A full and synthetic model for Asset-Liability Management in life insurance, and analysis of the SCR with the standard formula

Adel Cherchali; Aur\'elien Alfonsi; Jose Arturo Infante Acevedo

REVIEW 2 major objections 6 minor 22 references

A full and synthetic model for Asset-Liability Management in life insurance, and analysis of the SCR with the standard formula

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A synthetic ALM model shows shifted interest-rate models, not Hull-White, remain meaningful after regulatory shocks.

desk verdict A careful and practically motivated ALM model with a genuinely useful cash-flow-matching feature; the interest-rate-model comparison is clear but depends on a calibration convention that needs a robustness check before the conclusions are taken as general. read the letter →

arxiv 1908.00811 v1 pith:KTVZ43EE submitted 2019-08-02 q-fin.RM q-fin.PM

Aurélien Alfonsi , Adel Cherchali , Jose Arturo Infante Acevedo This is my paper · ORCID

classification q-fin.RMq-fin.PM MSC 91G3091G60

keywords ALMmodelSolvencycapitalrequirementStandardformulaCash-flowmatchingLiquiditygapSurrenderriskBookvalueProfitsharing

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper builds a synthetic asset-liability management model for participating life insurance contracts that tracks both market and book values, applies a French-style profit-sharing rule, and derives the crediting rate from a trade-off among the guaranteed rate, a competitor rate, and available profits. A distinctive feature is a bond portfolio of equally weighted maturities one through n, whose expiring nominal values fund surrenders. The authors use the model to compute the Solvency Capital Requirement with the Solvency II standard formula and argue that the choice of interest-rate model matters: after the regulator's yield-curve shocks, shifted models such as Vasicek++ recalibrate cleanly, while mean-reverting curve models such as Hull-White produce oscillating, unrealistic post-shock dynamics. They also show that cash-flow matching through the bond basket materially reduces the interest-rate SCR relative to a single-bond proxy, and that the optimal bond-ladder maturity depends on the interest-rate environment.

What carries the argument

The article's main device is the shifted short-rate model of Vasicek++ type, written $r_t = x_t + \phi(t)$, where $x_t$ is a mean-reverting Ornstein-Uhlenbeck process and $\phi$ is a piecewise-constant deterministic shift. The comparison of models relies on the convention that after a regulatory shock the parameters $(x_0, \theta, k, \sigma_r)$ stay fixed and only $\phi$ is recalibrated to the shocked yield curve, whereas Hull-White achieves the fit by adjusting its mean-reversion target $\vartheta(t)$; because Hull-White transmits changes through a damped integral of $\vartheta$, the required $\vartheta$ oscillates sharply, while the Vasicek++ shift changes smoothly. The second carrying mechanism is the equally weighted basket of bonds with maturities $1, \dots, n$, whose maturing nominal value is matched to the structural surrender rate, giving cash-flow matching without storing the trading history.

What would settle it

Calibrate the Hull-White model to the same shocked yield curve with a smoothing or full-recalibration convention, such as a piecewise-linear mean-reversion target or a re-estimated mean-reversion speed, and check whether the fitted curve still oscillates; if it does not, the paper's main modeling recommendation would not be supported.

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Extended reading notes

Core claim

The paper's central claim is that a faithful ALM model for life insurance needs to represent book and market values separately and needs a realistic crediting-rate rule, and once these are in place, the standard-formula SCR depends sensitively on two modeling choices: the short-rate model family and the maturity structure of the bond portfolio. Under the convention that a regulatory shock is implemented by re-fitting only the deterministic shift function, the shifted Vasicek++ model reacts to upward and downward regulatory shocks with stable shifts, whereas Hull-White, which mean-reverts to a parametric curve, requires a violently oscillating theta to hit the shocked curve, making post-shock valuations unreliable. The paper quantifies the consequence: in a 2% rate environment, the basket-of-bonds model gives a downward-shock SCR of 0.0078 and an upward-shock SCR of 0.0063, while a single-bond proxy gives 0.0113 and 0.0154, respectively. In a 0.5% environment, the optimal basket maturity moves from n=20 to n=12. A further finding is a discontinuity in the standard-formula aggregation factor when the upward shock becomes the binding one, which the paper argues is unfair and can be exploited by choosing bond maturity.

