Rates Radar. Dealing with negative rates. Interest Rate Strategy. 7 March For important disclosure information please see pages 9 and 10.

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1 Interest Rate Strategy Rates Radar Dealing with negative rates Like with an unexpected house guest who won t depart, markets will have to live with negative interest rates for the foreseeable future. In this note we answer key questions about the lower bound for markets, distortions and side-effects arising from rates below zero and valuation implications for interest rate products. 7 March 2016 In pursuit of the lower bound: Estimating the effective lower bound for interest rates is more of an art than a science. Considerations about cash holdings in the euro area and Switzerland as well as implications at banks leads us to believe that a soft border should emerge close to -0.75%. (p2) Distortions from negative interest rates: We summarise the main distortions and negative side effects arising from a misallocation of capital, problems for long-term savers as well as legal and technical frictions. (p4) Pricing implications: Interest rate models need to be adjusted when payment streams become uncertain and expected rates fall below zero. Floaters for instance behave more like zero-coupon bonds. We sketch hands-on pricing implications and how market participants need to adjust their toolkit. (p6) Six years beneath zero - more than a temporary aberration 1m EONIA swap and forwards in bp Analysts For important disclosure information please see pages 9 and 10. research.commerzbank.com / Bloomberg: CBKR / Research APP available Christoph Rieger Prof. Jessica James Michael Leister, CFA michael.leister@commerzbank.com Benjamin Schröder benjamin.schroeder@commerzbank.com

2 More than 10trn of global bonds trade with a negative yield and as the ECB is poised to move deeper into negative territory on 10 March, forwards don t expect positive rates until 2021 (frontpage chart). With negative interest rates thus becoming more relevant for a broader group of market participants, we explain the wider consequences as well as hands-on pricing implications for interest rate products where options and floaters are most affected. In pursuit of the lower bound To be sure, there is no fool proof scientific approach to derive the lower bound. The following considerations about the cost of holding cash and about banks lead us to conclude that the lower bound has not yet been reached at an ECB depo rate of -0.3% while the emergence of certain avoidance strategies in Switzerland suggests that a soft lower bound should emerge close to -0.75%. Cash considerations The euro pallet approximation The simplistic textbook assumption about the zero lower bound has been overhauled. The nonnegativity assumption of interest rates was based on the trivial consideration that physical cash carries a zero yield and investors could simply switch into banknotes to avoid negative interest rates. The practice, however, looks different. Those investors that are facing negative interest rates cannot easily convert their sizable holdings into cash, or are facing a significant cost or risk when doing so. The transportation and storage cost could still be manageable. For instance 5bn in 500 notes would fit on one euro pallet. The 15mn in annual interest rate savings (at - 0.3%) should thus easily pay for putting it into a strong room. The crucial variable in the calculation should be insurance. German media report that the Bavarian insurance chamber is offering coverage also for larger amounts for 1.50 per Including insurance tax, this would yield %. A look at the developments of banknotes in circulation and cash held by banks reveals though that the negative rates have not yet caused a hoarding of cash in the euro area (left-hand chart below). A practical obstacle keeping investors from larger-scale cash conversions is that physical cash in such large amounts does not provide the liquidity most money market investors need. Should rates become more negative though without chances of reversing soon, however, the cash option should become more relevant. Today, there are already signs of that happening in Switzerland (right-hand chart below), although the SNB denies this. In particular the CHF1,000 note has increased of late so that 62% of all Swiss cash was held in this denomination at the end of last year. Reports that a Swiss life insurer has approached banks and the SNB for a largerscale cash withdrawal also point in this direction. As there is no legal basis for the ECB to reject or even restrict banks requests for cash, it would have to overcome the practical limitation that so many banknotes are not available and cannot be printed easily - if need be via the issuance of new banknotes in denominations of 1bn. No trend-increase yet in euro cash in circulation Banknotes in circulation (trend-adjusted) and cash holdings by banks estimated from ECB statistics and monetary aggregates, in bn Bank cash Banknotes but demand for larger swiss franc notes on the rise! Swiss banknote circulation by denomination, indexed (2005=100) SFR 10 SFR March 2016

