sensitivity and scenario integrated technique under risk and uncertainty

XYZ company is considering a project for investment with the credential as:  life of the project =5 years, cost of the project Rs.100000/- scrap value =10000 and cost of capital 12% and applicable tax rate is 40%. The capacity of plant is 20000 units. The forecasted Operating details at the present/ideal/normal operating capacity and the forecasts likely extent of changes in variables under different economic scenario conditions are given below: (determine the Net present value of index rate when the uncertainty is confined only to operating results and when it is up to the cash inflows).

 cost and prices

Planned/ideal/normal conditions
& at 90% capacity utilization: 18000 units

forecasted scenario effects on costs, price as a proportionate change and volume in proportion of capacity utilization 

(Allocated to products and absorbed to units as per ABC) the costs other than the throughput and depreciation are of semi Fixed (SFC) and the changes forecasted of SFC are from the total of the respective variable and depreciation is fixed and the price and throughput are directly proportional to Volume.

Rs. per unit
(as per the

ABC

costing)

total Rs.
/Ratio

slump
scenario (- indicates reduction & +indicates increase of price and costs)

recession scenario (- indicates reduction & +indicates increase of price and costs)

boom scenario(- indicates reduction & +indicates increase of price and costs)

Volume / Level of Activity

90%

capacity

18000

0.5

0.6

1

Selling Price/Sales revenue

10

180000

-0.10

0.20

0.300

Through-Put Cost or Variable Cost-(VC)

4

72000

-0.25

-0.10

0.125

Batch Level Costs(SFC)

0.7

12600

-0.40

-0.30

0.150

 Product Level Costs-(SFC)

0.6

10800

-0.45

-0.12

0.120

Firm/Facility Level-(SFC)

0.5

9000

-0.33

-0.20

0.180

(allocated to products and not absorbed to units) customer Level-(SFC)

0.8

14400

-0.10

0.12

0.150

Project Book Costs (depreciation)-(FC)

1

18000

0

0

0.000

TFC(SFC+FC)

3.6

64800

 

 

 

TC

7.6

136800

 

 

 

Profit

2.4

43200

 

 

 

ROS on Ideal/Normal Revenues

0.240

 

 

 

Further, the occurrence of number of times of the scenarios during the life in random as: normal -1, the slump -1, recession-2 and boom-1 in its life. Inconsideration of the risks of scenarios you are requested to find out the certainty equivalent returns and net Present Index Rate (NPR) with details of sensitivity of operating returns to each variable to the above said scenarios effects on the net returns using suitable technique.

Solution:  on application of integrated technique (IPPPA):

Step-1:

Determination of ROS with IPPPA technique at the given IDEAL conditions/scenario:

particulars

formula

result

Investment/cost of the project

as given

100000

Contribution Margin(CM) Rs. Per unit

sp-vc

6

CM Ratio

CM/SP

0.6

Break Even Point (BEP) Rupees

TFC/CMR

108000

Break Even Ratio (BER)

BEP/S

0.600

Return on Sales (ROS / ΔROS)

ROS=(CMR*(1-BER)

0.24

Step-2:

The sensitivity constants are determined relating to each variable based on the present/initial/planned values as: volume= PVR*BER; SP=1, VC= VCR i.e. VC/SP; each component of SFC or FC: PVR*BER*(each component of SFC or FC /TFC) determined as follows.

Particulars of variables affected by the scenarios

for measuring their scenario change effects on the ideal/Normal  ROS

variable short name

Formula for  determining sensitivity constant as the given normal/Ideal basis

sensitivity
 constant 

(SC)

Volume of output & Sales

Volume

(CMR*BER)

0.36

Selling Price/Sales revenue

SP

SP/SP

1.00

through put cost or Variable Cost

VC

VCpu/SP

0.40

batch level costs

SFCa

(SFCa/TFC)*(CMR*BER)

0.07

(allocated to products and not absorbed to units) product level costs

SFCb

(SFCb/TFC)*(CMR*BER)

0.06

 (allocated to products and not absorbed to units) firm/Facility level

SFCc

(SFCc/TFC)*(CMR*BER)

0.05

(allocated to products and not absorbed to units) customer level

SFCd

(SFCd/TFC)*(CMR*BER)

0.08

Project Book Costs (depreciation)

FC

(FC/TFC)*(CMR*BER)

