Discussion Papers 2025

CIRJE-F-1241	"Super-Long Discount Rates for Insurers in Incomplete Markets with Bond Supply Control"
Author Name	Saito, Taiga and Akihiko Takahashi
Date	February 2025
Full Paper	PDF file
Remarks	Revised in July 2025

Abstract
This paper presents a multi-agent equilibrium model within an incomplete market framework, deriving 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 in this study 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. Using a convex dual problem approach, we derive expressions for the equilibrium interest rate and market price of risk associated with government bonds, categorized by their remaining time to maturity. Additionally, we associate exogenously determined dividend processes representing coupon payments from the government bonds categorized by time-to-maturity, which reflect market outstanding notional adjustments due to the central bank's bond purchases and government bond issuance, in order to derive the yield curve for discount rates in equilibrium. The main contribution of our study is the incorporation of changes in the supply of government bonds in the secondary market into the incomplete market equilibrium model. This approach incorporates the heterogeneous views of agents regarding fundamental risks, represented by Brownian motion, and constructs a model that explicitly calculates the impact of net supply changes of the government bonds on super-long discount rates. Finally, we examine how variations in the supply of government bonds impact the pricing of insurance products, including death benefits and pension annuities, through changes in the super-long discount rates. This analysis is essential for insurance pricing and the evaluation of government bonds on the asset side of insurance companies.
Keywords: Incomplete Market, Super-Long Discount Rates, Government Bond Supply, Insurance Pricing

Abstract

This paper presents a multi-agent equilibrium model within an incomplete market framework, deriving 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 in this study 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.

Using a convex dual problem approach, we derive expressions for the equilibrium interest rate and market price of risk associated with government bonds, categorized by their remaining time to maturity. Additionally, we associate exogenously determined dividend processes representing coupon payments from the government bonds categorized by time-to-maturity, which reflect market outstanding notional adjustments due to the central bank's bond purchases and government bond issuance, in order to derive the yield curve for discount rates in equilibrium.

The main contribution of our study is the incorporation of changes in the supply of government bonds in the secondary market into the incomplete market equilibrium model. This approach incorporates the heterogeneous views of agents regarding fundamental risks, represented by Brownian motion, and constructs a model that explicitly calculates the impact of net supply changes of the government bonds on super-long discount rates.

Finally, we examine how variations in the supply of government bonds impact the pricing of insurance products, including death benefits and pension annuities, through changes in the super-long discount rates. This analysis is essential for insurance pricing and the evaluation of government bonds on the asset side of insurance companies.

Keywords: Incomplete Market, Super-Long Discount Rates, Government Bond Supply, Insurance Pricing