Managerial Performance Evaluation with

Residual Income — Limited Investment

Budget and NPV-Maximization

Alwine Mohnen*

Department of Business Administration

University of Cologne

December 2003

* Department of Business Administration, University of Cologne, Herbert-Lewin-Strasse 2, 50931 Cologne, Germany, Phone: +49-221-6310, Fax: +49—221-5078, E-mail:Alwine.Mohnen@uni-koeln.de

Abstract

If the investment budget is limited, not all profitable projects can be re-

alized. Then the principal desires the agent to select the NPV-maximizing

set of projects. Residual income does not solve this problem if the agent

is impatient. Even the relative benefit cost allocation scheme proposed by

Rogerson does not provide a solution. In this paper we show how this goal

can be achieved under the same information structure as in Rogerson (1997)

or Reichelstein (1997). The result can be interpreted as an annuity benefit

cost allocation scheme. Periodic shifts of the cash flows are used, which lead

to a performance measure reflecting the NPV-ranking in each period.

1 Introduction

In many principal agent relationships the agent decides about investments

because he is better informed about the investment opportunities. Thus,

the principal’s aim is to achieve goal congruence between his selection of

projects and the agent’s. Accordingly, the principal’s objective is to mo-

tivate the agent to accept projects with an expected positive net present

value (NPV ) to maximize the value of the investment portfolio. This special

delegation relationship has been an important research field in recent years

(Reichelstein, 1997 and 2000; Reichelstein and Dutta, 1999a and 1999b and

also 2002; Wagenhofer, 1999; Dutta, 2003).

It is well known that this problem can be solved by introducing residual in-

come as a performance measure, because the present value of residual incomes

over a project’s entire life can be shown to equal the NPV (Preinreich, 1937

and 1938; Lücke, 1955). This consistency property is called ”weak” goal

congruence (Baldenius, Fuhrmann and Reichelstein, 1999:54). Even if the

investment budget is limited, residual income achieves goal congruence as

long as agent and principal have the same planning horizon and use identical

discount rates. In this case the desired NPV-maximizing projects will be se-

lected. What drives this result, is the consistency with the NPV. In the case

of an unlimited budget all projects with an ex ante expected positive NPV

will be undertaken. In the case of a constrained budget the agent ranks the

projects in the same way as the NPV does. The only condition that has to

be satisfied, is that the agent receives a constant share of all residual incomes.

Consequently, the present value of his bonuses for any project is the same

constant portion of each project’s NPV. Therefore he ranks the projects in

the same way as the NPV does and selects the NPV -maximizing investment

1

portfolio.

But often the agent is myopic, i.e. he has a shorter time horizon or uses a

higher discount rate. In an extreme case an impatient agent might just look at

the next period’s outcome. Due to the fact that the principal does not know

how impatient his agent is, periodic consistency between the performance

measure of any period and the NPV is necessary. For this purpose the

relative benefit cost allocation scheme as established by Rogerson (1997)

provides a specific solution. He finds an allocation rule for the investment

costs, resulting in all expected residual incomes of a project having the same

sign as the expected NPV. This property is called ”strong” goal congruence

(Baldenius, Fuhrmann and Reichelstein, 1999:55,58-60). In addition Rei-

chelstein (1997) demonstrates in a principal agent model that the residual

income in combination with Rogerson’s allocation rule is the unique (linear)

performance measure that achieves strong goal congruence in this context.

This performance measure results in the acceptance of all projects with an

expected positive net present value.

If, however, the investment budget is constrained, which is often the case, not

all profitable projects can be undertaken. The agent has to select a subset of

the possible projects. But if he has a shorter time horizon than the life span

of the longest project or a higher discount rate than the principal, his project

selection might not be NPV -maximizing. We demonstrate by using a coun-

terexample that the relative benefit cost allocation scheme fails to achieve

goal congruence in this case. It follows, that strong goal congruence generally

does not induce the NPV -maximizing investment selection. The principal

seeks to design a performance measure which creates goal congruence in a

way, that the agent should have an incentive to select the NPV -maximizing

portfolio of investment projects. Accordingly, our aim is to create a perfect

2

goal congruent performance measure:1 In each period the ranking of projects

by the performance measure residual income is identical to the ranking by

NPV.

In this paper we establish an allocation rule of cash flows, which obtains

a stronger goal congruence between the agent and the principal. We need

a performance measure, which ensures that it is optimal for the agent to

select the NPV -maximizing portfolio. Hence, we have to ensure periodic

consistency between the performance measure and the NPV -rankings.