0.10

TSFC:

(sum of SFC &FC)

0.36

Formulae for determination of impacts of variables individually on ideal/normal ROS as extent of effect or sensitivity using the sensitivity constants (SC) are:

1. Volume effect: (( SC* ΔVolR2) *(1+ ΔVolR1))+(ROS Idl.* ΔVolR1)

2. Price (SP) effect: ((SC* ΔSPR1) *(1+ ΔVolR1))

3. Variable Cost (VC) effect: (( SC* ΔVCR1)*(1+ ΔVolR1))

4. Concerned Individual SFC/FC effect: (( SC* ΔSFCR1or  ΔFCR1)*(1- ΔVolR2))+(1+ ΔVolR1)

5. To determine the sum extent or net extent of sensitivity from the ideal ROS:

(Sum of effects of Price and Volume) – (sum of the cost effects)

6. To determine the new ROS at the Ideal revenues after the net extent of sensitivity from the ideal ROS:

(Ideal ROS + net extent of effects)

7. To determine the new ROS at the respective scenario revenues after the net extent of sensitivity from the ideal ROS: ( scenario ROS at Ideal revenues(as determined f point 6 above)/ ((1+ ΔVolR1)*( (1+ ΔSPR1))

Interpretation of impacts of variables on ROS: a negative (-) value of price and volume and positive (+) value of costs indicates adverse and vice versa. The net impact is calculated as deducting the sum of costs effects from the sum of Price and Volume effects. The net negative indicate adverse and net positive indicates favorable. Finally, the net positive impact will be added to (net negative deducted from) the initial/Target ROS of target/ideal/normal revenues to determine the ROS after the impacts concerned of all at the variable. The ROS on the each scenario revenues individually be calculated by dividing the respective scenario net ROS on ideal revenues after the effects with the product of (1+ΔSPR1) and (1+ ΔVolR1) of the respective scenario. In the calculation of Δ Ratios 1 represents the limit-1 as the basis and 2 refers to limit-2 as the basis for calculating the ratio. Further the limit-1 is always the ideal/normal and limit-2 is the respective scenario value concerned.

A: when the uncertainty is confined to operating activities/results:

Step-3a:

forecasted scenario effects on costs, price as a proportionate change and volume in proportion of capacity utilization  as given

 sensitivity of ROS to the scenario changes in the variables (the extent of effect for a given change in each variable  to each of the distinct scenarios as the individual and net total  effect on the ROS of normal/ideal Revenues)

differential ratios of variable

slump
scenario

recession scenario

boom scenario

sensitivity
 constant

acronym of variable to represent the variable for their effect

slump
scenario

recession scenario

boom scenario

volume of sales

Ideal/normal

18000

18000

18000

scenario volume

10000

12000

20000

Vol

ΔVolR1

-0.444

-0.333

0.111

Vol

ΔVolR2

-0.800

-0.500

0.10

0.36

ΔVol

-0.267

-0.200

0.067

SP

ΔSPR1

-0.10

0.20

0.30

1.00

ΔSPR

-0.056

0.133

0.333

VC

ΔVCR1

-0.25

-0.10

0.125

0.40

ΔVCR

-0.056

-0.027

0.056

SFCa

ΔSFCRa1

-0.40

-0.30

0.15

0.07

ΔSFCRa

-0.028

-0.021

0.011

SFCb

ΔSFCRb1

-0.45

-0.12

0.12

0.06

ΔSFCRb

-0.027

-0.007

0.007

SFCc

ΔSFCRc1

-0.33

-0.20

0.18

0.05

ΔSFCRc

-0.017

-0.0100

0.009

SFCd

ΔSFCRd1

-0.10

0.12

0.15

0.08

ΔSFCRd

-0.008

0.0096

0.012

FC

ΔFCR1

0.00

0.00

0.00

0.10

ΔFCRe

0

0

0

ΔTFCR1∑ΔSFCR+ΔFCR

-1.280

-0.500

0.600

0.36

ΔTFCR

-0.0795

-0.0286

0.0387

net impact on ROS at the planned revenues

-0.187

-0.011

0.306

new ROS after the scenario effects on planned revenues

0.053

0.229

0.546

new ROS after the scenario effects on the respective scenario revenues

0.106

0.286

0.378

Step-4a:  expected returns on Ideal sales revenues: based on the information of random occurrences of the scenarios the Probability (P) is determined and the CEVROS at 3σ as:

scenario 

ROS

P(probability)

