department of economy and civil engineering risk … · suppose two investment projects ... between...

2009 SLOVAK UNIVERSITY OF TECHNOLOGY8

1 INTRODUCTION

The art of identifying and effectively managing risk in a timely manner has become an integral part of strategic management Property developers who are unable to identify the scope and consequences of related risks and who consequently fail to introduce efficient mechanisms to manage those risks clearly jeopardise the stability not only of their investments but also their overall success The level of risk rises in proportion to the level of the difficulty and scope of the whole investment Therefore these risks need to be examined predicted analysed and quantified as the investment is prepared and finalised Individual phases of the risk management process in property development are shown in Figure 1

2 PARAMETERS NECESSARY FOR RISK QUANTIFICATION AND RISK MODELLING

The quantification of risks is the part of risk analysis that describes and numerically evaluates the impact of potential threats The main goal of risk quantification is to estimate the frequency probability and significance of potential losses associated with a particular project In analysing risk property developers work with parameters that in most cases cannot be accurately measured Their values are often based on the estimates of experienced professionals For the purpose of risk quantification property developers need to determine the parameters as accurately as possible The following two basic parameters are used

J MATUacuteŠKA

RISK MODELLING IN THE PROPERTY DEVELOPMENT PROCESS

KEY WORDS

bull Risk bull Development process bull Gross revenue bull Analysis methodbull Sensitivity analysisbull Risk managementbull Property development

ABSTRACT

The property development process is accompanied by risks which influence this process in each and every phase For this reason property developers constantly endeavour to examine analyse and evaluate the given risks and take precautios to minimize them An essential part of examining the risks is risk analysis which helps property developers to measure such risks ie to quantify them To express the parameters of risk analysis property developers use different methods as a basis In regular practice there are two most commonly used groups of methods namely qualitative and quantitative methods On the one hand former are easier and faster to work with on the other hand they are more subjective The latter are based on the numerical calculation of risks The group of quantitative methods also comprises the sensitivity analysis shown in the concrete example as seen below

Jaacuten MatuacuteškaDepartment of Economy and Civil Engineering Management Faculty of Civil Engineering SvF STU Radlinskeacuteho 11 81368 Bratislava

Research field Risk Management in the Property Development Process

20092 PAGES 8 ndash 16 RECEIVED 15 10 2008 ACCEPTED 31 3 2009

20092 PAGES 8 mdash 16

9RISK MODELLING IN THE PROPERTY DEVELOPMENT PROCESS

a) probability of risk occurrence and b) estimated (expected) value of the loss associated with the risk

a) Probability of the occurrence of the risk is closely related to the term lsquodegree of riskrsquo Phenomena with highly probable losses are regarded as more risky than those with lower probabilities If we view risk as the likelihood of an unfavourable deviation from the expected and desired result then the degree of risk will be measured based on the probability of such unfavourable deviation occurring Probabilities of accidents in road air and sea transport offer one example Road transport is accountable for the highest share of accidents - about 60 - followed by sea and air transport with a 30 and 10 share respectively By applying this concept of probability as an unfavourable deviation from the desired result we perceive that the risk of accident rate in sea transport is higher than air transport but at the same time lower than road transport The higher the probability of an unfavourable event the higher the probability of a deviation from the desired result hence the higher our risk b) Estimated (expected) value of loss on an investment is based on two parameters probability of the occurrence of a loss and magnitude of the potential loss For example if the amount of 20 monetary units (MU) is subject to risk and the probability of loss is 03 then the probable value of the loss will be 6 MU An important factor here is time Both the value of a loss and the probability of an event the occurrence change over time The amount of probable loss Z(t) within the time interval lang0 T0rang can be calculated as

(1)

where

r(t) ndash risk function in time t expressed as the probability within the interval lang0 1rang

v(t) ndash loss function in time t (in practice a jump function is often used and assumes values of 0 or 1) [12]

Z(t) ndash magnitude of probable loss within the time interval lang0 T0rang which we attempt to optimise ie to find the minimum of the given function Z(t)

Our goal in Figure 2 is to minimise the volume under the Z(t) area The amount of vacation which is taken by a developerrsquos staff may serve as an example During a year vacations are usually taken in June July and August Therefore these months represent the highest loss for the company (bad management decisions cost of wages etc) The Z(t) function will therefore assume its highest values in the aforementioned months An integral part of risk quantification is to determine the statistical characteristics of risks The most widely used instruments for risk quantification include maximum expected return root mean square deviation coefficient of variation and variance The most common method used in analysing risk is based on the criterion of expected average profit as the indicator of estimated profitability combined with the variance or root mean square deviation of the profits as the indicator of risk

Expected returnTo compare the profitability of alternative investments within a given risk environment a new measure needs to be introduced - expected return E(X) - which is defined as the mean value of the returns

(2)

where

Risk management process in property

development

Identify objective of risk analysis

Analyse risks Evaluate risksDevelop and propose

measures

Identify risks Quantify risksSelect appropriate

risk analysis method

Fig 1 Overview of risk management processes in property development

Fig 2 Magnitude of probable loss Z(t) within the time interval lang0 T0rang

10 RISK MODELLING IN THE PROPERTY DEVELOPMENT PROCESS


X ndash random variable which is the attained return xi ndash the i-th attained return pi ndash probability of attaining the value of xi

