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Portfolio decision with a quadratic utility and inflation risk
Advances in Difference Equations volumeÂ 2018, ArticleÂ number:Â 366 (2018)
Abstract
This paper considers a portfolio selection problem with a quadratic utility of consumption, which is symmetric with respect to a bliss point. At bliss point, the utility function has its maximum value and further consumption lowers the utility. In the presence of inflation risk, we introduce an inflationlinked index bond to manage the inflation risk and derive explicit expressions for the optimal consumption and portfolios by applying duality method. Based on quantitative results, we see that inflationlinked index bond plays an important role in choosing consumption and portfolio rules.
1 Introduction
A quadratic utility function is widely applied in economic and finance theory. The main advantage of a quadratic utility function is its tractability, and this is the main reason why the quadratic function is applied as an objective function for various optimization problems in economics and finance. In particular, modern investment theory and practice are mainly based on the meanâ€“variance portfolio theory developed by Markowitz [7], where the analysis is based on the quadratic function of portfolio return. The analysts in practice apply the meanâ€“varianceefficient portfolios for asset allocation. Recently, Sharpe [9] showed that the meanâ€“variance portfolio allocation has similar results to that of the expected utility asset allocation if we consider the quadratic utility whose risk aversion is measured by the variance of portfolio return.
In this paper we consider a quadratic utility as an agentâ€™s preference. Similar to the traditional utility functions of constant relative risk aversion (CRRA) or constant absolute risk aversion (CARA), the law of diminishing marginal utility still holds for the quadratic utility. The marginal utility, however, can be negative when the consumption level is higher than a bliss point. In other words, above a bliss point, if the agent consumes more he/she reduces utility. Thus, the optimal level of consumption and value function should be constant if the wealth level is greater than the critical level at which the optimal consumption rate is the bliss point. With a quadratic utility function, investment pattern is quite different from that when using other utility functions for a sufficiently high wealth level. For example, for an agent with a quadratic utility investment in the risky asset is zero when the wealth level is larger than the bliss level (see, e.g., Koo et al. [5] or Rho et al. [8]).
Due to the negative marginal utility for higher consumption levels, the quadratic utility seems to be unrealistic. It is worth to note, however, that the quadratic utility is sufficient to explain the agentâ€™s economic behavior for the realistic level of wealth. Koo et al. [5] studied the quadratic utility in the presence of subsistence consumption constraint, and Rho et al. [8] investigated the effect of borrowing limits on the consumption and investment of the agent with a quadratic utility.
If the time horizon is long enough, it is persuasive to incorporate the inflation risk into consumption and portfolio selection problem. In our continuoustime model, the time horizon is infinite, and thus we expect that the inflation rate would significantly affect the longterm financial planning. To hedge the inflation risk, we introduce the inflationlinked index bond. In fact, many developed countries already have inflationlinked securities which are traded in the financial market. Especially, Treasury InflationProtected Securities (TIPS) which were introduced in 1997 have become the most traded bonds in the US bond market.
In this paper, we provide explicit expressions for consumption and portfolio rules by applying the duality approach. Due to the optimal consumption process, we have to separate the wealth into three regions as in Koo et al. [5] and Rho et al. [8]. In the third region, where the wealth level is above the bliss level, the optimal consumption stays constant as its maximum so the value function is also fixed. To make the value function constant, there should be no risky asset investment. This fact is welldescribed in Koo et al. [5] and Rho et al. [8]. In the presence of inflation risk, however, even though the value function in the third region is still constant, the optimal investment in an inflationlinked index bond is not fixed. To hedge the inflation risk, all the financial wealth is invested in an inflationlinked index bond and investment in other assets becomes zero in the third region. When the wealth level is below the bliss level, the role of inflationlinked index bond can be different according to market parameters. In particular, for sufficiently large growth rate of the price process, the demand for speculative motive is reflected only on the inflationlinked index bond. For a small growth rate of the price process, however, the demand for speculative motive is distributed into both portfolios of inflationlinked index bond and the risky asset.
