Month: October 2017

Portfolio Insurance and Black Monday, October 19, 1987

On the thirtieth anniversary of Black Monday, the stock market crash of October 19th and 20th in 1987, there have been mentions of “portfolio insurance” having possibly exacerbated the crash.

 

Portfolio insurance, in principle, is exactly what you might expect it to be: if you own a stock, Stock A, you insure it with a put option on Stock A.  Your position becomes equivalent to a call option on Stock A until the put option expires, with the price of this position being the premium of the put option when you bought it.

If you are managing a portfolio on behalf of clients, though, and you just need to insure the portfolio up to a certain date, after which, say, you hand over the portfolio, then to buy American put options to insure the portfolio would be unnecessary.  European put options would suffice.  So let’s suppose that we are only interested in European options.

In the article that I cite at the bottom (Abken, 1987), it seems that at the time, buying put options as insurance had a few issues.  This is assuming that the portfolio we want to insure is a stock index: the S&P 500 index.  The issues were:

  • Exchange-traded index options only had matures up to nine months
  • Exchange-traded index options had a limited number of strike prices
  • It’s implied that only American options were available (which we would expect have a premium over European options).

Thus, instead of using put options to insure the portfolio, the portfolio and put options are replicated by holding some of the money in the portfolio and some of it in bonds, Treasury bills, that we assume to provide us with the risk-free rate.

Without worrying about the math, the Black-Scholes equation gives us a way to represent our stock index S and put options P as:

$$S + P = S \cdot N_1 + K \cdot DF \cdot N_2$$

 

d

 

 

Source:

Abken, Peter A.  “An Introduction to Portfolio Insurance.”  Economic Review, November/December 1987: 2-25.

Link to articleArchived.

 

Value-added Tax and Sales Tax

(This is mostly a summary of and heavily borrowed from https://en.wikipedia.org/wiki/Value-added_tax, archived).
(The first three figures are taken from Wikipedia).

 

Comparing No Tax, Sales Tax, and VAT

Imagine three companies in a value chain that produces and then sells a widget to a consumer. The raw materials producer sells raw materials to the manufacturer for $1.00, earning a gross margin (revenue – Cost Of Goods Sold, COGS) of $1.00. The manufacturer sells its product, the widget, to the retailer for $1.20, earning a gross margin of $0.20. The retailer sells the widget to a non-business consumer (for the customer to use and consume) for $1.50, earning a gross margin of $0.30.

No tax example

Imagine we add a sales tax of 10%.  Sales tax applies only to the transaction with the final end-user, the consumer, i.e. the final transaction. So the consumer pays the retailer $1.50 + 10% sales tax = $1.65 to the retailer. The retailer remits the sales tax, $0.15, to the government.

Sales tax example

  • Only retailers remit collected sales tax to the government. Different regions, products, and types of consumers may have different sales taxes, so retailers are burdened with the maintenance of functions that process this for every different kind of sales tax.
  • There are cases (e.g. in the U.S., remote sales, i.e. cross-state or internet sales) where the retailer isn’t required to charge sales tax on its sales to consumers.  Instead, the consumer is responsible for remitting a use tax to the government on his or her remote purchases.
  • Only end-users pay the sales tax. Thus, someone who is an end-user has an incentive to masquerade as a business and purchase products for usage.
  • The government thus requires businesses (namely non-retailers, in this example) with the burden to prove, via certifications, that it is a business (and thus does not need to pay sales tax on the products it buys) and that it sells to other businesses (and thus does not need to charge and remit sales tax on products it sells).

