2.1 Model calibration

The five structural parameters ${\bf b} = \left[{\matrix{ {\kappa^{\cal P}} \hfill & {\theta^{\cal P}} \hfill & {\sigma^2} \hfill & {\kappa^{\cal Q}} \hfill & {\theta^{\cal Q}} \hfill \cr } } \right]^{\prime}$ are estimated using the generalized method of moments (GMM) method. We obtain the data from the European Central Bank (ECB) statistical data warehouse. The estimation window consists of daily Euro-area nominal government bonds of constant maturities of 3 months, 5 years and 10 years with triple A issuing ratings over the sample period September 6, 2004 to November 15, 2013.Footnote ³

Summary statistics are displayed in Table 1. The average 3-month rate is 1.62% with a standard deviation of 1.45%. The 5-year rate has a mean of 2.61% with standard deviation of 1.1%. The 10-year bond has the highest average rate but lowest volatility among the three. The augmented Dickey–Fuller test shows that we cannot reject the unit root hypothesis for any of the three series.

Table 1.

Summary statistics

Std, standard deviation; Δ, one-period lag; ADF, the augmented Dickey–Fuller unit root statistics with a 5% critical value of −2.86.

Mean and standard deviation of the sample series, mean and standard deviation of one-period lag and the autocorrelations of 3-month, 5-year and 10-year treasure yields denoting r _t, Y _t(5) and Y _t(10), respectively. The table reports the summary statistics of ECB data set for the sample period September 6, 2004 to November 15, 2013.

In Table 2, we report the parameter estimation results. The parameters are expressed in annual terms. The first two columns of Table 2 report the estimated values and standard deviations of structural parameters. The unconditional mean of spot rate, $\theta ^{\cal P}$, is about 1.615%. The annual speed of reversion $\kappa ^{\cal P}$ is 17.164%, which is associated with a half-life of the short rate of 4 $\lpar { = \lpar \ln \lpar 2\rpar /\kappa^{\cal Q}\rpar = \lpar \ln \lpar 2\rpar /17.164\percnt \rpar } \rpar$ years.Footnote ⁴ The spot rate has an instantaneous volatility of σ = 0.525%, which is associated with a long-term variance of 1.6 bp $\lpar { = \lpar \sigma^2/2\kappa^{\cal P}\rpar = 0.273\,{\rm bp}/17.164\percnt } \rpar$. Estimate for σ has a relatively lower standard error than other parameters indicating that spot-rate volatility can be estimated more accurately. The rest of Table 2 reports the correlation matrix of parameters. $\kappa ^{\cal P}$ and $\theta ^{\cal P}$ are negatively correlated, indicating that an increase in the long-term spot rate is associated with a decrease of the speed of reversion.

Table 2.

GMM estimates

The table reports the estimation results of the five structural parameters ${\bf b} = \left[{\matrix{ {\kappa^{\cal P}} \hfill & {\theta^{\cal P}} \hfill & {\sigma^2} \hfill & {\kappa^{\cal Q}} \hfill & {\theta^{\cal Q}} \hfill \cr } } \right]^{\prime}$ using GMM. The first two columns display the expected values and the standard deviations of the point estimates $\hat{{\bf b}}$. The remaining body displays the correlation matrix of $\hat{{\bf b}}$. We choose k = 20 as the number of lags applied in Newey West estimator. Estimations are based on the ECB data over the sample period September 6, 2004 to November 15, 2013.

Panel A of Table 3 reports the point estimates of λ ₀ and λ ₁, as well as the expected values of σλ ₀ and σλ ₁. Panel B presents the expected risk premia and volatilities of the nominal bonds with maturities of 5 years, 10 years and 20 years. Longer-term bonds have higher expected risk premia and are also associated with higher volatilities than bonds with shorter maturities.

Table 3.

Market price of risk and bond risk premium

The table presents the estimation results for λ ₀, λ ₁, σλ ₀ and σλ ₁ over the sample period September 6, 2004 to November 15, 2013. The table also reports the estimated risk premium and return volatility of 5-, 10- and 20-year bonds when r _t = 1%, using the estimated results displayed in Table 2.

2.2 Parameter uncertainty

In this section, we embrace the concept of parameter uncertainty to the bond pricing models introduced in Section 2. Following the framework of Anderson et al. (Reference Anderson, Hansen and Sargent2003) and Hansen and Sargent (Reference Hansen and Sargent2007), we assume that investors treat bond diffusion processes as approximate models for the unknown true state evolution. Similar to Maenhout (Reference Maenhout2004) and Shen et al. (Reference Shen, Pelsser and Schotman2019), we reduce the general model uncertainty problem to a first-moment parameter uncertainty problem by assuming that the investors only fear a subclass of parameter misspecification.