The expected return measures only the potential amount of the return (profit) Therefore it may only be regarded as an indicator of the return (profit) It does not say anything about the certainty of the achieved amount of the return (profit) or the risk associated with such a return (profit) In most cases as the following example shows the expected return cannot be the basis for making a decision as to whether a particular investment is appropriate Suppose two investment projects A and B with xi returns and pi probabilities of attaining such returns as shown in Table 1 below

The investor cannot make a decision based on the criterion of maximum expected return since the calculation (2) above yields expected returns of 1620 MU on both investments As Table 1 shows the return for Project A is less stable - hence more risky - than Project B This fact both justifies and explains the need to introduce an additional criterion (variance) to consider both the profitability and risk associated with each investment project

VarianceVariance is the measure of the dispersion of returns around their expected value It represents the range of potential deviations of the actual return from the expected return The more the actual return differs from the expected return the greater the deviation Variance σ2(X) is defined as the mean value of the square of the deviation of potential returns from the expected value E(X) ie

(3)

where X - random variable which is the attained return xi - value of i-th attained return pi- probability of attaining such return E - value of the maximum expected return on a given parameter during a definite period

As the expected return is expressed as an absolute value in monetary units we will use the root mean square deviation σ(X) in evaluating the results The root mean square deviation is defined as the principal square root of the variance Thus it has the same lsquodimensionsrsquo as the expected return It is evident that a risk-averse investor deciding between two investments with equal expected returns will choose the one with a lower variance of returns

Root mean square deviationThe root mean square deviation can be expressed as the square root of variance

(4)

The root mean square deviation is unquestionably the most common instrument used in measuring risk as it gives a result in the same units as the expected return This indicator may for example represent the maximum amount by which the price of an investment instrument might change during a defined period of time (eg one day) at a specific probability level Thus for an investment project with a mean value of 2000 MU and a root mean square deviation of 100 MU (plusmn 5) the minimum and maximum values will be 1900 MU and 2100 MU respectively Should the root mean square deviation assume higher values the mean value would then become unstable Though there exists a chance of a higher return potential losses would be far more painful

Mean value ndash Coefficient of variationThe coefficient of the variation expresses the level of risk σ per one unit of the expected return E thus

(5)

Suppose the returns on investments A and B as the random variables X and Y Based on the mean valuecoefficient of the variation rule a risk-averse investor will choose investment A if

The coefficient of the variation expresses the fluctuation of potential revenues with respect to the expected return The lower the ratio the more favourable the project seems to be At the same time as shown in the example of the two investments in Table 2 below the amount of the expected return E(X) which relates to the resulting ratio is also important Clearly investment B is more profitable than investment A as even the worst potential result for investment B is better than the return offered by alternative A

Table 1 Expected return on two investmentsA B E(Xa) E(Xb)

Xi Pi Xi Pi 100 02 1100 02 20 220

2000 08 1750 08 1600 1400Total 1620 1620



The following property developerrsquos decision about investments in three different markets may serve as an example of using these instruments Based on the experience of property development professionals the companyrsquos management has made estimates of the probability of making a profit on the sale of a residential building in three different markets as shown in Table 3 The property developerrsquos task is to determine which of these three markets is the best option for this particular investmentBelow is the calculation of the expected return during one period for all three markets

Market 1

MU

Market 2

MU

Market 3

MU

Below is the calculation of the variance for individual markets

Market 1

UM

Market 2

MU

Market 3

MU

Calculation of the root mean square deviation for each market Market 1

MU

Market 2

MU

Market 3 MU

Calculation of the coefficient of the variation for each market

Market 1

Market 2

Market 3

A summary of the calculations is shown in Table 4Based on the overall evaluation shown in Table 4 and Chart 1 it is clear that the best market for the property developer is Market 3 The decision to invest in Market 3 was made based on the highest expected return (26800 MU) at an acceptable level of risk ie 3054In the preceding paragraphs we have analysed risky investments without considering the element of time Investors usually make

Table 2 Comparison of investmentsInvestment A Investment B

Return xi Probability pi Return xi Probability pi

3 07 15 08E(X) σ(X) C(X) E(Y) σ(Y) C(Y)21 09 042 12 3 025

Table 3 Estimated probability of making profit on the sale of flats Market 1 Market 2 Market 3

SaleReturn (MU)

Proba-bility

Return (MU)

Proba-bility

Return (MU)

Proba-bility

Weak 15000 05 10000 03 17000 06Normal 20000 06 15000 07 22000 04

Excellent 25000 02 20000 03 26000 03



investment decisions for periods of two or three years One of the most basic instruments used in making such decisions are the criteria of net present value (NPV) and the internal rate of return (IRR) Both are time discounted methods of measuring the benefit of investments They are based on the assumption that we have an exact knowledge of the cost and return of investment alternatives

The net present value is the difference between the total expected cash flows in individual years over a projects lifecycle discounted to their present value using the applicable discount rate and initial cost of the investment ie