There are large strands of literature on the portfolio selection problem with inflation risk, and most studies consider the inflationlinked index bond to manage inflation risk. Fischer [3] investigated the demand for index bond and claimed its introduction. Later, Campbell and Viceira [2] and Brennan and Xia [1] studied the dynamic asset allocation in the presence of inflation and interest risks. Gong and Li [4] incorporated the subsistence consumption constraint into the model with inflation risk. Recently, Kwak and Lim [6] investigated the effect of inflation risk on lifeinsurance purchase and provided quantitative analysis of the role of an index bond.
This paper is organized as follows. In Sect.Â 2, we provide the financial market in the presence of inflation risk. We introduce an inflationlinked index bond to hedge the inflation risk. In Sect.Â 3, we state the optimization problem with a quadratic utility function and seek to derive explicit solutions to the optimization problem by applying the duality approach. In Sect.Â 4, explicit forms of the optimal consumption and portfolios are given, and we provide some implications of the results in Sect.Â 5. Finally, Sect.Â 6 concludes.
2 Model
In our continuoustime model, we assume that the financial market consists of a riskless asset (money market account), a risky asset (stock), and an inflationlinked index bond. Let us denote by \(B_{t}\) the price of the riskless asset at time t and assume that the riskless asset earns a constant rate of return \(R>0\) as follows:
The price of the risky asset at time t, denoted by \(S_{t}\), evolves according to the following geometric Brownian motion:
where \(\mu_{S}>R>0\) and \(\sigma_{S}>0\) are constants; \(W_{1,t}\) is a standard Brownian motion on a probability space \((\Omega , \mathcal{F}, \mathbb{P})\). Let us denote by \(I_{t}\) the price of the inflationlinked index bond at time t, which evolves according to the following equation:
where \(r>0\) is the real interest rate and \(P_{t}\) is the price level at timeÂ t. We assume that \(P_{t}\) proceeds according to the following geometric Brownian motion:
where \(W_{2,t}\) is another standard Brownian motion on the probability space \((\Omega , \mathcal{F}, \mathbb{P})\), which is independent of \(W_{1,t}\). Hence, \(W_{t}=(W_{1,t}, W_{2,t})\) is a 2dimensional Brownian motion on \((\Omega , \mathcal{F}, \mathbb{P})\). Note that if we define \(\widehat{W}_{t}\equiv \rho\,dW_{1,t}+\sqrt{1\rho^{2}}\,dW _{2,t}\) then \(W_{1,t}\) is correlated with \(\widehat{W}_{t}\) and their correlation coefficient is \(\rho \in (1,1)\), i.e., \(\langle dW_{1,t},d\widehat{W}_{t}\rangle=\rho \,dt\). Let \((\mathcal{F}_{t})_{t\geq 0}\) be the \(\mathbb{P}\)augmentation of the natural filtration generated by \(W_{t}=(W_{1,t}, W_{2,t})\).
Now let us denote by
Denote by \(X_{t}^{N}\) the nominal value of the wealth. Then we have
The inflationadjusted real wealth level \(X_{t}\) is given by \(X_{t}=X_{t}^{N}/P_{t}\). We assume that the nominal income rate \(y_{t}^{N}\) is proportional to the price level \(P_{t}\), i.e.,
for some constant \(y>0\). If we apply ItÃ´â€™s product rule to \(X_{t}^{N}/P_{t}\), we obtain the inflationadjusted real wealth process as in the following lemma.
Lemma 1
where \(c_{t}\equiv c_{t}^{N}/P_{t}\) is the inflationadjusted real consumption rate, and y is the inflationadjusted real income rate.
