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Consumption, Inflation Risk and Dynamic Hedging

Our study examines the behavior of a risk-averse investor who faces two sources of uncertainty: a random asset price and inflation risk. Both sources of uncertainty make it difficult to stabilize consumption over time. However, investors can enter risk-sharing markets, such as futures markets, to manage these risks. We develop a dynamic risk management model. Optimal consumption and risk management strategies are derived. It is shown that dynamic hedging increases an investor’s welfare in terms of the expected inter-temporal utility of consumption.


Introduction
The importance of risk management has inspired empirical and theoretical contributions to investment and consumption decision making under uncertainty. Most of the literature on economic risk and risk behavior dealing with investment, consumption and hedging decisions has incorporated the assumption that investors are concerned with random nominal wealth denominated in one currency. However, an investor's wealth may also change when domestic prices change due to unexpected domestic price inflation, for example. An asset holder should not neglect inflation risk because his or her consumption opportunity set also changes. Therefore, economic analysis of dynamic investment and consumption should be imbedded in a framework of inflation risk (Adam-Mueller, 2002;Battermann & Broll, 2001;Bodie, 1976;Broll & Wong, 2013;Eaton, 1980;Elder, 2004;Kawai and Zilcha, 1986;Nocetti and Smith, 2011).
The real value of assets will change with inflation.
Inflation causes money to decrease in value regardless of whether the money is invested (Blanchard, Amighini, & Giavazzi, 2013). Real wealth is therefore uncertain for two reasons: The value of accumulated assets is uncertain, and the overall price level is risky. However, the investor can step into futures markets in order to reduce the volatility of real wealth by hedging directly against asset price fluctuations and indirectly against inflation risk due to correlations. Our discussion illustrates a typical use of futures contracts.
In an excellent study, Nocetti and Smith (2011) use a very general framework to analyze how the price uncertainty of commodities affects the (precautionary) saving and welfare of consumers. Nocetti and Smith (2011) develop and solve a model with interesting specific features: recursive preferences, an infinite horizon with many goods, financial assets correlated with the growth rate of prices of multiple commodities. They study the impact of consumer preferences that disentangles risk aversion from preferences towards goods in the absence of risk. Furthermore, they discuss a class

Consumption, inflation risk and dynamic hedging
of recursive preferences, appropriately adapted to the case of multiple commodities, that disentangles intertemporal substitution preferences and risk aversion.
In addition to other authors such as Adam-Mueller (2002), Bodie (1976), Eaton (1980), Pelster (2014) and others we focus in this study on the risk-reducing role of futures markets and derive optimal hedge ratios and welfare in cases where an asset price and the price of a composite commodity are uncertain. The existence of risk-sharing markets and risk-reducing institutions in open economies significantly modifies the impact of price and return risk. The problem might be, however, that the economy does not have a complete set of markets, such that existing markets must typically serve several different functions simultaneously. The effect of risk on consumption and welfare will thus depend on whether there is a futures/forward market (for a detailed discussion, see, for example, Froot, Scharfstein, & Stein, 1993;Newbery & Stiglitz, 1981).
In our study, following the standard approach in the literature, we apply time-separable preferences (Briys, Crouhy, & Schlesinger, 1990;Broll, Clark, & Lukas, 2010). We demonstrate that the optimal dynamic hedging strategy can be decomposed in a varianceminimizing component and in a speculative component. In contrast to Nocetti and Smith (2011), in our model, the variance-minimizing component is preference free and depends only on the variance of the hedging instrument and on covariances. The speculative component is influenced by preferences, i.e., it depends on the investor's degree of relative risk aversion. 1 Introducing examples, we prove that the optimal consumption-wealth ratio may be greater or smaller than it is without the hedging instrument, depending on the magnitude of relative risk aversion. However, in any case, we show that the investor's overall welfare is greater with hedging. This analysis can also be applied to aspects of asset returns and the business cycle in the economy as well (Heer & Maußner, 2009).
The rest of this paper is organized into four sections.
In the first section, we present the model and show the risk-reducing role of futures markets. Then, we characterize investors' optimal hedging and consumption strategies. Furthermore, we employ a standard iso-elastic utility function to gain further economic insights. The final section concludes the paper. The appendix contains additional calculations.

