Fixed Income Portfolio Management in a Low Rate … · Seeyond is a brand of Natixis Asset...

Fixed Income Portfolio Management in a Low Rate Environment

Research paper #7

June 2016 // Document intended for professional clients.

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The Fixed income investment division implements an active fundamental management, where risk is taken into account at every stage of the investment process. It offers a collegial approach with sector teams specialised by market segment.The Fixed income investment division is supported by close to one hundred specialists, including asset managers, credit analysts, strategists, financial engineers and economists.With €252 bn under management1 and a track record of more than 30 years, this investment division has proven experience.

The Quantitative Research and Analysis team supports Natixis Asset Management’s fixed income investment teams by providing portfolio construction tools, quantitative model outputs, and valuation models for structured credit and derivatives. They help to calibrate the portfolio management process and the risk budgeting approach. 1. Source: Natixis Asset Management – 31/03/2016.

2. Seeyond is a brand of Natixis Asset Management.

3. Emerise is a trademark of Natixis Asset Management and Natixis Asset Management Asia Limited.

4. Mirova is a subsidiary of Natixis Asset Management.

NATIXIS ASSET MANAGEMENT Fixed income investment division

Publishing Director:Ibrahima KobarCo-chief Investment Officer,in charge of Fixed income investment division–Written by Quantitative Research and Analysis - Fixed income:Nathalie Pistre, PhD, Head of the team,Deputy Head of Fixed income investment divisionAxel BotteGuillaume Bernis, PhDChafic Merhy, PhD

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TABLE OF CONTENTS

I. AN ECONOMIC EXPLANATION TO LOW INTEREST RATES 5

1. THE DECLINE IN NATURAL INTEREST RATES 5

a. What is the natural interest rate? 5

b. Why have natural interest rates declined? The Laubauch-Williams approach 6

c. Potential drivers of the decline in natural rates 7

2. ECONOMIC ISSUES WITH LOW AND NEGATIVE INTEREST RATES 9

a. Through the zero bound: an unforeseen situation 9

b. Low interest rate policies: the mechanics of monetary stimulus 9

c. Risks to financial stability associated with low/negative rates 10

II. IMPLICATIONS OF LOW RATES ON VALUATION 10

1. TO NEGATIVE RATES AND BELOW: EMPIRICAL EVIDENCE

FROM HISTORICAL DATA 11

2. IMPLICATIONS OF LOW RATES ON THE PRICING OF LONG RATE INSTRUMENTS 13

a. Blunt SABR or Light SABR? 13

b. Shifting the distribution to negative rates 14

c. Models of the term structure 15

III. IMPLICATIONS OF LOW RATES ON RISK MODELLING 16

1. THE RISK-RETURN TRADE-OFF IN A LOW RATE ENVIRONMENT 16

a. The return 16

b. The risk 18

2. VOLATILITY IN A LOW YIELD ENVIRONMENT 18

a. What is a good risk model? 19

b. Modelling the volatility of factors 19

i. Normal volatility 19

ii. Lognormal volatility 19

c. When volatility is not enough 20

3. THE VOLATILITY OF TR AND THE ADEQUACY OF USUAL SENSITIVITY

MEASURES, DURATION AND DTS, IN A LOW YIELD ENVIRONMENT 22

a. Adjusting the volatilities 22

b. Adjusting the sensitivity 23

CONCLUSION 24

REFERENCES 25

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At the beginning of 2016, negative interest rates that would have been unconceivable a few years ago on the fixed income markets can be observed.This state of affairs is very difficult to sustain for certain investors and raises a few questions that we want to explore in this paper.

Low rate environment raises several issues on the modeling of both rates and volatilities. It also questions the validity of classic and widely used interest rates models. In a low rates environment, rates’ hikes are more likely to occur than rates tightening. It results in a pronounced skewness of the distribution of rates. The risk-return trade-off is hence understated when volatility is retained as a risk yard stake.

In the first part of this paper we examine the reasons of a decline in natural rates and the economic issues raised by low and negative interest rates.

In the second part of the paper we will see how financial models deal with this possibility and the implications of low rates on the pricing of long rate instruments.

The implications are particularly meaningful for risk and allocation models and raise some questions regarding the behavior of volati-lities when rates become negative as well as the accuracy of the approximation of the portfolio risk by its duration times the volatility of rates.

Traditionally, the volatilities of rates and spreads are assumed to be either Gaussian or lognormal. In the former case, volatilities are constant irrespective of the underlying level. In a lognormal framework, volatility is proportional to the level of the underlying whether rate or spread. Should the underlying fall to zero then its volatility will also drop to null. Besides, lognormal rates or spreads cannot be negative in which case volatility would be negative too!

We argue that volatilities do not drop to null when rates become negative. Instead, they reach an incompressible floor irrespective of the rate level. Adjusting the volatility of rates can be done either by shifting the distribution of rates into positive territory or by adding a normal noise to the unshifted lognormal distribution or even by allowing for jumps in the rates diffusion process. Moreover, approxi-mating the volatility of returns by the duration times the volatility of rates’ movements can be addressed by adjusting either the volatility of rates or the sensitivity of returns to a movement in rates.

This is what we show in the third part.

FIXED INCOME PORTFOLIO MANAGEMENT IN A LOW RATE ENVIRONMENT

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1AN ECONOMIC EXPLANATION TO LOW INTEREST RATES

In this section, we briefly review the theory of interest rates. Interest rates depend on population growth, productivity growth and consumer time preferences. The so-called natural rate is the level of interest rate consistent with full employment and stable inflation. Over the past few years, equilibrium interest rates appear to have declined substantially. Using a framework developed by Fed economist Thomas Laubach and FOMC member John Williams, we estimate factors behind the protracted fall in interest rates. However, unobserved factors seem to have accounted for a significant share of the decline in natural rates. The last subsection follows on from this finding as we discuss alternative explanations for the fall in natural rates.

1. THE DECLINE IN NATURAL INTEREST RATES

a. What is the natural interest rate?

The primary policy tool of many Central Banks is the real short-term interest rate. Policymakers aim to minimize fluctuations in the business cycle around some trend by setting interest rates at the appropriate level. The conduct of monetary policy requires a measure of a “natural” or “equilibrium” rate to determine the stance of policy. The natural rate is the level of real short-term interest rate consistent with the economy operating at full capacity. The natural rate is the interest rate that goes with output converging to potential and stable inflation. It represents a medium-run anchor used in monetary policy rules such as John Taylor’s (1993).