We start with a general setting of parameter uncertainty. Assuming that vector ${\bf b}$ is estimated with error. Let ${\bf b}_0$ be the true value of the structural parameters. Then ${\bf b}-{\bf b}_0$ represents a parameter misspecification vector that satisfies the following condition

$${\bf b}-{\bf b}_0\sim {\cal N}\lpar {0\comma \;{\bf \Sigma }} \rpar$$

where ${\bf \Sigma }$ is the variance–covariance matrix of ${\bf b}$. Given that ${\bf b}-{\bf b}_0$ is normally distributed, the following statistical inference holds

$$\lpar {{\bf b}-{\bf b}_0} \rpar ^{\prime}\,{\bf \Sigma }^{{-}1}\lpar {{\bf b}-{\bf b}_0} \rpar \le \kappa ^2$$

where $\kappa ^2 = CV_\alpha /N$ is a function of the critical value of the Chi-square distribution χ ²(5) at significance level α.

In the general setting, we need to deal with five-dimensional uncertainties. However, the covariance matrix Σ only has two eigenvalues (0.0334 and 0.0054) that are significantly different from zero. We therefore shrink the five-dimensional parameter uncertainty problem to a two-dimensional parameter uncertainty problem. In particular, we assume parameters that have the largest estimated standard deviations are exposed to parameter uncertainty and the rest parameters can be estimated precisely. Table 2 shows that, among the five parameters, $\kappa ^{\cal P}$ and $\theta ^{\cal Q}$ have the highest standard deviations for both sample periods. Therefore, these two parameters are most likely to be misspecified.

Let $\kappa _0^{\cal P}$ and $\theta _0^{\cal Q}$ be the true value of $\kappa ^{\cal P}$ and $\theta ^{\cal Q}$, respectively. Then let $\Delta \kappa ^{\cal P}$ and $\Delta \theta ^{\cal Q}$ be the estimation error of the two parameters:

$$\eqalign{\kappa ^{\cal P}-\kappa _0^{\cal P} = & \,\Delta \kappa ^{\cal P} \cr \theta ^{\cal Q}-\theta _0^{\cal Q} = & \,\Delta \theta ^{\cal Q}}$$

where $\Delta \kappa ^{\cal P}\ne 0$ and $\Delta \theta ^{\cal Q}\ne 0$ are also called distortion parameters. The rest three parameters are estimated precisely with their true values equal to the estimated values, σ ₀ = σ, $\kappa _0^{\cal Q} = \kappa ^{\cal Q}$ and $\theta _0^{\cal P} = \theta ^{\cal P}$. Therefore, function B(τ) is independent of parameter uncertainty under our setting, since $\kappa ^{\cal Q}$ can be estimated with a reasonable level of accuracy. As a result, only the drift term λ _t of the bond-price diffusion process (8) is exposed to parameter uncertainty. In other words, investors only consider the misspecification issue of the coefficients of the risk premium function λ _t, namely λ ₀ and λ ₁.

Define ${\bf \delta } = \left({\matrix{ {\lambda_0} \cr {\lambda_1} \cr } } \right)$ and let ${\bf \delta }_0$ be the true value of ${\bf \delta }$ with ${\bf \delta }_0 = \left({\matrix{ {{\lpar {\lambda_0} \rpar }_0} \cr {{\lpar {\lambda_1} \rpar }_0} \cr } } \right)$, where (λ ₀)₀ and (λ ₁)₀ represent the true values of λ ₀ and λ ₁, respectively. Then we transfer the perturbations to the risk premium parameters, λ ₀ and λ ₁:

$$\eqalign{\lambda _0 = & \underbrace{{\displaystyle{{\kappa _0^{\cal P} \theta ^{\cal P}-\kappa ^{\cal Q}\theta _0^{\cal Q} } \over \sigma }}}_{{{\lpar {\lambda_0} \rpar }_0}} + \underbrace{{\displaystyle{{\Delta \kappa ^{\cal P}\theta ^{\cal P}-\kappa ^{\cal Q}\Delta \theta ^{\cal Q}} \over \sigma }}}_{{c_0}} \cr \lambda _1 = & \underbrace{{\displaystyle{{\kappa ^{\cal Q}-\kappa _0^{\cal P} } \over \sigma }}}_{{{\lpar {\lambda_1} \rpar }_0}}\underbrace{{-\displaystyle{{\Delta \kappa ^{\cal P}} \over \sigma }}}_{{c_1}}}$$

where c ₀ = c _0t and c ₁ = c _1t are the two time-varying perturbation parameters with their values depending on instantaneous state variables and hedging horizon:

$${\bf c} = \left({\matrix{ {c_0} \cr {c_1} \cr } } \right)\quad {\rm and}\quad {\bf \delta }-{\bf \delta }_0 = {\bf c}$$

