With the comparison between the first-best and second-best solutions of manager’s effort, from Equations (33) and (15), we can observe that the second-best action chosen is less than the marginal rate of return, i.e., αSB* <s1 =α*FB . Consistent with the analysis of Holmstrom (1979), the second-best action is strictly inferior to the first-best action. Note that observing from the relationship of
1
*
* (1 SB) s
SB = +β ⋅
α (Equation (29)), we can find that ceteris paribus, the less negative the optimal hedge ratio, the higher the optimal level of effort. That is, if the stockholder desires to motivate the manager to work harder, he can decrease the hedge ratio in the compensation package, ceteris paribus, but it depends in part on how risk averse the manager is. Furthermore, if the optimal hedge ratio becomes un-hedged (i.e., βSB* =0), the optimal level of manager’s effort satisfies αSB* = s1 where the marginal rate of return s1 is exactly equal to the first-best action derived in section 3.2. That is, when the stockholder totally gives the whole share to the manager, alike the company is “sold” to the manager, the manager bears all risk.
But the incentive problem is internalized, and thus, the manager will choose the optimal level of effort equal to the first-best action (i.e., αSB* = s1 =α*FB). Besides, we can verify the illustration of Equation (32) that if the stockholder designs a risk-free incentive contract (i.e., the risk-free hedge ratio β =−1), the optimal level of effort the manager chooses is zero.
As for the comparison between the first-best and second-best solutions of the hedge ratio, from Equation (32), the second-best hedge ratio will be greater than –1.
That is, under the situation where the stockholder cannot perfectly infer the action chosen by the manager, the resulting optimal hedge ratio (βSB* ) (i.e., the second-best solution) will not be the risk-free hedge ratio (β =−1)(i.e., the first-best solution).
Otherwise, a risk and effort averse manager will have no incentive to work hard, and thus he will choose the minimal possible effort.
Finally, from Equations (17) and (34), the comparison between the first-best and second-best solutions of the fixed payment shows that the second-best fixed payment depends on more complicated factors, such as the minimum expected output without any effort, the cost of debt, the setting of the futures price of stocks, and the situation of the managerial labor market.
5. SUMMARY
In the Principal-Agent theory, the firm is considered as a nexus for contracting relationships, and does not have the unique objective for maximizing the company’s profit. Contrarily, both sides of the contracting relationship pursue self-utilities maximization, and therefore, this agency relationship breeds what is called the
“agency problem”. And further, the source of agency cost is that the agent would like to transfer the principal’s wealth to himself through his dysfunctional behavior.
The goal of the research attempts to explore another type of incentive mechanism to make the objectives of both sides coincident, so as to maximize the
firm’s profit. What the mechanism is to design incentive contracts involving a set of derivative financial commodities (e.g. stock futures), in light of the stockholder, in order to motivate the manager to choose the optimal level of effort. Specifically, we design an incentive contract involving “hedged futures”, putting emphasis on the hedge for the manager pursuing self-utility maximization. This emphasis is resulting from considering that a risk- and effort-averse manager trades off benefits from decreasing efforts with losses from taking risk when pursuing self-utility maximization. Thus, a rational manager prefers “utility-maximization hedge”, not just “risk-minimum hedge”. Constructing from Holmstrom’s (1979) framework, we derive the manager’s optimal level of effort, optimal hedge ratio, and the optimal fixed payment under the incentive contract involving “hedged futures”, including first best solution and second best solution. Following linear-exponential-normal (LEN) formulation of Holmstrom and Milgrom’s (1987) framework, we investigate the comparative static analysis after deriving the optimal solution, and obtained many meaningful economic implications.