Contrary to the assumption of a constant 2% real equilibrium rate used in The Taylor Rule, the natural rate is unlikely to be constant over time. In standard economic models, the natural rate responds to shifts in both preferences and technology. The equilibrium rate is derived from an optimal savings condition yielding a balanced output growth path. The real rate relates to economic fundamentals as follows:

where σ denotes the intertemporal elasticity of substitution in consumption (assu-ming a constant relative risk aversion utility function for consumer preferences), n is the rate of population growth, q is the rate of labor-augmenting change in technology and θ defines the rate of time preference. The natural rate is linked positively to productivity and population growth (trend output growth) and time preferences (higher rates go with higher preference for current consumption). The equilibrium real interest rate will fall with a higher elasticity of substitution in consumption.

Population growth implies increasing needs for productive capital (factories, airports, machinery equipment...). The failure to raise the available capital stock results in unemployment and declining output per capita under the assumption of unchanged technology. Capital formation requires savings. Businesses and households must thus forego current consumption out of current production. Higher population growth hence requires a larger pool of savings which is more likely attainable with high real interest rates.

eq. 1r = q + n + 1 θσ

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b. Why have natural interest rates declined? The Laubauch-Williams approach

Laubauch and Williams (2001, 2015) have developed a methodology to estimate the natural interest rate in the context of the US economy. Basically, the authors posit that the equilibrium rate is the level of interest rates that corresponds to no output gap (the output gap is the difference between current output and its potential) with inflation stable. In the following, we give a broad overview of the Laubach-Williams methodology. The equation below describes the relationship between real interest rates and the output gap.

where ỹt=100*(yt-y*t ) denotes the output gap between yt the logarithm of real GDP (growth domestic product), yt* the unobserved potential output (potential output is the level of output associated with no price tensions and efficient use of resources). In turn, r is the ex-ante real Fed Funds rate and r* is the unobserved natural interest rate. Uncorrelated errors Є1t and the lags of the output gap control for transitory shocks so that large shifts in the relationship between output gap and the real Fed Funds rate are traceable to changes in the natural rate. Ay(L) and Ar(L) denote lag operators.

Real rates are measured ex-ante. This requires a model of inflation expectations. The authors use core PCE (personal consumption expenditures) price index as the inflation reference. Inflation πt is determined by its lags, the output gap ỹt, relative prices xt to control for import price shocks (unrelated to the output gap) and an uncorrelated error term Є2t.

To reflect economic theory, rt* is determined by trend output growth gt, other factors zt and a serially uncorrelated error term Є3t as follows:

In addition, the authors allow for shocks in the level of potential output and its trend growth rate. Potential output and trend growth are modelled as random walk processes with serially uncorrelated errors which are also uncorrelated to innovations to zt (Є3t ).

Leaving aside discussions about estimation techniques, the Laubach-Williams approach yields the following results (see Chart 1.1 below). The estimation period spans from 1961 to 2014. Their estimates show two periods of significant declines in the natural interest rate. A moderate decrease in the natural rate occurred in the 20 years prior to the financial crisis that started in 2007. The natural rate came down from about 3.5% in 1990 to 2% in 2007. A second more abrupt downshift was observed during the 2008-2009 recession bringing the equilibrium real rate down to about 0%. The natural rate has remained in the vicinity of 0% over the subsequent five years.

The second chart (Chart 1.2) provides a comparison of market-based real rates derived from the TIPS market with LW estimates of the natural real rates. TIPS yields have been available since 1997 only. Market-determined real yields exhibit a similar downtrend as natural rate estimates.

eq. 2ỹ = A (L) ỹ +A (L)(r -r* )+Є t y t- r t- t- t

eq. 3π = B (L) π +B (L) ỹ +B (L)x +Є t π t-1 y t-1 x t 2t

eq. 4r* = cg + Zt t t

eq. 5Z* = D (L) Z + Єt Z t- 3t

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The Laubach-Williams approach reveals that the natural rate and potential output growth have been closely aligned through 2000 with unspecified factors having a modest contribution to changes in the natural rate. The natural rate has typically evolved within 1pp of trend growth albeit unspecified factors had temporarily lowered real equilibrium rates in the early 1990s. Since year 2000, the importance of the unspecified factors has increased. These factors have lowered equilibrium rates by up to 3pp in the wake of the crisis on LW estimates.

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natural rate Bloomberg Generic 10-year TIPS yieldBloomberg Generic 5-year TIPS yield

That being said, part of the decline in natural rates is still attributable to potential growth falling from a long-standing 3% clip to a ‘new normal’ of 2%. In the next section, we discuss the reasons behind the apparent fall in potential output growth.

c. Potential drivers of the decline in natural rates

As expressed in the model described above, the protracted decline in real interest rates stems from falling potential output growth and other unspecified factors. In this

Figure 1.2: Laubach-Williams (one-sided) estimates of the natural rate and market-determined TIPS yields

Source: Laubach and Williams 2001, 2015; Bloomberg

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natural rate g (trend growth rate)z (unobserved factors affecting natural rate)

Figure 1.1: LW estimates of natural rate, trend growth and unspecified factors affecting equilibrium rates

Source: Laubach and Williams 2001, 2015

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section, we briefly examine the key reasons behind the protracted fall in trend growth (most of which will persist) and shed some light on the unspecified factors in the LW model dampening equilibrium real rates.

According to Robert Gordon (2012), US potential output growth is faced with two major headwinds. Demography plays a central role in this analysis. The demographic dividend from the movement of women into the labor force until the late 1990s - a one-off event - is now in reverse motion. Aging also weighs on participation of wor-king-age population to the labor force. Lower participation reduces hours worked per capita in the economy. Furthermore, as life expectancy grows relative to retirement age, this headwind is likely to augment. For at least the next twenty years, GDP per capita is set to grow at a slower pace than output per hour (productivity). The second headwind relates to future productivity. Educational attainment in the US has levelled off around 1990. Over the past decade, higher education costs have skyrocketed lea-ding to outsized student debt burden (and indeed double-digit default rates on such debt). Elevated tuition costs deter low-income people from attending college at all. Net returns on human capital investments have indeed fallen. Other headwinds include globalization and its impact on wage-setting mechanisms, environmental regulations and issues related to debt deleveraging.

Larry Summers offers a different view focusing on the risk of secular stagnation. Secular stagnation should be associated with a decline in real interest rates. The business cycle can have a significant influence on longer-run prospects for economic growth. In other words, the trend in growth can be adversely affected by short-term developments. A financial crisis of the magnitude of the 2008-2009 financial crisis would appear to encompass those factors - left unspecified in the LW model - dampening the natural level of real interest rates in the past seven years. There likely exists a shadow-cast forward of the financial crisis – hysteresis in economists’ parlance – preventing the economy to re-equilibrate. If the natural rate has fallen to, say, negative 2%, the zero bound on nominal rates could also prevent the economy from operating at full employment and achieving strong growth (see section on negative rates below).

