Math Matters: Why Do I Need To Know This?Bruce Kessler, Department of Mathematics

Western Kentucky UniversityEpisode Fourteen

1 Annuities – Investment strategies

Objective: To illustrate how knowing the mathematics behind annuities canhelp us to understand and calculate the future value of investments. We alsoillustrate how small changes in our contributions now can cause a large differencein the future value of our investments. We develop the formula for the sum of ageometric 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 lastepisode of Math Matters, and we, we . . . it’s been a lot of fun. I enjoy getting the opportunityto talk about the applications of mathematics, and hopefully it’s been an educational processfor 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 foryou to know, so that you can kind of plot your course through life, planning your money andso 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 whatan annuity is. An annuity is any type of account, a savings account kind of thing, where youmake 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 onthis money that you put into the account. Examples would be IRA’s, individual retirementaccounts, 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, andthen it has some value when you’re done paying on it. And even little things, like ChristmasClub accounts where you, you put money in monthly and then in December you get the moneyout 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 tocalculate the value of these things, and to do that, lets talk about how these things wouldwork. You would typically think about in the timeline, and these are the are months, thisis segmented in months, so somewhere here you set this thing up, and then at the end ofthe month, typically out of your paycheck, you’ll make a contribution. So let’s assume we’remaking a hundred dollar regular payment into the annuity, which would be made at the endof the month, okay? Now, how much is that going to be worth when we’re done? Makessense it would be about $400, right? I’m making four $100 payments, so my answer shouldbe somewhere close to or slightly above $400. The way to kind of get to the future value ofthis after a period of time is to take that monthly rate, we’ve done this before, this is thecompound interest formula, it’s 100% of what you have, plus the monthly rate, and you’regoing to compound that interest, but, in this first month, you only draw one, two, three monthsof interest on it, so thats raised to the third power. The second payment only receives twomonths 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 youstarted 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 hardway, 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, Ineed a short cut to the answer. So what I’m going to do is, I’m going to show you a littletrick, this is a cool little trick, where we add up things. What we’re talking about, when weadd up things raised to an increasing power it, we call that a geometric series. And we havea 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 itup, 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 stunthere, I’m going to multiply S by r. Now what that does, is it raises the power by one. So I’mlining 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, andI can factor out that S. Over here the r1, the r2, the r to the whatever, even the rn canceleach other out. So, I’m just left with r0, which is 1, minus rn+1. And so to solve for S, Ijust divide through by the junk, standard little trick. It actually, the way I set it up, it endsup 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 alittle 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 backto this, where I’ve got all the different accounts, all the different payments put in, and thecompound interest that they, that they draw, I’m going to add those things up. So each one ofthose 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, thereis 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 1

2% 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 bigwad 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 along period of time. So, first let me say that the general formula, let me just go back, andI’ll take the one-hundred, make that a P, n is the number of payments, okay? So here’s theformula that I would use to calculate the value of the annuity after so many payments. Let’ssay for example that I pay a hundred dollars from each paycheck for thirty years. And I’llsay still I’m getting 6%, okay? So, taking those amounts, here’s the 100, there’s the 1

2%

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 changesto the percentage rate. One percent of $36,000 is $360. However, compound interest worksso that I draw interest on my interest, and that snowball effect, if I raise this up to 7%, asopposed to 6%, the end value, the future value of this account, is $142,330, basically, thats a41, or $42,000 increase. That – that’s dramatic, okay? So it shows how affected that is bythe interest rate. (Figure 5)

I’ve actually got a couple of summary pages that have those formulas on them, that willkind of summarize what we just talked about, and I’ll get us set-up for the next segment.