The results show that under the situation where the stockholder can perfectly infer the manager’s action, the optimal risk sharing contract can be effectively achieved by two ways. First, the optimal contract pays the manager a fixed payment to a minimal acceptable level of wage, W~ =F*=W
. Second, the optimal contract can be designed as a risk-free incentive contract (if and) only if the manager chooses the action desired by the stockholder. This risk-free incentive contract
{
FFB* =W+T,βFB* =−1}
can be effectively realized when the manager will be penalized substantially given any action selected other than the first-best. These two contracts both can achieve the optimal risk sharing since a risk neutral stockholder doesn’t not care about bearing all the risk, while a risk averse manager prefers bearing no risk, ceteris paribus.Moreover, with LEN formulation, the results indicate that given the contract offered, the first-best action chosen by the manager is equal to the marginal rate of return (i.e., αFB* =s1), and the optimal contract derived satisfies {βFB* =−1,
T s W
FFB = + 12 +
* ˆ (1 2) }. That is, analogous to the analysis without LEN form: first, the optimal contract can pay the manager a constant, ~* ˆ (1/2) 12
s W
W = + , the
function of the certainty equivalent of the manager’s reservation utility (Wˆ) and the marginal rate of return (s1), which is independent of the output and the manager’s effort. Second, the stockholder can design a risk-free incentive contract
{
FFB* ,βFB*}
involving the hedge ratio, βFB* =−1 (if and) only if the first-best action has been chosen. Besides, the optimal fixed payment FFB* to the manager depends on the certainty equivalent of the manager’s reservation utility (Wˆ), the marginal rate of return (s1), and the futures price of stocks (T ). When the certainty equivalent of the
manager’s reservation utility (Wˆ) or the futures price of stocks (T ) heightens, the optimal fixed payment compensated to the manager will build up. In addition, when the marginal rate of return contributed from manager’s effort (s1) raises, the optimal fixed payment to the manager will increase to motivate higher efforts and meanwhile, the optimal level of manager’s effort will also increase. This risk-free incentive contract
{
FFB* ,βFB*}
can be effectively achieved when the manager would be penalized substantially given any action selected other than the first-best, i.e.,1
* s
FB =
α .
On the other hand, under the situation where the stockholder cannot perfectly infer the manager’s action, if the stockholder would like to motivate the manager to increase his effort, as discussed by Holmstrom (1979), the optimal contract trades off the benefits of imposing risk to provide sufficient incentives with the cost of imposing risk on a risk averse manager. Furthermore, with LEN formulation, the results illustrate that the resulting optimal hedge ratio (βSB* ) (i.e., the second-best solution) will not be the risk-free hedge ratio (β =−1), depending on the coefficient of absolute risk aversion (c), the marginal rate of return (s1), and the variance of the distribution of the output (σ2). In addition, the optimal second-best action depends on the coefficient of absolute risk aversion (c), the marginal rate of return (s1), and the variance of the distribution of the output (σ2). Further, consistent with the analysis of Holmstrom (1979), the second-best action is strictly inferior to the first-best one (i.e., αSB* <s1 =α*FB ). However, if the stockholder “sells” the company to the manager, the incentive problem is internalized, and thus, the manager will choose the optimal level of effort equal to the first-best action (i.e.,
* 1
*
FB
SB s α
α = = ).
With the comparative analysis, the results indicate that when the marginal rate of return (s1) increases, the coefficient of absolute risk aversion (c) decreases, or the variance of the distribution of the output (σ2) decreases, the optimal level of effort selected by the manager will rise. Further, when the marginal rate of return (s1) increases, the optimal hedge ratio (βSB* ) will become less negative (i.e., shift toward un-hedging). This is consistent with Holmstrom’s (1979, corollary 1) discussion that when the marginal rate of return increases, the stockholder will provide more incentive to induce the manager to work harder. When the coefficient of absolute risk aversion (c) or the variance of the distribution of the output (σ2) increases, the optimal hedge ratio (βSB* ) will become more negative (i.e., shift toward risk-free hedge ratio). That is, when the manager is more risk averse or faces more uncertainty, the stockholder needs to reduce the risk of the compensation package, such as increasing the hedge ratio in the risky portfolio.
Besides, the resulting optimal fixed payment (FSB* ) depends on the coefficient of absolute risk aversion (c), the variance of the distribution of the output (σ2), the marginal rate of return (s1), the minimum expected output (s2), the debt cost (D), the futures price of stocks (T ), and the certainty equivalent of the reservation utility (Wˆ). When the minimum output (s2) decreases, the debt cost (D) increases, the futures price the company specifies (T) increases, or the certainty equivalent of the reservation utility (Wˆ) increases, the stockholder needs to compensate the manager higher fixed payment to ens ure that the manager can obtain exactly the reservation utility. Furthermore, our result notes that ceteris paribus, the less negative the optimal hedge ratio, the higher the optimal level of effort. That is, if the stockholder desires to motivate the manager to work harder, he can decrease the hedge ratio in the compensation package, ceteris paribus, but it depends in part on how risk averse the manager is.