Among the reasons for lower equilibrium real rates, diminished debt-financed investment demand appears to rank high. The legacy of excessive leverage and tighter regulation restricting financial intermediation after the crisis reduced demand for debt-financed investment. Deleveraging is akin to rising private-sector savings, not to mention the unwelcome rush to cut public deficits in the years following the financial crisis. The rise of new information technology businesses requiring little initial capital investment (and funded instead by venture capital) is one example leading to low debt-financed investments and lower equilibrium rates.

Rising income inequality appears to have some bearing on the level of interest rates. Emmanuel Saez of Berkeley University estimates that the top 1% of the income distribution in the US captured 52% of total real income gains between 1993 and 2008. Since high-income households tend to save more out of current income than poorer households, inequality is likely to have weighed on equilibrium real interest rates. The resulting saving glut indeed exerts downward pressure on natural interest rates. In the theoretical framework described in section 1, time preference may be lower.

Thirdly, the evolution of the relative price of capital goods (business equipment and machinery) could be another driving force for lower investment demand. From 1980 to 2015, the price of equipment good (measured by the equipment good investment deflator) relative to the GDP deflator has fallen by about 2% per annum. Cheaper equipment means that investment spending and associated borrowing can be lower.

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One final unspecified factor behind the decline in natural rate is the substantial move by global Central Banks to accumulate safe assets (through quantitative easing, foreign-exchange reserve management) in particular US Treasuries.

2. ECONOMIC ISSUES WITH LOW AND NEGATIVE INTEREST RATES

The opening section draws upon the theory of interest rates to make sense of low rates in the recent period. However, other factors play a considerable role in explaining the drift in interest rates since the financial crisis. These factors include unconventional monetary policy pursued by monetary authorities which we discuss in some detail in this section.

a. Through the zero bound: an unforeseen situation

In theory, the possibility to hold banknotes yielding no interest should prevent interest rates from falling below zero percent. However, the cost of storing value is not zero which gives scope for negative rates. These costs mainly pertain to transportation and warehousing services, insurance and security. Hence, holding physical currency in a safe has significant costs, similar to a negative interest rate. Also, hidden costs, such as the loss of convenience of transacting electronically and holding deposits, need to be taken into account. The effective lower bound on interest rates is therefore below zero. The unobservable ‘true’ lower bound should correspond to the penalty interest rate beyond which demand for paper currency soars as the costs of holding physical cash become less than the negative rate earned on deposits.

b. Low interest rate policies: the mechanics of monetary stimulus

Since 2008, many Central Banks have engaged in policies that had not been tried before. Non-standard measures took the form of outright interventions in bond markets and negative policy rates. Together with Central Banks commitment to keep rates low, unconventional monetary action has reduced nominal yields to zero or even negative levels in several (mostly European) countries. This is unpre-cedented. Even during the US Great Depression of the 1930s, nominal rates had always remained positive. Instead of paying interest, governments in the euro area, Switzerland, Denmark or Sweden are effectively being paid to borrow while investors are no longer compensated for bearing interest rate risk.

Monetary policy is said to be accommodative when policy rates are below natural rates. Central Banks can choose the degree of stimulus by setting short-term interest rates. Monetary stimulus works through five channels. The credit channel is the main transmission mechanism. By driving down the costs of financing, low-rate policies deter saving and encourage borrowing through bank lending and bond issuance. However, in the case of negative deposit rates, the ‘tax’ imposed on commercial banks can have unintended adverse effects including a reduced provision of credit to the real economy. Indeed, if retail deposit rates and lending rates do not fall proportionately to Central Bank rates, a negative rate policy will have little traction. The second channel works through asset price revaluation. Asset valuations rise as discount factors fall with Central Bank rate cuts. The one-off shift in asset prices in turn affects the real economy through wealth and confidence effects. Expectations of cash-flows on risky assets may increase as the economic outlook improves. Thirdly, low or even negative interest rates encourage portfolio rebalancing out of risk-free bonds into riskier assets. The rebalancing channel compress credit spreads and term premia in bonds facilitating lending. It encourages the chase for yield that is desirable at the early stage of an economic recovery. The fourth channel is the reflation mechanism. Rising inflation expectations should foster household and business spending while reducing the risk of a deflationary spiral in

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economies facing a debt overhang. This is thus a form of insurance policy against extreme economic outcomes. The fifth channel relates to currency depreciation. Depreciation should boost net exports and inflation via import prices. Obviously, not all countries can depreciate their currencies at the same time. Currency war is a zero-sum game. The exchange rate channel is therefore uncertain to have the intended effect on short-term growth.

c. Risks to financial stability associated with low/negative rates

The interest rate can also be thought of as the price of leverage. When interest rates are low, the pressure to reduce leverage is low. The incentive for public and private economic agents to reduce their debt diminishes as debt-service costs decrease. Thus low rates can, temporarily, distort analyses of debt sustainability. This is all the more the case if rates are negative.

As concern investors’ behavior in financial markets, asset valuations buoyed by low discount factors may become out of line with underlying productivity and growth potential. In this case, monetary policy decisions supplant economic fundamentals in their role to set asset valuations. It is important to note that monetary accom-modation is only about buying time and repairing balance sheets but it cannot be a substitute for economic reforms or corporate restructuring. This also makes financial markets more sensitive to changes in monetary policy. The ‘tapering’ announcement by the Fed’s Ben Bernanke in spring 2013 is one such recent example of a ‘Minsky moment’ (after Hyman Minsky’s theory of sudden stops in financial markets) in financial markets, when market participants came to recognize that valuations were distorted by unconventional monetary policy.

The longer-run issues with low and negative rates lie with the way financial institu-tions do business. Pension funds, insurance companies, money market funds and banks all face significant challenges in a low rate environment. Given negative deposit rates, retail banking faces a dilemma to pass on this cost or take a hit on margins. In extreme cases, economic agents could decide to pull holdings from their bank accounts and decide to use other forms of currency, including bitcoins. The financial system would be hit hard. In loan agreements however, negative rates have had little impact to date. In the euro area, interbank rates are not low enough to cause banks to actually pay interests to borrowers. In many instances, loan agreements include clauses to floor interest payments to zero. Pension funds and insurance companies are undoubtedly at risk in the current environment. Low interest rates inflate the present value of long-term liabilities, eating into the solvency of pension funds in particular. In turn, in this market environment, life insurance companies may have increasing difficulty to meet guaranteed return levels.

2IMPLICATIONS OF LOW RATES ON VALUATION

The first part shows that low rates and even negative ones are not incompatible with economic theory. We will see in this section how financial models deal with this possibility.

