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Risk Neutral Probability

Real world survival probabilities are very useful for mortality studies. However, they do not account for longevity risk - the financial risk that people live longer than expected and are thus not suitable for the pricing of insurance products. Hence, this module includes a function that transforms the real world (P world) survival function to a risk neutral (Q world) survival function according to a risk-neutral principle following Tang & Li (2021).

To introduce the transformation methods used by this function, the following simplified notation is used:

  • \(f^P(t)\) or \(f^Q(t)\): probability density function (pdf)
  • \(F^P(t)\) or \(F^Q(t)\): cumulative distribution function (cdf)
  • \(S^P(t)\) or \(S^Q(t)\): survival function

where \(S^P(t) = 1 - F^P(t)\) and \(S^Q(t) = 1 - F^Q(t)\). Note that these functions are applied on an individual aged \(x\) in year \(y\) for each simulation \(i\).

The first set of transformations are survival function distortions and are as follows:

  • Wang transform: \(S^Q(t) = 1 - \Phi(\Phi^{-1}(1 - S^P(t)) - \lambda) \quad (\lambda \geq 0)\)
    where \(\Phi(x)\) represents the cdf of a standard Gaussian distribution.

  • Proportional hazard transform: \(S^Q(t) = (S^P(t))^{\frac{1}{\lambda}} \quad (\lambda \geq 1)\)

  • Dual-power transform: \(S^Q(t) = 1 - \left(1 - S^P(t)\right)^\lambda \quad (\lambda \geq 1)\)
  • Gini principle: \(S^Q(t) = (1 + \lambda)S^P(t) - \lambda(S^P(t))^2 \quad (0 \leq \lambda \leq 1)\)
  • Denneberg's absolute deviation principle: \(S^Q(t) = \begin{cases} (1 + \lambda)S^P(t), & 0 \leq S^P(t) < 0.5 \\ \lambda + (1 - \lambda)S^P(t), & 0.5 \leq S^P(t) \leq 1 \end{cases} \quad (0 \leq \lambda \leq 1)\)
  • Exponential transform: \(S^Q(t) = \frac{1 - e^{-\lambda S^P(t)}}{1 - e^{-\lambda}} \quad (\lambda > 0)\)
  • Logarithmic transform: \(S^Q(t) = \frac{\log(1 + \lambda S^P(t))}{\log(1 + \lambda)} \quad (\lambda > 0)\)

The remaining transformations are pdf distortions and are given by:

  • Canonical valuation: \(f^Q(t) = \frac{e^{\lambda t} f^P(t)}{\sum^\infty_{s=0} e^{\lambda s} f^P(s)} \quad (\lambda > 0)\)
  • Esscher transform: \(f^Q(t) = \frac{e^{\lambda t} f^P(t)}{\mathbb{E}[e^{\lambda t}]} = \frac{e^{\lambda t} f^P(t)}{\sum^\infty_{s=0} e^{\lambda s} f^P(s)} \quad (\lambda > 0)\)

We note that for the pdf distortions (univariate canonical valuation and esscher transforms), the risk-adjusted pdfs are identical.

Survival Function Risk-Neutral Transformation

survivalP2Q(StP, method, lambda)

   Parameters:

     StP : matrix/array

       survival function under P-measure with survival time rows, cohort/year columns

       (and simulation number 3rd dimension)

     method : character

       the distortion risk measure to be used, see details for more information

     lambda : numeric

       parameter associated with the distortion risk measure

   Details:

       "wang": Wang Transform \((\lambda \geq 0)\)

       "ph": Proportional Hazard Transform \((\lambda \geq 1)\)

       "dp": Dual-power Transform \((\lambda \geq 1)\)

       "gp": Gini Principle \((0 \leq \lambda \leq 1)\)

       "dadp": Denneberg's Absolute Deviation Principle \((0 \leq \lambda \leq 1)\)

       "exp": Exponential Transform \((\lambda > 0)\)

       "log": Logarithmic Transform \((\lambda > 0)\)

       "canon": Univariate Canonical Valuation \((\lambda > 0)\)

       "esscher": Esscher Transform \((\lambda > 0)\)

   Returns:

     matrix/array of risk neutral survival function under specified Q-measure with survival time

     rows, cohort/year columns (and simulation number 3rd dimension)

   Usage:

# create survival function for an individual aged 55
AUS_male_rates <- mortality_AUS_data$rate$male
ages <- mortality_AUS_data$age # 0:110
old_ages <- 91:130
fitted_ages <- 76:90

completed_rates <- complete_old_age(AUS_male_rates, ages, old_ages,
                                    method = "kannisto", type = "central",
                                    fitted_ages = fitted_ages)

all_ages <- 0:130
surv_func <- rate2survival(completed_rates, ages = all_ages,
                           from = 'central', init_age = 55)

# convert from P to Q measure survival function
surv_func_Q <- survivalP2Q(surv_func, method = "wang", lambda = 1.5)

   References:

Sixian Tang & Jackie Li (2021) Market pricing of longevity-linked securities, Scandinavian Actuarial Journal, 2021:5, 408-436, DOI: 10.1080/03461238.2020.1852105