Next, we map the joint distribution of the general estimation-error vector ${\bf b}-{\bf b}_{\bf 0}\sim {\cal N}\lpar {0\comma \;{\bf \Sigma }} \rpar$ to vector ${\bf c}$ using the first order Taylor expansion. Therefore, vector ${\bf c}$ is also asymptotically normal with

(12)$${\bf c}\sim {\cal N}\lpar {0\comma \;\Omega } \rpar $$

where ${\bf c} = {\bf c}_t$ is the instantaneous distortion vector and $\Omega = {\bf J}\Sigma {\bf J}$, with ${\bf J}$ the Jacobian matrix of ${\bf \delta }$ with respect to structural parameter ${\bf b}$. We assume that there is an artificial agent (‘nature’) who makes an instantaneous decision on ${\bf c}_t$ such that the true value of bond premium perturbs from the point estimate. The distortions c ₁ and c ₂ shift the drift of the bond diffusion process (8) by a unit of Bσ(c ₀ + c ₁r _t). Hence they specify a set of alternative measures referring to different specifications of the stochastic process known as a Girsanov kernel.

The distortions are bounded by a time-homogeneous uncertainty set ${\bf S}$. A larger uncertainty set ${\bf S}$ indicates a higher degree of uncertainty aversion. To calibrate the bound of the specification error ${\bf c}_t$, we relate the distortions to the econometric parameter estimation error. We obtain a statistical constraint on ${\bf c}$ using the property that ${\bf {c}^{\prime}}\Omega ^{{-}1}{\bf c}$ is Chi-square distributed. This leads to the following uncertainty set ${\bf S}$:

(13)$${\bf S}: = \left\{{\left({\matrix{ {c_0} \cr {c_1} \cr } } \right)\vert {\left({\matrix{ {c_0} \cr {c_1} \cr } } \right)}^{\prime}\Omega^{{-}1}\left({\matrix{ {c_0} \cr {c_1} \cr } } \right)\le \gamma^2} \right\}$$

where $\gamma ^2 = CV_\alpha /N$ with $CV_\alpha$ denoting the critical value of χ ²(2) at significance level α and N is the sample size.

Our uncertainty set differs from the Hansen and Sargent framework in two ways. First, under their framework the perturbations are formally measured by relative entropy. The entropy term penalizes drift distortions that move away from the reference model. The relative entropy parameter plays the same role as a Lagrange multiplier of the Hamilton–Jacobian–Bellman equation for optimality. However, this Lagrange multiplier implicitly imposes a constraint on the following bound for the specification error:

(14)$${\rm {\opf E}}\left[{\sum\limits_{t = 0}^T {{{\bf {c}^{\prime}}\!\!}_t\cdot {\bf c}_t} } \right]\le \eta _0\comma \;$$

where η ₀ sets the average size of the potential model misspecification. The uncertainty set (14) is an aggregate-budget style present-value bound that restricts the distortions cumulatively over the decision horizon T. This present-value constraint (14) is time-inconsistent. However, our uncertainty set (13) is a time-homogeneous constraint that we can impose cleanly on nature's future actions without keeping track of what has happened in the past.

The second difference is related to the uncertainty set calibration. Under the Hansen and Sargent framework, the boundary parameter η ₀ is linked to relative entropy which in essence is a Lagrange multiplier over the misspecification bound (14). Hansen and Sargent (Reference Hansen and Sargent2007) use a Bayesian model selection problem (also called the detection error probability method) to calibrate this relative entropy parameter. The detection error probability method performs likelihood ratio tests under the two models based on available data. A higher degree of robustness leads to a lower detection error probability. We follow Shen et al. (Reference Shen, Pelsser and Schotman2019) by calibrating the boundary of the uncertainty set using statistical reference.

We choose a significance level of α = 5% at which the corresponding critical value of χ ²(2) is 5.99. We also assume that the minimum sample horizon for estimation is ${\cal N} = 200$ time units. Hence, as a benchmark, the misspecification bound is γ = 0.17 or γ ² = 0.03. Either a longer data set or a higher significance level would reduce the bound parameter γ and results in a smaller uncertainty set for c _t, which indicates a gaining confidence over the reference model.

