Professeur André Farber Théorie Financière 2005-2006 6. Analyse de projets d’investissement (2) Professeur André Farber

Investment decisions (2) Objectives for this session : A project is not a black box Timing: How long to invest? When to invest? Project with different lifes: Equivalent Annual Cost Tfin 2005 06 Capital budgeting (2)

A project is not a black box Sensitivity analysis: analysis of the effects of changes in sales, costs,.. on a project. Scenario analysis: project analysis given a particular combination of assumptions. Simulation analysis: estimations of the probabilities of different outcomes. Break even analysis analysis of the level of sales at which the company breaks even. Tfin 2005 06 Capital budgeting (2)

Sensitivity analysis Year 0 Year 1-5 Initial investment 1,500 Revenues 6,000 Variables costs (3,000) Fixed costs (1,791) Depreciation (300) Pretax Profit 909 Tax (TC = 34%) (309) Net Profit 600 Cash flow 900 NPV calculation (for r = 15%): NPV = - 1,500 + 900  3.3522 = + 1,517 Tfin 2005 06 Capital budgeting (2)

Sensitivity analysis 1. Identify key variables Revenues = Nb engines sold  Price per engine 6,000 3,000 2 Nb engines sold = Market share  Size of market 3,000 0.30 10,000 V.Cost =V.cost per unit  Number of engines 3,000 1 3,000 Total cost = Variable cost + Fixed costs 4,791 3,000 1,791 Tfin 2005 06 Capital budgeting (2)

Sensitivity analysis 2. Prepare pessimistic, best, optimistic forecasts (bop) Variable Pessimistic Best Optimistic Market size 5,000 10,000 20,000 Market share 20% 30% 50% Price 1.9 2 2.2 V.cost / unit 1.2 1 0.8 Fixed cost 1,891 1,791 1,741 Investment 1,900 1,500 1,000 Tfin 2005 06 Capital budgeting (2)

Sensitivity analysis 3. Recalculate NPV changing one variable at a time Variable Pessimistic Best Optimist Market size -1,802 1,517 8,154 Market share -696 1,517 5,942 Price 853 1,517 2,844 V.cost / unit 189 1,517 2,844 Fixed cost 1,295 1,517 1,628 Investment 1,208 1,517 1,903 Tfin 2005 06 Capital budgeting (2)

Scenario analysis Consider plausible combinations of variables Ex: If recession market share low variable cost high price low Tfin 2005 06 Capital budgeting (2)

Monte Carlo simulation Tool for considering all combinations model the project specify probabilities for forecast errors select numbers for forecast errors and calculate cash flows Outcome: simulated distribution of cash flows Tfin 2005 06 Capital budgeting (2)

Monte Carlo Simulation - Example Model Qt = Qt-1 + ut mt = m + vt CFt = (Qtmt - FC - Dep)(1-TC)+Dep Procedure 1. Generate large number of evolutions 2. Calculate average annual cash flows 3. Discount using risk-adjusted rate Notations Qt quantity mt unit margin FC fixed costs Dep depreciation TC corporate tax rate ut,,vt random variables Random number generation Random number Ri : uniform distribution on [0,1] Use RAND() in Excel To simulate  ~ N(0,1): NORMSINV(Rand()) Tfin 2005 06 Capital budgeting (2)

Simulated cash flows Tfin 2005 06 Capital budgeting (2)

Break even analysis Sales level to break-even? 2 views Account Profit Break-Even Point: Accounting profit = 0 Present Value Break-Even Point: NPV = 0 Tfin 2005 06 Capital budgeting (2)

Timing Even projects with positive NPV may be more valuable if deferred. Example You may sell a barrel of wine at anytime over the next 5 years. Given the future cash flows, when should you sell the wine? Suppose discount rate r = 10% NPV if sold now = 100 NPV if sold in year 1 = 130 / 1.10 = 118 Wait Tfin 2005 06 Capital budgeting (2)

Optimal timing for wine sale? Calculate NPV(t): NPV at time 0 if wine sold in year t: NPV(t) = Ct / (1+r)t Tfin 2005 06 Capital budgeting (2)

When to invest Traditional NPV rule: invest if NPV>0. Is it always valid? Suppose that you have the following project: Cost I = 100 Present value of future cash flows V = 150 Possibility to mothball the project Should you start the project? If you choose to invest, the value of the project is: Traditional NPV = 150 - 100 = 50 >0 What if you wait? Tfin 2005 06 Capital budgeting (2)

To mothball or not to mothball? Suppose that the project might be delayed for one year. One year later: Cost is unchanged (I = 100) Present value of future cash flow = 160 NPV1 = 160 - 100 = 60 in year 1 To decide: compare present values at time 0. Invest now : NPV = 50 Invest one year later: NPV0 = PV(NPV1) = 60/1.10 = 54.5 Conclusion: you should delay the investment + Benefit from increase in present value of future cash flows (+10) + Save cost of financing of investment (=10% * 100 = 10) - Lose return on real asset (=10% * 150 = 15) The analyzis of when to invest in a determist setting is based on A. Dixit and R. Pindyck Investment under Uncertainty, Princeton University Press 1994 (Chapter 5) The net present value at time 0 of an investment at time t is: NPV(t) = [V(t) - I] e-rt where V(t) = present value of future cash flows at time t I = cost of investment e-rt = discount factor (r is the continuously compounded interest rate) Taking the first derivative with respect to t gives: NPV’(t) = [V’(t) - rV(t) + rI] e-rt The terms between brackets represent the costs and benefits of delaying the investment: V’(t): variation in the value of future cash flows (>0) - rV(t): opportunity cost of waiting to invest (<0) rI: opportunity cost of investment (>0) Tfin 2005 06 Capital budgeting (2)

Equivalent Annual Cost The cost per period with the same present value as the cost of buying and operating a machine. Equivalent Annual Cost = PV of costs / Annuity factor Example: cheap & dirty vs good but expensive Given a 10% cost of capital, which of the following machines would you buy? EAC calculation: A: EAC = PV(Costs) / 3-year annuity factor = 24.95 / 2.487 = 10.03 B: EAC = PV(Costs) / 2-year annuity factor = 20.41 / 1.735 = 11.76 Tfin 2005 06 Capital budgeting (2)

The Decision to Replace When to replace an existing machine with a new one? Calculate the equivalent annual cost of the new equipment Calculate the yearly cost of the old equipment (likely to rise over time as equipment becomes older) Replace just before the cost of the old equipment exceeds the EAC on new equipment Example Annual operating cost of old machine = 8 Cost of new machine : PV of cost (r = 10%) = 27.4 EAC = 27.4 / 3-year annuity factor = 11 Do not replace until operating cost of old machine exceeds 11 C0 C1 C2 C3 15 5 5 5 Tfin 2005 06 Capital budgeting (2)