Basics of valuation Value = Sum of discounted cash flows Future cash flows have lower value. Discount rate R = time value of money Present value of a stream of cash flows: PV 0 = Σ t CF t /(1+R) t Perpetuity: C t = C, t = 1, 2, 3, … PV 0 = C/R Growing perpetuity (with const rate g): C t = (1+g) t-1 C PV 0 = C/(R - g) Stocks with dividends growing with const rate g PV 0 = Div 1 /(R-g)

Annuity: C t = C, t = 1,…,T PV 0 = (C/R) [1 – 1/(1+R) T ] Growing annuity (with const rate g) PV 0 = (C/(R-g)) [1 – (1+g) T /(1+R) T ] Bond with coupon C and face value F (at T) P 0 = (C/R) [1 – 1/(1+R) T ] + F T /(1+R) T We will use DCF to evaluate projects (together with other methods), mainly NPV = Σ t CF t /(1+R) t (where CF can be outflows too) 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 rules General criteria for investment analysis: It should focus on cash flows rather than accounting earnings It should place higher weight on earlier cash flows It should penalize the expected cash flows from riskier projects more heavily

For the time being we will abstract from the issues of how to account for risk and uncertainty – just assume a given discount rate Discounted cash flow techniques Net Present Value (NPV) criterion Net Present Value (NPV) criterion Internal Rate of Return (IRR) criterion Internal Rate of Return (IRR) criterion Nondiscounted cash flow techniques Payback Period (can be discounted) Payback Period (can be discounted) Accounting Rate of Return Accounting Rate of Return How to deal with: Mutually exclusive projects Mutually exclusive projects Capital rationing Capital rationing Projects with unequal lives Projects with unequal lives

NPV Rule Under no resource constraints, no mutual exclusive projects accept the project if NPV > 0 and reject if NPV 0 and reject if NPV < 0. When two projects are mutually exclusive and both have NPV > 0, accept the project with the higher NPV. Under resource constraints choose the combination of projects such that NPV is max, s.t. to the constraints. R is the opportunity cost of capital (or required return)

Value additivity property of NPV rule Definition: Projects A and B are independent if they dont affect each others cash flows Definition (Value Additivity): We will say that an investment rule satisfies value additivity if the following holds: if C is independent of A and B, then A is preferred to B A + C is preferred to B + C If X and Y are independent NPV(X+Y) = NPV(X)+NPV(Y) – hence, NPV rule satisfies value additivity

Quickie Enterprises Microprocessor Plant Cash Flows in $ million Year Cash Flow Present Value 0-$ $ NPV = $24.45 Calculating A Projects NPV- An Example

Internal Rate of Return (IRR) Rule Accept the project if IRR > R, otherwise reject it R is the opportunity cost of capital (or required return) R is the opportunity cost of capital (or required return) Among two mutually exclusive project choose the one with the higher IRR Isnt it equivalent to the NPV criterion? For a decision whether to accept or not a single project whose NPV is monotonically decreasing with the discount rate – yes. But in general – NO! IRR solves

Problems with IRR: Problems with ranking mutually exclusive projects (can be: NPV(A) > NPV(B), but IRR(A) NPV(B), but IRR(A) < IRR(B)) scale effect timing effect Does not satisfy value additivity principle (can be: IRR(A) > IRR(B), but IRR(A+C) IRR(B), but IRR(A+C) < IRR(B+C) Multiple IRR when some CF are negative Multiple IRR when some CF are negative NPV can be a positively sloping function of r. Then IRR is nonsense. NPV can be a positively sloping function of r. Then IRR is nonsense. Sometimes no IRR exists Sometimes no IRR exists

Example of IRR problems Year Project 1 Project 2 Project 3 PV 10% Project 10% IRR % % % % % NPV(1) > NPV(3), but IRR(1) < IRR(3) IRR(1) > IRR(2), but IRR(1+3) < IRR(2+3)

Incremental IRR rule When comparing two mutually exclusive projects A and B one can use incremental IRR rule: IRR(A-B) solves Accept the project if IRR(A-B) > R, otherwise reject it R is the opportunity cost of capital R is the opportunity cost of capital This approach solves the problem of ranking mutually exclusive projects if NPV(A-B)(r) is downward sloping. Otherwise, usual IRR problems.

Is IRR ever helpful? Useful to measure sensitivity of NPV to estimation error in the cost of capital A scale-independent measure of efficiency (useful to compare businesses of different scale, for comparison – no need to know the cost of capital) Aggregates the info about an investment into one number But anyway, if NPV is properly used then NPV is the best: max m NPV(m), s.t. I Ī, m M, where M is the set of all possible combinations of projects, Ī – resource constraint

Payback period rule How long does it take for a project to recover or pay back its initial investment? If recovery time < threshold – accept, otherwise – reject. Discounted Payback Period:

Disadvantages of DPP: Ignores the cash flows after the payback period (what if they are negative?) Ignores the cash flows after the payback period (what if they are negative?) Arbitrary standard for setting the period Arbitrary standard for setting the period Advantages of DPP: Simple Simple Measure of project liquidity Measure of project liquidity Measure (rough) of project risk Measure (rough) of project risk

Average Accounting Return rule AAR = Average Net Income / Average Investment (book value) Simple BUT ignores time value of money and is based on accounting income rather than cash flow. Simple BUT ignores time value of money and is based on accounting income rather than cash flow. Moreover, what is the target rate? Moreover, what is the target rate?

Project selection with resource constraints Resource: capital, premises, people, time, etc… The straightest way is: max m NPV(m), s.t. I Ī, m M, where M is the set of all possible combinations of projects, Ī – resource constraint But can be too complicated, hence…

Profitability Index Profitability Index: PI = PV of cash flows / Resource consumed Variations: PI = PV of cash flows subsequent to initial investment / I 0 PI = PV of cash flows subsequent to initial investment / I 0 PI = NPV / Resource PI = NPV / Resource Rule: rank the projects by the value of PI, then select projects starting from the highest PI until the resource is consumed Doing so you will approximately maximize NPV under the resource constraint if all projects are independent

Example: PI with a human resource constraint

Example (cont-d)

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