math

I was reading David Bach’s book “The Automatic Millionaire” on the flight back home from our Philly/NY trip - except for some little excitement (for the wrong reasons) while reading his views on real estate, the book almost put me to sleep (sorry Bach fans). May be it’s not Bach’s fault here - there is a lot of stuff on personal finance (by various authors) that make me yawn with increasing frequency after a dozen pages or so.

Anyways, back to the point of interest. In one particular section in the book (titled “Six Reasons Why Homes Make Great Investments“), Bach starts painting a very rosy picture of investing in real estate. While some of the reasons are OK, I was not comfortable with the manner in which a couple of them were explained. For example, in reason #2, he starts with explaining the concept of leverage and then jumps to calculating returns on investment. Here, he tries to show how real estate investment can yield handsome returns by using the power of leverage. This is what comes next:

Let’s say you are buying a $250,000 home with a down payment of 20 percent. What this means is that you’re putting in $50,000 of your own money and borrowing the remaining $200,000 from a bank. Since you’ve put in only one fifth of the purchase price, you’ve got five to one leverage. Now let’s say the value of the house increases over the next five years to $300,000. Given that you’ve put in only $50,000, the $50,000 increase in value means you’ve effectively doubled your money. This is the power of leverage.

What!?

Am I being too critical or that just doesn’t sound right? That’s not telling the whole story. What about the payments that you have made every month over those five years, isn’t that a part of your investment? A quick calculation using a generic mortgage calculator shows that by the end of five years, a $200,000 mortgage (at a reasonable 6% rate of interest) would cost about $58,000 in interest alone (plus some amount of principal is paid back too ~ but let’s ignore that for the time being). Shouldn’t such costs be taken into account before making claims about doubling money? In fact, for this particular set of numbers, if you sell the house after five years, you would be making a handsome net loss.

If you are not yet convinced, here are some specific problems with David Bach’s argument:

• Let’s extend Bach’s calculation a little further. What if you put down only $25,000 instead of $50,000? With his calculation that would give a 200% rate or return. For $10,000 down payment that would be a 500% rate of return. Man..isn’t buying a house really profitable?! At this rate, Bach could easily tell you that if you don’t put anything down you get $50,000 on a $0 investment - now, I don’t think there are many people who could divide by zero (someone once mentioned to me that Chuck Norris can), but if Bach continues with this argument, he might as well perform the division and declare an “infinite rate of return” on investments.

• The doubling money argument totally ignores any kind of interest rates (his logic is applicable only when you borrow money at 0% interest rate and don’t make any payments for five years ~ and no one’s ever going to give that kind of a mortgage). For someone who is trying to explain potential gains using leverage, this isn’t a good sign. Profitable financial leverage cannot be explained without explaining the difference between the rate imposed on borrowed money and the rate of return that you get when you invest the borrowed money. For example, if you borrow $200,000 at the rate of 6% and invest it for a 4% rate of return, then it’s not a very wise way of using financial leverage ~ you will be losing money in such a deal (of course, there are complications like cash flow considerations - when you rent whole or part of your property, but we will ignore such issues for the time being - because Bach does not consider them either in his explanation).

Bach keeps up with this questionable math in the next paragraph:

Over the last five years, many homes have doubled in price. Think what this means in terms of leverage. If you invested $50,000 in a $250,000 home five years ago and it’s now worth $500,000, you’ve made $250,000 on a $50,000 investment. In investment circles, that’s called a five-bagger - an amazing 500% return on your money.

What does that “500% return on your money” sound like to you?

*cough* real estate agent *cough*

What do the credit card numbers mean and how are they generated? I need to start with a disclaimer: Do not use any credit card numbers, except your own, to buy things off internet. It’s wrong and it’s illegal. The purpose of this post is *not* to create fraudulent workable card numbers. It is to explain the math and the science behind those numbers that most of us see day in and day out; and hence this post should be viewed from a purely academic perspective.

Typical credit card anatomy

Before we understand how credit card numbers are generated, here is a brief explanation of what a typical credit card number means.