Load-bearing premise

The paper's comparison of rate models rests on the convention that after a regulatory shock the main parameters stay fixed and only the curve-fitting shift is changed; if the whole model were re-calibrated instead, the oscillating behavior that disqualifies Hull-White might disappear.

Editorial extensions

If this is right

Insurers using Hull-White or other mean-reverting curve models to set the initial term structure may get unstable and economically meaningless post-shock valuations; shifted models such as Vasicek++ or CIR++ are safer choices for the standard formula.
The maximal bond maturity $n$ is a genuine capital-management lever: in the moderate-rate setting approximately $n=20$ minimizes the interest-rate SCR, while in the low-rate setting approximately $n=12$ is optimal, so the practice of setting $n=1/p$ is not always SCR-minimizing.
Cash-flow matching through a bond ladder reduces the interest-rate SCR substantially in the paper's runs (upward SCR falls from 0.0154 to 0.0063 and downward SCR from 0.0113 to 0.0078), because it avoids realizing latent gains or losses when surrenders must be paid.
The standard formula's mean-based aggregation does not reward the lower variance of the Basic Own-Funds distribution produced by cash-flow matching, and its discontinuous correlation factor makes the SCR jump when an upward shock becomes the binding one, which can be triggered by small changes in $n$.
In low-rate regimes, the additive shock factors produce crossing shift functions in Vasicek++ after about 30 years, so even a well-behaved shifted model can show counterintuitive long-maturity rate movements and non-monotone crediting rates.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: because the Hull-White oscillation is driven by fixing $(r_0, k, \sigma_r)$ and re-fitting only $\vartheta$, a robustness check with full recalibration or a different $\vartheta$ parametrization would test whether the paper's model recommendation survives.
Beyond the paper: the relative frequencies of the four crediting-rate cases A-D could be published as an ALM distress indicator, since cases C and D signal that the insurer is falling short of the competitor or guaranteed rate and thereby trigger dynamic surrenders.
Beyond the paper: the shift of the optimal basket maturity from roughly $n=20$ in the 2% regime to roughly $n=12$ in the 0.5% regime hints that a rate-dependent or dynamic bond-ladder policy could outperform any fixed choice of $n$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper builds a synthetic ALM model for French GAAP life insurance and uses it to compute SCR under Solvency II. What is actually new: a four-case crediting-rate rule that is closer to practice than the usual Grosen-Jørgensen rule, an equally weighted bond basket that matches cash flows without storing trade history, and a comparison of Vasicek++ versus Hull-White under EIOPA shocks. The model is coherent, and the balance-sheet recursions check out; Lemma 2.3 is correct, and the accounting treatment of the capitalization reserve and profit-sharing reserve is careful.

The bond basket is the most valuable piece. Investing equally in maturities 1 through n and letting the one-year bond cover expected surrenders is a simple way to get immunization without tracking the full trade history. The crediting-rate mechanism is also a real step beyond the usual minimal-rate rule, and the four cases give a useful diagnostic for ALM monitoring. The Vasicek++ versus Hull-White analysis is analytically grounded, not a pure simulation artifact: the oscillations in the Hull-White theta follow from equation (3.5) when the shifted-model phi has jumps and the theta is forced to be piecewise constant. That said, the central modeling recommendation is conditional on the recalibration convention in Section 3—keeping k and sigma fixed after the shock and re-fitting only the shift function. A piecewise-linear theta or a re-estimated k might tame the Hull-White oscillations, so the paper overstates when it says Hull-White is "meaningless" after the shocks. The authors themselves flag a puzzling artifact on the Vasicek++ side, the shift functions crossing after 30 years, which reinforces that the model-ranking conclusion is not fully robust.

The numerical section has softer spots. Parameters are hand-set rather than calibrated, and there is no sensitivity analysis except for k and gamma. SCR values in Table 11 are reported without error bars, even though the BOF means have confidence intervals; the propagation from BOF means to SCR is not discussed. The proxy model for "no cash-flow matching" is adjusted via equations (4.1) and (4.2) to match the initial loss, so the comparison is not a clean experimental control. These are fixable issues, not fatal flaws.