This means, if a project has a higher NPV than another, the residual in-

comes of the more profitable project exceed those of the less profitable one

in every period. The idea behind our solution is, that there exists an al-

location of the residual incomes over a project’s lifetime, which guarantees

that the agent chooses the NPV-maximizing investment portfolio even if he

looks only at a single period. For this purpose we look for an allocation rule

which "transfers" cash flows into accruals so that different projects are com-

parable. The determination of accruals, that means the determination of the

asset base and the allocation of benefits and costs, remains crucial for the

consistency between the performance measure and NPV -rankings. We show

that perfect goal congruence can be reached by using an annuity benefit cost

allocation scheme. The corresponding periodic shifts of the cash flows can be

interpreted as provisions or as receivables. Consequently our annuity benefit

cost allocation scheme allows a comparison of different projects, even if just

one period is considered. At last, it is very important to mention, that we

1Yet, Egginton (1995) required this kind of periodic consistency, a consistency betweenindividual ex ante residual incomes and NPV s. He finds a depreciation method, thatachieves periodic consistency for projects with equal life. But to reach this, he needs somuch information, that the NPV itself could be used. Therefore he argues, that residualincome does not satisfy the desired requirements.

3

now use the same information structure as Rogerson (1997) or Reichelstein

(1997) to achieve this goal.

Our performance measure leads us to a wider and more realistic field of

application. So far it was only possible to motivate the agent to accept

all projects with a positive NPV. Here we establish a performance measure

inducing the optimal selection of investment projects, that the NPV will be

maximized. This is necessary in the very realistic case of a limited investment

budget or for alternative project decisions.

The paper is organized as follows. In section 2 we shortly describe the results

by Rogerson (1997) and Reichelstein (1997) and demonstrate that they are

not sufficient to achieve the NPV-maximizing project selection. In section 3

a solution for these limitations is developed by using a special allocation rule

for accruals, the "perfect" and the "annuity" benefit cost allocation scheme

which attain perfect goal congruence. Our findings are summarized in section

4.

2 Recent results and their limits

2.1 The model

We analyze the following situation. An owner (principal) delegates his in-

vestment decisions to a better informed manager (agent). In t = 0 the agent

decides about the realization of investment projects. A possible project Pi

∈ P can be described by Pi = (ai,c1i, c2i, . . . cni) with ai being the initial cash

outflow and cti, t = 1, . . . , n the periodic cash flows during the project’s life

span of n periods. By P ⊂ Rn+1 we denote the domain of possible projects

at t = 0. All cash in- and outflows are received or issued directly by the

4

principal. Agent and principal learn the initial investment ai of a realized

project, but in t = 0 the expected cash flows E [cti] associated with a project

Pi are known only by the agent. Similarly to the models of Rogerson (1997)

and Reichelstein (1997), we assume that the cash flow in period t for project

i is given by

cti = xti · yi + εti (1)

where the noise terms εti are independently and identically distributed with

E[εti] = 0. Consequently the expected cash flows are E[cti] = xti ·yi.Withoutloss of generality the time horizon is the same for each possible project.2 The

parameter yi is called profitability factor and {xti}nt=1 reflects the temporaldistribution of cash flows. As in Rogerson (1997) or Reichelstein (1997) we

assume that the distributional parameters xti are known to both parties,

whereas yi is private information of the agent.

The owner’s aim is to maximize the NPV of the investment portfolio. His

ex ante expected utility equals the expected NPV of all realized projects:

E£UP¤=

Xi

nXt=1

µE [cti]

(1 + rP )t− ai

¶· I(Pi)

=Xi

E [NPVi] · I(Pi) (2)

with I(Pi) = 1, if project Pi is undertaken and I(Pi) = 0 if Pi is rejected.

The principal has a constant discount rate rP .

The agent has a utility function

E£UA¤= E

"nAXt=1

wt

(1 + rA)t

#2Note that of course if one project stops to generate (expected) cash flows at some

earlier date than n, then xti = 0 for the last periods before n.

5

which is (weakly) monotonically increasing in his wages wτ of all periods. We

assume that the agent is more impatient than the principal. For instance, he

uses a constant discount rate rA with rA > rP or principal and agent may

have different planning horizons nA, nP with nA < nP .3 We assume that

the planning horizon of the principal exceeds the life span of the possible

projects. rA and nA are hidden information to the principal. He only knows

the fact that the agent is impatient but he doesn’t know how impatient he

is.

Both riskneutral parties sign a linear compensation contract.4 The wage is

assumed to be linear in a performance measure Πt and hence

wt = Ft + st ·Πt. (3)

with Ft being a fixed amount in t, st a premium rate and Πt the performance

measure in t. The agent maximizes his utility and therefore he grounds his

investment decision on the present value of his ex ante expected bonuses.

Additionally, we allow a transformation of cash flows into accruals, which has

to be controllable by the principal. Accordingly, the transformation rule is

based on realized cash flows. As we accept the assumption that an accounting

system is able to transform ai and cti separately for each project, we describe

a general performance measure for a project i as follows:

Πti = αti · cti − βti · ai (4)

The allocation rules αti and βti have to be designed to provide the desired

3Reichelstein (1997:166) offers several explanations for the manager’s impatience.4As we achieve the first best solution, the assumption of a linear contract is no con-

straint.

