Fluctuation in the general price level of goods and services is reflected through the inflation rate. Despite rather small changes in the inflation rate over the past two decades, the last recession created the potential for price instability.
Before the Great Recession, inflation risk was typically associated with high inflation rates. Events such as the Great Inflation of the 1970s are notable for steep inflation rates and could explain, to some extent, the common misconception that links inflation risk to high inflation rates. Yet, lately, inflation risk has become much more symmetric, and a fear that inflation may fall too low has emerged (so-called deflation). For instance, some recent actions undertaken by the Federal Reserve (Fed) in the US have been motivated by this fear of deflation, and an important number of European countries have had deflation issues in the past few years. Deflation is one of the most challenging risks faced by central banks nowadays.
In the long-run, insurance contracts, reserves, investment policies, and pension liabilities are sensitive to unexpected inflation.1
Long-term liabilities could be much worse if the inflation rate suddenly changes (e.g., see Bohnert et al. [1
]). In addition, asset returns are tied up to the inflation, meaning that portfolio income could be quite dependent on the prevailing inflation rate.
Life insurers have an uncommonly large vulnerability to deflation risk: falling prices could lead to insufficient investment returns to support the rate credited to policyholders. Deflation is indeed “the worst possible world for a life insurance company” (see Mittelstaedt [2
]). Japan is a notable example of how deflation can impact the life insurance business. The historical Japanese inflation rate remained below zero for most of the last twenty years, and this episode of deflation provoked the bankruptcy of at least eight Japanese insurers since 1996.
Life insurers are also vulnerable to embedded optionalities in the design of their products. For instance, inflation-indexed annuities often provide additional protection that increases the paid amount in the case of positive inflation, but does not reduce the amount in the case of deflation. This added protection is similar to an inflation floor, as the annuitant is covered against decreases in the general price level of goods and services.
On a similar note, inflation-indexed defined-benefit pension plans also provide the same kind of protection to their participants: in many instances, a decrease of the Consumer Price Index (CPI) would not reduce the retirement benefit amount.2
As an increasing number of plan sponsors decide to de-risk their financial exposures by buying special annuities, they transfer their interest rate, longevity, and deflation risks to life insurance companies. These life insurers issue pension buy-in and buy-out annuities, similar to portfolios of inflation-indexed annuities, thus exposing themselves to additional deflation risk.
Oddly, inflation is only indirectly measured by insurance companies in general [3
], and Solvency II does not directly account for inflation risk. As deflation risk could have a significant impact on life insurers’ portfolios, it should be incorporated into the insurers’ enterprise risk management (ERM) framework consistently. Market-consistency is important, as an ERM framework must be able to capture the changing economic conditions. Further, new solvency regulations force insurance companies to assess their liabilities using “mark-to-market” approaches. As a matter of fact, market-consistency requires the use of models with sufficient degrees of freedom to integrate all relevant market data [4
]. In this spirit, this paper presents a market-based methodology for measuring deflation risk based on a discrete framework inspired by Fisher’s [5
] equation. The model accounts for the real interest rate, the inflation index level, its conditional variance, and the expected inflation rate. The real interest rate and the expected inflation rate both follow a first-order autoregressive (AR) model. The conditional variance follows the generalized autoregressive conditional heteroskedasticity (GARCH) model of Heston and Nandi [6
]. This approach is somewhat different from common practice, as we model both the real interest rate and the expected inflation separately.3
The dynamics used in this paper are in line with the inflation modelling literature. Fleckenstein et al. [7
] use a similar model to assess deflation risk, although their continuous-time model does not allow for heteroskedasticity in the index price level. A regime-switching first-order AR model for the expected inflation rate is proposed by Ahlgrim and D’Arcy [8
]. The authors’ model is a generalization of Wilkie’s [9
] inflation model, as it allows for different parameters, depending on the prevailing regime. Singor et al. [10
] consider a Heston [11
] type framework to model the inflation index, thus considering a dynamic variance process as is the case in this study. Ang et al. [12
] consider, among other things, autoregressive moving-average (ARMA) models and Phillips [13
] curve to model the inflation rate.4
As is commonly the case with tail risks, deflation is hard to quantify using classic econometric methods. For instance, Ang et al. [12
] use a large number of models based on time series of inflation to assess inflation forecasting, and conclude that these models perform weakly in forecasting the first moment of inflation. In addition, even though they conclude that survey data perform better, these provide little or no evidence of potential deflation risk.