Who this is for: actuaries and ALM practitioners, especially in the European life insurance space. It deserves a serious referee, and the editor should ask for standard errors on SCR estimates, a robustness section on the recalibration convention, and ideally code or data to reproduce the tables. With those revisions, I would be comfortable seeing it published.

Referee Report

2 major / 6 minor

Summary. The paper proposes a synthetic Asset-Liability Management (ALM) model for French general-account life insurance, tracking market and book values, a crediting-rate rule based on regulatory, competitor, and profit-sharing constraints, and a bond basket that matches a fraction of surrender cash flows. The model is used to compute the Solvency Capital Requirement (SCR) under the standard formula, for both the 2012 multiplicative shocks and the 2018 additive low-rate recommendation. Two main claims are made: (i) after regulatory interest-rate shocks, shifted models such as Vasicek++ are more meaningful than mean-reverting curve models such as Hull-White; and (ii) the bond-basket cash-flow matching materially reduces the interest-rate SCR, as shown by comparing the full model with a single-bond proxy model.

Significance. If the results are robust, the paper is useful for life insurers and regulators: it provides a practical ALM framework with the main accounting and behavioral features, and its cash-flow-matching analysis makes a decision-relevant point that the bond basket maturity can be used to reduce SCR. Strengths include a self-contained balance-sheet recursion with explicit steps, a correct and useful Lemma 2.3, the use of common random numbers for central and shocked scenarios, and a clear separation of the Vasicek++ and Hull-White parametrizations. The paper also usefully highlights a discontinuity in the standard-formula aggregation rule. The main limitation is that the central model-ranking claim is conditional on the chosen recalibration convention, and the cash-flow-matching conclusion depends on an ad hoc proxy model; both need robustness support before the quantitative recommendations can be taken at face value.

major comments (2)

[Section 3, Eqs. (3.3)-(3.5), Figures 1-4] The paper's central modeling recommendation—that Vasicek++ is more meaningful than Hull-White after EIOPA shocks—is established only under a specific recalibration convention. In Section 3, after a shock the authors keep (x0, theta, k, sigma_r) fixed in Vasicek++ and keep (k, sigma_r) fixed in Hull-White, re-fitting only phi_shock or theta_shock; since Eq. (3.5) connects the two classes only when phi is differentiable, Figure 1 compares a piecewise-constant shift with a piecewise-constant mean-reversion level, i.e., two different smoothness assumptions rather than two models. The authors themselves flag a counter-intuitive artifact on the Vasicek++ side in Figure 4 (the upward shock lowers the spot rate after about 30 years). To make the claim robust, the paper should show that the oscillations persist under alternative parametrizations (piecewise-linear theta, spline theta, or re-estimated k) or justify the piecewise-constant rule from regulatory practice. As written, the SCR differences in Figure 3 and Tables 10-11 are conditional on this convention.
[Section 4.2, Eqs. (4.1)-(4.2), Tables 10-11] The quantitative conclusion that cash-flow matching changes the SCR (Table 11) rests on the proxy model for a no-cash-flow-matching world, but the proxy is defined by ad hoc choices. The maturity n_p = n/2 is chosen "after numerical investigation", and Eq. (4.2) adjusts the bond position to force the same initial shock size; no sensitivity analysis is reported for n_p or for the approximation that the roll-over in the proxy model realizes no capital gain or loss. Since the magnitude of the SCR reduction is a central result, the authors should provide robustness tests (e.g., varying n_p, using an alternative single-bond model, or checking the no-CGL approximation) before claiming that the bond basket substantially reduces the SCR.

minor comments (6)