6

incentives. For the performance of a whole period follows

Πt =Xi

(αti · cti − βti · ai) · I(Pi).

In our analysis, the performance measure is determined by accruals. An

accrual accounting system defines performance in each period by matching

costs and benefits.

2.2 Residual income and relative benefit cost alloca-

tion schedule

In the main part of the recent literature it is assumed αti ≡ 1, ∀i, t (Rei-chelstein, 1997; Rogerson, 1997). That means, that all operative cash flows

directly equal accruals, hence no ”real“ allocation rule between operative cash

flows and accruals is used. Only the initial cash outflow ai is allocated by

βti over a project’s life span. We now summarize the results obtained in this

scenario and examine which "degree" of goal congruence can be obtained, if

the only difference between a cash flow based performance measure and one

based on accruals is the transformation of the investment cash outflow.

Weak Goal Congruence

In a world with a perfect capital market the principal’s aim is to realize all

projects with an expected positive net present value, that is, he wants the

agent to undertake all projects withPn

t=1E[cti](1+rP )t

−ai > 0. In such an environ-ment it is well known that residual income RI induces the right investment

decision, because this performance measure fulfills the weak goal congruence

property, i.e. the present value of the expected residual incomes equals the

expected NPV of each project,Pn

t=1E[RIti](1+rP )t

= E [NPV i]. This property is

also called conservation property of residual income and is well known in

7

literature. Unless the agent has the same time preferences as the principal

he realizes all projects with a positive expected NPV. Residual income is

defined as profit minus an interest charge for the assets employed. If αt = 1

residual income is defined by RIti = cti − dti · ai − rP ·Bt−1,i. It is calculated

by operative cash flows, cti, minus a depreciation charge, dti · ai, and minusan interest charge for the book value of the assets employed, rP ·Bt−1,i. The

depreciation schedule can be chosen freely as long asPn

t=1 dti = 1. Only the

interest charge for the book value Bt−1,i has to be accounted by rP ·Bt−1,i. Ac-

cording to the familiar notion of ”clean surplus accounting“ we assume that

the book value at the beginning of period t = 1 equals the investment cash

outflow. The book value in the following periods is reduced by depreciation

rates dti, these rates determine the depreciation for a project i in period t. In

accordance with the depreciation schedule the book value at the beginning of

period t equals Bt,i =¡1−Pt−1

τ=1 dτi¢ · ai. Consequently, our allocation rule

βti in the performance measure equation (4) can be described by

βti = −"dti + rP

Ã1−

t−1Xτ=1

dτi

!#(5)

Hence, it follows for our performance measure

ΠRIti = αti · cti − βti · ai

= cti − dti · ai − rP ·Bt−1,i.

That the conservation property holds for a single project was already proved

by Preinreich (1937, 1938) or by Lücke (1955). That this still holds for a set

of projects is shown in Reichelstein (1997:163):

8

Xi

nXt=1

ΠRIti

(1 + rP )t=Xi

NPV i.

This is still true for the expected values, because the expected values of

the random variables equal zero. If principal and agent apply the same

discount rate, the conservation property of residual income ensures weak goal

congruence, provided that all bonus coefficients are identical, i.e. st ≡ s. No

restrictions are imposed on the depreciation schedule, exceptPn

t=1 dti = 1.

Strong Goal Congruence

In a more general case of an impatient agent, i.e. in the case of an higher

discount rate of the agent rA ≥ rP or in the case of a shorter time horizon of

the agent nA ≤ nP , weak goal congruence is not sufficient to introduce the

desired investment decision. The present value of the bonus payments cal-

culated by an impatient agent does not generally equal the NPV . Rogerson

(1997:789-791) offers a solution for this scenario, the relative benefit cost al-

location schedule. He designs a cost allocation βti of the initial investment in

a way that it takes into account the intertemporal allocation of cash flows.5

βti =xtiPn

τ=1xτi

(1+rP )τ

(6)

The intuition behind this special allocation scheme is that costs are allocated

according to the periodic returns. That means in periods with high returns

(xti high), high portions of the investment costs are allocated and in periods

with low returns (xti low), low costs are considered and if the return is zero in

a period (xti = 0) no investment costs are allocated. The resulting modified

5Rogerson (1997) has to restrict the domain of possible projects by xt ≥ 0,∀t. Ageneralization of this problem can be found in Mohnen (2002).