• Out of the 16 numbers on a typical credit card, the set of first 6 digits is known as the issuer identifier number (read this for details), and the last digit is known as the “check digit” which is generated in such a way as to satisfy a certain condition (the Luhn or Mod 10 check). “Luhn check” is explained later in this post. The term sounds intimidating, but it’s really a very simple (and elegant) concept.
• Taking away the 6 identifier digits and 1 check digit leaves us with 9 digits in the middle that form the “account number”.
• Now, there are 10 possible numbers (from 0 to 9) that can be arranged in these 9 places. This gives rise to 10⁹ combinations, that is, 1 billion possible account numbers (per issuer identifier).
• With each account number, there is always an unique check digit associated (for a given issuer identifier and an account number, there cannot be more than one correct check digit)
• Amex issues credit cards with15 digits. The account numbers in this case are 8 digit long.

What is the “Luhn” or “Mod 10″ check?

In 1954, Hans Luhn of IBM proposed an algorithm to be used as a validity criterion for a given set of numbers. Almost all credit card numbers are generated following this validity criterion…also called as the Luhn check or the Mod 10 check. It goes without saying that the Luhn check is also used to verify a given existing card number. If a credit card number does not satisfy this check, it is not a valid number. For a 16 digit credit card number, the Luhn check can be described as follows:

1. Starting with the check digit, double the value of every second digit (never double the check digit). For example, in a 16 digit credit card number, double the 15th, 13th, 11th, 9th…digits (digits in odd places). In all, you will need to double eight digits.
2. If doubling of a number results in a two digit number, add up the digits to get a single digit number. This will result in eight single digit numbers.
3. Now, replace the digits in the odd places (in the original credit card number) with these new single digit numbers to get a new 16 digit number.
4. Add up all the digits in this new number. If the final total is perfectly divisible by 10, then the credit card number is valid (Luhn check is satisfied), else it is invalid.

When credit card numbers are generated, the same steps are followed with one minor change. First, the issuer identifier and account numbers are assigned (issuer numbers are fixed for a given financial institution, whereas the account numbers are randomly allocated - I think). Then, the check digit is assumed to be some variable, say X. After this, the above steps are followed, and during the last step, X is chosen in such a way that it satisfies the Luhn check.

This part is a bit confusing and takes some time to understand. However, don’t get stuck here…continue reading through the examples below and you will figure out what this is all about.

Credit card numbers valid or invalid?

Have you ever wondered if those numbers on the fake plastic or cardboard credit cards that come with the “preapproved” offers are real or imaginary? If they are not valid, how do you know it?…Just apply the Luhn check and all the those fake credit cards will invariably fail.Here is an example of a VISA credit card (look at the expiry date - 01/09 ..it’s still valid ! )

Note that the credit card number starts with “4″…so it is indeed a VISA issued credit card (VISA cards start with “4″ and MasterCard/Maestro cards start with “5″). Now, let us apply the Luhn algorithm to this card. To make it easier on you guys, I have created a schematic of the steps towards the Luhn check (below) for this card number 4552 7204 1234 5678:

• In this case, when we sum up the total, it comes to 61 which is not perfectly divisible by 10, and hence this credit card number is invalid.
• If such a credit card number is ever generated, the value of the check digit would be adjusted in such a way as to satisfy the Luhn condition. In this case, the only value of the check digit, that will create a valid credit card number, is 7. Choosing 7 as the check digit will bring the total to 60 (which is perfectly divisible by 10) and the Luhn condition will be satisfied. So the valid credit card number will be 4552 7204 1234 5677.

Let’s try another example, this time with a MasterCard.

Again, performing the Luhn check on this credit card number, we have:

• The total comes to 65 which is not perfectly divisible by 10. Hence this credit card number is invalid.
• In this case, a valid credit card number will result only if the check digit is 8. This will bring the total to 70 which is perfectly divisible by 10. So the valid credit card number will be 5490 1234 5678 9128.

Closing remarks

If I still have your attention, here are some additional thoughts. In the context of this post, by the term “valid”, I mean “mathematically valid”. A mathematically valid credit card does not mean a “working” credit card. The Luhn formula validates only the credit card number; it does not validate the expiry date and/or card security code (CVV, CVC). Plus, as discussed before, the 9 digit account number will yield 1 billion combinations; so the chances of getting a working credit card number are very remote. It should also be noted that, this validation is usually employed at the transaction end; which means that numbers that do not satisfy the Luhn check are not forwarded to the card issuer and the transaction is terminated. If you have a fake credit card which satisfies the Luhn check, it will go through at the transaction end, but the card issuer will most likely catch the mischief. So don’t go about trying to use these numbers to buy stuff.

