Discount Rate

 

The Entertainer

I am the entertainer
The idol of my age
I make all kinds of money
When I go on the stage
Ah, you've seen me in the papers
I've been in the magazines
But if I go cold I won't get sold
I'll get put in the back in the discount rack
Like another can of beans

What should be the rate in that discount rack?

If the universe is hyperbolic, perhaps the discount rate, how much the future is worth compared to the present, is a fundamental constant of that universe.  The odds of a choice, which is defined as1where 0 is not making that choice, is 0.5.  That means that at its median, 50% will have made the choice and 50% will not have made the choice.  Let’s call 0.5 the range, s, of the choice.  Then the square root of the variance according to a normal logistics distribution of choice is 0.5*√3/π.  This number is 0.9067, or “A bird in the hand is worth 1.1027 (its inverse) in the bush.”  This means that the discount rate is 1/.9069 -100% or 10.27%.  If the interest rate is less than 10.27%, the universe is telling you that the cost of investing, on average, will have a positive rate of return. If interest rate is greater than 10.27% then the universe is telling you that the cost of investing, on average, will have a negative rate of return.

This might explain why humans as a group get nervous when the interest rate of a loan is much higher than 10%  It is only when the repayment is in future inflated dollars that higher interest rates may be tolerable. This discount rate also assumes that the asset being acquired is fully productive during its life and it is not depreciating. If it is depreciating, then that should be subtracted from the discount rate.  However, if an asset can be sold for 5% of its present value after 20 years and produces full value over  those 20 years, it is only depreciating by 0.25% per year.

Quantum Entanglement

 

Spooky

In the cool of the evening
When everything is getting kinda groovy
I call you up and ask you
Would you like to go with me and see a movie
At first you say no you've got some plans for tonight
And then you stop and say, "alright"
Love is kinda crazy with a spooky little girl like you

Maybe it isn’t so spooky after all?

Given the Heisenberg uncertainty principle, and the superposition of particles, isn’t quantum action at a distance inevitable, not spooky.

For example, take a deck of cards.  Divide them into one deck of red cards and one deck of black cards.  Turn the two decks over and shuffle them so that you can’t tell which deck is which.  Pick two cards from one deck but do not look at them.  You do not yet know the color of the card. Now look at the first card.  If it is red then that means that every card in its deck was red.  Now if you look at the second card, it is ….surprise, surprise…. red.  There was no need for communication between the first and second card. Any distance separating the cards is irrelevant. By observing the first card, which had not yet been observed, you could be certain of the color of the second card.  This isn’t spooky, it’s statistics.

Schrodinger’s cat, before it is observed, has odds of being 50% alive or 50% dead.  But the unobserved odds have nothing to do with the cat.  The cat will be either 100% alive or 100% dead but you will not know until you open the box, observe it.  The odds are NOT the result.  The odds of a coin flip are 50% heads.  But the result of every coin flip will be either 100% heads or 100% tails. There is nothing spooky about that.

Inflation VIII

 

Catch The Wind

For me to love you now
Would be the sweetest thing
That would make me sing
Ah, but I may as well, try and catch the wind

Raising the Prime Interest rate to control inflation may be trying to catch the wind

In being asked to control inflation by raising its interbank interest rates, the Federal Reserve Bank is using its primary function, to serve as the lender of last resort for private banks, in order to try to control random events. In doing so, it may doing more harm than good. The interest rate hikes may have contributed to private bank failures, and preventing bank failures is precisely why the Federal Reserve was created in the first place.

Banks exist to take existing liquid value and convert it to long term but less liquid future streams of value. When that value, a currency/medium of exchange, is based on a commodity there is a problem. Using a commodity as the medium of exchange in economic transactions means that you have to store, safely maintain, and transport that commodity. When that commodity is gold, this is not a trivial issue. Gold is heavy. Banks started issuing notes that represented a certain amount of that commodity. But that imposed a burden of trust on parties when using those bank notes. Both the buyers and sellers in economic transactions have to trust the bank and the value represented by its note.

