Diversification: does it really fail, when you need it most?
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INVITED EDITORIAL
Diversification: does it really fail, when you need it most? Bernd Scherer1 Revised: 25 September 2020 / Accepted: 10 October 2020 / Published online: 29 October 2020 © Springer Nature Limited 2020
Abstract Diversification has been labeled as the “only free lunch in investment management.” This conventional knowledge has been under attack for many years as investors observed that correlations of risky assets have been rising during periods of increasing systematic risks. Higher correlations are painful, but they usually do not happen ceteris paribus. In times of very volatile markets, even high correlations will proof to be useful and diversification benefits might increase rather than decrease. What are the conditions for this to happen? Keywords Diversification · Market crisis · Security equivalent
Problem Diversification has been labeled as the “only free lunch in investment management.”1 This conventional knowledge has been under attack for many years as investors observed that correlations of risky assets have been rising during periods of increasing systematic risks. Higher correlations are painful, but they usually do not happen ceteris paribus. In times of very volatile markets, even high correlations will proof to be useful and diversification benefits might increase rather than decrease. What are the conditions for this to happen?
Framework Without loss of too much generality, we assume two risky assets, e.g., home equities and foreign equities. Both offer equal return ( 𝜇home = 𝜇foreign ) and equal volatility ( 𝜎home = 𝜎foreign = 𝜎 ). How large is the diversification benefit of an optimally diversified portfolio (minimum variance portfolio with w∗ = 21 ) versus an investment into the home market? We start by defining the utility difference ( ΔU )
Dr. Bernd Scherer is a Research Associate at EDHEC Risk Institute and executive board member for a German asset manager. * Bernd Scherer [email protected] 1
EDHEC Risk Institute, 393‑400 Promenade des Anglais, France
between these two portfolios for an investor with risk aversion 𝜆 as2
ΔU = Δ𝜇 − 𝜆Δ𝜎 2 Note that ΔU represents the security equivalent (risk-free return that would create the same utility difference) has a return dimension. We already assumed Δ𝜇 = 0 . How large 2 is Δ𝜎 2 = 𝜎minvar − 𝜎 2 ? From basic portfolio theory, we can express this as
( )2 1 2 𝜎2 Δ𝜎 2 = 𝜎minvar − 𝜎2 = 2 ( )( ( ) ) 1 1 1 2 2 𝜎 +2 1− 𝜌𝜎 2 − 𝜎 2 + 1− 2 2 2
(1)
From here it follows, that
Δ𝜎 2 =
1 2 𝜎 (𝜌 − 1) 2
and finally we can compute the benefits of diversification from
1 ΔU = −𝜆 𝜎 2 (𝜌 − 1) ≧ 0 2
(2)
1
Attributed to Harry Markowitz by Berntein (1998). The following section derives the results provided in Wilcox (1999), probably the most overlooked book in investment management. The framework looks at expected utility gains from diversification (which by definition are always positive), rather than realized gains. The later are highly sample dependent and noisy.
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Intuitively, this shows that a
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