Just to be clear on this, I don’t expect comments like these (check out the source of this comment):

hey. im hearing good things about your site! i need some money to jump start my poker career. Probably about 40-100$ would do. i dont have a credit card to use and it pisses me off because i know i could beat the majority of the people online. please help

If you intend to post such comments, at least be extremely funny.

So you think you can separate out valid and invalid account numbers now? Here are a couple of trial numbers for you:

• 5491 9469 1544 4923 - Valid or invalid? If invalid, what should have been the correct check digit to make it valid?
• 4539 9920 4349 1562 - Valid or invalid? If invalid, what should have been the correct check digit to make it valid?

Sudoku fans will quickly figure out multiple valid combinations of the above numbers. If you don’t want to do the math, here are some ready made valid (”test”) credit card numbers from Paypal.By the way, the Luhn check is also valid for debit card numbers.I am still in the learning phase with this topic and trying to further understand how people use (or misuse (?)) such information. If you have some insight in this matter, please feel free to share it with us.If you liked what you read above, go ahead and subscribe to this blog to get more updates. It’s easy - just click on one of the buttons below and get the feed.

There is a vast amount of literature on the Luhn algorithm and a quick Google search will enlighten you on how popular this topic is. If you don’t want to read all that, here are links to some interesting reading.

• Card Identification Features (pdf file) - extremely useful, must read.
• Credit Card Numbers and Credit Card Generator @ Graham King.
• Anatomy of Credit Card Numbers.
• Credit Card Number - Wikipedia.

VISA card image source: http://www.hkuaa.org.hk, MasterCard image source: http://www.nscs.org

Long time back, I wrote a post on the calculations behind the rule of 72 with regards to estimating the time it takes to double your initial investment. The focus of that post was to understand exactly how the rule of 72 works and whether it is accurate for all rates of return (ROR). I did not explicitly show the answer to question “how long will it take to double your money at a given rate of return?”. That’s what I will do in this post by giving you a lookup table…just look up the rate of return/interest on your investment and see how many years it will take to double the initial amount (without any additional contributions to the initial investments). Of course, you could use the rule of 72 as a rough estimate..but the table below is far more accurate, and I wanted to do this in spite of the rule of thumb.

To get this lookup table, I have used this relation between time and rate of return (read here for details on how this formula is derived from compound interest formula):

In this formula, t is the time in years and r is the rate of return (rate of interest) in decimal form (5.10% => 0.051). Since I have assumed that the frequency of compounding is annual, r becomes analogous to APY if you are considering a savings account (or a CD). Graphically, this is what the formula tells us about how much time it will take to double the initial investment:

Here is the corresponding lookup table (rounded to the nearest number of months):

Another quick and crude way to estimate the time is to “double rate and half time”- meaning, if you double your rate of return (interest rate), it will take half as much time to double your money. For example, at 1% interest rate it takes about 70 years to double your initial investment, so at twice the interest rate (2%) it will take about 70/2 = 35 years; and if you double it once more (4%) it will take about 35/2 = 17.5 years.

I should point it out (although mathematically it is very obvious) that the graphical representation is very instructive. A rise of rate of return from 1% to 4% (a difference of 3%) has a drastic effect on reducing the time (from about 70 years to about 17.5 years), however, a rise from 17% to 20% (again a difference of 3%) reduces the time from 4 years and 5 months to just 3 years and 10 months. Instinctively, I think the risk increases much more rapidly in going from 17% to 20% than in going from 1% to 4%. I am almost tempted to overlap the above graph with some kind of risk curve (towards pointing out some sort of a *comfort zone*), but I was not able to find enough information on risk-return relationship. I will appreciate any input in this matter.

I am also trying to generate a similar (but more useful) lookup table that will incorporate regular additional investments on top of the initial principal amount….that would prove to be an useful tool to convince some people to save money..as in “look…saving just $X.XX per month will double your money is Y.YY years!”