3 Everything considered, at the ECB depo rate of -0.3%, the lower bound has not yet been reached but given the available avoidance strategies, the ECB will likely be wary about cutting rates much further without changing the rate-setting mechanism. Due to the different monetary frameworks, the deposit rates are not readily comparable across countries. At the SNB rate of -0.75%, we seem to be getting close to the lower bound, and the lower bound should increase the more persistent negative rates are as the investment for avoidance strategies becomes more attractive. 1 We thus consider it likely that the ECB changes its rate-setting mechanism toward a tiered system with allowances when cutting rates further this week (see below). Bank constraints Margin pressure Besides thes cash considerations above, the lower bound for central bank rates also crucially depends on how the central bank implements negative rates and how other relevant lending rates are being affected. The negative central bank deposit rate only applies to the cash that banks deposit at the central bank. As far as banks don t pass on this negative rate to their depositors, a different constraint for central banks in imposing negative rates thus comes from the adverse implications on banks interest rate margins. As the liability side of banks balance sheets is less flexible to the downside (retail deposits for instance will unlikely incur negative rates given their higher regulatory value, certain wholesale deposits may be spared initially for relationship reasons), central bank rates below zero tend to weigh on banks profitability. At this stage, this is not yet visible in the data. ECB statistics on lending margins even reveal that margins have gone up in most core countries while coming down from higher levels in the periphery. This is what one should have expected given the still longer duration of the assets while most funding is usually tied to a floating rate. As funding rates hit the zero bound while the fixed asset rates are adjusted lower over time, banks interest rate margins will predictably come under pressure when ECB rates stay negative for longer. Allowances would shift the lower bound And the problem also looks set to get bigger as rates turn more negative. The penalty tax rate charged to banks will increase with the ECB poised to lower the depo rate further, while the tax base also continues to rise as excess reserves keep accumulating by some 60bn per month via QE (left-hand chart below). One way to mitigate the problem is via the introduction of allowances, i.e. a multi-tiered interest rate system where part of the reserves are exempt. 2 We have long argued that such a system can be designed in a way to still achieve the full market impact (i.e. lower money market rates, steeper curve, weaker euro). If the ECB were to adopt this approach it could cut the depo rate more aggressively. Increasing penalty tax rate and tax base for banks ECB deposit rate in % and excess reserves in bn, including forward projections Jan-14 Sep-14 May-15 Jan-16 Sep-16 Depo rate Excess reserves (rhs) with core banks being most affected Eurosystem excess reserves by county in bn GE FR NL* LX FI IT SP Source: National central banks, Commerzbank Research; *)estimate 1 According to the BIS the weighted average rate on bank reserves in Switzerland (-0.27%) is close to that of the euro area (-0.25%), see How have central banks implemented negative policy rates?, BIS Quarterly Review, March When comparing the reserves that are subject to the penalty rate to total bank assets, however, the Swiss system seems more restrictive. 2 For a comparison of the different systems, the impact on market functioning and the transmission beyond money markets, also see How have central banks implemented negative policy rates?, BIS Quarterly Review, March March

