Generalized Makeham’s formula and economic profitability

Abstract

This paper generalizes Makeham’s formula, allowing for varying interest rates and for a non-flat structure of valuation rates. An average interest rate (AIR) is introduced, as well as an average valuation rate (AVR), both of which exist and are unique for any asset. They can be computed either as principal-weighted arithmetic means or as interest-weighted harmonic means of period rates. Economic profitability of an asset or a portfolio of assets is captured by the spread between AIR and AVR, which has the same sign as the Net Present Value. This makes (i) AIR a more reliable tool for valuation and decision than the venerable Internal Rate of Return, and (ii) AVR a natural generalization of the cost-of-capital notion.