I have been thinking about this for quite some time. I think about this every time I hit the freeway and involuntarily increase my speed up to 5 ~ 10 miles per hour above the legal limit. I thought really hard about this when I got my last speeding ticket about 3.5 years ago for driving 19 mph above the limit. Finally, I have made up mind to write about it….just needed to flush those numbers out of my system. I will probably qualify for the smart-ass of the year award for this post. Whatever.

Speeding to save time?

Let’s look at some typical scenarios and see how much time we save by speeding to our destination. I have considered three cases that most of us would usually come across:

• Case 1: short distance, low speed limit - This applies for most in-town driving. Clearly, speeding in this case makes no sense at all. In the chart below, look at the time saved by going 20 mph above the limit…just 5 minutes ! Doesn’t seem like it’s worth it, does it? Plus, if you drive at 60 mph in a 40 mph zone, you will be noticed (by cops perhaps). Also, in some areas/cities, the traffic lights are synchronized in such way that, if you drive 10 mph over the limit, you will keep hitting red lights often.

• Case 2: medium distance, high speed limit - This applies more to situations when you are driving to a nearby town/city. Most of us will generally prefer the freeway for such distances and hence the 70 mph speed limit. If you are like me, you would probably start cruising at 5~10 mph over the speed limit. But look at the graph below. Going 10 mph above the speed limit is saving you just a little more than 5 minutes. You could potentially take less time by speeding more, but you need to look at the risk analysis (scroll down for that) before you do that.

• Case 3: long distance, high speed limit - This is about driving to a far away town/city. Alright ! so this time you can save 19.29 minutes by driving at 80 mph instead of 70 mph. This one sounds reasonable, doesn’t it? Well, if you look at it with only a slightly practical point of view, this doesn’t sound very promising either. You will save 19.29 minutes if you drive constantly at 80 mph for the full 180 miles. In all probability, over longer distances, your average driving speed may be well below 80 mph, considering general traffic, road conditions, traffic lights, etc. You may occasionally pump it to 80 mph, but then that’s not going to save you the full 19.29 minutes.

Speeding vs. risk, a trade-off
Risk is a inseparable part of the speeding package. To really make sense of whether speeding is worth the time saved, we should look at the risks involved. First, let’s look at the risk of getting a speeding ticket in the event that you have a speed measuring laser pointed at your vehicle. The graph shown below is just to give you a feel of how the risk changes with speeding. Note that, there are different laws in different states and this may not be specifically apply to you. More details about state-specific laws can be found here.

Btw, if you are mathematically inclined, I have used a sigmoid function to draw this graph and assumed that the chance of getting a ticket, when driving 10 mph above limit, is 50%. This particular graph is for a 70 mph speed limit, but you can apply it to any speed limit. OK, all this is a bit geeky, but stay with me here. The point is to show that about 5~6 mph above the speed limit you probably won’t get pulled over for speeding (some states even have official tolerances in this zone). However, beyond this, your luck will start vanishing very rapidly. If you desperately want to speed, your best chances of not getting caught are in Zone 1 and Zone 2 (when you are not going more than 10 mph above the speed limit).

Now, go back to the charts that showed the time gain and look up the time gained by driving just 10 mph above the limit in all of them. Do you still find it worth?

More risks and costs

If you are still not convinced, let me throw some hackneyed reasons at you. May be, against the above background, these things will make a greater effect here.

• Accidents: risk to life - You must have heard this a zillion times. Speeding is just not safe. Speed limits are put up for a reason…they are not just there to frustrate the hell out of you. Some people probably must have spent their lives trying to determine the optimal speed limits for certain roads. Show them some respect.

• Fuel: cost of increased consumption - Speeding costs money. Here is a some sort of a quantification by the government (here is the source):
  

  As a rule of thumb, you can assume that each 5 mph you drive over 60 mph is like paying an additional $0.20 per gallon for gas.

• Tickets: cost of fines - Most fines hover in the range of $70 to $150 when you are speeding between 0-20 mph over the limit. Plus, there are court fees and other associated costs (like a driving safety course fee) involved when you get a ticket.

• More monetary loss: cost of insurance - You won’t be very happy with your insurance premiums if you have a couple of tickets sticking in your driving record. Of course, all tickets don’t lead to this, but you should keep this in mind.

