The first fair valuation model we will present, is a model developed by Erwin Charlier and Ruud Kleynen (Charlier and Kleynen 2005). This model incorporates stochastic interest rates and applies the whole term structure to determine the fair value of L&P contracts and simulate intertemporal values of the assets and liabilities to investigate the financial position of L&P companies in any point in time. These two modifications from the GJ-model, makes the model more realistic and applicable to find the fair value of L&P contracts in a situation where policyholders are bought out of their guarantees prior to maturity. In the following summary of the Charlier and Kleynen (CK) model, we will only look at it from policyholders’ point of view.
9.2.1 Model Characteristics
The contracts are issued at time t=0 and matures at t=T; hence the values of the L&P contracts at time intervals between 0, 𝑇 are interesting. At time 0, policyholders pay a contribution of 𝐿!! to an L&P company, that is credited a base return, roffered, each period. The offered return represents the
interest rate guarantee, i.e. the minimum return credited to policyholders account each period. At the expiration date of the contract, the policyholder receives a minimum payment of 𝐿∗!, giving the relationship:
𝐿∗! = 𝐿 !
!𝑒!!""#$#%!
(22)
In addition to the base return, policyholders are also entitled to a share of the profits, i.e. a participating contract. The model assumes that L&P companies consist of shareholders and
policyholders, so the value of the company at time 0, A0, consist of shareholders fair value E0, and
policyholders fair value, 𝐿!!. Policyholders’ proportion of the total value of the company is represented by the leverage ratio, 𝛼.
9.2.1.1 The Contract at Maturity
If the value of assets does not cover the pension savings at maturity, policyholders receive the remaining value of the assets. Shareholders are left with no obligations and receive nothing, due to their limited responsibility.
59 When the value of the assets reaches the break-even point, policyholders receive their policy
reserves with a return equal to the offered return: 𝐿∗! = 𝛼𝐴
! → 𝐴! = 𝐿∗!
𝛼
(23)
If the value of the assets is above this break-even point, the profit is looked at as bonus where policyholders have a claim on a fraction, 𝛿, of the surplus they are entitled to:
𝐵! = 𝛿𝛼𝑚𝑎𝑥 0, 𝐴!−𝐿∗! 𝛼
(24)
Here the bonus has a lower limit of 0. Consequently, the final value policyholders receive is the initial contribution compounded by the offered return, and a bonus element if the value of the company exceeds the break-even point:
𝐿! = 𝐿∗!+ 𝐵 ! (25) 𝐿!= 𝐿!!𝑒!!""#$#%!+ 𝛿𝛼𝑚𝑎𝑥 0, 𝐴 !− 𝐿∗! 𝛼 (26) 9.2.2 Option View
Like in the GJ-model, the payout to policyholders could be looked at in an option view:
𝐿! = 𝐿∗!− 𝑚𝑎𝑥 0, 𝐿∗!− 𝐴! + 𝛿𝛼𝑚𝑎𝑥 0, 𝐴!− 𝐿∗!
𝛼
(27)
The second term corresponds to a short European put option with the value of the assets as underlying and the strike price equal to 𝐿∗!. This shorted put option represents the situation where the value of assets is less than the policy reserve, and policyholders receive the remaining value of the assets. The last term represents the situation where the value of the assets is higher than the break-even point; hence policyholders participate in the profit of the company. In an option view, this relates to a long European call option on the assets with strike price equal to the break-even point.
9.2.2.1 The Contract at Time t=0
When the contract value at maturity is found, the next step is to find the value of the contract at time 0:
𝐿! = 𝐿∗!𝑃(0, 𝑇) − 𝑃𝑢𝑡 𝐴!, 𝐿∗! + 𝛿𝛼𝐶𝑎𝑙𝑙 𝐴!, 𝐿∗!
𝛼
(28)
Here the value of the policy reserves is discounted with the price of a ZCB at time 0, with the time to maturity as the pension contract, T.
60 9.2.2.2 Intertemporal Values of the Contract
Eventually, the intertemporal values between time 0 and maturity can be calculated, resulting in: 𝐿!= 𝐿∗!𝑃(𝑡, 𝑇) − 𝑃𝑢𝑡 𝐴
!, 𝐿∗! + 𝛿𝛼𝐶𝑎𝑙𝑙 𝐴!, 𝐿∗!
𝛼
(29)
In this formula, P(t,T) is the price of a risk-free ZCB at time t with time to maturity of T-t and the value of the assets at time t. When the general valuation framework is established, we will next shortly mention the models used to evaluate the evolution of P(t,T), 𝑃𝑢𝑡 𝐴!, 𝐿∗! and 𝐶𝑎𝑙𝑙 𝐴
!,!!
∗
! .
9.2.3 Asset Modeling
In order to find the value of L&P contracts with guarantees, we need to look at how A(t) and P(t, T) evolve over time. Like in the GJ-model, the value of the asset portfolio is assumed to evolve like a Geometric Brownian motion. To look at the evolution of the price of a ZCB, we need to look at a term structure model that explains the behavior of the short-term interest rate. The most common approach is the Vasicek model. When finding the value of the European call- and put option, the framework for pricing options proposed by Black and Scholes (1973) and Merton (1973) could be used. The Vasicek model and the framework of Black and Scholes and Merton will not be further examined in this thesis.
By using stochastic interest rates as input when finding the fair value of the contracts, this model produces a more realistic estimate of the fair value of the guarantees than the GJ model does with fixed interest rates. However, the model could be improved further to reflect the true world better, which is done in a fair valuation model developed by Anna Rita Bacinello, where periodical payments are considered as well as mortality. Because of these additional elements, the model is more complex than the two others and will due to this be simplified and explained shortly in the following.