Now let’s take away the 10% sales tax and add a 10% VAT. The raw materials producer charges $1.00 + a $0.10 VAT, which is 10% of the $1.00 value they added to the product) to the manufacturer. It remits the $0.10 VAT to the government. The manufacturer sells its product to the retailer for $1.20 + 10% or $0.12: $0.10 of which is the VAT that the raw materials producer charged the manufacturer and is now getting “paid back” by this transaction and the remaining $0.02 of which is 10% of the value added to the product by the manufacturer, which is $1.20 – $1.00 = $0.20, and remits the $0.02 to the government. The retailer sells its product to the customer for $1.50 + 10% of $1.50 or $0.15 ($0.12 of which is VAT that it paid to the previous 2 companies in the value chain and is now being “paid back” by this transaction and the remaining $0.03 of which is 10% of the value that the retailer added to the product, $0.30) and remits $0.03 to the government.

VAT example

  • From the consumer’s viewpoint, nothing has changed. End-users still pay the same $1.65 for the widget.
  • The government earns the same $0.15 as it earned with sales tax.  But instead of receiving all of it from the final transaction between the retailer and the consumer, it earns it in bits of [10% * each value added by each company in the value chain], which are $0.03, $0.02, and $0.10.
  • From the perspective of each business, they’re charged VAT by companies that they purchase from and they charge VAT to companies/consumers that purchase from them.  When they’re charged VAT on purchases ($0.12 for the retailer in the example), they are effectively charged the VAT of all companies that are further down the value chain from them ($0.10 for the raw materials producer and $0.02 for the manufacturer).  When they charge VAT on their sales, they effectively charge VAT for the value they added ($0.03) plus the VAT of all companies that are further down them in the value chain ($0.12).  The difference, the VAT for the value they added, is remitted to the government ($0.15 – $0.12 = $0.03).  So when a company purchases and is charged VAT, they are effectively “in the red” for that amount of VAT ($0.12) until they can sell their product up the value chain and charge that amount of “downstream” VAT + the VAT they own on the value they added ($0.12 + $0.03 = $0.15).  Then with that sale, they get “refunded” the portion of VAT that they paid before ($0.12) and remit the remainder ($0.03) to the government.
  • All buyers, whether they’re a consumer or a business, pay the VAT. So there is no incentive for anyone to masquerade as anyone else (e.g. a consumer to masquerade as a business).
  • All businesses process VAT (charged VAT on products they buy, charge VAT on products they sell, and pay VAT to the government) so all businesses are burdened with the maintenance of functions that process this.
  • Because businesses are charged VAT when they buy products and remain “in the red” that amount of VAT until they can sell those products, they are incentivized to make sure they charge VAT on the products they sell in order to make up that VAT they were already charged (e.g. the manufacturer paid the raw materials producer $0.10 of VAT so it’s incentivized to charge VAT on the products it sells to the retailer to make sure to make up for that $0.10). Because everyone is incentivized to charge VAT on their buyers, “everyone collects the tax for the government.”
  • This symmetry where everyone in the VAT system charges and is charged VAT doesn’t exist when it comes to cross-border trade, which is discussed below.

 

