As of today, the ECB is entering unchartered territory after having lowered the overnight deposit rate to -0.1%. Negative interest rates have been a subject of inquiry for me throughout my career. Having been a strategist for the better part of my professional life, my focus was typically on detecting relative value/arbitrage opportunities. Here is a short run-down of typical encounters I had with this topic.

## Yen Carry Trade of 1998

In November of 1998, I was a sales person in a Listed Derivatives group, looking out for dislocations in derivatives space. Back then, the market was not as efficient as it is today (lacking the computer power of high-frequency trading) and with a good understanding of the Futures Delivery Option and some Excel skills, one could extract a few basis points here and there. At that time, because of negative interest rates in Japan, an interesting situation had occurred. One could buy Japanese Yen (and sell US Dollars) as of December 16, 1998 at an exchange rate which can be locked in with December ’98 Dollar-Yen Currency Futures and sell the Yen as of March 17, 1999 at the rate locked in through the March ’99 Dollar-Yen Futures, earning an implicit interest rate of LIBOR+16 bps on the Dollar investment. This is under the assumption that one is able to hold the Yen interest-free in a cash account. This trade is an investment into a risk-free asset (cash) and the only counterparty or credit risk is versus the Clearing Organization of the Futures Exchange. At the time, LIBOR+16 bps were actually about 100 bps above a comparable risk-free investment into Treasury Discount papers. This is probably as close to the textbook-definition of “arbitrage” as it gets, and certainly counts as an example for the somewhat weaker definition of “relative value”.

The risk of this Yen carry trade was that the Yen could not be deposited at a zero interest rate rather than at a negative interest rate. In fact, I remember trying to quantify the costs of depositing cash in large quantity. One can put money into a bank deposit box (to avoid negative interest rates on checking accounts), but those boxes are not insured. We tried to get a sense of how much an insurance on valuables in a deposit box would cost, and also, how much it costs to pay for an armored vehicle to transport the money back and forth. In the end, the relative value proposition was not attractive enough for our clients to enter the trade in size, but it was a good foretaste about how creative one has to become, once interest rates turn negative.

## Repo distortions of 2003

Five years later, while being a Derivatives Strategist with another bank, client focus turned on the U.S. repo market. August to November of 2003, the U.S. got its own first taste of negative interest rates as some U.S. Treasury securities experienced negative repo rates. In a repo transaction, money is lent to another party for an agreed-upon interest rate (repo rate); the short-term loan is essentially collateralized by high-quality securities. According to rules in place at the time, if the recipient of the collateral is not able to return the collateral on time (maybe because he sold it and is not able to buy it back in the market due to a squeeze), the loan would prolong for the time until the collateral is handed back. As a penalty for the so-called “fail”, the loan no longer generates interest income. Typically, this is incentive enough for the recipient of the collateral to make delivery on time (in order to avoid giving an interest-free loan).

However, when interest rates are negative, being put into a loan contract at 0% is a good deal. Thus, what started to happen was that “strategic fails” started to make sense. Based on a loan with an initially-set negative (repo) interest rate, the lender (of money) would strategically fail to return the collateral at the end of the term-repo contract, and then benefit from a more attractive interest rate (0%, instead of negative) going forward. It was clear that at this point additional penalties had to be implemented for fails. Indeed, guaranteed-delivery special collateral Repo contracts with negative interest rates had been introduced for the first time.

This was another eye-opener what negative interest rates can do to rational market behavior. It reminds me of how you can force money at a zero percent interest rate on your government (i.e., a risk-free investment) by strategically over-estimating your expected income in the current tax year (e.g., claiming to expect capital gains from investments abroad), triggering the IRS/Finanzamt to demand estimated tax payments throughout the year (which will net out in the following year once the tax declaration/Steuererklärung is submitted).