• The cost of lost peace of mind: There are other intangible costs that go beyond monetary values, like stress while driving, stress after getting a speeding ticket, overall loss of time if you are required to go to court and stuff, etc.

You could reduce some/all speeding ticket associated risks by using radar detectors and such, but those things won’t warn you of an impending accident risk on account of your speeding.

Or you could just relax, drive within speed limits, and enjoy your driving. I think I am going to try doing just that.

Infinite speed limit image source: www.continuum2.com

A few days ago, I posted this problem:

You have two job offers, from two different companies (Company A and Company B) with the same starting salary, $50,000 per year. Company A gives you an offer which says “We will increase your salary by $500 at the end of every six months” and Company B counters that offer by saying “We will increase your salary by $2000 at the end of every year“. Considering all other factors equal, which company would you choose?

This seemingly dry and uninteresting problem generated some interest over the last couple of days. After a few comments on the post and after some interesting discussion with Yan of ProBargainHunter.com, I made a long comment on the situation…by far the longest comment ever (sucks that it had to be on my own post!). Just sharing the comment (with minor edits) which reflects my thoughts on the problem. Please feel free to point out flaws in the arguements.

[click to continue...]

This trivial problem is in the spirit of one of my earlier posts in which I mull over how people are losing very basic mathematical skills. Here is the problem statement:

You have two job offers, from two different companies (Company A and Company B) with the same starting salary, $50,000 per year. Company A gives you an offer which says “We will increase your salary by $500 at the end of every six months” and Company B counters that offer by saying “We will increase your salary by $2000 at the end of every year“. Considering all other factors equal, which company would you choose?

Hint: Are these offers really different…or it’s just jumbling of the words to create an illusion?

There, jog your grey cells a bit with it.

Don’t kill all the fun by using calculators or excel sheets. Do it mentally and make a note of how much time it takes. Btw, I have just reworded the problem from an old book of puzzles; so just in case you have the book, the answers may not be the same.

[This is the second article in the guest blogging experiment at this blog. If you are interested in contributing as a guest blogger feel free to contact me]

So you’ve heard that you can make money blogging.

How real is that potential to generate extra monthly cash without leaving the comforts of your home? And how much money can you really make? If you have done your research, you will find out that there is a lot of “white noise” out there. How do you filter the information available on what works and what doesn’t? It is every beginner blogger’s dream to take that first step to financial freedom.

Why don’t we get started then? Let us set a target of $600.00 a month from your blog starting January 2007. Isn’t that just a nice new year’s resolution? What can you do with all that money…. Good heavens!

Let’s make this a reality in three steps!

Step 1 - Payperpost.com

Average monthly revenue towards the goal - $480.00

Daily time spent - 20 minutes / 2 posts at approximately $8.00 each

Payperpost connects Bloggers with Advertisers. Once you register, you can check daily on available opportunities that you can blog about. Each opportunity will pay you between $5 - $20 approximately. You can take two opportunities maximum a day, just make sure they are not one after another. There has to be at least a post in between them. Each sponsored blog post must remain on your blog for 30 days. In your account section of Payperpost it counts down how many days before payday. And it’s really cool seeing all those “You’ve Got Cash” emails from Paypal!

Step 2 - Google Adsense
Average monthly revenue towards the goal - $75.00 / $2.50 a day

Google Adsense is a very good source of revenue. However, there are a couple of factors that you need to pay attention to, in order to make money. Targeting $2.50 a day doesn’t really seem much, but you need to understand how you can make that money in less clicks - 3 clicks vs. 10 clicks.

Diva tips in optimizing your blog:

• Generate an overall theme for your blog and stick to it. The more theme oriented your blog is, the better success it will have - wedding, relationships, cars, dogs…. etc.
• Blend the color of your Google Ads to your blog. In other words, the Google ad background and border must be the same color as the spot where you are placing it on your blog. Use the same text color for Ad text, post title color for Ad Title and Ad Url.
• Each post title must be clear and straight to the point. Notice I didnt make the post title of this post as WANT TO MAKE MONEY ON THE INTERNET? Do you know how many posts out there with that title?
• Strategize your post - how much revenue can you expect per clicks if you talk about the weather (advertiser’s maximum bid $.50), visa (advertiser’s maximum bid $2.00), lexus (advertiser’s maximum bid $1.15), crystal cruise (advertiser’s maximum bid $3.00), or find love (advertiser’s maximum bid $0.62). How do I know this? This is based on overture’s listing. Why don’t you try it next time you make a post?
• Last but not the least, don’t be shy in posting your adsense. Google allows you three widgets. Make it work….