(6)

where Ct ndash net present value of cash flows at the end of the t-th period

I0 ndash initial cost of investment k ndash discount rate (expected return on investment) n ndash period of project lifecycleUnder the assumption that the goal of the company is to maximise shareholder value the companyrsquos decision based on the NPV criterion will be as follows If NPV is positive the project is acceptable if NPV is negative the project is rejected andif there are several investment options the project with the highest NPV is chosen Internal rate of return (IRR) is the discount rate R at which the net present value of the net cash flows Ct equals its initial investment cost I0 Thus the IRR is equal to the value of R that expresses the projectrsquos internal rate of return as calculated by the following equation

(7)

When evaluating several mutually independent investment projects both criteria provide the same results ie lead to the same investment decision (ierejecting or accepting the project) The use of IRR is based on the following analysis The project is acceptable if it has a positive NPV ie if

(8)

In such a case the IRR of the same project equals the value of R achieved by substituting equation (7) into (8) Thus

then R gt k

If the net present value of the project is positive R will be higher than k If the net present value equals zero then R is equal to k Finally in the case of a negative net present value R is lower than k Unlike NPV which may vary depending on the selected discount rate the projectrsquos IRR remains fixed The criterion for making a decision using IRR is therefore based on the comparison of the discount rate k and the minimum desired rate of return RIf R gt k the project is acceptableIf R lt k the project is rejectedThus the project is acceptable if IRR gt kProblems can occur when using the above two profitability measures in the case of mutually dependent investment projects as the NPV and IRR methods can yield differing project rankings These differences may result from the scope of an investment (which is

Table 4 Results calculated based on given parameters in three different markets MARKET 1 MARKET 2 MARKET 3Maximum expected return E

24500 19500 26800

Variance (in thousand SKK)

57325 41325 67032

Standard deviation (SKK)

757132 642845 818730

Coefficient of variation C ()

3090 3296 3054

Chart 1 Results calculated based on given parameters

Maximum expected return E

Variance σ2

Root mean square deviation (SKK) σ

Coefficient of variation ()

MARKET 1 MARKET 2 MARKET 3

Risk Quantification Instruments



not taken into account by IRR) the timing of cash flows or differing reinvestment assumptions NPV is generally regarded as a better criterion since unlike IRR it reflects the amount of profit expressed in absolute terms and not only profitability

3 SENSITIVITY ANALYSIS

Sensitivity analysis appears to be the most commonly used project evaluation method

Task Suppose a property developerrsquos investment of SKK 100 million for a period of two years t = 1 2 with a zero initial cost and an expected rate of return at 15 which already includes inflation The developerrsquos best estimates of cash flows are summarised in Table 5 Gross revenue R represents the cash flow from the investment over a period of one year

By this analysis the company is aiming for the best possible estimate of its revenues and cost Based on the resulting estimates NPV and IRR are calculated Subsequently the sensitivity of NPV to potential errors in gross revenue and individual cost item estimates is examined The following example explains the principle of sensitivity analysis First we will make the companyrsquos estimate of the projectrsquos NPV

(9)

where Tc - tax

and

R = 100 million C1 = 42505 million C2 = 212 million C3 = 403 million C4 = 36345 million (SKK) then

-

-

- - = 081100 16275 - 081

42505 16275 - 081 212 16275 - 081 403 16275 - 081 36345 16275 = 1318275 - 56033 - 2791 - 5312 - 4791 = 1975 million (SKK)

Ceteris paribus the new estimate of the net present value NPVα will be

(10)

where α = 1215In equation (10) the numerator R will be stepwise substituted by the values C1 C4 provided in Table 5

i = 1 4

It is clear that ifα gt 0 rArr NPVα gt NPV α lt 0 rArr NPVα lt NPV

Due to increasing prices we will only suppose the case α gt 0 ie NPVα gt NPV for α = 1215

Table 5 Structure of the investmentrsquos cost and cash flowsGross revenue R 100 millionCostLand C1 42505 millionEngineering C2 212 millionProject design C3 403 millionContractor C4 36345 millionTotal cost Σ Ci 85 millionPre-tax annual return 15 millionTax 19 on SKK 15 million 285 millionNet annual return 1215 million

Chart 2 Impact of the change in α on the gross revenue and cost items C1 C4



The objective of the analysis will be attained if the property developer does not decrease the 15 expected rate of return If the individual cost items change the NPV of the project will change as well Gross revenue depends on the inflation rate which according to the National Bank of Slovakia (NBS) is currently at 2-3 Thus ceteris paribus NPV will grow as described in Table 6 ndash ie from SKK 1975 million to SKK 3950 million If the value of the considered cost C1 to C4 grows at a rate of 2-3 such growth will be significantly lower than the growth at a gross revenue of R since C1 to C4 represent lower values hence their changes will be lower as well If the prices of individual cost items were to grow at different rates we would first adjust NPV for the given α and the appropriate cost item for NPVα We would then calculateNPVα with respect to changes in the next cost item Once we incorporated all the changes the resulting NPVα would give us the net present value of the whole property development process With the currently assumed two year investment period a situation might ensue where

the price of land grows by as much as 50 and the cost items C1 to C4 by a maximum of 2-7 on an annual basis Our calculation is based on the data shown in Table 7The following holds for R at a growth rate of α = 3 (see Table 7)NPVα = 1975 + 395048 = SKK 2373048 millionFor other calculations NPV will be replaced by NPVα As per Table 7 we will set NPV growth as followsC1 ndash land (50 annual growth α1 = 05) 27985 (Table 7)C2 ndash engineering (2 annual growth α2 = 002) 0055 (Table 7)C3 ndash project design (4 annual growth α3 = 004) 0212 (Table 7)C4 ndash contractor (7 annual growth α4 = 007) 3350 (Table 7)