Proof
We apply ItÃ´â€™s product rule to get
Rearranging (2.2) with \(\pi_{0,t}+\pi_{1,t}+\pi_{2,t}=1\) yields the inflationadjusted real wealth process (2.1).â€ƒâ–¡
Let us define
an exponential martingale
and the pricing kernel
Then by the Girsanovâ€™s theorem, for any fixed T, there exists an equivalent martingale probability measure \(\widetilde{\mathbb{P}}\) defined by \(\widetilde{\mathbb{P}}(A)=\mathbb{E}[\xi_{T} \mathbf{1} _{A}]\), \(A\in \mathcal{F}_{T}\), and under the new probability measure \(\widetilde{\mathbb{P}}\),
are independent standard Brownian motions. Applying ItÃ´â€™s lemma to the product of \(e^{rt}\) and \(X_{t}\) yields
Integrating (2.3) from 0 to \(T>0\) we obtain
If \(X_{t}\), \(t>0\), is lower bounded, the righthand side of (2.4) is a lowerbounded local martingale under \(\widetilde{\mathbb{P}}\), hence a supermartingale under \(\widetilde{\mathbb{P}}\). We then obtain
and if we take the limit \(T\rightarrow \infty \) and recover the physical measure by applying Bayesâ€™ rule, the static budget constraint given in (2.5) is transformed into the following inequality:
3 Optimization problem
We consider the agentâ€™s expected utility maximization problem where the utility function is quadratic as follows:
Notice that the quadratic utility function has its global maximum at \(\bar{c}\equiv 1/2Q\) which is called a bliss point. Obviously, the utility function is symmetric with respect to that point, which means the utility is increasing for \(c<\bar{c}\) and decreasing for \(c>\bar{c}\) and the values of the utility at \(\hat{c}=c<\bar{c}\) and \(1/Q\hat{c}\) are equivalent. Moreover, the constant Q reflects the agentâ€™s risk aversion. In particular, a larger (or smaller) Q implies a smaller (or larger) marginal utility if the consumption rate is smaller than the bliss point. In the meanâ€“variance analysis, Q can be regraded as a risk aversion parameter measured by the variance of wealth.
Definition 1
We call \(\boldsymbol{c}\equiv (c_{t})_{t\geq 0}\) and \(\boldsymbol{\pi }\equiv (\pi _{0,t},\pi_{1,t},\pi_{2,t})_{t\geq 0}\) the consumption and portfolio process, respectively. We call \((\boldsymbol{c},\boldsymbol{\pi })\) an admissible policy at x if

(a)
\(X_{t}\) evolves according to (2.1), \(X_{0}=x\), and \({X_{t}>\frac{y}{r}}\), \(t \geq 0\),

(b)
c is a measurable, adapted nonnegative process, and \(\int_{0}^{t} c_{s}\,ds<\infty \), for all \(t\geq 0\) a.s.,

(c)
Ï€ is a measurable, adapted process and \(\int_{0}^{t} ( \pi^{2}_{0,s}+\pi^{2}_{1,s} )\,ds<\infty\) for all \(t\geq 0\) a.s.
Denote by \(\mathcal{A}(x)\) the set of all admissible policies atÂ x. Then our optimization problem is stated as follows.
Problem 1
The agent wants to maximize the expected utility by optimally choosing the consumption and portfolio processes. In other words,
where
subject to the budget constraint (2.6); \(\beta (>0)\) is the time preference of the agent.
Let us define a convex dual function \(\widetilde{u}(\lambda )\) as follows:
Then the first order condition implies that the maximizer c of (3.1) is given by
Thus, for any \((\boldsymbol{c},\boldsymbol{\pi })\) and \(\lambda >0\), we have
where \(\lambda_{t}=\lambda e^{\beta t}H_{t}\). Note that \(\lambda_{t}\) has its dynamics as
If we define the dual value function JÌƒ as
then by Feynmanâ€“Kacâ€™s formula, \(\widetilde{J}(\lambda )\) solves the following ordinary differential equation (ODE):
Let us denote by \(n_{+}\) and \(n_{}\), respectively, the positive and negative roots of the equation
Then by the growth condition, we can obtain the general closedform solution to the ODE (3.5). The next proposition summarizes the result.
