One way to avoid (or, at least drastically reduce) this problem is to either send or store multiple copies of the information. Say you send the message “BILL JOHN” 3 times. One of the times the error converts this to “KILL JOHN”, but two of the times the message transmits successfully. Then you know that most likely the intended message was “BILL JOHN”. Similarly, your bank probably stores your account information of multiple storage locations to avoid this and other problems (such as, catastrophic data loss due to a fire).  However, redundancy is not very efficient, and may be overkill. Further, it may not be always possible to apply (say the communication takes a long time and repeat communication is not possible).

Another solution is to add a small amount of extra information to the message being communicated or stored to validate the correctness of the data. This is, in effect, redundancy, but is often a more space-efficient form of it. Its purpose is primarily to detect errors in the message data, not to handle catastrophic destruction of data (which is the primary purpose of full redundancy). The most common way to detect errors in transmitted data is by the use of parity bits. The data is first divided into blocks of bits, say 7 or 15-bit blocks. Next, for each block of bits a parity bit is added, thus making the block size 8 or 16 bits. For example, if the original data is “10110101000101010″, and the block size is 7 +1 parity bit, then the data is first extended to make it a multiple of 7. So, say the message is changed to “000010110101000101010″. Then, it is broken into 7-bit blocks – “0000101″, “1010100″, “0101010″. And finally, for each 7-bit block one more bit is added. Say, it is added to the right end. The parity bit is dependent on the data bits. The parity bit is set to a 1 or a 0 depending on the number of 1s in the message. If there are an odd number of 1s in the message, the parity bit is set to 1 to make the total parity of the 8-bit block even. I am assuming even parity in this example; the parity may also be set up to be odd. Going back to the example, if there are already an even number of 1s in the 7-bit data block the parity bit is set to 0. The recipient knows what the block size is, where the parity bit appears, and what kind of parity (Even or Odd Parity) is being used. This is part of a pre-established communication agreement. For example, “0000101″ has 2 1s, that is, an even number of 1s. The parity bit is set to 0 and appended to these 7 bits to make the transmission block, “00001010“. Upon transmission, say there is a bit flip and the message becomes “00101010″. The receiver can then figure out that the parity, which is supposed to be even is now odd, and therefore indicates an error. The receiver can then request a retransmission. Notice that the parity bit itself may be transmitted in error, and that triggers an error as well (even though in reality the data bits were all transmitted fine). The size of the block is set to be small enough so that the probability of a bit flip is very very low. More importantly, the probability of two flips is infinitesimally small. Notice that if there are two flips, however, a single bit parity scheme as described above will not work. There is then a need for a more involved parity scheme. If the channel or storage medium has a propensity for unidirectional flips (that is, say it can only flip a 1 to a 0, bit never a 0 to a 1), then a counter may be maintained along with the data to count the number of 1s in the data block. If the number of 1s changes, then the receiver can detect an error. It is a little tricky once you realize that the number of 1s counted must include the number of 1s in the counter itself! The counter bits need to be transmitted as well, and are subject to bit flips as well.

In some situations, there may not be a second copy to rerequest the data upon detecting the error. Or in case of a communication channel, it may take too long to request a replay. And regardless of the application, it seems mathematically challenging to come up with a way to send a message such that not only is the error detected, it is also corrected by the receiver. This was the challenge that Richard Hamming took up an solved in an ingenious way. Here I would like to think through the same problem and come with a solution, which will then help us think through some of Hamming’s thoughts in the 1930s and 40s.  We will stick to the single bit error scenario. We can always reduce the block size sufficiently that only a single bit error can occur with any meaningful probability.

The first insight that led to the discovery of the Hamming code seems to be that some extra meta data (just like parity) needs to be transmitted in addition to the main data. In the presence of this extra data the receiver must be able to identify a bit flip in any bit in the transferred message (including a bit flip in the meta bits).  How do we uniquely identify a bit that flipped? We need to identify the position of the bit that flipped. In a message with d bits of data and p bits of meta data, the total number of transmitted bits is d+p. To identify a unique position where the flip may have occurred, we need log2(d+p) bits. The genius of Hamming was in recognizing that the bits being transmitted should be grouped into several groups such that each bit was a member of a unique set of groups. Further, the binary representation of the bit positions was the simplest way to identify the groups.

Say the message were 10 bits long. That is, d+p=10. The 10 positions can be represented as:

0001 – message bit 1
0010 – message bit 2
0011 – message bit 3
0100 – message bit 4
0101 – message bit 5
0110 – message bit 6
0111 – message bit 7
1000 – message bit 8
1001 – message bit 9
1010 – message bit 10

Notice that we start from 0001, and not 0000. This is because we need to make sure all numbers have 1 (or 0). This means to represent 8 positions (d+p=8), we actually need 4 bits. So the number of groups = log2(d+p+1). Further, notice that the binary representation of every message-bit-position has 1s in unique power-of-2 positions. This is obvious. After all, that is how we represent the positions uniquely. But the insight Hamming was able to draw was that each of the representations (left column above) with a 1 in a certain power-of-2 position could be clubbed together into a group. That is, in the example above, message bits with a 1 in the 2^0 position could be grouped into Club 0. Message bits with a 1 in the 2^1 position could be grouped into Club 1. Notice that some of the numbers which were members of Club 1 were also members Club 0 (for example message bit 3). Similarly, Club 2 and Club 3 could be formed. Further, exactly 4 clubs were needed to identify every bit (including the meta bits uniquely).

0 | 0 | 0 | 1 | – message bit 1  – Club 0
0 | 0 | 1 |0 | – message bit 2 – Club 1
0 | 0 | 1 | 1 | – message bit 3 – Club 0,1
0 | 1 | 0 | 0 | – message bit 4 – Club 2
0 | 1 | 0 | 1 | – message bit 5 – Club 0, 2
0 | 1 | 1 | 0 | – message bit 6 – Club 1, 2
0 | 1 | 1 | 1 | – message bit 7 – Club 0, 1, 2
1 | 0 | 0 | 0 | – message bit 8 – Club 3
1 | 0 | 0 | 1 | – message bit 9 – Club 0, 3
1 | 0 | 1 | 0 | – message bit 10 – Club 1, 3

Notice that no two bit-position-representations can belong to the same set of Clubs. After all, every bit position is represented uniquely in binary, and a 1 in the binary representation corresponds to a “key” to a certain club. No two binary representations are the same and so no two bit-positions can belong to the same club combination.

