Expected Utility Theory on General Affine GARCH Models

ABSTRACT

Expected utility theory has produced abundant analytical results in continuous-time finance, but with very little success for discrete-time models. Assuming the underlying asset price follows a general affine GARCH model which allows for non-Gaussian innovations, our work produces an approximate closed-form recursive representation for the optimal strategy under a constant relative risk aversion (CRRA) utility function. We provide conditions for optimality and demonstrate that the optimal wealth is also an affine GARCH. In particular, we fully develop the application to the IG-GARCH model hence accommodating negatively skewed and leptokurtic asset returns. Relying on two popular daily parametric estimations, our numerical analyses give a first window into the impact of the interaction of heteroscedasticity, skewness and kurtosis on optimal portfolio solutions. We find that losses arising from following Gaussian (suboptimal) strategies, or Merton's static solution, can be up to $2.5\mathrm{\%}$ and 5%, respectively, assuming low-risk aversion of the investor and using a five-years time horizon.

KEYWORDS:

• Dynamic portfolio optimization
• affine GARCH models
• non-Gaussian innovations
• IG-GARCH model
• expected utility theory
• wealth-equivalent loss

Disclosure statement

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

Notes

1 It is possible to derive a set of HN-GARCH parameters matching the first two moments of a given IG-GARCH parameter set (Christoffersen, Heston, and Jacobs 2006). For this pair of models, the market price of risk in the IG-GARCH coincides with the parameter λ of the HN-GARCH model in (15a(15a) $\begin{array}{rl}{X}_t-X_{t-1}& =r+\lambda h_t+\sqrt{h_t}{z}_t,\end{array}$(15a) ).

2 It is known (Escobar-Anel, Gollart, and Zagst 2022) that the optimal strategy in the HN-GARCH model converges to the continuous-time Heston solution as of Kraft (2005) for $\mathrm{\Delta }\to 0$ under certain conditions. Thus, imposing these requirements and taking the limit $\eta \to 0$, our optimal solution approaches the continuous-time Heston strategy.

3 The same conclusions are obtained with the ML estimates from IG Set 2.

4 Note that the conclusion for the alternative set of parameter estimates from IG Set 2 is different, replicating the finding by Escobar-Anel, Gollart, and Zagst (2022) in the HN-GARCH framework. In these settings, however, the fraction of wealth invested in the risky asset is significantly smaller.

5 Note that the parameter λ of this artificial HN-GARCH calibration coincides with the market price of risk $\lambda =\nu +{\eta }^{-1}$ of the IG-GARCH. The notation thus is consistent.

6 Neither Equation (25(25) $\nu +\frac{\sqrt{(1-2E_{t+1,T}^{\ast }a{\eta }^4)}}{\eta \sqrt{(1-2\eta \gamma {\pi }_t-2E_{t+1,T}^{\ast }c)}}={\pi }_t-\frac{1}{2},$(25) ) nor the recursive formula for $E_{t,T}^{\ast }$ depend on r or w.

7 The same if true for the relation of IG Set 2 and HN Set 2, of course.

8 The annualized volatility is even higher in the estimation in IG Set 2, suggesting that the effect of η on the conditional skewness is really propagated via the formula ${\mathcal{S}}_t[X_{t+1}-X_t]=3\eta (h_{t+1}{)}^{-1/2}$.

9 Note that the opposite is true for the alternative parameter set (see Appendix A.1.3), where the IG-GARCH suggests the largest fraction invested in the risky asset, followed by the HN-GARCH. This also coincides with a reversed order concerning realized return moments.

10 In contrast to the setting before, the estimate for w is now negative. We note that this can lead to negative values for the conditional variance in case the previous value is close enough to zero. Referring to Christoffersen, Heston, and Jacobs (2013), we adjust our parameter setting by imposing w = 0.

11 Indeed, w<0 can be problematic w.r.t. the positivity of the conditional variance. We one more refer to Christoffersen, Heston, and Jacobs (2013) and the adjustment w = 0.