Proposition 1
The dual value function is given by
where \(D_{1}\) and \(D_{2}\) are given by
We assume that the following inequality holds.
Assumption 1
Under this assumption, we can show that the constants \(D_{1}\) and \(D_{2}\) are positive.
Lemma 2
The following inequalities hold:
Proof
We borrow the idea from Koo et al. [5]. Note that the constant \(D_{1}\) can be rewritten as
so it is enough to show that the value in the parenthesis is positive. Let us define functions \(f(n)\) and \(g(x)\) by
and
respectively. Then, for \(x\in (n_{}, n_{+})\), \(g(x)\) is positive and strictly decreasing. Thus, we have \(g(0)>g(1)>0\), which implies
or equivalently,
By AssumptionÂ 1, we also have
Therefore, by combining these two inequalities, we obtain
from which we have \(D_{1}>0\).
To show the positivity of \(D_{2}\), first note that \(D_{2}\) can be rewritten as
Similarly to the proof of the positivity of \(D_{1}\), let us define function \(k(x)\) by
Then for \(x>n_{}\), \(k(x)\) is positive, strictly decreasing and convex. Further, convexity implies
or equivalently,
Therefore, we can show that
and consequently \(D_{2}>0\).â€ƒâ–¡
4 Solution
In this section we provide the closedform solution to Problem 1. From the duality relation, we can recover the primal value function \(V(x)\) through
where \(\widetilde{J}(\lambda )\) is given in PropositionÂ 1. Notice that from the optimal consumption rate in (3.2), \(\lambda =0\) corresponds to the bliss level of consumption cÌ„. Let us denote by xÌ„ the wealth level at \(\lambda =0\), which is then defined by
Moreover, if we denote the wealth level corresponding to \(\lambda =1\) by xÌƒ, it is also given by
Thus, we have to separate the wealth level into three regions, which are \((y/r, \tilde{x})\), \([\tilde{x}, \bar{x})\), and \([\bar{x}, \infty )\). The next theorem provides the value function on each region. Before we proceed, let us define
and
Theorem 1
The value function of Problem 1 is given by
where \(\lambda^{**}\) and \(\lambda^{*}\) solve \(x=h_{2}(\lambda^{**})\) for \(\frac{y}{r}< x<\widetilde{x}\) and \(x=h_{1}(\lambda^{*})\) for \(\widetilde{x}\leq x < \bar{x}\).
Proof
If we substitute the dual value function \(\widetilde{J}(\lambda )\) given in PropositionÂ 1 into the duality relation in (4.1), the relations between the inflationadjusted real wealth level x and its dual variables \(\lambda^{*}\) and \(\lambda^{**}\) can be obtained from the first order conditions. It is obvious that with the inflationadjusted real wealth level x above xÌ„, which corresponds to the bliss level of consumption cÌ„, it is optimal to consume at a rate cÌ„, and consequently the value function constantly equals to
â€ƒâ–¡
It is worth to note that the first region, \(y/r< x<\tilde{x}\), corresponds to \(\lambda >1\) and in that region the optimal consumption rate is zero. Thus, the agent endures zero consumption until when the wealth level is greater than xÌƒ. The optimal consumption rate is asymmetric with respect to the bliss point even though the utility function is symmetric. We summarize the optimal consumption and portfolios in the next theorem.