For cost items C1 to C4 the NPV will grow by

= 081(27985+0055+0212+3350) =

08131602=25597

Table 6 Sensitivity analysis of the projectrsquos NPV with respect to deviations αEstimate error Revenue R Land Engineering Project design Contractor

α

R C1 C2 C3 C4

000 1975000 1975000 1975000 1975000 1975000001 210668251 2030972 1977792 1980307 2022860002 2238365 2086943 1980583 1985614 2070720003 237004753 2142915 1983375 1990920 2118580004 2501730 2198887 1986167 1996227 2166440005 263341255 2254858 1988958 2001534 2214300006 2765095 2310830 1991750 2006841 2262160007 289677757 2366802 1994542 2012148 2310020008 3028460 2422773 1997333 2017454 2357880009 316014259 2478745 2000125 2022761 240574001 3291825 2534717 2002917 2028068 2453600011 342350761 2590688 2005708 2033375 2501460012 3555190 2646660 2008500 2038682 2549320013 368687263 2702631 2011292 2043988 2597180014 3818555 2758603 2014083 2049295 2645040015 395023765 2814575 2016875 2054602 269290002 4608650 3094433 2030833 2081136 293220003 59254753 3654150 2058750 2134204 341080004 7242300 4213866 2086667 2187272 388940005 85591255 4773583 2114583 2240340 4368000



If NPVR is the change in the net present value of the gross revenue and NPVC1C2C3C4

is the change in the net present value with respect to C1 C4 for a relevant α then

Thus the net present value of this property development project is negative and the internal rate of return (IRR) is lower than the expected rate of return of 15 Therefore the management of the property development project has two options either to decrease the expected return on capital or to request the investment company to increase the gross revenue R to a level corresponding to a positive net present value (NPV)

5 CONCLUSION

The art of mastering risk management in the property development

process represents a successful investment for all property developers Risk management embraces also risk analysis part of which is risk quantification itself The risk quantification enables property developers to measure risks and consequently to quantify their value The risk quantification is of cardinal importance for property developers especially when evaluating the risks of any investment project The use of statistical parameters such as the expected return variance root mean square deviation and mean value ndash coefficient of variation may serve as an example In addition to the above mentioned statistical parameters a sensitivity analysis method is frequently used as well This method helps property developers to determine the estimate of the total revenue and cost for any concrete project as accurately as possible By means of the net present value and its sensitivity to potential errors in gross revenue and individual cost item estimates property developers decide on possible investment realization

Table 7 Sensitivity analysis of the projectrsquos NPV with respect to deviation α for individual items

Estimation error

α R C1 C2 C3 C4

000 000000 000000 000000 000000 000000001 131683 055972 002792 005307 047860002 263365 111943 005583 010614 095720003 395048 167915 008375 015920 143580004 526730 223887 011167 021227 191440005 658413 279858 013958 026534 239300006 790095 335830 016750 031841 287160007 921778 391802 019542 037148 335020008 1053460 447773 022333 042454 382880009 1185143 503745 025125 047761 43074001 1316825 559717 027917 053068 478600011 1448508 615688 030708 058375 526460012 1580190 671660 033500 063682 574320013 1711873 727631 036292 068988 622180014 1843555 783603 039083 074295 670040015 1975238 839575 041875 079602 71790002 2633650 1119433 055833 106136 95720003 3950475 1679150 083750 159204 143580004 5267300 2238866 111667 212272 191440005 6584126 2798583 139583 265340 2393000



REFERENCES

[1] McCutcheoon J J ndash Scott W F An Intoduction to the Mathematics of Finance Clays Ltd St Ives plc1993 ISBN 0 7506 0388 7

[2] Levy H ndash Sarnat M Capital Investment and Financial Decision Prentice Hall 1993 SBN 0-13-115585-7

[3] Pinda Ľ - Starečkovaacute A Redukci rizika poisteniacutem Vydavateľstvo Ekonom Bratislava 2006 ISBN 80-225-2153-3

[4] Saunders AFinancial Institutions Management Irving McGraw-Hill 1997 ISBN 0-256-15367-1

[5] Zuacutebkovaacute M Marketing nehnuteľnostiacute In Trh nehnuteľnostiacute a developeacutersky proces STU Bratislava 2007 ISBN 978-80-227-2661-0 strana 70

[6] Miles E Berens Real Estate Development Principles and Process 3d edi Washington DC Uli ndash Ther Urban Land Institute 2000

[7] Seldin M Swenk R H Real Estate Investment Strategy The American University 1970

[8] Francis JC Management of Investment West Publishing Company StPaul 1994 ISBN 0-314-02822-6