In the light of this issue, assessing deflation risk would require more than just past inflation rate data. Fortunately, a considerable amount of inflation-dependent securities are available, and their prices shall inform us on the likelihood of deflation over a given time horizon. Inflation-indexed bonds are securities that protect the investors’ purchasing-power by adjusting the principal based on changes in an inflation index. Most of the developed countries issue these bonds, yet US Treasury Inflation-Protected Securities (TIPS) are the most liquid ones, with about $500 billion in issuance. Inflation swaps are also available for most developed countries. In the US, they were introduced at the same time as TIPS, in 1997. The notional size of the US inflation swap market is estimated to be hundreds of billions [14
]. A parallel market for inflation option exists; it started in 2002 with the introduction of caps and floors on the realized inflation rate. According to Fleckenstein et al. [7
], the inflation option market is sufficiently liquid: active quotations are available since 2009 for a broad range of maturities and strikes.
To estimate our model, multiple types of data source are used, since inflation index levels would only capture the average expected inflation rate and not the tails of the index level distribution. Characterizing the entire index level distribution is of paramount importance in this study, because deflation risk is related to the left tail of this distribution. Thus, in addition to inflation index levels, we include nominal risk-free bonds, inflation swaps, and inflation options to the model estimation, as these securities contain relevant information about the tails of the distribution. The four types of data used in this paper are similar to what other inflation studies have utilized. For instance, Ang et al. [12
] and Ahlgrim and D’Arcy [8
] use time series of realized inflation rates, which is technically the same as using the index level.5
Inflation option market prices were used by Singor et al. [10
], and Kitsul and Wright [15
]. Fleckenstein et al. [7
] employed swap prices and option data, as in this study.
An estimation method based on the unscented Kalman filter (UKF) is considered. All physical and risk-neutral parameters are estimated using likelihood maximization in a unique stage. The proposed method contrasts with other methods used in the literature (i.e., calibration techniques and multi-stage estimations). On one hand, calibration methods such as the one used by Singor et al. [10
] select the model parameters based on a given day’s available prices. Even though this methodology can give a set of parameters that is consistent with market prices, it is impossible to know whether these are robust in time. In addition, only risk-neutral parameters can be inferred, since physical parameters are not directly involved in the pricing of derivatives. On the other hand, multi-stage methods, such as the one presented by Fleckenstein et al. [7
], can be inadequate to capture the interaction among various model parameters, and also lack formal proofs of their effectiveness.
Another interesting consequence of using the UKF-based approach is that the expected inflation rate is forward-looking, as it is based on the current market prices. The model adapts readily to new market conditions, and the expected inflation rate is then consistent with investors’ expectations.
To estimate the model, US inflation data are used. The unconditional annualized long-term trend of expected inflation is about 1.8%, which is consistent with the Fed desired target range for inflation—between 1.7% and 2.0%. The evolution of the expected inflation is dependent on the current economic conditions: for instance, during the pre-recession era, the average expected inflation was about 3.4%, whereas its average is about 0.8% in the post-recession period. Deflation risk changes over time, with average 1-year deflation probabilities of 4%, 18%, and 29% for the pre-recession, recession, and post-recession eras, respectively.
