Guarantee valuation in Notional Defined Contribution pension systems

The notional defined contribution pension scheme combines pay-as-you-go fi- nancing and a defined contribution pension formula. The return on contributions is based on a notional rate which is linked to an external index set by law, such as the growth rate of GDP, average wages, or contribution payments. The volatility of this return may introduce a pension adequacy problem in the system and therefore guarantees may be needed. Here we focus on the guarantee of a minimum return on the contributions made to the pension scheme and we calculate its price in a utility indifference framework. We obtain a closed-form solution in a general dependence structure with exponential preferences and in presence of stochastic short interest rates