Basics of valuationBasics of valuation

Value = Sum of discounted cash flowsValue = Sum of discounted cash flowsFuture cash flows have lower value. Discount rate RFuture cash flows have lower value. Discount rate R

= = time value of moneytime value of moneyPresent value of a stream of cash flows:Present value of a stream of cash flows:

PVPV00 = = ΣΣtt CF CFtt/(1+R)/(1+R)tt

Perpetuity: CPerpetuity: Ctt = = C, t = 1, 2, 3, …C, t = 1, 2, 3, …

PVPV00 = C/R = C/R

Growing perpetuity (with const rate g): CGrowing perpetuity (with const rate g): Ct t = (1+g)= (1+g)t-1t-1CC

PVPV00 = C/(R - g) = C/(R - g)

Stocks with dividends growing with const rate gStocks with dividends growing with const rate g

PVPV00 = Div = Div11/(R-g)/(R-g)

Annuity: CAnnuity: Ctt = = C, t = 1,…,TC, t = 1,…,TPVPV00 = (C/R) [1 – 1/(1+R) = (C/R) [1 – 1/(1+R)TT ] ]

Growing annuity (with const rate g)Growing annuity (with const rate g)PVPV00 = (C/(R-g)) [1 – (1+g) = (C/(R-g)) [1 – (1+g)TT/(1+R)/(1+R)TT]]

Bond with coupon C and face value F (at T)Bond with coupon C and face value F (at T)PP00 = (C/R) [1 – 1/(1+R) = (C/R) [1 – 1/(1+R)TT] + F] + FTT/(1+R)/(1+R)TT

We will use DCF to evaluate projects (together We will use DCF to evaluate projects (together with other methods), mainly NPV = with other methods), mainly NPV = ΣΣtt CF CFtt/(1+R)/(1+R)tt (where CF can be outflows too)(where CF can be outflows too)Two big issues:Two big issues: How to compute cash flows?How to compute cash flows? What is the proper discount rate?What is the proper discount rate?

Investment decision rulesInvestment decision rules

General criteria for investment analysis:General criteria for investment analysis:It should focus on It should focus on cash flowscash flows rather than rather than accounting earningsaccounting earnings

It should place higher weight on earlier cash It should place higher weight on earlier cash flowsflows

It should penalize the expected cash flows It should penalize the expected cash flows from riskier projects more heavilyfrom riskier projects more heavily

For the time being we will abstract from the issues For the time being we will abstract from the issues of how to account for risk and uncertainty –of how to account for risk and uncertainty –just assume a given discount ratejust assume a given discount rate

Discounted cash flow techniquesDiscounted cash flow techniques Net Present Value (NPV) criterionNet Present Value (NPV) criterion Internal Rate of Return (IRR) criterionInternal Rate of Return (IRR) criterion

Nondiscounted cash flow techniquesNondiscounted cash flow techniques Payback Period (can be discounted)Payback Period (can be discounted) Accounting Rate of ReturnAccounting Rate of Return

How to deal with:How to deal with: Mutually exclusive projectsMutually exclusive projects Capital rationingCapital rationing Projects with unequal livesProjects with unequal lives

NPV RuleNPV Rule

Under no resource constraints, no mutual exclusive Under no resource constraints, no mutual exclusive projects accept the project if NPV > 0 and reject if NPV < projects accept the project if NPV > 0 and reject if NPV < 0.0.When two projects are mutually exclusive and both have When two projects are mutually exclusive and both have NPV > 0, accept the project with the higher NPV.NPV > 0, accept the project with the higher NPV.Under resource constraints choose the combination of Under resource constraints choose the combination of projects such that NPV is max, s.t. to the constraints.projects such that NPV is max, s.t. to the constraints.

