Impact of Dual Stock Holding and Stochastic Income on the Investor’s Remuneration Package ()
1. Introduction
In this paper, we derive the return for an investor who is rewarded with company stock number of
units for managing two non-traded geometric Brownian motion risk assets
and also receives stochastic income which accrues at the rate
, by trading in options. We want to investigate to what extent a stock based compensation for an investor who also earns stochastic income at time
that outperforms the strictly stock units compensation scheme.
This study was motivated by the work of Henderson [1] who examined a dynamic portfolio choice for an investor receiving a stream of income rate over a finite investment horizon. The income rate in [1] was stochastic and it was imperfectly correlated to the stock, and the investor could not replicate income risk with the stock and bond assets alone. This led to an investigation that how non-tradibility of the income affects optimal allocation of wealth between the stock and bond. Henderson [1] concluded that this approach allowed more flexibility as most of the parameters could be chosen without resorting to vigorous estimation techniques.
On the other hand, Choon and Manoyios [2], used a similar income rate as Henderson [1], examined the hedging of a stochastic stream in an incomplete market and concluded that when the income rate is modified by
, then there is reliability in the results.
Other authors who have studied the impact of stochastic income are; Kronborg [3], who considered the optimal consumption and investment problem for an investor endowed with a deterministic stochastic income. The others are Doctor and Offen [4] who incoorporated stochastic income and considered an optimal problem for an investor for different utilities. The study [4] described how an investor can adjust the Merton portfolio through an interpolating hedging demand, in reaction to the stochastic income. Wang, et al. [5] studied optimal consumption and savings with stochastic income subject to elastic inter temporal substitution, and concluded that higher risk aversion increases savings and lowers the consumption.
Other works that motivated the approach used in this paper are Chen [6] and Huisman, etal. [7]. Both ( [6] [7] ) applied the method of value matching and smooth pasting conditions to find the optimal investment threshold and the investment value function of an investor. As a sequel to Chen [6] and Huisman, etal. [7], we want to find the optimal investment thresholds of
for
, and the expected investment value function for an investor who is compensated with a number of company stock units
.
This paper is organized as follows: In Section 2 we formulate the model which is analyzed and simulated in Section 3. Section 4, we calculate the investors remuneration package.
2. The Model
Consider an investor who manages two company assets
assumed to evolve as;
(1)
where
is a standard Brownian motions,
are constants and
is a random jump-amplitudes. We interpret
to mean that whenever there is a jump, the value of the process before the jump is used to ensure that one jump does not make the underlying asset worthless, and
is an independent Poisson jump processes with jump rate
, that is,
describes the jump times of
, with
and we assume that the processes
and
are independent, implying that
, and
(2)
Suppose the investor is also endowed with a non-negative and non-traded continuous time income rate
at time t. The income rate is stochastic and state variable
evolves as follows:
(3)
where
and
is the correlation coefficient with
, and the process
is a Brownian motion defined on
probability space,
and
are constants. For
, the filtration
is not generated by
, implying that the market is incomplete and the claim cannot be perfectly hedged via
.
Remark 2.1. Further, we notice that, when
, then the stochastic income behaves like a riskless asset, and when
then the stochastic income is related to the stock.
However, the stochastic income defined in Equation (2) not bounded, as such the process
grows uncontrollably large. Therefore, we assume that it is bounded below at the stochastic income rate
, where
. For
, the investor receives an income stream and for
he receives nothing. This choice of
holds because the process
is continuous, non negative, allows no arbitrage and allows flexibility in modeling as most of the parameters can be estimated [8].
Remark 2.2.
· It is interesting to observe that the stochastic income can be thought as a future value payment or claim that pays the future value
at terminal time T; by taking the sum of the income stream which pays at a rate of
from time t to terminal time T.
· We also observe that n gives the weight to
, but does not change the behavior of the function itself. That is why our study differs from Choon and Manoyios [2] who defined
(We refer the reader to Appendix 9 for more information).
In the next section, we find the investors optimal investment threshold and investment value function.
3. The Investors Investment Value Function
Let
, defined in Equation (1) be company assets for which the investor is rewarded
units in each stock at terminal time
for the management of these stocks. He is not allowed to exercise the right to trade these units before
avoid inside trading and related arbitrage opportunities (see Kovaleva [9] ). Although the investor could under certain circumstances short sell these units, the units remain purely non-traded assets. The reward of assets is then embedded in a remuneration package to be determined later depending on performance.
The objective of this investor is to optimize the company stock value which ultimately increases his wealth through rewards of
units of shares. Because he cannot trade his assets he is subjected to risks such as market fluctuation and other company risks that affect the magnitude of the reward.
Let
(4)
be the total price of the investors stock at time t in the market, where
are the total stock output, a is a given constant,
are the company assets and
to ensure that
is positive. We denote the investment value function of the investor for the risk assets
by
.
Using the standard real option method, the Bellman equation for the value function can be expressed as
(5)
where we assume
to be a continuous time discount rate that ensures that stock is exercised within a finite period of time. Using Itos Lemma on Equation (5), we obtain
(6)
Note that
implying that if
ever goes to zero, the value functions remain zero. This removes arbitrage opportunities.
We look for a solution of Equation (6) of the type
(7)
where
are positive constants. Substituting Equation (7) into (6) yields the following characteristic equation for
:
(8)
Equation (8) simplifies to
(9)
and has solution:
McDonald and Siegel [10] have argued that the condition
ensures that
so that the solution is well defined, as such we ignore the other solution of
for
since it gives a negative solution.