When a currency, the medium of exchange, is a commodity that creates problems when groups hoard that commodity, corner the market. Then only economic transactions by those parties can be accommodated, which is precisely the opposite of the purpose of a medium of exchange. Trading by barter requires that both parties want what the other has and value their goods the same way. Since you can’t always find such parties, medium of exchanges were used to measure the value of goods. When the medium of exchange is a commodity, like gold, it is a commodity currency, and as mentioned when groups hoard that commodity and will not participate in exchanges, then the number of economic transactions declines. That is why fiat currencies were developed that are not finite and the market can not be cornered. The fiat currency should be sufficient to allow economic transactions to take place, including any increases in value, and you have to trust that the party issuing that fiat currency. If they set it to an amount that is more than the economic transactions it must support,  such as with the Weimar Republic, hyperinflation can result. That is why national economies usually set their currency to the value of their economy, e.g. M2. If that was the only economic transactions that needed to be considered, then that should suffice. The problem trade between nations with their own currencies.

During World War II, it was realized that gold as the international trading medium of exchange,  could not accommodate international trade because almost all of the world’s gold was in the Untied States and an international fiat currency did not exist. During Bretton Woods, John Maynard Keynes proposed the creation of Bancor as the international fiat trading currency. The problem was there was as yet no trusted group to issue such a fiat currency. John Foster Dulles and the United States prevailed in an argument that the US Dollar, while a fiat currency domestically, would be convertible into gold for international trade, a commodity currency.

This dual status of the US Dollar, a fiat currency domestically and a commodity currency internationally, was maintained until the Nixon Shock of 1971. At that time, the US Dollar was became a fiat currency at home AND in international trade. However the dollars in circulation were only created to support the value of the US domestic economy, M2. There was no recognition of the value in international trade that was supported by the US Dollar. As domestic  US Dollars competed with international US Dollars, the result was inevitable as the high inflation of the late 20^th Century. Arguably the fiat value of  the US Dollar should be M2 PLUS the USD used in international trading, MI. International trading has been increasing by 6-8 percent per year and the USD used in international trade, which according to the Society for Worldwide Interbank Financial Telecommunications, SWIFT, is approximately 50% of that trade. Thus the fiat value represented by the USD should be the current USD plus the change in M2 plus 50% of the change in all currency in MI. Measuring inflation over the last year includes changes in product inflation AND changes in currency inflation over the last year. While Year Over Year inflation has been declining and has been  as recently as 2020 been about 2%, long term inflation in  the CPI since 1971 has been about 4% per year. The pervasive inflation is because in those periods where random product inflation is zero or negative, there still has been an increase in currency inflation.

Arguably the Federal Reserve should only be concerned with currency inflation. However by setting the US Dollar only to accommodate the changes in M2, this ignores any changes in MI. It is proposed that the money supply of the US Dollar should recognize its use in MI. In this case, currency, medium of exchange, inflation should be zero. The problem is that inflation also includes product inflation. The Federal Reserve Bank tries to address product inflation by changing its Prime Inter-Bank Interest rate. In all honesty it can not do this. At best it can time-shift supply and demand, but it can not create or remove this supply or demand. The problem is that banks turn existing liquid value into illiquid future values. Withdrawals, if any, are from liquid reserves. If the withdrawals exceed the liquid reserves, then long term illiquid assets may be sold at a loss to turn them into liquid assets. If there are no buyers, or the loss is too great, then the bank fails.

A billionaire investor, e.g. Peter Thiel, withdraws his deposits/value from a bank, e.g. Silicon Valley Bank. While the bank has already converted his liquid value into streams of illiquid future values of a greater amount. However when each investor withdraws, that investor is allowed to withdraw liquid assets from the bank's reserves rather than receiving the converted illiquid value. The result may be a bank failure. The action precipitating the withdrawal may have been reduction of the future value of the stream of illiquid value because of the raising of interest rates. By continuing the fiction that the deposits have not already been converted and allowing them to be withdrawn from liquid reserves, the very act of raising the interest rates may have precipitated the bank failure that the Federal Reserve was created to prevent. You can change the impact of numerous random events. i.e. climate, setting the house odds. You can not control every random event, i.e. weather, each roll of the dice. Trying to do so is as futile as trying to catch the wind.

Normal

 

A Wonderful Guy

I'm as corny as Kansas in August,
I'm as normal as blueberry pie.
No more a smart little girl with no heart,
I have found me a wonderful guy!

You might be normal, but are you hyperbolically normal?

A hyperbolic normal distribution is proposed to be one in which 1/3  of the observations, outcomes, fall within the median ± σ and 100% of the outcomes fall within the median ± 3σ.  As in any normal distribution, the median will be equal to the mean, but the mean can be computed without ordering the observations.  

