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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...

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  • $\begingroup$ Is this rate the nominal or the real rate? I'm wondering if real rates could ever possibly be negative, wouldn't this contradict the fact that money has time value? $\endgroup$
    – Vim
    Nov 26, 2018 at 12:24
  • $\begingroup$ @vim Right. So this would be the nominal interest rate as appears in arbitrage proofs as in e.g. Shreve Stoch Calc for Finance I, p. 2. There he says: ~ ‘r usually > 0, but only r > -1 is required’... We don’t usually consider inflation in arbitrage arguments (Should we?). $\endgroup$
    – Don Slowik
    Nov 26, 2018 at 13:59
  • $\begingroup$ You shouldn't include inflation in arbitrage arguments and real rates have been negative. Nominal rates have been negative and we haven't had deflation in the EU, at least officially. $\endgroup$
    – Bob Jansen ♦
    Nov 26, 2018 at 14:30
  • $\begingroup$ I see that inflation is a non-issue in using arbitrage to set up equivalent portfolios and thereby pricing. It may be an issue in deciding if some money making scheme is worthwhile over time. $\endgroup$
    – Don Slowik
    Nov 26, 2018 at 15:16

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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.

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

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

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  • $\begingroup$ So within practical market considerations (that may even include inflation; see@vim above), small deviations from strict arb free limits, in this case r > 0, happen. But how can Shreve state only r > -1 is required? I think if r ~ -0.8 or so, mattresses are big enough. $\endgroup$
    – Don Slowik
    Nov 26, 2018 at 14:19
  • $\begingroup$ I don't fully understand your comment, might it be a new question? $\endgroup$
    – Bob Jansen ♦
    Nov 26, 2018 at 14:30
  • $\begingroup$ Well if I have proven that r < 0 results in arb and is therefore disallowed, the fact that r < 0 exist would be an illustration that some of the arb assumptions are being stretched. E.g. borrowing 100 to make 0.5 in one year at r=-.05 is not worth the risk a mouse eats the bill. But if r=-.80, I would take that risk to make the 80 per year. We will never see r so small that people actually start trading this strategy on it. Any real arb strategy has to deal with all the issues, size of safe mattresses, as you point out. There may be another Why r > -1? question. $\endgroup$
    – Don Slowik
    Nov 26, 2018 at 15:30
  • $\begingroup$ I think that is outside of the scope of this question and deserves a new question. $\endgroup$
    – Bob Jansen ♦
    Nov 26, 2018 at 15:51
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  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.

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  • $\begingroup$ 1. Yes, if the bank allows deposits at r = 0 while I can borrow at r < 0. $\endgroup$
    – Don Slowik
    Nov 27, 2018 at 14:10
  • $\begingroup$ It seems to me that inflation does not enter the picture if you are using no arbitrage to set up an equivalent portfolio to price a derivative. It may enter the picture when you have identified an arbitrage opportunity, but only if the trading strategy requires an initial investment up front - then ur profit must beat inflation. The strategy I suggest in the OP requires no initial capital and is therefore immune to inflation - yes the buying power of the profit is reduced but you still realize a free profit. $\endgroup$
    – Don Slowik
    Nov 27, 2018 at 14:18
  • $\begingroup$ @DonSlowik: The point there is that if you start adding security to your mattress in order to lower the risk of just losing the money to theft, the thing you will be building is a bank. So for the same reasons banks aren't offering zero, you won't obtain zero. $\endgroup$
    – Phil H
    Nov 27, 2018 at 14:55
  • $\begingroup$ @DonSlowik: The inflation comment was just to point out that real rates have been negative for some time, so any investment strategy which even comes out flat in real terms would beat deposits, even when rates are positive. People treat negative rates like some floor, when it's both nonexistant and arbitrary. $\endgroup$
    – Phil H
    Nov 27, 2018 at 14:57
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There is no mattress under which you can park $100m this is your answer in a nutshell. Switzerland is a very good example.

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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.

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    $\begingroup$ Your 4th paragraph reminded me of this article where the 'How?' section refers to '‘zero-interest-rate bonds’, or coins and banknotes as they’re more commonly called..' as allowing negative rates to provide an arbitrage opportunity. It also points out that electronic coinage could be programmed to decay with time removing the arb. $\endgroup$
    – Don Slowik
    Dec 5, 2018 at 2:13
  • $\begingroup$ This is indeed an interesting and thoughtful article! Thanks for sharing. $\endgroup$
    – g g
    Dec 5, 2018 at 10:46
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Check out Reverse-Repos. What you are describing is exactly what is done in high finance.

Let's say I have 100 mln Swiss Francs. I then buy a Repo-Eligible Bond that yields more than -0.75%/2. Then I post this bond as a collateral for an overnight loan and will have to pay a negative 0.75% (which means I get paid). I immediately invest the proceeds in the same bond. Lets assume the bond yields 0%:

100mln bond at 0% 100mln loan at -0.75% 100mln bond at 0%

This will leave me with a positive 0.75% on my investment. Only thing I will have to do now is hedge credit and duration risk.

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  • $\begingroup$ How to reconcile "buy a Repo-Eligible Bond that yields more than -0.75%/2" and "Lets assume the bond yields 0%:"? $\endgroup$
    – Confounded
    Jun 21, 2019 at 16:45
  • $\begingroup$ What I was saying that you should look to buy a bond that yields more than -0.375% (half of the current CHF Repo Rate). --> x > -0.375% Then you post this bond as a collateral and you will have to pay -0.75% for the equivalent credit you recieve (which means you get paid 0.75%). --> y = -0.75% if you now invest this proceedings in the same credit, you will hold: 2*x agains 1*y which by definition is positive. If you are able to get your hands on a bond which yields significantly more than -0.375% then your yield will increase correspondingly. $\endgroup$
    – R. Steigmeier
    Sep 3, 2019 at 15:33

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