Figures 4-6 show how the changes in the parameter values affect the growth option exercise level VG, the risk-switching point before the exercise of the optionVS0, the default level before the exercise of the option
VB0, the leverage ratioLR, the managerial equity ownershipβ, and the agency costsAC. The dashed lines denote the results under thefirst-best scenario, and the solid ones denote the results under the managerial compensation maximization scenario. Unless otherwise noted, all parameters are base case configuration.
7.1
Option Scaling Factor
θ
Figure 4 shows the changes in the base case results when the growth option scaling factor θ is varied from 0.1to 1.0. In Panel A, observe that when θ is lower than 0.13, the managers exercise the option immediately. After the exercise, they employ the lowest-risk level. As such, the firm enjoys high tax benefits and low expected bankruptcy costs, but has no growth option. Although it is possible to motivate
the mangers to delay the exercise by giving them larger ownership in the firm’s equity, such a level of ownership induces them to take on too much risk. Because the size of the growth option is small, it is optimal to let the managers destroy the value of the growth option and keep low risk.
Whenθis above 0.13,VG increases withθ, because the value of the option becomes a significant part of the totalfirm value, so it is optimal to give the managers large enough ownership in thefirm’s equity so that they choose higherVG. However, notice thatVG in the managerial compensation maximization case is lower than that of the first-best case, indicating the overinvestment problem.
In Panel B, for small θ, VS0 in the managerial compensation maximization case is lower than that in the first-best case, indicating that thefirm under the control of the managers has much higher risk than the firm under the first-best scenario. The difference betweenVS0 in the two cases becomes smaller asθ becomes larger.
In Panel C, VB0 initially increases but finally decreases with θ, and because the wealth constraints binds, it is proportional to thefirm’s debt.
In Panel D, for θ<0.13, the leverage ratio is very high because thefirm gives up the growth option and takes on very low risk. However, for θ ≥0.13, leverage becomes much lower and is very similar to that in thefirst-best case.
In Panel E, for θ < 0.13, the optimal β is low because it is optimal to gives the managers small ownership in the firm’s equity so that they exercises the option immediately. However, forθ ≥0.13, the optimalβinitially decreases in order to reduce the incentives of the managers to delay the investment and take on high risk. But asθbecomes larger, the optimal β increases in order to motivate the managers to delay the exercise of the growth option and take on higher risk.
In Panel F, forθ<0.13, the agency costs are relatively high because the managers destroy the growth option by exercising the option immediately. For θ ≥ 0.13, the agency costs decreases with θ because the investment and risk management policies under the two scenarios converge as θ increases. In any case, the highest agency costs are only 0.8percent, indicating that thefirm value under the managerial
compensation maximization scenario is very close to that under thefirst-best scenario.
7.2
High Risk Level
σ
HIn Figure 5, we vary the high risk level σH from0.2to0.4, keepingσL constant. Panel A shows that the managers increase VG as σH increases. However, VG under the managerial compensation maximization case is lower than that in the first-best case, indicating the overinvestment problem.
In Panel B, for a low value of σH, VS0 in the managerial compensation maximization case increases with σH, but for a high value ofσH, it decreases withσH. For a very lowσH, thefirm chooses very low
VS0 to increase the option value. AsσH increases, the marginal costs of bearing risk become high, so the firm increaseVS0to reducefirm risk. However, asσHbecomes much larger, the value of the growth option increases significantly, and the marginal costs of bearing risk become low, so the managers lowerVS0 to increase risk.
Panel C shows thatVB0is a decreasing function ofθ. Because the wealth constraints bind,VB0reflects the leverage ratio.
Panel D shows that the leverage ratio in the managerial compensation maximization case is very similar to that in thefirst-best case, suggesting that thefirm in both scenarios has similar risk.
In Panel E, for a lowσH, the optimalβ is high. This is because there is little room for the managers to influencefirm risk, so the only role the optimalβ plays in this situation is to motivate the managers to exercise the option optimally, so it needs to be high to counter the overinvestment problem. As σHgets larger, the marginal costs of increasing risk are larger than the marginal costs of reducing risk, so the optimal β decreases to motivate the managers to lower firm risk. However, as σHbecomes very large, increasing risk becomes less costly, so the optimal β increases.
Panel F shows that the agency costs are generally very low; the highest are only 0.22 percent. A small discrete jump at σH = 0.22occurs because, for lower σH, the managers behave like equityholders
by choosingVS1=VB1, but for higherσH, the managers behave like the agents in the first-best scenario by choosingVS1=∞. This creates a divergence betweenfirm risk in the first-best and in the managerial compensation maximization case, which results in relatively high agency costs.
7.3
Payout Rate
δ
In Figure 6,δis varied from0.01to0.10.Panel A shows that the managers decreaseVG whenδincreases. Observe that they overinvest in the growth option, as VG in the managerial compensation maximization case is lower than that in the first-case. However, overinvestment decreases withδ.
In Panel B,VS0declines withδin both cases, but in the managerial compensation maximization case,
VS0is lower than in thefirst-best case, indicating that thefirm controlled by the managers has higher risk. In Panel C, the default threshold VB0 declines withδ in both cases. Because it is the level at which the wealth constraints bind, it reflects the debt level.
In Panel D, leverage declines withδin both cases, reflecting decreasingfirm risk. However, the leverage in the managerial compensation maximization case is slightly lower than that in thefirst-best case.