The groups/clubs thus formed are:

Club 0 – message bits 1, 3, 5, 7, 9
Club 1 – message bits 2, 3, 6, 7, 10
Club 2 – message bits 4, 5, 6, 7
Club 3 – message bits 8, 9, 10

Now, Hamming must have realized that he has created clubs with careful membership such that a unique member could be located if we knew which clubs he belonged to. So, a flipping bit can be located if that flip can identify all the clubs the bit belongs to. The clubs must start with some common property, and the flip must change that property for each club that it belongs to. This would help identify the clubs in which the flipped bit has membership, and by that unique combination, we can identify the flipped bit.

That unique property that Hamming gave each club was parity. Since no two clubs has the same membership, each club needed at least 1 unique parity bit. Since we want to keep the number of parity bits as small as possible to keep the message overhead as small as possible, the smallest solution would need as many parity bits as clubs. And we know that for transmitting d+n bits, we need log2(d+p+1) bits to represent the positions and, therefore, log2(d+p+1) clubs. This leads to the following observation:

The minimum overhead bits/parity bits = log2(d+p+1). But we assumed that the number of parity bits = p.

p >= log2(d+p+1)

Now we know how many groups, which bits make up the groups and the minimum number of parity bits. In the above example with d+p=10, p=4. Thus for 6 data bits we need to transmit 4 parity bits. But we must recognize that the parity bits cannot be simply lumped together into any casually selected positions. This is because without careful selection we may not have at least 1 parity bit per group. For example if the first 4 bit positions of a 10 bit message are used for parity, there would be no parity bits in Club 3. If we instead used the last 4 bits in a 10-bit message for parity, each club would have at least one parity bit, but all 3 of Club 3’s members would be parity bits. This creates a dependency on filling out the parity bits on the sender side. First parity bit 7 would have to be established based on Club 2 which has 3 data bits and only 1 parity bit. Then the remaining parity bits in Club 0 and Club 1 would have to be established, followed finally by the parity for Club 3. Hamming realized that it would be much easier to ensure that one parity bit belonged to 1 club, no more no less. All the members of a club are data bits except the one parity. This helps speed up setting the parity bit – there is no dependency and fixed order in which the parity bits need to be established. The bit positions in the message which belong t0 only 1 club are positions with a single 1 and all else 0s. These are also the power-of-two bit positions. Bit position 1, 2, 4 and 8. By making these bits parity bits, there is another advantage. At the receiving end, once parity is established for the various clubs, if it turns out that club 2 and club 1 are not maintaining parity, then to identify the bit position which flipped all that is needed is to calculate 2^2 + 2^1 = 6. Bit at position 6 is the only bit which is a member of Club 2 and 1 and no other club.

References:
I found this video of a class, held in University of New South Wales, Australia, and taught by Professor Richard Buckland, to be very useful in helping me understand how Hamming Codes work. The Hamming Code material starts at minute 8:00. http://www.youtube.com/watch?v=kE7V7UI4jpk

http://www.justiceharvard.org/

It contains some fascinating discussions on morals, philosophy, rights and justice. Professor Sandel has a very interesting teaching style, where he almost helps the students discover right vs. wrong, rather than just teaching it to them. Also, he has an extraordinary delivery style – careful and sincere. It is clear that he is actively participating in the discussion himself; he tailors the material such that it conveys all the crucial points while at the same time allowing the journey to these crucial points to be shaped by the students in the classroom. The class itself is comprised of over a thousand students, hanging on to every word from the teacher, and is a sight to behold.

A must watch. More accurately, a must think.

The mathematical constant shows up in strange places. Moreover, its significance is not as easy to grasp as that of the other famous constant, , because there is no easy physical object in whose context to imagine it. For example, is the ratio of the circumference to the diameter of a circle. Yes, it is irrational, but if you can get over that mystery (or ignore it for the time being), it is straightforward to imagine what is. Every circle does seem to have a certain circleness, which makes them all look the same. It is intuitively not hard to agree with the hunch that every circle has an unchanging ratio between the circumference and the diameter; and it makes sense to keep that ratio handy and give it a name.

The constant is considerably more elusive. It appears, at first, to be a number you would not go hunting after. You just happen to stumble upon in during one of your mathematical excursions; it seems interesting enough that you then pick it up and put in in your pocket for some potential use later. After stumbling upon the same thing along other mathematical excursions, in hindsight, it does seem to be something rather useful. Something you should have gone looking for.

Jacob Bernoulli, sometime in the late 1600s, stumbled upon this constant when he was trying to calculate the maximum compound interest that can be earned on an investment by compounding continuously.

If you invest 1 dollar in an account that earns 100% annual interest, then after 1 year the value of your investment will be 2 dollars assuming the compounding (interest calculation) happens once during the year. Assuming you do not touch the money and let it grow, by the end of 2 years, the money would have doubled again and you’ll have 4 dollars in all. Now, imagine the bank decided to compound every 6 months instead of once a year. The annual interest rate remains the same, 100%. 50% is applied at the end of 6 months and another 50% is applied at the end of the year. The 1 dollar grows to 1.50 after the first 6 months. For the second half of the year 1.50 dollars grow in value. At the end of the first year, the total value in your account is 1.50 + (1.50*50%), that is, 2.25 dollars. Not bad! Just by compounding every 6 months you made 2.25 instead of 2. Now you start thinking (as, perhaps, did Jacob Bernoulli). What if the bank compounded every 3 months? What if the bank compounded every month? Every week? Every day? Every second? Every picosecond? A similar question is what Jacob Bernoulli asked himself in the late 1600s. It turns out that this calculation results in an infinite series, which when added up closes in on 2.718281…This means if your bank was really generous and compounded continuously, your 1 dollar would become 2.718281 dollars at the end of the year. Quite a swing there – between 2 and 2.71, wouldn’t you say? And this given that the interest rate is unchanged between the two scenarios. (Aside: In general, pay some attention to how often your investments are being compounded, not just to the the annual rate of return.)