In this paper, we follow Holmstrom and Milgrom’s (1987) approach and introduce linear-exponential-normal (LEN) formulation since it’s a more tractable formulation of the agency model. In the future research, it could be extended to other types of formulation to investigate the differences between various formulations. With regard to applying derivative financial commodities in incentive contracts for solving agency problems, the comparison between various derivative financial commodities could be done in the future research, such as options and futures. Particularly, because futures should be exercised at the expiration date, but options don’t have to be exercised at the expiration date, it may be possible that the optimal level of effort the manager chooses may be higher under an incentive contract involving “futures” than that involving “options ”. This could be strictly explored in the future research.
APPENDIX
Here, we shall illustrate that the risk-free effect of the incentive contract involving “hedged futures”. Definitions and assumptions are the same as those in section 3.1. Particularly, the manager’s compensation package is assumed to be
P F
W~ ~
+
= where P~
is the cash flow of the risky portfolio, and F is the fixed payment. Let the cash flow of the risky portfolio be P~ B~ S~
β +
= , where B~
is the gain from futures, β is the hedge ratio, and S~
is the end period of stock value.
Further, the gain from futures is assumed to be B~=S~−T
, where T is the futures price of stocks. And the end period of stock value is assumed to be S~(α)=u~(α)−D in which D denotes the debt cost, which indicates that the end period of stock value is equal to the whole company’s value after paying the money borrowed. And
note that the number of company’s stock is assumed to be one share.
Substituting these definitions into the manager’s compensation package, we can re-express the compensation package as:
{ }
[ ] (1 )( )
W% = + = + +F P% F B% βS% = +F +β u%−D −T
That is, as discussed by Hemmer (1993), the compensation package can be expressed as risky portfolios related to the company’s output for providing incentives combined with a fixed payment that can be chosen to guarantee a minimal acceptable level of expected compensation. Note that such a contract involving “hedged futures” can be re-expressed as a linear function of the output.
In addition, when β =−1 is held, the compensation package paid to the manager is the fixed payment, i.e., W~ =F
, in which involves no risk. Therefore, such β is called the risk-free hedge ratio, and thus, an incentive contract involving the hedge ratio β =−1 is called a risk-free incentive contract.
Moreover, under linear-exponential-normal (LEN) formulation defined in section 3.2, more economic implications can be derived. Specifically, the output u~
is assumed to be a normal distribution and for simplifying computation, the expected value of u~ is assumed to be a linear function of the manager’s efforts.
Mathematically, the output function is expressed as u%=s1α+ +s2 ε% where s1 represents the marginal rate of return from the manager’s efforts, s2 represents, on average, the minimum output of the company when the manager makes no efforts, and ε~ denotes the state of nature with zero mean. Let σ2 denote the variance of u~, i.e., ε~~N(0,σ2) and thus, u~~N(s1α s+ 2,σ2). Note that the manager’s action is assumed to have an impact on the expected value but not the variance of the output.
As described above, one unit cash flow of risky portfolio can be re-expressed as:
( ) (1 )( )
P u% % = +B% βS%= +β u%−D −T (A.1) where E P u % %( ) (1 = + β)(s1α +s2 −D)−T (A.2) Var P u% %( ) = +, (1 β)2×σ2 (A.3) The risky portfolio is a linear function of the observed output, where the manager’s effort affects the mean but not the variance of the risky portfolio. With the discussion of Equations (A.2) and (A.3), we can investigate the range of the hedge ratio. Here, as described in section 4.1, more effort is assumed to increase the
expected value of the portfolio, i.e., first-order stochastic dominance. Thus, by differentiating the expected value of the portfolio (Equation (A.2)) with respect to manager’s effort, the resulting first-order condition satisfies(1+β)⋅s1 ≥0.
Since the marginal rate of expected return is assumed to be increasing (i.e., 0
)
( >
′α
u ), the marginal rate of return (s1) is positive with the result that the hedge ratio is no less than -1 (i.e., 1+β ≥0). Further, observing the variance of the portfolio (Equation (A.3)), we can obtain the range of hedge ratio is from 0 to -1.
That is, the stockholder can provide an incentive contract involving the hedge ratio ranging from un-hedging (i.e., β =0) to, at most, the risk-free hedge ratio (i.e.,
−1
= β ).
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