The challenges of modelling negative interest rates

November 03, 2020 General

Plaatje blog negatieve rentes min

In 2009, the Swedish Central Bank (SCB) introduced negative interest rates for the first time in history (Christophe Madaschi, 2017). The European Central Bank (ECB) followed in 2014 and introduced negative interest rates to boost the economy during the aftermath of the global economic crisis. Since changes in interest rates affect a bank’s profitability, it is of utmost importance to have well performing interest rate models. However, not all interest rate models are able to deal with negative interest rates. In this blog, we stress the importance of good interest rate models, review various candidate models and provide solutions in case traditional models do no longer function with negative interest rates.

Negative interest rates

The idea behind the introduction of negative interest rates was twofold. First, a negative interest rate stimulates economic consumption, since consumers will be less inclined to save money. Secondly, funding for companies becomes cheaper and therefore more attractive, and will stimulate economic production. Combined, these two effects should lead to an economic growth stimulus.

Importance of appropriate interest rate models

For all banks, but in particular for banks with a lot of retail deposits, interest rate models are needed that can deal with a negative interest rate policy (NIRP). Banks with such models can create a competitive advantage because appropriate interest rate models contribute to sound risk management, and also because they are able to better adapt their business to changes in the interest rate environment.

The importance of appropriate interest rate models is also recognized by regulators. In July 2018, EBA guidelines on IRRBB (EBA/GL/2018/02) were finalized as part of Pillar II of Basel’s capital framework. These guidelines discuss Interest Rate Risk in the Banking Book (IRRBB), i.e. the risk that adverse movements in interest rates affect the earnings and the economic value of an institution. The guidelines state that in low interest rate environments, institutions should also consider negative interest rate scenarios. Therefore, it is not only the competitive advantage aspect, but also the regulatory requirements that stress the need for interest rate models that can deal with a NIRP.

It is important that the interest rate models are aligned with its purpose. For the IRRBB, different interest rate models are needed than for example the trading book. For the trading book it might be preferred to use a more complex interest rate model in order to accurately capture the volatility smile. The drawback of these complex models is that more simulations are needed to compute the present value of derivatives. When institutions hedge on a frequent basis, the question arises whether these complex models are the most convenient. A less accurate model, which is easier to use, can be preferred in this situation.

What interest rate models are commonly used and what are their limitations in a negative interest rate environment?

To forecast interest rates, in practice often short-rate models are used. Short-rate models are mathematical models that model the future evolution of the short-rate over time. However, in the current negative interest rate environment, some of these models lose their functionality or their predictive power. The specifications of these models can for example be restricted to allowing only positive current short-rates or assume a log-normal distribution for the interest rates, which results in a lower bounded interest rate of 0%. These model characteristics prevent the interest rate models to predict negative interest rates in a negative interest rate environment. However, some of these restrictions can be circumvented, as we will discuss.

One of the short-rate models that has problems with modelling the evolution of interest rates in a negative interest rate environment, is the Cox-Ingersoll-Ross (CIR) model. The CIR model uses the square root of the current short rate as input to forecast interest rates. Although imaginary interest rates sound like an interesting concept, practically speaking the square root of negative short rates is not defined.

The horizon for which short-rate models provide accurate predictions, can be relatively short. Therefore, for predictions on longer horizons, other types of interest rate models might be preferred. An often preferred model is the LIBOR Market Model[1], which models a set of forward rates for different periods that are directly observable in the market. A downside of the LIBOR Market Model is that it assumes that the forward rates are log-normally distributed. The log-normal distribution restricts the forecasted interest rates to be positive, resulting in the fact that negative interest rates can’t be predicted by this model.

What adjustments can be made to the models to make them suitable under a NIRP?

Some interest rate models, such as the LIBOR Market Model and the Cox-Ingersoll-Ross (CIR) model, are not able to forecast negative interest rates. For these interest rate models, the model specification needs to be adjusted before they can be used to properly model the interest rate movements in a negative interest rate environment.

The non-negativity limitation of the LIBOR Market Model can be solved by introducing a shift in the log-normal distribution, which allows for negative interest rates. For the CIR model an additional parameter can be added to the model specification to ensure that the current short-term interest rate is shifted to a positive value, such that the square-root transformation is defined (Orlando, Mininni & Bufalo, 2019).

A potential drawback of this adjustment for the LIBOR Market Model is that, since the log-normal distribution is shifted, the distribution is still capped by a negative boundary. How to implement the shift also introduces expert judgement and possibly frequent recalibrations in a declining interest rate environment. This also is a drawback for this adjustment related to the CIR model, since the shift must be large enough to ensure that the current short rate becomes positive and in a declining interest rate environment this shift needs to be constantly larger.

What model alternatives are better equipped to deal with a NIRP?

A popular choice in the current low interest rate environment is to use shadow rate term structure models. These models use a shadow short rate rather than the actual short rate . The actual short rate is defined as:

that is, the short rate equals the shadow rate, if this is above the lower bound , while the actual short rate remains at the bound if the shadow rate is below the bound. In order to account for negative interest rates, the lower bound can be set to a negative time-dependent value. Shadow rate models can capture the decline in yield volatility far better than a popular benchmark model that ignores the presence of a lower bound. (Lemke & Vladu, 2017).

Other popular choices are regime switching interest rate models. By means of regime switching techniques, different models are used depending on the environment at hand: negative, low, normal or high interest rates. This has the advantage of always using a suitable model, whatever the level of the interest rate may be. A drawback of regime switching models is that the model increases in terms of complexity. Traditional short rate models are easier to interpret and are often arbitrage free and provide closed form formulas. The increase in complexity of the regime switching interest rate models makes it more difficult to calibrate the model on the market data.


The NIRP affects the profitability of a bank, and therefore suitable IR models are needed that can deal with this situation. Not only is this desired from a competitive advantage perspective, but it is also a regulatory requirement. While some models are still valid, other models need adjustments either by changing their specification, or combining models into one model using regime switching techniques.

At RiskQuest, we are well aware of the effects of the NIRP, both on banks and on interest rate models. We are also mindful of the importance to align the complexity of the model with its purpose, where complexity should be balanced with frequency of use. We are ready to advise and provide the best possible IR models for each situation and tailor this for specific needs.

For more information about IRRBB, negative interest rates or specific interest rate models, please contact Lino Kragtwijk (consultant) or Nick Verbaas (consultant)

If you want to join our RiskQuest team, please check our current job openings here

[1] Note that the LIBOR Market Model can also be used to model other interest rates than the LIBOR