The effects of deflation risk are assessed in the life insurance industry. In this spirit, the distribution of a fictitious life insurer’s future payments in present value is investigated. On average, the proposed inflation model yields risk measures that are higher than the ones obtained with competing models. The right tail of the future payments distribution is fatter, as low inflation scenarios are taken into consideration, and this is according to the current market conditions. The importance of modelling adequately embedded optionalities is also shown; an inflation-indexed annuity for which the paid amounts do not decrease when the inflation index level declines is used. The results show that deflation could have a notable impact on life insurers when inflation protections are embedded in the product design.
The contributions of this paper are manifold. First, a modelling framework is constructed to capture the desired empirical facts, such as correlation between the real interest rate and the expected inflation, as well as conditional heteroskedasticity. This model allows for derivative prices in closed or semi-closed form solutions. Second, forward-looking and market-consistent assessment regarding deflation can be done using the framework. Option data are useful in capturing the tails of the expected inflation distribution, and the filter-based methodology allows for dynamic risk assessments that are consistent with market participants’ expectations. This is of utmost importance, as new solvency regulations require insurers to assess their risk in a market-consistent way. Finally, our last contribution is to apply various models—full and nested frameworks—to assess, in the context of a life insurer, the importance of the two core components of the nominal interest rate: the real interest rate and the expected inflation. Based on two different exercises, we show that the full framework allows for more flexibility and a better understanding of deflation risk.
The remainder of the paper is organized as follows. Section 2
introduces the inflation framework. The estimation procedure is explained in Section 3
. In Section 4
, the estimation procedure is applied to US inflation data. Section 5
shows examples of how deflation risk can affect the life insurance industry. Finally, Section 6
3. Estimation Method
In this study, the real interest rate, the expected inflation rate, and the conditional variance of the inflation index level are latent variables.13
It is thus challenging to infer these unobservable variables. Moreover, in presence of latent variables, estimation can be somewhat more complicated. Filtering techniques are useful in these situations.14
It is possible to link the observable quantities to the latent ones through a state space representation. The latter is useful to filter the real interest rate, the expected inflation rate and the conditional variance of the inflation index level by associating them to noisy security prices and the inflation index level. It is then possible to recover state variables’ estimates based on investors’ current expectations. Note that security prices contain forward-looking information, which allows for a better understanding of the three latent variables. Further, both - and -parameters are estimated in a one-stage procedure.
The state propagation equations are given by Equations (2
), (4), and (5). These equations explain how the latent variables evolve in time. The measurement equations show how the unobservable variables are linked to observable quantities. The first measurement equation is related to the inflation index level. It is given in Equation (3
is observed. Then, the market security prices are incorporated in the measurements to capture the latent variables’ true nature and the wedge between
The bond measurement equation is
is the market T
-year zero-coupon bond price at time t
is a centred Gaussian random variable of standard deviation
. To maintain positive prices, a logarithmic transformation is applied. Note that this measurement equation is linear in
The swap measurement equation is
is the market T
-year inflation swap price at time t
is a centred Gaussian random variable of standard deviation
. This measurement equation is also linear in
Regarding option data, we would like to minimize the relative implied volatility root mean square error in the spirit of Christoffersen et al. [21
] and Ornthanalai [22
]. However, since the real interest rate, the expected inflation rate, and the conditional variance are not predictable variables, we cannot use the method directly. Instead, we model the relative implied volatility error as a Gaussian random variable. The option measurement equation is therefore given by
is the market Black and Scholes implied volatility of a time t
cap option of strike price K
and maturity T
is the model Black and Scholes implied volatility for a time t
cap of price C
, strike K
, and maturity T
is a centred Gaussian random variable of standard deviation
is given by
are three parameters to be estimated.15
Homoskedastic normally distributed errors could be restricting. Therefore, we allow for heteroskedasticity in option errors by letting the standard deviation be a function of the data; i.e., the absolute value of the strike K
and the maturity of the option T
. This formulation is similar to the standard deviation of the option errors proposed in Bardgett et al. [23
]. Note that for inflation floor, the rationale is the same: we simply replace cap prices by floor prices.