The model
Consider a risk-averse individual, called "the investor, " who has accumulated real wealth W defined as where S denotes the spot market price of assets, A is the stock of assets accumulated, and P represents the overall price level. Both S and P are stochastic. Applying Itô's lemma, the change in the investor's real wealth can be written as The investor can take a short or long position on the futures market of volume H (a positive/negative H denotes a short/long position). To enter the futures market costs nothing, but from thereon, the investor's margin account has to be continuously adjusted due to changes in the futures price F. Moreover, the investor can consume at the rate Cdt, where C denotes the rate of real consumption.
C can be broadly defined as a composite good. Thus, the last term of equation (2), the change in the value of accumulated assets at any instant, given P S / , is which says that in addition to price changes, real wealth changes due to real consumption and hedging.
Hence, real wealth accumulation (2) becomes We assume that the evolution of S, P, and F can be de- Eq. (5) describes the stochastic real wealth accumulation equation.

Optimal consumption and risk management
The investor's objective is to maximize the expected present value of utility of consumption over his plan- is assumed to be time separable. Parameter β denotes the investor's constant rate of time preference. Equation (6) shows that the investor ultimately cares about real consumption; he or she tries to reduce the volatility of wealth as expressed by (5) Proposition 1. Maximization of expected present value of utility leads to optimal consumption and hedging where ^ denotes an optimal value.
Eq. (8a) is a standard optimality condition and equates the marginal utility of consumption U C to the marginal utility of wealth, W V . Solving this equation for C yields the investor's rate of real consumption. Eq. (8b) describes the investor's optimal hedge ratio ( implies a short hedge, long hedge).
The first term on the right-hand side of (8b) represents the preference-free, variance-minimizing hedge ratio and can be understood as follows. On one hand, hedging is used to reduce the variance

Examples
Example 1 Let us consider the simple case of a deterministic evolution of the price level by setting . The optimal hedge ratio is Whereas the speculative component remains unchanged, the variance-minimizing hedge rate simplifies, as it depends only on the covariance between the spot asset price and the futures price (and, of course, on its variance). A positive correlation thus implies a short hedge to minimize the variance of wealth. If the asset price and the futures price are also perfectly correlated, i.e., if they obey the same Wiener process ( and, thus, The variance-minimizing component then equals The ultimate purpose of hedging is to stabilize the evolution of real wealth, enabling the investor to smooth her consumption profile and to increase her . To observe this more formally, in the next example, we consider a particular utility function.
Example 2 To obtain more insight into the hedging strategy and its effects, and for analytical convenience, from now on, we assume that the investor's utility function is iso-elastic and that her planning horizon is infinite. The maximization problem becomes denotes the investor's degree of relative risk aversion. In case of 0 = γ , the utility function is logarithmic. Because the planning horizon is infinite, the utility function is additively separable in time, and the involved stochastic processes (4) do not directly depend on time, the value function ) (⋅ V can be expressed in terms of wealth W solely. 2 As shown in the appendix, the value function V, optimal consumption, and the hedge ratio, are where (9d)