This also indicates that the minimum number of outcomes must be 3 or a hyperbolic distribution will be abnormal.  For example for two outcomes, e.g. a two player game, a choice/transition/phase change, will have a variance, σ² , but it will be hyperbolically abnormal because while it passes the 100% test, it fails the 1/3 test.  By contrast three‑or‑more‑players will always pass  both the 100% and the 1/3 test.  The variance, σ²,  of a two outcome game is 0.277777 which makes the range, σ, 0.166667.  If choice one has a value of 1 and choice two has a value of 2, then the median and the mean are both 1.5.  However while both outcomes pass the 100% test in that they are within 1.5  ± 3*(0.166667), they fail the 1/3 test in that neither outcome is within 1.5 ±  0.166667.  By contrast, a three player outcome: 1, 2, or 3; has a variance, σ², of 0.111111, or 1/9, which means that σ=1/3.  This distribution has a mean of 2 and a median of 2.  One hundred percent of the outcomes are within 2 ± 1, and 1/3 of the outcomes are within 2 ± (1/3).

The hyperbolic skew is proposed to be the ratio of the mean and the median.  When the ratio is greater than 1, the distribution  favors higher outcomes.  When the ratio is less than 1, it favors lower outcomes.  A hyperbolic normal distribution is one where this ratio is between 0.75 and 1.5 . In these cases, the observations will pass both the 100% and 1/3 test but the lowest observation will also not be less than 0.  When the median is equal to the mean of course the hyperbolic skew is 1.

It is not surprising that the minimum number of outcomes in a hyperbolic normal distribution is 3. In game theory there is a different strategy for playing two-player games and three‑or‑more‑player games.  If there are only two players in a game, then there have to be three outcomes: e.g. win, loss, and tie.  Having only two outcomes is abnormal.  If there are only two outcomes, you can not tell if the outcome is due to chance or the winner is better than the loser.

Bank Runs II

 

Honey in the Honeycomb

There's money in the savings bank
And I personally guarantee
If there's honey in the honeycomb
Then, Baby, there's love in me

So what is a bank?

A bank is a time machine.  It takes value that has been accumulated in the past and turns it into a stream of future values.  The problem is that it follows the laws of entropy such that the process is not reversible, while runs on banks pretend that it is reversible.  A run on the bank is an attempt to pull the accumulated value from the bank, without any acknowledgement that the value has already been converted.  You can’t do that.  As George Bailey famously explained in the run on the bank in the movie It’s a wonderful Life.

“you're thinking of this place all wrong As if I had the money back in a safe. The money's not there.  Your money's in Joe's house.

(to one of the men)

Right next to yours. And in the Kennedy house, and Mrs. Macklin's house, and a hundred others. Why, you're lending them the money to build, and then, they're going to pay it back to you as best they can. Now what are you going to do? Foreclose on them?” 

When you withdraw money from a bank, the bank it actually makes that withdrawal from its reserves to prevent the  need to foreclose.   Only when the withdrawals exceed its reserves, does the bank have to think about foreclosing. Just because you don’t see the sausage being made, does not mean that it is not sausage.

Hyperbolic Statistics

 

Natural Gift

You don't have to be a genius to find
All the hidden potential deep in your mind
You don't have to know about nuclear physics
Know all the formulas and vital statistics
You don't have to be an intellectual, you don't have to be a scientist
To use your natural gifts, you got natural gifts, yeh
Use your natural gifts, you got natural gifts

Coming soon to a math journal near you.  Hyperbolic statistics!

The first moment of observations about the origin is:

m₁=(1/n) ∑f_i x_i,  , summed from i=1 to k

where the index i is the i^th grouped observation, f_i is the frequency of that observation, (e.g. if there are 3 observations of 2, this makes 3 the frequency of the group of observations that is 2), k is the number of groups, and n is the number of observations. This is also the formula for the arithmetic, computed, mean, x̄ , often called the average. The average is more properly defined as the centrality of the normal. The median is the centrality, but in a normal distribution the mean is also equal to the median. Therefore saying the average is the mean is only true when the observations are also normally distributed.

