T. B. Sprague

1. Introduction.

The object of this work is to show that the effective rate of interest of loans practically can be calculated with so great an accuracy that all practical demands regarding accuracy are more than satisfied. As will be seen from section 5, the effective rate of interest can be very quickly calculated by use of inverse linear interpolation for ordinarily occurring cases with a result which in general is correct with 3, 4 or more figures all according to practical demands. These methods can generally be used for all types of loans.     

Two Paradigms for The Market Value of Liabilities

Abstract

Asset/liability management (ALM) theory and practices of insurers have matured and developed from early applications to guaranteed investment contracts (GICs) to all annuity and insurance products today. An important and logical next step of inquiry is the definition of, and calculation procedures for, the market value of an insurance liability. Because all ALM strategies have as their goal the management of some value of assets in relation to some value of liabilities, this inquiry will provide at last a canonical basis for ALM: the management of relative market values.

To set the stage for this exploration, the theory and application of pricing in a complete market are reviewed, as are the practical limitations of this theory in the real, and far from complete, financial markets. The notion of an ad hoc pricing model is developed, and examples are reviewed and critiqued. These models, though imperfect compared with pricing in a complete market, bridge the gap between pricing theory and practice.

The current state of the liabilities market is also discussed, and this market is seen to naturally split into a “long” and a “short” submarket. Of particular interest is the theoretical possibility of these markets becoming broad-based, deep and active, and the conclusions are relevant to the issue of long/short price equalization.

Two paradigms are then explored for defining and subsequently calculating an insurance liability market value. A “paradigm” is a generalized model or framework for accomplishing the task at hand. Each paradigm reflects observable market trading activity, however infrequent, and each is based on methods of valuation consistent with finance-theoretic approaches that are routinely used for the market valuation of assets.

In addition, each paradigm allows for a sequence of ad hoc valuation methodologies, which differ in the extent to which various risks are explicitly modeled versus judgmentally reflected in a risk spread. These paradigms are discussed and contrasted, and arguments made for the potential evolution of the respective values if a “liability” market began trading actively. Practical constraints on the realization of this evolution are also noted.

The last section of this paper discusses a host of considerations related to the application of option-pricing theory to insurance company liabilities.     

Consumption and Portfolio Choice over the Life Cycle

This article solves a realistically calibrated life cycle modelof consumption and portfolio choice with non-tradable laborincome and borrowing constraints. Since labor income substitutesfor riskless asset holdings, the optimal share invested in equitiesis roughly decreasing over life. We compute a measure of theimportance of human capital for investment behavior. We findthat ignoring labor income generates large utility costs, whilethe cost of ignoring only its risk is an order of magnitudesmaller, except when we allow for a disastrous labor incomeshock. Moreover, we study the implications of introducing endogenousborrowing constraints in this incomplete-markets setting.