Non-Recombining Trinomial Tree

We will now present another possible, and in fact implementable solution to the shortcom-ing of our proposed model in relation to systematic uncertainty in time. More specifically, we will introduce a way of modeling incremental systematic uncertainty with time, through a non-recombining trinomial tree, which is equivalent to modeling systematically dependent cash flows. Furthermore, it is consistent with our proposed model, as cash flows have a beta of zero within states. To get an idea of the underlying intuition behind this extension, we begin by considering some informal arguments, before giving an example through a simulation of the process that the market follows in our proposed model. We then present a concrete valuation using the extension, which builds upon the stylized example from section 8.2.1.

Consider the fact that the amount of systematic risk in a cash flow is determined by its co-movement with the market. In many cases, it then makes sense that for any given initial cash flow value, the level of the cash flow over time, should evolve with the market’s level over time.

Consequently, should the market take on a path in time where it reaches strikingly low levels, so should the cash flows (at least if they are risky). For example, if the market experiences threeLow states in a row, then for many projects, the cash flow received in the thirdLow state will be lower than the cash flow that would have been received if the market had experienced two Mid states and then one Low state. To be more concise, we are saying that the level of the cash flow in a given state and in a given year, is not only determined by the market state, but also by the level of the cash flow in the previous year. Hence, the evolution of the market can determine the evolution of the cash flows, and cash flows are thus systematically dependent.

Furthermore, consider how uncertainty might be related to time. It is not unreasonable to think that we will have a wider distribution of possible levels of the market for points in time far out into the future, than for points of time in the near future. In fact, if we assume that the market evolves like a geometric Brownian motion (GBM), the notion of a widening distribution for the market is mathematically consistent, since the volatility of the natural logarithm of the market’s level is proportional to the square root of time. This can be proven mathematically or simply seen by conducting many simulated paths of a Brownian motion (or GBM). However, the process that the market follows in our proposed model is not an ordinary GBM, it is a regime shifting stochastic process. More specifically, we have that each year, the market can change drift and volatility parameters depending on what state we are in (which is defined by the backward looking 1-year holding period return). In other words, the market switches between three GBMs that have different volatility and drift parameters. Hence, to give an accurate example of how uncertainty increases with time (assuming the market evolves like in our model), we will simulate a large number of possible paths for this process.

To do so, we use an approach that is somewhat similar to the simulation approach that we used

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over 10 years. For any given path, we assume that we are in theMid state at time zero, with an initial value of the market index of 100, and simulate the first year as GBM with the conditional parameter estimates corresponding toMid from table 5.5. We use the estimatedntdparameter from section 5.1.2 as the number of time steps per year, such that one time step is∆t= 1/257 years. We then use the 1-year backward looking simulated holding period return to define what state we are in (similar to what we did when subsetting the data in section 5.1.4). The next year of our stochastic process is then simulated as a GBM with drift and volatility parameters corresponding to the defined state. This process is reiterated until we have simulated 10 years of the process. The procedure is described in detail in appendix .1 and .2.

After simulating 10,000 paths using the algorithms described above, we plot the paths together with the 5th, 50th and 95th percentile paths throughout time. The percentile paths are calcu-lated by finding the respective percentile of all the paths at every point in time and aggregating this into one ’percentile path’. We plot the paths in figure 8.7, where we also plot the logarith-mic index values. We plot the latter to give a normalized view on the simulated data, as the level of the paths tend to deviate substantially with time.

Figure 8.7: 10,000 simulations of the regime shifting stochastic process. The red lines indicate the 5th and 95th percentiles respectively. The green line indicates the 50th percentile. The plot on the LHS show the index values through time, while the plot on the RHS shows normalized index values through time (lognormal). The index value at time zero is 100

As can be seen from the plot on the left-hand side of figure 8.7, the paths form a pattern that resembles a lognormal distribution (or perhaps a mixture lognormal distribution). Furthermore, we can see that the width of the distribution increases with time. This shows that uncertainty is in fact increasing with time for the underlying process that we assume that the market follows

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in our model. Moreover, we can see that the positive drift causes there to be certain extreme positive deviations, i.e. paths which almost explode upwards. However, by examining the per-centile paths, we can see that there seems to be relatively larger concentration of mass in the left tail of the distribution. Considering the results from section 6.3.3, this should be expected.