[9] Johnson F H Real Estate Principles and Practices (third edition) Merril Publihsing Company Bell and Howell Company Columbus Ohio 43216 1983

[10] Smejkal V Rais K Řiacutezeniacute rizik MBA Grada Publishing as Praque 2003 ISBN 80-247-0198-7



a) probability of risk occurrence and b) estimated (expected) value of the loss associated with the risk

a) Probability of the occurrence of the risk is closely related to the term lsquodegree of riskrsquo Phenomena with highly probable losses are regarded as more risky than those with lower probabilities If we view risk as the likelihood of an unfavourable deviation from the expected and desired result then the degree of risk will be measured based on the probability of such unfavourable deviation occurring Probabilities of accidents in road air and sea transport offer one example Road transport is accountable for the highest share of accidents - about 60 - followed by sea and air transport with a 30 and 10 share respectively By applying this concept of probability as an unfavourable deviation from the desired result we perceive that the risk of accident rate in sea transport is higher than air transport but at the same time lower than road transport The higher the probability of an unfavourable event the higher the probability of a deviation from the desired result hence the higher our risk b) Estimated (expected) value of loss on an investment is based on two parameters probability of the occurrence of a loss and magnitude of the potential loss For example if the amount of 20 monetary units (MU) is subject to risk and the probability of loss is 03 then the probable value of the loss will be 6 MU An important factor here is time Both the value of a loss and the probability of an event the occurrence change over time The amount of probable loss Z(t) within the time interval lang0 T0rang can be calculated as

(1)

where

r(t) ndash risk function in time t expressed as the probability within the interval lang0 1rang

v(t) ndash loss function in time t (in practice a jump function is often used and assumes values of 0 or 1) [12]

Z(t) ndash magnitude of probable loss within the time interval lang0 T0rang which we attempt to optimise ie to find the minimum of the given function Z(t)

Our goal in Figure 2 is to minimise the volume under the Z(t) area The amount of vacation which is taken by a developerrsquos staff may serve as an example During a year vacations are usually taken in June July and August Therefore these months represent the highest loss for the company (bad management decisions cost of wages etc) The Z(t) function will therefore assume its highest values in the aforementioned months An integral part of risk quantification is to determine the statistical characteristics of risks The most widely used instruments for risk quantification include maximum expected return root mean square deviation coefficient of variation and variance The most common method used in analysing risk is based on the criterion of expected average profit as the indicator of estimated profitability combined with the variance or root mean square deviation of the profits as the indicator of risk

Expected returnTo compare the profitability of alternative investments within a given risk environment a new measure needs to be introduced - expected return E(X) - which is defined as the mean value of the returns

(2)

where

Risk management process in property

development

Identify objective of risk analysis

Analyse risks Evaluate risksDevelop and propose

measures

Identify risks Quantify risksSelect appropriate

risk analysis method

Fig 1 Overview of risk management processes in property development

Fig 2 Magnitude of probable loss Z(t) within the time interval lang0 T0rang







(3)




(4)



(5)




Xi Pi Xi Pi 100 02 1100 02 20 220

2000 08 1750 08 1600 1400Total 1620 1620




Market 1

MU

Market 2

MU

Market 3

MU


Market 1

UM

Market 2

MU

Market 3

MU


MU

Market 2

MU

Market 3 MU


Market 1

Market 2

Market 3




3 07 15 08E(X) σ(X) C(X) E(Y) σ(Y) C(Y)21 09 042 12 3 025


SaleReturn (MU)

Proba-bility

Return (MU)

Proba-bility

Return (MU)

Proba-bility

Weak 15000 05 10000 03 17000 06Normal 20000 06 15000 07 22000 04

Excellent 25000 02 20000 03 26000 03





(6)



(7)


(8)


then R gt k



24500 19500 26800


57325 41325 67032


757132 642845 818730


3090 3296 3054



Variance σ2












(9)

where Tc - tax

and


-

-

- - = 081100 16275 - 081

42505 16275 - 081 212 16275 - 081 403 16275 - 081 36345 16275 = 1318275 - 56033 - 2791 - 5312 - 4791 = 1975 million (SKK)


(10)


i = 1 4










= 081(27985+0055+0212+3350) =

08131602=25597


α

R C1 C2 C3 C4

000 1975000 1975000 1975000 1975000 1975000001 210668251 2030972 1977792 1980307 2022860002 2238365 2086943 1980583 1985614 2070720003 237004753 2142915 1983375 1990920 2118580004 2501730 2198887 1986167 1996227 2166440005 263341255 2254858 1988958 2001534 2214300006 2765095 2310830 1991750 2006841 2262160007 289677757 2366802 1994542 2012148 2310020008 3028460 2422773 1997333 2017454 2357880009 316014259 2478745 2000125 2022761 240574001 3291825 2534717 2002917 2028068 2453600011 342350761 2590688 2005708 2033375 2501460012 3555190 2646660 2008500 2038682 2549320013 368687263 2702631 2011292 2043988 2597180014 3818555 2758603 2014083 2049295 2645040015 395023765 2814575 2016875 2054602 269290002 4608650 3094433 2030833 2081136 293220003 59254753 3654150 2058750 2134204 341080004 7242300 4213866 2086667 2187272 388940005 85591255 4773583 2114583 2240340 4368000