To estimate the model, we follow a simple filtering approach. The unscented Kalman filter of Julier and Uhlmann [24
] is applied, since some measurement equations are nonlinear.16
The UKF handles the non-linearity and approximates the posterior state density using a set of deterministically chosen sample points. These points capture the mean and covariance of the Gaussian state variable, and when propagated through the measurement equations, it captures the posterior mean and covariance of the observations accurately up to the second order. According to Christoffersen et al. [26
], the UKF may prove to be a good approach for a number of problems in fixed income pricing with nonlinear relationships between the state vector and the observations. In addition, many researchers used the UKF to infer the latent states in financial applications.17
Note that the filter allows quasi-likelihood function computation in a somewhat direct manner.18
Hence, quasi-maximum likelihood estimation is done by numerical optimization. Overall, the parameters to be estimated are θ
. A single set of parameters is obtained by this estimation method.
5. Impacts of Deflation on Life Insurance
The risk of deflation is real and could be rather important, as shown in Section 4
. In this section, we focus on the implications of deflation risk on a fictitious life insurer. Indeed, deflation risk can have a material impact on the insurance industry. Life insurance is threatened by deflation risk, as negative inflation could have a significant impact on the prevailing nominal interest rates, and could thus increase the present value of the payments to be made in a distant future. Generally, the guaranteed interest rate is fixed for the whole term of the insurance contract. Thus, an inflation rate decline would make the guaranteed rate too high, as it cannot be reduced in general. Life insurers would therefore guarantee more interest than what they can earn. Additionally, inflation-indexed insurance policies and annuities might include implicit floor options. As a matter of fact, this kind of inflation protection would hurt life insurers in the eventuality of deflation.
A good example of the impact of deflation risk on life insurers is Japan: the nominal 10-year Japanese government bond yield dropped from a level of 8% in the early 1990s to less than 2% over the past decade. In addition, its historical inflation rate remained below zero for most of the last two decades. As a consequence of this extreme economic environment, eight Japanese insurers have gone bankrupt since 1996. For an analysis of deflation in the context of Japanese life insurers, see Hoshi and Kashyap [35
Few papers discuss the role of deflation on life insurers. Ahlgrim and D’Arcy [8
] acknowledge that the life insurance industry might be more affected by uninterrupted deflationary pressures, and that deflation makes it difficult to earn promised rates. Additionally, Dorfman et al. [36
] find that deflation could have strong negative effects on life insurers while using a simple model for deflation. In their framework, the deflation risk is captured through a flat interest rate term structure that can decrease to lower levels.
5.1. Impacts on the Discount Rate
In this subsection (and the next), the focus is put on annuities. To this end, we consider a simple structure to assess the role of deflation risk on the present value of a life insurer’s future payments. The insurer portfolio is a collection of immediate annuities for which the payments are made at the end of each year until the moment of death.32
Thus, the present value at time t
of the life insurer’s future payments is given by
is the number of annuities sold,
is the time to death of individual i
is the annual amount paid to the i
th individual, and
is the time s
insurer discount rate for this line of business.
To simulate the life insurer portfolio, some additional assumptions are made. We assume that this portfolio contains only females aged 65. The mortality in the portfolio is random, and follows a Gompertz model fitted to US data. The parameters estimated by Pflaumer [37
] are used in this study.33
The annual payment is set to
, and the number of life insurance sold is
. The annualized discount rate
used by the life insurance company is assumed to be 2% over the nominal risk-free interest rate
The purpose of this simple exercise is to assess the implications of deflation on life insurers without having to rely on complex assumptions. Complicated assumptions regarding mortality or the insured population could be easily taken into account, although it is not the purpose of the current study: deflation risk is our main concern.