We solve for the value function in Equation (6) subject to the following boundary conditions:
(10)
(11)
(12)
Condition (10) simply indicates that the value will be 0 if
, while condition (11) and (12) are the value matching and the smooth pasting conditions to ensure that
is optimal and
can be maximized when the investment is at the threshold
. The parameter
in Equation (11) is a unit cost and
is the total cost with respect to the stock output.
Note that for a positive value function
and
we obtain
. In simple terms the value matching condition can be seen as the net pay of this investor, where
gives gross value.
We can write the value that the investor can invest only once as
(13)
where the investment optimal thresholds
are to be determined. According to Henderson [8] the solution
implies that the investor can invest immediately and
implies that investing now is subject to risks due to unforeseen market conditions such as volatility, etc.
From Equation (13) we get the investment threshold as:
(14)
(15)
(16)
Now, to obtain the optimal stock output
(we use (14), (15) into (7)), and the optimal investment threshold
, we utilize the value matching and smooth pasting condition of the investor with
to obtain
(17)
The expected optimal investment value function of the investor given the optimal levels of
is given by:
(18)
Clearly, when
, for
,
, this avoids any arbitrage opportunities. Nevertheless an alternative and generalized methodology can be used to obtain (18) as shown in the Appendix (1).
Proposition 3.1. The investment value function
increases as
.
Proof. We sketch the proof briefly. Since
then substituting
into Equation (18) we obtain
(19)
which is linear in both
and
. Hence, for either or both
and
the function
increases
, increases.
We can conclude from proposition (3.1) that the value of the investor increases with the number of shares received.
It is worth noting that, unlike the Black-Scholes formula, the investor has two criteria to meet: first he targets an output
and secondly, he works towards a higher prices for the stocks,
, which in turn increases his stock share
.
Simulation of the Investment Value Function
So far, we have looked at
in general terms. Now in Figure 1 we want to simulate
by varying the values of
, to demonstrate Proposition 3.1. The parameter
and
are speculated and have no market value but are used to demonstrate Proposition 3.1. For these simulations, we have used;
,
,
and
.
Figures 1(a)-(d) illustrate how the value function increases for various values
Figure 1. Illustration of how the total investment value function
increases for various values of
. (a) For
:
and for
:
; (b) For
:
and for
:
; (c) For
:
and for
:
; (d) For
:
and for
:
.
of
. From Figure 1(a) and Figure 1(b), we see that an increase in the investors allocation of
stock units from
to
increases the total investment value. While from Figure 1(c) and Figure 1(d), we see that a decrease in the allocation of
stock units from
to
, still brings an increase in the total investment value. Nevertheless, the graphs generally suggest that, as long as units from both stocks are nonzero (i.e.
and
), the investment growth will outperform the reward where the investor is rewarded with units from one stock only.
Figure 2 shows that the higher
, the higher the return for the investor. However, the investor must ensure that
.
It must be pointed out that having these stock units does not necessarily guarantee that the investor will get good returns as company performance can be affected by a number of factors like the market and company risks that are not considered in this study [9].
4. The Investor’s Remuneration Package
In this section, we have evaluated the investors remuneration package, and further more investigate the impact of stochastic income on the remuneration package.
A remuneration package can be seen as the investors benefits or rewards when assets are converted in monetary value at time
. The generalized
Figure 2. Observation of total investment value function
when
. (a) For
:
and for
:
; (b) For
:
and for
:
.
Figure 3. Impact of stochastic income rate on the remuneration package.
expected value function that constitutes the remuneration package of the investor is given by
(20)
where
is the salary, and monetary returns from the stock units and the inflow value from the stochastic income.
Then Figure 3 gives a general overview of the impact of stochastic income on the remuneration package (given by Equation (20)). Some of the parameter values we used are;
,
,
,
,
,
,
,
and
. These values were chosen arbitrarily and they give an illustration that support our conclusions.
From Figure 3, we observe that the presence of stochastic income increases the investors wealth. Note that the larger the rate of receiving the stochastic income, the bigger net remuneration for the investor. That is, for
and
we see the presence of stochastic income, while for
indicates the absence of stochastic income. For more reading on what the significance of n refer to remark (2.2) and Appendix (6.2).
5. Conclusion
In this paper we derived and analysed the investment value function and remuneration package of an investor who is rewarded company stock units
and also receives stochastic income. From our results, we have concluded that the investor has two objectives; first to satisfy the company requirement that
and secondly to increase the number of shares
given to him as rewards for ensuring that
. The stochastic income has the effect of increasing the investor’s wealth but it is the rate at which he receives the stochastic income which matters. At this point, our results and all our analyses can hold in the n-dimension, and we leave that open to the reader for further study.
Acknowledgements
I gratefully acknowledge the funding received from the Simmons Foundation (US) through the Research and Graduate Studies in Mathematics (RGSMA) in Botswana International University of Science and Technology. Many thanks go to my father in heaven who made sure that this paper is completed.
Appendix
Alternative Method of Section 3
This alternative, is a generalized version that one can work with. Here we define our variable in terms in matrix form. Hence we redefine our HJB equation as
(21)
Then our boundary conditions are as follows
(22)
(23)
(24)
where
,
,
and
. Then value is equal to
(25)
where
. Therefore from (23), (25) and equating the
of both equations we obtain threshold S as
(26)
(27)
which will yield into an optimal threshold
and optimal stock output
:
Then the investment value function for
is
(28)
Impact of n on the Income Rate
We looked at an investor with income rate
, and we assumed that
, as indicate in Section 2. Figure 4 shows that indeed when n is a positive integer the income rate produces positive results, while for
no income is received and when n is negative gives negative income, as such violate the condition of income rate (of being bounded below at zero). This generally shows that n play as the intensity of
and it is not restricted only on
as indicated by the works of [2].
Figure 4. Impact of n on the income rate.