## Cass means business Faculty of Actuarial Science and Statistics (2004)

### BibTeX

@MISC{Ballotta04cassmeans,

author = {Laura Ballotta and Laura Ballotta},

title = {Cass means business Faculty of Actuarial Science and Statistics},

year = {2004}

}

### OpenURL

### Abstract

Alternative framework for the fair valuation of participating life insurance contracts

### Citations

57 | P.Sebastiani,“Learning Bayesian networks from incomplete data
- Ramoni
- 1997
(Show Context)
Citation Context ...es that we need to find the parameter h solving r = ln ML (1 + h, 1) − ln ML (h, 1) , or, making use of the Lévy-Khintchine formula, ∫ a − r − xν (dx) + σ2 2 + σ2 ∫ h + R e R hx (e x − 1) ν (dx) = 0. =-=(13)-=- It can be easily checked that this last expression corresponds to the martingale condition (8) for the choices G = −σh and H (t, x) = ehx (see also equation A8 in the Appendix). Consequently, ˆPh [N ... |

38 | Addendum to “Analytic and Bootstrap Estimates of Prediction Errors in Claims Reserving - England - 2001 |

34 |
Maximum entropy sampling and optimal Bayesian experimental design
- Sebastiani, Wynn
- 2000
(Show Context)
Citation Context ... as: A (t) = A (0) e L(t) , A (0) = P0, where {L (t) : t ≥ 0} is a Lévy motion with finite activity under the real probability measure P, i.e. ∫ t ∫ L (t) = at + σW (t) + x (N (ds, dx) − ν (dx) ds) , =-=(3)-=- where • W is a standard P-Wiener process; 0 R • X1, X2, ..., is a sequence of i.i.d. random variables with density function f (dx), modelling the size of the jumps in the Lévy process; • N is an homo... |

26 | Valuation of guaranteed annuity conversion options - Ballotta, Haberman - 2003 |

24 |
Mixture Reduction via Predictive Scores
- Cowell
- 1996
(Show Context)
Citation Context ...essed by a constant. Therefore, the martingale condition characterizing the equivalent probability measure ˆ P can be expressed as ∫ a − r − R xν (dx) + σ2 2 ∫ − σG + R (e x − 1) H (t, x) ν (dx) = 0. =-=(8)-=- However, equation (8) shows that the market is incomplete, as in general there are infinitely many ways of choosing G and H so that (8) is satisfied, which means that ˆ P is not unique and the market... |

21 | The Fair Valuation Problem of Guaranteed Annuity Options: The Stochastic Mortality Environment Case - Ballotta, Haberman - 2003 |

17 | The use of exogenous knowledge to learn Bayesian Networks for incomplete databases
- Ramoni, Sebastiani
- 1997
(Show Context)
Citation Context ...ˆ Ph-martingale. This is obtained by determining the parameter h as solution of e −rt S (t) = Êh ( ) −ru e S (u) |Ft , t < u, or, equivalently, S0 = Êh ( ) −rt e S (t) = S0e −rt ( ) t MX (1 + h, 1) . =-=(12)-=- MX (h, 1) The application of this procedure to the market model proposed in section 3, implies that we need to find the parameter h solving r = ln ML (1 + h, 1) − ln ML (h, 1) , or, making use of the... |

15 | 2006): Guarantees in With-Profit and Unitized With-Profit Life Insurance Contracts: Fair Valuation Problem in Presence of the Default Option - Ballotta, Haberman, et al. |

14 | On the Forecasting of Mortality Reduction Factors - Renshaw, Haberman |

13 | Parameter Learning from Incomplete Data Using Maximum Entropy II: Application to Bayesian Networks - Cowell - 1999 |

12 | Modelling and Valuation of Guarantees in With-Profit and Unitised With-Profit Life Insurance Contracts - Haberman, Ballotta, et al. - 2003 |

12 |
A Note on D-optimal Designs for a Logistic Regression Model
- Sebastiani, Settimi
(Show Context)
Citation Context ...= P0 α with k=0 { ∑T −1 = P0 α k=0 e −rk (1 − α) k T∏ −k t=1 ∏ e −rk (1 − α) k T −k t=1 V M t (0) = Ê [ e −r (1 + rP (t)) ] Ê [ e −r (1 + rP (t)) ] + e −rT (1 − α) T V M t (0) + e −rT (1 − α) T } } , =-=(4)-=- = Ê [ e −r ( 1 + rG + (βrA (t) − rG) +)] = e −r (1 + rG) + Ê [ e −r ( βe L′ ) ] + (1) − (β + rG) . (5) The term L ′ denotes an independent copy of the Lévy process L. Equation (5) shows that the poli... |