That number, 2.718281…, shows up in such seemingly disparate scenarios that it may be hard to see the connection between those scenarios. I recall that the way this constant was introduced to me in school was entirely different from the discussion above. I was taught that , where is a constant, is the only function whose derivative is equal to itself. That is, the slope of the curve is the value of the curve. Wow, I thought. That is quite a curve. I now feel that though that was correct, it may not have been the best way to introduce ; it does not say why is the only function that satisfies this property. Further, it is not explained why a function that does satisfy this property should even be of the basic form , where a is a constant.

I decided therefore to think through two things. First, I wanted to understand why the only function whose slope at a given is the same as its value. I wanted to approach this problem by using the knowledge of this special property, and then trying to come up with the function which would fit this requirement. Second, I wanted to tie together the two seemingly different arenas where e shows up – continuous compounding and as the base of the curve whose slope is equal to its value. There are other areas where shows up also. For example, if the probability of winning a wager is 1 in , and a man places bets, then the probability that he wins at least one bet is . This question is related closely to the hat check problem (also known as Derangements). also shows up in Euler’s Identity. I will need to do more digging before I attempt to understand, intuitively, why shows up in those scenarios so I may not tackle these in much depth in this article.

Why is the only function whose slope at a given is the same as its value?

Before I looked at this particular issue and attempted to uncover the function which gave me this special property, I decided to start small. I decided to look at some other canonical functions that might have been interesting. For example, “What is a function whose value, , is equal to ?”, “What is a function whose value is equal to ?”, “What is a function whose value is equal to ?”. The first question is easy. It defines . The curve for this is a straight line at a angle through the origin. The second one is easy as well. It defines . A parabola passing through the origin, (1,1) and (-1,1). The third question is the easiest of all, but the one that gets us closest to the question we are going after. It defines ! That is silly. Any function will fit. Notice, however, that in the third question the value of the function is dependent on the value of the function and not directly on . This starts to highlight the significance of the question we are trying to answer. We want to discover a function, . There is something quirky going on here. The right hand side is the derivative of the left hand side. So which comes first? It seems like you cannot really know the slope at , until you define . But then, you need the slope of to define because its value is the slope of . It is a bit confusing. The trick is to realize that this function is continuous. So the value of the slope of will be pretty close to the value of the slope of , the slope of the curve just prior to x. (This recursion also vaguely points to an upcoming infinite series.)

Now, let us try to find the function which satisfies the interesting property, . In words, this means the rate of growth of this function at a given point is equal to the value of the function at that point. More on that in a bit. Let us use the fact that the function is continuous and therefore, , the slope of the function at is given by the following.

Since we know that the function we are trying to uncover has the special property that , we can substitute instead of in the previous equation, to get the following.

This can be rewritten as

which in turn can be rewritten as

I will substitute with (this is unfortunately one of those things which you do after you have worked out the steps and realize that doing something like this may make things a little cleaner to understand).

So we get,

Swapping the right and left sides, we get the following.

The above equation represents in terms of . Using steps similar to the ones shown above if we express in terms of we get the following.

Extending this process times we get the following.

If is large enough that then the above equation becomes very interesting.

Notice that is still not a power to any base. So is still not an exponential curve, at least it does not look so. It does seem like it may expand into a sum of polynomials with the largest polynomial being of degree . A polynomial, however large in degree, grows slower than an exponential curve (as long as the degree is finite). However, note that is not a finite integer. It is an infinitely large number because is infinitesimally small. So there is still hope that this sum of polynomials may catch up to an exponential curve. Now, we apply a trick which changes the nature of this curve. A sum of polynomials becomes an exponential curve, and the goes up to a “position of power”. We simply reapply the equation to the and the term in the above equation to get the following.

Now, say we represent by a new variable, . Then the above equation can be written as follows.

which in turn can be written as

And now we start seeing the exponential form emerge. The is the power to which must be raised to evaluate .

This term, , when expanded out (remember that is infinitely large) hones in on a constant, 2.718281. And that is why, a special name is given to this term (). The right hand side thus becomes .

This takes the general form , where , the value of the function at . In other words we have proved that a function that satisfies the property that the curve’s value at any point is equal to the slope of the curve at that point follows the general form of an exponential curve.

How is continuous compounding related to the base of the exponential curve whose slope is equal to its value?

Now that we understand the general structure of the exponential curve and its special property, slope equals value, let us revisit the issue of continuous compounding. How is that related to this curve? It is intuitive once we realize that the rate at which money grows at any point in time is directly proportional to the amount of money in the account at that point in time. This is the same as saying the slope of the curve at a given point is directly proportional to the value of the function at that point. The interest rate plays the role of tempering the proportionality. For example, if the interest rate is 100%, the value of the continuously compounding account after units of time ( years) is . Here is the amount in the account at the beginning (value of the function at ). If the interest rate is 8%, then the value at time years is . Once we see that the two phrases, “slope of a curve is proportional to its value” and “rate of growth of something is proportional to the amount of that something”, are saying the same thing, it is clearer that these two ways of defining or discovering are equivalent.

Other places shows up

Another place shows up is called Derangements, as in the opposite of arrangement. A popular example used to explain this issue is the hat check problem. Say there are people who check in at a party, and the butler takes their hats as they enter the party. There are boxes to hold the hats, one per guest. The boxes have names on them to identify which guest’s hat should go into it. The butler, however, does not know the names of the guests. He puts the hats into those boxes randomly. When the number of hats (and boxes) is large, the probability that no hat is in the right box is . I have been able to work this out mathematically, but I still have not been able to understand this intuitively. Maybe the reason shows up here has nothing to do with the properties of . Maybe it just so happens that the same summation, which adds up to e, , shows up here.

Solution to Hat Check Problem: An Outline

The probability that all hat are in wrong boxes = 1 – probability that at least one hat is in the right box

= 1 – (prob. that the hat is in the right box OR prob that the hat is the right box OR … OR probability that the hat is in the right box)

Note that for the hat to be in the right box the first i-1 hats have to avoid that box. Therefore,

Probability that the hat is in the right box = (probability that the hat has not taken up the right box of the hat).(probability that the hat has not taken up the right box of the hat)…(probability that the hat has not taken up the right box of the hat).(the hat does take up the right box)

Using this, the probability that all hat are in wrong boxes

where,

Putting this back in the equation for the probability we are looking for, we get,

now, replacing by ,

The second term goes to as increases. So we are left with,

I do not know why shows up here. But, well, it does. Yet another place where shows up with unabated vigor is when representing complex numbers, and, of course, in that famous Euler’s equation that brings together 5 of the most interesting quantities in mathematics, . I will go into exploring that another time.