The value of bond premium, −B(τ)σλ, is governed by the instantaneous spot rate r _t and is subject to parametric misspecification. Figure 1a presents the 20-year nominal bond risk premia, −B(20)σλ, for a realistic range of spot rates. The solid dotted line plots the 20-year bond risk premia as a function of r _t using the point estimates $\hat{\lambda }_0$ and $\hat{\lambda }_1$. For example, when spot rate is 1%, the associated 20-year bond risk premium is 4.6%, which is consistent with Table 3. The dotted lines surrounding the point estimates of bond premia represent the upper and lower boundaries of the perturbed bond prima driven by the distortions c ₀ and c ₁ with $\lcub {c_0\comma \;c_1} \rcub \in {\bf S}$. The bond risk premium is not sensitive to B(τ), since B(τ) is independent of parametric misspecification issues. Bond risk premium is increasing with the spot rate. In short, misspecification of MPR results in a perturbation of the bond premium of around 200 basis points either downward or upward.

Figure 1.

Bond premia. The figure presents the 20-year bond risk premia for different spot rates under different choices of λ ₀ and λ ₁. The ambiguity of MPR constrained by set ${\bf S}$ is shown in (13) with γ ² = 0.03. Panel 1a presents the minimum and maximum values of bond premia within the feasible region for different r. The point estimate premia $\hat{\lambda }$ are located in between the two extremes. Panel 1b plots the feasible region of bond premia for different choices of λ ₁ under the unconditional expectation of spot rate.

The uncertainty set ${\bf S}$ has a circular shape in the {c ₀, c ₁} space with zero in the center, but it turns to an ellipsoid shape when mapping to the ${\bf \delta }$ space. Figure 1b plots the uncertainty set that maps the perturbation vector ${\bf c}_t$ to the {B(20)σλ, λ ₁} space. The uncertainty set is centered by the point estimate $\lcub {B\lpar {20} \rpar \sigma \hat{\lambda }\comma \;{\hat{\lambda }}_1} \rcub$ with r = 2%, where $\hat{\lambda } = \hat{\lambda }_0 + \hat{\lambda }_1r$. The center of the uncertainty set shows that given the point estimates, $\hat{\lambda }_1 = {-}27.04$ and $\hat{\lambda }_0 = {-}0.31$, the associated 20-year bond premium equals to 6%. The very narrowly shaped ellipsoid spaced is mainly driven by the large variation between the point estimates of λ ₀ and λ ₁.

3.1 Dynamic replication strategy

The agent with wealth X _t at time t = 0 aims to replicate an ultra-long-dated liability with payoff equal to one at a long end maturity time T by investing in two liquid fixed-income instruments: a long-term zero coupon bond with maturity τ ₂, where τ ₂ < T; and a short-term bond. Let the long-term bond maturity be the same as the term of market last liquid point. The hedging horizon T is assumed beyond the term of the last liquid point of the financial market. The replication strategy is constructed based on dynamic hedging and the agent is only interested in eliminating the downside risk. Therefore the hedging criterion is defined over the expected shortfall [1 − X _T]⁺ at time T. This hedging criterion allows the agent to replicate the ultra-long-term cash flow as much as possible by active trading using the minimum amount of wealth. If the agent is not aware of parameter uncertainty, the hedging portfolio relies fully on the point estimator $\hat{\delta } = \left({\matrix{ {{\hat{\lambda }}_0} \cr {{\hat{\lambda }}_1} \cr } } \right)$ and we define such specification-error-free trading strategy as a naive policy. The naive dynamic optimization problem is defined as

(15)$$\mathop {\min }\limits_{w_{t\colon 0 \le t \lt T}} {\rm {\opf E}}\lsqb {{\lpar {1-X_T} \rpar }^ + {\rm \mid }{\cal F}_t} \rsqb $$

The fraction of wealth allocated to long-term bonds with maturity τ ₂ at time t is indicated by w _t and leverage is allowed. The law of motion of wealth is given by

(16)$$dX = \lpar {r-wB\lpar {\tau_2} \rpar \sigma \lambda } \rpar Xdt-wB\lpar {\tau_2} \rpar \sigma XdW^{\cal P}$$

where we omit the subscripts t.

For the dynamic trading strategy (15), we omit the effect of rollovers which occur when reinvesting wealth from matured bonds into newly issued bonds of the same feature. We assume that the invested fixed-income securities are continuously issued and are liquidly traded during the entire decision horizon. Even when the remaining hedging horizon is shorter than then the marketable term, the investor still keeps on active trading. If we stop the active trading as soon as the market becomes complete when the remaining maturity of the liability equals to maturity of long-term bond, then the expected shortfall hedging criterion would not be minimized and the required initial capital for hedging long-term liabilities can be higher.

