Vrftjkry

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.