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Risk Analysis in InvestmentValuation: an Introduction
Emanuele [email protected]
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Discounted Cash Flow
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Discounted Cash Flows
n Investment must be characterized in terms of Timing
n Qualitative Analysis: j=1mn What are the types of cash flows that investors have rights to
receive
n Quantitative Analysis (Forecast): xitn Estimation of the cash flow value
t
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Decision Criteria
n Net Present Value (NPV) :
n Where T is the valuation horizon, t is the CF timing, m isthe number of equity cash flow types
n Internal Rate of Return (IRR)
1
1( )1
mt
jTj
tt e
x
Vk
=
=
=+
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The Decision Making Process
Yes/N
oValuationCriteria
Decision SupportModel
Inputs
SensitivityAnalysis
Assumptions
Clemen (1997,
Ch.1)
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66
Sensitivity Analysis Uses
n Model Output:
n Model Correctness
n Stress Test
n Equilibrium stability and changes (Samuelson
Comparative Statics)
n Parameter Importance
)X,...,X,f(XY n11=
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Sensitivity Analysis
CorrectnessTest
ImportanceMeasures
RiskAnalysis
UncertaintyAnalysis
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Importance Measures
Is it possible to quantify the relevance of each factor in themodel?
What assumption influences the outcome of the
decision the most?
Is it possible to quantify the relevance of groups ofassumptions?
Sensitivity Analysis methods have been developed withthe purpose of answering those questions
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Uncertainty Analysis
Is it possible to quantify the degree of confidence of thedecision maker in the model results?
What uncertain factor influences uncertainty the most?
Where should one focus to reduce uncertainty in theModel results?
Global Sensitivity Analysis methods have beendeveloped to answer these questions
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Risk Analysis Insights
How is risk distributed among factors
What assumption to monitor in order to reduce risk?
Sensitivity analysis results share a risk analysis interpretation
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Risk Analysis Insights
How is risk distributed among factors
What assumption to monitor in order to reduce risk?
Sensitivity analysis results share a risk analysis interpretation
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The Differential Importance Measure
n Let f(x) be a function differentiable at x0. Provided thatf(x0) is not orthogonal to dx, then the importance of xsat x0 is defined as [Borgonovo and Peccati (2004),(2006)]:
n Ds(x0,dx) is the fraction of the differential of f(x0) that isassociated with parameter xs.
=
==n
1j
j
1
j
s
1
s
x
s1
s
dx)x(f
dx)x(f
df
fd)dx,x(D 1
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DIM Main Properties
n H1: Uniform Changes n H2: Proportional Changes
=
=n
1j
1
j
1
s1
s
)x(f
)x(f)x(D1
j,idxdx ij = )ji(j,idx
dx
i
j=
=
=
n
1j
1
j
1
j
1
s
1
s1
s
x)x(f
x)x(f)x(D1
)dx,x(D)dx,x(D
1m
1js
1
s,...,s,s jm11
==
3) DIM accounts for the way parameters are varied
1) Additivity:
2) Sum: 1)dx,x(D)dx,x(D1
n
1j
s
1
S j==
=
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Application to the financialanalysis of a Project FinanceInfrastructure Project
A Parking Lot
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Large Projects Valuation: Criteria
n Sponsors Side:
n or
n Lenders Side
n or
( )= +
=N
ii
e
e
i
k
CFNPV
1 1
jj
j
jPI
FCFDSCR
+=
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Identification of Key Drivers
n Complex Non-linear models (Van Groenendaal (1998),
Kleijnen and Van Groenendaal (1997), (2002)).n Absence of analytical expression:
n Sensitivity Analysis (SA) plays a crucial role in unveilingModel dependence on the parameters
n SA essential in fully exploiting model information
n Sensitivity Analysis method should be:n Quantitative and Model freen Enable to evidence effect of multiple factors (interactions)n Avoid top-down or a priori parameter selection, since one
runs the risk of excluding relevant parameters
n High Number of parameters leads ton Computational issues (i.e. high computational cost [Kleijnenand Van Groenendaal (2002)] )
n Issue in communication of the results when high number ofparameters
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Parameter Importance in Investment Evaluation
n Investment criterion is a function of the exogenous variables(factors) that influence the project economnics [Borgonovo andPeccati (2006)]:
n These factors become input parameters (X) for the model
n Parameter Influence is given by:
Since in a financial model parameters have different dimensions.