5 CONCLUSION




Estimation error

α R C1 C2 C3 C4

000 000000 000000 000000 000000 000000001 131683 055972 002792 005307 047860002 263365 111943 005583 010614 095720003 395048 167915 008375 015920 143580004 526730 223887 011167 021227 191440005 658413 279858 013958 026534 239300006 790095 335830 016750 031841 287160007 921778 391802 019542 037148 335020008 1053460 447773 022333 042454 382880009 1185143 503745 025125 047761 43074001 1316825 559717 027917 053068 478600011 1448508 615688 030708 058375 526460012 1580190 671660 033500 063682 574320013 1711873 727631 036292 068988 622180014 1843555 783603 039083 074295 670040015 1975238 839575 041875 079602 71790002 2633650 1119433 055833 106136 95720003 3950475 1679150 083750 159204 143580004 5267300 2238866 111667 212272 191440005 6584126 2798583 139583 265340 2393000



REFERENCES

















(3)




(4)



(5)




Xi Pi Xi Pi 100 02 1100 02 20 220

2000 08 1750 08 1600 1400Total 1620 1620




Market 1

MU

Market 2

MU

Market 3

MU


Market 1

UM

Market 2

MU

Market 3

MU


MU

Market 2

MU

Market 3 MU


Market 1

Market 2

Market 3




3 07 15 08E(X) σ(X) C(X) E(Y) σ(Y) C(Y)21 09 042 12 3 025


SaleReturn (MU)

Proba-bility

Return (MU)

Proba-bility

Return (MU)

Proba-bility

Weak 15000 05 10000 03 17000 06Normal 20000 06 15000 07 22000 04

Excellent 25000 02 20000 03 26000 03





(6)



(7)


(8)


then R gt k



24500 19500 26800


57325 41325 67032


757132 642845 818730


3090 3296 3054



Variance σ2












(9)

where Tc - tax

and


-

-

- - = 081100 16275 - 081

42505 16275 - 081 212 16275 - 081 403 16275 - 081 36345 16275 = 1318275 - 56033 - 2791 - 5312 - 4791 = 1975 million (SKK)


(10)


i = 1 4










= 081(27985+0055+0212+3350) =

08131602=25597


α

R C1 C2 C3 C4

000 1975000 1975000 1975000 1975000 1975000001 210668251 2030972 1977792 1980307 2022860002 2238365 2086943 1980583 1985614 2070720003 237004753 2142915 1983375 1990920 2118580004 2501730 2198887 1986167 1996227 2166440005 263341255 2254858 1988958 2001534 2214300006 2765095 2310830 1991750 2006841 2262160007 289677757 2366802 1994542 2012148 2310020008 3028460 2422773 1997333 2017454 2357880009 316014259 2478745 2000125 2022761 240574001 3291825 2534717 2002917 2028068 2453600011 342350761 2590688 2005708 2033375 2501460012 3555190 2646660 2008500 2038682 2549320013 368687263 2702631 2011292 2043988 2597180014 3818555 2758603 2014083 2049295 2645040015 395023765 2814575 2016875 2054602 269290002 4608650 3094433 2030833 2081136 293220003 59254753 3654150 2058750 2134204 341080004 7242300 4213866 2086667 2187272 388940005 85591255 4773583 2114583 2240340 4368000






5 CONCLUSION




Estimation error

α R C1 C2 C3 C4

000 000000 000000 000000 000000 000000001 131683 055972 002792 005307 047860002 263365 111943 005583 010614 095720003 395048 167915 008375 015920 143580004 526730 223887 011167 021227 191440005 658413 279858 013958 026534 239300006 790095 335830 016750 031841 287160007 921778 391802 019542 037148 335020008 1053460 447773 022333 042454 382880009 1185143 503745 025125 047761 43074001 1316825 559717 027917 053068 478600011 1448508 615688 030708 058375 526460012 1580190 671660 033500 063682 574320013 1711873 727631 036292 068988 622180014 1843555 783603 039083 074295 670040015 1975238 839575 041875 079602 71790002 2633650 1119433 055833 106136 95720003 3950475 1679150 083750 159204 143580004 5267300 2238866 111667 212272 191440005 6584126 2798583 139583 265340 2393000



REFERENCES














Market 1

MU

Market 2

MU

Market 3

MU


Market 1

UM

Market 2

MU

Market 3

MU


MU

Market 2

MU

Market 3 MU


Market 1

Market 2

Market 3




3 07 15 08E(X) σ(X) C(X) E(Y) σ(Y) C(Y)21 09 042 12 3 025


SaleReturn (MU)

Proba-bility

Return (MU)

Proba-bility

Return (MU)

Proba-bility

Weak 15000 05 10000 03 17000 06Normal 20000 06 15000 07 22000 04

Excellent 25000 02 20000 03 26000 03





(6)



(7)


(8)


then R gt k



24500 19500 26800


57325 41325 67032


757132 642845 818730


3090 3296 3054



Variance σ2












(9)

where Tc - tax

and


-

-

- - = 081100 16275 - 081

42505 16275 - 081 212 16275 - 081 403 16275 - 081 36345 16275 = 1318275 - 56033 - 2791 - 5312 - 4791 = 1975 million (SKK)


