Evaluaing Projecs under Uncerainy March 17, 4 1 Projec risk = possible variaion in cash flows 2 1
Commonly used measure of projec risk is he variabiliy of he reurn 3 Mehods of dealing wih uncerainy in projec evaluaions Sensiiviy analysis Risk adjused MARR Probabiliy rees Mone Carlo simulaions 4 2
Example showing how use of risk-adjused MARRs can lead o he wrong decision (from Sullivan e al, Engineering Economy, (11h ed), p. 445) The Alas Corporaion is considering wo alernaives, boh affeced by uncerainy o differen degrees, for increasing he recovery of a precious meal from is smeling process. The firm s MARR for is risk-free invesmens is 10% per year. Alernaive End-of-year, k P Q 0-160,000-160,000 1 120,000 20,827 2 60,000 60,000 3 0 120,000 4 60,000 60,000 Because of echnical consideraions, Alernaive P is hough o be more uncerain han Alernaive Q. Therefore, according o he Alas Corporaion s Engineering Economy Handbook, he risk-adjused MARR applied o P will be 20% per year and he risk-adjused MARR for Q has been se a 17% per year. Which alernaive should be recommended? 5 Soluion A he risk-free MARR, boh alernaives have he same PW of $39, 659. Wha o do? All else equal, choose Q, because i is less uncerain (hence less riskier) han P. Bu now, do a PW analysis, using Alas Corporaion s prescribed risk-adjused MARRs for he wo opions: PW P = -160,000 + 120,000 (P/F, 20%, 1) + 60,000 (P/F, 20%, 2) + 60,000 (P/F, 20%, 4) = $10,602 PW Q = -160,000 + 20,827 (P/F,17%,1) + 60,000 (P/F,17%, 2) + 120,000 (P/F,17%,3) + + 60,000 (P/F,17%, 4) = $8575 Hence according o his mehod we would choose P. In oher words, using he risk-adjused MARRs makes he more uncerain projec, P, look MORE aracive han Q!! 6 3
Soluion (cond.) From Sullivan e al, p.447 7 Example of probabilisic analysis Consider he simple decision wheher o make a new invesmen, when here is uncerainy abou he duraion of demand. I0 = 6800 R = 7000/yr M = 0 + (n-1)1000 i = 20% N IN 1 1600 2 800 3 400 4 The probabiliy of demand for he service provided by his asse persising for: 1yr 0.1 2yr 0.2 3yr 0.3 4yr 0.4 Quesion: Should his invesmen be made? 8 4
Example (coninued) Quesion: Should his invesmen be made? LAC = I o ( A / P,20%,N) - I N ( A / F,20%, N) + 0+ 1000( A /G, 20%, N) N LAC Levelized OI (P/A,20%,N) PW of O.I. Probabiliy 1-9560 -2360 0.833-2132 0.1 2-7542 -542 1.528-828 0.2 3-6997 3 2.106 6 0.3 4-6864 136 2.589 352 0.4 1yr p=0.1 PW = -2132 px PW = -213.2 Yes 2yr p=0.2 PW=-828 p x PW = -166 3yr p=0.3 PW = 6 p x PW = 2 4 yr p = 0.4 PW = 352 p x PW = 141 Expeced Value = -$236 No 0 9 The Problem of Invesmen Timing Example 1: Uncerainy over Prices Widge facory Iniial cos = I = $1600 Annual operaing cos = 0 Producion rae = 1 widge per year Curren widge price = $ Price nex year (and forever afer): $300 wih probabiliy 0.5 $100 wih probabiliy 0.5 =0 =1 =2 0.5 P1 = $300 P2 = $300 Po = $ 0.5 P1 = $100 P2 = 100 Assume ineres rae of 10%/yr Quesion: Should he firm inves now, or should i wai for 1 year and see wheher he price of widges goes up or down? 10 5
Example 1 (cond.) Since expeced price of widges is always $, he NPV of an invesmen now is NPV = -1600 + Â 0 = -1600 + 2 = $600 (1+ 0.1) Thus i migh seem sensible o go ahead. Bu wha if we wai unil nex year? Then we would decide o inves only if he price goes up. If he price falls, i would make no sense o inves. The NPV in his case is given by: È -1600 3300 850 NPV = (0.5) È -1600 300 + Â = 0.5 Î Í + = = $773 Í Î 1.1 (1+ 0.1) =1 1.1 1.1 1.1 So if we wai a year before deciding wheher o inves in he facory, he projec NPV oday is $773. Clearly i is beer o wai han o inves righ away. If we had no choice, and eiher had o inves now or never, we would obviously choose o inves, since his would have a posiive NPV of $600. Bu he flexibiliy o choose o pospone he decision and inves nex year if he marke price is righ is worh somehing. Specifically, i is worh 773-600 = $173. In oher words, we should be willing o pay up o $173 more for an invesmen opporuniy ha is flexible han one ha only allows us he choice of invesing now or never. This is he value of flexibiliy in his case. Sill anoher way of saying his is ha here is an opporuniy cos of invesing now, raher han waiing. 3/17/04 Nuclear Energy Economics and 11 Example 2 -- Uncerainy over coss We can consider wo differen kinds of cos uncerainy a. Suppose ha I = $1600 oday, bu ha nex year i will increase o $2400 or decrease o $800, each wih a probabiliy of 0.5. (The cause of his uncerainy could be sochasic flucuaions in inpu prices, or regulaory uncerainies.) The ineres rae is again 10% per year Quesion: Should we inves oday or wai o decide unil nex year? As before, if we inves oday he NPV is given by: NPV = -1600 + Â = -1600 + 2 = $600 (1 0.1) 0 + If we wai unil nex year, i will be sensible o inves only if he invesmen cos falls o $800, which happens wih a probabiliy of 0.5. In his case he NPV is given by: È -800 È -800 NPV = + Â = 0.5 ÎÍ + =1 + 1.1 + 2 (0.5) Î Í 1.1 = $636 (1 0.1) (1 0.1) so once again i is beer o wai han o inves immediaely. 12 6
Example 2 -- cond. a. Suppose, alernaively, ha here are uncerainies over how much i is going o cos o complee he projec ha can only be resolved by acually doing i. You don know for cerain how much i is going o cos unil you complee i. Le s say ha his uncerainy akes he following form: To build he widge facory you firs have o spend $1000, and ha here is a 50% probabiliy ha he facory will hen be complee, and a 50% probabiliy ha you will have o spend anoher $3000 o complee i. Assume ha he widge price remains consan a $, and ha he ineres rae is 10%. A firs blush, he invesmen would make no sense. The expeced cos of he facory is: 1000 + 0.5.(3000) = 2500. Â = 2 And since he value of he facory = 1.1, we migh conclude ha i makes no NPV = -1000 + (0.5) Â = + $ 100 1.1 =0 =0 sense o proceed. Bu his ignores he addiional informaion ha is generaed by compleing he firs phase of he projec, and ha we can choose o abandon he projec if compleion requires an exra $3000. The rue NPV is: Since he NPV is posiive, one should inves in he firs sage of he projec. 13 7