P*ROS

Ideal/planned/normal

0.240

0.2

0.0480

slump

0.053

0.2

0.0106

recession

0.229

0.4

0.0914

boom

0.546

0.2

0.1091

mean

(sum of p*ROS)  

0.2591

Stn.Dev. σ

Stndv of (p*ROS)

0.04431

certainty equiv valet returns (CEVROS)

(Mean ROs) – ( 3*σ =3*0.04431 =0.1329) =(0.2591-0.1329)

(CEVROS)

0.1262

Step-5a:  The determination of net Present Rate (NPR) as follows:

CEVROS

determined above

0.1262

tax at 40%

as given

0.0505

ROSAT

(CEVROS-T)

0.0757

ITR

(IDEAL-S/I)

1.8

ROIAT

(ROSAT*ITR)

0.1363

add depreciation rate

% of Depreciation

0.18

CIFR

ROIAT+CIFR

0.3162

PV annuity at KI 12%

as per the table

3.605

PVCIFR

(PVA*CIFR)

1.1403

PV factor yr5 at 12%

as per the table

0.567

scrap rate

as given

0.10

PVCIF of scrap

scrap*PV'5

0.057

Total PVCIFR

(PVIFR+PVIF Scrap)

1.1973

certainty equivalent Net present value of cash inflow index rate (CEVNPVIR)

(1-PVIFR)

0.1973

B: when the uncertainty is confined to operating activities/results:

Step-3b & 4b:

forecasted scenario effects on costs, price as a proportionate change and volume in proportion of capacity utilization  as given

 sensitivity of ROS to the scenario changes in the variables (the extent of effect for a given change in each variable  to each of the distinct scenarios as the individual and net total  effect on the ROS of normal/ideal Revenues)

differential ratios of variable

slump
scenario

recession scenario

boom scenario

sensitivity
 constant

acronym of variable to represent the variable for their effect

slump
scenario

recession scenario

boom scenario

volume of sales

Ideal/normal

18000

18000

18000

scenario volume

10000

12000

20000

Vol

ΔVolR1

-0.444

-0.333

0.111

Vol

ΔVolR2

-0.800

-0.500

0.10

0.36

ΔVol

-0.267

-0.200

0.067

SP

ΔSPR1

-0.10

0.20

0.30

1.00

ΔSPR

-0.056

0.133

0.333

VC

ΔVCR1

-0.25

-0.10

0.125

0.40

ΔVCR

-0.056

-0.027

0.056

SFCa

ΔSFCRa1

-0.40

-0.30

0.15

0.07

ΔSFCRa

-0.028

-0.021

0.011

SFCb

ΔSFCRb1

-0.45

-0.12

0.12

0.06

ΔSFCRb

-0.027

-0.007

0.007

SFCc

ΔSFCRc1

-0.33

-0.20

0.18

0.05

ΔSFCRc

-0.017

-0.0100

0.009

SFCd

ΔSFCRd1

-0.10

0.12

0.15

0.08

ΔSFCRd

-0.008

0.0096

0.012

FC

ΔFCR1

0.00

0.00

0.00

0.10

ΔFCRe

0

0

0

ΔTFCR1∑ΔSFCR+ΔFCR

-1.280

-0.500

0.600

0.36

ΔTFCR

-0.0795

-0.0286

0.0387

net impact on ROS on the ideal/normal revenues

Ideal/normal

-0.187

-0.011

0.306

Scenario ROS after the scenario effects on the ideal/normal revenues

0.24

0.053

0.229

0.546

Tax @ 40%

0.096

0.0212

0.0916

0.2184

Scenario ROSAT after the scenario effects on the ideal/normal revenues

0.144

0.0318

0.1374

0.3276

ITR (IDEAL-S/I)

1.8

1.8

1.8

1.8

ROIAT

0.2592

0.05724

0.24732

0.58968

add depreciation Rate

0.18

0.18

0.18

0.18

CIFR

0.4392

0.23724

0.42732

0.76968

Probability

0.2

0.2

0.4

0.2

P weighted CFR

0.08784

0.047448

0.170928

0.153936

Mean of CIFR

0.08784+0.047448+0.170928+0.153936=0.4602

Step-5b:  expected returns on Ideal sales revenues: based on the information of random occurrences of the scenarios the Probability (P) is determined and the CEVCIFR at 3σ as:

(Mean of Cash Inflow Rate) Mean of CIFR

0.4602

Stn.Dev. σ of scenario CIFR

0.0576

Certainty equivalent CIFR: (Mean ROs) – ( 3*σ =3*0.0576 =0.1727) =(0.4602-0.1727)

0.2874

as per the PV Annuity  table @12% by the end of 5th year

3.605

(PVA*CIFR)

1.0362

as per the PV table @12% 5th year

0.567

Scrap rate: (scrap/I)

0.1

Scrap Rate*PV'5th yr.

0.057

(PVIFR+PVIF Scrap)

1.0932

NPICIFR (Net present Index of CIFR) (1-PVIFR)

0.0932

Verification of the ROS results (if required):

PARTICULARS

ideal/normal conditions
& at 90% capacity utilization: 18000 units

forecasted scenario effects on costs, price as a proportionate change and volume in proportion of capacity utilization 

verification of results

Rs. per unit
(as per the

ABC

costing)

total Rs.
/Ratio

slump
scenario (- indicates reduction & +indicates increase of price and costs)

recession scenario (- indicates reduction & +indicates increase of price and costs)

boom scenario(- indicates reduction & +indicates increase of price and costs)

slump
scenario

recession scenario

boom scenario

Volume/ level of activity

90%

capacity

18000

0.5

0.6

1

10000

12000

20000

Selling Price/Sales revenue

10

180000

-0.10

0.20

0.300

90000

144000

260000

through put cost or Variable Cost-(VC)

4

72000

-0.25

-0.10

0.125

30000

43200

90000

batch level costs

12600

-0.40

-0.30

0.150

7560

8820

14490

 product level costs-(SFC)

10800

-0.45

-0.12

0.120

5940

9504

12096

 firm/Facility level-(SFC)

9000

-0.33

-0.20

0.180

6030

7200

10620

(allocated to products and not absorbed to units) customer level-(SFC)

14400

-0.10

0.12

0.150

12960

16128

16560

Project Book Costs (depreciation)-(FC)

18000

0

0

0.000

18000

18000

18000

TFC(SFC+FC)

64800

 

 

 

50490

59652

71766

TC

136800

 

 

 

80490

102852

161766

Profit

43200

 

 

 

9510

41148

98234

ROS on ideal/normal revenues

0.240

 

 

 

0.053

0.229

0.546

ROS on respective scenario revenues

scenario revenue based ROS

0.106

0.286

0.378

Conclusion: it is to conclude that the IPPPA tool -2 is not only effective for decision making on operating aspects in progression but also provide the platform for capital budgeting decisions of long-term as a system of operating integrated investment decision analysis, being meaningfully unifying certainty based advanced capital budgeting techniques of sensitivity, scenario, break-even and integrated with the certainty equivalent technique used for uncertainty conditions. Therefore it is a compact technique applicable for both certainty and certainty conditions analysis. It is measuring the sensitivity/elasticity of returns with reference to the ideal/standard/normal/budgeted sales revenues enable to standardize the analysis to assess the effects/impacts of changes in any one or more variables at predicted different scenario conditions directly as the extent of the effect on the results more importantly in terms of proportions rather than the absolute amounts. Further, it is enabled to assess the effectiveness of the capital budgeting decision being assessed the returns & cash flows directly as a proportion to the Revenues of a standard reference and similarly the cash inflows to such investments all through the process of analysis.  It is able to relieve the misinterpretation of NPV at times of comparing the projects/proposals having distinct outlay/cost of capital/life periods and the assumption of IRR of deemed reinvestment of earlier period cash inflows at the IRR.

Further, the IPPPA Tool-1 is able to provide the important data of operating and investment risk premium analysis for operating and capital budgeting decisions under uncertain conditions. The application of proportion measurement is able to integrate the investment analysis with the operating market risk analysis viz. the demand, supply, competition analysis of managerial economic theory for strategic product life cycle analysis. Following cognizant hypothetical firm, data enable to provide the insights for the application of both the IPPPA tools for capital budgeting decision analysis for the product life cycle.

Note: if the uncertainty is applicable to cash-inflows then the probability will be applied at the cash inflows after taxes i.e. at the point of ROIAT+ depreciation.