Super-Long Discount Rates for Insurers in Incomplete Markets with Bond Supply Control

Abstract

This paper presents a multi-agent equilibrium model within an incomplete market framework, introducing a super-long discount rate for insurance companies while incorporating the dynamics of government financing and central bank operations in purchasing government bonds.
The financial market under examination includes dividend-paying securities that represent the market value of outstanding government bonds, categorized by their time to maturity. The model considers the optimal consumption and portfolio problems of agents, who possess varying risk sentiments and heterogeneous time preferences for consumption.
Applying a convex dual problem approach, we derive expressions for the equilibrium interest rate and market price of risk associated with government bonds, segmented by their remaining time to maturity. Additionally, we associate exogenously determined dividend processes, which reflect notional adjustments due to the central bank’s bond purchases, government bond issuances, and coupon payments, in order to derive the yield curve for government bonds in equilibrium.
Our study’s contribution lies in incorporating changes in the supply of government bonds in the secondary market into the incomplete market equilibrium model. This approach reveals the heterogeneous perspectives among agents regarding fundamental risks, represented by Brownian motion, and constructs a model that explicitly calculates the impact of net supply changes on super-long discount rates.
Lastly, we examine how variations in the supply of government and central bank bonds impact the pricing of insurance products, including death benefits and pension annuities, through changes in the super-long discount rates. This analysis is vital for insurance pricing and the evaluation of government bonds on the asset side of insurance companies.