9

residual income is

ΠRIti = cti − xtiPn

τ=1xτi

(1+rP )τ

· ai. (7)

The allocated periodic investment costs can be split into a depreciation and

an interest charge (for example Reichelstein, 1997:168), so that the perfor-

mance measure is still a kind of residual income. Under the assumption of

all distribution parameters being positive, it can be shown easily that for the

relative benefit cost allocation schedule the expected residual income in each

period of the life span is positive (negative), only if the expected NPV is

positive (negative) (Reichelstein, 1997):

E£ΠRIti

¤= E [cti]− xtiPn

τ=1xτi

(1+rP )τ

· ai

=xtiPn

τ=1xτi

(1+rP )τ

·Ã

nXτ=1

xτi(1 + rP )τ

xti · yixti

− ai

!

=xtiPn

τ=1xτi

(1+rP )τ

·

nXτ=1

xτi · yi(1 + rP )τ

− ai| {z }

E[NPVi]

Therefore, the manager has an incentive to invest in all projects with a

positive expected NPV, even if he looks at just one period. In contrast to

the case with identical discount rates and identical planning horizons of agent

and principal, the depreciation schedule is now determined uniquely by the

relative benefit cost allocation scheme. According to a positive contribution

to the agent’s utility in each period, the bonus coefficients can vary between

periods if the expected NPV is positive,

To summarize these results, we list the properties of the performance measure

10

residual income:

• If the allocation rule βti is determined by −£dti + rP

¡1−Pt−1

τ=1 dτi¢¤,

the performance measure equals residual income which ensures the con-

servation property. The present value of all residual incomes equals

the NPV. Under the assumptionPn

t=1 dti = 1 weak goal congruence is

achieved.

• Residual income achieves strong goal congruence if the relative benefitcost allocation scheme, βti =

xtiPnτ=1

xτi(1+rP )

τ, is used and if every distri-

bution parameter is nonnegative, xti ≥ 0. Hence, residual income is apositive multiple of the NPV in each period. Consequently, an impa-

tient agent realizes all projects with an expected positive NPV, too.

These properties are achievable in the ex ante situation, which is the relevant

view to induce the desired investment decision. So in a perfect capital market

goal congruence between agent and principal is reachable because the agent

has an incentive to realize all projects with an ex ante expected positive

NPV.

2.3 Limits of the relative benefit cost allocation scheme

Now, we look at nearly the same situation, but we assume a limited budget,

so that not all projects can be realized. To keep things simple, we consider

a case with just two possible projects, i = 1, 2. Due to the limited budget,

only one of them can be realized. Generally, in a case with a limited budget

the principal wants the agent to maximize the expected NPV by his project

selection. With regard to our simplification with only two possible projects,

it is the principal’s aim to realize the project with the higher (expected)

11

NPV. Whether goal congruence is still achieved, depends on the cash flow

structure of the two projects.

Proposition 1 Given xt1 = xt2,∀t, goal congruence is achieved. The agentselects the project with the higher expected NPV.

Proof

E [RIti] = xtiyi − ai · xtiPnj=1

xji(1+rP )j

=xtiPn

j=1xji

(1+rP )j

nXj=1

xji(1 + rP )j

yi − ai| {z }

E[NPV i]

q.e.d.

If xt1 = xt2,∀t, the same fraction of each E [NPV i] , i = 1, 2, is reflected in

every period. Thus, the agent selects the project with the higher expected

NPV , whatever his time preference is.

If, however, xt1 6= xt2 this is obviously no longer the case. It is instructive to

consider an example (see Pfaff, 1999:67-68 for another example).

Example:

Each project requires an initial investment of 70,000. We assume that agent

and principal have a discount rate of 10%.

12

Project 1

t E [ct1]E[ct1](1+r)t

xt1 βt1 · a1 E£ΠRIt1

¤ P E[ΠRIt1 ]

(1+r)t

1 15,000 13,636 0.15 14,255 745 677

2 15,000 12,397 0.15 14,255 745 1,293

3 20,000 15,026 0.20 19,007 993 2,039

4 25,000 17,075 0.25 23,759 1,241 2,887

5 25,000 15,523 0.25 23,759 1,241 3,658

Project 2

t E [ct2]E[ct2](1+r)t

xt2 βt2 · a2 E£ΠRIt2

¤ P E[ΠRIt2 ]

(1+r)t

1 20,000 18,182 0.21 19,066 934 849

2 25,000 20,661 0.26 23,833 1,167 1,814

3 20,000 15,026 0.21 19,066 934 2,515

4 15,000 10,245 0.16 14,300 700 2,993

5 15,000 9,314 0.16 14,300 700 3,428

Over the whole planning horizon of 5 periods project 1 has a higher E [NPV ]

than project 2. Namely, E [NPV 1] = 3, 658 is higher than E [NPV 2] =

3, 428. Assume the agent gains a fixed share of the residual incomes. If the

planning horizon of the agent is shorter than 5 periods he will not decide

according to NPV -maximization. Due to the fact that he is only interested

in the present value of his expected bonuses he selects project 2 which has

a lower expected NPV than project 1. For instance, if he only looks at the

next 3 periods the present value of the expected residual incomes is 2,039 for

project 1 and 2,515 for project 2. Thus he selects the second one. Only if his

planning horizon is 5 periods he will select the NPV-maximizing project 1.