(10)


i = 1 4










= 081(27985+0055+0212+3350) =

08131602=25597


α

R C1 C2 C3 C4

000 1975000 1975000 1975000 1975000 1975000001 210668251 2030972 1977792 1980307 2022860002 2238365 2086943 1980583 1985614 2070720003 237004753 2142915 1983375 1990920 2118580004 2501730 2198887 1986167 1996227 2166440005 263341255 2254858 1988958 2001534 2214300006 2765095 2310830 1991750 2006841 2262160007 289677757 2366802 1994542 2012148 2310020008 3028460 2422773 1997333 2017454 2357880009 316014259 2478745 2000125 2022761 240574001 3291825 2534717 2002917 2028068 2453600011 342350761 2590688 2005708 2033375 2501460012 3555190 2646660 2008500 2038682 2549320013 368687263 2702631 2011292 2043988 2597180014 3818555 2758603 2014083 2049295 2645040015 395023765 2814575 2016875 2054602 269290002 4608650 3094433 2030833 2081136 293220003 59254753 3654150 2058750 2134204 341080004 7242300 4213866 2086667 2187272 388940005 85591255 4773583 2114583 2240340 4368000






5 CONCLUSION




Estimation error

α R C1 C2 C3 C4

000 000000 000000 000000 000000 000000001 131683 055972 002792 005307 047860002 263365 111943 005583 010614 095720003 395048 167915 008375 015920 143580004 526730 223887 011167 021227 191440005 658413 279858 013958 026534 239300006 790095 335830 016750 031841 287160007 921778 391802 019542 037148 335020008 1053460 447773 022333 042454 382880009 1185143 503745 025125 047761 43074001 1316825 559717 027917 053068 478600011 1448508 615688 030708 058375 526460012 1580190 671660 033500 063682 574320013 1711873 727631 036292 068988 622180014 1843555 783603 039083 074295 670040015 1975238 839575 041875 079602 71790002 2633650 1119433 055833 106136 95720003 3950475 1679150 083750 159204 143580004 5267300 2238866 111667 212272 191440005 6584126 2798583 139583 265340 2393000



REFERENCES















(6)



(7)


(8)


then R gt k



24500 19500 26800


57325 41325 67032


757132 642845 818730


3090 3296 3054



Variance σ2












(9)

where Tc - tax

and


-

-

- - = 081100 16275 - 081

42505 16275 - 081 212 16275 - 081 403 16275 - 081 36345 16275 = 1318275 - 56033 - 2791 - 5312 - 4791 = 1975 million (SKK)


(10)


i = 1 4










= 081(27985+0055+0212+3350) =

08131602=25597


α

R C1 C2 C3 C4

000 1975000 1975000 1975000 1975000 1975000001 210668251 2030972 1977792 1980307 2022860002 2238365 2086943 1980583 1985614 2070720003 237004753 2142915 1983375 1990920 2118580004 2501730 2198887 1986167 1996227 2166440005 263341255 2254858 1988958 2001534 2214300006 2765095 2310830 1991750 2006841 2262160007 289677757 2366802 1994542 2012148 2310020008 3028460 2422773 1997333 2017454 2357880009 316014259 2478745 2000125 2022761 240574001 3291825 2534717 2002917 2028068 2453600011 342350761 2590688 2005708 2033375 2501460012 3555190 2646660 2008500 2038682 2549320013 368687263 2702631 2011292 2043988 2597180014 3818555 2758603 2014083 2049295 2645040015 395023765 2814575 2016875 2054602 269290002 4608650 3094433 2030833 2081136 293220003 59254753 3654150 2058750 2134204 341080004 7242300 4213866 2086667 2187272 388940005 85591255 4773583 2114583 2240340 4368000






5 CONCLUSION




Estimation error

α R C1 C2 C3 C4

000 000000 000000 000000 000000 000000001 131683 055972 002792 005307 047860002 263365 111943 005583 010614 095720003 395048 167915 008375 015920 143580004 526730 223887 011167 021227 191440005 658413 279858 013958 026534 239300006 790095 335830 016750 031841 287160007 921778 391802 019542 037148 335020008 1053460 447773 022333 042454 382880009 1185143 503745 025125 047761 43074001 1316825 559717 027917 053068 478600011 1448508 615688 030708 058375 526460012 1580190 671660 033500 063682 574320013 1711873 727631 036292 068988 622180014 1843555 783603 039083 074295 670040015 1975238 839575 041875 079602 71790002 2633650 1119433 055833 106136 95720003 3950475 1679150 083750 159204 143580004 5267300 2238866 111667 212272 191440005 6584126 2798583 139583 265340 2393000



REFERENCES


















(9)

where Tc - tax

and


-

-

- - = 081100 16275 - 081

42505 16275 - 081 212 16275 - 081 403 16275 - 081 36345 16275 = 1318275 - 56033 - 2791 - 5312 - 4791 = 1975 million (SKK)