4 but uneven North-South impact on banks could give rise to political considerations Notwithstanding, political considerations are probably slowing the Governing Council s excitement about allowances though as the excess reserves are distributed very unevenly between North and South - with the banks in the North benefiting disproportionately from the introduction of allowances (also see Ahead of the Curve, 18 February, page 7/8). At the same time, peripheral banks net interest rate margins are likely suffering more from further rate cuts given the lower duration of their loan portfolios (in Spain, for instance, variable-rate mortgages are more common). A bolder ECB rate cut alongside allowances could thus ease the burden on core banks while not helping or even hampering the profitability of peripheral banks. Distortions from negative rates Below we provide a brief overview of the main general distortions arising from negative interest rates before diving deeper into the technical pricing aspects of the rates products that are most affected. 1. The largest longer-term drawback of negative rates (and unconventional monetary policies in general) is the formation of asset price bubbles. With solvent borrowers being paid to borrow and mortgage rates at historic lows, higher debt burdens will eventually end in tears. Granted, the collapse in equities and credit valuations in recent months and in particular the sell-off in Japan after the BoJ joined Club Neg has silenced these warnings. For the time being we also see no evidence in the data that investors are irresponsibly moving down the credit curve (also see Big Picture, 18 and 25 February). Yet, once volatility subsides and investors will need to focus again on yield enhancement instead of capital protection, this should change. Our economists have also identified signs of overheating in the German property market (Ahead of the Curve, 29 January). 2. A less talked about longer-term drawback from negative rates could be the demise of the euro as reserve currency. At present, the ECB may welcome the weaker euro from foreigners abandoning euro assets. As foreign central banks that cannot invest in negativeyielding assets may not return quickly when yields turn positive again, negative rates are jeopardising the benefits the ECB has from issuing a reserve currency. 3. When borrowers rejoice, savers are suffering. With close to 80% of German government paper carrying negative yields (left-hand chart below), in particular pension funds and life insurers are struggling to meet their interest rate promises without taking undue risks. Although guarantees have been lowered consecutively (e.g. to 1.25% for new life insurance contracts in Germany, right-hand chart below) and pension schemes are being moved from defined benefit to defined contribution schemes, the average guarantee rate for outstanding policies in Germany still stands at around 3%! Granted, the ECB would argue that it isn t its fault. In absence of its unconventional policies, nominal long-end yields would have been driven even lower by weaker economic growth, lower inflation and flight-to-quality. Given the questionable impact of many measures and fears about the side-effects (see Japan), however, the verdict of the cost-benefit analysis is still out. 80% of German sovereign paper carries negative yield Yields falling faster than rate guarantees Euro area sovereign bonds and bills with negative yields German life insurer guarantee rate and average sovereign yield (Umlaufrendite) in % % % % FR GER ITL ES NL BEL AT FIN IRL paper with negative yields share of outstanding marketable debt (rhs) 20% 0% yield guarantee 4 9 March 2016

5 4. When rates dip deeper into negative territory and are being passed on to a broader group of people, avoidance strategies will lead to various distortions in payment behaviour and cash holdings. Already today there are reports about Swiss Cantons abandoning their deferral penalties. The cash withdrawals discussed above nevertheless lead to an inefficient use of resources.as banknotes are being printed purely for the purpose of storing cash, the firms involved in the printing, logistics and storage are flourishing, but the economic benefits beyond this industry are questionable. Legal standards keep evolving for bonds, loans, derivatives and collateral 5. Legal obstacles are still evolving as more interest rate products are being affected by negative rates. We are not aware of any negative coupons on bonds. Besides the expected legal challenges when an obligation to pay is being imposed on bondholders 3, technical and credit considerations render negative coupons very unlikely when issuers need to collect payments from their investors. It is therefore common practice that payments in floating rate notes are floored at zero (also see next section). For loans, the situation is less clear-cut. While zero floor language has widely been adopted in LMA-based agreements, it can be argued that the omission of such clauses could lead to a reversal of the payment obligation once the client margin has been eaten up by the increase in reference rate. 4 For deposits that are governed by the German civic law (BGB), 488 is being interpreted in a way that negative rates are not allowed in existing deposits. 5 In interest rate derivatives, negative payments (i.e. payer positions turning into receiving positions and vice versa) are commonly accepted as ISDA defines the floating rate relative to an index and negative fixed rates are also required upon inception for the swap to work. As a side-note, the asymmetry between zero-floored loans and floaters on one side and interest rate derivatives that are not floored on the other side means that interest rate swaps become less useful for hedging purposes. In terms of collateral derivatives trading, negative rates are not applied to posted collateral unless this has explicitly been agreed in the contracts. An increasing number of derivative market participants have opted to sign the ISDA 2014 Collateral Agreement Negative Interest Rate Protocol, making negative rates payable on collateral should both parties sign the agreement. On one hand other market participants have chosen not to sign the protocol mail due to financial impact considerations, on the other hand there are relevant documentation clusters (e.g. DRV / BSA) for which no such protocol exists. As a consequence lots of bilateral discussions between banks are ongoing with respect to modifying existing documents to include negative CSA rates. The bottom line is that a level playing field has not yet been established. 6. When it comes to IT systems, banks were quick to adjust their front-office infrastructure to cater for negative rates; however, an automated end-to-end handling including financial accounting still constitutes an issue for a number of institutes. Beyond the banks, however, many corporates appear to have so far been reluctant to accept that negative rates could become a permanent feature, requiring larger IT investments. It is our understanding that most SAP reporting solutions established by corporates so far are not capable of flexibly adapting to negative revenues and claims. Hands-on pricing implications When the sensitivity of a floater converges to the sensitivity of a zero-coupon bond Evaluating so-called linear fixed income products (i.e. bonds or interest rate swaps where the payment stream is known in advance) remains straightforward as any payment stream can be discounted to a present value with any positive or negative discount rate. When it comes to socalled non-linear products (i.e. options where payments are subject to a probability distribution contingent on future rate levels) or floating rate notes (FRNs) that are floored at zero (coupons are zero when reference rate plus margin are negative), however, traditional pricing models need to be adjusted because the commonly used log-normal probability distribution of interest rates does not allow for negative interest rates. In the extreme, when rates are very negative, FRNs for instance behave like zero-coupon bonds - with a correspondingly high price sensitivity (modified duration) as the probability of receiving 3 In contrast to many jurisdictions in continental Europe, English law has no principles against payments being imposed on bond holders, see Slaughter and May: Financing Briefing, June Also see Slaughter and May: Financing Briefing, June See for instance 9 March