The nominal interest rate is given by the full model: the nominal interest rate is thus defined as the sum of the real interest rate
and the expected inflation rate
, as in Equation (1
). In addition to the full model, three other models are used to compare the impact of the modelling assumptions on the risk measures of the fictitious life insurer. First, a model with stochastic real interest rate and deterministic inflation rate is used: the so-called anticipated inflation rate (AI) is calculated from the full model. This inflation rate is defined as the conditional expected value of the future inflation given today’s economic conditions. It is therefore deterministic, but consistent with market participants’ expectations. Second, the nominal interest rate (NIR) model presented in Section 4.3
is employed. In this framework, the sum of the real interest rate and the expected inflation rate is modelled directly through a univariate process (instead of a bivariate process, as in the full model). Third, the generalized Wilkie (GW) model described in Section 4.3
is used. As the latter does not model the real interest rate, we assume that the annualized real risk-free interest rate is constant and equal to 0.65%.3435
At this point, it is relevant to stress that the application does not focus on pricing. Instead, it focuses on risk assessment and management.36
We focus our attention at the impact of deflation upon two risk measures: the value-at-risk (VaR) and the conditional tail expectation (CTE). These risk measures are important to set up reserves and understand the risk profile of the insurer’s portfolio. Remember that, in our setting, the nominal interest rate is not known. It is rather a random variable that has an impact on the tails of the future payments distribution. Specifically, decreases in the nominal interest—or in its constituents, the real interest rate and the expected inflation—create scenarios that would increase the present value of the portfolio, thus rising the risk measures.
shows an example of four histograms of the life insurer’s portfolio, in present value (5 February 2014). The four inflation assumptions are used: the full model (top left panel), the anticipated inflation assumption (top right panel), the nominal interest rate model (bottom left panel), and the generalized Wilkie model (bottom right panel). The full model allows for more dispersion than the other cases—especially the AI model and the GW model—leading to a larger VaR and CTE at a confidence level of 95%. The VaR(95%) is 21.9 when the full model is used. It is 17.2, 19.9, and 15.6 for the AI, the NIR, and the GW models, respectively. The right tail of the AI’s distribution is very thin, yielding a CTE(95%) of only 19.3. As AI considers a stochastic real interest rate and a deterministic inflation rate, we can assume that a (market-consistent) deterministic inflation rate is not flexible enough to capture the tail risk adequately in the insurer’s portfolio. The right tail of the GW’s distribution is also thin: a CTE(95%) of 17.2 is associated to this model. Under this framework, the expected inflation is stochastic, but the real interest rate is deterministic. Therefore, to capture the tail behaviour of the insurer’s portfolio, we also need a stochastic real interest rate.
The expected inflation and the real interest rate may be modelled jointly via the nominal interest rate model, as is commonly done in practice. The NIR model yields a higher CTE(95%) than AI and GW. The full model and the nominal interest rate model histograms are also somewhat similar, even though the risk measures of the full model are slightly larger. This shows that modelling the sum of the expected inflation and the real interest rate through a unidimensional process could most probably be enough if one is only interested in the average behaviour of the future changes in the discount rate. Yet, the risk measures are not exactly the same, and separating the real interest rate and the expected inflation can have an impact on the specific behaviour of the risk measures, which depend explicitly on the levels of
. Figure 6
displays risk measures for different levels of real interest rate and expected inflation, and for the four different inflation assumptions. These surfaces show the marginal impact of changes in the real interest rate and in the expected inflation. The AI surfaces’ level is lower than the one of the full model, implying thinner tails for AI (27% and 38% lower on average for the VaR and the CTE, respectively). This observation is consistent with the results of Figure 5
. For the generalized Wilkie model, the real interest rate has no marginal impact, as the GW model does not account for this dimension. Overall, the level of the GW surfaces is lower than the one of the full model (20% and 33% lower on average, respectively). On average, the full model VaR and CTE surfaces’ level is about 5% and 6% higher than the one of the NIR model, respectively.