Theorem 2
The optimal wealth, consumption, and portfolios are given as follows:

(1)
For \(\frac{y}{r}< X_{t}<\widetilde{x}\),
$$ \textstyle\begin{cases} {X_{t}=n_{} D_{2} (\lambda_{t}^{**})^{n_{}1}\frac{y}{r} } , \\ {c_{t}^{*}=0}, \\ {\pi_{0,t}^{*} X_{t}= \{ \frac{\theta_{1}}{\sigma_{S}}+ ( \frac{ \rho }{\sigma_{S}}\frac{1}{\sigma_{P}} ) \frac{\theta_{2}}{\sqrt{1 \rho^{(2}}} \} n_{}(n_{}1)D_{2}(\lambda_{t}^{**})^{n_{}1} }, \\ {\pi_{1,t}^{*} X_{t}= ( \frac{\theta_{1}}{\sigma_{S}}\frac{\rho \theta_{2}}{\sigma_{S}\sqrt{1\rho^{2}}} ) n_{}(n_{}1)D_{2}( \lambda_{t}^{**})^{n_{}1}}, \\ \pi_{2,t}^{*} X_{t}=X_{t}\pi_{0,t}^{*} X_{t}\pi_{1,t}^{*} X_{t}, \end{cases} $$where \(\lambda_{t}^{**}\) solves \(X_{t}=h_{2}(\lambda_{t}^{**})\).

(2)
For \(\widetilde{x}\leq X_{t}<\bar{x}\),
$$ \textstyle\begin{cases} {X_{t}=n_{+}D_{1}(\lambda_{t}^{*})^{n_{+}1}+\frac{1}{2Q ( \theta _{1}^{2}+\theta_{2}^{2}+\beta 2r ) }\lambda_{t}^{*}\frac{y}{r}+ \frac{1}{2Qr}} , \\ {c_{t}^{*}=\frac{1\lambda_{t}^{*}}{2Q}}, \\ {\pi_{0,t}^{*} X_{t}= \{ \frac{\theta_{1}}{\sigma_{S}}+ ( \frac{ \rho }{\sigma_{S}}\frac{1}{\sigma_{P}} ) \frac{\theta_{2}}{\sqrt{1 \rho^{2}}} \} \{ n_{+}(n_{+}1)D_{1}(\lambda_{t}^{*})^{n _{+}1} \frac{1}{2Q ( \theta_{1}^{2}+\theta_{2}^{2}+\beta 2r ) } \lambda_{t}^{*} \} }, \\ {\pi_{1,t}^{*} X_{t}= ( \frac{\theta_{1}}{\sigma_{S}}\frac{\rho \theta_{2}}{\sigma_{S}\sqrt{1\rho^{2}}} ) \{ n_{+}(n_{+}1)D _{1}(\lambda_{t}^{*})^{n_{+}1}\frac{1}{2Q ( \theta_{1}^{2}+\theta _{2}^{2}+\beta 2r ) }\lambda_{t}^{*} \} }, \\ \pi_{2,t}^{*} X_{t}=X_{t}\pi_{0,t}^{*} X_{t}\pi_{1,t}^{*} X_{t}, \end{cases} $$where \(\lambda_{t}^{*}\) solves \(X_{t}=h_{1}(\lambda_{t}^{*})\).

(3)
For \(X_{t}\geq \bar{x}\),
$$ \textstyle\begin{cases} {c_{t}^{*}=\frac{1}{2Q}}, \\ {\pi_{0,t}^{*} X_{t}=0}, \\ {\pi_{1,t}^{*} X_{t}=0}, \\ \pi_{2,t}^{*} X_{t}=X_{t}. \end{cases} $$
Proof
From TheoremÂ 1, \(X_{t}=h_{2}(\lambda_{t}^{**})=n _{} D_{2} (\lambda_{t}^{**})^{n_{}1}\frac{y}{r}\) for \( \frac{y}{r}< X_{t}<\widetilde{x}\). We use ItÃ´â€™s lemma for \(X_{t}\) along with \(d\lambda_{t}^{**}/\lambda_{t}^{**}=(\beta r)\,dt \theta_{1}\,d{W}_{1,t}\theta_{2}\,d{W}_{2,t}\) to obtain
where the last equality comes from rearranging (2.1). Comparing diffusion terms in (4.3) and (4.4), we obtain the optimal portfolios \(\pi_{0,t}^{*} X_{t}\) and \(\pi_{1,t}^{*} X_{t}\). Moreover, \(\pi_{2,t}^{*} X_{t}\) can be found from the relation \(\pi^{*}_{0,t}+ \pi^{*}_{1,t}+\pi^{*}_{2,t}=1\). The optimal inflationadjusted real consumption rate \(c^{*}\) is such that makes the first equality in (3.3) hold as an equality, i.e., (3.2) is satisfied. For \(\frac{y}{r}< X_{t} \leq \widetilde{x}\), \(\lambda_{t}^{**}\geq 1\), hence \(c^{*}=0\).