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2 Fair division – Estate settlement

Objective: To illustrate how some very simple arithmetic can be used to dividean 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. Perhapsyou’ve inherited some money, or you’ve inherited a house, or a boat, or things like this, andyou have siblings, so you have to share it. Well, some things share very well. Money, youcan take a wad of money, if there’s three of us, we divide by three, we each take that amountof the money, and we go with it. It’s not so easy when I talk about things like houses, andboats, and so forth. You can cut them into thirds, but then you’ve destroyed the value ofthem. You can sell things, but, thats not always a good option either. If it’s the old familyhome, you don’t want to sell the old family home, you know, you want to, someone needsto hold on to the possession of that in the family. And so the question becomes how do youdivide things up so that everyone feels like they are getting their fair share of the estate. Andit turns out there’s a nice little mathematical algorithm for doing this that I’m going to, I’mgoing 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 includesa 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 mehold on to that cash for just a moment, and let’s talk about how we can divide the estate upfairly, 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 completesecrecy, and I’ll show you later, that it actually works to their advantage to bid on it’s valueto them, not some kind of inflated value. So, we go through, and all the heirs bid on theitems, 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 theywould consider to be a fair share of the estate. And the way we do that is we add up thevalue of everything according to what they called the value. So for Heir 1, we take the valuesthat they gave here, add those up, and then the $75,000 is worth $75,000 to everybody, so weadd that up, and that makes the estate worth $168,000 – to this heir. We’re going to shareit equally, so we divide by three, and that gives us what we call their “fair share,” accordingto 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, butthey’re not identical, and that’s kind of important. But this is what every one of them shouldconsider 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 bethe most valuable item of the three of you, so you get that item. Now in our case here, thatwould mean that Heir 1 gets the house, Heir 2 bid the highest with $7,000 for the boat, andHeir 3 bid highest on the car, so they get the car, okay? You get those items. Now here’sthe problem: this house is valued at, actually by all of them, at much more than their fairshares. 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 everybodydown to the fair share, okay? So the first person, the house is worth more than their fair

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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 theirfair share, so we’re going to make up that difference in cash, okay? So, Heir 2, when wesubtract, they would receive $48,000, paid from the estate. That’s removed from the amountof cash we have on hand. And Heir 3 received the car I think. That difference in their fairshare, what they consider their fair share, and the cars value, what they consider the carsvalue, is $47,000, so we take that out of the estate, pay that to them. And when we’re alldone with this, we still have a little bit of money left over. There’s nine thousand dollars leftout of the cash. (Figure 4)

And so the next step is easy. Divide 9,000 by three, give each of the people left, give eachof them what’s left, $3,000 to each of the heirs. So what does that really give, what do theyend up with, that’s kind of the point of this. The first heir gets the house, which they valueat $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 theirfair share. The second heir got the, I think it was the boat, received some cash, and receivedthe additional cash. That additional cash is over and above what they considered to be theirfair share. And then the third heir, the same thing happens. This is the system, becauseeverybody ended up getting more than what they considered to be fair. Everyone should behappy with this arrangement. Which is nice, that’s a good system, that’s a good way to kindof 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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caught, and I’ll show you how that works. Let’s say that the first heir really, really, reallywanted the house. So they gave this exorbitant price, this exorbitant bid on the house. Eventhough they really thought it was valued at $85,000, they bid $200,000 on it, okay? Nowwatch how this effects their totals. Their fair share now is going to go up, right when Ichange this to $200,000, the value goes up here, and then when you divide by three, it goesup. 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 changesnow though, is how much is paid to the estate. If this is two-hundred thousand, this is nowa 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 morecash on hand for the estate, okay? (Figure 7 and 8) Everything else is as it was, but whatchanges is, instead of taking the amount $9,000, dividing by three, and getting $3,000, nowwhen 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 thoughtit was actually valued at. They end up with a lot less than the others. And what they’vedone is they cheated themselves out of, they did get the item they wanted, but they cheatedthemselves 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 themathmatics of it. I’ve summarized the algorithm for this in a couple of summary pages, andI’ll get set up for the last little segment. (Figure 10)