3.2 Robust hedging

The robust control method is formulated by a min–max expected utility framework which according to Hansen and Sargent (Reference Hansen and Sargent2007) resembles a two-player zero-sum game. The investor who fears parametric misspecification takes the role of min-player and seeks a robust-hedging policy that minimizes the expected shortfall at maturity T under the worst-case scenario decided by the artificial agent. The artificial malevolent agent takes the role of max-player and makes a decision on distortions ${\bf c}_t$ given the min-player's choice. The equilibrium of the game gives us an instantaneous robust hedge. The robust optimization problem is given by

(17)$$\mathop {\min }\limits_{w_{t\colon 0 \le t \lt T}} \mathop {\max }\limits_{c_t\in {\bf S}} \;{\rm {\opf E}}\lsqb {{\lpar {1-X_T} \rpar }^ + {\rm \mid }{\cal F}_t} \rsqb $$

the artificial agent makes an instantaneous decision on c ₀ and c ₁ bounded by the time-homogeneous constraint ${\bf S}$, as thus controls the risk premium parameters λ ₀ and λ ₁.

5.1 Optimal portfolio choice

Figure 3 plots the optimal allocation to the 20-year (τ ₂ = 20) nominal bonds, w _t, as a function of the investment horizon. The rest amount, 1 − w _t, allocates to the short-term bonds. We do not limit the opportunity for shorting in the money market for two reasons. First, in many countries pension funds are allowed to leverage. For example, the big-six Canadian pension funds increased their average leverage to 24% in 2017, from 19% in 2009.Footnote ⁵ Second, we aim at constructing an optimal bond portfolio that can replicate the ultra-long-term liability risk as much as possible when funds are underfunded and when the market last-liquid-point is not long enough to reach the hedging horizon. Such a replication portfolio can not only help investors measure the non-tradable liability risk, but also provide investors a minimum funding requirement that guarantees the future cash-flow commitment via dynamic trading. One can also use this minimum funding requirement with a τ-year maturity to calculate the corresponding τ-year discount rate y(τ).

Figure 3.

Optimal bond portfolio for different horizons. The figure plots 20-year bond weights against different investment horizons when current funding ratio is 80% (a) and 110% (b) for different hedging policies. The present spot rate is 2%. The weights are based on three different hedging policies: naive hedge, robust hedge and delta hedge. Results are based on 10,000 draws of predictive bond distribution.

Figure 3 displays three different hedging policies: (1) a naive policy that relies purely on the point-estimate based investment opportunity set; (2) a robust policy that considers parametric uncertainty and (3) a delta-hedging policy that eliminates the model-implied ultra-long-term liability risk without considering speculative potential when confronting with solvency risk.

Delta Hedging The delta of a long-dated liability is defined as the rate of change of the price of liability with respect to the price of the underlying τ ₂-year bond, which is given by

(19)$$\Delta = \displaystyle{{\partial P\lpar {0\comma \;T} \rpar } \over {\partial P\lpar {0\comma \;\tau_2} \rpar }} = \displaystyle{{B\lpar T \rpar } \over {B\lpar {\tau_2} \rpar }}\comma \;$$

assuming the Vasicek model is valid for all maturities. We use Δ as a benchmark to compare with some dynamic hedging strategies that consider the solvency conditions. Let funding ratio, a fraction of the current wealth level X ₀ and a hypothetical present value of liability P(0, T), be the measure of the solvency position of a fund. If the funding ratio is lower than one, then this fund has a higher expected value of liability than of its assets, which implies under-funding.

We consider two funding scenarios in Figure 3. Figure 3a presents the three hedging policies when the fund is underfunded with a funding ratio of 80%. Figure 3b shows the results when the fund is not facing a solvency issue with a funding ratio of 110%.

Regardless of solvency positions, Figure 3 demonstrates two risk management incentives. First, long-term investors have more holdings on the long-term bonds than short-term investors, which is consistent with financial literature (e.g., Campbell and Viceira, Reference Campbell and Viceira2001).Footnote ⁶ As shown in both figures, allocation to 20-year bond is increasing with the hedging horizon at a diminishing rate for all hedging policies since intertemporal hedging component B(τ) increase with hedging horizon. For example, the delta hedge Δ = 1,  1.5 and 1.8 for long-term investors with investment horizons of 20, 40 and 60 years, respectively. Second, short-term investors put less weights on the long-term bonds compared to long-term investors. Pension funds that have a shorter hedging horizon are usually more mature than those with a longer-term hedging horizon, and are also associated with a shorter liability duration. Benzoni et al. (Reference Benzoni, Collin-Dufresne and Goldstein2007) and Lucas and Zeldes (Reference Lucas and Zeldes2006) argue that mature funds should have a lower level of risk-taking because their projected liabilities are more akin to those of a short-duration bond rather than of risky financial instruments, as the cash flows resulting from promised pension benefits become less uncertain for mature funds. In addition to these risk incentives, Figure 3 also shows that delta hedge is independent of funding status. As shown in (19), delta hedge is simply a function of τ and is irrelevant to the funding ratio. As a result, delta investors do not have incentives to take extra risks to offset mismatched liabilities when the fund is underfunded.