Low rate (and more generally yields) environment raises several issues on the modeling of both rates and volatilities (implied and historical). It also questions the validity of classic and widely used interest rates models. In a Section A, we inves-tigate the compatibility of the hypothesis of negative swap rates with empirical

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data. Recently, short term swap rates, such as the 3 year tenor, have hit negative levels. What is the range where these swap rates can evolve? Is this phenomenon likely to happen on longer tenors, which remain, for the moment, positive? These questions will be studied in Section A. However, even if not frequently observed in historical data, negative long rates have been embedded in the market prices of certain instruments (e.g. swaptions), especially in a context of high implied volatility. We show in Section B how this feature questions the modelling of interest rates and requires adjustments in classical models.

1. TO NEGATIVE RATES AND BELOW: EMPIRICAL EVIDENCE

FROM HISTORICAL DATA

From the point of view of the (risk-neutral) valuation of interest rate derivatives two questions need to be addressed. What is the negative lower boundary for short tenor swap rates? Does long tenor swap rates historical data contain any evidence of potential negative rates? If not, the use of pricing models with negative rates could be questionable. Indeed, the risk-neutral probability is equivalent to the historical probability. Therefore, if there is a positive probability of negative rates under one of these probability measures, then, the other measure must put a positive weight on negative rates too. Equivalently, if the probability of an event is null under one of these measures, then it has a null probability under the other.

The form of empirical distributions does not seem to behave as normal distributions. Rather than resorting to the solutions displayed in Section B, we prefer working with Jacobi processes, which are Markov diffusion processes living in a bounded open interval. Even if their use in finance is not extremely developed – due to the complexity of the zero-coupon formula – they provided, in this context, a good fit on historical data. The reader can refer to Delbaen and Shirakawa (2002) for details about Jacobi process, and their use in the pricing of interest rate derivatives. The Jacobi process has the particularity to have, first, a mean-reverting feature, second, a local volatility which is equal to 0 at the extreme points of the interval. Therefore, when the rates are low, the volatility effect fades away and the process is driven by the mean-reverting feature. To this extent, this model can be seen as an extension of the model proposed by Cox, Ingersol and Ross (1985), with which it shares a certain number of mathematical properties.

As an example, we have considered the 3 year Euro swap rates, from October 11th 2011 to January 26th 2016. On this period the minimum is reached on January 22th 2016, with a value at some -0.09%. Using the historical distribution, we fit the parameters of a Jacobi process, defined on the open interval ]-0.2% ; 1.8%[. In this framework, we are able to compute, at any point of our sample, the 5% and 95%-percentiles on a week, which stem from the dynamics of the diffusion. Then, we compare the 1 week variations of the sample to the theoretical percentiles. Chart 2.1 displays these quantities1 as functions of the swap rate. We have sub-tracted the spot level of the rate from the percentile, in order to compare directly the variations with the percentiles. This graph also shows the specific form of the Jacobi process percentiles as a function of the spot.

It appears that the proportion of variations outside the confidence interval is in line with the level of the percentiles. We have 9.1% outside the interval, for a theoretical value of 10% (5.3% below and 3.8% above). This shows that, for this rather short term swap rates, the historical data is compatible with a diffusion admitting negative rates greater than -0.2%. This still works with more negative lower boundaries, but

1. The percentiles are computed with a Gaussian approximation based on the real first 2 moments of the Jacobi process. Numerical experiments with Monte Carlo simulations show that, in the case of such short time horizons, the approximation is quite accurate.

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the fit is less accurate and the proportion of observations outside the percentiles moves away from 10%. For instance, with a lower boundary of -0.25%, we observe 8.8% of increments outside the percentiles. Not surprisingly, this change is due to the proportion below the lower percentile (negative part of the distribution), which decreases with the lower boundary.

The same numerical application on longer rates (10 year for instance) does not seem to provide diffusion parameters compatible with negative rates: theoretical quantiles are much larger than the empirical ones. It implies that only a small amount of observation is outside the confidence interval (around 6.5%). Besides, when allowing negative rates, the fit of the Jacobi distribution is less efficient than for positive rates only. Chart 2.1.1 displays the empirical sample and the theoretical percentiles for the 10 year swap rate, with a lower boundary of 0%.

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95%-percentile 5%-percentile 1 week variations

Figure 2.1: 1 week variations of the 3 year Euro swap rate and theoretical 95% and 5%-percentiles on 1 week, minus the spot level of the rate

Source: Natixis Asset Management. Sample from 11/10/2011 to 26/01/2016

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95%-percentile 5%-percentile 1 week variations

Figure 2.1.1: 1 week variations of the 10 year Euro swap rate and theoretical 95% and 5%-percentiles on 1 week, minus the spot level of the rate


Thus, the negativity of 3 year swap rates is compatible with moderately negative lower bounds (say between -0.2% and 0.3%). The occurrence of negative rates for longer maturities – not observed in the sample – does not seem likely with the type of diffusion involved (e.g. for the 10 year swap rates).

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2. IMPLICATIONS OF LOW RATES ON THE PRICING OF LONG

RATE INSTRUMENTS

Even if the negativity of swap rates (especially 10 year swaps) is rather uncommon in the historical distribution, the market prices reflect this possibility. This was par-ticularly glaring in April 2015, with rates at their minimum and skyrocketing implied volatilities. When such a situation occurs, the modelling of interest rate by the mean of classic models may not reflect properly what is priced by the market prices.

a. Blunt SABR or Light SABR?

First of all, the possibility of long term negative rates – say 10 year Euro swap rate – questions the classical modelling of (vanilla) swaptions. It triggers issues concerning the hedging of low strike floors, for instance those embedded on notes indexed on long term rates. The model generally used to quote the volatility was the SABR model, introduced by Hagan et al. (2002). This stochastic volatility model was mainly favored by practitioners for two main reasons:

> It offers closed form formula for implied volatility approximations within the classic Black & Scholes framework

> It gives a straightforward P&L explanation in terms of its internal parameters

The SABR model, however, is based on a Constant Elasticity Volatility (CEV) model with a stochastic lognormal volatility. The CEV model only allows negative rates in the normal case, which is one of the two polar limits of the CEV model (the other is the lognormal case). Besides, the approximation in the implied volatility formula may induce arbitrage in the model for very low strikes or long date options, as dis-cussed in Berestycki, Busca and Florent (2004) and Doust (2012). The limitations of the SABR model for low strikes raise several issues in the construction of the implied distribution of swap rates, which is at the core of the Constant Maturity Swaps (CMS) pricing.

In the past few months, the sharp decrease of swap rates (between June 2014 and April 2015 the 10 year Euro swap rate fell from 1.6% to 0.45%) questioned the validity of this framework. It became more crucial to take into account the implied low floor options embedded in many floating rate notes. Notes with coupons indexed on CMS are popular among insurance companies, which are generally looking for long term rates indexations. Basically, a CMS indexation consists in receiving a swap rate of a given maturity – say 10 years – on different future dates. One of the simplest type of pay-offs consists in a coupon equal to “10 year swap rate + margin”, capped and floored at given levels. Let us denote by m the margin and by K the floor rate. Basically, it means that the note holder has a floor option on the 10 year swap rate, with strike K-m. Some of these notes, issued a few years ago, have implied floors with negative or slightly positive strikes. The banks, which have structured these deals, need to hedge these options which have an increasing value in the context of low interest rates. This accounts for a change in the structure of the smile. A part of the distribution, and therefore of the options values, has moved from high strikes to small strikes.