Step 3 - Tumri Cornerstore

Average monthly revenue towards the goal - $45.00 / $1.50 per day
Never heard of Tumri? Well, you have now. I wish I can say that they are the best kept hidden secret on the net, but they are not. They are a new company and only few blogs are carrying their cornerstore ads.

What makes them special? Tumri takes the Amazon concept a step or two forward. They have nice and exciting flash widgets that display 4 - 5 products at the same time! It draws products from Amazon, Overture, Ebay, and your favourite store! When you are targeting impulse buyers on your site, you need to have a look-at-me-i’m-special presentation that just shouts BUY ME! And that is what Tumri brings to your blog.

Tumri is not like Amazon that you will have to wait until someone buys your product before you generate income. Tumri will pay you for clickthroughs! Here is an excerpt from my email interview with Jason Ng at Tumri :

• We do not violate the Google Adsense terms as we are not ‘contextual’; we do not scan the content of the page. We have behavioral based targeting where we optimize revenue based on user behavior and publishers will be able to completely control what is displayed within the widget.

• The commission that we will payout to you will be based on a combination of click based revenue offers and order based revenue offers. Click based or CPC (cost per click) commission occurs when a valid click is made on any CPC based product offer. Order based commission or CPO (cost per order) occurs when a lead/click is generated from your site to one of our merchant partners and an order is made. You will receive a percentage of that total order amount. After you have selected the product categories that you want to display in the widget, we will optimize the CPC and CPO product offers based on the user behavior so as to maximize your commission.

Payperpost - $480 + Google Adsense $75 + Tumri Cornerstore $45.00 = equals $600! That’s $7200 in your pocket in a year. Can you make more? Of course. Please make me green with envy….

So, who’s game to join The Tao 600 challenge starting January 2007? What an awesome way to start the New Year, don’t you think?

About the author: Imaginary Diva has been a regular reader at Money, Matter, and More Musings for a long long time. Click here to visit her blog. That’s just one of her blogs..she writes on more than half a dozen others.

A few days ago, I posted a YouTube video which posed a very basic problem relating to an investment. The purpose was to get people to revisit some very basic algebra. Over time, most of us have started relying heavily on calculators and computers and have often shown tendencies towards loosing these basic mathematical skills.

In all there were 6 unique comments and all had the right answer. However, the time taken to solve the problem varied between 10 seconds (?) and 8 minutes. The people who gave correct answers are: Tom, Pete, Keith and Teri, Jon, and Dimes. I would like to assume that Trent had the right answer but did not post it. Thanks guys for participating.

Now the solution (by one possible easy method)

Essentially, the problem statement was: A person invests a total of $7200. Part of that is invested at 4% annual rate of return and the rest of it at 5% return. If the return from each investment is the same, find the income from each investment.

Let the amount invested at 4% be P; the remaining amount = (7200 - P) is invested at 5%
The return on 4% investment = P *4/100;
The return on 5% investment = (7200-P) * 5/100;
Since these returns are equal, our equation is:

Now, 72 is directly divisible by 9 ( 9 * 8 = 72 ), so no need for a calculator here. After dividing by 9, we have:

Now, the return on this P at 4% = 4000 * 4/100 = 160;
the return on the remaining amount at 5% = 3200 * 5/100 = 160.
This is your final answer.

You could do this in many different ways and equally simpler ways. I would really love it if people do this without using calculator/excel solvers. Solve some simple problems once in a while to stay in good algebraic-shape.

All you wannabe investors and money crunchers, post your answers to this little investment problem in the video. It is very basic math, so don’t shy away from it. To make it a bit more interesting, please mention the time it took for you to solve it. If you are unable to solve it or you think it is very difficult, mention that in your comment. Also, let me know if you think that the problem makes no sense. All those who give correct answers will be mentioned in a forthcoming post. Good luck.

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Just mention the answer and the time it took you to get that, don’t post in your solution procedure…that may take the fun away for the rest of us
P.S: That’s not my voice in the video !