6 any variable coupons converges to zero. This of course has important implications as money market funds who often own FRNs were not set up to deal with high mark-to-market risks. The higher price-sensitivity of floaters may thus be an unwelcome side-effect of negative rates. A positive property, on the other hand, is a positive convexity of zero-floored floaters as the duration increases when rates fall while it decreases when rates rise. Interest rate models have always been a slightly special case Interest rate valuation models The development of option pricing models deserves some attention at this point. In 1900, Louis Bachelier published a model to value stock options in his thesis at the Sorbonne. It assumed that market returns were normally distributed, and was ahead of its time, to say the least 6. But it ignores discounting and assumes that rates can go negative, thus is not very suitable for stock options. Subsequently development continued slowly, until the groundbreaking work of Black and Scholes in 1973 opened up the financial option market as never before. 7 Generalised Interest Rate Model Summary of features from popular models σ t is a volatility, and can be constant or varying r is a rate, like a spot rate or a t forward rate β controls the distribution of rates. If β=0, rates are normally distributed, if β=1 rates are log-normally distributed dr t = θ r,t dρ + σ t r t β dw t θ allows for effects like time r,t, dependence or mean reversion W t is a random variable Source: Commerzbank Research Interest rates, however, have always been a slightly special case. Their behaviour is quite distinct from that of stocks or commodities; they have a stable range which they rarely exceed and tend to revert to. Additionally, it is desirable to preserve the arbitrage-free nature of the term structure. Like stocks and bonds, however, until recently it was assumed that they would not take negative values 8. Thus from the start, they have had slightly modified models to account for these unique features. The early models, like Merton (1973) and Vasicek (1973) tended to model the short interest rate. As the area developed, models expanded their scope to include the forward rate curve (the Heath-Jarrow-Morton type models). Later developments allowed more realistic behaviours to be represented (allowing flexible distributions, for example, or introducing correlations between observed market variables). Below we give a general descriptive formula to show how interest rate models tend to work. To represent the bounded nature of interest rates, models tend to include either some kind of mean reversion (eg Vasicek, Cox-Ingersoll-Ross, Hull-White), or to have a lognormal distribution (eg Black-Derman-Toy, Black-Karasinski). Sometimes an additional equation is needed to represent the model; for example, the Chen model has a stochastic mean and volatility which are separately defined, and these are referred to as multi-factor models. 6 The equations he developed were published independently 5 years later in Einstein s celebrated paper which explains Brownian motion (1905). It is believed neither were familiar with the other s work at the time. 7 Though Black and Scholes (1973) was the revolutionary paper, much important work was done between this and Bachelier s publication. Sprenkle (1961) adapted Bachelier s work using lognormal returns, Boness (1964) introduced discounting, and Samuelson (1965) allowed the option to have different risk levels than the stock. Merton, a student of Samuelson, worked on the theory with Black and Scholes, and their model is often referred to as the Black-Scholes-Merton model. 8 Switzerland s brief experience in 1979 of negative rates was assumed to be a momentary aberration. 6 9 March 2016

