it’s not really very useful, okay? I couldn’t disagree with
that attitude more, okay? The problem sometimes is not in the mathematics, it’s in our
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Figure 6, Segment 3
Figure 7, Segment 3
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understanding of other sciences, or other applications, the actual content where it could be
applied, the area where it could be applied.
You know, I always think back, a great analogy for this would be the space program, okay?
The space program really kind of was driven initially by this fear of the Soviet Union dropping
bombs on us from space and all this, but even after we had kind of won the cold war, we
continued with our space program. People said now why would, why do we care about that
now, you know, we won. But you have to look at the benefits that have come from that
space program. Cell phones are a direct benefit, cell phones were actually developed through
technology developed by NASA, in getting people to the moon and things like that. Velcro
was designed after stick-tights in nature. But where did they use that? Well, they use it in
space capsules because you can’t lay a pencil down in space, it floats off. So they needed a
little piece of velcro, and a little piece of velcro, and they’d stick it to stuff, okay? So, it
really, it’s not always the mathematicians fault, or the mathematics’ fault that we don’t see
the application. Sometimes we need to educate ourselves in other areas, okay?
Terry Wilcutt – I heard him say this one time – Terry Wilcutt is an astronaut, who
graduated from Western with a degree in mathematics, we’re very proud of him – he said
“Mathematics is the common language of the sciences.” And that’s true. If you’re working
in the science area, and you understand mathematics – sometimes it would be very difficult
for a biologist to talk to a geologist, for example, they may not have common things there –
but a mathematician, in some sense, in some way could probably talk to both of those people,
as long as we’re working on some kind of a problem together. I really would encourage you
not to give up, even if you’ve had a rough time this first time through. There’s very cool stuff
out there, and it just takes a little more education sometimes to find it.
I’ve summarized a few of those little comments that I talked about, and I’ll flash those
pages up, then we’ll come back and wrap things up.
Closing
Well, I want to thank you for tuning in. I hope you’ve checked out several of the episodes
this semester. I hope they’ve been helpful to you. I hope they’ve been educational, and I hope
they have served your purposes, helped you in class, and helped kind of expand your horizons
as far as what math is good for. There’s a lot of things out there.
I still would like to hear your input on things, and would encourage you – we have a
webpage that we’ll flash up in a few minutes – use me as a resource if you wish to, to tackle
things about applications, and – you know I’m always looking for new ideas around these
lines. So, I want to to thank you for your time. I think I’ve done all the damage that I can
do, and I hope to see you again.
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Summary page 1, Segment 3
Summary page 2, Segment 3
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