By Thomas Luther
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Additional resources for Aktien: Erfolgreiche Strategien an der Börse
By assigning an appropriate probability description of the scenario space, we can move from conditional probability distributions to an unconditional distribution, which ‘summarizes’ both components of risk, namely the uncertainty risk and the risk of random ﬂuctuations. This topic will be focussed in Chapter 7 while dealing with the assessment of longevity risk. 16. 6 Conditional probability distributions of the random present value of the life annuity. Types of life annuities In the previous sections we have dealt with an immediate life annuity in arrears, that is a life annuity whose ﬁrst payment is due one period (one year, according to our assumptions) from the date of purchase, while the last payment is made at the end of the period preceding the death of the annuitant.
Allowing for randomness provides us with a tool for assessing the ‘risk’ inherent in a life annuity portfolio or a pension plan. 9(a) and (b), random ﬂuctuations affect the portfolio behaviour, and these are caused (in this example) by the randomness in the number of survivors throughout time. 11. 12 M20 Statistical distributions of M5 and M20 . 2). 10(a) and (b) suggest measures which can be used for assessing the riskiness of a life annuity portfolio in terms of the ‘dispersion’ of the fund Zt .
A measure of the mutuality effect. when moderately old ages are involved (say, in the interval 65–75), the values of θ are rather small. In such a range of ages, they could be ‘replaced’ with a higher yield from investments (provided that riskier investments can be accepted), and so, in that age interval, a withdrawal process could be preferred to a life annuity. Conversely, as the age increases, θ reaches very high values, which obviously cannot be replaced by investment yields. So, when old and very old ages are concerned, the life annuity is the only technical tool which guarantees a lifelong constant income.