Next, we dive deep into the two funding scenarios. When funds have insufficient wealth to cover their liabilities (see Figure 3a), both naive and robust investors look for speculation opportunities to catch up the mismatch. As a result, both hedging policies are above the delta hedge over the investment horizon. In other words, delta strategy may effectively eliminate the long-term interest-rate risk but does not help to meet the long-dated commitments when facing an insufficient solvency status. We summarize two takeaways from Figure 3a. First, when hedging horizon is shorter than the last liquid point (T ≤ 20), robust investors would choose an identical hedging portfolio as naive investors. This is because when investment horizon falls within the last liquid point, the financial market becomes complete. As a result, concerns about parametric uncertainty vanish. For example, both naive and robust investors with 5-year hedging window would choose to put about 45% of their current wealth to the 20-year long-term bonds to minimize the asset-liability mismatch at a funding ratio of 80%.

Second, when hedging horizon is beyond the market last liquid point of 20 years, naive and robust-hedging policies start to diverge. Investors who worry about misspecification issues take a riskier hedging position by investing more long-term bonds than naive investors. This is because robust investors are afraid that the point estimate on the 20-year bond premium can be over-estimated. In their view, the point-estimate implied hedging strategy is not taking enough risk to offset the downside risk. Hence, a more aggressive hedging portfolio is appreciated by robust investors. The gap between the robust and the naive policies is increasing with hedging horizon, indicating that concerns about the impact of parametric uncertainty on long-term bond returns are accumulating over the investment horizon. Remark, the concavity-shaped naive policy at the long end of hedging horizon is likely to be caused by numerical issues.

At first glance, the second takeaway from Figure 3a contradicts the existing literature. For instance, both Maenhout (Reference Maenhout2004) and Shen et al. (Reference Shen, Pelsser and Schotman2019) show that robust investors prefer a more conservative portfolio than naive investors. Essentially, robust investors among all studies (including ours) have concerns about an over-estimated risk premium implied by the reference estimation. As a result, they will employ alternative opportunity sets surrounding the point-estimate one and take one of the perturbed opportunity sets that brings the worst case scenario to their objective functions. Therefore, investment recommendations for robust agents are mainly driven by the underlying objective functions. For example, in Maenhout (Reference Maenhout2004)'s framework, robust investors aim to maximize their terminal wealth. Concerns about parametric uncertainty make them believe that the referenced investment strategy overstate the expected returns of risky assets. Therefore, robust investors adjust their risk-return tradeoffs of risky assets downward and also reduce their risk holdings accordingly. In Shen et al. (Reference Shen, Pelsser and Schotman2019)'s framework, investors aim to minimize an expected shortfall objective function that involves a stochastic liability constraint. In this case, robust investors not only worry about an over-estimated expected stock return, but also an under-estimated expected liability return in the meantime. Therefore, a conservative portfolio is appreciated by robust investors. Different from either studies, investors under our framework are facing a guaranteed commitment, which increases robust investors' incentives to take more risks to offset the over-estimated risk premium.

In Figure 3b, we plot the three hedging policies when funding ratio is greater than one. When market is complete (T ≤ 20), the three hedging policies are identical.Footnote ⁷ This is because when the current funding ratio is above one, downside risks fade away. Neither robust nor naive investors have incentives to take extra risks, since one can fully replicate the long-dated payoff by constructing a bond portfolio via delta hedging. When hedging horizon reaches beyond the market last liquid point, naive policy still closely follows the delta hedge. This is because when over funding, the likelihood of facing a shortfall is relative small via dynamic hedging. As a result, naive investors' hedging component is dominated by the intertemporal hedging component B(T), which matches the delta hedge. To measure the long-term interest rate risk, naive investors extrapolate the yield curve beyond the last liquid point based on the point estimates. Robust investors, however, would extrapolate a set of yield curves and employ the one that creates highest expected shortfall by the end of the hedging horizon. Therefore, robust investors have more incentives to take extra risks to hedge potential solvency risk than naive investors even if the fund is now over-funded.