RR is the opportunity cost of capital (or required return) is the opportunity cost of capital (or required return)

T

tt

t IR

CFNPV

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Value additivity property of NPV rule Value additivity property of NPV rule

DefinitionDefinition: Projects A and B are : Projects A and B are independentindependent if they don’t if they don’t affect each other’s cash flowsaffect each other’s cash flows

Definition (Value Additivity)Definition (Value Additivity): We will say that an : We will say that an investment rule satisfies investment rule satisfies value additivityvalue additivity if the following if the following holds:holds:if C is independent of A and B, thenif C is independent of A and B, thenA is preferred to B A is preferred to B A + C is preferred to B + C A + C is preferred to B + C

If X and Y are independent NPV(X+Y) = NPV(X)+NPV(Y) If X and Y are independent NPV(X+Y) = NPV(X)+NPV(Y) – hence, NPV rule satisfies value additivity– hence, NPV rule satisfies value additivity

Quickie Enterprise’s Microprocessor PlantQuickie Enterprise’s Microprocessor Plant

Cash Flows in $ millionCash Flows in $ million

Year Cash Flow PVIF@12%Year Cash Flow PVIF@12% Present Value Present Value

00 -$400-$400 1.00001.0000 -$400.00-$400.00

11 100 100 0.89290.8929 89.29 89.29

22 110 110 0.79720.7972 87.69 87.69

33 120 120 0.71180.7118 85.41 85.41

44 130 130 0.63550.6355 82.62 82.62

55 140 140 0.56740.5674 79.44 79.44

NPV =NPV = $24.45 $24.45

Calculating A Project’s NPV- Calculating A Project’s NPV- An ExampleAn Example

Internal Rate of Return (IRR) RuleInternal Rate of Return (IRR) Rule

Accept the project if IRR > R, otherwise reject itAccept the project if IRR > R, otherwise reject it RR is the opportunity cost of capital (or required return) is the opportunity cost of capital (or required return)

Among two mutually exclusive project choose the one Among two mutually exclusive project choose the one with the higher IRRwith the higher IRR

Isn’t it equivalent to the NPV criterion? For a decision Isn’t it equivalent to the NPV criterion? For a decision whether to accept or not a whether to accept or not a singlesingle project whose NPV is project whose NPV is monotonically decreasing with the discount rate – yes. monotonically decreasing with the discount rate – yes. But in general – NO!But in general – NO!

IRR solves 0)1(1

0

T

tt

t IIRR

CF

Problems with IRR:Problems with IRR: Problems with ranking mutually exclusive Problems with ranking mutually exclusive

projects (can be: NPV(A) > NPV(B), but projects (can be: NPV(A) > NPV(B), but IRR(A) < IRR(B))IRR(A) < IRR(B))

scale effectscale effecttiming effecttiming effect

Does not satisfy value additivity principle (can Does not satisfy value additivity principle (can be: IRR(A) > IRR(B), but IRR(A+C) < be: IRR(A) > IRR(B), but IRR(A+C) < IRR(B+C)IRR(B+C)

Multiple IRR when some CF are negativeMultiple IRR when some CF are negative NPV can be a positively sloping function of r. NPV can be a positively sloping function of r.

Then IRR is nonsense.Then IRR is nonsense. Sometimes no IRR existsSometimes no IRR exists

Example of IRR problemsExample of IRR problemsYearYear Project 1Project 1 Project 2Project 2 Project 3Project 3 PV factor PV factor

@ 10%@ 10%1+31+3 2+32+3

00 -100-100 -100-100 -100-100 1.0001.000 -200-200 -200-200

11 00 225225 450450 0.9090.909 450450 675675

22 550550 00 00 0.8260.826 550550 00

ProjectProject NPV @ 10%NPV @ 10% IRRIRR

11 354.55354.55 134.5%134.5%

22 104.55104.55 125.0%125.0%

33 309.09309.09 350.0%350.0%

1+31+3 663.64663.64 212.9%212.9%

2+32+3 413.58413.58 237.5%237.5%

• NPV(1) > NPV(3), but IRR(1) < IRR(3)

• IRR(1) > IRR(2), but IRR(1+3) < IRR(2+3)

Incremental IRR ruleIncremental IRR rule

When comparing two mutually exclusive projects A When comparing two mutually exclusive projects A and B one can use incremental IRR rule:and B one can use incremental IRR rule:

IRR(A-B) solvesIRR(A-B) solves

Accept the project if IRR(A-B) > R, otherwise reject itAccept the project if IRR(A-B) > R, otherwise reject it RR is the opportunity cost of capital is the opportunity cost of capital

This approach solves the problem of ranking mutually This approach solves the problem of ranking mutually exclusive projects if NPV(A-B)(r) is downward sloping. exclusive projects if NPV(A-B)(r) is downward sloping. Otherwise, usual IRR problems.Otherwise, usual IRR problems.