In this context, it can be interesting for investor to remove the floor of such notes or to accept negative floors in order to increase the yield of their investment. This amounts to sell "out-of-the-money" floor options, which have a value while remaining a remote risk. Indeed, it has to be observed that, in all this section, we are dealing with implied distributions of the interest rates. For the moment, no negative 10 year rate has been observed, and the forward 10 year interest rates are positive.

14


Chart 2.2 displays the implied distributions of a 10 year CMS rate, in 5 years, in the normal and lognormal models. The 2 distributions are calibrated so that they yield the same price for a caplet of maturity 5 years on this CMS rate, with a strike equal to 40% of the money. With the normal distribution, the CMS has an implied probability of being negative equal to some 8.6%. Chart 2.2 also represents the displaced lognormal distribution, introduced below.

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

-4.0% -2.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0%

Normal Log-Normal Displaced log-normal

Figure 2.2: Normal and lognormal implied distributions of a 10 year CMS rate, in 5 years

Source: Natixis Asset Management

b. Shifting the distribution to negative rates

In order to deal with the pricing of vanilla products under low rate environment, simple modifications of the usual framework can be applied, which does not hinder the tractability of the original pricing models. Let us consider the simple case of a Black & Scholes model. We consider the forward of a certain security with matu-rity T, denoted by Fwd(t,T), with 0≤t≤T. In the standard Black & Scholes model, the diffusion of the forward rate is given by F(0) = Fwd(0,T)>0 and

Here W represents a standard Wiener process, under the T-forward risk-neutral probability, and σ>0 is the constant Black & Scholes volatility of the forward rate. This equation admits a (strong) solution, which is, almost surely, positive (exponential martingale) and centered on Fwd(0,T) under the T-forward risk-neutral probability. In order to take into account negative rates, it is possible to use a so called displaced diffusion or shifted lognormal process. Let G(t) := Fwd(t,T)+θ, with θ>0, and

Here, γ>0 represents the volatility of the shifted forward. Therefore, the forward can be negative at any time t with probability P(t)

We have denoted by Φ the cumulated distribution function of the normal law.

eq. 60 ≤ t ≤ T ,

A dF (t) = σF(t)dW(t)

eq. 70 ≤ t ≤ T ,

A dG (t) = γG(t)dW(t)

eq. 8P (t) = Ф ln + tθ

Fwd(0,T ) + θ γ²2

γ�t

15


In the displaced diffusion framework, the option pricing is equivalent to an option pricing under the original model up to a change of strike. Another consequence of the displaced diffusion is to add skew to the Black & Scholes model. The displaced diffusion was originally used for the purpose of skew modelling. See, for instance, Brigo & Mercurio (2006).

c. Models of the term structure

A rich academic literature is dedicated to the construction of stochastic models of the term structure. These models are mainly designed to price contingent claims, which depend on the whole interest rate curve and its evolution. Basically, they can be divided into three categories:

> Short-term rates dynamics. Classic example are Vasicek (1977), and Cox, Ingersol and Ross (1985), or Black and Karasinski (1991)

> Forward rates dynamics. The founding model is given by Heath, Jarrow and Morton (1992). It is generally used in its linear, Gaussian and Markov form (LGM model)

> Market instruments dynamics. For instance Libor and swap market models, introduced in Brace, Gatarek and Musiela (1997)

The short term and forward rates are linked by the following relation, which provides the price of a zero-coupon of maturity T, at time t:

Here, E{.} is the expectation operator under the (spot) risk-neutral probability and Ft , t≥0, denotes the filtration, which represents the available financial information at each date.

Some models postulate positive short-term rates, such as Cox, Ingersol and Ross (1985), or Black and Karasinski (1991). Other specifications, based on Gaussian models, admit negative rates (Vasicek (1977), Heath, Jarrow and Morton (1992)). On the other hand, the market models specifies lognormal dynamics for market instruments -Libor rates, in Brace, Gatarek and Musiela (1997) – and therefore exclude negative market rates.

Let us consider the LGM model calibrated on the 10 year swaption, at the money, with an expiry in T0=5 years. Let Tn= T0 +10 years = 15 years be the maturity of the underlying swap. Under the T0–forward probability, for any t ≤ T0, we can compute the probability of the forward swap rate S(t,T0 , Tn) to be negative. Basically, using a classic approximation of the swap rate, we can say that S(t,T0 , Tn) is negative if, and only if, B(t, Tn) > B(t, T0).

In the LGM model, a choice for the volatility of the zero-coupon of maturity T is

Let us assume that the instantaneous volatility σ is constant and the mean-rever-sion of the volatility β is equal to 0.05 year-1. For the following, numerical example, the valuation date is June 9th 2015. The underlying at the money volatility of the corresponding swaption – in a Black & Scholes framework- is equal to 37.7% year -1. For t=2 years, the probability of having a negative swap rate S(t,T0 , Tn) is equal to 0.9%. For t=4 years, it is equal to 6.0%, and for t=5 years, it is equal to 9.1%. In this example, the curve is rather steep between 5 and 10 years, and we have a forward rate around 2.04%:

The field of risk-neutral probabi-lity models is more deeply impacted by the low/negative rates environ-ment than the field of historical pro-bability model. Indeed, the market prices can significantly weigh the occurrence of negative rates, even for long term rates. This context requires some adjustment in the pricing methodology, because nega-tive rates are generally excluded from many widely used interest rate models. Moreover, some of these models are less accurate for low rates (such as SABR models). Last, embedded options in many financial products (such as floors on coupons of structured notes) introduce a sen-sitivity of these products to the way negative rates are handled. Howe-ver, straightforward modifications, such as distribution shifting, can adapt this traditional framework to the current situation of rates under risk-neutral probability.

eq. 9B (t,T ) := E{e } = exp tT

F-�t r(s)dsT

-�t f(t,s)ds|

eq. 10y(t,T ) = σ (t) 1-e -β(T-t)0 ≤ t ≤ T,

Aβ

16


eq. 11�515 f(0,s)ds � 2.04%x10

These results only depend on the interest curve and the implicit volatility of the swaption. In particular, the choice of β>0 has no influence on the result (because of the calibration procedure). But it has to be observed that the volatility of the same instrument was much higher in March and April 2015, with peaks above 80%.

