*2.2. Adaptive Strategies*

In contrast to deterministic strategies where the asset allocation depends only on time, adaptive strategies allow the asset allocation to depend on the prevailing state of the investment portfolio. Since we search for the optimal controls over all portfolios with the same wealth after cash injection (*W*+*i* ), this means that *pi* = *pi*(*W*+*i* , *t*+*i* ). With an adaptive strategy, it can be optimal to withdraw cash from the portfolio (Cui et al., 2014; Dang and Forsyth, 2016). We denote this optimal cash withdrawal as *ci* ≡ *<sup>c</sup>*(*<sup>W</sup>*<sup>−</sup>*i* + *qi*, *ti*). Since we only allow cash withdrawals, *ci* ≥ 0. The control at action time *ti* now consists of the pair (*pi*, *ci*), i.e., after withdrawing *ci* from the portfolio, rebalance so that the fraction invested in the risky asset is *pi*.

For explanatory purposes, let us consider first consider a dynamic (multi-period) MV criterion with a specified desired value of *E*[*WT*] = *Wd*. The problem to be solved can be stated as

$$\text{subject to} \begin{cases} \min\_{\{(p\_0, c\_0), \dots, (p\_{M-1}, c\_{M-1})\}} & \text{Var} \left[ W\_T \right] = E \left[ W\_T^2 \right] - W\_{d\_r}^2 \\\\ \text{At horizon } T; E \left[ W\_T = S\_T + B\_T \right] = W\_{d\_r} \\\\ \text{Between action times } t \notin T; (B\_t, S\_t) \text{ follow processes (2), (3),} \\\\ \text{At action times } t \in T\_1; \\\ W\_i^+ = S\_i^- + B\_i^- + q\_i - c\_i, \\ S\_i^+ = p\_i W\_i^+, \quad B\_i^+ = W\_i^+ - S\_i^+, \\ p\_i = p\_i \left( W\_i^+, t \right), 0 \le p\_i \le L\_{\text{max}}, \\ c\_i = c\_i \left( W\_i^- + q\_i, t \right) \; c\_i \ge 0. \end{cases} \tag{7}$$

A criticism of the pre-commitment MV problem (7) is that it is *time inconsistent*. In other words, the investor has an incentive to deviate from the strategy computed at time zero (Basak and Chabakauri, 2010). However, in order to solve problem (7), we can use the embedding technique (Li and Ng, 2000; Zhou and Li, 2000). Consider a control set

$$P = \left\{ \left( p\_i \left( \mathcal{W}\_i^+, t\_i^+ \right), c\_i \left( \mathcal{W}\_i^- + q\_i t\_i \right) \right), i = 0, \dots, M - 1 \right\} \dots$$

Informally, if *P*∗ is an optimal control for problem (7), then there exists a *W*∗ such that *P*∗ is also the optimal control for the following problem:

$$\begin{aligned} & \min\_{\{(p\_0, c\_0), \dots, (p\_{M-1}, c\_{M-1})\}} \quad E\left[\left(W^\* - \mathcal{W}\_T\right)^2\right], \\ & \text{subject to} \begin{cases} \text{At horizon } T \colon E\left[\mathcal{W}\_T = S\_T + B\_T\right] = \mathcal{W}\_{t'} \\ \text{Between action times } t \notin \mathcal{T} \colon (B\_t, S\_t) \text{ follow processes (2), (3),} \\ \text{At action times } t \in \mathcal{T}\_l; \\ & W\_i^+ = S\_i^- + B\_i^+ + q\_i - c\_{i'} \\ \quad S\_i^+ = p\_i W\_i^+, \quad B\_i^+ = W\_i^+ - S\_i^+ \\ & p\_i = p\_i \left(W\_i^+, t\right), 0 \le p\_i \le L\_{\text{max}}, \\ & c\_i = c\_i \left(W\_i^- + q\_i, t\right) \; c\_i \ge 0. \end{cases} \end{aligned} (8)$$

Problem (8) can be solved using dynamic programming methods.<sup>11</sup>

<sup>11</sup> If problem (7) is not convex, there may be solutions to problem (8) that are not solutions to problem (7). However, these spurious solutions can be eliminated in a straightforward way (Dang et al., 2016; Tse et al., 2014).

As noted above, it is optimal to withdraw cash from the portfolio under some conditions (Cui et al., 2012; Dang and Forsyth, 2016). Let

$$Q\_{\ell} = \sum\_{j=\ell+1}^{j=M-1} e^{-r\left(t\_j - t\_{\ell}\right)} q\_j \tag{9}$$

be the discounted planned future contributions to the DC account at time *t*-. If

$$\left(\mathcal{W}\_i^{-} + q\_i\right) \supset \mathcal{W}^\* e^{-r(T - t\_i)} - Q\_{i\nu} \tag{10}$$

then the optimal strategy is to

(i) withdraw cash *ci* = *<sup>W</sup>*<sup>−</sup>*i*+ *qi* − *W*<sup>∗</sup>*e*<sup>−</sup>*<sup>r</sup>*(*<sup>T</sup>*−*ti*) − *Qi* from the portfolio; and

(ii) invest the remainder *W*<sup>∗</sup>*e*<sup>−</sup>*<sup>r</sup>*(*<sup>T</sup>*−*ti*) − *Qi*in the risk-free asset.