12 |
First-order Optimal Designs for Non Linear Models
- Sebastiani, Settimi
- 1996
(Show Context)
Citation Context ... e −r (1 + rP (t)) ] Ê [ e −r (1 + rP (t)) ] + e −rT (1 − α) T V M t (0) + e −rT (1 − α) T } } , (4) = Ê [ e −r ( 1 + rG + (βrA (t) − rG) +)] = e −r (1 + rG) + Ê [ e −r ( βe L′ ) ] + (1) − (β + rG) . =-=(5)-=- The term L ′ denotes an independent copy of the Lévy process L. Equation (5) shows that the policy reserve can be decomposed into a sequence of one year riskless zero coupon bonds and one year call o... |

11 | 2003a): Pricing Guaranteed Life Insurance Participating Policies with Annual Premiums and Surrender Option - Bacinello |

11 | A Bayesian Generalised Linear Model for the Bornhuetter-Ferguson Method of Claims Reserving - Verrall - 2001 |

11 |
Coherent Criteria for Optimal Experimental Design
- Dawid, Sebastiani
- 1996
(Show Context)
Citation Context ...licyholder’s overall claim at expiration can be summarised as follows: { A (T ) if A (T ) < P (T ) C (T ) = P (T ) + γR (T ) otherwise, or, in a more compact way: C (T ) = P (T ) + γR (T ) − D (T ) , =-=(2)-=- where D (T ) = (P (T ) − A (T )) + . Applying risk-neutral valuation, the market value of the policyholder’s claim is: with C (0) = V P (0) + γV R (0) − V D (0) , V P (0) = Ê [ e −rT P (T ) ] , V R (... |

11 | Robust Parameter Learning in Bayesian Networks with Missing Data
- Sebastiani, Ramoni
- 1997
(Show Context)
Citation Context ...um to be zero. In this case, the Radon-Nikod´ym derivative is η (t) = dˆ PM dP G2 (t)− = e−GW 2 t , where G solves the martingale condition ∫ a − r − xν (dx) + σ2 ∫ − σG + 2 R R (e x − 1) ν (dx) = 0, =-=(9)-=- and ˆ PM denotes the equivalent martingale measure resulting from this approach. Under these assumptions, we obtain that ˆPM [N (t) = n] = E [ ] η (t) 1(N(t)=n) = E [η (t)] E [ ] 1(N(t)=n) = P [N (t)... |

11 |
Sampling without Replacement in Junction Trees
- Cowell
- 1997
(Show Context)
Citation Context ...; h) Hence, the pricing formula for the fair value of the policy reserve is V P ⎧ ⎨ (0) = P0 ⎩ α ∑T −1 e −rk (1 − α) k [ e −r ∞∑ (1 + rG) + n! k=0 +e −rT (1 − α) T } n=0 e −λµh+1 (λµh+1) n f (n; h) . =-=(15)-=- 5 Numerical results: analysis of the price biases In this section we use the results obtained above to analyze the differences in the contract value implied by the Lévy process setting proposed, and ... |

11 | When Learning Bayesian Networks from Data, using Conditional Independence Tests is Equivalant to a Scoring Metric - Cowell - 2001 |

10 | Multiple State Models, Simulation and Insurer Insolvency - Haberman, Butt, et al. - 2001 |

10 | A Cash-Flow Approach to Pension Funding - Khorasanee - 2001 |

10 | The Income Drawdown Option: Quadratic Loss - Gerrard, Haberman, et al. - 2004 |

9 | Lee-Carter Mortality Forecasting, a Parallel GLM Approach, England and Wales Mortality Projections - unknown authors - 2002 |

9 | Application of Frailty-Based Mortality Models to Insurance Data - Butt, Haberman |

9 | Optimal Premium Pricing in Motor Insurance: A Discrete Approximation. (Will be available - Gerrard, Glass - 2003 |

9 | The Neighbourhood Health Economy. A systematic approach to the examination of health and social risks at neighbourhood level - Mayhew - 2002 |

9 | Kaishev V.K and Krachunov R.S. Optimal Retention Levels, Given the Joint Survival of Cedent and Reinsurer - Ignatov - 2003 |

9 | Finite time Ruin Probabilities for Continuous Claims Severities. Will be available in - Dimitrova, Ignatov, et al. - 2004 |

9 | Application of Stochastic Methods in the Valuation of Social Security Pension Schemes - Iyer - 2004 |

9 | Lee-Carter Mortality Forecasting Incorporating Bivariate Time Series - unknown authors - 2003 |

9 | Khuen Y.Y. and Verrall R.J. Modelling Operational Risk with Bayesian Networks - Cowell - 2004 |

9 | Alternative Framework for the Fair Valuation of Participating Life Insurance Contracts - Laura - 2004 |

9 |
Some Results on the Derivatives of Matrix Functions
- Sebastiani
- 1995
(Show Context)
Citation Context ... contractually specified guaranteed annual policy interest rate. In this discussion, we ignore lapses and mortality. Hence, the policy reserve is defined as P (t) = αP 1 (t) + (1 − α) P (t − 1) , α ∈ =-=(0, 1)-=- , P (0) = P0, where P 1 (t) is the unsmoothed asset share such that P 1 (0) = P0, P 1 (t) = P 1 (t − 1) (1 + rP (t)) , { rP (t) = max rG, β A (t) − A (t − 1) A (t − 1) and rG and β ∈ (0, 1) are the g... |