Even though the surfaces associated with the full model and NIR are tilted in the same direction—the risk measures increase when either the real interest rate or the expected inflation decrease—a marginal decrease in each of these two dimensions is not equivalent. Interestingly, for a given level of
, the average VaR (CTE) of the full model is similar to the average VaR (CTE) of the NIR model. This implies that decomposing the nominal rate into its core components is important to capture market participants’ expectations with respect to each source of risk. According to Figure 6
, a marginal decrease of 1% in the real interest rate has the same impact as a marginal decrease of 1% in the expected inflation rate for the NIR model by construction. For the full model, this is not the case: a marginal decrease of 1% in
is different when compared to a marginal decrease of 1% in
shows the time series evolution of the VaR(95%) and the CTE(95%), computed using the four modelling assumptions. The four models are able to capture the dynamic nature of the risk, especially during the last NBER recession: there was an increase in both the VaR and the CTE during 2009. The full model yields risk measures that are larger than the other three models. Indeed, for AI and GW, this is not a surprise: these frameworks do not adequately capture the future evolution of the nominal interest rate, as one core component—either the real inflation rate or the expected inflation—is not modelled through a stochastic process. Even if the values used in the computation are market-consistent, a deterministic assumption is unable to capture the risk associated with low nominal rates. The NIR model does a better job; yet, the risk measure estimates using NIR are lower than the ones of the full model. A couple of reasons could explain these differences. First, the tails of the expected inflation distribution are better captured by the full model, as inflation caps and floors are used in the estimation of this model. Second, the separation of the real interest rate and the expected inflation brings a better understanding of the core constituents of the nominal rate, as these two components do not have the same marginal impact on the risk of the insurer’s portfolio.
In summary, the nominal interest rate needs to be fully stochastic to adequately capture the right tail of the insurer’s future payments distribution. For instance, AI and the GW model are not adequate, as an essential component of the discount rate is not modelled explicitly. Modelling the sum of the real interest rate and the expected inflation—instead of each component individually—captures the average behaviour of the risk in the right tail. Yet, the individual impact of each component does not seem to be symmetric, as shown in the top panels of Figure 6
. Accounting for the real interest rate and the expected inflation separately allows for a better understanding of the portfolio’s risk.
5.2. Impacts of Embedded Optionalities
Life insurers are exposed to deflationary pressures, as falling prices could lead to insufficient investment returns to support the rate credited to policyholders. Yet, this is not the only channel of transmission for deflation risk. Inflation-indexed protections could also make insurers vulnerable to deflation. This kind of insurance policy is constructed to cope with the risk of purchasing-power erosion. In practice, the paid amounts of inflation-indexed products follow some inflation index, such as the CPI.
Inflation-indexed annuities often provide additional protection that increases the paid amount in the case of positive inflation, but does not reduce the amount in the case of deflation. This added protection—similar to an inflation floor—covers the annuitant against decreases in the CPI level.37
This kind of protection—which is common in practice—is risky for life insurers, as deflationary pressures would increase the value of the embedded inflation floors.
Even though these inflation-indexed products are available, they are far less popular than equity-indexed products in the general population [38
]. Yet, there is still demand for similar inflation-indexed products through reinsurance channels. For example, an increasing number of defined-benefit pension plan sponsors decide to de-risk their financial exposures by buying pension buy-in and buy-out annuities. These special annuities transfer the risks borne by pension funds to life insurers. In the case of inflation-indexed pension benefits, the life insurer is exposed to the dependence between the inflation index level and the nominal interest rate. As most of these inflation-indexed pension plans do not reduce the retirement benefit amount in case of deflation, the life insurer thus bears additional deflation risk in a similar fashion to the indexed annuities described above—a consequence of the embedded inflation floor option.
The focus in this exercise is still on annuities. The portfolio is a collection of annuities for which the indexed paid amount is made at the end of each year until the moment of death. The insurer increases the paid amount using the inflation index when it is favourable for the annuitant. Thus, the present value at time t
of the life insurer’s future payments is given by
The term is a combination of the protection, , and the embedded inflation floor paying the maximum between and zero.