(10)


i = 1 4










= 081(27985+0055+0212+3350) =

08131602=25597


α

R C1 C2 C3 C4

000 1975000 1975000 1975000 1975000 1975000001 210668251 2030972 1977792 1980307 2022860002 2238365 2086943 1980583 1985614 2070720003 237004753 2142915 1983375 1990920 2118580004 2501730 2198887 1986167 1996227 2166440005 263341255 2254858 1988958 2001534 2214300006 2765095 2310830 1991750 2006841 2262160007 289677757 2366802 1994542 2012148 2310020008 3028460 2422773 1997333 2017454 2357880009 316014259 2478745 2000125 2022761 240574001 3291825 2534717 2002917 2028068 2453600011 342350761 2590688 2005708 2033375 2501460012 3555190 2646660 2008500 2038682 2549320013 368687263 2702631 2011292 2043988 2597180014 3818555 2758603 2014083 2049295 2645040015 395023765 2814575 2016875 2054602 269290002 4608650 3094433 2030833 2081136 293220003 59254753 3654150 2058750 2134204 341080004 7242300 4213866 2086667 2187272 388940005 85591255 4773583 2114583 2240340 4368000






5 CONCLUSION




Estimation error

α R C1 C2 C3 C4

000 000000 000000 000000 000000 000000001 131683 055972 002792 005307 047860002 263365 111943 005583 010614 095720003 395048 167915 008375 015920 143580004 526730 223887 011167 021227 191440005 658413 279858 013958 026534 239300006 790095 335830 016750 031841 287160007 921778 391802 019542 037148 335020008 1053460 447773 022333 042454 382880009 1185143 503745 025125 047761 43074001 1316825 559717 027917 053068 478600011 1448508 615688 030708 058375 526460012 1580190 671660 033500 063682 574320013 1711873 727631 036292 068988 622180014 1843555 783603 039083 074295 670040015 1975238 839575 041875 079602 71790002 2633650 1119433 055833 106136 95720003 3950475 1679150 083750 159204 143580004 5267300 2238866 111667 212272 191440005 6584126 2798583 139583 265340 2393000



REFERENCES
















= 081(27985+0055+0212+3350) =

08131602=25597


α

R C1 C2 C3 C4

000 1975000 1975000 1975000 1975000 1975000001 210668251 2030972 1977792 1980307 2022860002 2238365 2086943 1980583 1985614 2070720003 237004753 2142915 1983375 1990920 2118580004 2501730 2198887 1986167 1996227 2166440005 263341255 2254858 1988958 2001534 2214300006 2765095 2310830 1991750 2006841 2262160007 289677757 2366802 1994542 2012148 2310020008 3028460 2422773 1997333 2017454 2357880009 316014259 2478745 2000125 2022761 240574001 3291825 2534717 2002917 2028068 2453600011 342350761 2590688 2005708 2033375 2501460012 3555190 2646660 2008500 2038682 2549320013 368687263 2702631 2011292 2043988 2597180014 3818555 2758603 2014083 2049295 2645040015 395023765 2814575 2016875 2054602 269290002 4608650 3094433 2030833 2081136 293220003 59254753 3654150 2058750 2134204 341080004 7242300 4213866 2086667 2187272 388940005 85591255 4773583 2114583 2240340 4368000






5 CONCLUSION




Estimation error

α R C1 C2 C3 C4

000 000000 000000 000000 000000 000000001 131683 055972 002792 005307 047860002 263365 111943 005583 010614 095720003 395048 167915 008375 015920 143580004 526730 223887 011167 021227 191440005 658413 279858 013958 026534 239300006 790095 335830 016750 031841 287160007 921778 391802 019542 037148 335020008 1053460 447773 022333 042454 382880009 1185143 503745 025125 047761 43074001 1316825 559717 027917 053068 478600011 1448508 615688 030708 058375 526460012 1580190 671660 033500 063682 574320013 1711873 727631 036292 068988 622180014 1843555 783603 039083 074295 670040015 1975238 839575 041875 079602 71790002 2633650 1119433 055833 106136 95720003 3950475 1679150 083750 159204 143580004 5267300 2238866 111667 212272 191440005 6584126 2798583 139583 265340 2393000



REFERENCES
















5 CONCLUSION




Estimation error

α R C1 C2 C3 C4

000 000000 000000 000000 000000 000000001 131683 055972 002792 005307 047860002 263365 111943 005583 010614 095720003 395048 167915 008375 015920 143580004 526730 223887 011167 021227 191440005 658413 279858 013958 026534 239300006 790095 335830 016750 031841 287160007 921778 391802 019542 037148 335020008 1053460 447773 022333 042454 382880009 1185143 503745 025125 047761 43074001 1316825 559717 027917 053068 478600011 1448508 615688 030708 058375 526460012 1580190 671660 033500 063682 574320013 1711873 727631 036292 068988 622180014 1843555 783603 039083 074295 670040015 1975238 839575 041875 079602 71790002 2633650 1119433 055833 106136 95720003 3950475 1679150 083750 159204 143580004 5267300 2238866 111667 212272 191440005 6584126 2798583 139583 265340 2393000



REFERENCES













REFERENCES











department of economy and civil engineering risk … · suppose two investment projects ... between...

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