The same is true, if the agent uses a higher discount rate even though his

13

planning horizon is longer than 5 periods. If his discount rate is at least 26%

he will not choose project 1, because the E [NPV ] of project 1 is 2,440 and

of project 2 is 2,441 for a discount rate of 26%.

From this counterexample we can conclude:

Proposition 2 In general, the relative benefit cost allocation scheme fails to

achieve perfect goal congruence.

Evidently, an impatient agent does not in every case select the project with

the highest expected NPV because he does not take into account the resid-

ual incomes of each period or he discounts them with a higher discount rate.

The relative benefit cost allocation scheme ensures that the expected perfor-

mance measure has the same sign in each period as the expected NPV of

each project, but it does not ensure that the same portion of each project’s

expected NPV is reflected in each period. Consequently, an incentive to

choose the NPV-maximizing portfolio is not induced.

The present value of the agent’s premiums of a project i is given by

nXt=1

s · E £ΠRIti

¤(1 + rAt)t

= s ·nXt=1

1

(1 + rA)t·µ

xtiPnτ=1 xτi · (1 + rP )−τ

· E [NPV i]

¶= s ·

Pnt=1 xti · (1 + rA)

−tPnτ=1 xτi · (1 + rP )−τ

·E [NPV i] , (8)

if the premium rate is constant. With regard to the project selection out of

the domain P, the factorPn

t=1 xti·(1+rA)−tPnτ=1 xτi·(1+rP )−τ shows that it is generally not true,

that an impatient agent (with either a higher discount rate or a shorter time

horizon6) selects the same subset of profitable projects as the principal does,

even if the premium rate is constant.

6Then the factor equalsPnA

t=1 xti·(1+rA)−tPnτ=1 xτi·(1+rP )−τ .

14

The above mentioned example demonstrates that in the case of a limited

investment budget or in the case of competing projects the postulation of

strong goal congruence in the sense of same signs of residual income and NPV

in each period and for each project is not strong enough to induce the NPV-

maximizing range of projects, if the agent is impatient. The application of

both, residual income and the allocation rule relative benefit cost allocation

scheme, is so far restricted to the selection of all projects with a positive

expected NPV. However, the investment budget is often limited or you have

to select between alternative projects. Hence, it is important to ask the

question of whether a different way of constructing a performance measure

can be found that induces the right choice of investment projects.7

3 Perfect Goal Congruence

3.1 Definition

It is our aim to construct a perfect goal congruent performance measure under

the same informational assumptions as in Reichelstein (1997) or Rogerson

(1997). This aim seems to be unreachable, as Reichelstein (1997:168-169)

shows, that the relative benefit cost allocation schedule in (6) is the unique

method for achieving goal congruence if the discount rate of the agent is

unknown and higher than the principal’s one. However, Reichelstein and

Rogerson solely regard a special kind of residual income in their analysis.

They both assume indirectly that αti ≡ 1. Consequently all operative cash7Egginton (1995:213f) analyzes the ”maintainable” residual income. He uses a depre-

ciation charge that gives a constant ex ante residual income in each year of a project’slife. But it provides periodic consistency only for projects of equal life, it cannot copethe ranking of unequal lives. The most critical point of his method is the informationstructure. To control the performance measure the principal has to know the NPV of aproject. In that case, the principal could directly base the agent’s premium on the NPV.

15

flows cti equal accruals8 in the same period, equation (4) is then restricted to

ΠRIti = cti−βti ·ai. Obviously, only the investment cash outflow ai is allocated

by using an allocation rule βti. In fact, under the informational assumptions

by Rogerson and Reichelstein there is no reason why the operative cash

flows cti should not be reallocated over a project’s entire life span. The cash

flows cti are verifiable. An allocation rule based on realized cash flows is not

manipulable as far as the allocation of ai is not. Regarding accounting rules

in practice, we also find some reallocation rules for realized cash flows, for

example provisions. Therefore, we will now show that the problem of project

selection can be solved if we allow for a reallocation of operative cash flows.

We are able to find a performance measure that motivates the agent to select

the NPV -maximizing set of projects.

We define perfect goal congruence as follows:

Definition 1 Given E[NPVi] > (<)E[NPVj], perfect goal congruence is

achieved if and only if E[ΠRIti ] > (<)E[Π

RItj ], ∀i, j, t.

If the agent has to decide between alternative projects, a perfect goal con-

gruent performance measure induces the right investment decision.