This is optimal because in this case *E* -(*W*∗ − *WT*)<sup>2</sup>= 0, which is the minimum of problem (8).

We refer to any cash withdrawn from the portfolio as *surplus cash* in the following. For the sake of discussion, we will assume that surplus cash is invested in the risk-free asset, but does not contribute to the calculation of the mean and variance of terminal wealth.

Allowing cash withdrawals prevents penalization of wealth paths such that *WT* > *W*<sup>∗</sup>, which can result in forcing the optimal strategy to lose money if market gains are good, which is clearly an undesirable outcome (Cui et al., 2012; Dang and Forsyth, 2016). We remark that in practice, this withdrawal can be virtual, i.e., any amount of wealth satisfying equation (10) is simply invested in the risk-free asset, and and the surplus cash is not taken into account when computing the optimal fraction to invest in equities. See Dang and Forsyth (2016) for more detail on this. In fact, if we use continuous rebalancing, then the optimal strategy is such that Equation (10) is never satisfied (Vigna, 2014). In the discrete rebalancing case, the generation of surplus cash is a low probability event.

This target-based approach of problem (8) provides a reasonable objective on its own (Menoncin and Vigna, 2017; Vigna, 2014, 2017). Solving (8) minimizes quadratic shortfall (QS) with respect to *W*<sup>∗</sup>, so we will refer to the resulting strategy as the *QS optimal* strategy below. However, this becomes even more compelling when we recall that the solution is also pre-commitment MV efficient. The solution then simultaneously minimizes *two* risk measures: variance around the desired *E* [*WT*] and QS with respect to *W*<sup>∗</sup>, as seen at time zero.

We emphasize that the fact that the pre-commitment MV policy is time inconsistent is irrelevant since we take the point of view that we are seeking the QS optimal control, from problem (8). Since the QS problem (8) can be solved using dynamic programming, the controls are trivially time consistent. The fact that the QS problem gives rise to time consistent controls, whereas the MV problem (7) is time inconsistent, is due to the fact that we fix *W*∗ for the QS problem, for all time. At time zero, the MV problem controls and the QS problem controls are the same for *W*∗ computed at time zero. At later times, this correspondence holds only if we allow *W*∗ to change as a function of time. However, using a fixed *W*∗ is intuitively reasonable for DC pension plan saving (Menoncin and Vigna, 2017; Vigna, 2014).

We note that there are techniques for forcing a time consistent constraint for the MV problem (7) (Bjork and Murgoci, 2010, 2014; Bjork et al., 2014; Wang and Forsyth, 2011). However, we prefer the target based QS approach since it is relatively easy to communicate to end user investors (Menoncin and Vigna, 2017; Vigna, 2014). In addition, forcing the time consistent constraint can have result in non-intuitive strategies with strange features (Bensoussan et al., 2019; Wang and Forsyth, 2011).

We formulate problem (8) as the solution of a nonlinear Hamilton–Jacobi–Bellman (HJB) partial integro differential equation. See Dang and Forsyth (2014) for details concerning the numerical solution. Given an arbitrary value of *W*<sup>∗</sup>, we can solve problem (8) for the optimal control, which we denote by *<sup>P</sup>*<sup>∗</sup>(*W*<sup>∗</sup>). Given the optimal control, cumulative distribution functions are easily found using Monte

Carlo simulation. However, we seek the solution to problem (7), which is expressed in terms of a specified expected value *E*[*WT*] = *Wd*. We determine the value of *W*∗ for problem (8) which satisfies the constraint *E*[*WT*] = *Wd*. We enforce this by a Newton iteration, whereby each function evaluation requires a solution of an HJB equation.

### **3. Data and Parameter Estimates**

The underlying stochastic model outlined above in Equations (2), (3), and (4) involved a constant risk-free rate *r* for the bond component and a double exponential jump-diffusion for the equity component. Estimation of the parameters of these equations follows the methods described in Forsyth and Vetzal (2019). These procedures are summarized briefly here for convenience. Readers interested in additional details are referred to Forsyth and Vetzal (2019).

We use monthly US data obtained from the Center for Research in Security Prices (CRSP) for the period 1926:1 through 2015:12.<sup>12</sup> Our base case uses the CRSP value-weighted total return index (which includes all distributions for all domestic equities trading on major US exchanges), along with the CRSP 3-month Treasury bill (T-bill) index. The original data are in nominal terms, but we convert them to real terms using the US CPI, also obtained from CRSP. We use real indexes since investors with long-term savings objectives such as funding retirement should concentrate on real (not nominal) wealth goals. For some tests, we use alternative underlying assets: the CRSP equal-weighted total return index (which invests the same amount in each component security, rather than weighting by market capitalization) and a 10-year US Treasury bond (T-bond) index.

