Merton Portfolio Choice with CRRA Utility

The Portfolio Problem and Its Assumptions

In the finite-horizon Merton portfolio problem, an investor chooses the fraction of wealth held in a risky asset, with the remaining wealth invested at a constant risk-free rate. The objective here is terminal utility only: there is no intermediate consumption. The fraction is unconstrained, so short positions and leverage are allowed. This makes it possible to identify directly how the risky asset’s excess drift, volatility, and the investor’s curvature of utility determine the optimal fraction.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). The measure \(\mathbb{P}\) is the physical probability measure; no change to a risk-neutral measure is made. Suppose the risky asset satisfies \(dS_t/S_t=(r+\lambda)\,dt+\sigma\,dB_t\), where \(r\) is the constant risk-free rate, \(\lambda\) is the risky asset’s excess drift, and \(\sigma>0\) is its volatility. If \(\pi_t\) is the fraction of wealth in the risky asset, wealth \(X_t\), starting from \(X_t=x>0\) at time \(t\), satisfies

The first coefficient is the wealth drift; the coefficient of \(dB_s\) is the wealth diffusion coefficient. Controls \(\pi_s\) are progressively measurable. The Ito integrals below are defined for progressively measurable integrands \(H_s\) in \(\mathcal{L}^2\), meaning \(\mathbb{E}\int_t^T H_s^2\,ds<\infty\). In particular, require \(\mathbb{E}\int_t^T(\pi_sX_s\sigma)^2\,ds<\infty\), and require the drift integral to be well-defined. We consider positive wealth and controls for which the terminal utility expectation and the verification expectations used below are finite.

Deriving and Solving the HJB Equation

For a smooth value function, the dynamic programming principle over a short time interval gives the Hamilton-Jacobi-Bellman equation. For fixed fraction \(\pi\), the wealth generator acting on a twice differentiable function \(f\) is \((r+\pi\lambda)xf'(x)+\tfrac12\pi^2\sigma^2x^2f''(x)\). Thus the terminal-value HJB equation is

Here the supremum is over all real fractions. To solve the equation, use the homogeneity suggested by power utility and set \(V(t,x)=A(t)x^p/p\), with \(A(T)=1\). The derivatives are \(V_t=A'x^p/p\), \(V_x=Ax^{p-1}\), and \(V_{xx}=A(p-1)x^{p-2}\). Substitution yields

Define \(h(\pi)=r+\pi\lambda-\tfrac12(1-p)\pi^2\sigma^2\). Because \(1-p>0\) and \(\sigma>0\), this is a strictly concave quadratic, so it has a unique maximizer.

Proof. Differentiating \(h\) gives \(h'(\pi)=\lambda-(1-p)\sigma^2\pi\). Its unique zero is \(\pi^*=\lambda/((1-p)\sigma^2)\); since \(h''(\pi)=-(1-p)\sigma^2<0\), this point is the unique maximum. Completing the square gives \(h(\pi)=k-\tfrac12(1-p)\sigma^2(\pi-\pi^*)^2\), where substitution of \(\pi^*\) gives \(k=r+\lambda^2/(2(1-p)\sigma^2)\). The HJB equation for the proposed form becomes \(A'/p+kA=0\), or \(A'=-pkA\). With terminal condition \(A(T)=1\), solving this scalar differential equation gives \(A(t)=\exp(pk(T-t))\). Substituting this \(A\) into the equation verifies the proposed HJB solution and its terminal condition. \(\square\)

The factor multiplying \(h(\pi)\) in the HJB equation is \(A(t)x^p\), which is positive for every \(x>0\), including when \(p<0\). Therefore maximizing \(h\) indeed maximizes the HJB expression for both positive and negative \(p\). The restriction \(p\ne0\) is needed for the stated power-utility formula; logarithmic utility is a separate limiting case.

