From the category archives:

# interesting numbers

## Things You Should Know About Percentage Traps

by on October 31, 2007

The issue of percentages comes up almost every time we talk about any kind of numerical data. This is especially true in the investment world where people generally tend to talk in terms of percentage gains and losses - rather than the absolute values. That’s simply because it’s easier to interpret financial changes on a common scale of 100. However, there are certain pitfalls that one should be aware of when dealing with percentage values. Here’s a quick (and elementary) look at some of these traps.

#### Absence of a time frame

When it comes to investments, simply stating percentage gain/loss is not nearly enough. There is something else that MUST go along with the percentage data: it is the duration or time frame. Without the time frame, most financial percentage figures are effectively meaningless. For example, simply saying “Get a 100% return on your investment” doesn’t make any sense -100% return in how much time? 69 years 8 months (corresponding to 1% annual return)? or 3 years 10 months (corresponding to 20% annual return)?

So, whenever you judge the performance of a stock (or compare two stocks), make sure that you are aware of the time frame before you start comprehending numbers like “300% increase”! *cough* penny stocks emails *cough*

#### Arithmetic mean against geometric mean

Let’s consider this statement “Our picks have yielded 25% average annual returns over the past two years“. Sounds genuine - after all it mentions the time frame clearly. But let’s run through an example and see how this can be pretty deceptive.

• Initial investment: \$100
• Value of investment after one year: \$200 (100% return this year)
• Value of investment after second year: \$100 (50% loss this year)
• Net gain/loss amount over the two years: \$0

Now, when you calculate “average annual return” using arithmetic mean, you are basically taking the average of 100% and -50%, giving you a 25% average annual return. But obviously, since your net gain amount is \$0 (or 0% over two years), the simple arithmetic mean is not really telling you the entire story in this case.

This is where geometric mean comes into picture. Click here to learn how to calculate geometric mean of a given set of percentage values (including negative percentages). If you apply the geometric mean for the above example, you will end up with 0% average rate of return - which is correct. This rate of return - calculated using geometric mean - is known as “annualized return“.

Another way to look at it is that geometric mean takes into account the effect of compounding, whereas arithmetic mean does not.

I won’t be surprised to find a lot of proponents of volatile stocks using arithmetic mean to make their case of higher returns. So make sure you are aware about the difference between “average return” or arithmetic mean and “annualized return” or geometric mean.

#### Irrelevant time frames

Sometimes, data is presented with reference to a time frame (perhaps with careful consideration to the geometric mean) but this time frame may be largely misleading. In fact, I still don’t have a good explanation of what makes an “appropriate” time frame, but I am working on it.

For example, let us consider the share price of the S&P 500 exchange traded fund (SPY) over various time spans, going backwards from yesterday:

• 1 year return: 10.99%
• 5 year return: 69.69% [11.16% annualized (geometric) return; average (arithmetic) annual return of 13.94%]
• 10 year return: 62.83% [5% annualized (geometric) return, average (arithmetic) return of 6.28%]

Now, it is obvious that the annualized return differs if you choose different time windows over the history of the share. Depending on how you want to make your case, you could choose one of these time windows and go crazy with it. For example, you could pick the 5% annualized return for the last 10 years and blast the ETF’s performance, or you could pick the 11.16% return for the last 5 years and try to make a generalized statement like “… gives almost a 12% return on an average“.

Additionally, in all probability, people who want to downplay the 10 year performance will use geometric mean to make their case (geometric mean is always lower than the arithmetic mean), and people who want to glorify the 5-year performance, will use the arithmetic mean.

Readers should be wary of such pick-and-choose explanations. Try to find data that is most relevant to your investment time frame.

#### The disconnect between percentages and amounts

These types of examples are rarely seen in the investment world (I haven’t seen any), but are often observed in blog monetization circles. Here is an example:

My advertising income increased by 300%

Yeah right! It was 5 cents yesterday and today it’s 20 cents.

Obviously, in this case, the percentage value creates a greater boasting impact than the actual dollar value - so it might be used as a good marketing ploy to sell ideas/products.

It helps to have this scaling effect in your mind when you compare your income/net worth with others. For someone with \$1,000,000 in hand, 10% means \$100,000; whereas, for someone with \$1,000 in hand, it’s just \$100 (think in terms of buying power).

This becomes very obvious when you think in terms of percentage discounts on clearance products. What’s more attractive, a “50% off” sale on a \$2 item you need or a “5% off” sale on a \$100 item you need?