The high volatility and negative drift in the Low state, makes it relatively more probable to remain inLow than to transition toMid orHigh. The larger concentration of mass can be seen more easily from the normalized plot on the right-hand side in figure 8.7. Here we can also see the kink in the 5th percentile path after year 1 more clearly. The kink results from the fact that in the 5th percentile, we have transitioned to the Low state from the Mid state, which causes more abrupt and downward skewed movements.

Nevertheless, if the distribution of the market becomes wider with time, then according to our previous argumentation, it is natural that for many projects, the cash flow distribution also becomes wider with time. Hence, we aim at providing an extension that can incorporate this argument in the pricing of an asset. To do so, we take inspiration from the fact that it is possible to create a binomial approximation of Brownian motion. As already mentioned, we introduce a non-recombining trinomial tree. The intuition behind is as follows: If we have three possible states to transition to at every time step, then for every additional time step we add into our non-recombining trinomial tree, the number of paths increase. Specifically, the number of paths will be 3^t, where t is the number of time steps in our tree. Because the amount of High and Low transitions will grow exponentially with t, we have that the further we go out into the tree, the lower our lowest market levels will be, and the higher our highest market levels will be. This is similar to the growing width of the path distribution seen in figure 8.7.

Now consider the following, if we let our cash flows be systematically dependent, such that they grow by some amount when transitioning to High, decline by some amount when transitioning to Low, and perhaps do not change when transitioning to Mid. Then the further we go into the future, the higher the highest possible levels of our cash flows will be and the lower the lowest possible levels of our cash flows will be. To see this more clearly, consider the cash flow projections on the left-hand side of figure 8.8.

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t=0 t=1 t=2 t=3

40 60 80 60 80 100

80 100 120

t=0 t=1 t=2

60 114.8

80 60 154.6

100 80 194.4

100

80 80 80 266.8065 223.9 154.6

100 100 100 266.8065 283.302 194.4

120 120 120 266.8065 342.704 234.2

100

100 120 194.4

120 140 234.2

140 274

80 100 120 100 120 140 120 140 160

Present Value Calculations Cash Flows

t=0 t=1 t=2

t=0 t=1 t=2 t=3

Figure 8.8: Valuation in a non-recombining trinomial tree

As can be seen from figure 8.8, the distribution of future cash flows is widening with the number of time periods, i.e. together with the distribution of the future market. Also, notice that the initial cash flows are the same as in the example from section 8.2.1. However, in this example, our cash flows increase with 20 if we transition to High, remain unchanged if transition toMid, and decrease with 20 if we transition to Low. We note that the ’rules’ we have used for our projections are made to keep the example as simple and tractable as possible. However, in principle, one could most certainly model more complex and different systems of future cash flows.

Even though we have an increased variation in cash flows with time, it is not necessarily clear what implication this has for the present value of a project. Hence, we investigate further.

Let us first begin with explaining how the present value is calculated in our non-recombining trinomial tree. An easy approach to doing so is to relate back to the real option example in section 8.1. Here, we worked our way backwards through a system when calculating present

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values. We can do exactly the same for our trinomial tree. To see this, consider the two grey boxes on the right-hand side in figure 8.8. The uppermost box illustrates the calculation of the present value of our project if we stand inLow at time 2, after being in Low at time 1. Firstly, we can see that we have a vector with the value 80 for all elements. This vector denotes the cash flow received in Low at time two. Note that the only reason the cash flow is organized in vector form is because we would like to compute the total present value (current cash flow + PV of future cash flows) through matrix algebra. Furthermore, we calculate the present value of the three possible cash flows we can receive at time 3, which are determined by where we stand today. After adding the current cash flow to the present value of the future expected cash flow, we obtain a vector with three present values¹. We emphasize that since we are currently standing in Low, only the Low present value should be considered. This present value is then stored in the new Low time 2 node, which can be seen in the lowermost of the grey boxes (in red). If we repeat this exercise for all possible time 2 states, and work our way backward in the tree (i.e. doing the same for time 1 and 0), we arrive at the present value for the project.

