*2.1. Deterministic Glide Paths*

TDFs generally use a deterministic glide path, where the asset allocation depends only on time. In our case, this would imply *pi* = *p*(*t*). One case is a *linear glide path*, with

$$p\_i = p\_{\text{max}} + \frac{t\_i \times (p\_{\text{min}} - p\_{\text{max}})}{T},\tag{5}$$

where *p*max and *p*min are parameters. Note also that a constant proportion strategy can be viewed as a deterministic glide path with *p* = const. for all action times *ti* ∈ T1.

Between action times, the amounts the investor has in the risk-free and risky assets follow the processes (2) and (3), respectively. Recalling that *qi* is a cash contribution, at action times prior to the horizon date (i.e., *ti* ∈ T1), we have

$$\begin{aligned} \mathcal{W}\_i^+ &= S\_i^- + B\_i^- + q\_{i'}\\ S\_i^+ &= p\_i \mathcal{W}\_i^+ \\ B\_i^+ &= (1 - p\_i) \mathcal{W}\_i^+ \end{aligned} \tag{6}$$

In the case of deterministic glide paths, closed form recursive expressions for the mean and variance of terminal wealth *WT* are developed in Forsyth and Vetzal (2019). The cumulative distribution function (CDF) for *WT* is computed using a Monte Carlo method. In our numerical tests below, we compare all strategies by fixing expected terminal wealth. Since we have closed form expressions for the mean, we determine the glide path parameters using a Newton iteration in order to enforce this condition.<sup>10</sup>

<sup>8</sup> Since the investor rebalances her portfolio discretely, insolvency could also occur if *L*max > 1 in the special case of the model where jumps are ruled out (*λ* = 0), i.e., the value of the risky asset follows geometric Brownian motion.

<sup>9</sup> More precisely, suppose that insolvency occurs at time *t*, i.e., *St* + *Bt* < 0. Letting *t*<sup>+</sup> be the instant after *t*, then *Bt*+ = *St* + *Bt* and *St*+ = 0.

<sup>10</sup> For example, we can exogenously specify *p*min and find the value of *p*max which generates the desired expected terminal wealth via Newton iteration. Alternatively, we can exogenously set *p*max and numerically find the appropriate value of *p*min.