3.2 A class of perfect goal congruent performance mea-

sures

The definition of perfect goal congruence seems to be difficult to fulfill. In

contrast to Reichelstein (1997) or Rogerson (1997) we consider the decision

between projects and not the decision about separable projects. Our aim is to

make the possible projects ex ante comparable in the performance measure of

8Cost or benefits, in frameworks of Reichelstein (1997) or Rogerson (1997) benefitsbecause they assume only positive operative cash flows.

16

each period. In our model it would be sufficient, if the expected performance

measure is a positive portion of the expected NPV, but the portion of the

expected NPV has to be the same for all projects. Hence, we have obtained

the following simple result:

Lemma 1 If

E[Πti] = γt · E[NPVi], (9)

with γt > 0,∀i, t, is achieved, perfect goal congruence as defined in the lastsection is obtained.

The factor γt merely depends on the period index t and is independent of

the project index i.9 This means, that in period t, the same share γt of each

expected NPV equals the expected performance measure, E[Πti], of each

possible project Pi.10 This must be true for every period. Hence, if we were

able to construct a performance measure fulfilling this property, we would

achieve a class of goal congruent performance measures.

We now demonstrate, that the principal indeed has enough information to

ensure E[Πti] = γt ·E[NPVi]. First, we transform equation (9) into:

E[Πti] = γt · E[NPVi]

= γt ·Ã

nXt=1

xti · yi(1 + rP )t

− ai

!

= γt · yi ·nXt=1

xti(1 + rP )t

− γt · ai. (10)

9Reichelstein (1997) and Rogerson (1997) require that E[Πti] = γti · E[NPVi] withγti > 0. This difference also illustrates why it is impossible to reach perfect goal congruenceif only the initial cash outflow is allocated over time.10Also equation (8) induces, that the principal has to ensure the same portion of each

expected NPV as performance measure in each period. The principal does not know thediscount factor of the agent, so he can just ensure that xtiPn

τ=1 xτi·(1+rP )−τ is the same factorfor each project in each period.

17

To assure that E[Πti] = γt ·E[NPVi] for some vector of positive constants γt,

we have to show that there are values for αti and βti, such that the expected

value of our performance measure defined in equation (4), E[Πti] = αti ·E [cti] − βti · ai, equals γt · E[NPVi] in every period. Therefore,

αti ·E [cti]− βti · ai, has to equal equation (10):

αti · E [cti]− βti · ai = γt · yi ·nXt=1

xti(1 + rP )t

− γt · ai, ∀(yi,ai) ∈ R2.

This is equivalent toÃαti · xti − γt ·

nXt=1

xti(1 + rP )t

!· yi− (βti − γt) · ai = 0, ∀(yi,ai) ∈ R2. (11)

Since yi and ai can vary for each possible project Pi ∈ P, the two coefficientsfor yi and ai, the terms in brackets, have to be zero, in order for the left-hand

side of (11) to be identically equal zero. Hence,

αti · xti − γt ·nXt=1

xti(1 + rP )t

= 0

and

βti − γt = 0.

From these equations we can directly infer that a perfect goal congruent

performance measure can be constructed as follows:

αti =γtxti·

nXt=1

xti(1 + rP )t

(12)

and

βti = γt (13)

18

for an arbitrary vector of γt. We call this class of allocation rules, (12)

together with (13), "perfect" benefit cost allocation rules.

The principal ex ante knows the distribution of the expected cash flows. So he

can setup an allocation rule αti based on the distributional parameters and he

can choose a parameter γt > 0. Hence, we obtain the following proposition.

Proposition 3 Perfect goal congruence is achieved, if the agent is rewarded

on the basis of the performance measure, Πti = αti · cti − βti · ai, where theallocation rule, αti =

γtxti·Pn

t=1xti

(1+rP )twith γt > 0, is used to reallocate

operative cash flows into accruals and the allocation rule, βti = γt, is used to

allocate the investment cost.

A perfect goal congruent investment decision is induced, because the agent

bases his investment decision on his expected bonuses. Independently of the

premium rate st he maximizes his utility function, if he chooses the NPV -

maximizing subset of projects.

In view of the fact that the performance measure should be a residual income

ensuring that weak goal congruence still holds, we have to examine whether

the conservation property holds. The following corollary shows that we can

indeed find values for γt such that this objective is obtained.

Corollary The perfect benefit cost allocation rule results in a residual income

as a performance measure if and only ifPn

t=1γt

(1+rP )t= 1.

ProofnXt=1

E [Πti]

(1 + rP )t= E [NPVi]

⇔

19

nXt=1

γt · yi ·³Pn

t=1xti

(1+rP )t

´− γt · ai

(1 + rP )t=

nXt=1

xti · yi(1 + rP )t

− ai

⇔

yi ·Ã

nXt=1

xti(1 + rP )t

!·Ã

nXt=1

γt(1 + rP )t

− 1!=

ÃnXt=1

γt(1 + rP )t

− 1!· ai

⇔nXt=1

γt(1 + rP )t

= 1.

q.e.d.