References and Acknowledgments

My discussions with my friend Mani, and my investigations on the internet led to this write up. I found the Wikipedia site very useful to tie together the seemingly unconnected ways in which the constant shows up in real life. The BetterExplained.com website has a very good illustrated tutorial for those who want to understand the concept of in simple terms.

Wikipedia.com – http://en.wikipedia.org/wiki/E_(mathematical_constant)

BetterExplained.com – http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/

Reference for the population figure: http://www.google.com/publicdata?ds=wb-wdi&met=sp_pop_totl&tdim=true&dl=en&hl=en&q=world+population

Figure1: Basic plan showing the material required

We have two 4′x8′ (4 feet by 8 feet) raised beds, which we use for vegetable plants. Kavita has been asking me to either buy or build a trellis for her climbing plants (cucumbers, tomatoes and eventually some types of squash and gourds). I read several websites online and decided to build a simple trellis using PVC piping. It took one trip to the local Home Depot, and then about 2 hours of work. The cost for the material was under $10 (I already had all the tools needed).

We decided to build one and test it out before getting carried away and building more. We decided that we would roughly want the trellis to be 4 feet wide by 5 feet high. At the Home Depot we did some quick calculations based on the basic design we had in mind and came up with a total of about 29 feet of PVC tube. The calculation is shown in Figure 1.

Figure 2: Materials and Tools for the project (4-way, + shaped, PVC connector missing)

The PVC pipes are sold in 10′ pieces. We got 3 pieces. We also got some string (I tried polypropylene string since I did not know any better, we’ll see how that works out).

Figure 2 shows most of the material and tools. The one caveat is, since I took this picture after completing the project, the one 4-way 1/2″ PVC pipe connector used at the center of the frame is missing from the picture. I had extra connectors of the other type, so I could use them for the picture. Also, one other thing that is missing from the picture is a power drill and drill bits. I used a 3/16″ drill bit to drill evenly spaced holes in the pipes to draw the string through, to create a framework.

Figure 3: Trellis plan with the stringing shown

Figure 4: Framework ready, stringing is yet to be completed

The spacing between the holes and how the trellis is supposed to look eventually is shown in Figure 3.

The thing that took the most time was measuring and marking the PVC pipes, cutting them to the right size with the saw, then measuring and marking the locations for the holes for the string to go through and then drilling the holes with the power drill. Since the PVC pipe keeps rolling about, making it stable before drilling is important. I just used an old rag to wrap around the pipe in order to hold it somewhat still.

Once the pieces were all ready, putting the trellis together took less than 10 minutes. There was no need for glue, since the connectors fit quite snugly. Figure 4 shows the trellis laid out on the lawn (with only one piece of string drawn through, Kavita will work on getting the rest of the mesh this evening).

We are not sure how well this will hold up, how long it will last etc. I will update the post with some pictures on the trellis in action, later.

To sign up I had to fill a form at the National Marrow Donor Program website. Then, about 4 days later I received a testing kit in an envelope by regular mail. The testing kit basically contained 4 swabs to collect cheek cells with. I did that. It took about 2 minutes. (Aside: All that “collecting cheek cells” involves is opening your mouth and rubbing the cotton swabs against the inner cheek for 10 seconds. This collects some of the cheek cells on the swab. it is not visible to the eye, but apparently that is all they need. ) Then I put the swabs back in the return envelope they send, and posted it. That is it. The entire process was free and streamlined.

When I try to donate blood with Red Cross I typically need to wait 1 year after a trip to India. This rule often seriously curtails my ability to donate blood. But then, maybe blood donation is more prevalent and since there are only a handful of types it is easier to find a match quickly on demand. It is probably more expensive to collect blood and then test each unit of blood separately for potential risk factors based on where the donor has been in the last 1 year. With marrow, the matches are so rare that the first step is to keep a large database of potential donors, and upon a match the appropriate tests can be conducted to rule out any potential problems with the donor. Typically only 1% of the people on the potential-donor database are actually ever contacted for a donation. And the donation itself is a pretty safe and quick process. It may require you to take a day off from work, and be under local anesthesia when the marrow is being drawn. You need to be between 18 and 60 years old. The donated marrow regenerates in your body in a few weeks.

I encourage you to register, a small but important, painless and free first step. And if you are the 1% who eventually (years down the line) gets a call asking for a donation, I am pretty sure you will feel good about it.

Taking an Interest in Loan Mathematics

Let us first try to understand, in simple terms, the philosophy of any loan process. In particular, I’ll focus on the home loan process. You intend to purchase a house. Why purchase, and not continue to rent? Well, that is an interesting question in its own right. But, to not get distracted, let us say you got tired of the rent going up each year, or moving every few years, or actually figured that it was economically better to buy rather than rent. So, you decide to buy a house. You need money. You go to the bank (a mortgage lender). Say you want $200K ($200 thousand). The bank gives you the money at a certain interest rate, say, 5%. What does that really mean? Here in the US, the norm is to calculate the remaining balance every month. The 5% is actually 12*0.4166%, where 0.4166%, or 0.004166, is the monthly interest rate. That is, after the first month, the outstanding balance is $200K + $200K*0.004166. In other words, because the bank did you a favor by giving you $200K which you did not have, it wants $200K*0.004166, which is $833.33. As a quick aside, notice that because the interest is calculated monthly, the annual interest rate is, in reality, greater than the 5% we started with. The real annual interest rate would be 1.004166^12=0.0511, or in other words 5.11%. Nevertheless, the common practice is to quote this as 5% base interest rate, and that is fine, as long as we know what it means.