Similarly, for \(\widetilde{x}\leq X_{t}<\bar{x}\), \(X_{t}=h_{1}(\lambda _{t}^{*})=n_{+}D_{1}(\lambda_{t}^{*})^{n_{+}1}+\frac{1}{2Q ( \theta _{1}^{2}+\theta_{2}^{2}+\beta 2r ) }\lambda_{t}^{*}\frac{y}{r}+ \frac{1}{2Qr}\), and we obtain
from which we obtain optimal portfolios, and \(c_{t}^{*}\) can be obtained from (3.2).
Lastly, the case for \(X_{t}\geq \bar{x}\) can be derived by letting \(\lambda_{t}^{*}\rightarrow 0\) in the case where \(\widetilde{x}\leq X_{t}<\bar{x}\).â€ƒâ–¡
For \(y/r< X_{t}< \tilde{x}\), the optimal investments in the risky asset and the inflationlinked index bond is monotone in wealth but, for \(\tilde{x}\leq X_{t} < \bar{x}\), the optimal investments in the risky asset and the inflationlinked index bond have their extreme values at Î»Ì‚, which is given by
If \(\frac{1}{\sigma_{s}} ( \theta_{1}\frac{\rho \theta_{2}}{\sqrt{1 \rho^{2}}} ) >0\), we can check that \(\pi_{1,t}^{*} X_{t}^{*}\) is maximized at \(X_{t}=\hat{x}\), where
Furthermore, from a similar analysis in Koo et al. [5], we see that, for \(\tilde{x}\leq X_{t}\leq \hat{x}\), \(\pi_{1,t}^{*} X_{t}\) is increasing in \(X_{t}\) and, for \(\hat{x}< X_{t}<\bar{x}\), \(\pi_{1,t}^{*} X_{t}\) is decreasing in \(X_{t}\) and approaches zero as \(X_{t}\) gets close toÂ xÌ„. Note that we have the opposite results for \(\pi_{0,t}^{*} X _{t}\). Specifically, for \(\tilde{x}\leq X_{t}\leq \hat{x}\), \(\pi_{0,t}^{*} X_{t}\) is decreasing in \(X_{t}\) and, for \(\hat{x}< X _{t}\leq \bar{x}\), \(\pi_{0,t}^{*} X_{t}\) is increasing in \(X_{t}\) and approaches zero as \(X_{t}\) gets close to xÌ„.
5 Discussion
In this section, we discuss the quantitative results obtained from the optimization problem and exhibit implications through some numerical illustrations. Let us rewrite the optimal policies as follows.
Remark 1
The optimal consumption rate in TheoremÂ 2 can be rewritten as
Similarly, the optimal portfolios can also be rewritten as
respectively, where \(y_{t}^{*}\) solves \(X_{t}=h_{1}(\lambda_{t}^{*})\).
Let us call \((y/r, \tilde{x}), [\tilde{x}, \bar{x})\), and \([\bar{x}, \infty )\) as regions 1, 2, and 3, respectively. In region 3, as we expected, the optimal consumption rate is \(1/2Q\) at which the quadratic utility function has its maximum. Interestingly, we also have a constant consumption in region 1. The main reason why the agent consumes nothing in region 1 might be that the agent wants to get out of the lower wealth level as soon as possible. This effort could be compensated as additional consumption in region 2.
