Last updated 10-9-02


Puzzle: How the Euro can make you rich

Suppose you have two accounts, one in Dutch Guilders (NLG) and one in Euro (EUR). If you transfer money from one account to the other, the amount will automatically be converted using the rate of 2.20371 guilders per euro. Note that the amount will in general have to be rounded off. For instance, suppose you transfer 4 guilders from the NLG account to the EUR account. The bank will round off EUR 1.815209... to 1.82. When you transfer back EUR 1.82, the bank will round off NLG 4.0107522 to NLG 4.01. Thus you have made a profit of 1 cent! The profit per transaction is 0.5 c.

You can do better than that. Suppose you transfer NLG 4 seven times. The balance of the EUR account will be 7 times EUR 1.82 = EUR 12.74. If you transfer that back you end up with NLG 28.08. The profit is 8 c with only eight transactions so that's already 1 c per transaction! Now here's the puzzle:

What is the least upper bound of the profit per transaction you can achieve with a scheme as above.

Here are the rules:


If you can figure this out, you could also have a go at this:

Prove that no valid scheme can have a profit per transaction equal to the least upper bound referred to in the puzzle above.

In practice

You could try to send a large number of transfer orders to your bank automatically.

However, it is unlikely this will actually work. Your bank would probably limit the maximum number of transactions per day. Also, how much does it cost you to email the bank every time? Will you be flooded with bank statements? Does the bank charge you for every transaction? And anyway, in 2002 the whole thing is over of course ...

Capital gains tax

Irish capital gains tax has to be paid on profits made on any trades. If assets are held for at least one year, the initial amount paid may be multiplied by a factor reflecting inflation in each year. This is called indexation allowance.

Losses can be carried forward and offset against gains in subsequent years. There is also a tax free allowance. For simplicity, assume that this is already used in each year.

Capital gains tax is due only when assets are actually disposed of. There are some special cases which should also be regarded as a 'disposal':

Options scheme

Suppose an investor holds certain shares and wants to avoid capital gains tax. Assume that a wide range of call and put options are available with those shares as underlying value. The idea is to buy a number of call options and write the same number of put options, both with an exercise price equal to the current share price, and one year until expiration. If it is not closed before expiration, this position gives the same profit (or loss) as holding the shares, except that one does not lose the interest on the money needed to buy the shares. Thus the net price has to be roughly equal to the interest on the underlying value. This also follows from the Black-Scholes option price model, as can be derived easily (the price is equal to the interest rate plus higher order terms, representing interest on interest etc.). If one applies this method in a number of consecutive years, and to a range of different stocks, losses in some years can be offset against profits in other years.

To summarize, the strategy amounts to holding the following position:

Below we will compute the value of x, the underlying value of the long call-short put combination which eliminates the effect of capital gains tax.

The following variables will be used:

Rates are expressed as fractions (e.g. r=0.04 means interest is 4%). Fees are not taken into account.

Capital gains tax is computed as follows:

If the initial investment is y and the profit is z, then tax is u(z-y(1+a)).

Note that if p < a, no tax is due so the scheme does not apply. Consider the case p > r+ca. The put options are worthless at expiration and a profit is made on the long call-short put combination (the call options may or may not be profitable separately). Assume that the shares are sold when the options expire and no dividends are paid.

Leaving out any quadratic and higher order terms in r we get the following results:

Now equate the final amount to 1+p:

x(p-u(p-r-ca)) = u(p-a)


x = u(p-a)/(p-u(p-r-ca)).

Note that for r=a=0, x = u/(1-u) = 1/(1-u) - 1, which is obvious: this is equivalent to investing 1/(1-u) and paying a tax of u over the profit, resulting in a total profit of (1-u)p/(1-u) = p.

Typical values of the variables are r=0.04, u=0.2, a=0.03, p=0.08 and c=0.2. In this case x=1/7.32=0.137.

Unfortunately, the value of x depends not only on the interest rate but also on the return on the investment! In particular, if p is only slightly bigger than r, the scheme will not work. However, for large p, r and a are relatively small and x will be close to u/(1-u).