We now investigate how sensitive the two optimal hedging policies are in response to the changes of instantaneous funding ratio over different hedging horizons. Figure 4 displays the optimal asset allocations to the 20-year bonds for naive (see Figure 4a) and robust (see Figure 4b) hedging policies in three-dimensional graphs as functions of a reasonable range of the funding ratio and the investment horizon. We can draw three conclusions from Figure 4. First, as also observed in Figure 3, when the hedging horizon is shorter than the market last liquid point, robust investors make the same investment decision as naive investors, for liabilities can be fully replicated in a complete market. Second, when the hedging horizon is beyond the market last liquid point, robust investors would choose a more aggressive risk position when funding ratio is below 1 while above 0.2, due to their concerns about an over-estimated bond premium. This feature is also observed in Figure 3. Third, when funding ratio is close to zero, the robust policy recommends a more moderate risk position than the naive policy for ultra-long-term investors. In fact, the robust policy in this case is very close to a delta hedge. This is because when funding status is extremely low, the likelihood of fully replicating long-term liabilities is close to zero even with an extremely aggressive portfolio. As a result, funding status in this extreme case barely plays a role in the investment decision. Thus the resulting robust-hedging portfolio is very similar to a delta hedge.

Figure 4.

Optimal portfolio under different funding ratios (1). The figure plots the optimal portfolio of 20-year bond as a function of investment horizon and current funding ratio with spot rate equal to 2%.

Figure 5 displays both naive (see Figure 5a) and robust (see Figure 5b) hedging policies over the hedging horizon under three different funding-ratio scenarios. In addition to the conclusions drawn from Figures 3 and 4, we highlight some additional features of the two policies. First, a lower funding ratio is associated with a riskier hedging position. This property holds for both policies. For example, a naive investor with a 40-year investment horizon would allocate 210% of the assets to 20-year bonds when the funding ratio is 80% but would reduce to 140% when the funding ratio is 110%. As discussed before, the robust policies take higher risk positions than the naive policies. Second, naive hedging policies start converge to the delta hedge when hedging horizon is approaching 80 years, while robust policies increase their risk positions with hedging horizon at a diminishing rate. This is because when facing an ultra-long-term hedging horizon, naive investors are more confident than robust investors in terms of the point-estimate implied risk-return opportunity set and they find delta hedge with a very small speculation component is enough to cover the ultra-long dated commitment. Robust investors, however, have a much lower reward expectation given the same level of risk positions as naive investors.

Figure 5.

Optimal portfolio under different funding ratios (2). The figure summarizes the key insight of Figure 4 under three reasonable funding ratio levels.

Figure 6 presents the optimal allocation to the 20-year bond under three different spot-rate scenarios when the instantaneous funding ratio is 80%. We start with extracting a common feature that holds for both naive (see Figure 6a) and robust (see Figure 6b) hedging policies. Both policies indicate that a higher spot rate is associated with a higher allocation to the 20-year bonds when hedging horizon is beyond the market last liquid point. This is because the risk-return trade off of the 20-year bond increases in response to the interest rate hikes. For instance, as shown in Figure 1a, r = 0.5% is associated with a naive 20-year bond premium of 3% and r = 2% is associated with a naive bond premium of 7%. In other words, long-term bonds become more attractive when the spot rate is high. As a result, a higher allocation to long-term bond is desired when interest rate is high and vice versa. Next, we discuss the difference between the two hedging policies in response to the changes of interest rates. The robust policy differs from the naive one in the sense that the robust policy is less sensitive to the spot rate. As can see in Figure 1a, the perturbed bond-premium curve is flatter than the naive one indicating that the worst case scenarios that robust investor follow are less sensitive to the movements of spot rates than the reference scenario. Over all, resilience against macro-risk factors is a desired feature of the robust policy.

Figure 6.

Robust policy under different spot rates. The figure plots naive (a) and robust (b) optimal bond portfolios as a function of spot rate and hedging horizon when current funding ratio is 80%.

Figure 7 plots the optimal realized expected shortfalls in a three-dimension space as a function of the instantaneous funding ratio and the investment horizon. In general, the robust expected shortfall (see Figure 7b) is larger than the naive one (see Figure 7a) with given investment horizon and when funding ratio is low. When hedging horizon is <20 years, the difference between the two expected shortfalls implied by the two hedging policies are relatively small. As explained in Figure 3, robust and naive investors would choose an equal hedging portfolio when hedging horizon is shorter than the last liquid point. However, robust investors are pessimistic about the expected bond premium, which results in a higher level of expected shortfall by the end of hedging horizon. When investment horizon goes beyond the last liquid point, the difference between the two expected shortfalls implied by the two hedging polices enlarges. This is because robust investors' fear for over-estimated bond premia increases with hedging horizon, as shown in Figure 2.