Figure 1 provides plots of monthly real returns for the 3-month T-bill, 10-year T-bond, value-weighted total return, and equal-weighted total return indexes. For comparability, all four indexes are plotted with the same vertical axis scale. As expected, the two equity indexes exhibit much higher volatility, with occasional months having returns of large magnitude. This provides a measure of support to our modelling assumptions which assume a constant interest rate but a jump diffusion specification for the equity index. It is also interesting to observe that there was extremely high equity market volatility during the 1930s. By contrast, volatility during the period following the financial crisis that began in 2007 was comparatively mild.

Figure 2 graphs cumulative real returns for the four investment indexes. The vertical axis uses a logarithmic scale. This enhances visibility over time, as otherwise the dramatic growth in the equity indexes over the latter part of the sample obscures the behaviour of those indexes during the earlier periods and renders the behaviour of the two Treasury indexes all but invisible. All four indexes begin at a value of 100 at the start of 1926. The equal-weighted index ends up with the highest value. The historical outperformance of equal-weighting has been attributed to such portfolios having higher exposure to value, size, and market factors (Plyakha et al., 2014). It is also interesting to observe that the 10-year T-bond index had higher cumulative returns than the 3-month T-bill index, but that was entirely due to the post-1980 period: prior to then, the longer maturity T-bond index offered no cumulative advantage over the T-bill index.

Table 1 presents parameter estimates. A threshold method (Cont and Mancini, 2011) was used for the jump diffusion model. These parameter estimates were originally provided in Forsyth and Vetzal (2019), and are reproduced here for convenience. The estimates using the value-weighted equity market index imply an expected real annual return of close to 9%, about 3% lower than the corresponding value for the equal-weighted index. Of course, the price to be paid for this difference is higher risk. The equal-weighted index shows higher diffusive volatility (*σ*). Since jumps are expected to occur on average every 1/*λ* years, the equal-weighted index tends to have jumps a bit more often. Conditional

<sup>12</sup> More precisely, our calculations are based on data from Historical Indexes, c 2015 Center for Research in Security Prices (CSRP), The University of Chicago Booth School of Business. Wharton Research Data Services (WRDS) was used in preparing this work. This service and the data available thereon constitute valuable intellectual property and trade secrets of WRDS and/or its third-party suppliers.

on a jump occurring, it is much more likely to be a downward jump in each case. Average jump magnitudes are 1/*η*1 for upward jumps and 1/*η*2 for downward jumps, and these are both larger for the equal-weighted index. A similar comment applies to the standard deviation of the jump size since this is equal to the mean for the exponential distribution. Turning to the bond market indexes, Table 1 shows that the long run average real annual return for the 10-year T-bond index was just over 2%, while that for the shorter maturity index was around 80 basis points. Of course, these higher returns are accompanied by higher volatility, as indicated by the top two plots of Figure 1.

**Figure 1.** Monthly real returns for US investment indexes, 1926–2015.

**Figure 2.** Cumulative real returns for US investment indexes, 1926–2015.

**Table 1.** Annualized parameter estimates based on real monthly data from 1926:1 to 2015:12. These values originally appeared in Forsyth and Vetzal (2019) and are reproduced here for convenience. Parameters for the equity market indexes were estimated using the threshold technique of Cont and Mancini (2011). The average returns for the bond indexes were calculated as log[*B*(*T*)/*B*(0)]/*T*, where *B*(*t*) denotes the index level at the time *t*.


### *3.1. Robustness to Parameter Estimation*

Our main purpose for calibrating the parameters for the stochastic processes (3) is to determine a control strategy (i.e., fraction in risky asset at rebalancing dates). Consequently, our concern is with the effect of calibration errors on the computed strategy, rather than minimizing fit errors in an econometric sense. In Forsyth and Vetzal (2017a) and in Dang and Forsyth (2016), an extensive study of the effect of parameter ambiguity is carried out. In particular, the parameters for stochastic process (3) were determined using maximum likelihood, and threshold techniques with various parameters. Robustness of the strategy (in the synthetic market) was tested by using Monte Carlo simulations with different parameters than were used in computing the optimal strategy. For example, the strategy was computed and stored, assuming parameters determined from the threshold strategy. Then, this strategy was tested using Monte Carlo simulations, in a synthetic market driven by a stochastic process with parameters determined using maximum likelihood. In other words, the parameters used to compute the strategy were misspecified. In all combinations of methods, the results were robust to this type of parameter misspecification.

However, the real test of our strategies is their performance on bootstrapped resampled tests. In the bootstrap tests, we make no assumptions about the stochastic process followed in the historical market. However, in all our bootstrap tests, the adaptive quadratic shortfall strategy (computed using the estimated market parameters) outperforms the glide path and constant proportion strategies.