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Is IRR ever helpful?Is IRR ever helpful?

Useful to measure sensitivity of NPV to Useful to measure sensitivity of NPV to estimation error in the cost of capitalestimation error in the cost of capitalA scale-independent measure of efficiency A scale-independent measure of efficiency (useful to compare businesses of different scale, (useful to compare businesses of different scale, for comparison – no need to know the cost of for comparison – no need to know the cost of capital)capital)Aggregates the info about an investment into Aggregates the info about an investment into one numberone numberBut anyway, if NPV is properly used then NPV is But anyway, if NPV is properly used then NPV is the best:the best:maxmaxmm NPV(m), s.t. NPV(m), s.t. II ĪĪ, m, mM,M,where M is the set of where M is the set of allall possible combinations possible combinations of projects, of projects, ĪĪ – resource constraint – resource constraint

Payback period rulePayback period rule

How long does it take for a project to recover How long does it take for a project to recover or “pay back” its initial investment?or “pay back” its initial investment?

If recovery time < threshold – accept, If recovery time < threshold – accept, otherwise – reject.otherwise – reject.

Discounted Payback Period:Discounted Payback Period:

01

0

PB

tt ICF

0)1(1

0

DPB

tt

t IR

CF

Disadvantages of DPP:Disadvantages of DPP: Ignores the cash flows after the payback Ignores the cash flows after the payback

period (what if they are negative?)period (what if they are negative?) Arbitrary standard for setting the periodArbitrary standard for setting the period

Advantages of DPP:Advantages of DPP: SimpleSimple Measure of project liquidityMeasure of project liquidity Measure (rough) of project riskMeasure (rough) of project risk

Average Accounting Return ruleAverage Accounting Return rule

AAR = Average Net Income / Average AAR = Average Net Income / Average Investment (book value)Investment (book value)

Simple BUT ignores Simple BUT ignores time valuetime value of money and is of money and is based on based on accounting incomeaccounting income rather than cash flow. rather than cash flow.

Moreover, what is the target rate?Moreover, what is the target rate?

Project selection with resource Project selection with resource constraintsconstraints

Resource: capital, premises, people, time, Resource: capital, premises, people, time, etc…etc…

The straightest way is:The straightest way is:

maxmaxmm NPV(m), s.t. NPV(m), s.t. II ĪĪ, m, mM,M,

where M is the set of where M is the set of allall possible possible combinations of projects, combinations of projects, ĪĪ – resource – resource constraintconstraint

But can be too complicated, hence…But can be too complicated, hence…

Profitability IndexProfitability Index

Profitability Index:Profitability Index:PI = PV of cash flows / Resource consumedPI = PV of cash flows / Resource consumedVariations:Variations: PI = PV of cash flows subsequent to initial investment / IPI = PV of cash flows subsequent to initial investment / I00 PI = NPV / ResourcePI = NPV / Resource

Rule: rank the projects by the value of PI, then Rule: rank the projects by the value of PI, then select projects starting from the highest PI until the select projects starting from the highest PI until the resource is consumedresource is consumedDoing so you will Doing so you will approximatelyapproximately maximize NPV maximize NPV under the resource constraint if all projects are under the resource constraint if all projects are independentindependent

Example: PI with a human resource Example: PI with a human resource constraintconstraint

Example (cont-d)Example (cont-d)

ShortcomingShortcoming

You will never meet the resource You will never meet the resource constraint precisely constraint precisely can happen that the can happen that the selected combination does not maximize selected combination does not maximize NPVNPV