Sales Tax versus VAT

  • Most countries (166 out of 193 countries) in the world use VAT. The US uses sales tax and is the only one to do so in the OECD.
  • Asymmetry creates perverse incentives: In sales tax, only retailers charge sales tax and remit sales tax to the government. Consumers want to masquerade as businesses, retailers are not especially incentivized to make sure that their buyers are charged sales tax, and if there are ways to get around sales tax (remote sales, which include cross-state sales and online sales; wholesaling to consumers ), retailers and consumers might want to do that. In the US, retailers don’t need to charge sales tax to consumers buying in a state in which the retailer doesn’t have a physical presence.  In this case, there is a use tax charged on the consumer to make up for this, but compliance of use tax is low.  (Source, archived.)  Estimates of sales tax lost due to remote sales in 2012 varied up to a high of USD $23 billion where total retail sales that year (excluding food sales because many states don’t charge sales tax) was around USD $350 billion a month, i.e. around USD $4.2 trillion that year.  (Archived, archived, archived.)
  • In VAT, all businesses process VAT the same way, so there are no such perverse incentives.
    • The big exception to this is cross-border trade, i.e. imports and exports.  Governments have a choice of whether to charge VAT on goods it exports and goods it imports and whether to charge differently for every country it trades with.  This creates a huge potential for asymmetries in the VAT system.
  • Imports and Exports:
    • Sales Tax countries charge sales tax on imported goods if and when they reach the end-user.  If the imported good is exported again, then it hasn’t reached an end-user, and thus is never sales taxed.
    • Both sales tax and VAT are consumption taxes – the purpose is to tax consumption.  This is why the sales tax doesn’t tax a good that is imported and then exported without being consumed in the country.  VAT accomplishes this as well, but also would ideally keep the cross-border trade situation simple and sensible when dealing with other VAT countries or sales tax countries.
    • In order to have as symmetric and fair a system of cross-border trade, VAT countries generally:
      • Do not charge VAT goods that are exported.  When a good is exported by an exporter, the government refunds the exporter the entire VAT that it paid on its cost of goods sold purchases;
      • Charge VAT on goods that were imported on that good’s first subsequent sale that occurs after importation for the full sale value (not just the value added by the importer, which is [sale price – cost of goods sold], but the full [sale price]). I.e. after the importer imports the good, when that importers sells that good, that sales transaction is VAT-taxed for the full sale price.  This is assuming that the imported good is being sold to another domestic company and not being immediately exported.
      • Note that if an imported good is exported, the government does not receive any VAT.  This is the same as in sales tax and it accomplishes what a consumption tax is supposed to do (which in the case of a good that is imported and then exported without being consumed in the country is to not tax the good).  Each company in the value chain plays its usual part in the VAT system, but the last one, the exporter, is refunded by the government all the VAT it paid on its purchases of cost of goods sold.

