Capital Rationing

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• Section VI
• Capital Rationing

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Section Highlights

• Capital rationing
• Linear programming
• Shadow prices and the cost of capital
• Integer programming
• Goal programming

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Capital Rationing
A
Capital Constraint
B
C
Cost of Capital
Internal Rate of Return
D
E
Cost of Capital
F
G
H
Funds
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Managerial Issues Latent in Capital Rationing

• Capital rationing is synonymous with rejection of
  some investments having positive net present
  value
• Management must decide that true funds rationing
  - that is, a ceiling on funds made available - is
  preferable to security issuance
• Opportunity loss is sustainable

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Operational Issues Under Capital Rationing

• The problem is to find the optimum combination of
  investments
• Maximize NPV from available investments by
  combinatorial techniques
• Simple ranking by the NPV index may or may not
  suffice

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Solution Techniques

• Linear Programming (LP)
• Integer Programming (IP)
• Goal Programming (GP)
• Each has characteristics that are sometimes
• strengths and sometimes weaknesses.

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Illustration
Policy of rationing limits available funds to
2000
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Illustration Ranking by NPV
Choose B and funds are
exhausted
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Illustration Ranking by EPVI
Choose A C and funds are exhausted
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NPV Maximization Problem
Funds Availability Period 1 2 Period 2 1.5
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Linear Programming (LP)

• Mathematical model of a realistic situation
• Optimal solution can be found with respect to the
  mathematical model
• Tailor-made for solving capital budget problems
  when resources are limited
• Cannot be used with large indivisible projects

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Linear Programming (LP)General Form
J Investments
Subject to funds constraint
AIT PV of outlay required in budget
period T
BT Budget (funds) ceiling
T Planning horizon
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NPV Maximization Problem 1
Funds Availability Period 1 2 Period 2 1.5
Max NPV 5.25 XA 3.00 XB
Subject to Period 1 2.0 XA 1.0 XB lt
2.0 Period 2 1.0 XA 1.0 XB lt 1.5 0 lt
XA, XB lt 1
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Two Investment LP Solution
Period 1 Funds Constraint
III
XA 0.5 XB 1.0
II
Period 2 Funds Constraint
I
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• Summary of Net Present Values of Alternative
  Investment Selections
• (In Packet)

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Shadow Price Opportunity Cost of Funds

• Definition Increment to NPV per unit of
  increment to funds
• Method of derivation
• Increment funds available
• Find new optimal solution
• Determine new sum of NPV
• Find increment to NPV over previous optimum
• Measure increment to NPV relative to increment of
  funds

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Calculating Shadow Price

• Optimal solution
• XA 0.5 XB 1 NPV 5.625
• Increment period 1 funds available from 2.0 to
  2.25
• New optimal solution
• XA .75 XB .75 NPV 6.1875
• Gain in NPV
• 6.1875 - 5.625 0.5625
• Shadow price 0.5625 / 0.25 2.25

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Two Investment LP Solution
Relaxed Period 1 Funds Constraint
XA 0.75 XB 0.75
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Calculating Shadow Price

• Increment period 2 funds available from 1.5 to
  1.75
• New optimal solution
• XA .25 XB 1.5 NPV 5.8125
• Gain in NPV
• 5.8125 - 5.625 0.1875
• Shadow price 0.1875 / 0.25 0.75

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Two Investment LP Solution
XA 0.25 XB 1.50
Relaxed Period 2 Funds Constraint
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Interpreting Shadow Prices

• Shadow prices account for total value of the
  optimum solution in terms of unit values of funds
  constraints
• Constrained Shadow Total
• Constraint Value x Price Value
• Period 1 2.0 2.25 4.500
• Period 2 1.5 0.75 1.125
• 5.625

Implication We can find the contribution of
each periods constrained funds to net present
value of a capital budget.
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Using Shadow Prices

• Shadow price on constrained funds is the cost of
  capital because it is a statement of the
  opportunity loss in NPV imposed by funds
  rationing - unique to rationing circumstance.
• Alternatively, shadow price is marginal cost of
  capital equal at the optimum to marginal NPV.
  Hence, use it to measure value of more funds.

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Critiquing the Optimal LP Solution

• Optimal solution XA 0.5 XB 1.0
• In LP there will be at least as many fractional
  projects as there are funds constraints
• Adjustment for the indivisibility problem
• Some investments are divisible
• Rounding up or down
• The optimal solution may change
• Shadow prices will change - marginal values are
  intrinsic to structure of opportunities

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Integer Programming (IP)

• Solution values are 0 or 1 integers
• Investment interdependencies are readily
  accommodated. Simpler models assume that cash
  flows of investments are independent.
• Mutually exclusive investments
• Prerequisite (or contingent) investments
• Complementary investments
• Solution time
• Doubtful meaning of shadow prices

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Integer Programming (IP)General Form
J Investments
Subject to funds constraint
AIT PV of outlay required in budget
period T
BT Budget (funds) ceiling
where XI 0,1
T Planning horizon
The zero-one condition is the sole distinction
from LP
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Integer Programming Solution
Integer Lattice Point
Supplementary Linear Constraint
Linear Constraint
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Integer Programming and Mutual Exclusivity