Now, continuing with our example, in the first month, the bank wants you to pay $833.333 as interest accrued over that month. Say you paid exactly $833.33. The outstanding balance at the beginning of the second month would then be exactly 200K again. And at the end of that second month, the interest would be $833.33 again. Say you pay the bank $833.33 again. The outstanding balance at the beginning of the third month will again be $200K. This pattern could go on endlessly. You may argue that this looks like renting. Every month you pay the rent. You don’t see any of that money. But with buying there is a fundamental difference. After 10 years of doing the above, that is, paying $833.33 each month, you decide to sell the house. The house itself, typically, appreciates in value. Say the value of the house is now $300K. You sell and make a $100K profit. You paid 10*12*$833.33 over the 10 years, which, coincidentally, comes out to exactly 100K. What that means is you basically lived in a house for 10 years for free (of course you did pay property taxes, painted the house a couple of times, bought a lawn mower, replaced light bulbs, and took care of the house in general). But overall, it sounds like a pretty sweet deal.

One word we all glossed over in this discussion is “typically”. Home prices “typically” appreciate. The bank does not gloss over that word. If the value of the house drops, say 10 years later, the value of the house drops to 150K. You have paid your “rent” for 10 years, and are ready to sell. The bank wants its 200K back, since you never paid any principal all these years. The sale, however, would only fetch you 150K. The bank has the title (ownership document) to the house. It will not let you sell. It says give us 50K first, then sell for 150K, and gives us that 150K as well. You do not have 50K to give to the bank. The loan is foreclosed – the bank keeps the title to the house, but the bank does not like this situation. The bank now owns the house. But the house is worth only 150K. The bank does not want to be in the business of selling a house, especially one that won’t bring them their original 200K back.

To prevent this scenario, the bank employs two interesting tactics. Firstly, it does not let you pay only the interest of $833.33 each month. It requires you to pay off some of that principal on top of the interest. Secondly, the amount of principal you pay atop the interest is calculated such that the loan is guaranteed to be paid off in a certain “term”. Further, to keep the payment terms simple for the customer, the total payment each month remains unchanged. It is important to recognize that calculation of interest depends only on the interest rate. The calculation of the monthly payment which includes both the interest and some piece of the principal requires the notion of the “term”. The payment has to at least be the interest due that month. It is a bit more than that each month because of the principal paid. (In reality it ends up being even more because you pay a part of the annual property taxes, hazard insurance etc. each month � but we can ignore that for this discussion). Paying a bit of principal each month causes the outstanding balance reduce each month; this causes the interest payment to reduce a bit each month. This allows you to pay off even more principal each month, and that cascading effect finally ends exactly when the term runs out. Throughout this period, as mentioned earlier, the actual monthly payment does not change. The reduction in interest is compensated for by the increase in principal payment, which in turn reduces the outstanding balance and causes the next month’s interest payment to reduce even further. This constant monthly payment (interest + principal) is carefully calculated to achieve this effect. “Term” is the number of months the loan is supposed to be fully paid off by. The shorter the term, the better the rates, in general, because to pay off a loan faster (shorter term), you have to pay greater amounts each month. So there needs to be an incentive for you to pay more money each month to the bank. And that incentive is the lower rate. Otherwise, wouldn’t you rather go with the longer term, pay less to the bank each month and invest that leftover in the stock market?

By forcing you to pay a bit of principal each month, the bank is earning less interest each month. But the good news is that, if after 10 years, you decide to sell the house and the value of the house is only 150K instead of the 200K you bought it for, the bank risks less. You have already paid off about 40K. So the loss for the bank is only 10K instead of 50K, if it had allowed you to only pay the interest each month. So in other words, the bank wants you to pay principal each month not to help you reduce your interest payments, but rather to help it stave off any chance of losing money on the house if prices fall.

Black Magic – Calculating the Monthly Payment

When you talk to a mortgage banker on the phone, you will notice that they like to quickly tell you that your monthly payment would be some x amount. They use the phrase, “run the numbers”, with some pride. It is interesting and empowering to understand how the monthly payment is actually calculated.

We have all the information we need. At the beginning, we have a 200K loan. Let us use L to indicate this “Loan Amount”. Say, the monthly rate, which is 0.004166 in our example, is represented by c. Say, n represents the term, n months. Say, P represents the monthly payment. We want to determine P ourselves, instead of depending on our loan officer to tell us that information.

After the first month, the outstanding balance is:
L + L*c – P
= L(1+c) – P
This is because the loan amount L increases by the monthly interest amount, L*c, but then we make the payment of P. This is the outstanding balance for the second month.
At the end of the second month, the outstanding balance is:
{L(1+c) – P}(1+c) – P
= L(1+c)^2 – P(1+c) – P
= L(1+c)^2 – P{(1+c)+1}
At the end of the third month, the outstanding balance is
= {L(1+c)^2 – {P(1+c)+P}} (1+c) – P
= L(1+c)^3 – P {(1+c)^2 + 1+c) + 1}
If you are still with me, you may start seeing a pattern emerge. After n months, the outstanding balance will be:
= L(1+c)^n – P {(1+c)^(n-1) + (1+c)^(n-2) + … + (1-c)^2 + (1+c) + 1}
Which, can be rewritten as
= L(1+c)^n – P {1 + (1+c)^2 + (1+c)^3 + … (1+c)^(n-1)}
Now. The punch line. After n months, we *know* that the outstanding balance should be 0. So
0 = L(1+c)^n – P {1 + (1+c)^2 + (1+c)^3 + … (1+c)^(n-1)}
P {1 + (1+c)^2 + (1+c)^3 + … (1+c)^(n-1)} = L(1+c)^n
P = L(1+c)^n/{1 + (1+c)^2 + (1+c)^3 + … (1+c)^(n-1)}
There you go. That is P, your payment each month. Phew! Done? Well, almost. The denominator in the above calculation is not Excel-friendly. Remember, you want this to go into a spreadsheet that can help you with decision making. The number of terms depends on n. Not good. Let’s try to find a closed form solution for the denominator. Thankfully, it is not hard. Notice that the denominator is of the form:
1 + a + a^2 + a^3 + … + a^(n-1)
where I replaced 1+c with a. Let us call the above sum X.
X = 1 + a + a^2 + a^3 + … + a^(n-1)
Adding a^n to both sides (as an aside, this kind of intuition is the reason Kavita hates math)
X + a^n = 1 + a + a^2 + a^3 + … + a^(n-1) + a^n
Shamelessly using some more of that darned intuition, we extract out a common factor, a, from the last n terms to get to:
X + a^n = 1 + a {1 + a + a^2 + … + a^(n-1)}
But notice that the stuff inside the {} is precisely what we defined X to be. So:
X + a^n = 1 + a*X
X + a*X = 1 + a^n
X * (1 + a) = 1 + a^n
Therefore,
X = (1-a^n)/(1-a)
Replacing a with (1+c),
X = (1-(1+c)^n)/(1-(1+c))
X = (1-(1+c)^n)/(-c)
X = ((1+c)^n – 1)/c
Finally, substituting this into the equation for P:
P = L.c.(1+c)^n/{(1+c)^n – 1}
Now we are seriously done with this calculation.

