SURELY THE WEAK WILL PERISH, AND ONLY THE FITTEST SURVIVE (Robert W. Service)–MATH THAT OUGHT TO BE TAUGHT


I bought a car 10 years ago for $10,000 and financed it at 2% for 5 years, because I could get more on the money by investing it than the loan was worth, much as I hate loans for cars.  The salesman told me the payments would be $250 a month.

“You’re wrong,” I said, in about 3 seconds.

“That’s what the computer said,” he replied, as if he were quoting Genesis or a Sura from the Qu’ran.

“It should be about $190,” I replied.  “Go back and do it again, or I leave.”

He left, and returned, about 5 minutes later.  “Sorry,” he said, a little sheepishly.  “Your payments are $187.”

Was I unusually smart?  No.  Anybody with a calculator can do this.  I just happened to estimate, and I work with numbers a little better than the average person.

Here is my thinking, and notice the simple assumptions I make:  Suppose I pay nothing.  $10,000@2% is $200 of interest a year.  Yes, it is compounded and a little more, but learn to estimate and not worry so much about the damned exactness you need.  There is a time and a place for estimation, and it is being lost in teaching today.  I learned by computing batting averages on my favorite baseball players and by estimating cost per ounce, back when these things weren’t available.  It’s a lost art, and I use the word art exactly as it is meant to be.  I don’t need eight decimal places, like the math teacher who used the calculator to find the tangent of 67 degrees to 8 places, when I gave a decent estimate (about 2.25) in 5 seconds.  It is 2.35.  This isn’t a post about trig, but maybe I should do one on the uses of it, too.

Back to the car.

Suppose I don’t pay for 5 years.  My loan has “ballooned” to about $11,000.  Yes, it is a little more than that, but not much.  Five years,  60 months * $250/month is $15,000, far over $11,000.  $250/month is $1000 in 4 months, and 4 into 60 is 15. 

Alternatively, $11,000/60 is like 1100/6, and I know 1200/6 is 12/6  with two zeros attached, or $200.  The payments must be below $200, and not much.  I estimated between  $180-$190/ per month, and I rounded it upwards.  I saved $750 a year in payments.  Not knowing and using math is a tax that not only the Republicans aren’t going to repeal, by their refusal to fund education adequately, they are ensuring many people pay for it.  That is shameful, but again, I need to get back to the subject.

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I listen to the Suze Orman show, often amazed at the financial stories people tell.  The latest was from a woman in Texas who had buried $300,000 in her backyard, and had it there for at least 20 years, because her grandparents had done this.  The woman said, rightly, that interest rates now were near zero.  Suze countered by saying  that the interest on that in the past 20 years would have been $50-60 thousand.

“I don’t think so,” replied the woman.

Based on what?  I wondered.  She doesn’t think so?  This is not a debate about who is going to win the wild card playoffs or the Super Bowl.  This is about pure math, where there is a pure answer.

The average interest rate in the past 20 years has been, by a quick look at a graph of CD interest rates, about 2.5%.  That is the worst possible CD.  It is not what would be gained by a 5 year CD, for example, which would have been at least 4%.

If she puts $15,000 a year away, she will make $375 a year using simple interest, which is not what is paid.  Interest is continuously compounded.  The second year, she has another $15,000, and she now doubles her interest, but that is an underestimate, too, so she has made $1125 in interest.  This now becomes even more in the third year, and the fourth, and so on.  There are online calculators that are fast at computing what she would have after 10 years of doing this at any interest rate you want, any way of compounding, and any way of depositing the money.  Suze was right.  The woman would have made at least $50,000 in interest during this time.

The woman might not “think so,” but she is wrong.  Burying money in a backyard, of course, is a horrible idea, since once it is gone, it is gone.  Survivalists believe big government will take everything; they would do well to worry about little people, like other survivalists, who watch what they are doing.  Survival of the fittest goes to those who are fitter in math.  That is today’s story.

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Here are a few mathematical rules you need to deal with finance.  They can all be proven, so to say, “I don’t think so,” requires you to show me a mathematical proof why it is not so, not assume that repetition establishes validity, a sadly common approach to things that don’t appear to make sense.  This approach is well on the way to ruining the country, but that, too, is a story for another time.

Rule of 72:  

72/interest rate in %  = years it takes to double money, debt, or any other growth rate. 

