Robust portfolio choice with stochastic interest rates

Abstract

We determine the optimal investment strategy for an ambiguity-averse investor in a setting with stochastic interest rates. The investor has access to stocks, bonds, and a bank account and he is ambiguous about the expected rate of return of both bonds and stocks. The investor can have different levels of ambiguity aversion about the two types of risky assets. We find that it is more important to take model uncertainty about the stock dynamics than model uncertainty about the bond dynamics into account. Furthermore, the investor’s ambiguity increases his hedging demand. Consequently, the bond/stock ratio increases with his ambiguity and implies less extreme positions in the bank account. Altogether, our model yields portfolio allocations which are more in line with what is implementable in practice. Finally, we demonstrate that neglecting model uncertainty implies significant losses for the investor.

Appendices

Appendix A: Optimal investment strategy

Substituting $u^*$ from (3.8) into the HJB Eq. (3.5) yields

$$\begin{aligned} 0 = \sup _{\pi }&\left\{ \tilde{V}_t + \tilde{V}_W W \left( r + \pi _t^\top \sigma \lambda - \tilde{V}_W W \pi _t^\top \sigma \varPsi \sigma ^\top \pi - \pi _t^\top \sigma \varPsi \tilde{V}_r \bar{\sigma }_r \right) \right. \nonumber \\&\textstyle \quad +\, \tilde{V}_r \kappa \bar{r} + \tilde{V}_r \bar{\sigma }_r^\top \lambda - \tilde{V}_r \tilde{V}_W W \bar{\sigma }_r^\top \varPsi \sigma ^\top \pi -\tilde{V}_r^2 \bar{\sigma }_r^\top \varPsi \bar{\sigma }_r - \tilde{V}_r \kappa r \nonumber \\&\textstyle \quad +\, \frac{1}{2} \tilde{V}_{WW} W^2 \pi ^\top \sigma \sigma ^\top \pi + \frac{1}{2} \tilde{V}_{rr} \bar{\sigma }_r^\top \bar{\sigma }_r + \tilde{V}_{Wr} W \pi ^\top \sigma \bar{\sigma }_r \nonumber \\&\textstyle \quad +\, \frac{1}{2}\text{ tr } \left( \tilde{V}_W^2 W^2 \sigma ^\top \pi \pi ^\top \sigma \varPsi + \tilde{V}_W \tilde{V}_r W \sigma ^\top \pi \bar{\sigma }_r^\top \varPsi \right) \nonumber \\&\textstyle \quad \left. +\, \frac{1}{2} \text{ tr } \left( \tilde{V}_W \tilde{V}_r W \bar{\sigma }_r \pi ^\top \sigma \varPsi + \tilde{V}_r^2 \bar{\sigma }_r^\top \bar{\sigma }_r \varPsi \right) \right\} . \end{aligned}$$

(6.1)

The first-order condition with respect to $\pi$ implies that a candidate for the optimal investment strategy is given by

$$\begin{aligned} \pi ^* \!=\! \left( \tilde{V}_{WW}W \sigma \sigma ^\top \!-\! \tilde{V}_W^2 W \sigma \varPsi \sigma ^\top \right) ^{-1}\tilde{V}_W \left( - \sigma \lambda \!-\! \frac{\tilde{V}_{Wr}\sigma \bar{\sigma }_r}{\tilde{V}_W} \!+\! \tilde{V}_r \sigma \varPsi \bar{\sigma }_r \right) . \quad \quad \end{aligned}$$

(6.2)

Substituting $\pi ^*$ into the HJB Eq. (6.1) yields a partial differential equation (PDE) for $\tilde{V}$. If this PDE has a solution, $\tilde{V}(W,r,t)$, such that the strategy $\pi$ is well-defined, it follows from a verification theorem that the strategy is optimal and that the function $\tilde{V}(W,r,t)$ equals the indirect utility function.¹⁷ By using the linear homogeneity in wealth, which we have ensured by the choice of $\varPsi$ given in (2.17), we make the following conjecture for the solution to the HJB equation in (6.1)

$$\begin{aligned} \tilde{V}(W,r,t) = \frac{W^{1-\gamma }}{1-\gamma } \, \exp \left\{ \left( 1-\gamma \right) \left( A(T-t)+B(T-t)r\right) \right\} , \end{aligned}$$