9 |
Coolen F.P.A. Guidelines for Corrective Replacement Based on Low Stochastic Structure Assumptions
- Newby
- 1997
(Show Context)
Citation Context ...an call option in equation (5) . Under the framework set out in this section, it follows that [ ÊM e −r ( βe L′ ) ] + (1) − (β + rG) = ÊM { [ ÊM e −r ( βe L′ ) ∣ + ∣∣∣ (1) − (β + rG) N ′ ]} (1) = n . =-=(10)-=- Consequently, if y is a standardized Normal random variable, the inner expectation in the previous equation can be rewritten as [ ÊM e −r ( βe rn− v2 ) ] + n +vny 2 − (β + rG) = e −λ(µ−1)+n ln µ f (n... |

9 |
Approximations for the Absorption Distribution and its Negative Binomial Analogue
- Newby
- 1997
(Show Context)
Citation Context ...(n) n! e −λµ (λµ) n whilst the fair value of the policy reserve is V P ⎧ ⎨ (0) = P0 ⎩ α ∑T −1 e −rk (1 − α) k [ e −r (1 + rG) + k=0 +e −rT (1 − α) T } n! ∞∑ n=0 f (n) ; e −λµ (λµ) n n! f (n) ] T −k . =-=(11)-=- 4.2 Policy fair valuation: the Esscher measure In the previous section, we derived a valuation formula under the assumption that the jump risk is not priced, as in Merton (1976). In terms of CAPM ass... |

9 |
Risk Based Optimal Designs
- Sebastiani, Wynn
- 1997
(Show Context)
Citation Context ... martingale condition (8) for the choices G = −σh and H (t, x) = ehx (see also equation A8 in the Appendix). Consequently, ˆPh [N (t) = n] = E [ ] η (t) 1(N(t)=n) = P [N (t) = n] e n ln µh−λt(µh−1) , =-=(14)-=- with h2 hµX+ µh = e 2 σ2 X; 13and (( [ ] kL(t) k r− Êh e |N (t) = n = e σ2 2 −∫R ehx (ex ) −1)ν(dx) t+nµX+nhσ2 ) X + k2 2 (σ2t+nσ2 X) , (see equation A9 in the Appendix) which implies that, conditio... |

9 |
Optimal Overhaul Intervals with Imperfect Inspection and Repair
- Dagg, Newby
(Show Context)
Citation Context ...) = V P (0) + γV R (0) − V D (0) , against the payment of an initial (single) premium P0, as seen in section 2, then the no-arbitrage combinations of contract parameters must be such that C (0) = P0. =-=(16)-=- Since the market parameters, like the volatility of the reference portfolio or the frequency with which jumps occur in the economy, i.e. λ, are in general not under the control of the life insurance ... |

9 | A Characterisation of Phase Type Distributions - Wolstenholme - 1997 |

9 | A Comparison of Models for Probability of Detection (POD) Curves - Wolstenholme |

9 | FINEX : Forensic Identification by Network Expert Systems - Cowell - 2001 |

8 |
Fair pricing of life insurance participating contracts with a minimum interest rate guaranteed
- Bacinello
- 2001
(Show Context)
Citation Context ...P-Brownian motion. The one year call option embedded in the policy reserve has value Ê [ e −r ( βe ( r− σ2 ) A +σA 2 ˆ W ′(1) − (β + rG) ) +] . (6) Applying the Black-Scholes formula to (6) (see also =-=Bacinello, 2001-=-, and Miltersen and Persson, 2003, for similar results), we obtain Ê [ e −r ( βe ( r− σ2 ) A +σA 2 ˆ W (1) − (β + rG) ) +] = βN (d1) − e −r (β + rG) N (d2) , where d1 = ln β β+rG + ( σA r + σ2 A 2 ) ;... |

8 |
Moments and Generating Functions for the Absorption Distribution and its Negative Binomial Analogue
- Newby
- 1996
(Show Context)
Citation Context ... d1 − σA. Consequently, the value of the policyholder’s account at inception is V P { ∑T −1 (0) = P0 α k=0 +e −rT (1 − α) e −rk (1 − α) k [ e −r (1 + rG) + βN (d1) − e −r (β + rG) N (d2) ] T −k T } . =-=(7)-=- 4 Option pricing in a jump-diffusion economy Consider now the more general case in which the driving process is described (in the real world) by equation (3). In order to calculate the fair value of ... |

4 | Pension Funding and the Actuarial Assumption Concerning Investment Returns - Owadally - 2003 |

4 | Bayesian Experimental Design and Shannon Information - Sebastiani, Wynn - 1997 |

3 | This number issued to Ben - S, Karlsoon - 2004 |