[Table 6] In the sdown row for maturity t=7, the entry is "39%" without a minus sign; this appears to be a typo for "-39%", given the adjacent values and the text.
[Table 10] The 95% confidence interval for the central Basket BOF is printed as "[0.0206,0.02010]"; the upper bound "0.02010" is likely a typo for "0.0210" or a misprinted precision.
[Section 2.2.3, after Eq. (2.9)] The model assumes MV_t > 0 and explicitly does not model the case MV_t <= 0; because the paper studies regulatory stress scenarios, the authors should either model this boundary case or report its probability of occurrence in the simulations.
[Tables 8-9 and 12-13] All model parameters are hand-set rather than calibrated to market data; the paper should state more explicitly that the numerical SCR values are illustrative and that conclusions such as the optimal n = 20 versus n = 12 are parameter-dependent.
[Figure 4 and surrounding text] The text says the shift functions cross after 30 years, but the right-hand panel shows the constant-rate example; the crossing for the low-rate Vasicek++ model is visible in the left and middle panels, so the figure labels and caption should be clarified.
[Table 11] The SCR values in Table 11 are reported without Monte Carlo error bars; given the nontrivial sampling noise visible in the BOF confidence intervals in Table 10, adding standard errors to the SCR differences would improve the comparability of the basket and proxy results.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Vasicek++ versus Hull-White ranking and the SCR results follow from explicitly stated recalibration rules and pre-chosen parameters, not from fitted outputs or self-citations.

full rationale

No load-bearing step in this paper derives its target result from its own fitted constants, and no central claim is justified by a self-citation chain. The ALM model equations in Sections 2–3 are constructed from explicit balance-sheet identities, legal reserve rules, and the standard-formula shock tables; the crediting rate determination is a decision rule, not an identity fitted to output. The interest-rate model comparison is an analytical consequence of the stated recalibration convention: after a shock, the authors freeze the mean-reversion parameters and recalibrate only the piecewise-constant shift functions, then show that Hull-White produces oscillating shifts while Vasicek++ does not. This is a clearly disclosed modeling convention, not a claim that the result is forced by the data or by a self-referential definition. The Vasicek++ versus Hull-White equivalence in Eq. (3.5) is taken from the external textbook Brigo-Mercurio [8], and the paper explicitly notes that the two parametrizations cease to be equivalent once piecewise-constant shifts are imposed, so the comparison is coherent rather than circular. The numerical SCR analysis in Section 4 uses parameters chosen before simulation and common random paths across central and shocked scenarios, so the SCR values are not fitted to the modules they are used to compute. The proxy-model comparison in Section 4.2 explicitly equalizes the initial shock size via Eqs. (4.1)–(4.2), controlling for the first-order effect rather than fitting the final SCR, and the remaining BOF/SCR differences reflect the cash-flow-matching mechanism. The paper itself flags the post-30-year sign crossing of the Vasicek++ shifts as puzzling, which is an honest limitation of the convention rather than a circular justification. The only substantive weakness, the one-sided recalibration convention, is an assumption about how a modeler applies regulatory shocks; it affects robustness and interpretation of the model ranking, but it is not circular because the comparison does not presuppose the conclusion through a parameter fitted to that conclusion. The paper is self-contained against institutional benchmarks (EIOPA tables, delegated regulation), and no step reduces to its own inputs by construction.

Assumptions & free parameters 17 free parameters · 9 assumptions · 0 invented entities

The central results rest on institutional accounting rules, standard pricing models, and many hand-set ALM parameters. None of the model parameters is estimated from market data, and the model introduces no new physical or financial entity; the competitor rate and reserves are institutional objects.

free parameters (17)

Equity allocation weight ws = 0.05 (moderate), 0.08 (low)
Constant target equity weight; chosen so SCReq and SCRint are of the same order, not estimated from data.
Bond allocation weight wb = 0.95 (moderate), 0.92 (low)
Complement of ws; hand-set allocation.
Equity volatility sigma_S = 0.1
Hand-set Black-Scholes volatility.
Interest rate volatility sigma_r = 0.01
Hand-set Vasicek volatility.
Mean reversion speed k = 0.2
Hand-set mean reversion speed; also varied in Figure 3.
Correlation gamma = 0
Hand-set correlation between equity and rate Brownian motions; varied in Figure 8.
Central rate level r0 and theta = 0.02 (moderate), 0.005 (low)
Hand-set initial and long-term short rate levels.
Participation rate pi_pr = 0.9
Profit-sharing rate; chosen above the French legal minimum of 0.85.
Guaranteed rate rG = 0.015 (moderate), 0 (low)
Minimum guaranteed crediting rate; hand-set.
PSR distribution fraction bar_rho = 0.5
Fraction of profit-sharing reserve distributed; motivated by the 8-year French redistribution rule.
Bond basket maximum maturity n = 20 (moderate), 10 (low)
Maturity of the equally weighted bond ladder; object of the cash-flow matching study.
Maximum dynamic surrender DSRmax = 0.3
Hand-set cap on market-driven surrenders.
Massive lapse threshold alpha = -0.05
Hand-set spread below which surrenders hit DSRmax.
Triggering surrender threshold beta = -0.01
Hand-set spread below which dynamic surrenders begin.
Structural lapse rate p = 0.05 (moderate), 0.1 (low)
Hand-set minimum surrender rate.
Time horizon T = 30 years
Run-off horizon chosen for the simulations.
Number of Monte Carlo paths N = not reported
Number of paths used in the simulations is not stated, which weakens the reported SCR precision.