Consumption, inflation risk and dynamic hedging
Term â describes the expected real rate of return on wealth, and b measures the variance of the Wiener process for real wealth accumulation when optimal hedging (9c) is chosen. 3 The following observations can be made.
Eq. (9c) confirms that when the optimal hedge ratio is smaller (in absolute value), the investor's relative risk aversion ( 1 − γ ) is greater, and the variance of the futures price is higher. It is worth noting that in case of an infinite planning horizon combined with an iso-elastic utility function and stochastic processes described by geometric Brownian motions, the optimal hedge ratio is time invariant, i.e., the hedge ratio is static. From (9b) in combination with the time-invariant optimal hedge ratio, entering â and b , it follows that the optimal consumption-wealth ratio is time invariant as well.
Depending on the sign of γ , from equation (9b), we see that an increase in the expected rate of return on real wealth â increases or lowers the consumptionwealth ratio. There are two opposite effects at work.
First, according to the income effect, a higher â increases consumption, thus increasing W C / . Second, due to the substitution effect, a higher expected rate of return encourages wealth accumulation, thus lowering W C / . Which effect dominates depends on γ . We also observe that the expected real rate of return â depends on the optimal hedging strategy and on the expected growth rate of the futures price, i.e., ( Inserting (9c) and (9d) into (9b) and simplifying, we obtain for the optimal consumption-wealth ratio Because the first two terms on the right-hand side can be identified as the optimal consumption-wealth ratio without hedging possibilities, i.e., ( ) hedging without W Ĉ , we can write (9b′) Looking at (9b′), we note that the consumption-wealth ratio may be greater or lower than in the case without hedging. Again, this depends on γ . In the case of the logarithmic utility function, 0 = γ and the consumption-wealth ratio is unaffected by hedging and is given Welfare It is important to note that it is not the investor's objective to maximize utility by means of choosing the highest possible consumption-wealth ratio. Ultimately, the investor only cares about the utility of real consumption C. Most importantly, given initial real wealth , ) 0 ( W we can infer from (9a) in combination with (9b′) that regardless of the specific value of , at time zero, hedging always leads to an inter-temporal utility that is higher than in the case without hedging. 4 We can thus summarize that in general, the introduction of hedging against inflation risk increases the investor's expected utility, i.e., hedging without as long as clearly lowers the expected rate of return on real wealth â , as seen from eq. (9d). An increase in 2 P σ results in a higher variance parameter b of real wealth accumulation and, perhaps a little surprisingly, raises the expected rate of return on real wealth. The reason for this is that due to the convexity of real wealth P A S W / ≡ in P, an increase of P ∆ , say, reduces real wealth by less than a fall of magnitude P ∆ − increases real wealth; hence, real wealth increases on average. As we have already mentioned, changes in â and b cause income and substitution effects, and their overall effects on the consumption-wealth ratio depend on the elasticity of utility γ . Most interestingly, the optimal hedging strategy ĥ is unaffected as long as only P µ and 2 P σ change, as they do not appear in (9c). This does not mean that the investor will

Conclusions
In a continuous time framework in which an investor faces real wealth risk, we derived the investor's optimal hedging and consumption strategy. We showed that the optimal hedge ratio can be decomposed in a preferencefree variance-minimizing component and in a speculative component, depending on the investor's degree of relative risk aversion. We discussed the interaction between optimal dynamic hedging, correlation, the expected growth rate of the futures price, volatility on the futures market, and the sign of the optimal hedge ratio, See, e.g., Dixit and Pindyck (1994) and Turnovsky (2000). The Bellman differential equation is then an ordinary one, allowing it to be solved analytically.

Bellman equation
The Bellman equation of the optimization problem is is calculated using Itô's lemma: which is equation (7). Maximization of the right-hand side gives rise to the two first-order conditions from which (8a) and (8b) immediately follow.

Iso-elastic utility function
Inserting this utility function and the optimized values Ĉ and ĥ into the Bellman equation (A2), , thus dropping the t V -term on the right-hand side, and rearranging yields with â and b defined as in the text. This is an ordinary differential equation. The solution of the differential equation (A3) is achieved by trial and error. Looking at (A3), it is quite natural to postulate a solution of the where the coefficient δ has to be determined (see Turnovsky (2000), chapter 15). Equation (A4) implies Substituting (A4) and (A5) into the first-order conditions (A3.1) and (A3.2), using which is equation (9c).

Solution of the Bellman equation
Next, we solve (A6.1) for W C  . From (A7), we obtain β = W C / both with and without hedging because the optimal hedging strategy enclosed in â and b does not influence the optimal consumption-wealth ratio (A7). However, it can be shown that