=
=n
j
jj
ss
s
xxv
xxvxD
1
11
11
1
)(
)()(1
)(xvV =
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Project Contractual Structure
Operation&
Mantenance Contract
Merchant Sale
EngineeringProcurementConstruction
Contract
SPC
InsuranceContracts
ShareholderAgreement
LoanAgreement Pr
ice
Marke
t
Risk
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Cash Flows Estimation
RevenuesNet of Turnover Tax
LESS Operating Expenses
EBITDA
LESS Long-term loans interest
Subdebt Interests
Interest Income
EBTDA
LESS Depreciation
EBT
LESS Taxes
Net Profit
LESS Dividends
Equals Retained Earnings
Legal Reserve
Income Statement
Revenues Netof TurnoverTax
LESS OperatingExpenses
LESS Taxes
Operating Cash Flows
Plus Trapped cash Previous year
Cash Before Capex
LESS Capital Expenditures
Plus Equity Injections
Plus Subdebt Injection
Plus Principal Injections
Cash Flows Available for interest payment
Less Debt Interests
Cash Flows Available for principal payment
LESS Principal Repayment
equals Cash FlowsAvailable forSDinterest
LESS ShareholderInterests
Equals CashFlows AvailableforSDPrincipal
LESS Shareholder PrincipalLESS IOE
equals Cash flow available for dividends
Dividends
Equals Trapped cash
Cash Flow Statement
Balance
Sheet
Assets
Current Assets
Cash
Inventory
RiceivableLong Term Assets
Total Assets
Liabilities
Current Debt
Equity
Retained Earnings
Legal Reserve
Shareholder Debt
Debt
Total Liabilities
n Project economic life projection
n High level of accuracy for multi-million deals
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2020
ModelStructure
InputsAuxiliaryCalculatio
ns
Finstat
s
Results
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Methodological Approach
n Number of parameters: 428
n Input Categories Defined by Financial Model:n Revenuesn Construction Costsn Fiscaln
Financialn Macroeconomicn Opex
n Valuation Criteria: NPV and
n Grouping on three levels
n Level 3: individual inputs (428)n Level 2: 17 groupsn Level 1: 6 groups
n Statistical Comparison through Savage Score corr. Coeff.
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Equity NPV,Sponsors side
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Parameter ranking (Level 3)
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NPV: Level 1
1.00
.00
.00
.00
1.
11
.11
.11
Opex Constr.
Costs
Infl Rev Ass Fisc. Financing
NPV: factors grouped into main categories1
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NPV: Level 2, 17 categories
%1
%11
%11
%11
%11
%11
%11
Opex Parcheggi Autom Amm GiOcc Perc. Occ. mesiOccupaz Ass.Fiscali k
NPV: Importanza gruppi di ipotesi (Livello )1
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Summary on NPV results
1. Revenue parameters are the most importantones
1. ke plays a significant role
1. Leverage is not among 20 most relevantinputs.
1. 60 Parameters were non-influential: modelcorroboration
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Debt Service Coverage Ratio:Lenders side
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DSCR: Level 3
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DSCR: Level 1
.111
.000
.000
.000
.000
.111
.111
.111
.111
Opex Constr.
Costs
Infl Rev Ass Fisc. Financing
DSCR: Level
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DSCR: Level 3
%1
%11
%11
%11
%11
%11
%11
O pe x A ule P arc h. V ia b. V . A uto m. Infl. A m m. Ta riffe G iO cc N ro t P e rc .
Occ.