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3 Wrap-up

The last thing I’d like to do with you today, since it’s our last episode, is I would like to kindof wrap things up, and end with some encouraging words. Now I realize, at this stage of thegame, you’re probably getting ready for finals, you may be getting ready for finals, and youmay even be a little sick of mathmatics at this point if you’re finishing up the semester. ButI would like to encourage you to follow up, and take some more mathmatics. And I have acouple of reasons for telling you this. We’ve already demonstrated that we could handle somevery tough situations with the mathmatics that we find in the general math courses. We’velooked, we spent a semester looking at things like that. It’s also true though, that you canget into some fairly routine situations that have tough mathmatical answers, and that’s thekind 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 thisequation, well, you’ve probably realized by now that that’s a line. It’s a linear equation, sothe graph of that is a line, and the solutions are all the points on that line. That’s prettyeasy, a graphing calculator can do that for us. (Figure 1) However, if I ask the questionthis 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). Thoseare integer solutions, they’re not always, not everything on the line though will work for thisanswer. And so we’re getting into a field of mathematics called discrete mathematics, andthat requires some further study, although that’s a very, that’s a very simple kind of question.(Figure 2)

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Here’s another one. Let’s say I want exact solutions to this quadratic equation. Nowyou folks have probably had, know the quadratic formula, some of you even, I had somekids yesterday singing me, singing me the quadratic formula, which I thought was prettycool. 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 takeanyone of those powers up above 2, if I make that 2 a 3, that becomes a very difficult thingto 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 prettydifficult problem, because I don’t have a cubic formula. I have a quadratic formula. I knowhow to solve linear things, but something like that is actually kind of tough. We can talkabout 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 biglong number that is the product of two primes. It’s easy to multiply the primes together. It’svery difficult to divide that composite number into the two primes, and that’s, a lot of ournational 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 makethat x3, make that y4, all of the sudden, this has become an extremely difficult problem tosolve 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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these things, and I can numerically isolate that point, and there are other techniques fordoing that. But exact solutions are very difficult for non-linear systems. So, these are thekind of things that can pop up, and, and in day-to-day applications, I mean not everythingis 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 bakerythat makes five different types of fudge – you know it sounds like something you would askout of just being naive – how many different ways can we fill a package with twelve piecesof 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 lotof ways to do this. That is a very difficult question to answer. That’s also something we’dhandle in the field of mathematics called discrete mathematics. We have counting techniquesfor doing things like that. But it’s a little difficult. If I go back and say, well okay, let me puta restriction on that. Let’s make sure we have one of each type in the package, then it getsjust 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 theanswer could be up in the thousands, ten-thousands, whatever it is, okay? So this is somefairly 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 wouldencourage you to continue to study math is, you know, I think we jump the gun sometimeswhen we’re studying, and we make this assumption that because we can’t immediately findsome applications for things, that it’s not really very useful, okay? I couldn’t disagree withthat attitude more, okay? The problem sometimes is not in the mathematics, it’s in our

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understanding of other sciences, or other applications, the actual content where it could beapplied, 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 droppingbombs on us from space and all this, but even after we had kind of won the cold war, wecontinued with our space program. People said now why would, why do we care about thatnow, you know, we won. But you have to look at the benefits that have come from thatspace program. Cell phones are a direct benefit, cell phones were actually developed throughtechnology developed by NASA, in getting people to the moon and things like that. Velcrowas designed after stick-tights in nature. But where did they use that? Well, they use it inspace capsules because you can’t lay a pencil down in space, it floats off. So they needed alittle piece of velcro, and a little piece of velcro, and they’d stick it to stuff, okay? So, itreally, it’s not always the mathematicians fault, or the mathematics’ fault that we don’t seethe 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, whograduated 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 workingin the science area, and you understand mathematics – sometimes it would be very difficultfor 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 younot to give up, even if you’ve had a rough time this first time through. There’s very cool stuffout 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 thosepages 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 episodesthis semester. I hope they’ve been helpful to you. I hope they’ve been educational, and I hopethey have served your purposes, helped you in class, and helped kind of expand your horizonsas 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 awebpage that we’ll flash up in a few minutes – use me as a resource if you wish to, to tacklethings about applications, and – you know I’m always looking for new ideas around theselines. So, I want to to thank you for your time. I think I’ve done all the damage that I cando, and I hope to see you again.

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