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In the actuarial literature, it has become common practice to model future capital returns and mortality rates stochastically in order to capture market risk and forecasting risk. Although interest rates often should and mortality rates always have to be non-negative, many authors use stochastic diffusion models with an affine drift term and additive noise. As a result, the diffusion process is Gaussian and, thus, analytically tractable, but negative values occur with positive probability. The argument is that the class of Gaussian diffusions would be a good approximation of the real future development. We challenge that reasoning and study the asymptotics of diffusion processes with affine drift and a general noise term with corresponding diffusion processes with an affine drift term and an affine noise term or additive noise. Our study helps to quantify the error that is made by approximating diffusive interest and mortality rate models with Gaussian diffusions and affine diffusions. In particular, we discuss forward interest and forward mortality rates and the error that approximations cause on the valuation of life insurance claims.

For the most part of the 20th century, actuaries used to model the lifespan of an insured stochastically, but relied on a mere deterministic prognosis of capital returns and mortality rates. The past has shown that these assumptions can vary significantly within one contract period. Especially in recent years, financial markets have experienced increased volatility, and life expectancies have risen in many developed countries at an unforeseen rate. For two decades, actuaries have been increasingly using stochastic interest rate models, and since the last decade, stochastic mortality rate models have become more and more popular. On the one hand, the stochastic approach helps to better capture systematic risk by offering the possibility to calculate confidence intervals. On the other hand, investors and insurance regulators nowadays call for a market-consistent valuation of the insurers assets and liabilities, so that concepts from financial mathematics have entered actuarial science, where the market value of a claim is calculated as a risk-neutral expectation in a specific stochastic model.

In financial mathematics, stochastic modeling of interest rates is a common approach, and a great variety of different interest rate models can be found in the literature. The majority of authors models the interest rate with the help of diffusion processes. Typical short rate models are, for example, the Vasicek model and the Cox–Ingersoll–Ross model. The evolution of forward rates is often modeled within the Heath–Jarrow–Morton framework. Readers may refer to [

The uncertainty about future mortality probabilities always drew the attention of life insurance actuaries. However, since recently, actuaries have been increasingly using stochastic models for the mortality rate. A very popular approach is to adopt the well-known models from interest rate theory, such as the Vasicek or the Cox–Ingersoll–Ross model. A more recent development is the concept of forward mortality rates, which are frequently modeled by a Heath–Jarrow–Morton framework similarly to forward interest rates. References that use diffusion process are, e.g., [

In the present paper, we study the asymptotics of diffusions with affine drift and a general noise term to corresponding diffusions with an affine drift term, but an affine noise term or additive noise. In the additive case, we end up with Gaussian processes, that is, the finite-dimensional projections are multivariate, normally distributed. The affine case is a generalization of the additive case and allows for better approximations, but the corresponding distributions are not always analytically tractable. We introduce a reasonable approximation concept and calculate bounds for the absolute moments of the approximation errors. Furthermore, we calculate the error that the approximating processes imply when valuating insurance claims. Our basic notion is to study asymptotics relative to the size of the noise term; the smaller the noise term, the better the approximation. Our results help to better assess the quality of Gaussian and affine approximations for interest and mortality rate models.

After introducing a basic life insurance model in

Assume that the lifetime of a policyholder is described by a nonnegative random variable

Let

Suppose that we are at time

In this section, we discuss in some generality the approximation of diffusions with affine drift and a general noise term by corresponding Gaussian diffusions with an affine drift term and additive noise.

Let

the column vector

the mapping,

the mapping

there exists a constant

These assumptions are sufficient to ensure that the stochastic differential equation according to (

In the insurance applications that we have in mind, the distribution of

For the proof, see

A natural question is whether approximating

All in all, we can conclude that in the case that interest and mortality rates are diffusion processes of the form of Equation (

For the proof, see the details of the proof of Theorem 3.2.

In the previous section, we simplified the diffusion process

For the proof, see

While

Suppose that the spot rates

For the proof, see

We now give a numeric example that illustrates the asymptotics of the Gaussian and the affine diffusion approximation. Let

Expectation function

Relative difference of

One-path simulation for (

Valuation w.r.t. the Single Simulated Path | ||||
---|---|---|---|---|

true value | 0.37988 | 0.43749 | 0.50274 | 0.57575 |

expectation approximation | 0.65644 | 0.65644 | 0.65644 | 0.65644 |

(relative error) | 0.72804 | 0.50046 | 0.30573 | 0.14016 |

Gaussian approximation | 0.39773 | 0.44968 | 0.50926 | 0.57768 |

(relative error) | 0.04700 | 0.02785 | 0.01296 | 0.00336 |

affine approximation | 0.41127 | 0.45185 | 0.50731 | 0.57635 |

(relative error) | 0.08264 | 0.03283 | 0.00910 | 0.00105 |

Monte Carlo simulation for the expectations of (

Valuation w.r.t. 10,000 Simulations | ||||
---|---|---|---|---|

true value | 0.68733 | 0.67571 | 0.66712 | 0.65827 |

expectation approximation | 0.65644 | 0.65644 | 0.65644 | 0.65644 |

(mean absolute deviation) | 0.16151 | 0.13076 | 0.09186 | 0.04839 |

Gaussian approximation | 0.70483 | 0.68155 | 0.66873 | 0.65835 |

(mean absolute deviation) | 0.10065 | 0.05486 | 0.02272 | 0.00500 |

linear noise approximation | 0.70280 | 0.68152 | 0.66875 | 0.65839 |

(mean absolute deviation) | 0.04802 | 0.02486 | 0.00863 | 0.00113 |

we have

the continuous function,

the mapping,

the continuous function

Let now

By using Inequalities (

Because of nice analytical features, in the actuarial literature, stochastic models for interest and mortality rates often have the form of Gaussian diffusions with an affine drift term and additive noise. The Gaussian diffusion framework is justified with the argument that it would reasonably approximate the true development of interest and mortality rates. We studied approximation errors if appropriate models for interest and mortality rates have the form of diffusions with affine drift, but a general noise term. We calculated theoretical bounds for absolute moments and valuation formulas, showing the speed of convergence. A numerical study illustrates the approximation error, confirming the theoretical results. Our results indicate that approximation errors are reasonable if the noise term is not too large. In particular, that might be the case when modeling the mortality intensity, since demographic changes of mortality typically happen slowly, but steadily.

We generalized the Gaussian diffusion approximation with additive noise by introducing an affine diffusion approximation with an affine noise term. We found that with an affine noise term instead of additive noise, the speed of convergence can be improved from

All in all, we identified a large class of diffusion processes that can be well approximated by Gaussian diffusions with additive noise or by affine diffusions with affine noise terms. As a consequence, we can say that modeling interest and mortality rates by Gaussian diffusions is well justified if there is evidence that the true developments of future interest and mortality rates have the form of diffusion processes with an affine drift term and an arbitrary, but small, noise term.

In the present paper, we measured approximation quality by point-wise moment errors and by mean absolute errors for some specific insurance claims. Future research should consider also other approximation measures.

The author thanks two anonymous referees whose comments greatly helped to improve the paper.

The author declares no conflict of interest.