Let us try to use L=200K, c=0.004166 and n=360 (a 30-year term, which is quite common in the US), and calculate P, your monthly payment. P comes out to $1073.64. The interest component is $833.33, and the principal is $1073.64 – $833.33 = $240.31. Because you pay off a tiny bit of the 200K principal, the outstanding balance at the beginning of the second month is $200000 – $240.31 = $199759.69. The interest for the second month is therefore going to be lesser than $833.33. In fact, it is $832.33. This $1 we pay less in interest goes towards the principal, which increases from $240.31 in the first month to $241.31 in the second. Looking a few months into this process the interest payments are $833.33, $832.33, $831.33, $830.32, $829.30, etc., and principal payments are $240.31, $241.31, $242.32 etc.

Click on the thumbnails above to see the monthly and cumulative payment schedules. The first figure shows how much interest, principal and total payment needs to be made each month. The second figure translates that to a cumulative amount, that is, at any given point in time it tells us how much interest, principal and total payment you would have made. It is interesting to note from the first figure that in the first few years the bank makes most of the money it expects to make on the house (the interest tails off during the later years). The second figure shows that by the end of the loan term, you’d pay about 200K in interest!

Does Overpayment Make Sense?

At this point it is important to understand that by paying off the $240.31, $241.31, $242.32 etc. principal each month the benefit you are getting is in terms of reducing the interest you pay each month. By actually being vested in the house, that is, by owning that piece of the house, you do not get any direct benefit; when the house sells, its value will not depend on how much of the house you actually own. Think of it like this – the principal payments you make are investments where the rate of return is determined by the reduction in the interest payments.� Let us take an example. Say, somehow, you convince the bank to allow you to pay only the interest, $833.33, each month. You take the difference between your bank-determined payment of $1073.64 and your negotiated payment of $833.33 and invest it ($1073.64 – $833.33 = 240.31) in the stock market at 10% annual rate of return. Either way, after 10 years, we’d have invested $240.31*12*10 = $28,837.2. Since the investment is accruing a rate of return each month, we need to carefully calculate how much profit we make (I use an Excel spreadsheet to do this, however, we could use a closed form expression similar to the one we developed above). At a 10% rate of return, we make about $21,000. If we put this same $241.31 into the principal payment each month, after 10 years, our profit (the interest savings compared to the case where we do not pay any principal payment each month) is $8,478. Of course, the actual profit by investing is reduced by the tax you need to pay on that profit. Regardless, it is still a sweet deal to invest the money in the stock market, provided you can guarantee the 10% return on investment. Even if we assume a safe 6% rate of return (after taxes and everything), we stand to make $10,900 in the stock market vs. the $8,478 we “make” by putting it into the house. In any case, this is a moot point, since the bank will not allow you to make interest payments only. What this discussion is intended to drive home is that it may not make sense to overpay above the monthly payment of $1073.64, unless you intend to stay at the house for a shorter term. If you stay for a shorter term in the house, then the stock market rate of return may be too risky, whereas the paying into the house guarantees a certain rate of return.

The figures above show monthly and cumulative payments for a 15 year loan term – that is a loan for which the monthly payment has been calculated such that is supposed to be paid off in full in 15 years. Typically, 15 year loans have a slightly better rate than a 30 year loan, to give you the incentive to give up more of your cash each month in payment. However, since I am continuing to use a 5% interest rate to plot these curves, these really indicate how your overall payment time line changes if you overpay each month. The overpayment amount is basically the difference between the monthly payment shown in this figure and the minimum monthly payment shown in the previous section. As you can see here, even in the first month you pay as much towards principal as interest, and secondly, by the end of the loan term, you pay only about $80K in interest. The advantage of this scheme is that you are required to only pay in accordance with the 30-year term, but you may choose to overpay if you wish to reduce your interest payments. That way, if you occasionally miss your overpayment target, that is fine as long as you pay the minimum payment for that month. That said, like we discussed above, it may still make sense to not overpay if you can invest that money instead.

Does Refinancing Make Sense?

Now that we have understood some of the nuances of the loan process, let us consider how to make a refinancing decision. Refinancing is the process of getting a new loan in order to pay off an existing loan. If this were a free process, that is, there were no cost of refinancing, the decision would have been very simple. If the new interest rate is better than the old interest rate refinancing would make sense. However, there is, typically, a cost involved. The question then changes to how long do you need to stay in the same house after refinancing to recoup the cost of refinancing. Let us take an example. Say you currently have a 5% loan with $200K outstanding, and a different lender offers a 4% loan, with a $2000 closing cost. The current monthly payment is $1073.64. The new monthly payment is $954.83. Since the interest rate is lower, you�ll likely be paying less interest each month with the new loan. So over time the cumulative interest you pay the bank may be lesser with the new loan. For this example, after 1 year the total interest paid with the current loan is about $9900, whereas with the new loan it is $7900. This is about $2000 in savings in 1 year just from the interest rate reduction. Since the interest you pay is like the fees the bank charges for its services, you have found a low-fee option. So in 1 year you have overcome the $2000 cost of refinancing. From year 2 onward you stand to gain by doing this refinance. (Note: I am ignoring the fact that interest is tax-free money, that is, you get the taxes you paid on the interest in your next year’s tax returns. The absolute savings from interest reduction are therefore about 20 to 25% lesser than the savings I am quoting here and in the spreadsheet. It is easy to fix that though, if you choose to. Instead of saving $2000, you’d have actually only saved $1500 if you fall in the 25% tax rate bracket.)

Now, let us see what happens to the rest of the money you are paying each month, the principal. With the current loan the cumulative principal payment after 1 year is
$2950. With the new loan the cumulative principal paid in 1 year is $3522. That is, you own more of the house. But this is not important by itself. Yes, you own more of the house, but the net is you converted some cash into a bit of house. If you had not owned any of the house you’d have been left with cash which you could have invested and actually grown it. The house grows or falls equally in value regardless of whether you are invested in it or not.