72/time it takes to double in years=interest rate in %.  

It is that easy.  Interest rates of 1% double money in 72 years.  Population growth of 4% doubles population in 18 years.  That should be scary, but I’ve been harping on that issue for 40 years with no success.  We have no children, and it won’t likely be our problem when nature readjusts matters.  I do digress a lot when I discuss this stuff.  My apologies.

Rule of 110:  Tripling time.  At 4% interest, money triples in 27.5 years.

Rule of 40:  Halving time, increase of 50%.  30% credit card debt interest rate increases the debt 50% in 1.3 years.

Actual return on a real estate investment:

You buy it for $50,000 and sell it for $100,000 8 years later.  You double your money in 8 years, and 8 into 72 is 9, so you make 9% interest.  Right?

No, wrong.

You buy it for $50,000 and pay $2000 in various fees.  During this time, you pay property taxes of $1000 a year, so your total cost is about $60,000.  Notice how I use the word “about.”  Math is a pure science, but estimation is valuable.

You sell it for $100,000 and pay $7000 in fees.  Your capital gain is $50,000, and let’s say you pay 15% on it.  That is $5000 + half of $5000 (15% is 10% + 5%; 10% of anything is the number minus the last digit.  Five per cent is half of 10, so a capital gain of $43,000 is 10%  $5000 (lose last digit) + half of it $2500, or $7500.

Your net on this is $100,000-$7000-$7500=$85,500

We’re going to do this without a calculator.  The money paid was $60,000 and you got 85,500.   Half of $60,000 is $30,000, and the two added together are $90,000.  True, that isn’t $85,500; the latter is about half as much but not quite, so the rate of return will be somewhat lower than the Rule of 40 will tell me.

Use the Rule of 40.  40/time=interest rate OR   40/interest rate=time

We have time.  40/8=5%, which is higher than the actual rate of return, because we assumed the gain was 50%, when it was in fact closer to 40%.  The actual rate of return is 4.3%, far below the 9% that the real estate people will tell you and in the ballpark of the predicted 5%.  In the real world, where I live, there are things called closing costs, property taxes, insurance, repairs, and capital gains taxes.  They have to be factored in, too.  You may or may not get a tax deduction for the payments.  In my world, I’d get rid of them for second homes or mortgages over $250,000.  But again, I digress.  I do a lot of that, when people ask how we are going to balance the budget.

I can prove all of this using the interest formula.  I know it and used a calculator to check the various rules many years ago.  What I wanted to show here is that estimation can be useful, a calculator may be used for the estimation, should one wish, but one can become very, very close to the actual rate of return, close enough to be useful.  Does it really matter whether the rate of return was 4 or 5% vs. 9%?  You decide. If it does, then use the exact formula.  It works.  But if you want an estimate, you are capable of doing it yourself.

Here’s how to determine good rates of return.  Google “calculator.”  Put in the amount you made and divide it by the amount you paid.

You pay in 80,000 and 6.5 years later you have $100,000.  Enter 100,000 and divide it by 80,000.  Then press the button called “ln” and “ans” and you will get 0.223.  Divide that by 6.5, and that is your rate of return.

Next  press ln and then 2.  You will get 0.693.  Converting decimals to percentages gives us 69.3, and we use the rule of 72, not 69.3, because 72 is evenly divisible by 2,3,4,6,8,9, and 12.

Press ln and 3, and you get 1.098.  Change to percent, and that is where the rule of 110 comes from.  Want the time for 3.5 x the amount of money?  ln 3.5 and you get 1.25.  That is the rule of 125, should you want that.  I showed that to a young man who wanted to be a stockbroker but couldn’t understand why he was learning crazy formulas.  Nobody had ever told him about these rules or the fact with a calculator, you could really do some really neat stuff. He walked out of the room a different person from the one who walked in.

“I don’t think so,” is a bad argument in math.  Don’t use it, unless you know what you are talking about.  Leave it for the social scientists, when we decide whether long term health insurance will save costs, factoring in net present value, likelihood of hospitalization, effectiveness of preventive care, and using multivariate analysis.  There are far more assumptions they have to make, like whether people can really see primary care doctors when they want to, or whether Americans are too impatient to wait and will go to urgent care.  This affects the results, but the math is still helpful.  But for earning power of money, and the cost of debt, learn the rules.  They work.

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