3IMPLICATIONS OF LOW RATES ON RISK MODELLING

Rates and spreads volatilities are traditionally assumed to be Gaussian, i.e. constant irrespective to the underlying level, or lognormal. In a lognormal framework, volatility is proportional to the level of the underlying whether rate or spread. Should the underlying fall to zero, then its volatility will also drop to null. Moreover lognormal rates or spreads cannot be negative in which case volatility will be negative too! This paradigm has been challenged by the recent evolution of rates and spreads. The implications of these observations are particularly important in the case of risk models. In fact low yield environment does not necessary mean low risk. In a first section we examine the risk return trade off when rates level reaches their lower boundary and question in section B the adequacy of volatility as a risk measure. In section C we adress adjustments to conduct in order to cope with the underesti-mation of risk. The first improvement affects the volatility of the risk factors, while the second concerns the sensitivity to those risk factors.

1. THE RISK-RETURN TRADE-OFF IN A LOW RATE ENVIRONMENT

When rates are low, the risk-return trade-off is affected. The distribution of future rates movements become skewer since rates’ hikes are more likely to occur than rates’ tightening. This pronounced asymmetry in rates’ distribution hampers the volatility of rate as a risk measure.

a. The return

The total return over the period TRt→t+1t→t+1 of a bond bought at a yield rt can be approximated by the following equation:

where Dur denotes the duration of the bond and dr the magnitude of variation of the yield. The probability distribution of the yields’ movements is asymmetric since the latter are floored by rfloor. rfloor denotes the threshold under which rates cannot drop. This floor is typically null, though in current market conditions it moved to negative area as argued in section I. Yields’ movements may thus vary inside a semi bounded interval: and the total return ranges between . While the potential gain is bounded, potential loss is theoretically unlimited. It is the well-known asymmetry of return of bonds inherited from the asymmetry of

eq. 12TRt→t+1 � - Durt x drt→t+1 + rtPrice return Carry return

TRt→t+1Є ∞; rt (1+Dur)- Dur x rfloor

drt→t+1Є[-(rt -rfloor );+∞[TRt→t+1Є ∞; rt (1+Dur)- Dur x rfloor

drt→t+1Є[-(rt -rfloor );+∞[

17


their yields. Still when yields level is fairly high, it is not uncommon to assume a symmetric distribution for the sake of simplicity. However, when yields approach their floor, little room is left for narrowing, and widening movements are more likely. Carry component becomes negligible, if not negative, and may not offset an adverse movement. Yields behave as if they were truncated: only upside move-ments are allowed. This also holds for each of the yield’s traditional components, i.e. the reference rate yt and the spread St , with rt=yt+St.

Assuming that rates movements follow an elliptical distribution2 over the long run, one can check on chart 3.1 how the distribution of rates shifts to the right increasing the likelihood of rates hike and consequently the probability of occurrence of nega-tive total return, when the truncation becomes more important. The asymmetry of the conditional distribution becomes more pronounced as the truncation threshold increases. One can notice the impact of truncation threshold on skewness of rates as depicted in chart 3.2. The higher the truncation threshold, the skewer the dis-tribution is, boosting thus the likelihood of higher rates. In this simple framework, the lower the yields are the higher the probability of an adverse movement will be. Still the volatility does not capture this risk.

0%2%4%6%8%

10%12%14%16%18%20%

-4,0% -3,0% -2,0% -1,0% 0,0% 1,0% 2,0% 3,0% 4,0% 5,0% 6,0%

ratesr|r>-2%r|r>-1%r|r>0%

Figure 3.1: Distribution of truncated rates with initial rate level = 1%, volatility of the rates = 1%. r|r>rfloor denotes the conditional distribution of rates above rfloor


0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

r r|r>-3% r|r>-2% r|r>-1% r|r>0%

Skew

ness

Figure 3.2: Skewness of rates as a function of the truncation threshold with initial rate = 1 %, volatility of the rates = 1 %


2. Elliptical distributions are symmetric ones. They are widely used in risk assessment and allocation models. Refer to Meucci (2005).

18


b.The risk

The impact of truncation on volatility can be analytically computed in the case of Gaussian distributions. The variance of a truncated Gaussian variable u→N(μ,σ2) is given by the following equation:

eq. 13σ2(u|u > c) = σ2 1+ M - -M 2 -c-μσ

c-μσ

c-μσ

where M(c)= (c)(c)Ф

Ф

(.)

(.)

σ(u|u>c)<σ(u)

denotes the ratio of Mill, i.e. the ratio of the density

M(c)= (c)(c)Ф

Ф

(.)

(.)

σ(u|u>c)<σ(u)

to the cumulative Gaussian distribution

M(c)= (c)(c)Ф

Ф

(.)

(.)

σ(u|u>c)<σ(u)

evaluated at truncation threshold C. It is worth mentioning that the volatility of the truncated distribution is lower than that of the unconditional one, i.e.

M(c)= (c)(c)Ф

Ф

(.)

(.)

σ(u|u>c)<σ(u), which bears out the assertion that volatility understates the risk of low yield bonds3.

Following that, one can compute the volatility of the truncated distribution of rates, i.e. the local conditional volatility, and compare it to the longer term unconditional one. This is done on chart 3.3. The volatility of the one week variation of the 3 year Euro swap rate is about 39bp/y from 11/10/2011 till 26/01/2016. Over that period, we witnessed a drop of the 3 year swap rate from 1.69% to -0.08%. Should rates remain above -0.20% then only a movement above -12bp is authorized. The vola-tility of the conditional distribution is thus reduced by 4bp/y. In fact the difference between the lower boundary of rates and their actual level defines the cutoff threshold for the conditional distribution. Should rates pursue their tightening and reach their lower boundary, assumed at -0.20%, then the annualized volatility should drop till 22bp. It is interesting to note that the volatility does not drop to null even if rates fall below -0.20%.

0

5

10

15

20

25

30

35

40

-0,20% -0,15% -0,10% -0,05% 0,00% 0,05% 0,10% 0,15% 0,20%

vola

tility

[bp/

y]

cutoff threshold

Volatility of truncated rates for various thresholds

Figure 3.3: The annualized volatility of weekly variations of the 3 year Euro swap rate as a function the cutoff below which the rate distribution is truncated


2.VOLATILITY IN A LOW YIELD ENVIRONMENT

Given equation (1) one can deduce that the volatility of TR can also be approximated by the following equation:

3. Teilletche (2015) argues that skewness “can be deemed as a sensible risk measure”.

In a low rates environ-ment, rates’ hikes are more likely to occur than rates tighte-ning. It results in a pronounced skewness of the distribution of rates. The risk-return trade-off is hence understated when volatility is retained as a risk yard stake.

eq. 14σ(TRt→t+1 )≈ Durt x σ(drt→t+1 )

19


As previously explained, in a low yield environment, given the asymmetry of yield’s distribution, volatility would underestimate the downside risk. Which raises a number of questions: what is a good risk model? How to model σ(drt→t+1 )

σ(TRt→t+1 )

σ(TRt→t+1 )� Durt x σ(drt→t+1 )

in such a context? How

σ(drt→t+1 )

σ(TRt→t+1 )

σ(TRt→t+1 )� Durt x σ(drt→t+1 )

behaves as rates approach their floor level? Does the approximation

σ(drt→t+1 )

σ(TRt→t+1 )

σ(TRt→t+1 )� Durt x σ(drt→t+1 ) still hold?

a. What is a good risk model?