The second moment about the about the computed mean, often called the variance, is:

m₂=(1/n) ∑f_i (x_i- x̄)² summed from i=1 to k

The square of the Standard Deviation is also often defined as the variance. In flat Euclidean space, the Standard Deviation, S.D., is: 

Euclidean S.D.=square root(∑f_i (x_i- x̄)²)/(n-1))

However this is only true in Euclidean space since it relies on Pythagoras’ Theorem for a hypotenuse. On a non-Euclidean surface, this is not the correct formula for the sum of squares. For example, the shortest distance between two points, a, and b, on the spherical surface of the Earth is the Great Circle Distance. According to Pythagoras’ theorem for a spherical surface, where R is the radius of the surface, e.g. the Earth, this is R*cos^-1(cos(a/R) *cos (b/R)). When the distance between points a and b is very small compared to the Radius, e.g. of the Earth, of the surface, then this is virtually indistinguishable from the traditional Pythagorean theorem.

If the surface is hyperbolic, not spherical or flat, then the shortest distance between two points is
cosh^-1(cosh(a)*cosh(b)). If a is defined as the summation of the deviations about the mean, and b is defined as 0, then the Hyperbolic Standard Deviation is

(cosh^-1 (cosh( (∑f_i (x_i-x̄))/n)))^(1/2)

while the Euclidean, flat, S.D is as defined before.

When the number of observations, n, is very large, and the sum in the second moment is not zero then there is virtually no difference between the hyperbolic S.D. and the flat, Euclidean traditional S.D. This does not mean that the difference is not real, just that in many applications there is no observable difference between the hyperbolic S.D. and the traditional S.D. Even when the sum of the square of the differences between the observations and the computed mean is virtually indistinguishable from zero, the difference between the traditional and hyperbolic Standard Deviation is virtually indistinguishable for large n , as shown in the figure below.

With a roll of a traditional six-sided die, there are six possible outcomes, 1 though 6, which if the die is not loaded should follow a normal distribution. The mean outcome is 3.5. The median outcome is 3.5.  The Euclidean Standard Deviation is 1.7.  This requires that according to the 68/95/99 rule for normal distributions that 99.7% of the outcomes should fall between the mean minus 3 SD and the mean plus 3 SD. According to the traditional SD, this requires that 99.7% of the die roll outcomes should fall between -0.6 nd 7.6.  While this is true, a more useful metric might say that 100% of the outcomes fall between 1 and 6. This requires the variance, σ², to be .694 and the square root of the variance, σ, to be .833. Then 100% of the observations fall between the mean ± 3σ. The Hyperbolic Standard Deviation of a six-sided die role is .34.  According to the Hyperbolic SD, this requires that 99.7% of the die roll outcomes should fall between 2.48 and 4.56, which is also the incorrect variance. 

The reason that the square of a Standard Deviation might not be σ², the true variance, is because of error. The true mean is not necessarily the computed mean because the computed mean can contain error.

x̄=μ+ ε¯

x̄           =           The computed mean, (1/n) ∑f_i x_i,  , summed from i=1 to k;

μ            =          The true mean;

ε¯          =           The mean error, (1/n) ∑ ε_i,  , summed from i=1 to k. 

The moments about the computed mean will only have non-zero values if the computed mean is NOT equal to the true mean.  This is because  (1/n) ∑f_i (x_i- x̄)^r summed from i=1 to k is equal to zero for every moment r when there is no error.  If there is no error, then the square root of the second moment about the mean should not be solved using Euclidean mathematics, etc. The computed Euclidean Standard Deviation added the Bessel adjustment, n-1, in order that the square of the Euclidean Standard Deviation be closer to the True Variance. The Bessel adjustment is only necessary if the Standard Deviation is computed using Euclidean geometry. It should be computed using non-Euclidean hyperbolic geometry. If the computed mean is the true mean, then the variance should be other than the square of the Standard Deviation.

It is suggested that the Standard Deviation for a normal distribution where the mean error is zero,  appears to be 0.  This is also not the variance squared, σ^2, but this is the limit of the Euclidean Standard Deviation when the number of observations approaches infinity.  This suggests that the 100% of the values of a traditional six-sided die, where this hyperbolic SD is 1.90,  occurs between -2.2 and 9.2.  Thus it is suggested that the square of the Standard Deviation is NOT the variance.  It is suggested instead that the high observation and the low observation be identified.  If the distribution is negatively skewed then the computed mean minus the lowest value minus .003 divided by 3 is the square root of the variance, σ.  If the distribution is positively skewed, then the highest value minus the computed mean minus  minus .003 divided by three is the squared root of the variance. If the skew can not be determined, then the maximum of these values should be taken as the square root of the variance, 