Cross-border VAT

  • The reason VAT countries don’t tax their goods upon export is because sales tax countries don’t tax their goods upon export, so this keeps that part of the trade symmetrical.  This also prevents any case of a good being double-taxed during a cross-border trade.
  • The reason VAT countries tax their import goods (subsequent to the import transaction) is because if that good is going to be consumed in the country, not VAT-taxing it would mean the good would be untaxed during and after its cross-border transaction and thus have an advantage over similar competing domestic goods at this and every following point on the value chain since domestic goods have been VAT-taxed up to this point and will be VAT-taxed on all following points on the value chain.
  • The reason why an imported good’s subsequent sales transaction is VAT-taxed its full sales price instead of just the value added by the importer is because:
    • If the good is only taxed by its value-added amount instead, this still is not enough to offset the disadvantage of domestic goods (which have been VAT-taxed for all value that has been added to the product up to that point, not just the value added by the last company to sell it).
    • The government of the country in which the good is consumed ought, in principle, to capture the entire consumption tax on the good.  Thus, when the good went across the border and the exporting country’s government refunds the VAT to its exporter, the good is effectively “untaxed” at this point (the exporting country’s government has refunded all previous VAT on it and the importing country’s government has yet to tax any of the value that has so far been added by producers of the exporting country).  By VAT-taxing it by its full sales price after importation, the government of the importing country captures the VAT that the exporting country’s government refunded to its exporter or “resets” the VAT to where it ought to be at this point in the value chain for itself.
  • Missing trader fraud/Carousel fraud: A type of fraud that exists when a good is imported into and then exported out of a VAT country without the good being consumed in the country.  Since cross-border trade is a point of asymmetry in the VAT system, it makes sense that this is where fraud occurs.
    • Company A imports a good legitimately, paying the exporter EUR 100 for the good.  This transaction is VAT-free.
    • Company A sells the good to Company B for EUR 110.  This transaction is VAT-taxed its full sales price (since it is the transaction subsequent to the good being imported and also is not being immediately exported), and Company A owes the government this VAT.  Note that the good in this transaction is “new” to the country and thus the government has not received any VAT from this good further down the value chain (that VAT has been collected by the exporter’s government and refunded back to the exporter).  If the VAT is 20%, that 20% is charged on the full EUR 110 sale price of the transaction (not the value added by Company A, which is EUR 10).  The total price of the transaction with VAT to EUR 132 and the government expects to be remitted a VAT of EUR 22 from Company A for this sales transaction.
    • Company B exports the good.  This transaction is VAT-free.  Furthermore, Company B has paid Company A a VAT of EUR 22, so it is entitled to a refund of EUR 22 from the government as the good is being exported and not consumed in the country.
    • Company A disappears or goes bankrupt without paying the VAT (of EUR 22) on the sale of goods by Company A to Company B.  This is key to the fraud because Company A is supposed to pay VAT for the full sales price of its sale of the good (20% * EUR 110 = EUR 22), not just VAT of the value added (20% * EUR 10 = EUR 2).  In the diagram above that depicts cross-border trade, the retailer would owe the government $0.15 of taxes, not $0.03 as in the diagrams that are further above that don’t depict cross-border trade.  By Company A disappearing, the government is losing the VAT of all value that has been added to the good by companies from this point and all the way down the value chain.
    • With no fraud occurring, the government is supposed to earn 0 tax: charge VAT starting from the importer selling the good to domestic companies but then refund all that VAT to the exporter at the end who exports the good since the good is not to be consumed in the country.  But instead, in this case where Company A disappears, the government has lost EUR 22 by refunding Company B, the exporter, for the VAT that it paid on its purchase of the good.
    • In reality, if the importer (Company A) and the exporter (Company B) are working together, the good may never even physically leave the port, and is imported, sold, and exported only on paper.  If there are many companies in between Company A and Company B (e.g. it could instead by A -> X -> … -> Z -> B), it could be difficult for the government to prove any wrongdoing by Company B as the link between A and B will be weak (Company B may even be innocent in some cases where the only fraud is Company A disappearing without paying its VAT) and thus the government will be obligated to refund the VAT to Company B that it paid on its purchases.
    • According to sources found in Wikipedia, it’s estimated that the UK annually lost around GBP 2 billion in 2002-2003 (archived) and between GBP 2 billion and GBP 8 billion annually for the years (archive) between 2004 and 2006 due to this kind of fraud.  Total UK retail sales (archived) for these years was around GBP 250 million to GBP 300 million.  For the EU, 2008 estimates were EUR 170 billion lost (archived) due to this type of fraud.  Total EU-27 retail turnover in 2010 was around EUR 2.3 trillion (archived).
    • In the above link to a BBC article from 2006, it says that in order to combat the losses from this fraud, the government is implementing a new system where:

      Under the new rules the last company to sell on goods like mobile phones – such as a retailer – will be responsible for paying the VAT.

      So it sounds like the portion of VAT that the government is “missing” from the imported good will be paid by the retailer instead of the importer – in other words, it’s a bit like the sales tax system.  In this system that’s described, if a good is imported and then exported, the amount that the government would refund the exporter will be smaller than in the previous system, making the fraud much less damaging.  And if the retailer disappears without paying the taxes it owes, that’s the same as a retailer disappearing in a sales tax system without paying taxes, or a raw materials producer in a VAT system disappearing without paying taxes.  (These cases are a much simpler sort of tax evasion and unlike the missing trader/carousel fraud where the importer disappearing and not paying the “missing” chunk of VAT that the government is owed is in combination with the exporter that is “refunded” that chunk of VAT from the government, even though the government never received that chunk of VAT from the missing trader.)

Instead of taxing the importer the “missing” VAT, tax the retailer the “missing” VAT

In Country A, the government refunds the manufacturer/exporter the VAT that it paid on its purchases.  In Country B, the importer is charged VAT only on its value added, and that VAT is remitted to the government.  The retailer is charged VAT on its value added (10% * $0.20 = $0.02) and on the “missing” VAT from the value added to the product prior to importation, which is $1.20 (the price that the importer paid the manufacturer/exporter), so that comes to 10% * $1.20 = $0.12.