• Investment J is an element of a set of mutually
  exclusive investments
• If we take one within the set, we do not take
  others
• XC XE lt 1
• If we must adopt one or the other
• XC XE 1

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Integer Programming and Delayed Investments

• Suppose we wish to consider delaying investment D
  by 1 or 2 years
• D immediate
• D 1 year delay (lower NPV)
• D 2 year delay (still lower NPV)
• XD XD XD lt 1

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Integer Programming and Prerequisite Investments

• If B cannot be accepted unless D is also
  accepted, we say D is prerequisite or B is
  contingent on D
• XB lt XD
• Suppose acceptance of A is contingent on
  acceptance of either C or E
• XA lt XC XE
• Suppose acceptance of A is contingent on
  acceptance of both C and E
• 2XA lt XC XE

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Integer Programming and Complementary Investments

• Suppose investments B and D are synergistic. If
  B and D are both accepted, NPV of each increases
  by 10.
• Create a new investment Z that is a composite of
  B and D. Include it as a free-standing
  investment in the problem and its optimization
  but also write
• XB XD XZ lt 1
• This precludes duplication in the event that
  Z is included.

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Integer Programming and Shadow Prices

• Shadow prices are not available from an IP
  solution. In IP investments must be thought of
  as coming in indivisible units. Therefore, we
  cannot speak of marginal profit contribution of a
  small change in available funds.
• Absence of continuity in IP destroys shadow
  prices.

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Goal Programming (GP)

• Recognizes pluralistic decision environment of
  capital budgeting
• Extends LPs uni-dimensional objective function
  to multidimensional criteria
• Operationally, deviations from several goals are
  minimized according to priority ranking
• GP requires assignment of priorities to goals
  plus
• Relative weights assigned to goals on the same
  priority level

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Goal Programming (GP)Components

• Economic (hard) constraints identical to LP
  constraints
• Goal (soft) constraints represent policies and
  desired levels of various objectives
• Objective function minimizes weighted deviations
  from the desired levels of various objectives
• Goal constraints employ deviational variables
  indicating that a desired goal is overachieved or
  underachieved

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Deviational Variables

• Assume minimizing objective function
• Achieve a minimum level Min D-
• Do not exceed Min D
• Approximate as closely as possible Min (D -
  D-)
• Maximize value achieved Min (D- - D)
• Minimize value achieved Min (D - D-)
• Priority levels are P and P1 gtgt P2 gtgt P3 etc.

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Goal Programming Specification of Objectives

• Need priority levels (ordinal) on which goals
  will be optimized
• Need relative weights for goals when there are
  two or more on the same priority level
• Need deviational variables

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Goal ProgrammingSolution Process

• State options feasible within economic (hard)
  constraints (funds)
• Solve within original constraints on priority
  level 1 - this reduces feasibility region
• Solve on priority level 2 and find another
  feasibility region
• Continue until solution has been found on all
  priority levels

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Two Investment GP Problem

• Cash Outflow
  Units of Mgt.
• Investment Year 1 Year 2
  Supervision
• 1 25 20 5
• 2 40 15 16
• Available 30 20 10
• Net Income
• Investment NPV Year 1 Year 2 Year 3
• 1 14 10 11 12
• 2 60 6 8 11
• Goal Levels 6 8 10

Net Income goals have P1, P2 and P3 respectively
and NPV is P4
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Two Investment GP Problem

• Objective function
• Min P1D-1 P2D-2 P3D-3 P4(D-4 - D4)
• Choice of Max NPV is arbitrary starter only
• Achieve this goal first and then overachieve
• Economic Constraints
• 25X1 40X2 lt 30 Funds Year 1
• 20X1 15X2 lt 20 Funds Year 2
• X1 lt 1 X2 lt 2

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Two Investment GP Problem

• Goal Constraints
• 10X1 4X2 D-1 - D1 6 NI Year 1
• 11X1 7X2 D-2 - D2 8 NI Year 2
• 12X1 11X2 D-3 - D3 10 NI Year 3
• 14X1 60X2 D-4 - D4 10 NPV

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• Goal Programming Graphs
• (In Packet)

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Goal Level Summary

• Optimal Solutions
• LP GP
• X1 0 X1 0.415
• Goal X2 0.625 X2 0.49
• NI Year 1 2.5 6.11
• NI Year 2 4.375 8.00
• NI Year 3 6.875 10.37
• NPV 37.5 35.21

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GP Solution and Sensitivity

• Iterative process
• Set priorities and relative weights
• Obtain optimal solution
• Depending on degree of consensus, perform a
  sensitivity analysis
• Vary priorities and weights, solve again
• Determine effect or lack thereof on optimal values

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GP Solution and Sensitivity

• If sensitivity analysis shows little or no impact
  on decision variables, then stop
• If sensitivity analysis shows extensive impact of
  varying priorities, try for another consensus