Hedging quantitative easing

Abstract

Arguably the greatest concern surrounding quantitative easing is its potential for expanding the money supply at a rate which outstrips the rate of growth in national output. This will almost surely lead to greater uncertainty in inflationary expectations and this, in turn, can have adverse consequences for stock prices. Our analysis employs the hedging procedures which underscore the Fundamental Theorem of Asset Pricing in conjunction with stochastic processes for stock prices and the money supply to design hedging strategies against potential downside movements in stock prices caused by the uncertainty in inflationary expectations associated with rapid monetary growth.

KEYWORDS:

• Quantitative easing
• money to stock price ratio
• fundamental theorem of asset pricing
• hedging

JEL CLASSIFICATIONS:

• C22
• C32
• E41
• G12

Acknowledgements

The authors gratefully acknowledge the constructive criticism and suggestions by the Editor and referees on an earlier draft of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The most famous, if not most eloquent, statement of this hypothesis is due to Friedman (1970a, 24): ‘Inflation is always and everywhere a monetary phenomenon in the sense that it is and can be produced only by a more rapid increase in the quantity of money than in output.'

2 The average annual rate of growth in the M2 monetary aggregate over the previous twenty years ending 31 December 2019 was a little under 6% (https://fred.stlouisfed.org/series/M2NS).

3 There is, of course, no traded security with a value equal to the money to stock price ratio. It is therefore important to emphasise that one can replicate the value of the money to stock price ratio by forming and then continuously rebalancing a portfolio comprised of a long investment in the quantity of money in circulation of $\log \left[\mathrm{cM}\left(t\right)\right]$ and a short investment in the stock price index of $\log \left[\mathrm{cP}\left(t\right)\right]$, where $c$> 0 is an arbitrary fractional constant. The value of this portfolio will then be $\log \left[g\left(t\right)\right]=\log \left[\mathrm{cM}\left(t\right)\right]-\text{log}\left[\mathrm{cP}\left(t\right)\right]=\log \left[\frac{M\left(t\right)}{P\left(t\right)}\right]$ or the logarithm of the money to stock price ratio.

4 Equation (6) of the text shows that the derivative security will have a non-trivial value at maturity when $1-\exp \left\{\frac{\left({\rho }_{12}{\sigma }_1{\sigma }_2-{\sigma }_2^2\right)}{\sqrt{{\sigma }_1^2{\sigma }_2^2\left(1-{\rho }_{12}^2\right)}}y-x\right\}>0$ or equivalently, when: $\begin{array}{r}\exp \left\{\frac{\left({\rho }_{12}{\sigma }_1{\sigma }_2-{\sigma }_2^2\right)}{\sqrt{{\sigma }_1^2{\sigma }_2^2\left(1-{\rho }_{12}^2\right)}}y-x\right\}<1\end{array}$ Taking logarithms across this latter expression will show this is equivalent to $\frac{\left({\rho }_{12}{\sigma }_1{\sigma }_2-{\sigma }_2^2\right)}{\sqrt{{\sigma }_1^2{\sigma }_2^2\left(1-{\rho }_{12}^2\right)}}y-x<0$ or $x>\frac{\left({\rho }_{12}{\sigma }_1{\sigma }_2-{\sigma }_2^2\right)}{\sqrt{{\sigma }_1^2{\sigma }_2^2\left(1-{\rho }_{12}^2\right)}}y$ as stated in the text.

Additional information

Notes on contributors

Adrian Melia

Adrian Melia has published papers on stochastic processes applied to finance as well as empirical finance. He has published papers which investigate alternative modelling processes for asset prices and the impact they have on abnormal return calculations, the calculation of logarithmic returns under Constant Elasticity of Variance stochastic processes as well as the impact of investor sentiment on corporate announcement returns. His most recent paper on the topic of investor sentiment ‘Firm-level investor sentiment and corporate announcement returns’ was published in the Journal of Banking and Finance.

Xiaojing Song

Xiaojing Song has published papers which investigate the role that cultural disparity plays in cross-border takeovers; alternative stochastic process for modelling asset prices and the impact they have on abnormal return calculations; the empirical validity of the Fama and French Asset Pricing Model and, the calculation of logarithmic returns under Constant Elasticity of Variance stochastic processes.

Mark Tippett

Mark Tippett has published papers which apply the methods of numerical mathematics to assess the accuracy and reliability of information summarized in corporate financial statements. He has also combined numerical mathematics with the theory of stochastic processes to estimate the convex relationship which exists between the market value of equity on the one hand and current earnings and the book value of equity on the other. Much of his research work in these areas has been summarized in his book entitled Principles of Equity Valuation (Routledge, 2012).

John van der Burg

John van der Burg has published papers on stochastic processes applied to finance with a particular focus on models to describe and predict cash flows and cash holdings. His most recent paper on this topic is ‘A Hyperbolic Model of Optimal Cash Balances' which was published in the European Journal of Finance. A related paper entitled ‘Distributional properties of the book to market ratio and their implications for empirical analysis’ is also forthcoming in the European Journal of Finance.