A good risk model should allow for accurate forecasting of the selected risk mea-sures (TE, Value at Risk or Expected Shortfall) over the investment horizon. These measures are historical estimation of the future risk level. An annual ex ante TE of 100bp means that if the portfolio composition remains unchanged, then the TE computed over a year horizon should be around 100bp.

The validity of risk models and consequently of related allocation models relies on their ability of forecasting the future risk based on previous data with the implicit axiom that the future resembles the past.Thus, it is of primary importance to iden-tify the law of motion of the market in order to achieve an accurate risk projection.

If in physics it is possible to identify absolute constancy, e.g. the speed of light upon which stable laws are derived (E=mc2), finding an equivalent for the speed of light in finance is practically impossible. We seek market invariants whose dynamics are mildly stable, i.e. risk factors that are locally stationary and useful for projecting the P&L of the portfolio. Failing that, practitioners settle for computing the product of sensitivity by the volatility of risk factors, mainly rates and spreads for fixed income securities.

b. Modelling the volatility of factors

Traditionally, we distinguish two main approaches for modelling each of these factors, i.e. rate and spread. Either their absolute changes or their relative or log are modelled, leading respectively to the normal model and lognormal model4.

i. Normal volatility In the normal volatility model, the volatility of changes in rates is considered to be locally stationary. The volatility of a bond would then be directly linked to that of rates. The volatility of rates is assumed to be independent of the rate level, e.g. should rates widen from 2% to 4% their volatility remains the same. Rates dyna-mics are governed by the following equation:

with K a positive scalar and a random variable of unit variance, which leads to the above mentioned constancy of rates volatility, i.e. σ(dy)=k . In a benchmarked process, an active exposure of 1 year of duration would generate the same TE irrespective of rates level. Similarly, if OASD is retained as a credit risk yardstick, then spread volatility is assumed to be the same whether spreads are low or high.

ii. Lognormal volatilityAlternatively, some argue5 that a bond’s risk is directly proportional to its interest rate level. The volatility of rates is thus proportional to rates level:

4. The terms normal and lognormal are not related to the Gaussian distribution which is by no means necessary since randomness can be modelled by a Student distribution or any other suitable probability law.

5. Douady (2013).

eq. 15dy = ĸ.ε

eq. 16= �.η σ(dy) = �.ydyy

20


with denoting a positive scalar and η a random variable of unit variance. The corollary of this assumption is that risk should drop to null as rates approach zero. Modelling the relative changes is particularly useful for spreads, since the volatility of spread variation is found to be proportional to spread level. We can conclude that the volatility of their relative variation is stable in time6 and that the DTS would then be a superior measure of sensitivity to spread risk. However, lognormal processes do not allow for negative rates or spreads. Such conclusions are not compatible with current data where rates and spreads are low without observing a proportional drop in their volatilities. We represent on Chart 3.4 a normal and a lognormal diffusion with similar para-meters, i.e. same initial value and a one year volatility of 1% for both frameworks. We notice that lognormal dynamics lead to higher volatilities (resp. lower) than those obtained under normal assumptions for forecasting horizons above (resp. under) one year.

0,00%

0,20%

0,40%

0,60%

0,80%

1,00%

1,20%

1,40%

1,60%

1,80%

2,00%

0,00 0,50 1,00 1,50 2,00 2,50

Vola

tility

Years

Normal Log Normal

Figure 3.4: Normal vs lognormal volatility as a function of time horizon, with the one year volatility level of 1% for both specifications


c. When volatility is not enough

One of the main reasons to set aside the lognormal framework (such as in Black and Karasinski (1991) for the short rate or in Brace, Gatarek and Musiela (1997) for the swap rate) is the fact that, as the rate goes down to zero, the volatility (which is a linear mapping of the rate) must go down to zero. Empirical observation shows that this it is not the case, as displayed by Chart 3.5. In this chart, we see the evolution of the rates and the trailing empirical volatility coefficient (100-business days). Thus, we see that this series is not compatible with a lognormal process, especially when the rates sharply decrease after November 2014. This last period of the sample clearly shows a change of regime if the volatility behavior.

We have proposed, in Section II-A, a different diffusion process to fit this series and in particular another form of volatility. In this subsection, we propose to keep the lognormal framework, and, therefore, the (almost surely) positivity of the rates, but to add jumps. This means that the lognormal behavior still exists but, when the

6. Ben Dor et al. (2005).

21


rate goes down to zero, the occurrence of jumps can produce a certain amount of variance, which is not compatible with the variance stemming from the diffusion part of the process. The low rate environment seems to be compatible with a non-vanishing variance, when we accept to exit the settings of continuous processes.

0

0,2

0,4

0,6

0,8

1

1,2

0,00%

0,50%

1,00%

1,50%

2,00%

2,50%

3,00%

26-fé

vr-11

14-se

pt-11

1-avr-

12

18-oc

t-12

6-mai-

13

22-no

v-13

10-ju

in-14

27-dé

c-14

15-ju

il-15

31-ja

nv-16

18-ao

ût-16

Vola

tility

(p.a

.)

rate

(P.a

.)

10Y EUR Swap rate Trailing Log-normal Volatility

Figure 3.5: 10 year Euro swap rate and Trailing empirical lognormal volatility coefficient


Basically, a simple lognormal jump-diffusion process can be given as follows:

where the Tn, n ≥ 1 represent the times of jump of a Poisson process N, inde-pendent from W, and the δn, n ≥ 1 are random variables, independent and identically distributed, independent from N and W, and almost surely greater than -1. We have fit the coefficient of this jump-diffusion on the same sample of 3 year Euro swap rate. For this purpose, we remove the extreme parts of the distribution of the log-increments (upward and downward). Then, we compute the parameters μ and σ, which drive the Brownian part of the process, on the remaining part of the distribution, in order to minimise the gap between the empirical percentiles and the theoretical ones. Chart 3.6 displays the empirical percentiles of the Brownian part in the jump-diffusion process, the theoretical percentiles of a Brownian motion, and the empirical percentiles of a pure diffusion process. We can see the accuracy of the jump-diffusion approach: its empirical percentiles perfectly fit the theoretical ones. The empirical percentiles for the pure diffusion show significant discrepancies with the theoretical ones. The volatility coefficient σ in the jump-diffusion model, is equal to 34% year-1, whereas the volatility coefficient in the pure diffusion case is equal to 49% year-1. The difference is explained by the jumps, the intensity of which is around 26 year-1 (The expected number of jumps is of 26 by a year). The expected size of the jumps given by the δn, n ≥ 1, is 11%, when δn is positive, and -10%, when δn is negative. The estimated probability of positive jumps is similar to the estimated probability of negative jumps. It can be noticed that the presence of jumps put more weight on extreme events than a simple lognormal diffusion process. This type of model can also be used, precisely for this reason, to fit a smile in the case of option pricing.