Thus, weak goal congruence is obtained.

An important issue is, that the premium rate can vary over time to ensure

perfect goal congruence. The reason behind this result is, that the ranking

of the projects in each period by the expected performance measure is the

same as the ranking by NPV. It does not matter whether the premium rates

vary between periods.

3.3 Annuity benefit cost allocation scheme

To compare projects over time, it is useful to ensure a constant expected

performance measure for each project over it’s entire life span. Then the

same constant portion of a project’s expected NPV is allocated over the

entire life time of a project. If the performance measure is still a residual

income such a constant expected value has to equal the expected annuity

20

of a project according to the conservation property of the residual income

concept. Consequently, the value of a project is equally allocated across all

periods and the value contribution of each project is correctly reflected over

the project’s life span. The following allocation rule provides a solution to

achieve this constant value contribution.

Proposition 4 If γt =1Pn

t=1γ

(1+rP )t, ∀t, then the expected performance mea-

sures equals the expected annuity.

ProofnXt=1

γ

(1 + rP )t= 1

⇔γ =

1Pnt=1

γ(1+rP )t

⇔γ = ArP

where ArP denotes the annuity factor, ArP =³Pn

t=1γ

(1+rP )t

´−1= (1+rP )

n·rP(1+rP )n−1 .

For the expected performance measure follows:

αti · E [cti]− βti · ai =ArP

xti·

nXt=1

xti(1 + rP )t

·E [cti]−ArP · ai

= ArP ·Ã

nXt=1

xti · yi(1 + rP )t

− ai

!= ArP ·E[NPVi]

= E[ANi]

with ANi as the annuity of project i.

q.e.d.

21

We call this allocation rule the annuity benefit cost allocation scheme.

Perfect goal congruence is achieved in this special case of the annuity benefit

cost allocation scheme because the expected performance of each project

in each period equals the expected annuity of a project. In the case of

two projects having the same expected NPV, the agent will be indifferent

because both expected performance measures are equal in each period. In

the case of projects with different expected NPV, the expected bonus of a

more profitable project is higher than that of a less profitable one. It follows,

that the agent has an incentive to select the set of NPV -maximizing projects

independently of his time preference.

The designed performance measure is a residual income, because the con-

servation property still holds. The allocation of the investment costs can

be separated into a depreciation charge and an interest charge for the book

value. The depreciation schedule is uniquely fixed by the investment alloca-

tion rule βti. Strong goal congruence is also reached, because every perfect

goal congruent performance measure has the property of strong goal congru-

ence, too.

The annuity of a project reflects the profitability of a project in each period,

because it is a uniform distribution of the NPV over a project’s life span.

To calculate this figure or its expectation, you normally use the NPV resp.

E[NPV ] and the annuity factor. Accordingly, it seems to be impossible to

create such an annuity performance measure without having all information

about a project. For example, one might think that the principal ex ante

needs to know all expected cash flows of a project to induce the right incen-

tives. He can do this, however, even though he has not all this information,

because it is sufficient to have the information about the distribution para-

22

meters. It is not necessary to know the profitability parameters. It follows

that we donot have to change the information structure, it remains the same

as in Reichelstein or Rogerson.

3.4 Application in an accounting system

So far the principal is able to calculate the perfect or the annuity benefit cost

allocation scheme. Thus, he has to instruct the transformation of the original

distributional parameters of the cash flows into the distributional parameters

of the accruals.11 In our model we have found a solution, but how can it be

used in practice? Residual income is a measure usually calculated in an

accounting system.12 Therefore we now "translate" our allocation rule into

"accounting terms".

By using periodic shifts of the original cash flows, we can obtain the desired

distribution of accruals. To illustrate the annuity benefit cost allocation rule

we construct an annuity distribution. We have a look at the following original

cash flow distribution (x1i, x2i, . . . , xki, . . . , xk+l,i, . . . , xni) of a single project.

We assume that all distributional parameters equal the annuity distribution

xAi = αti · xti= ArP ·

nXt=1

xti(1 + rP )t

,

except xki and xk+l,i. Therefore, one of the two parameters is less than xAi

and the other one is higher than xAi.

11Dutta and Zhang (2002) examine how various revenue recognition principles affect theincentive properties of performance measures based on aggregated accounting data.12See Dutta and Reichelstein (2003) for the use of alternative accrual accounting rules

from an incentive and control perspective.

23

If in period k the original distributional parameter is less than xAi we can

"transfer" cash flows which originally belong the period k + l to period k.