If the good is exported, refund the exporter as usual

If the good is imported into Country B and then exported without reaching a consumer, the amount of VAT that the importer is charged is only on its value added (10% * $0.10 = $0.01).  Thus, the amount of VAT that the government refunds the exporter is only that amount ($0.01).  If the importer disappears, the government only loses $0.01, which is 10% of the value added by the importer, not 10% of the full sales price of the product at this point.  Furthermore, if the importer and exporter work together and the importer sells the good to the exporter at a much higher price (raising the value added and the potential VAT that the government is supposed to refund the exporter), the exporter still needs to legitimately export the product in order to qualify for the refund.  It’s theoretically possible for the exporter to operate at a loss to make this possible, but this might raise an additional red flag that the government would become suspicious of, raising the risk of doing the fraud.


 

Economic Impact of Sales Tax versus VAT

Back to the diagrams for no tax, sales tax, and VAT:

No tax

Sales tax

VAT

Although only the consumer actually pays the tax in the end, as prices are raised at other transaction points, there is some friction that will discourage those transactions by some amount.

One can also think of this as an overhead cost for the businesses involved.  In the no tax and sales tax situations, the manufacturer buys goods at $1.00 and sells them at $1.20.  In the VAT situation, the manufacturer buys goods at $1.10 and sells them at $1.32, which minus the VAT remitted to the government becomes $1.30.  In both cases, the manufacturer earns a profit of $0.20, but there is an overhead of $0.10 in the VAT case.  An extreme analogy is: if you are a company that makes a profit of $1 on each good you sell, would you rather buy goods for $2 and sell them for $3 to earn your $1 profit or buy goods for $1,002 and sell them for $1,003 to earn your $1 profit?  Surely the former is easier and has less friction.

Back to the economic interpretation: if the demand and supply curve of a transaction point in a no tax situation is this:

 

then by adding a tax to the transaction, the price is increased.  For convenience, we add a second supply curve that is “supply + tax.”  While the end result is the same if we left the supply curve alone and added a “demand – tax” curve instead, the supply + tax curve more conveniently takes the hypothetical price of a product sold (the supply curve) and then adds the hypothetical price + tax of a product sold (the supply + tax curve).  What one can also do instead of drawing a new curve is take the vertical distance of the final tax per product and “fit it in between” the two curves from the left side of the diagram, and the end result will be the same.

 

(The diagram says “Consumer Surplus” but since this may represent a business-to-business transaction, it’d be clearer to just say “Purchaser Surplus.”)  So a tax will cause less quantity to be transacted, a higher post-tax price, some government tax revenue, lower purchaser and producer surpluses, and some deadweight loss.  Unless the government tax revenue is spent in a way that can overcome that deadweight loss (e.g. spending on things that have positive externalities), we have an inefficient outcome.

So in the VAT system, businesses are contending with higher prices (which is like more overhead) and lost quantity transacted compared to the sales tax system.  This is another cost that the VAT system pays (in addition to the missing trader/carousel fraud) in order to have a “symmetric” system where almost everyone in the value chain pays and collects VAT.

 

Testing MathJax-LaTeX

At first, we sample $$f(x)$$ in the \(N\) ($N$ is odd) equidistant points around \(x^*\):

\[
f_k = f(x_k),\: x_k = x^*+kh,\: k=-\frac{N-1}{2},\dots,\frac{N-1}{2}
\]

where \(h\) is some step.

Then we interpolate points \((x_k,f_k)\) by polynomial

\begin{equation}
\label{eq:poly} \tag{1}
P_{N-1}(x)=\sum_{j=0}^{N-1}{a_jx^j}
\end{equation}

Its coefficients \(a_j\) are found as a solution of system of linear equations:
\begin{equation} \label{eq:sys} \tag{2}
\left\{ P_{N-1}(x_k) = f_k\right\},\quad k=-\frac{N-1}{2},\dots,\frac{N-1}{2}
\end{equation}
Here are references to existing equations: \ref{eq:poly}, \eqref{eq:sys}.
Here is reference to non-existing equation \eqref{eq:unknown}.