Comparing risks is the very essence of the actuarial profession. This chapter offers mathematical... more Comparing risks is the very essence of the actuarial profession. This chapter offers mathematical concepts and tools to do this, and derives some important results of non-life actuarial science. There are two reasons why a risk, representing a non-negative random financial loss, would be universally preferred to another. One is that the other risk is larger, see Section 7.2, the second is that it is thicker-tailed (riskier), see Section 7.3. Thicker-tailed means that the probability of extreme values is larger, making a risk with equal mean less attractive because it is more spread and therefore less predictable. We show that having thicker tails means having larger stop-loss premiums.We also show that preferring the risk with uniformly lower stop-loss premiums describes the common preferences between risks of all risk averse decision makers. From the fact that a risk is smaller or less risky than another, one may deduce that it is also preferable in the mean-variance order that is ...
Analysis and Applications in the Social Sciences, 2004
ABSTRACT Multiple linear regression is the most widely used statistical technique in practical ec... more ABSTRACT Multiple linear regression is the most widely used statistical technique in practical econometrics. In actuarial statistics, situations occur that do not fit comfortably in that setting. Regression assumes normally distributed disturbances with a constant variance around a mean that is linear in the collateral data. In many actuarial applications, a symmetric normally distributed random variable with a variance that is the same whatever the mean does not adequately describe the situation. For counts, a Poisson distribution is generally a good model, if the assumptions of a Poisson process described in Chapter 4 are valid. For these random variables, the mean and variance are the same, but the datasets encountered in practice generally exhibit a variance greater than the mean. A distribution to describe the claim size should have a thick right-hand tail. The distribution of claims expressed as a multiple of their mean would always be much the same, so rather than a variance not depending of the mean, one would expect the coefficient of variation to be constant. Furthermore, the phenomena to be modeled are rarely additive in the collateral data. A multiplicative model is much more plausible. If other policy characteristics remain the same, moving from downtown to the country would result in a reduction in the average total claims by some fixed percentage of it, not by a fixed amount independent of the original risk. The same holds if the car is replaced by a lighter one.
The UvA-LINKER will give you a range of other options to find the full text of a publication (inc... more The UvA-LINKER will give you a range of other options to find the full text of a publication (including a direct link to the full-text if it is located on another database on the internet). De UvA-LINKER biedt mogelijkheden om een ...
Multiple linear regression is the most widely used statistical technique in practical econometric... more Multiple linear regression is the most widely used statistical technique in practical econometrics. In actuarial statistics, situations occur that do not fit comfortably in that setting. Regression assumes normally distributed disturbances with a constant variance around a mean that is linear in the collateral data. In many actuarial applications, a symmetric normally distributed random variable with a variance that is the same whatever the mean does not adequately describe the situation. For counts, a Poisson distribution is generally a good model, if the assumptions of a Poisson process described in Chapter 4 are valid. For these random variables, the mean and variance are the same, but the datasets encountered in practice generally exhibit a variance greater than the mean. A distribution to describe the claim size should have a thick right-hand tail. The distribution of claims expressed as a multiple of their mean would always be much the same, so rather than a variance not depen...
In the recent actuarial literature, several proofs have been given for the fact that if a random ... more In the recent actuarial literature, several proofs have been given for the fact that if a random vector X(1), X(2), …, X(n) with given marginals has a comonotonic joint distribution, the sum X(1) + X(2) + … + X(n) is the largest possible in convex order. In this note we give a lucid proof of this fact, based on a geometric interpretation of the support of the comonotonic distribution.
Comparing risks is the very essence of the actuarial profession. This chapter offers mathematical... more Comparing risks is the very essence of the actuarial profession. This chapter offers mathematical concepts and tools to do this, and derives some important results of non-life actuarial science. There are two reasons why a risk, representing a non-negative random financial loss, would be universally preferred to another. One is that the other risk is larger, see Section 7.2, the second is that it is thicker-tailed (riskier), see Section 7.3. Thicker-tailed means that the probability of extreme values is larger, making a risk with equal mean less attractive because it is more spread and therefore less predictable. We show that having thicker tails means having larger stop-loss premiums.We also show that preferring the risk with uniformly lower stop-loss premiums describes the common preferences between risks of all risk averse decision makers. From the fact that a risk is smaller or less risky than another, one may deduce that it is also preferable in the mean-variance order that is ...
Analysis and Applications in the Social Sciences, 2004
ABSTRACT Multiple linear regression is the most widely used statistical technique in practical ec... more ABSTRACT Multiple linear regression is the most widely used statistical technique in practical econometrics. In actuarial statistics, situations occur that do not fit comfortably in that setting. Regression assumes normally distributed disturbances with a constant variance around a mean that is linear in the collateral data. In many actuarial applications, a symmetric normally distributed random variable with a variance that is the same whatever the mean does not adequately describe the situation. For counts, a Poisson distribution is generally a good model, if the assumptions of a Poisson process described in Chapter 4 are valid. For these random variables, the mean and variance are the same, but the datasets encountered in practice generally exhibit a variance greater than the mean. A distribution to describe the claim size should have a thick right-hand tail. The distribution of claims expressed as a multiple of their mean would always be much the same, so rather than a variance not depending of the mean, one would expect the coefficient of variation to be constant. Furthermore, the phenomena to be modeled are rarely additive in the collateral data. A multiplicative model is much more plausible. If other policy characteristics remain the same, moving from downtown to the country would result in a reduction in the average total claims by some fixed percentage of it, not by a fixed amount independent of the original risk. The same holds if the car is replaced by a lighter one.
The UvA-LINKER will give you a range of other options to find the full text of a publication (inc... more The UvA-LINKER will give you a range of other options to find the full text of a publication (including a direct link to the full-text if it is located on another database on the internet). De UvA-LINKER biedt mogelijkheden om een ...
Multiple linear regression is the most widely used statistical technique in practical econometric... more Multiple linear regression is the most widely used statistical technique in practical econometrics. In actuarial statistics, situations occur that do not fit comfortably in that setting. Regression assumes normally distributed disturbances with a constant variance around a mean that is linear in the collateral data. In many actuarial applications, a symmetric normally distributed random variable with a variance that is the same whatever the mean does not adequately describe the situation. For counts, a Poisson distribution is generally a good model, if the assumptions of a Poisson process described in Chapter 4 are valid. For these random variables, the mean and variance are the same, but the datasets encountered in practice generally exhibit a variance greater than the mean. A distribution to describe the claim size should have a thick right-hand tail. The distribution of claims expressed as a multiple of their mean would always be much the same, so rather than a variance not depen...
In the recent actuarial literature, several proofs have been given for the fact that if a random ... more In the recent actuarial literature, several proofs have been given for the fact that if a random vector X(1), X(2), …, X(n) with given marginals has a comonotonic joint distribution, the sum X(1) + X(2) + … + X(n) is the largest possible in convex order. In this note we give a lucid proof of this fact, based on a geometric interpretation of the support of the comonotonic distribution.
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