7 In general, a log-normal distribution fits observed rates better than an explicit mean reversion factor, and has historically been preferred as it naturally precludes rates going below zero, as shown below. We see that the normal model will allow rates below zero, for suitable mean and volatility, but the lognormal will not. The SABR model is currently accepted as bestin-class; and a slight modification means it can handle negative rates The SABR 9 model; state of the art The SABR model, developed by Hagen et al, first came on the scene in 2002, and was immediately popular. It allows volatility to itself be a stochastic variable, and additionally introduces a correlation between the forward rates and the volatility. Finally, the parameter β, which defines the distribution, is also a variable which may be fitted to market data. This flexibility means that explicit mean reversion becomes unnecessary. Its evolution is defined as below: df t = σ t F t β dw t dσ t = αα t dz t dw t dz t = ρρρ Here F t is the forward rate, σ is the volatility, W and Z are random processes, α is the volatility of the volatility (sometimes called the volvol) and ρ is the coefficient of correlation. However one fly in the ointment remains; the parameter β is forced to be zero if rates are negative or close to negative! Thus removing one of the attractive pieces of flexibility of the model. The solution is the SABR model with an additional displacement factor s, as below: df t = σ t (F t + s) β dw t Normal vs lognormal distributions Probability distribution from SABR model with displacement Lognormal distribution has a hard stop at zero Data from Jan 2016, modified to show flexible tail Normal Lognormal 6,000 5,000 4,000 3,000 2,000 1, % -0.46% -0.43% -0.41% -0.38% -0.36% -0.33% -0.31% Source: Commerzbank Research Source: Commerzbank Research This has now become the best market practice to accommodate negative or near-negative rate environments, and gives flexible distributions, as illustrated below. We mentioned earlier that many contracts apply a legal cap or floor at zero to the contract. It was probably envisaged that these clauses would only be invoked in the case of severe market disruption, but the effect upon the contract is to lend option-like characteristics to a previously linear product. 9 SABR stands for Stochastic Alpha, Beta, Rho 9 March

8 Commonly used but unsophisticated models give anomalous results in the current environment A real-life example An EIB FRN matures in Considered with a, discount margin of -2.9 bp, and an assumed rate of % for 3m EUR Libor, it is valued by Bloomberg at As it is an FRN, we would expect almost no sensitivity to changes in Libor. However, if the Bloomberg yield analysis tool is used to change the assumed rate by 1 bp to %, with the discount margin kept constant, the price changes to a move of over 8 bp. This is a bizarrely high sensitivity for such an FRN to show one would expect it to be close to zero, and instead it is close to that of a zero coupon bond! To investigate what is going on, we plotted the change in price for a +1 bp change in assumed rate for the bond, over a range of assumed rates, keeping the discount margin fixed, finding the results below. We see that the usual minimal sensitivity to Libor changes abruptly when rates go negative. What is going on? In fact, this graph is a function of the Bloomberg calculation method; to calculate the change in value, it assumes a flat forward rate curve given by the input Libor level 11. This is unrealistic as the forward curve expects fixings to rise back to above-zero levels after some time, currently in We now add to the graph a similar test trade but the sensitivity is calculated using the SABR model. Instead of a flat change in a constant rate out to the horizon, the whole yield curve is given a parallel shift, and additionally, curve volatilities are derived from current market rates. We see that the change in sensitivity is far smoother and more as we would expect. We have also added a displacement to the SABR model so rates can go below zero. Note that the correct way to model the FRN is to add a floor (a swaption) whose effects are only noticeable when rates are close to negative values. Price sensitivity to +1bp change in rates, in Jan 2016, using Bloomberg and SABR model with added displacement EIB FRN (expires 2024, ISIN XS ) modelled with swaption with strike zero Change in FRN value for 10 bp change in rate 0.00% -0.02% -0.04% -0.06% -0.08% -0.10% -0.50% -0.30% -0.10% 0.10% 0.30% 0.50% Rate level BBG SABR Negative rates have had, and will continue to have, unexpected effects in different places. But the careful use of intelligent valuation models means that even products which acquire unforeseen option-like qualities may still be valued and traded Data from end Jan This method is taken from the 1986 book Floating Rate Instruments by Frank Fabozzi, Probus Publishing, page We would like to acknowledge Roberto Ricci and Sergio Dutra from Commerzbank s quantitative teams, whose generous help greatly improved this paper. 8 9 March 2016

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