A no/yes, heads/tails, off/on transition which occurs at μ is a normal distribution. The transition is from whatever choice is assigned a value of zero to whatever choice is assigned a value of 1. Then the median choice is  0.5, the mean choice is 0.5,  the true variance σ² is 0.028, and the square root of that variance, σ, is 0.167.  According to the rule of normal distributions this requires that:

  0.3% of the transitions from 0 to 1 will be made by μ-3σ, or μ-0.500;

  5%  of the transitions from 0 to 1 will be made by μ-2σ, or μ-0.333;

32%  of the transitions from 0 to 1 will be made by μ-σ, or μ-0.167;

50%  of the transitions from 0 to 1 will be made by μ;

68%  of the transitions from 0 to 1will be made by μ+σ ,or μ+0.167;

95%  of the transitions from 0 to 1 will be made by μ+2σ, or μ+0.333;

99.7% of the transitions from 0 to 1will be made by μ+3σ, or μ+0.500. 

Mathematically there is no difference between a transition that happens at μ+3σ and one that happens at μ-3σ. Or Biblically, the Parable of the Workers in the Vineyard (Matthew 20: 1–16). It is wrong to say that there is no variance in choice, transitions, even when there is no error. Mathematically there is a definite non-zero variance with every choice.

Intolerance

 

Humpty Dumpty

And all the king’s horsemen and all the king’s men
Couldn’t put Humpty together again

Wait, that’s not right, is it?

My wife and her sister have a small business on Etsy. On the items that they sell is a cloth book of the children’s nursery rhyme Humpty Dumpty. My wife’s sister designs the fabric, and my wife sews the fabric. The line above is how it appears in their fabric book. A disgruntled customer was upset because in her opinion it should  have been “all the king’s horses”,  not “all the king’s horsemen.”  She was offered a refund including shipping both ways if she retuned the book. She refused. She wanted the fabric to be reprinted and the book to be  changed. This was not possible. She wanted the book to be no longer sold to others because it contained an “error” and left a bad review because of that. In other words, she wanted her opinion to be imposed on others.

This sounds like someone who is concerned with the letter of the law and not justice, the spirit of the law. I bet that customer also thinks she is being a good Christian which makes her better than other religions, and non-Americans who don’t speak English and are not as white as her. Who is going to tell her that that Jesus Christ was a Jew, didn’t speak English, didn’t live in the United States and was not as white as her?

She also should realize  that her preferred wording is not the only way that the nursery rhyme has even appeared. According to Wikipedia the first appearance of the Poem in 1797 was

Four-score Men and Four-score more,
Could not make Humpty Dumpty where he was before.

In 1810 it appeared as

Threescore men and threescore more,
Cannot place Humpty dumpty as he was before.

Elsewhere in that poem, “Humpty” was spelled “Humpti” and “sat” was spelled as “sate” but that is another issue.

The current version did not appear until 1882.

The poem was thought to be a riddle, where Humpty Dumpty is an egg, which is speculated as Dutch slang for Egg. Which reminds of an English coworker who triggered a spit take by me when he made the comment that he spent the weekend "knocking up" an old girlfriend. He meant that he “was visiting”,  not what I thought he meant. Two great civilizations separated by a common language indeed! The motto for the state of my birth is “Hope” which came from a misunderstanding by the early English settlers that the nearby Native American village was Mount Hope. The natives were saying “mantoup” in their language

From Wikipedia

In 1996, the website of the Colchester tourist board attributed the origin of the rhyme to a cannon recorded as used from the church of St Mary-at-the-Wall by the Royalist defenders in the siege of 1648.In 1648, Colchester was a walled town with a castle and several churches and was protected by the city wall. The story given was that a large cannon, which the website claimed was colloquially called Humpty Dumpty, was strategically placed on the wall. A shot from a Parliamentary cannon succeeded in damaging the wall beneath Humpty Dumpty, which caused the cannon to tumble to the ground. The Royalists (or Cavaliers, "all the King's men") attempted to raise Humpty Dumpty on to another part of the wall, but the cannon was so heavy that "All the King's horses and all the King's men couldn't put Humpty together again".

The poem has also been advanced as a story based on the Laws of Thermodynamics because once broken an egg, Humpty Dumpty, could not be but together again because its entropy has increased.

In any event, on this night when the 95^th Oscars are to be awarded, it is only a story anyway. If someone objects to how you are telling a story and calls you illiterate, they are probably projecting and confirming that they are illiterate, not you. Besides to be “woke” shouldn’t it be "horsepersons" any way. J