So, there you have it. Pay more attention to percentages in future.

## Can You Make Sense Of This? 6000% Inflation, 600% Interest Rate, And A Booming Stock Market?

by on September 26, 2007

A few days ago, this news on BBC just wouldn’t let me concentrate elsewhere (source):

Zimbabwe’s annual inflation rate slowed in August to 6,592.8% from July’s record of 7,634.8%, according to the Central Statistical Office (CSO).

At the end of August, President Mugabe introduced jail terms of up to six months for anyone caught trying to raise prices or wages.

What? 6592.8% inflation! That’s crazy. I think you can actually feel the money becoming lighter and less valuable with the minute. To keep up with this rate of inflation, a person earning \$10,000 now, would have to earn \$659,280 a year from now in order to maintain his/her lifestyle!

Everything becomes 65.92 times more expensive within a year. If your salary doesn’t increase for whatever reason, you stand become 65.92 times poorer than the previous year without spending a dime!

Man, that just doesn’t sound quite right.

We are so used to a relatively *stable* economy, that we have come to accept certain ideas by default (sort of, “taken for granted” - although we pretend that we look at the historical data and try to predict the future) - for example, inflation rate around 2~4%, interest rates between 0~6%, stock market returns between 8% and 12%, etc. - it almost gives a feeling of complacency at times.

I wonder how our investing minds would react to a Zimbabwe-like economy. If, by some misfortune, we are ever caught in this type of an economic atmosphere, I wonder if we will have the ability to think outside the box.

Anyways, to see how other characteristics of this 6000% inflation economy look like, I checked up some data on the Reserve Bank of Zimbabwe. Here is what I got:

• Exchange rate: 30,000 Zimbabwe Dollars (ZWD) = 1 US Dollar (can currencies fall any lower?)
• Interest rate: Overnight Rate (analogous to Federal Funds Rate in US) = 600% (at this interest rate you will arithmetically “double” your initial investment in about 4 months! - although that doubled amount will only be a fraction of it’s original *value* in the same amount of time.)

It’s obvious that the inflation will override the growth of funds in a savings account by about 10 times (assuming if you get a full 600% on the savings account) - makes a simple savings account an invalid option to park your money, even at that interest rate.

Next, I tried to look for some data on the Zimbabwe stock market. I recently explained how the interest rate cycle and market sentiments go hand-in-hand. With that picture in mind, the 600% interest rate in the background, and with the knowledge that the economy was falling apart, I was expecting terrible scene at the stock market.

However, I was in for a big surprise here. This is what I found about the Zimbabwean stock market (source):

Zimbabwe is in the middle of an economic disintegration, with GDP declining for the seventh consecutive year, half what it was in 2000. Ever since President Mugabe’s disastrous land-reform campaign, the country’s farming, tourism, and gold sectors have collapsed. Unemployment is said to be near 80%.Yet something odd is happening.

The Zimbabwe Stock Exchange (the ZSE) is the best performing stock exchange in the world, the key Zimbabwe Industrials Index up some 595% since the beginning of the year and 12,000% over twelve months. This jump in share prices is far in excess of increases in consumer prices. While the country is crumbling, the Zimbabwean share speculator is keeping up much better than the typical Zimbabwean on the street.

This is probably going to cause a paradigm shift in my thinking about how stock markets react to economic conditions - I was expecting the graph to be exactly opposite! On the basis of conventional investing knowledge in US, there is no way anyone could have predicted that kind of stock market performance - in an economy that’s just out of control, and with lending rates that should be absolutely out of whack with that kind of inflation.

Perhaps, more than the money equations, it’s the geo-political situations that affect the market sentiments. Perhaps, there is more to stock markets than what my tiny brain can comprehend at the moment. Perhaps, it’s just the randomness that people talk about. Whatever.

Yeah.. whatever … I am just glad not to be in Zimbabwe .. I think I would have resorted to gold smuggling by now.

Stuff like this keeps reminding me of Warren Buffet’s ovarian lottery concept.

## A Little Pizza Problem

by on September 5, 2007

Here is a 5-minute pizza workout for you.

A quaint little pizzeria sells the following specialty pizzas at the prices mentioned:

• 9″ pizza: \$8.00
• 13″ pizza: \$12.00
• 16″ pizza: \$15.00
• 19″ pizza: \$26.00

You have been given \$26 for buying some pizzas and you are asked to “make the best of it”. If you get some change back from your order, it is yours to keep.