Of course, the situation is simplified here. For instance, buying the options also means losing interest. Dividends also change the situation. A limitation is the fact that one option contract corresponds to 100 shares, so if one only has a small number of shares, say 200 or 300, it is not possible to use the precise computed value of x. On the other hand, since the value of p can not be determined in advance, the precision of x is not very important anyway.

A similar strategy would be just to buy more shares. Theoretically, both ways should be equivalent. There are some differences. For example, when borrowing money with shares as collateral, the maximum credit percentage allowed is independent of the stock in question (typically 60%), while the margin requirement for derivatives depends on the underlying value. Also, the fact that the indexation allowance does not apply to short positions has some influence. The fees are negligible and are probably about the same.

Note that there is no options exchange in Ireland! Therefore, this does not work with shares listed only in Ireland.

Dutch capital tax

NB: The tax system was changed in 2001. Therefore this scheme is no longer applicable.

In the Netherlands, one has to pay a certain percentage of one's capital as tax (it was recently changed from 0.8% to 0.7% p.a.). There is quite a large allowance. For the sake of argument, let us assume the allowance is about EUR 100000.

If you own shares, you may round off the price of those shares to integer Euro values, so EUR 12.55 becomes EUR 12, for instance. I am not sure if this also applies if the share price is less than one Euro (probably not, let us assume that no round off is allowed then).

Here's the trick: set up a mutual fund that does nothing else than issue shares of EUR 1.99 and keep the invested money on an ordinary account. Suppose a client has a capital of EUR 1 million. He would normally have to pay EUR 6300 per year (0.7% of EUR 900000). Give the client a EUR 1800000 loan to invest in the shares. He can then buy 1800000/1.99=904522 shares, on December 31. His total capital is then still EUR 1000000, but from a tax point of view it is actually EUR:

The 'normal' capital has a book value of, say, EUR 900000. The 'shares' are valued at EUR 1800000, but because the share price may be rounded off to EUR 1.00, their value for tax purposes is only EUR 904522. The 'loan' is a debt of EUR 1800000, so adding it all up we obtain a total capital of EUR 104522, only EUR 4522 more than the allowance, leaving a tax bill of only EUR 31.65 instead of EUR 6300.

On January 2, the client sells the shares again (note that all markets are closed on New Year's Day). You have to make a profit yourself of course, so you charge e.g. EUR 2000 for transaction and administration costs and interest on the loan. The client still has an advantage of almost EUR 4300. Can you give a client a loan of EUR 1800000 if he only has EUR 1000000? Depending on the type of assets he has, you probably can. It is customary to allow 60% credit on a stock portfolio, 70% on bonds and more than that on property. Cash can of course be used directly, but it may be more convenient to give 100% credit on it instead. Suppose the client's portfolio is such that you can give him a loan of EUR 750000. He buys shares for this amount, and then you can give him EUR 450000 credit again with those shares as collateral (60% of EUR 750000). This can be repeated a number of times, to obtain a loan of EUR 1800000 in total.

(An (easy?) puzzle: what is the largest amount you can reach?)

Some advice

A common misconception about investing money is that there are investments without risks. No matter how you invest your money, there is always a risk of losing it. Even if you keep the money on a standard current account, you can lose it because the bank can go bankrupt, or through inflation.

Keeping your money in a bank account is of course relatively safe. It is very unlikely that any major European bank will go bankrupt in the next few years, and you don't have to worry much about inflation.

However, you never know what might happen in the future and many people have been eqully confident about their financial situation when it turned out that things were about to go wrong very dramatically.

There is no need to be too pessimistic and afraid, though. If you have some money that you don't need for a while and you are not one of those people who have sleepless nights about losing their money, it is commendable to invest in assets that are likely to generate a higher return than a savings account, especially when intrest rates are very low.

Some tips:

Do not think you can get rich quickly. You can if you take big risks and are very lucky but it is unlikely you will succeed.

Invest for the long term.

Do not bet everything on one (wrong) horse.

