Gfrktyl

Given a security having price $1$ at time $t=0$, and price $1+r$ in all world states at $t=T$, i.e., a risk-free bond with interest rate $r$. At $t=0$, short this and also buy an other risk-free bond. At $t=T$, sell the bond and use the $1+r$ to pay back the loan. So this strategy has no arbitrage even if $r < 0$. However if you take the unit of currency obtained from shorting the bond at $t=0$ and stuff it safely under a mattress until $t=T$, it will then be worth more than the $1+r$ you must pay the bank at $t=T$ if $r<0$. So with $0$ invested at $t=0$ a risk-free profit(arbitrage) of $|r|$ is realized at $t=T$ if $r<0$.

So it seems that the existence of a safe mattress provides a natural floor of $0$ to the risk-free rate...

As a practical aside on a large scale, I have heard the rumours of European banks and even a consortium of banks considering plans to build an ultra secure deposit facility for cash, and also the ECBs push back for doing so based on an unwillingness to actually provide physical currency. I have never heard about the actual realisation of any of these rumours and I suspect due to the following reasons...

What is the cost of building, operating and controlling a secure facility for cash? I have no idea, but I would suspect if you planned on housing at least a billion physical Euros it would cost in excess of 10 million Euros to construct, nevermind security overheads and the risk of transporting large quantities of Euros back and forth between destinations must account for some average loss due to criminality. So even for two or three years the pickup of the 0.4% carry due to interest rate differential would probably not recoup this cost.

Added to that the availability of physical cash to make the endeavour worthwhile doesn't exist. The ECB balance sheet shows in 2017 there is 1.2Tr banknotes in circulation roughly EUR2,500 per citizen, while there is 1.9Tr (average of ~2bn per institution) on electronic account central bank accounts. So given all these broad numbers it doesn't seem to be a practical, flexible or profit generating endeavour to undertake. For large amounts of cash there are simply no other ways of allocating cash therefore, either than to central bank deposits, short term government bonds, or collateralised repo, all at negative rates.

On a smaller scale, bank institutions offer the financial compensation scheme, and often (in order to attract cash capital for their regulatory capital ratios) banks will offer accounts for these smaller deposits at zero percent anyway, so it is the same effect as a mattress and safer. As a smaller investor you are often subject to large bid offer spreads, no economies of scale on brokerage costs and so it all comes out in the wash.

What seems like arbitrage on paper, probably ends up being a big waste of time.

Indeed, interest rates have been below zero and your logic appears sound.

My conclusion: safe mattresses that are large enough don’t exist.

1. A bank is the mattress you speak of


2. They all have a credit risk


3. Even in the absence of taxes on deposits, rates have still been negative in Euroland for some time, in a multi-trillion Euro market

There is no magic mattress, and no zero floor. In reality inflation has been higher than base rate for years in Europe, resulting in an implicit negative real rate long before the headline rate was negative.

The only real mechanical problem with negative rates is that a negative coupon requires the holder to pay, so they always set the coupon at or above zero and floor inflation-linked bonds. The yield can still be negative with a positive coupon.

There is no mattress under which you can park $100m this is your answer in a nutshell. Switzerland is a very good example.

I like this question a lot! It shows how prejudiced and short sighted we all tend to be and how far financial risk management has to go to cover all bases. Furthermore it is a nice exercise in reconciling market theory with market practice.

A few years (say 10) ago most people considered a floor at r>=0 as being rather obvious. Textbooks stated the possibility of negative rates as major drawback of Gaussian short rate models. This was based on "obvious" arbitrage arguments as the OP made. But practice has shown that this kind of arbitrage is not possible, it is based on lazy thinking.

To reconcile what is going on with theory, imagine a mid-sized institutional investor with 1bn in interest rate sensitive investment. Bank deposits are a no go because banks will refuse the investment since they are charged with negative rates as well. So cash is the obvious way out. But 1bn in cash raises some serious practical questions. How do you transport it? Where do you store it? What about fire and vermin? Is there insurance? All this is not clear but there is one certain thing: Cash is expensive. Swiss pension funds (being charged up to -75bps!) seriously considered this move. Rumor has it that a few larger ones had already scouted out decommissioned army bunkers in the Swiss Alps for storage. (See attached article in German). But this never materialised, which shows that the cost of holding lots of cash is larger than 75bps. I estimate this cost somewhere between 1% to 2%, depending on a lot of factors, such as the size of the operation.

So does this establish a floor at say 2% by arbitrage? No, this would again be lazy thinking. Such a move would create a counter reaction from the central bank. It would either outright refuse to give you the cash (there have been legal opinions in what jurisdictions this is possible or not), issue small bills only or - most radical, most effective - outlaw cash and move to electronic money only.

Well, does this show there is no floor? You guessed right: No, you can't conclude there is a floor. Because faced with outrageous negative rates (for example -10%), investors would consider alternatives, even though they would not match their investment profile. For example, they might start buying gold. Then again, government can restrict your ability to purchase and own gold, then those funds would move to copper and so and so on.

Lesson learned: When stuff hits the fan, a lot can and will happen, that seemed impossible before.