Which pizzas will you order?

Note: “make the best of it” is a little open ended requirement, but it will be more fun that way.

## The Worst Credit Card I Have Ever Seen

by on August 31, 2007

I often wonder about the irony of subprime lending - whether it’s made available through mortgage or through credit cards or through any other source of credit. I mean, people with bad credit are the very people who should be staying away from subprime borrowing. These people are already in the subprime category because of financial troubles and then there are these stupid (and very likely, unaffordable) subprime products targeted at them - which have the potential to throw them deeper down the financial hole.

Here is one such stupid subprime product - the Continental Finance MasterCard. It’s hard to believe that this piece of plastic comes with an initial credit limit of just \$53 after a long list of fees!

Below are some of the amazing rates and fees that go along with this card.

• Account setup fee: \$99
• Program participation fee: \$89
• Annual fee: \$49
• Account maintenance fee: \$120 (charged @ \$10/month)
• Purchase APR: 19.92%
• Authorized user fee: \$30 (great! seems like \$53 credit is a bit too much for a single person to handle)
• Credit limit increase fee: \$25 (and you don’t even have to ask for it!)
• Each Credit Limit increase will be \$100.00, subject to a maximum Credit Limit of \$2,000.00. Each increase will appear on your Account no later than one (1) month after you have qualified for such increase. At the time of each Credit Limit increase, a \$25.00 Credit Limit Increase Fee, which is a FINANCE CHARGE, will be charged to your Account.

You need to call these people and ask them to stop; otherwise, they are automatically going to increase the limit by \$100 each time and charge you the \$25 fee.

• Internet payment fee: \$4 for each authorized internet payment. I just don’t get this - why are people with bad credit charged for paying their bills online? .. probably to make sure that they don’t start paying their bills automatically or something?

And, here is how the \$53 initial credit limit appears:

Your initial Credit Limit will be \$300.00 and you agree to pay the following fees, which will be billed to your Account and will appear on your first monthly Billing Statement: a one-time Account Processing Fee of \$99.00, a one-time Program Participation Fee of \$89.00, a monthly Account Maintenance Fee of \$10.00 and an Annual Fee of \$49.00. Your available credit after these charges will be \$53.00 at Card issuance.

Here is an YouTube video that points fingers at this same credit card for some of the reasons mentioned above:

I agree that people with bad credit pose the risk of potential losses for the credit issuing company, but this a bit too much for a credit line worth just \$300. This is where the irony strikes me - if you make it difficult for people to pay back what they borrow, it’s only going to increase the chances that they will never pay back what they borrowed. Isn’t there a risk assigned to this perspective?

If it’s so much risk then don’t issue credit cards to people with bad credit (I have similar thoughts with regards to subprime mortgage - it’s a self-inflicted disaster on part of both, lenders and borrowers). Just offer secured credit cards and lend against a collateral.

By the way, there is another not-so-obvious trap hidden in such credit cards for people with bad credit. If you apply for a particularly horrible “bad credit” credit card and start building your credit history with this card, then some time down the line, when you think you have built sufficient credit and no longer need this horrible card, you are going to face a problem - your credit history is going to take a hit if you close this account. And, if you don’t close it down, the annual fees are going to create a leak in your pocket for quite some time to come.

So much for the cost of bad credit and so much for people with bad credit wanting a credit card (which creates the demand for such products in the first place).

If we ever have a competition for the worst credit card in the world, do you know some good competitors for the one I mentioned above?

## Fill In The Blanks - Only If You Are An Adult

by on August 5, 2007

Here is an interesting exercise for those who are more than 18 years old.

All of the incomplete statements below are taken either from PBS.org or from certain articles in New York Times [stuff published between 2000 and 2004]. Fill the blanks to the best of your knowledge.

• On the Internet, ______ is one of the few things that prompts large numbers of people to disclose their credit card numbers. According to two Web ratings services, about one in four regular Internet users, or 21 million Americans, visits one of the more than 60,000 _____ sites on the Web at least once a month â€” more people than go to sports or government sites.
• None of the corporate leaders of AT&T, Time Warner, General Motors, EchoStar, Liberty Media, Marriott International, Hilton, On Command, LodgeNet Entertainment or the News Corporation â€” all companies that have a big financial stake in _______ and that are held by millions of shareholders â€” were willing to speak publicly about the _______ side of their businesses.
• Based on estimates provided by the hotel industry, at least half of all guests buy ______.
• At home, Americans buy or rent more than \$4 billion a year worth of _______.
• The ______ industry rakes in more money than pro-football, basketball and baseball combined. Americans spend more money on ________ than on movie tickets and the performing arts. [via NPR.org]

All answers are related to each other. There have been enough hints already in the above text, so no more hints.