Now that we have clarified the present value calculation in our trinomial tree, we continue by comparing the present values from figure 8.8 with present values calculated by the same method as in section 8.2.1, i.e. the method that is used in the unextended version of our proposed model and which Banz & Miller use. Note that the present value in the trinomial tree is calculated by using the same state price matrix as in 8.2.1. Moreover, note that the cash flow evolution in the trinomial tree is defined such that the conditional cash flow expectation of a state at any point in time equals the state-contingent expected cash flows from 8.2.1, i.e. 80, 100 and 120.

This can be easily seen by taking the average of all cash flows for a state at a given point in time. As can be seen from table 8.1, even though the cash flow expectations are the same, the valuation using the trinomial tree has a higher risk-adjustment.

Trinomial Tree State Price Matrix

P VLow 266.8065 280.3774

P V_{M id} 266.8065 280.3774 P VHigh 266.8065 280.3774

Table 8.1: Comparison of present values calculated using a non-recombining trinomial tree and a standard state price matrix

To provide some explanation of the mechanics behind the higher risk adjustment, we show how the two most ’extreme’ paths are discounted, i.e the two scenarios of only High transitions or only Low transitions in all of the three years. We start by using the state price matrix from our examples (where 0.43 is the state price for the Low states and 0.22 is the state price for the High states), to show the discount factors for the extreme scenarios.

1Note that the present values are equal for all departure states in the constructed example. This is because we did not include any state preferences in the state price matrix used in the example. This is simply done to make the analysis easier. Moreover, it ensures that the expected cash flows in the example remains unchanged when we move further into time since the probabilities are always 1/3

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Discount Factor (onlyLow states) 0.43³ = 0.0911 Discount Factor (onlyHigh states) 0.22³ = 0.0106

Table 8.2: Discount factors for extreme scenarios

As can be seen from table 8.2 the discount factor is higher for the extreme Low scenario, than the extreme High scenario. This is equivalent to a lower discount rate for the extreme Low scenario than for the extremeHigh scenario, which is rather intuitive, as we already know that the Low states have higher state prices (less discounting) than the High states. In our non-recombining trinomial tree, this effect is compounding with time, which is analogous to any other DCF approach. Note however, that this effect is modeled through our cash flows and we deliberately model the cash flows so that the expected cash flow remains unchanged (always 100) over time. This allows us to conclude that the reduced present value in the example is in fact due to the additional discounting because of increased systematic risk with time. More specifically, it is achieved through letting the cash flows evolve in the manner that we described above. This means that depending on how cash flows are modeled, their systematic uncertainty can be increasing, decreasing, both, constant, and/or non-linear.

Usefulness in Applied Capital Budgeting

The non-recombining tree allows for flexible and dynamic modeling of cash flows, which can be useful for applied capital budgeting purposes. Firstly, by letting the cash flows evolve with some factor (or constant) depending on the market states, we allow for increasing systematic variation in cash flows with time. This is likely to be the case for many assets. Secondly, the tree allows for very specific scenarios contingent on the paths of the market, i.e as introduced with the real option example. However, a drawback to the tree is the fact that it probably has to be implemented using a computer algorithm for more than three periods. Moreover, we have no closed form solution for the terminal value of a project. Therefore, one would have to set some sort of threshold where one stops growing the tree, i.e. where the present value of an additional cash flow is close to zero, to calculate the value of a perpetual cash flow stream. This is not technically difficult, but it does impede the applicability of the model, as it becomes less tractable for the average practitioner. On the other hand, we believe that in today’s computerized world, this is less of an issue. After all, practitioners already use complex tools in their daily day activities, such as when estimating market betas with regression software packages.

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