In Panel E, the optimalβ increases withδ. Whenδ is low, the optimalβ is also low to motivate the managers to take on low risk. Whenδis high, it is optimal for thefirm to take on high risk, so the optimal β is also high.
In Panel F, the agency costs initially increase withδ for a lowδ, but eventually declines withδ for a high δ.However, the agency costs are very low in general, with the highest being only0.12percent.
7.4
Managerial Equity Ownership
β
In Figure 7, we assume that the managerial equity ownership β is determined exogenously by factors outside our model and then explore how VG, VS0, VB0, the leverage ratio LR, the yield spread Y S, and the agency costsAC change in response to the changes inβ.
Panel A shows that whenβ is lower than 20percent, the managers overinvest in the growth option. However, as β increases, the managers’ incentives to overinvest decrease, as shown by an increase inVG. Whenβ increases beyond20percent, the mangers behave like equityholders and underinvest in the growth option.
In Panel B, when β is 0, the managers’ choice of VS0 is 0. This is because when β is very low, the managers have incentives to exercise the option very quickly and thus destroy the value of the growth option, so thefirm’s debt level must be low to make the managers’ compensation sensitive to a change in thefirm value. In this case, whenβ is0,the optimal leverage should also be0.Since thefirm has virtually no debt, it has no default risk. The managers in such afirm chooseVS0= 0in order to increase the value of the growth option and their own compensation.
Whenβ is between0and5percent, the managers’ incentives to overinvest become weaker, so it is now optimal to let thefirm take on higher debt to take advantage of tax benefits. However, higher debt brings higher default probability, so the managers increaseVS0 in order to protect his ownfixed payments.
When β is between 5 and 23 percent, the managers’ interests are more aligned with those of the equityholders. Consequently, they lowerVS0to increasefirm risk. However, they still use risk management sinceVS0> VB0.
Whenβis above23percent, the managers stop using risk management altogether and always maintain the high risk strategy, that is, they chooseVS0=VB0.
Panel C showsVB0 as a function of β. The default level is proportional to thefirm’s leverage because the wealth constraints bind in both cases.
In Panel D, the leverage ratio under the managerial compensation maximization case is0whenβ is0. Whenβ is between0and 5percent, it increases withβ.
Whenβ is between5and 23percent, leverage decreases with β, reflecting that the managers employ an increasingly risky strategy. A small downward jump in leverage at β = 11percent is caused by the
change in theVS1from ∞toVB1.
Whenβis above23percent, the managers have incentives to take on so much risk that if default risk is to be optimally contained, leverage must be reduced much further. As a result, thefirm will have little tax benefits. In this case, it is better to increase leverage to keep the tax benefits high, and let the managers choose the highest risk strategy, that is, let them chooseVS0=VB0. Because the managers cannot choose a more risky strategy than this, the debt level indirectly controls the managers’ ability to increase risk. The optimal leverage ratio is25percent and remain constant at this level untilβ= 1.
In Panel E, for β close to 0, the yield spread increases sharply with β due to a sharp increase in leverage. The spread increases more gradually asβ approaches11percent. It then decreases slightly when β is between 11 and 23 percent. Whenβ is higher than 23 percent, the yield spread remains stable at about100 basis point.
Panel F shows that the agency costs are lowest whenβ is determined optimally at7percent and that they may increase substantially when β is not optimally chosen. Notice that the agency costs under the current parameter setting are much higher whenβ is very small than when it is very large.
7.5
Empirical Implication
Our model offers some important empirical implications that are related to corporate investment and risk management. First, the model suggests that, except for firms that are managed by equityholders, the larger the size of the growth option, the higher is firm risk. This is because when the size of the growth option is large, it has a large share in the total firm value, and it is optimal to keepfirm risk relatively high to protect the value of the option. Our results thus suggest that empirical tests of the determinants of hedging may want to consider the size of the growth option as one of the explanatory variables.
Furthermore, our model predicts that there should be a positive relation between the riskiness of the firm’s asset and the value of the underlying assets, that is, when the asset value is high, thefirm may want
firms should increase risk by taking bets infinancial markets only when they have comparative advantage.
However, in our model, firms increases risk by changing the volatility of the underlying asset to increase the value of the option to expand. In this respect, the positive relation between volatility of thefirm’s asset and the value the assets should be robust empirically even amongfirms that do not have any comparative advantage but have the flexibility to change the riskiness of their operation.
Next, the model indicates that the larger the size of the growth option, the smaller is the firm’s leverage. This negative relation is suggested also in Barclay, Morellec, and Smith (2003). Our model complements theirs in that it expands the condition under which the relation holds. In their model, debt causes underinvestment because it reduces free cash flow available for investment. Hence, the more the growth optionsfirms have, the higher are the underivestment costs of debt. Therefore, the optimal amount of debt declines with the number of the growth options. On the other hand, our model suggests that even iffirms can raise enough cash for investment by issuing additional equity without any costs, the negative relation between the size of the growth option and debt level still exists because firms with large growth option tend to have high risk, which depresses the optimal level of debt.
Finally, our model highlights the role of managerial equity ownership to counter mangers’ incentives to overinvest. Controlling for risk, we expect tofindfirms with potentially severe problem of overinvestment to have higher level of managerial equity ownership. Although, the model can generate the results that are consistent with the findings by Tufano (1996) and Schrand and Unal (1998) that hedging increases with managerial equity ownership, it demonstrates that the relation between firm risk and managerial equity ownership can be non-linear and cautions against a simple linear regression test for such a relation.