Apparently, AT&T and General Motors sold their relevant stakes in this issue to Comcast and News Corporation, respectively; so, they don’t seem to be involved in this matter any more.

## What Is Gap Insurance? Who Needs It And Who Does Not? - A Graphical Interpretation

by on August 3, 2007

I have had many questions on the issue of gap insurance in the past and never really found a satisfactory answer. The most common answer to the question: “What is gap insurance?” - something that a car salesman or a website will give you - is this:

Gap insurance closes the gap between what your auto insurance company thinks your car is worth (if your car is stolen or totaled) and what you owe the finance company.

Now, technically, that is a correct definition, but it doesn’t really help you visualize the concept very well. Plus, such a definition doesn’t help in answering any other related questions on gap insurance; for example, “Do you need gap insurance on used cars?”, “Do you need gap insurance on leased cars?”, “In what other situations is this insurance not necessary?”, etc.

In this post, we will try to visualize the idea of gap insurance with the help of some graphs, and then try to extend it to various situations to address other commonly asked questions about gap insurance.

• What exactly is the “gap”? and why does this gap needs to be insured?

Let us follow a hypothetical purchase of a 2007 Buick Terraza, one of the worst depreciating vehicles out there. The high depreciation of this vehicle will allow us to see the “gap” clearly. Here is how the value of this vehicle depriciates over time (data from Edmunds.com):

The values in the table above are what insurance companies are going to follow for a *regular* insurance. So, if you total the car within a few days after driving it off the dealer’s lot, you will get only about \$24,190 from the insurance company. If you total it after Year 1, you will get \$19,860 and so on.

Now, let us assume that you financed this vehicle with \$0 down and 6% APR auto loan for 60 months. Your monthly payments (according to Bankrate.com) will be \$623.54. Based, on this monthly payment, and accounting for the interest per year, here is how the amount you owe the financing company will look like:

Now, let us plot these values together to get the following graph and observe how the gap looks like.

The shaded portion is the “gap” in the term “gap insurance”. This is the gap that needs to be insured. For example, if you total your Buick Terraza after the 1st year, your regular insurance will give you only \$19,860, whereas, your financing company will ask you to shell out \$26,551 to clear off your debt. There is a gap of \$6,691 between these to figures and you need gap insurance if you don’t want to pay this gap amount.

• How does the gap look when you lease a vehicle?

You saw above that financing the car (at 6% APR) created a gap of \$6,691 after the first year. Instead, if you had leased the vehicle, for say 60 months (usually, it’s less than 60 months, but this will help us understand it better) , your monthly payments would have been much lower (probably around \$350 ?) - which means that you would have owed more (than the financing option) after the first year, and therefore the gap would have been larger. This is shown in the graph below.

Also you can see that the gap exists for the entire period of the lease. That’s why you MUST have gap insurance at all times when you are leasing a vehicle.

Another point to note is that the *residual value* as defined by most leases is usually higher than the actual depreciated value of the vehicle. This will be the case, whatever the length of your lease is. This means the gap will ALWAYS exist for a leased vehicle.

• Do you need gap insurance if you are financing a used vehicle?

From the above graphs, it is obvious that the gap is maximum when you just drive off a dealer’s lot with a brand new vehicle and then it gradually decreases over time. It eventually goes to zero within a few years (3 years in the above example).

Generally, gap insurance won’t be offered on used vehicle at all; but just in case it is being offered (people can offer you anything to extract money from you), you won’t really need it for vehicles that are more than a few (2 ~ 3) years old. Depreciation slows down after the first year, and in all probability, for a vehicle that is about 3 years old, your monthly payments will be more than the loss in the vehicle’s value. So, you will never hit a point where you owe more than the value of your vehicle (or more than what your regular insurance will pay you).

The only exception to this is when you have been sold a used vehicle for a price that’s way above it’s depreciated value (not a good deal to begin with) - and you completely financed it without any down payment. At that time, for a brief initial period, you will be owing more than the value of the vehicle.