But there is one more component to this equation other than the interest and principal. The overall monthly payment has reduced from $1073.64 to $954.83. That is a freeing up of $118.81 each month to be invested as you choose. Even if this was invested conservatively in a 6% rate of return investment, you end up with $1472 at the end of the first year. This is money that would not have been available at all with the current loan. So in fact, at the end of year 1, you have save $2000 + $1472, the former coming from the interest savings and the latter from investing the cash freed up. This means, the $2000 cost of refinancing will actually be made up even sooner than 1 year. Given the above 6% assumption it is more like 7 months. If you plan to live in this house for 7 months or more, go for the refinancing

The figure above shows the time to recoup the cost of refinancing, considering only the interest savings and also considering the case where the overall reduction in monthly payment can be invested at 6%.

Acknowledgements

My understanding of the issues involved in refinancing, in particular, and mortgages, in general, is based upon my going through this decision-making process recently. Much of this understanding was developed during discussions with my friends Gordie and Srini. If you find flaws in my understanding please let me know. Some online resources that helped me were http://www.mtgprofessor.com/formulas.htm , http://en.wikipedia.org/wiki/Refinancing and http://en.wikipedia.org/wiki/Mortgage.

Here is what I learnt about the various types of plastics. The following list is from Wikipedia’s article on Plastics.

1. PET (PETE), polyethylene terephthalate: Commonly found on 2-liter soft drink bottles, water bottles, cooking oil bottles, peanut butter jars.
2. HDPE, high-density polyethylene: Commonly found on detergent bottles, milk jugs.
3. PVC, polyvinyl chloride: Commonly found on plastic pipes, outdoor furniture, siding, floor tiles, shower curtains, clamshell packaging.
4. LDPE, low-density polyethylene: Commonly found on dry-cleaning bags, produce bags, trash can liners, and food storage containers.
5. PP, polypropylene: Commonly found on bottle caps, drinking straws, yogurt containers, Lego building blocks.
6. PS, polystyrene: Commonly found on “packing peanuts”, cups, plastic tableware, meat trays, take-away food clamshell containers
7. OTHER, other: This plastic category, as its name of “other” implies, is any plastic other than the named #1–#6, Commonly found on certain kinds of food containers, Tupperware, and Nalgene bottles.

Three main things I learnt are:

1. The biggest difficulty with recycling plastics is that these types tend to get mixed up when people try to keep the plastics aside for recycling, and separating them is a labor intensive problem – often not worth the trouble – and thus, unfortunately, sending all this plastic, irrespective of the good intentions of recycling them, into landfills.

2. PET (Coke and Pepsi bottles) is very different from PVC (Gatorade bottles and my problem at hand, shower curtain liners), both of which are different form HDPE (milk jugs and detergent bottles)! PVC if mixed with PET, can ruin not only the batch of PET that is being recycled, it can even hurt the recycling machine! So be careful with recycling PVC.

3. Polystyrene (PS), which is used in “styrofoam” to-go food takeout boxes, is typically not recycled because it is not cost-effective to do so (maybe because of how cheap it is and also because of the food contamination that goes with the territory). It almost definitely ends up in landfills, and only centuries of wait may see its end. Along the way, during those centuries, who know how much damage it causes to the living things that inadvertently ingest it.

So, all in all, recycling is not as easy as dumping anything plastic into the recycling bin; it is important to be aware of the differences between plastics. It is important also to educate ourselves about how to ensure that what we think we are recycling actually gets recycled. Styrofoam is best avoided. Maybe we need to leave repeated comments on restaurant websites that they should avoid using styrofoam-based to-go boxes, and instead switch to paper (like Chinese-takeout buckets).

I am not sure how well I will be able to follow all of my own suggestions. The intention here is to start thinking about these issues.

A couple of other interesting websites I found:

1.  http://www.townofcary.org/depts/pwdept/recycling/trivia.htm

2.  http://www.greensangha.org/PVCaction.PDF

Now, the question still remains. What do I do with the used PVC shower curtain liner? I don’t know if Cary recycles PVC. I need to find out and act accordingly. And maybe it is best to stop buying these liners, and instead switch to a cloth-only liner or some bio-degradable variety.

The spark of a creator’s idea, must be weighed and studied for the sustainability and fesibility of that which the idea generates. Once the creative force is assured of the sustainability and usefulness of its creation, it must also analyze the destructibility of the creation. Only when the idea passes both these tests – useful when in existence and destroyable when not useful – is the creation sustainable. To be destroyed does not mean to make it go away or vanish. Being destroyed here means to change form. The death of one is the birth of another. The death of a wine glass when it slips from your hand and shatters is the birth of a hundred pieces of glass. The death of those hundred pieces in a kiln if the birth of liquid silica, which dies to takes up another form when shaped into a glass window.

Sunstainable creation is dependent on reliable destruction, which in turn is dependent on future creation. In our lives nowadays, I wonder if the creator’s dependence on the destroyer is being slowly forgotten. Things are getting created with no concern for its destructibility (and often with no concern even for sustainability). Creation is driven by sustainability and usefulness, which is fine. However, the second part of the pre-creation analysis, destruction, is becoming only a secondary concern.

A case in point is plastics. Plastics are almost irreplaceable in certain situations. However, its usage cycle has overflowed its equilibrium bounds. The ease of creating plastics and the convenience of sustaining plastics have together overpowered the responsibility of destroying them. The durability of plastics, which is often a big positive, makes it equally hard to destroy. And when used in scenarios where such durability becomes a liability, the benefits of plastics are questionable. Wikipedia’s article on Plastics has this, somewhat scary, line. “Due to their relatively low cost, ease of manufacture, versatility, and imperviousness to water, plastics are used in an enormous and expanding range of products, from paper clips to spaceships. They have already displaced many traditional materials, such as wood; stone; horn and bone; leather; paper; metal; glass; and ceramic, in most of their former uses.” Notice that all the things that plastics have replaced are either natural or biodegradable, or both. I agree that there are some organic palstics in nature, and there are some man-made biodegradable plastics; however, the point I am trying to make is not hijacked by either of these. From wall-to-wall carpets to the teacups used by chai-wallahs in roadside dhabas, from ziploc bags to microwaveable idli-plates, plastics have slowly but surely taken over our lives. In this takeover, not only has the senseless overuse of plastics created a dangerous imbalance in the natural world, it has paralyzed us into a state of helplessness compliance. Plastics have destroyed the destroyer.