Similar to the consumption rate, the portfolios can also be described according to the different regions. We consider for the case where \(\theta_{1}\rho \theta_{2}/\sqrt{1\rho^{2}}>0\). Then, the investment in the risky asset, \(\pi_{1,t}X_{t}\), is always nonnegative, and the investment in the riskless asset (or bank account), \(\pi_{0,1}X_{t}\), is always nonpositive. Note that \(y/r\) is the present value of the future income stream, called human capital. We also call the sum of financial wealth \(X_{t}\) and human capital as the total capital, which is always nonnegative.
Now we can describe the optimal portfolios in more details. In region 1, the investment in the inflationlinked index bond consists of two components, which are the demand for hedging inflation risk and an additional investment which is proportion to the total capital. Since \(n_{}<0\), the additional investment is positive if \(\theta_{2}>0\), which implies the speculative or myopic demand due to the excess return on the inflationlinked index bond. Moreover, under the condition that \(\theta_{1}\rho \theta_{2}/\sqrt{1\rho^{2}}>0\), the investment in risky asset also represents myopic demand which is proportion to the total capital. We can verify that the total myopic demand in the risky asset and inflationlinked index bond is borrowed from the bank account.
Similar to the portfolios in region 1, in region 2 the total wealth is invested in the inflationlinked index bond to hedge the inflation risk and there exists an additional investment in that bond. The myopic demand, however, is not proportional to the total capital anymore. We see that the second term in (5.4) represents the myopic demand and it is proportional to the difference between xÌ„ and current wealth. Note that xÌ„ is the wealth level at bliss point. Thus, in region 2, xÌ„ plays a role of target wealth and a proportion \((\bar{x}X_{t})\) is invested as myopic demand. We can also confirm that the optimal investment in the risky asset given in (5.3) consists of two components, and the first one, which is a myopic demand, is proportional to \((\bar{x}X_{t})\).
Under AssumptionÂ 1, we have \(n_{+}<2\). Thus, the second term of (5.3) in region 2 is positive. Since the coefficient \(D_{1}\) is determined by the boundary condition at xÌƒ, we can describe the second term as the hedging demand for not falling into region 1. Again, we can verify that, except for the inflation risk hedging demand, the total investment in the risky asset and the inflationlinked index bond is borrowed from the bank account as we can see in the portfolio \(\pi_{0,t}X_{t}\) in (5.2).
In region 3, the investment in the risky asset and bank account becomes zero and all the financial wealth is invested in the inflationlinked index bond. This is a quite different result from the case with other types of utility function such as CRRA or CARA. Since the quadratic utility has a bliss point and the value function is flat for wealth levels that are larger than bliss level of wealth, there should be no speculative motive, and thus, there is no myopic demand in asset allocation. In particular, we have the myopic demand in both the risky asset and the inflationlinked index bond as explained before. They converge to zero in region 3. Moreover, the hedging demand for not falling into the region 1, which is the last term of the portfolio (5.3) in regionÂ 2, also converges to zero in region 3. This is because the financial wealth is large enough. Recall that in region 2, except for the inflation risk hedging demand, all the myopic and hedging demand is borrowed from the bank account. Thus, the investment in the bank account, \(\pi_{0,t}X_{t}\), should be also zero and, as a result, an inflationrisk hedging demand exists only in region 3. To sum up, in the presence of inflation risk and the inflationlinked index bond, there exists no demand in the risky asset or in the bank account. Instead, only inflation risk hedging demand remains. Neither myopic demand nor other hedging demand is necessary.