Figure 7.

Optimal expected shortfall (1). The figure plots the expected shortfall as a function of hedging horizon and funding ratio under naive (a) and robust (b) policies when spot rate is 2%.

To take a closer look at Figure 7, in Figure 8 we plot the two hedging-policy implied expected shortfalls together, over the instantaneous funding ratio. We present two scenarios: when hedging horizons is 40 years (see Figure 8a); and when T is 60 years (see Figure 8b). We summarize two features of the hedging-policy driven expected-shortfall functions from Figure 8. First, the robust expected shortfall is much higher than the naive one when the instantaneous funding ratio is low. For example, when the instantaneous funding ratio is 0.5, naive investors with a 40-year hedging horizon believe that the point-estimate driven hedging strategy can fully replicate the long-term cash flow. The corresponding expected shortfall at T = 40 is close to zero. However, robust investors are much more pessimistic about the hedging performance, as their expected shortfall is 0.35 unit higher the naive expectation. The difference between the two expected shortfall lines reaches maximum at a certain point and starts to reduce when the funding ratio deviates from that certain point. For example, when T = 60 the largest gap between the two expected-shortfall curves occurs when instantaneous funding ratio is 0.22. The gap reduces to zero either when the current funding ratio approaches to zero or when it is beyond 0.55. In short, the expected shortfall at T implied by different hedging policies can be substantially different, especially for long-term investors. In other words, ignoring parameter uncertainty may result in an underestimated shortfall risk. Second, the maximum gap between the two policy-implied expected shortfalls increase with the investment horizon. The increasing gap is driven by the accumulative fear of the underestimated long-term bond premia.

Figure 8.

Optimal expected shortfall (2). The figure plots the robust expected shortfall against the naive one as a function of funding ratio when hedging horizon is 40 years (a) and when the horizon is 60 years (b).

5.2 Robust term structure

An investor is interested in the optimal strategy that guarantees the liability with the lowest required initial wealth X*. Such an optimal strategy is called the super-hedging strategy if the guaranteed commitment can be fully replicated. Once the minimum wealth X* is determined, one can derive implicitly a T-period discount rate, since X* can also be interpreted at the price of a T-period zero-coupon bond. Therefore, the hedging-strategy driven T-period discount rate is given by

(20)$$y\lpar T \rpar = {-}\displaystyle{1 \over T}\ln X^\ast $$

In Section 5.1 we have shown replication portfolios with and without the preference for parameter uncertainty. These portfolios are constructed based on a minimum expected-shortfall hedging criteria which aims at replicating non-tradable ultra-long-term liability risk as much as possible when market is incomplete. In our framework, a super-hedging strategy does not exist. There is always a small probability of underfunding, for the interest rate process is Gaussian. Instead, we introduce an approximate risk-free rate obtained via the minimum wealth requirement that targets an expected-shortfall threshold of less than S,

(21)$$y\lpar T \rpar = {-}\displaystyle{1 \over T}\ln X_S^\ast \comma \;\quad {\rm with}\;S = {\rm {\opf E}}_0\lsqb {{\lpar {1-X_T} \rpar }^ + {\rm \mid }X_0 = X_S^\ast } \rsqb $$

where $X_S^\ast$ is the minimum wealth requirement that guarantees a targeting shortfall level of S. The lower S is, the more initial wealth is required to hedge against the targeting shortfall risk, hence the lower y(T) will be. The lowest level of y(T) is simply the underlying term-structure model that by definition guarantees zero expected shortfall.

Figure 9 presents the yield curves driven by different hedging policies, different targeting shortfall levels and different spot rates. As a benchmark, we plot the two hedging-policy based yield curves together against the Vasicek yield curve. We highlight two crucial insights from Figure 9. First, the robust yield is always lower than the naive yield regardless of the current spot rate or the shortfall target. Due to the concerns about an under-estimated bond premium, a robust investor requires more initial wealth to meet the shortfall target. Second, both policy-based yield curves are higher than the Vasicek curve when the spot rate is low (Figure 9a and b). This implies the model-based yield curve will present a higher value of liability and hence will require more initial wealth to meet the long-term commitment than the hedging-policy driven curves. The model-based curve implicitly assumes a zero-shortfall requirement, and hence has a lower yield curve compared with the other two curves which allow for positive expected shortfalls. We also find that the Vasicek yield curve overlaps the naive curve when the spot rate is high if S is sufficiently small (Figure 9c).

Figure 9.

Robust yield curve. The figure plots policy-based yield curve again the benchmark Vasicek yield curve under different spot rate and shortfall target. The benchmark yield curve by definition has zero expected shortfall.