información General, descripción de operaciones y desarrollo
ACTOS POSTERIORES A LA CONSTITUCIÓN DE LA EMPRESA
8.3.1
Setup
Given the characterizations of the problem, the starting point is to calculate the shareholder’s payoff given that he has already decided to hire a manager. In this situation the payoff to the shareholder depends on two stochastic vari- ables, the level of the cashflow (Yt) and the level of the manager’s wealth (Wt).
The manager’s wealth determines the level of effort he will exert and thus the eventual investment cost. In this sense the investment cost is stochastic from the shareholder’s point of view. Thus there are parallels between this and the real options analysis of projects where the cash flow and investment cost are stochastic.6
Once the manager has been hired the shareholder waits for the manager to invest. Given that shareholder’s value function now depends on the man- ager’s wealth, it can be derived using the same methodology used for the manager’s problem. Following the same procedure used for the manager, Equation (3.8) becomes the following
S(W, Y, t) = e−βdtE[S(W (t + dt), Y (t + dt), (t + dt)] (8.1)
5When the shareholder hires a manager, he gives up the value of the project if he
managed it himself.
where this differs from Equation (3.10) in that the shareholder’s consumption and/or investment do not affect his valuation of the cash flow and he is not maximising the function himself. The general value function (Equation (3.9)) thus becomes7 βS(W, Y, t) = E[Stdt + SwdW + SydY + 1 2SwwdW 2+1 2SyydY 2+ S wydY dW ] (8.2) The interesting feature here is that Wt is the manager’s wealth and thus dW
is defined in exactly the same way it was in the manager’s problem. Therefore the manager’s consumption and (portfolio) investment decisions impact the evolution of the Wt and thus the payoff to the shareholder. Substituting
in dW and dY from Equations (3.4) and (3.7) and simplifying gives the differential equation that the shareholder’s payoff from hiring a manager must satisfy: βS(W, Y, t) = St+ Sw(rW + π (µm− r) − C) + Syµy +1 2Sww(πσm) 2+1 2Syy φ 2 + ρ2σ2 y + Swy(πρσyσm)) (8.3)
We already know from Equation (6.2) that the shareholder’s payoff when the manager invests is PM[α] = (1 − α) Y r + µ − ρσyΦ r2 − I[ˆe] (8.4) Therefore if the manager invests the shareholder receives the payoff defined by Equation (8.4) , otherwise he receives the solution to Equation (8.3). This can be represented formally as
SM(W, Y, t) = (
PM[α] if α(G[Y ] − I[ˆe]) ≥ VRO(W, Y, t)
Solution to (8.3) if α(G[Y ] − I[ˆe]) < VRO(W, Y, t)
7Note that this is not technically a Bellman equation since the shareholder does not
We have already solved PM[α], G[Y ], ˆe and VRO(W, Y, t) when examining
the manager’s problem in Chapters 4, 5 and 7. Therefore all that is required to calculate SM(W, Y, t) is to solve the differential Equation (8.3).
8.3.2
Solution
Because Equation (8.3) does not involve any policy functions that must be solved, the problem can be solved using standard finite difference methods. Given that the technique is standard, it is not reproduced here. Before continuing it is worth repeating the parameters that we will use as the base case for this section.
Table 8.1: Modified base case parameters for Chapter 8
γ = 1 r = 0.1 µm = 0.15 µy = 0.1 A = 100 λ = 1 W = 1 κ = 0.1
β = r σm = 0.1 σy = 0.3 ρ = 0.0 B = 80 θ = 0.1 Y = 10
Note that unlike the rest of Part II, we are assuming that σm = 0.1. This
is because we are interested in examining the implications of the investment “spike”.
Figure 8.1 plots the shareholder’s value function in the first date (t = t1 = 0)
as well as the date at which the investment option expires (t = tn = 10).
We can see from Figure 8.1 that the standard real options intuition still holds in that the shareholder’s payoff is smoothed out when delay is possible. However, one has to keep in mind that the manager is making the decision for the shareholder. The obvious distinction between the two graphs is that when t = tn the graph is kinked as opposed to being relatively smooth.
This occurs because there is no longer any opportunity to delay investment and thus the project value is simply the payoff from immediate investment.8 Conversely, at the first date, where the manager still has significant scope to
delay investment, the shareholder receives value from the manager delaying the investment decision. This is evidenced by the smooth shape of the graph along the Y axis. The reasons behind the smoothness is that even when the project would have a negative immediate payoff, there is value in waiting to see if the project improves. The graphs also have the expected shape when one looks along the W axis in that the payoff increases as W decreases and then hits an asymptote. This is due to the facts that the manager’s effort is higher for a lower W and that there is a limit on how much effort can reduce the investment cost.