The only model that can be used to compare the full model is the generalized Wilkie model, as it is the only one modelling the inflation index as a full-fledged variable.38 Figure 8
shows the time series evolution of the VaR and the CTE for both models.39
To some extent, the full model as well as the GW model are able to capture the changes in the risk as economic conditions unfold. Yet, the risk measures of the full model capture the increased deflation probabilities in the post-recession era, whereas the GW’s risk measures are much more stable in time. The GW model is not able to capture deflation risk in the post-recession era, as the risk measures are back to their pre-recession levels.
The risk measures computed with the full model are higher than the ones of the GW model. As seen in Section 4.4
, the deflation probabilities are not well captured by the GW model. Three main reasons might explain this: (1) the use of option in the model estimation; (2) the conditional heteroskedasticity of the inflation index noise terms; and (3) a fully stochastic nominal interest rate.40
By using information coming from nominal bonds and inflation derivatives, we are thus able to break down the contributions of each component of the nominal interest rate and understand their specific impact. This is of utmost importance, as the future payments of life insurers depend on both the inflation index level and the nominal inflation rate.
The approach used in this study is different from the common practice that combines both the real interest rate and the expected inflation by modelling the nominal rate directly. Dividing these two effects allows for more flexibility and permits a better understanding of the underlying economic risk factors. It is also the only alternative to account for embedded inflation options such as the floors depicted in this exercise.
6. Concluding Remarks
For the past 20 years, inflation remained virtually constant. However, the recent recession unveiled the potential for unexpected changes in the price level.
The longstanding nature of insurance activities exposes insurers to deflation risk. Dynamic and market-consistent estimates of insurers’ risk profile is of paramount importance to capture the changing economic conditions. To this end, a market-based methodology for assessing deflation risk is presented in this study. The proposed model accounts for the real interest rate, the inflation index level, its conditional variance, and the expected inflation rate. Pricing formulas for nominal bonds, inflation swaps, and inflation options are available in closed or semi-closed form expressions.
An estimation method based on the UKF was adopted. It captures the dynamics of the three latent variables. Multiple types of data source were utilized in the estimation: the use of inflation options—along with nominal risk-free bonds, inflation index levels, and inflation swaps—allows us to characterize the tails of the inflation index level distribution more appropriately. Additionally, the methodology incorporates the information as it becomes available in the market, leading to a market-consistent forward-looking assessment of deflation risk. This is important for proper estimation of deflation risk and its time-varying nature.
Then, we used US inflation data to estimate the framework. The evolution of the three latent variables is quite dependent on the market conditions. For instance, the filtered expected inflation rate decreased during and after the last recession, implying increased probabilities of deflation. Overall, the average 1-year deflation probabilities for the full model are estimated to be 18% and 29% for recession and post-recession eras, respectively.
The effects of deflation risk are analyzed in the context of life insurance. In general, the full model yields larger risk measures than the ones of the three nested models. We show that the nominal interest rate needs to be fully stochastic to capture the right tail of the insurer’s future payments distribution. Additionally, modelling the core components of the nominal interest rate separately—in opposition to the common practice of modelling the nominal interest rate directly—allows for a better understanding of the portfolio’s risk profile. By directly modelling the nominal interest rate, one cannot capture the asymmetric marginal impact of changes in both the real interest rate and the expected inflation, as shown in Figure 6
, for instance.
Some life insurance products rely on the complex dependence structure between the inflation index level and the nominal interest rate. In these cases, modelling the core components of the nominal interest rate separately is the only alternative. For instance, the inflation-indexed annuities are impacted by both the inflation index level and the nominal interest rate, as investigated in this study.
Market-consistency links the likelihood of deflation scenarios to current market expectations. This study shows how deflation impacts the risk measures of a fictitious life insurer: the increases shown in the last section stress the need to include deflation in risk modelling and management—especially nowadays as deflation risk is more uncertain than during the last two decades.