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Financial Markets and Portfolio Management

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17-06-2020

Behavioral portfolio insurance strategies

Authors: Marcos Escobar-Anel, Andreas Lichtenstern, Rudi Zagst

Published in: Financial Markets and Portfolio Management | Issue 4/2020

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Abstract

Portfolio insurance strategies that ensure a certain minimum portfolio value or floor such as the Constant Proportion Portfolio Insurance (CPPI) and the Option-based Portfolio Insurance are economically important and widely spread among the banking and insurance industries. In distress and volatile market environments, investors such as pension funds have a need to insure their portfolios against downside risk in order to meet certain future payments or liabilities. Non-anticipated shocks or negative interest rates, jumps, crashes, or overnight trading restrictions in stock prices could drop pension fund portfolios below desired levels (present value of pension obligations) making them underfunded with pension assets to pension liabilities ratio below 100%. In particular, within the current low interest rate environment, a high number of pension funds happen to be underfunded which is a severe practical problem. Because of such scenarios, there is a need for an investment strategy which covers both the case of funded and underfunded portfolios. This article introduces a novel strategy which generalizes the CPPI approach. It has the overall target of guaranteeing the investment goal or floor while participating in the performance of the assets and limiting the downside risk of the portfolio at the same time. We show that the strategy accounts for behavioral aspects of the investor such as distorted probabilities, a risk-averse behavior for gains, and a risk-seeking behavior for losses. The proposed strategy turns out to be optimal within the Cumulative Prospect Theory framework by Tversky and Kahneman (J Risk Uncertain 5(4):297–323, 1992).

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The market consists of one risky and one risk-free asset with parameters \(\mu = 5\%\), \({\sigma = 20\%}\), \(r = 1\%\) and planning horizon \(T = 1\) year. Moreover, \(V_\mathrm{CPPI}(0) = 10\) and \({\alpha _{T} = 90\%}\) which lead to \(F = 9\); the multiplier is set to \(m = 4\). In particular this implies the starting exposure \(E(0) = 4.36\) to the active risky asset and thus \({{\hat{\pi }}_\mathrm{CPPI}(0) = \frac{E(0)}{V_\mathrm{CPPI}(0)} = 43.6\%}\).

Note that a change in the interest rate is not covered by the model assumptions and thus considered as an external shock of the model parameters.

\(\lambda _{DI}^{(u)}(t)\) is selected optimally based on a given portfolio optimization problem that is introduced later. More information is provided in Definition (GBPI).

In Sect. 4.2 we show the formula for \(\lambda _{\mathrm{Put}(V_{u},B,T)}^{(u)}(t)\) and explain how the put option can be hedged.

The optimal final payoff for any nonzero reference wealth B can simply be obtained by adding B to the optimal terminal wealth when B is forced to zero, while replacing v by \(v - {\mathbb {E}}[{\tilde{Z}} B]\).

Our analytical derivations work out only when applying the same \(\alpha \) above and below the reference point, nonetheless the theory holds and can be solved numerically for different \(\alpha \) values.

\(\liminf \nolimits _{v \rightarrow \infty }{\left( \frac{-vU''_{+}(v)}{U'_{+}(v)}\right) } = 1-\alpha > 0\).

Moreover, the phenomenon which describes that ‘losses loom larger than gains’ (Kahneman and Tversky 1979), widely known as loss aversion, is satisfied for any \(\beta _{-} > 1\).

It holds \({\mathbb {E}}\left[ U_{+}\left( (U'_{+})^{-1}\left( \frac{{\tilde{Z}}}{w'_{+}(F_{{\tilde{Z}}}({\tilde{Z}}))}\right) \right) w'_{+}(F_{{\tilde{Z}}}({\tilde{Z}}))\right] = \alpha ^{\frac{\alpha }{1-\alpha }} \cdot {\mathbb {E}}\left[ {\tilde{Z}}^{-\frac{\alpha }{1-\alpha }} \cdot w'_{+}(F_{{\tilde{Z}}}({\tilde{Z}}))^{\frac{1}{1-\alpha }}\right] < \infty \) iff \({{\mathbb {E}}\left[ {\tilde{Z}}^{-\frac{\alpha }{1-\alpha }} \cdot w'_{+}(F_{{\tilde{Z}}}({\tilde{Z}}))^{\frac{1}{1-\alpha }}\right] < \infty }\).

One can show theoretically that \(\inf _{c>0} k(c) = 0\) when the distortion by Jin and Zhou (2008) is additionally applied on the negative part under power utility; hence an alternative distortion is inevitable.

Unfortunately, \(w_{-} \equiv \text {id}\) is impossible as the problem turns ill-posed automatically. Further notice that a Prelec distortion with parameter \(\gamma _{-}=0.6\) is indirectly applied within the BPI numerical case study; \(w_{-}\) does not affect the form of the solution, but determines \(c^{*}\) and \(v_{+}^{*}\).

This CPPI strategy coincides with the GBPI strategy in Definition (GBPI) when the probability distortion is selected as identity function (\({a_{+} = b_{+} = 0}\)). The risky relative portfolio \({\hat{\pi }}_\mathrm{CPPI}(t)\) for a portfolio value \(V_{f}(t)\) is given by \({{\hat{\pi }}_\mathrm{CPPI}(t) V_{f}(t) = \frac{1}{1 - \alpha } \left[ V_{f}(t) - e^{-r (T-t)} B\right] \cdot {\hat{\pi }}^{G}}\).

For comparison of the underfunded GBPI strategy, a CPPI strategy on \({\hat{V}}_{u}(t)\) is chosen with multiplier \(\frac{1}{1-\alpha }\), i.e., the relative portfolio \({\hat{\pi }}_\mathrm{CPPI}^{(u)}(t)\) for the value \({\hat{V}}_{u}(t)\) is given by \({{\hat{\pi }}_\mathrm{CPPI}^{(u)}(t) {\hat{V}}_{u}(t) = \frac{1}{1-\alpha } \left[ {\hat{V}}_{u}(t) - e^{-r(T-t)}B\right] \cdot {\hat{\pi }}^{G}}\).

The total strategy for comparison consists of a CPPI strategy to the initial value \({v + \mathrm{Put}(V_{u},B,0,T)}\) and a short put option on \(V_{u}\) with exercise price B and maturity T which finances the CPPI. For the allocation only the CPPI part is visualized (denoted by \({\hat{\pi }}_\mathrm{CPPI}^{(u)}(t)\)), for the wealth process the value of the CPPI minus the value of the put option (denoted by \(V_{CPPI-Put}(t)\)) is plotted.

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Title: Behavioral portfolio insurance strategies
Authors: Marcos Escobar-Anel
Andreas Lichtenstern
Rudi Zagst
Publication date: 17-06-2020
Publisher: Springer US
Published in: Financial Markets and Portfolio Management / Issue 4/2020
Print ISSN: 1934-4554
Electronic ISSN: 2373-8529
DOI: https://doi.org/10.1007/s11408-020-00353-5

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