In many uses of plastics, they are certainly replaceable by other, more responsibly created, products. We, the users and sustainers of plastics, should vote down the creators’ decision to create them by reducing the use of plastics where possible (take your cloth grocery bags with you when you go shopping, use glass or steel dinner ware at home and paper or corn-based plates at picnics). When usage is not avoidable, we can restore the balance somewhat by paying due homage to the destroyer (use plastic that is recyclable and recycle the plastic that you use). A moments thought before consumption can not only help restore some balance in the cycle of creations and destruction, it can also help restore a sense of control over our destiny.

On my part, I make it a point to visit the Shiva temple of Cary once a month. It is a large, airy temple, with the added convenience of a drive-through pradakshina (the act of revenential, clockwise, walking around a Hindu temple’s central structure). Each time I go, the priest walks up to my car, greets me, and asks, “What do you have?”. Upon telling him about my problems, he points me to the correct deity to go pay my respects to. His utterance may seem strange for temple-talk, “Go to number 4″, but what he really means is, “Deity no. 4 will rid you of all your troubles and send you home free and uplifted”. The temple, for some strange secular reasons, likes to identify itself with a small, unadorned, non-ostentatious, green sign with white lettering that reads, “Cary Recycling Center”.

I agree with most of Dr. Raghunathan’s observations though I find that some of the arguments could be further strengthened by factual details. I also felt that the article does not point out one factor which can elevate the status of sports in India – media. I see many children and young people nowadays finding no outlet for their natural instincts to exercise their bodies and minds via sports. Instead much of their energies are being channeled towards unimaginative, creativity-sapping shows on the TV and the inescapable din created by the industry that religion is becoming. Lot of it is because in population heavy urban areas, there are not enough facilities and open spaces for the young to play. In rural areas where there are open areas and, arguably, time at hand for kids to explore sport, facilities and, more importantly, awareness are severely lacking. There is no incentive to try to be great at a sport.

The young minds of the country, with no reason or facilities to go out an play, are increasingly being moulded by what the see on the TV. Some get influenced by the western media and the western culture they see on the TV and blindly jump on to that bandwagon. The remaining shun the western influence so much that they lean to the other extreme, and get swept by fundamentalist religious rhetoric. In either case, the TV influences their ability to think for themselves. Instead of making the youth broad-minded in their approach to cultures, science and entertainment, it makes them confused, at first, and, dogmatic and narrow-minded, eventually.  Religion in India is losing its real meaning; it has become a service industry. It is encroaching the airwaves via loud speakers and TV and radio shows. It is encroaching every free piece of land, which should have been left as play areas for children, by temple construction projects. It is sucking up every rupee anyone can and cannot spare to feed its furnace of the self-fulfilling prophecies. It is taking over people’s ability to think. Its misinterpretation and misrepresentation over the years is reaching a point where sense and rationality no longer prevails, and democracy itself, that fairest social system, might no longer be able to see right from wrong. The opium of the masses, as Karl Marx called it, is finally taking over the sanity of a democracy. If a majority in a democracy are disillusioned, I wonder what keeps a democracy from self-destruction? The economic prosperity that India now enjoys is a great opportunity to keep religious and other extreme influences in check by clearly communicating to the masses the true reasons for this economic upturn; this upturn is in spite of the religious fervor gripping the country, not because of it.

Coming back to the topic on hand, I think the media is best equipped to extricate the population from such extremes and let some sense prevail. Sports is a great leveler and is one of the best ways to bring a nation together. The media has a huge role it can play to help kids play in their free time, rather than watch soap operas and immerse themselves in religious discourses. Here is a snippet from an email I wrote on this topic to some of my family members.

“Media, especially television, has a tremendous control over the nation’s psyche. Media can make or break national opinions. The current fascination with game shows, song-and-dance competitions or soap operas is by and large a media created state of mass-hypnosis. The same power of the media can get people to start appreciating the importance of physical exercise and sport. More importantly, it can be an enabler for pulling in interest and, therefore, money into regional and national sporting events. An example is the recent city-based cricket leagues. An entire industry, a multi-billion dollar enterprise, can be built around sports in India. It is an largely untapped market. Doordarshan’s depressing, half-hearted, monopolistic attempts at covering sporting events in India makes people even more reluctant to try sports professionally. Sports is real-life drama and entertainment. It needs professionally trained media-men to bring out that drama on screen. Once that excitement, that tension, and the drama can be conveyed, the audience will pay attention. This might require bringing out the personal backgrounds of the players, their histories, their stories of hardships and determination to the people. Once people are latched, competition increases both in the sport and the coverage of the sport. The advertising revenue starts to flow in. With money flowing in, there is a feed back effect. More people want to take part in sports, more people want to watch sports, more people want to cover sports and make money. For example, why do we prefer Harsha Bhogle to Sanjay Manjrekar in the cricket commentator’s box, and why do we prefer Star Sports to Doordarshan for sports coverage? Quality. People can perceive quality differences. Similarly, why is Praveen Kumar, the cricketer, evoke national interest? Because of his background as a wrestler from a small town. These small-town heroes are the media pets. They help catch the public’s fancy. “If it can be him…it can be me!”, they aspire.

In India, shooting, archery, wrestling, boxing are relatively easy to make popular. The sports persons have to be cast into media demi-gods… only then will people know their names and their existence, let alone pay much attention. When the drama of sports is discovered and conveyed by the media to the audiences, even shooting can be made into a heart-racing, edge-of-the-seat thriller.”

One thing that I should clarify is “Why should children play sports – a rather mundane, pointless exercise?”, “Why is winning medals at the Olympics that important?”, “Why not let the entire nation, instead, spend all their energy and time learning their place in God’s scheme of things?”. The answer, I think, is because they will discover their God, their place in the scheme of things, their goals, better, by participating in sports and playing their heart out, than by cracking coconuts, lighting agarbattis, exchanging bananas or watching TV.