PostiD iStmesiOc c Pos tMotAs s.Fisc.Ass.Fin/k k
DSCR Level
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Observations
1. Revenue assumptions are the most important ones
1. The cost of debt plays a relevant role (simmetric ke)
1. Leverage is significant (8th)
1. Equity relevant parameters (ke and retention ratio are noninfluential)
1. Income taxes influence NPV but little DSCR
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Cross Comparison
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Individual Parameter Ranking
Rank
NPV Parameter
Rank
DSCR Parameter
1 Nr. Of parking slots from year on1 1 Nr. Of parking slots from year on1
1 Daily occupation days from year on1 1 Daily occupation days from year on1
1 Tariff for first two hours 1 Tariff for first two hours
1 Rotation number for the first hours1 1 Rotation number for the first hours1
1 Percentage of Occupation of the First hours1 1 Percentage of occupation for the first two hours after 1 year
1 ke 1 kd
1 Tariff after the first two hours 1 Rooms construction costs
1 Rotation number after the first hours1 1 Leverage
1 VAT on Revenues 1 Tariff after first hours1
00 kd 1 Rotation number after first two hours
11 Night Tariff 1 Percentage of Occupation after two hours from year on1 1
Rotation Number 11 VAT on Revenues
Percentage of nightly occupation 11 Night Tariff
00 Rooms construction costs 11 Number of Night Rotation
111 Geological Inspection Cost 11 Percentage of Night Occupation after year1 111 Days payables for electricity connection costs 111 Cost for workplace set up
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Ranking Analysis
X
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Level 3
n Financial Structuring
1
.00
.00
.00
.00
1
.11
.11
.11
Opex Constr.
Costs
Infl Rev Ass Fisc. Financing
NPV vs DSCR: Parameter Group Importance
NPV
DSCR
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NPV vs DSCR, Level 2
%1
%11
%11
%11
%11
%11
%11
OPEX
ROOMS
PARKING
GREEN
AUTOM
INFLATION
AMORTIZ
TARIFF
DAYS
ROTATIONS
PERC.OCC
NrofPLACES
OCC
TIME
Slotsforbikes
FISCAL
FINANCIAL k
NPV vs DSCR importance of categories11
NPV
DSCR
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Conclusions (1)
n Sensitivity Analysis plays a key role to deepen theunderstanding of the investment financial performance,since an analytical expression of the valuation criterion isnot available.
n Large complex modelsn
High Number of Parameters Computational Costsn Analysis Communication Results Synthesis1
n Methodologyn Use of the differential importance measures enables to
solve the above issues thanks to the additivity property
n Numerical Computation: Algorithm based on Cauchyssequence convergence criterionn Statistical Comparison for Ranking: Savage Score
Correlations and Rank Correlations
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Conclusions (2)
n Financial Model with 428 parameters
n Synthesis of the results grouping parameters inthe corresponding categories
n Results have enabled us to:
n Identify key drivers for sponsors (NPV) and forlenders (DSCR) wihtout a-priori screening
n Individual parameter ranking needs not to bethe same
n Cost of Equity/Debtn Leverage
n different attention in negotiation
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Uncertainty and Risk
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Hertz (1964)
n Of all the decisions that business executives must make, noneis more challenging andhas received more attention - thanchoosing among alternative capital investment opportunities.What makes this kind of decision so demanding, of course, isnot the problem of projecting return on investment under any
given set of assumptions. The difficulty is in the assumptionsand in their impact. Each assumption involves its own degree -often a high degree - of uncertainty; and, taken together, thesecombined uncertainties can multiply into a total uncertainty ofcritical proportions. This is where the element of risk enters,
and it is in the evaluation of risk that the executive has beenable to get little help from currently available tools andtechniques.
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Cash Flows are uncertain
n Cash flows depend on exogenous variableswhich are not known at the moment theinvestment is evaluated.
n The valuation criterion then becomes a random
variable
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A direct quote from an early classic
n It should be recognized that, as defined, P isactually a random variable rather than aconstant.
n Hillier (1963), MS.
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An ExampleFrom Van Horne (1967), Capital Budgeting Decisions Involving Risky Investments,Management Science
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How do we value this investment?
n Expected present value:
n One needs to discount, instead of pointestimates, the expected values of the cash flows
n The equation is obtained by the additivity of theexpectation operator.
1
1
[ ]
[ ]
( )1
mt
jTj
tt e
E x
E V
k
=
=
=
+
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Solution
n See excel spreadsheet model
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Software for Uncertainty Analysis
n @Riskn Monte Carlo
n Latin Hypercube Sampling
n Best Fitn Distribution Fitting
n Crystal Ball
n SIMLAB
n Global SA