assumptions (9)

standard math Assets are priced under a risk-neutral measure Q with a standard Brownian motion; equity follows Black-Scholes with drift r_t.
Section 3; standard no-arbitrage pricing assumptions used without derivation.
standard math Zero-coupon bond prices in Vasicek++ and Hull-White use the closed-form formulas from Brigo and Mercurio.
Section 3; accepted background results for affine short-rate models.
domain assumption French GAAP reserves (MR, CR, PSR) and Solvency II BOF/BEL definitions, plus the EIOPA shock tables, are taken as given institutional rules.
Sections 2.1, 2.4, and 2.5; the model is built to comply with these rules.
ad hoc to paper Policyholders exit uniformly on (t,t+1) and are paid the guaranteed rate pro-rata (Eq. 2.7).
Modeling simplification that directly affects cash-flow matching and surrender payments.
ad hoc to paper Book values are updated by proportional reduction rather than FIFO (Eqs. 2.10 and 2.16).
Stated approximation in Section 2.2.3 that avoids storing the trading history.
ad hoc to paper For interest-rate shocks, recalibration changes only the deterministic shift function while keeping other model parameters fixed.
Section 3; this premise drives the Vasicek++ versus Hull-White conclusion and is not derived from data.
domain assumption The portfolio is in run-off with no new policies, and the terminal liquidation follows Section 2.3.
Solvency II recommends run-off for SCR calculations, and the terminal procedure is part of the model design.
ad hoc to paper The market value of assets is assumed positive; the MV_t less than or equal to 0 case is noted but not modeled in detail.
Section 2.2.3, Step 3; the unresolved case could matter in extreme stress scenarios.
ad hoc to paper The proxy model in Section 4.2 approximates the no-cash-flow-matching world with a single bond and adjusts the position to match the shock size.
The proxy baseline uses n_p = n/2 chosen by numerical investigation, and Eq. (4.2) enforces equal initial shocks.

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Pith. "Pith review of A full and synthetic model for Asset-Liability Management in life insurance, and analysis of the SCR with the standard formula." pith.science (2026). https://pith.science/paper/KTVZ43EE

@misc{pith2026190800811,
  author       = {Pith},
  title        = {Pith review of: A full and synthetic model for Asset-Liability Management in life insurance, and analysis of the SCR with the standard formula},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KTVZ43EE}},
  note         = {Machine review of arXiv:1908.00811}
}

read the original abstract

The aim of this paper is to introduce a synthetic ALM model that catches the main specificity of life insurance contracts. First, it keeps track of both market and book values to apply the regulatory profit sharing rule. Second, it introduces a determination of the crediting rate to policyholders that is close to the practice and is a trade-off between the regulatory rate, a competitor rate and the available profits. Third, it considers an investment in bonds that enables to match a part of the cash outflow due to surrenders, while avoiding to store the trading history. We use this model to evaluate the Solvency Capital Requirement (SCR) with the standard formula, and show that the choice of the interest rate model is important to get a meaningful model after the regulatory shocks on the interest rate. We discuss the different values of the SCR modules first in a framework with moderate interest rates using the shocks of the present legislation, and then we consider a low interest framework with the latest recommandation of the EIOPA on the shocks. In both cases, we illustrate the importance of matching cash-flows and its impact on the SCR.

Figures

Figures reproduced from arXiv: 1908.00811 by the authors.

