Portfolio Modeling with Sustainability Constraints

Abstract

As mentioned in Chap. 1 following the events of the world-wide financial crisis over the periods 2007–2009, the risk profile of some assets changed drastically and many assets exhibited large losses. These events have reinforced thinking about proper portfolio models that not only avoid large losses, but also allow to impose some constraints. This chapter will introduce standard static portfolio models that however also impose some constraints. Asset accumulation through saving or consumption decisions, will not be discussed in this chapter, so we will assume that the funds are given and only asset allocations have to be made. More generally, portfolio decisions under constraints are important for practitioners that invest on behalf of institutions with some ethical or social guidelines. For example, investment decisions of wealth funds, pension funds or university endowments, are often supposed to follow multiple guidelines and procedures rather than only choosing one procedure, such as an optimizing procedure without constraints. This is a point emphasized by Danthine and Donaldson (2005). They note that one step corresponds to the choice of instruments, another decision corresponds to the country or sector allocation or the choice of specific individual assets based on available information—to all of them maybe some constraints attached.

Appendix

3.1.1 Forecasting the Monthly Consumer Price Inflation

We use a monthly consumer price index (CPI) of the U.S. to forecast our expected consumer price inflation. First, we express U.S. monthly consumer price inflation (CPI) as the first difference of the log of the consumer price index. Second, before we compute our forecast, we examine whether U.S. monthly consumer price inflation contains a unit root and more specifically whether monthly inflation is covariance stationary rather than trend stationary over our estimation monthly sample period 1980:01–2014:03. For robustness, we conduct three unit root tests, namely the Augmented Dickey-Fuller test (which we refer to as ADF), the Phillips-Perron test (which we refer to as PP) and the Elliot-Rothenberg-Stock DF-GLS test (which we refer to as ERS). The ADF test statistic (with constant and trend)$= -9.33$ has a p-value = 0.00 and this test rejects the null of a unit root however it rejects to trend stationarity rather than covariance stationarity. Plotting the data for U.S. monthly consumer price inflation exhibits no trend, as a result we examine the ADF test statistic (with constant)$= -8.50$ which has a p-value = 0.00 and this shows that monthly inflation is covariance stationary. In addition, the ADF test statistic (with no constant and no trend)$= -5.33$ has a p-value = 0.00, confirming that monthly inflation is covariance stationary. Next, the PP test statistic (with constant and trend)$= -11.12$ has a p-value = 0.00, PP test statistic (with constant)$= -10.85$ has a p-value = 0.00 and PP test statistic (with no constant and no trend)$= -8.23$ has a p-value = 0.00. All the Phillips-Perron tests also confirm that monthly inflation is covariance stationary. Next, the ERS test statistic (with constant and trend)$= -3.22$ and the critical values are − 3. 48, − 2. 89 and − 2. 57 at a 1, 5 and 10 % level of significance respectively. Using the ERS test statistic (with constant and trend), we fail to reject the null of a unit root at a 1 % level of significance however we reject the null of a unit root at a 5 and 10 % level of significance. In addition, the ERS test statistic (with constant)$= -1.37$ which makes us to fail to reject the null of a unit root because the test statistic is less than all the relevant critical values which are − 2. 57 at a 1 % level.

Nevertheless, based on the Augmented Dickey-Fuller test, the Phillips-Perron test and the Elliot-Rothenberg-Stock DF-GLS test, the predominant finding is that U.S. monthly consumer price inflation is covariance stationary and hence can be estimated using OLS and be used for forecasting purposes. Our next point of departure is to examine the correlogram of U.S. monthly consumer price inflation and approximate the data generating process by evaluating different specifications. The best specification suggests an ARMA(2,2) process for U.S. monthly consumer price inflation. The parameters are estimated using OLS with Newey-West HAC consistent standard errors. The parameter estimates and corresponding standard errors which are in the parentheses, are as follows:

$$\displaystyle\begin{array}{rcl} \varDelta P_{t}& =& \mathop{0.00004}\limits_{(0.00001)} +\mathop{ 1.24}\limits_{(0.12)}\varDelta P_{t-1} -\mathop{ 0.26}\limits_{(0.11)}\varDelta P_{t-2} +\varepsilon _{t} \\ & & -\mathop{0.72}\limits_{(0.12)}\varepsilon _{t-1} -\mathop{\text{}0.28}\limits_{(0.12)}\varepsilon _{t-2} \\ R^{2}& =& 0.33\text{, }D - W\text{ }stat\text{ } = 1.99,\text{ }S.E.\text{ }of\text{ }regression = 0.003,{}\end{array}$$

(3.22)

where Δ P _t is U.S. monthly consumer price inflation and ɛ _t are the moving average components. Our estimated parameters have Newey-West HAC consistent standard errors and all the parameters are significant at all relevant levels of significance, however we continue to conduct residual diagnostic tests. We test for serially correlated residuals from our ARMA(2,2) process using the Ljung-Box (Q) test. The test statistic which is the Q-statistic of particular lag terms, has a null hypothesis that there is no autocorrelation in the residuals up to which ever relevant lag terms.