Do not borrow money to invest. In principle the expected gain may be more, but because it can go terribly wrong in a falling market, this high gain may be lost entirely.

The higher the expected return, the higher the risk (must be). Do not believe in schemes which promise high returns with low risks.

If you have a lot of time, attempt high risk investments. If you don't have time or don't want to wait (for instance, you want to invest for your retirement, which is only 5 years away), do not take too much risk. If it goes wrong, you will not have enough time to recover.

10 ways to fool your investors

Internet companies tend to be only good at one thing: marketing. Below are some of the tricks they use to hide their disastrous financial position.

For a long time, it was expected that their turnover would continue to increase exponentially long enough for them to make a profit eventually. Few people realised that this can not go on very long. For instance, suppose a company has 100 employees and this number doubles every year. After 26 years the company would employ the entire population of the world!

Some internet companies will only start making a profit when a billion people use their site every day ... The purpose of most of the tricks is to make it look like turnover is much bigger than it is in reality. It is not that bad to be losing money, as long as your turnover is big enough to let a reasonable improvement in the operating result turn your loss into a profit.

1 Book future income in the current year

If you sign a contract with a supplier for the next 5 years, book the entire income as turnover for this year.

On the long run, this works only if you sign more and more contracts every year.

2 Give huge discounts

Suppose you sell something for a 100 dollars. You say that the normal price is 200 dollars, and you are giving a 100 dollar discount. You book a 200 dollar turnover and 100 dollars for 'marketing costs'. Thus your turnover seems twice as much as it really is.

3 Book fake turnover

Suppose you produce windscreen wipers. You sell them for 5 dollars, but your production costs are 15 dollars. Thus your loss is huge compared to your turnover. However, instead of just the windscreen wipers you book the entire car as turnover (say, 20000 dollars). This way it seems like your loss is only one 1000th of your turnover and a slight improvement will turn your loss into a profit.

4 Depreciation

It is very hard to estimate how fast assets depreciate. Letting your assets hold on to their value a bit longer may improve your balance sheet.

5 Valuation of assets

Similarly, you can pretend your assets are worth more to begin with, especially things like databases.

6 Swapping banners

If you put a banner from another company on your website, and that company puts a banner from your company on their website, you can pretend you charge each other a million dollars for this and add that to your turnover, even though no money was ever exchanged.

7 Considerations

Send a free brochure about your company to IBM and then announce you are considering co-operation with IBM. More generally, you can of course 'consider' lots of other things ... The fact that none of these are realistic can easily escape the attention of investors.

8 Announcing your results

Suppose you make a profit of two million dollars. Send out a press release saying that your expected annual profit will be only 1 million dollars. A few weeks later, announce that your profit is a 100% higher than expected. With a bit of luck, no-one will remember that 6 months earlier, you actually indicated that you expected a 10 million dollar profit.

9 Offering free services

Offer free services but list the companies you offer these to as customers.

10 Impressive customers

Sell a paper clip to a Microsoft employee and then boast that Microsoft is one of your customers.

Financial psychology: the EURO exchange rate

The cost of conversion to the Euro amounts to many billions (of Euro, pounds, guilders ...). Cash registers have to be adjusted, ATM's, computer systems, a lot of people have to work overtime, etc.

There would be one way to cut the cost considerably. I have never heard anyone mention it, though it seems fairly obvious to me and I think one may draw interesting conclusions from the fact that no-one would ever even suggest it.

It's very simple: consider the separate nations that participate in the EMU. About 60 million use the French Franc, 15 million use the Dutch Guilder, etc. The largest nation is obviously Germany, with almost 80 million people. What if these would not have to change anything except the denomination of their currency?

A huge amount of money would be saved if the Euro would be worth exactly one Deutsche Mark, and even more if Europe would simply use the Deutsche Mark instead of their own currencies.

It would not be fair if the Germans would be the only people to benefit from this. One way around it would be to estimate the benefit, and let Germany pay a reasonable part of that benefit to all the other states.

Clearly, no politician would dare to suggest something like this.

Why not?