That means that the recognition of these cash flows in the accounting system

occurs in an earlier period than their realization. In this case we have to

consider an interest effect. Assume we need ξ · xk+l,i, 0 ≤ ξ ≤ 1, in k to

reach xAi. In period k the distributional parameter is calculated by xki+

ξ · xk+l,i = xAi. Then in period k + l we have to subtract ξ · xk+l,i · (1 + rP )l

because of the interest effect between these two dates.

In the opposite case, i.e. in period k+ l the original distributional parameter

is less than xAi, we "shift" cash flows from period k to a future period k+ l.

Then the recognition in the accounting system is later than the realization

of the cash flows. Hence, in k we need to reserve the present value of the

necessary amount. For example, we need ν · xki, 0 ≤ ν ≤ 1, in period k + l,

that xk+l + ν · xki = xAi. Therefore in period k an amount ν · xki · (1 + rP )−l

has to be blocked, that xki − ν · xki · (1 + rP )−l = xAi.

Generally, it is possible to construct an annuity distributional vector xAi by

using period shifts between multiple periods. If, for example, the surplus

over xAi in a former period is not high enough in a single period to add up

xti to xAi, shifts over more than one period are possible:

xAi = xti +t−1Xk=1

ξtk · xki · (1 + rP )t−k.

In accounting terms this equals the release of provisions. In former periods

the provisions are built up. Contrary, in the case of high returns in the

24

current period, xτi > xAi we can use

xAi = xτi −nX

l=τ+1

ντl · xli(1 + rP )l−τ

,

allocating the surplus of period τ reasonably over the following periods with

distributional parameters less than xAi. In accounting terms, this is booking

provisions. With opposite signs these periodic shifts can be interpreted as

the booking of receivables and their release. This can be phrased as follows:

If in an early period the distributional parameter is higher than xAi, we can

build up provisions and release them in periods with an original distributional

parameter less than xAi. In the case of a distributional parameter less than

xAi we can enter receivables in the books, which are released in later periods

with higher expected returns. This shows, that in an accounting system

our allocation rule can be established by the principal. Importantly, but

contrary to general use in practice, the periodic shifts have to heed the timing

effect, i.e. they have to take into account the (imputed) interests. Since we

pay attention to these interest yields, the present value of the original and

the modified distributional parameters is still the same. The conservation

property still holds.13

At this point it is obvious why we do not need the assumptions that xti > 0

which has be to imposed by previous models. In these models only the allo-

cation of the investment cash outflow is considered. Here, we allow for the

allocation of the operative cash flows, too. Hence, if the present value of the

distributional parameters of a project is positive, there exists an allocation

rule for the investment costs and the operative cash flows, so that the distrib-

utional parameters of the corresponding accruals are all positive. Therefore,

13For a detailed proof in a similar context, see Mohnen (2002:148f).

25

it is not necessary anymore to assume positive original distributional para-

meters of the cash flows.

4 Conclusion

In situations with a limited budget or alternative projects, the principal

wants the agent to select the NPV -maximizing investment portfolio. In these

situations the residual income in combination with the relative benefit cost

allocation scheme does not induce the same project ranking as NPV. Here,

we have shown that by using a special transformation of the cash flows into

accruals, the ex ante expected residual income equals the expected annuity in

each period. The agent selects the NPV -maximizing set of projects, because

the expected residual incomes of different projects reflect the ex ante NPV -

ranking of the projects, even if just one period is considered. It is sufficient

to look at this performance measures to rank projects.

We obtain the annuity residual income by an allocation scheme which we call

annuity benefit cost allocation rule. This rule transforms the cash flows into

"annuity accruals" by using periodic shifts. The applied periodic shifts can

be interpreted as accounting methods used in practice. There it is common

use to set up provisions and to use receivables, generally to smooth earnings

over several periods. In financial accounting these methods are restricted by

law, but in managerial accounting they are not.

Concerning the information structure, it is good news, that we can use the

same model setup as seminal papers like Reichelstein (1997) or Rogerson

(1997). The principal needs only information about the distributional pa-

rameters of the operative cash flows. Ex ante this information is enough to

ensure a perfect goal congruent performance measure. The rational behind

26

this is, that it is sufficient to transform the distributional parameters into

a perfect, particularly into an annuity allocation. Then the agent has an

incentive to select the NPV -maximizing projects no matter what his time

preferences are.

As a further result, we have shown, that the premium rate can vary over

time because in each period the agent enhances his utility by selecting the

NPV -maximizing portfolio of projects.

In practice our result may seem a bit difficult to implement. However, in-

come smoothing is well established and here we do not use it for financial

reporting, but only for managerial control and performance payment pur-

poses. Moreover, the annuity performance measure may be a solution to

derive rules to handle the so called "EVA-Bonusbank".14

14See, for instance, Stewart (1991) or Ehrbar (1998).

27

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