• Are there any cases in which you may not require gap insurance EVEN if you finance a new vehicle?

Yes. Depending on the term of you car loan and your initial down payment, it is possible that you may not require the gap insurance on a financed new vehicle. Let’s discuss two simple cases below.

• Case 1: You make a large down payment. For the example above, suppose you made a 25% down payment and availed an auto loan for just \$24,190. You are essentially taking care of most of the initial depreciation. In this case, you will probably never owe more than what your regular insurance will give you if you total your car. This is shown in the graph below.

• Case 2: You make a small down payment and your loan term is very short. In this case, the gap is small and is very rapidly reducing towards zero. You will be taking some amount of risk, but it will be a very short term risk. For our example of the Buick Terraza, this is shown in the graph below for a 10% down payment and a 24 month loan term. The gap reduces to zero in less than six months.

• Do you need gap insurance if you buy a vehicle using cash?

It is important to note that the gap is essentially defined ONLY if you are financing (or leasing) a vehicle. If you are using cash, you don’t owe anything to anyone, and there is no gap and hence, gap insurance doesn’t hold any meaning. This is how the graph looks like when you pay cash for a new vehicle:

What other things should you look out for when choosing a gap insurance?

• First issue - you don’t really need gap insurance for very long periods of time if you are financing your vehicle. It depends on your loan terms and conditions. In the above example, you need gap insurance for only about 3 years. If your loan period is longer (72 months or 84 months), or if you are buying a vehicle that’s worse in terms of depreciation, then you may probably need a slightly longer coverage. So keep that in mind before you jump for the longest coverage.
• Second important issue - watch out for the deductible. If the gap insurance has very high deductible, then it essentially defeats it’s own purpose. The purpose is to protect you from shelling out a large sum of money for a car that has been totaled (or stolen) ~ so that you would have enough cash in hand to arrange for an alternate vehicle if you need one. High deductible defeats this protection. Plus, sometimes (in case of cars that depreciate very slowly) the cost of the gap insurance and the deductible together might be greater than the gap itself. Beware of such situations.
• Of course, before you buy a separate gap coverage, you also need to make sure that it is not covered by your existing insurance company and/or included in the lease payments itself.

Hopefully, things are clearer by now. Remember, we followed one specific example (2007 Buick Terraza) for this post; the numbers will be different for different cars, car loans, and leases, but the general concept will be the same.

Feel free to suggest additions/modifications to this write-up. If you like it so far, make sure to bookmark this post so that you can explain the concept to your friend who is all eager to buy gap insurance for a 6 year old Toyota Avalon.

## Interesting Facts And Confusing Thoughts About The American Poor

by on August 1, 2007

Just came across some interesting numbers on American poverty, through this report: “Understanding Poverty in America“. In the write-up below, I am presenting some interesting highlights from the 2004 report (which is based on the Census data from 2002).

To better appreciate the facts, it is important to view them in light of the poverty thresholds (income levels below which a person or a family is considered “poor”) for 2002. Towards that, here are some of the 2002 poverty thresholds from Census.gov.

• Single person: \$9,183
• Two person household: \$11,756
• Three person household: \$14,348
• Four person household: \$18,392

In 2002, according to the US Census Bureau, there were about 35 million “poor” Americans - people who were below those thresholds.

Now, let’s move on to the *typical* characteristics of these poor people, as mentioned in the report [by the way, always keep in mind that "typical" does not mean "all" - it is just a statistical average, and there are always some data that do not fit in this average].

The following are facts about persons defined as “poor” by the Census Bureau, taken from various government reports:

• Forty-six percent of all poor households actually own their own homes. The average home owned by persons classified as poor by the Census Bureau is a three-bedroom house with one-and-a-half baths, a garage, and a porch or patio.
• Seventy-six percent of poor households have air conditioning. By contrast, 30 years ago, only 36 percent of the entire U.S. population enjoyed air conditioning.
• Only 6 percent of poor households are overcrowded. More than two-thirds have more than two rooms per person.
• The average poor American has more living space than the average individual living in Paris, London, Vienna, Athens, and other cities throughout Europe. (These comparisons are to the average citizens in foreign countries, not to those classified as poor.)
• Nearly three-quarters of poor households own a car; 30 percent own two or more cars.
• Ninety-seven percent of poor households have a color television; over half own two or more color televisions.
• Seventy-eight percent have a VCR or DVD player; 62 percent have cable or satellite TV reception.
• Seventy-three percent own microwave ovens, more than half have a stereo, and a third have an automatic dishwasher.