## Limitations of Bond Math

Also in 2003, many of us went back to Bond Math 1.0.1, to realize that negative interest rates, just like negative probabilities, are something most pricing/evaluation models had a hard time to deal with. Just input a negative interest rate into a Black-Scholes Option Pricing framework and watch the model blow up. The lognormal distribution is inapplicable when interest rates go below zero. For some market participants, the problem was even more basic: Their input screen would not even allow for negative numbers to be processed. It was the Y2K-equivalent for the financial markets, with IT groups, backoffices and front-end developers struggling to enhance existing systems.

The devil was in the detail. Assume, for example, Treasury securities with a negative coupon. Not only would the government have to go through the administrative nightmare of “collecting” the negative interest payment from the buyer of the security, also would the government face “credit risk” vis-à-vis the buyer of its securities, as far as the negative coupons are concerned. Because principal and interest payments can be stripped apart, the principal payment at bond maturity would not necessarily act as collateral for the negative coupon payments.

Most mathematical “problems” could be dealt with eventually, if only by letting go of old believes, such as that interest rates, probabilities, swap spreads etc. could not be negative. In fact, there are now publications on this subject (e.g. “Negative Probabilities in Financial Modeling” by Burgin/Meissner).

For me most relevant as a Derivative Strategist was the move away from models assuming interest rates to be lognormally distributed towards those based on normal distributions. At the time, we already used so-called “mixture of distribution” models, which would price a derivative first with the assumption of lognormality and then again with a normal assumption, finally defining the fair-value price to be that of a mixture of both results. For example, we would find that an option is best priced with a 50-50% mix. Soon, the mix was close to 100% normality vs. 0% lognormality. Eventually, we had a situation we called “super-normality” in which you have an above-100% assignment of the normal model and a sub-zero% assignment of the lognormal model. Super-normality assumes a distribution with even fatter tails (mostly on the left side, representing negative interest rates) than a normal distribution.

## Negative Swap Spreads of 2008

Related to negative interest rates is the phenomenon of negative swap spreads, materializing in the U.S. in 2008. Swap spreads, in a simplified explanation, measure the difference between an uncollateralized inter-bank loan and a risk-free investment. Banks lend to each other in the interbank market at LIBOR, and the extension of the LIBOR curve for longer maturities is called swap curve. The Treasury yield, on the other hand, is the interest for a risk-free investment (as the U.S. government has the power to avoid default by printing money – this is not true for all national governments in Europe, though). Thus, one expects the swap curve to always trade above the U.S. Treasury curve, and swap spreads to be positive as a consequence. In 2008, selected swap spreads turned negative, what was initially perceived as inconceivable, or even mathematically impossible. Responsible for this phenomenon was the aftershock of the Lehman crisis and the expected increase in Treasury supply. Also, the funding market was impaired, so that it became increasingly difficult to “arbitrage out” this dislocation (a relative-value trade would have required to purchase a 30-year Treasury, pay fixed on a 30-year swap and to fund the Treasury in the repo market for up to 30 years, or at least until swap spreads would have widened out again; few market participants had enough balance sheet for such transactions).

Those are only a few examples of how I encountered negative interest rates, and its effects, in the past. In each and every case, they led to questioning of conventional wisdom held by market participants, allowed for fascinating discussions and led to improved models/frameworks. For those reasons alone, I am sure we are about to enter another interesting episode that will intellectually challenge our (perceived) understanding of the interest rate dynamics.

## Torben Wiesner

Interesting article. However, I wonder why negative interest rates are not recognized in the mathematical models. If I have a look at the basics (e. g. http://www.insight-things.com/distribution-of-stock-return), then -10%=0.9 should be applicable, shouldn’t it?

## Fidelio Tata

Hello Torben und thanks for your comment!

Indeed, when pricing stock options (as in your link), the interest rate is not needed for the distribution modelling of the underlying (and is merely used for discounting purpose of future cash flows). But for interest-rate option pricing, interest rates enter into the model twice. For discounting, again, but then also to explicitly define the underlying. For example, entering „-10%“ as the strike rate of a swaption, when swap rates are modeled to be lognormaly distributed between zero and infinity, will not yield meaningful results.

Maybe I should have been more specific and make clear that I discuss the case of an interest-rate option.

Best, Fidelio