Figure 8.1: Payoff from hiring a manager (SM(W, Y, t)) at t = t1 and t = tn
(a) t = t1= 0 (b) t = tn= 10
Calculated using the base case parameters
It is important however to note that the undiversified manager is choosing an investment policy that maximises his utility. This will not necessarily co- incide with the policy that would maximise the market value of the project rights. In fact, it turns out that in some situations the manager waits too long to invest from the shareholder’s perspective - that is, the manager waits when investing immediately would give the shareholder a higher payoff. This is illustrated by comparing the shareholder’s payoff from hiring a manager (SM(W, Y, t)) and the payoff to the shareholder if the manager invested im-
mediately (PM[α, Y ]). This comparison is shown in Figure 8.2
Figure 8.2: SM(W, Y, t) vs PM[α, Y ] and PCAP M[Y ] vs FCAP M(Y, t)
6 7 8 9 10 11 12
-20 20 40
Calculated using the modified base case parameters and W = 0. The solid blue line represents the payoff from hiring a manager (SM(W, Y, t)), the dashed green line represents the payoff to the shareholder if
the manager invested immediately (PM[α, Y ]), the solid red line is the CAPM valuation of the project rights (FCAP M(Y, t), i.e. option) and the dashed brown line is the CAPM valuation of the cash flow
(PCAP M[Y ]).
In Figure 8.2 the solid blue curved line represents SM(W, Y, t) and the dashed
green line represents PM[α, Y ]. For comparative purposes we have also in-
cluded the CAPM valuation of the cash flow (PCAP M[Y ]) as the dashed
brown line and the CAPM valuation of the project rights9 (FCAP M(Y, t)) as
the solid red curved line. We can see that from just after Y = 7 the payoff from the manager investing is greater than the payoff to the shareholder from the manager delaying. Therefore in this situation the shareholder would be better off if the manager invested immediately. Even though he would prefer that the manager invested in this situation, he may still be better off than if he managed the project himself. This is illustrated by the fact over the range of Y we examine, SM(W, Y, t) exceeds FCAP M(Y, t) .
The fact that the shareholder’s payoff function (SM(W, Y, t)) is actually below the payoff from immediate investment (PM[α]) for a wide range of Y , while
not unexpected, is interesting in and of itself. Figure 8.2 is plotted at W = 0 which under the base case parameters is in the area where the manager’s investment “spike” occurs (see Chapter 7). Recall that in the “spike” region the manager delays investment beyond what a standard real options model predicts. Thus it makes sense that in this region the manager is delaying investment beyond what is optimal from the shareholder’s perspective. To confirm this logic we re-plot Figure 8.2 for values of W outside of the “spike” region. Figure 8.3 does so for W = −50 and W = 50. The solid dark blue line is SM(−50, Y, t) while the solid brown line represents SM(50, Y, t). Given that
the manager’s optimal effort is a decreasing function of his wealth, it is not surprising that we find that SM(−50, Y, t) > SM(50, Y, t). More importantly
we find that in both cases the shareholder’s value function from hiring a manager does not fall below the immediate payoff from investing. Thus for values outside of the “spike” region the manager doesn’t appear to delay investment beyond what is optimal from the shareholder’s perspective. It is also easy to show that as other parameters are changed in a way that reduces the spike10 we get a similar result. This occurs because the smaller the spike
is, the less likely the manager is to delay investment beyond what is optimal from the shareholder’s perspective.
It is useful to recall the results of Miao and Wang (2007) at this point. They found that relative to the risk-neutral model, investment would be delayed but that this is a second order effect. Our model is slightly different in that the point of comparison is a shareholder who uses the CAPM and incurs a monitoring cost. The assumption of the CAPM does not make a significant difference and thus (ignoring the monitoring cost) when W is very large or very small the effort decision is no longer a factor and thus we should obtain the same results as their model.
The only direct comparison that can be made between the shareholder’s in- vestment threshold absent a manager and the manager’s investment thresh-
10For example setting A = B or increasing σ
Figure 8.3: SM(W, Y, t) vs PM[α, Y ] for W = −50 and W = 50 6 7 8 9 10 11 12 Y -20 20 40 Payoff
Calculated using the modified base case parameters. The dark blue line is SM(−50, Y, t) while the green
line represents SM(50, Y, t).
old is when W is very large and thus the manager exerts no effort. This ensures that the investment cost is the same for the shareholder and the manager. For the base case parameters we find that the investment thresh- olds are the same for the manager and the shareholder. However, if the project’s volatility (σy) is increased we get the second order effect of Miao
and Wang (2007) in that the manager delays investment beyond what the shareholder would. In addition, if the monitoring cost (κ) is increased (which the shareholder bears but the manager does not), the manager invests later than the shareholder would in the absence of a manager. Intuitively, the monitoring cost makes the shareholder want to invest earlier to avoid the cost of monitoring, while idiosyncratic volatility makes the manager want to invest later for the reasons outlined in Miao and Wang (2007).