Verification with Ito’s Formula

A formal HJB solution is not by itself a proof that a control attains the value. The next result verifies this candidate for a stated class of controls. Define this class to include controls for which the wealth stays positive, \(\mathbb{E}[U(X_T)]\) is finite, and, when Ito’s formula is applied to \(V(s,X_s)\), the drift is integrable and the resulting stochastic integral is a true martingale. Concretely, require its integrand \(A(s)X_s^p\pi_s\sigma\) to be progressively measurable with \(\mathbb{E}\int_t^T A(s)^2X_s^{2p}\pi_s^2\sigma^2\,ds<\infty\). The Ito isometry then gives a square-integrable martingale, and its conditional expectation of each future increment is zero.

Proof. For an admissible control, apply Ito’s formula to \(V(s,X_s)\). The function is \(V(s,x)=A(s)x^p/p\), with partial derivatives \(V_s=A'(s)x^p/p\), \(V_x=A(s)x^{p-1}\), and \(V_{xx}=A(s)(p-1)x^{p-2}\). The second-order Ito correction is \(\tfrac12V_{xx}(s,X_s)(X_s\pi_s\sigma)^2\,ds\). Using \(dB_s\,dB_s=ds\), \(dB_s\,ds=0\), and \(ds\,ds=0\), the full differential is

The drift is nonpositive because \(h(\pi_s)\le k\). The stochastic integral is a true martingale by the stated square-integrability condition, not merely a local martingale. Integrating and taking expectations therefore gives \(\mathbb{E}[V(T,X_T)]\le V(t,x)\). Since \(A(T)=1\), \(V(T,X_T)=U(X_T)\), so every such control has expected terminal utility at most \(V(t,x)\).

For \(\pi_s=\pi^*\), the drift vanishes identically because \(h(\pi^*)=k\). Under this constant control, wealth is geometric Brownian motion: \(X_s=x\exp\{(r+\pi^*\lambda-\tfrac12(\pi^*\sigma)^2)(s-t)+\pi^*\sigma(B_s-B_t)\}\). Its positive and negative real moments of every fixed order are finite on a finite horizon, since \(B_s-B_t\) is normally distributed and a normal variable has finite exponential moments. In particular, the required expectation of the squared stochastic-integral integrand is finite. The stochastic integral is thus a true square-integrable martingale, and taking expectations in the Ito identity gives \(\mathbb{E}[U(X_T)]=V(t,x)\). The upper bound is attained, proving optimality. \(\square\)

Interpreting the Optimal Fraction and Avoiding a Sign Error

The fraction \(\pi^*=\lambda/((1-p)\sigma^2)\) is proportional to excess drift and inversely proportional to relative risk aversion and variance. Its sign is the sign of \(\lambda\): positive excess drift leads to a long position, while negative excess drift leads to a short position. Greater volatility or greater relative risk aversion reduces the magnitude of the fraction. When \(\lambda>0\), this lowers the fraction; when \(\lambda<0\), it raises the fraction numerically toward zero. It is not generally correct to say that more volatility always lowers the signed fraction.

Common mistake. A frequent algebra error is to leave an extra factor \(1/p\) in the Ito drift after substituting the candidate value function. The time derivative contributes \(A'X^p/p\), while the generator contributes \(AX^ph(\pi)\). Since \(A'=-pkA\), their sum is \(AX^p(h(\pi)-k)\), with no remaining \(1/p\). For example, in the preceding example at \(\pi=1\), the correct drift is \(-0.09AX^{0.5}\), not \((-0.09/p)AX^{0.5}=-0.18AX^{0.5}\). Substitution into the displayed Ito differential confirms the correct coefficient.

The constant-fraction conclusion depends on the model assumptions: constant \(r,\lambda,\sigma\), a single risky asset driven by one Brownian motion, power terminal utility, and no constraints on \(\pi\). The derivation does not include intermediate consumption or portfolio bounds. Within these assumptions, the HJB solution gives both the value’s dependence on current wealth and the allocation rule; the verification argument explains when the formal solution truly equals the control problem’s value.