The Q-statistic probabilities are adjusted for two moving average terms. We find Q-statistic p-values for the third and fourth lag to be p-value = 0.03 and p-value = 0.10 respectively. However, for the fifth lag until the tenth lag, the p-values are greater than p-value = 0.12, which leads us to fail to reject the null hypothesis of no serially correlated regression residuals. We also test for serially correlated residuals using the Breusch-Godfrey serial correlation test. The BG (10) test statistic = 14.63 and has a p-value = 0.15 and we fail to reject the null hypothesis of no residual serial correlation. Based on the serial correlation tests, the parameter estimates are efficient. Next, we test for heteroscedasticity in the residual terms using White’s test, where White (6) test statistic (without cross terms) = 19.51 has a p-value = 0.002 and White (20) = 93.50 has a p-value = 0.00 and based on these we reject the null of no heteroscedasticity (residuals are not homoscedastic). In addition, we use the ARCH test for heteroscedasticity where ARCH(6) test statistic = 40.15 has a p-value = 0.00 and hence both tests reject the null hypothesis of no heteroscedasticity in the regression residuals. The regression residuals are heteroscedastic, however the standard errors of our regression are Newey-West heteroscedastic and autocorrelated consistent standard errors and this is accounted for in the t-statistics and p-values associated with the parameter estimates which are all significant at a 1, 5 and 10 % level of significance. Next, we examine whether the regression residuals and the regressors in the ARMA(2,2) model are correlated. We regress the regression residuals on the regressors in the ARMA(2,2) model and this produces insignificant parameter estimates and an R-squared of 0.008. As a result, the regression residuals and regressors are uncorrelated and hence the parameter estimates are unbiased in the finite sample we have at hand. We test for structural breaks and changes in the regression and structural changes in the parameter estimates using the Chow break point test. We use two dates which we consider as being relevant, firstly August 1987 (1987:08) which coincides with the confirmation of Alan Greenspan as a successor to Paul Volcker. Secondly we use February 2006 (2006:02) which is consistent with President Bush’s appointment of Ben Bernanke as a member of the Federal Reserve Board of Governors and as Chairman. We report both the F statistic and log likelihood ratio statistic for the Chow break point test. Using 1987:08 and 2006:02 under the null of no breaks at both specified breakpoints, we find the F-statistic (10,393) = 3.5358 with p-value = 0.0002 and the Log likelihood ratio statistic (10) = 35.1494 with p-value = 0.0001. Using the dates jointly 1987:08 and 2006:02, the F-statistic and log likelihood ratio statistic show that there is structural break at both specified breakpoints. Next, we check for stability (and hence invertibility) of the estimated model, that is do all the roots of the autoregressive component lie outside the unit circle. The autoregressive part has two roots which are λ ₁ = 3. 74 and λ ₂ = 1. 03 and these lie outside the unit circle and this shows that the estimated ARMA(2,2) process is stable. This finding is consistent with the unit roots tests under which we find that U.S. monthly consumer price inflation is covariance stationary. The above noted tests suggests our estimated ARMA(2,2) model is reliable however it is subject to a problem that the model is not constant over time because of structural breaks and hence this relationship may change over time. We only use the ARMA (2,2) model to forecast inflation for only one period ahead (1 month ahead) rather than multiple periods ahead and because we only compute a short horizon forecast, we consider our forecast procedure to be reasonable and reliable. Nevertheless, the one period ahead expected (forecasted) U.S. monthly consumer price inflation for each month over the period 1980:03–2014:02 is computed as follows:

$$\displaystyle{ \varDelta P_{t+1}^{e} = 0.00004 + 1.24\varDelta P_{ t} - 0.26\varDelta P_{t-1} - 0.72\mathop{\varepsilon }\limits^{ \wedge }_{t} - 0.28\mathop{\varepsilon }\limits^{ \wedge }_{t-1} }$$

(3.23)

which is the 1 month ahead U.S. consumer price inflation we use in computing expected real returns on all assets but only over the period 1983:02–2008:07 and $\hat{\varepsilon }_{t}$ and $\hat{\varepsilon }_{t-1}$ are the regression residuals used as a proxy for the moving average terms. We compare the in-sample mean squared error (MSE) of the forecast and the out-of-sample mean squared error (MSE) of the forecast which correspond to different sample periods which are 1980:04–2014:03 for the in-sample forecast and 1980:03–2014:02 for the out of sample forecast. The forecast errors are virtually the same with the in-sample MSE = 0.0000076 and the out-of-sample MSE = 0.00000767. Finally, we annualize our forecasted U.S. monthly consumer price inflation because this maintains consistency with the annualized nominal yields or interest rates on each respective asset and also allows us to compute monthly real returns that are annualized.

3.1.2 Capital Allocation Line and Efficient Frontiers

In this section of the appendix we report diagrams/figures corresponding to estimated portfolios. Figure 3.2 reports the capital allocation line, its intersection with the tangency portfolio and the efficient frontier where this corresponds to the mean-variance utility portfolio model in the unconstrained framework.

Figure 3.3a, b report the efficient frontier corresponding to the mean-variance quadratic programming model and Fig. 3.3a corresponds to allowing short selling whereas Fig. 3.3b corresponds to a no short selling constraint.

Figure 3.3c reports the efficient frontier corresponding to the mean-variance quadratic programming model with the individual lower and upper bound constraints imposed and Fig. 3.3d reports the efficient frontier corresponding to the mean-variance quadratic programming model with both individual lower and upper bound constraints and both combination lower and upper bound constraints.