(6.3)

where $A(\cdot )$ and $B(\cdot )$ are deterministic functions. By substituting the relevant derivatives of $\tilde{V}$ into the HJB equation it follows that our guess solves the equation if

$$\begin{aligned} A(\tau )&= \left( \bar{r} - \frac{\lambda _1 \sigma _r }{\kappa \left( \gamma +\theta _1 \right) }- \frac{\left( \gamma +\theta _1-1 \right) \sigma _r^2}{2 \kappa ^2 \left( \gamma +\theta _1 \right) } \right) \left( \tau -b(\tau ) \right) + \frac{\left( \gamma +\theta _1-1\right) }{4 \kappa \left( \gamma +\theta _1\right) } \, \sigma _r^2 b(\tau )^2\nonumber \\&\quad + \frac{1}{2\left( \gamma + \theta _1 \right) } \, \lambda _1^2 \tau + \frac{1}{2\left( \gamma + \theta _2 \right) } \, \lambda _2^2 \tau , \end{aligned}$$

(6.4)

$$\begin{aligned} B(\tau )&= \frac{1}{\kappa }\left( 1-e^{-\kappa \tau } \right) = b(\tau ). \end{aligned}$$

(6.5)

The optimal investment strategy now follows from (6.2), and the worst-case measure accepted by the investor follows from (3.8).

Appendix B: Calibration of parameters

We calibrate the model to historical estimates of mean returns, standard deviations, and correlations for the US market. We consider a time period of 15 years, which is the time period of data available on US TIPS, which we need to determine the real return on bonds. The real return on the stock is constructed using real stock prices on S&P 500 taken from the homepage of Robert Shiller.¹⁸ The data period runs from January 1996 to January 2011. The annualized average real return on the stock approximately equals $\mu _S = 0.06$ with a standard deviation of $\sigma _S= 0.20$. As an approximation for the short-term interest rate we use the real 1-year interest rate also stated on Shiller’s homepage. The average real US short-term interest rate then equals $\bar{r}^\mathbb{P }=0.014$ and has a volatility of $\sigma _r = 0.022$. The correlation between the short-term interest rate and the stock index equals $-0.09$, that is, the correlation between the stock and bond market equals $\rho = 0.09$. If we know the standard deviation of the return on a bond, we can determine the speed of mean reversion in the short rate by solving

$$\begin{aligned} \sigma _B(\tau ) = \frac{1}{\kappa }\left( 1-e^{-\kappa \tau } \right) \sigma _r. \end{aligned}$$

To get an estimate of the standard deviation we use US TIPS represented by Barclays US Govt Inflation-linked Bond Index for which data is available from January 1997 to January 2012 on Bloomberg. We get a standard deviation of $\sigma _B = 0.06$. Hence, for a maturity of 10 years this gives us an estimate for the speed of mean reversion in the interest rate of $\kappa =0.36$. To get an estimate of the market price of interest rate risk, $\lambda _1$, we can use that

$$\begin{aligned} \lambda _1 = \frac{\kappa }{\sigma _r}\left( y_{\infty } + \frac{\sigma _r^2}{2 \kappa ^2} - \bar{r}^\mathbb{P } \right) . \end{aligned}$$

However, this implies that we need an estimate of the long-term real rate. As an approximation for the long-term real rate we use the inflation-indexed long-term average yield from the Federal Reserve H.15 Statistical Releases.¹⁹ We use a time period of only 8 years (from 2003 to 2011) due to the availability of the data, and we get that the average inflation-indexed long-term average yield equals $y_{\infty }=0.02$. It then follows that $\lambda _1 = 0.12$. Finally, we need an estimate of the market price of risk factor $\lambda _2$, which is given by

$$\begin{aligned} \lambda _2 = \frac{\psi -\rho \lambda _1}{\sqrt{1-\rho ^2}}. \end{aligned}$$

The Sharpe ratio follows from $\psi = \frac{\mu _S-\bar{r}^\mathbb{P }}{\sigma _S} = 0.2$, from which we finally get that $\lambda _2 = 0.22$. Our parameters are comparable to the estimated parameters given in Munk and Sørensen (2010). The parameter estimates are summarized in Table 1.