Volatility of rates (and spreads) are traditionally consi-dered to be either Gaussian or lognormal. While in the latter volatility is proportional to the underlying level, it is assumed to be constant in the former. Both modeling approaches are challenged when rates drop to null or become negative. One elegant alternative consists in allowing for jumps in the tra-ditional diffusion approach in order to gain accuracy while fitting the historical distribution.

eq. 17r(t)=r(0)xe x [1+δn]µxt+σWt- σ

2t12 ∏

Tn≤t

22


-3

-2

-1

0

1

2

3

4

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Jump Diffusion Theoretical Pure Diffusion

Figure 3.6: Empirical percentiles for the Brownian part of the jump diffusion, the pure lognormal diffusion and the theoretical Brownian motion.

Empirical distribution fit on the 10 year Euro swap rateSource: Natixis Asset Management. Sample from 11/10/2011 to 26/01/2016

3. THE VOLATILITY OF TR AND THE ADEQUACY OF USUAL

SENSITIVITY MEASURES, DURATION AND DTS, IN A LOW YIELD

ENVIRONMENT

For small figures of spreads, the volatility of spread is not null. Thus the propor-tionality law is not observed for low spreads. Volatility seems to decrease as spreads tighten until reaching a floor below which the proportionality law is no longer observed.

Volatility seems to be “normal” below a threshold and “lognormal” above it. Bond credit risk is thus proportional to its spread as long as the latter is above the thres-hold. Should the spreads drop below the threshold, volatility will flatten as it reaches an incompressible floor. Similar results were found by Douady (2013)7 in the case of rates. One explanation for the existence of a floor stems from the assumption that spread volatility is not driven solely by changes in risk but results from various noises caused by pricing errors and other frictions in the market.

How to address this issue?

Addressing this issue can be achieved in two ways not mutually exclusive: by adjusting volatilities or by adjusting sensitivities, i.e. DTS.

a. Adjusting the volatilities

> Shifting the lognormal distribution into negative territory, i.e. by translating rates by a fixed negative number. Hence, rates variations are proportional to rates level plus a positive constant f and rates volatility is proportional to rates level plus that constant:

Would rates drop to zero their volatility will not be null but equal .f.

7. Douady (2013)

eq. 18dy

(y+f ) =�.η σ(dy)=�.(y+f)

23


> Adding a normal noise to the unshifted lognormal distribution. Hence, the risk factor volatility is assumed to be the sum of two noises: a normal one that acts as a structural incompres-sible volatility level when rates approach zero and a lognormal one increasing when rates widen and dominating the first term.

with . This modelling is found to better fit actual data for

rates and spreads8, although the structural volatility level is found to be different from one market to another.

> Transforming rates by applying an inverse-call likewise function. The rationale behind this approach that finds its theoretical root in the work of Black (1995) consists in considering rates as call options on some shadow rates. The approach has been recently generalized by Meucci and Loregina (2014). It tolerates negative rates and allows for a smooth shift from normal to lognormal regimes.

b. Adjusting the sensitivity

The standard expression of DTS assumes proportionality of volatility of TR to spread level. This implies that volatility of TR should drop to null as spreads approach zero. As a consequence DTS might understate risk when spreads are very low. In order to address this issue, one should floor DTS when spread are below a certain threshold Sfloor:

This correction implies that DTS remains the appropriate sensitivity measure for spreads above Sfloor. Should spreads drop below Sfloor, then the sensitivity switches to spread duration and the volatility from lognormal to normal regime. Sfloor can be estimated by maximum likelihood techniques as explained in Desclée et al. (2015).

Approx imat ing the volatility of returns by the duration times the volatility of rates’ movements raises the question of the accuracy of this approximation when rates become negative. This issue can be addressed by adjusting either the volatility of rates or the sensitivity of returns to a movement in rates. Sensitivity can be floored by a threshold level precluding breaches in the approximation “r isk equals sensit iv i ty times volatility”. Adjusting the volatility of rates can be done either by shifting the distribution of rates into positive territory or by ading a normal noise to the unshifted lognormal distribution.

8. Refer to Ben Dor et al. (2005, a), Ben Dor et al. (2005, b) and Douady (2013).

eq. 19dy = y.�.η+χ.ε σ(dy)= ��2.y2+χ2

eq. 20DTS = OASD x max(s,sfloor)

χ > 0

24


CONCLUSIONWe have seen that there are several reasons explaining the negativity of interest rates today. This has become a common situation which can last for a long time.

As a consequence, asset managers have to deal with these low levels of rates and to adapt both pricing and risk management methods and measures to this never-before-seen situation.

We have shown that straightforward modifications, such as shifting the distribu-tion, adding floors to the commonly used risks indicators or adjusting the volatility of rates, can help improving commonly used mathematical/statistical models and increasing their efficiency to low rates environments.

Even if this is not the subject of this research paper there is also lots of work on the portfolio management side. Portfolio managers work on how to create value and performance in a world where carry is very low and even negative on certain assets. Different strategies can be put in place such as relative value or slope strategies , in order to have non negative performances in a negative rates world.

25


REFERENCES

Ben Dor, A., L. Dynkin., P. Houweling, J. Hyman, E. van Leeuwen and O. Penninga. 2005, a. A New Measure of Spread Exposure in Credit Portfolios, Lehman Brothers Fixed Income Research.

Ben Dor, A., L. Dynkin and J. Hyman. 2005, b. DTS – Further insights and applicability, Lehman Brothers Fixed Income Research.

Berestycki, H., J. Busca and I. Florent. 2004. “Computing the implied volatility in stochastic volatility models.” Communications on Pure and Applied Mathematics, No. 57(10), pp. 1352–1373.

Black, F. 1995. “Interest rates as options”. In Journal of Finance, No. 50.

Black, F. and P. Karasinski. 1991. “Bond and option pricing when short rates are lognormal.” Financial Analysts Journal, No. 47, pp. 52-59.

Brace, A., D. Gatarek and M. Musiela. 1997. “The Market Model of Interest Rate Dynamics.” Mathematical Finance, No. 7(2), pp. 127-154.

Brigo, D. and F. Mercurio. 2006. “Interest Rate Models - Theory and Practice.” Springer-Verlag Berlin Heidelberg.

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