Here is what the report says about the food/hunger situation of the poor people:

When asked, some 89 percent of poor households reported they had “enough food to eat” during the entire year, although not always the kinds of food they would prefer. Around 9 percent stated they “sometimes” did not have enough to eat because of a lack of money to buy food.

And, about the general financial situation:

Some 70 percent of poor households report that during the course of the past year they were able to meet “all essential expenses,” including mortgage, rent, utility bills, and important medical care.

The report later concludes:

The typical American defined as “poor” by the government has a car, air conditioning, a refrigerator, a stove, a clothes washer and dryer, and a microwave. He has two color televisions, cable or satellite TV reception, a VCR or DVD player, and a stereo. He is able to obtain medical care. His home is in good repair and is not overcrowded. By his own report, his family is not hungry and he had sufficient funds in the past year to meet his family’s essential needs. While this individual’s life is not opulent, it is equally far from the popular images of dire poverty conveyed by the press, liberal activists, and politicians.

Honestly, the data had me confused about my definition of poverty. Irrespective of how low the income is, if a person (or a family) is able to enjoy most of the things that an average family can enjoy, is able to get proper nourishment, and is able to meet all “essential expenses”, how can such a person be termed as “poor”?

These facts bring up some food for thought. Here is what I have been thinking:

• Can people really be defined as “poor” based on the income criteria? In the same breath, if income is not a good criteria for defining the “poor” - then it cannot be a good criteria for defining the “rich”.
• If you look at all the things that a “typical” poor person is enjoying - and then look at the poverty threshold (income level) - it seems to me that poor people might be doing things frugally. Of course this doesn’t say whether they are more (or less) frugal than rich people.
• It can also be construed that poor people are being too consumerist and in fact, are *poor* in the first place because they try to buy too much on too little income. However, “too little” is a very relative term - and it doesn’t make sense to put any dollar amount income levels here.
• Is a person, who is \$2,000,000 in net debt, but living in a mansion and earning \$100,000 a year, richer or poorer than someone who has a positive net worth but is making ends meet with great difficulties (barely able to provide food, education, shelter, etc) ?
• So who exactly classifies as poor? a person facing financial hardships? but hey, that can happen to people with high income and bad spending habits.

I still don’t have any conclusive answers yet… if you have any, please enlighten us.

A summary of the report referenced above can be obtained through this link (pdf file).

## Are You Spending More On Stuff Than Millionaires?

by on July 31, 2007

The other day, I came across some interesting data on spending habits of millionaires from the book “The Millionaire Next Door” by Stanley and Danko. It was about how much money millionaires spend on items like suit clothing, shoes, watches, and cars. I am presenting this data in a tabular form below - “typical” values are highlighted in green; for example, a typical millionaire spends about \$140 on a pair of shoes.

It’s a pity that the authors do not explicitly mention the time frame of the survey (although, there is some generic information about their surveys towards the end of the book), the number of millionaire participants, or the age distribution of the participants included in the survey - but what is presented should be enough to make (some) people think about their spending habits. Of course, all surveys have their limitations, so a bit of skepticism won’t be out of order (keep in mind that the book was first published in 1996 and the surveys were conducted around that time, so there will be some inflationary effect on the numbers by now).

Anyways, assuming that the survey satisfactorily reflects the millionaire population in the US (around 8.9 million households), if you find yourself placed in a higher percentile group (more than 10%) in the above tables, then there are probably hundreds of thousands of millionaires who have less expensive tastes than you. Try to visualize this whenever you feel like contemplating on your spending habits.

I (fortunately, so far) fall in the lowest spending percentile on all four counts. Here are some specific numbers for my case:

• Most expensive suit clothing: \$99 (plus tax) suit jacket from a JC Penny sale; bought 3 years ago.
• Most expensive shoes: \$70 leather shoes; bought 6 years ago.
• Most expensive wristwatch: \$9 + change (don’t laugh); bought more than 6 years ago.
• Most expensive car: \$12,200; bought a couple of weeks ago.

What about you? have you been spending more than the typical millionaires on the items mentioned above?

## Is David Bach’s Math Misleading?

by on May 31, 2007

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*

## How To Generate *Valid* Credit Card Numbers

by on April 12, 2007

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. Itâ€™s wrong and itâ€™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.

Resources and References

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.

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