**Figure 1.** Calibrated piecewise constant functions t 7→ ϕ shock(t) (left) and t 7→ ϑ shock(t) (right) after the upward and downward shocks specified in [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗

**Figure 2.** Simulations after the upward shock on interest rates described in Table 6. Left: mean of [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗

**Figure 3.** SCR values with Vasicek++ and Hull and White models in function of [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: Calibrated piecewise constant functions t 7→ ϕ shock(t) (left) and t 7→ ϑ shock(t) (middle) after the upward and downward shocks specified in [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]

**Figure 5.** Figure 5: Left: Empirical distribution of the cases A, B, C, and D determining the crediting rate, [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]

**Figure 6.** Figure 6: Before and after the equity shock of 39%. Evolution of the mean crediting rate [PITH_FULL_IMAGE:figures/full_fig_p027_6.png]

**Figure 7.** Figure 7: Before and after the downward and upward shocks on interest rates. Evolution of the [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]

**Figure 8.** Figure 8: Values of the different SCR modules in function of [PITH_FULL_IMAGE:figures/full_fig_p028_8.png]

**Figure 9.** Figure 9: Left: mean value of the Basic Own-Funds in function of [PITH_FULL_IMAGE:figures/full_fig_p029_9.png]

**Figure 10.** Figure 10: Empirical BOF distributions with the proxy model and the original (Basket) model with [PITH_FULL_IMAGE:figures/full_fig_p030_10.png]

**Figure 11.** Figure 11: Before and after the downward and upward shocks on interest rates. Evolution of the [PITH_FULL_IMAGE:figures/full_fig_p033_11.png]

**Figure 12.** Figure 12: Mean value of the Basic Own-Funds in function of [PITH_FULL_IMAGE:figures/full_fig_p033_12.png]

Discussion (0). Continue with ORCID to comment.

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Julien Vedani, Nicole El Karoui, Stéphane Loisel, and Jean-Luc Prigent. Market inconsisten- cies of market-consistent european life insurance economic valuations: pitfalls and practical solutions. European Actuarial Journal, 7(1):1–28, Jul 2017. 35

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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Asset-liability management for long-term insurance business

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Fair pricing of life insurance participating policies with a minimum interest rate guaranteed.ASTIN Bulletin, 31(2):275–297, 2001

Anna Rita Bacinello. Fair pricing of life insurance participating policies with a minimum interest rate guaranteed.ASTIN Bulletin, 31(2):275–297, 2001. 33

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The eﬀects of a low interest rate environment on life insurers

Elia Berdin and Helmut Gründl. The eﬀects of a low interest rate environment on life insurers. The Geneva Papers on Risk and Insurance - Issues and Practice, 40(3):385–415, Jul 2015

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A stochastic forward-looking model to assess the proﬁtability and solvency of European insurers

Elia Berdin, Christoﬀer Kok, and Cosimo Pancaro. A stochastic forward-looking model to assess the proﬁtability and solvency of European insurers. ICIR Working Paper Series 21/16, Goethe University Frankfurt, International Center for Insurance Regulation (ICIR), 2016

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Tim J. Boonen. Solvency ii solvency capital requirement for life insurance companies based on expected shortfall.European Actuarial Journal, 7(2):405–434, Dec 2017

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Main Determinants of Proﬁt-Sharing Policy in the French Life Insurance Industry.Geneva Papers on Risk and Insurance - Issues and Practice, 43(3):420–455, July 2018

Fabrice Borel-Mathurin, Pierre-Emmanuel Darpeix, Quentin Guibert, and Stéphane Loisel. Main Determinants of Proﬁt-Sharing Policy in the French Life Insurance Industry.Geneva Papers on Risk and Insurance - Issues and Practice, 43(3):420–455, July 2018

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Solvency ii’s market risk standard formula: Howcredibleistheproclaimedruinprobability? Journal of Insurance Issues, 38(1):1– 30, 2015

Alexander Braun, Hato Schmeiser, and Florian Schreiber. Solvency ii’s market risk standard formula: Howcredibleistheproclaimedruinprobability? Journal of Insurance Issues, 38(1):1– 30, 2015

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Interest rate models—theory and practice

Damiano Brigo and Fabio Mercurio. Interest rate models—theory and practice. Springer Finance. Springer-Verlag, Berlin, second edition, 2006. With smile, inﬂation and credit

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