A Mathematical Formulation of the Valuation of Silver ETFS: The Role of Investor Sentiment

Abstract

Silver ETFs are exchange-traded funds of silver mining stocks. This paper posits that investor sentiment toward risk influences silver ETF returns. The paper creates formulations of invest sentiment toward risk with risk-averse investors investing modestly in silver ETFs with optimal returns at the intersection of the Arrow-Pratt coefficient of risk-aversion and silver ETF returns. Moderate risk-takers make revised estimates of silver returns as modeled by a Laplace transform, which intersects with the Bessel function of silver ETF returns to yield the optimal returns. Risk-takers follow an exponential distribution of continuously rising return expectations, which yield an optimal return upon intersection with the Bessel function of silver ETF returns. Theoretical formulations were subjected to empirical validation using silver ETF returns regressed on proxies for investor sentiment, also known as investor risk aversion. The paper concludes with theoretical implications and practical implications. Theoretically, the study advances knowledge by presenting mathematical formulations of investor sentiment’s varying influence on silver ETF returns. From a practical standpoint, ETF managers can vary investor recommendations about investing in silver ETFs, depending upon whether investors are risk-averse, moderate risk-takers, or risk-takers. There is a paucity of similar studies with both a mathematical emphasis and empirical validation.