We consider the following market parameter set to for numerical illustrations:
This parameter set is consistent with Brennan and Xia [1], Koo et al. [5], and Kwak and Lim [6], and we have \(\theta_{1}\rho \theta_{2}/\sqrt{1\rho ^{2}}=0.2052>0\). FigureÂ 1 shows the optimal consumption and portfolios in the presence of inflation risk. For the given parameter set, \(y/r=25\), \(\tilde{x}=23.8\) and \(\bar{x}=58.3\). So regions 1, 2, and 3 are given by \((25, 23.8), [23.8, 58.3)\), and \([58.3, \infty )\), respectively. As we can see, the consumption rate is zero in region 1, and is strictly increasing and concave in wealth in region 2. In region 3, however, it stays constant. The investment in the risk asset is always nonnegative and has the global maximum value at \(x=\hat{x} (=6.526)\), which is defined in (4.7). Specifically, the investment in the risky asset increases with wealth for \(x\in (y/r, \hat{x})\) and decreases with wealth for \(x\in (\hat{x},\bar{x}]\). It approaches zero as wealth gets close to xÌ„. Investment in the bank account has exactly the opposite pattern compared to the investment in the risky asset. The investment in the inflationlinked index bond, however, increases with wealth in all the regions and it has the same value as the wealth when \(x\geq \bar{x}\).
Now we consider another set of market parameters such that \(\theta _{1}\rho \theta_{2}/\sqrt{1\rho^{2}}<0\). With other parameters fixed, let us consider the case of \(\mu_{S}=0.5, \mu_{P}=0.035\) and \(\rho =0.3\). Then we have \(\theta_{1}\rho \theta_{2}/\sqrt{1\rho ^{2}}=0.044<0\), \(\tilde{x}=23.36\), and \(\bar{x}=58.3\). Therefore, regions 1, 2, and 3 are given by \((25, 23.36)\), \([26.36, 58.3)\) and \([53.8, \infty )\), respectively. Note that xÌƒ and xÌ„ have similar values as in the case with market parameters of Fig.Â 1. FigureÂ 2 illustrates the optimal consumption and portfolios with these parameters. As we can see, the optimal consumption rate is similar to that of Fig.Â 1. The portfolios, however, have quite different results. In particular, the investment in the risky asset is not positive anymore, and it shows a similar shape to the investment in the bank account. Instead, compared to the case of Fig.Â 1, the investment in the inflationlinked index bond has much larger values in region 2. In fact, the negative sign of \(\theta_{1} \rho \theta_{2}/\sqrt{1\rho^{2}}\) implies a high mean return of the inflationlinked index bond, so more speculative demand would be required. Thus, in regions 1 and 2, the agent borrows money from the bank account or takes a short position in the risky asset to finance, and all the borrowed money is invested in the inflationlinked index bond. In other words, the investment in the inflationlinked index bond consists of two different demands, which are inflation risk hedging demand and myopic demand. We observe that the myopic demand does not appear in other portfolios. Note that in region 3, the investment in the inflationlinked index bond has the same value as the wealth level.
6 Conclusions
In this paper, we study the optimal consumption and portfolio decision with a quadratic utility in the presence of inflation risk. For a hedging instrument for inflation risk, we introduce an inflationlinked index bond which is directly linked to the price process. Depending on market parameters, the investment in the risk asset and in the inflationlinked index bond exhibit quite different patterns. For the lifetime financial planning, the inflation risk is easily ignored but it may frustrate an agent which consumes and invests optimally. Therefore, it is worthwhile to look into how an agent manages the inflation risk in the presence of an inflationlinked index bond through lifetime utility maximization with a quadratic utility function.
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Acknowledgements
We are grateful to Yong Hyun Shin for helpful comments and suggestions.
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Lim was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF2014S1A5A8018920, NRF2017R1E1A1A03071107). Lee was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF2016R1D1A1B03933406) and by the Research Grant of Kwangwoon University in 2018.
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Lim, B.H., Lee, HS. Portfolio decision with a quadratic utility and inflation risk. Adv Differ Equ 2018, 366 (2018). https://doi.org/10.1186/s1366201818341
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DOI: https://doi.org/10.1186/s1366201818341
Keywords
 Martingale duality method
 Portfolio selection
 Quadratic utility
 Inflation risk
 Inflationlinked index bond