Dr. Maddah ENMG 400 Engineering Economy 08/05/09. Chapter 18 Sensitivity Analysis and Expected Value Decisions

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1 Dr. Maddah ENMG 400 Engneerng Economy 08/05/09 Chapter 18 Senstvty Analyss and Expected Value Decsons Senstvty analyss Engneerng economy estmates of parameters such as costs and other cash flow are only an approxmaton of realty. The realzed future value of a parameter wll be generally dfferent from ts estmated value. Senstvty analyss attempts to measure the effect of ths uncertanty (varablty) n parameters estmates. Senstvty analyss dentfes parameters that have the most mpact on an economc decson. That s, t dentfes the parameters wth the hghest senstvty. Senstvty of a parameter s the effect of changng the parameter value on the economc crtera such as PW value. Graphcal senstvty analyss Senstvty analyss s often done graphcally by plottng the economc crtera as a functon of a parameter. The plot s over the range where the parameter s most lkely to vary. A flat plot ndcates nsenstvty. 1

2 That s, the parameter has lttle effect on the economc decson. No need to get very precse estmates for ts value. A hghly varable plot ndcates hgh senstvty. That s, the parameter has a sgnfcant mpact on economc decson. Estmatng ts value should be handled wth care. Example Example 18.1 gves PW vs. MARR for a project. Year Cflow 0 -$80,000 1 $25,000 2 $23,000 3 $21,000 4 $19,000 5 $17,000 6 $15,000 7 $13,000 8 $11,000 9 $9, $7,000 MARR PW(%) 10% $27, % $11, % -$ % -$10, $-PW Ex PW vs. MARR $30, $25, $20, $15, $10, $5, $0.00 -$5, % 10% 20% 30% -$10, $15, MARR In ths example, PW s hghly senstve to MARR. 2

3 Senstvty analyss wth several parameters When several parameters are beng analyzed for senstvty, A spder plot can be used. Ths s a plot of the economc crtera as a functon of percent changes from the most lkely estmates of parameters. Example of spder plot (Fg 18.3). In ths example, ROR s o Insenstve to ndrect cost (flat curve), and labor cost; o Moderately senstve to materal cost and captal; o Hghly senstve to sales volume and sales prce. 3

4 Senstvty analyss wth three estmates A common approach s to base senstvty analyss on three estmates for a parameter: pessmstc, most lkely, and optmstc estmate. Example 18.3 performs senstvty analyss of annual cost for three alternatves based on three estmates of parameters. In ths example, parameters are assumed to change together for the pessmstc, most-lkely and optmstc scenaros. Ths s also known as scenaro analyss. Strategy Frst Cost SV AOC Lfe AW Alternatve A Pessmstc -$20,000 $0 -$11, $19,327 Most lkely -$20,000 $0 -$9, $14,548 Optmstc -$20,000 $0 -$5, $9,026 Alternatve B Pessmstc -15, , $12,640 Most lkely -15,000 1,000-3, $8,229 Optmstc -15,000 2,000-2, $5,089 Alternatve C Pessmstc -30,000 3,000-8, $19,601 Most lkely -30,000 3,000-7, $13,276 Optmstc -30,000 3,000-3, $8,927 Accordng to the plot below, alternatve B s better than alternatves A and C under all scenaros. Selectng Alternatve B s a good choce. 4

5 P ML O Varablty and probablty When a cash flow (parameter) estmate s hghly uncertan and the economc decson s hghly senstve to t, a probablstc cash flow analyss s useful. Probablty of an event (here the cash flow taken on a value) s typcally defned as the long run fracton of tme where the event happens. If the event repeats several tmes, then one can use hstorcal date to estmate the probablty. That s, probablty s estmated as a frequency. E.g., one can estmate that the probablty that the energy cost n August exceeds $1,000 s 0.2, f ths cost exceeds 1,000 n 20% of past years. 5

6 However, n several stuatons, especally n one-tme engneerng projects, not enough hstorcal data s avalable. E.g., for the constructon of an exotc tower, the experenced cvl engneer estmates that there s a 95% chance (.e. a 0.95 probablty) that the cost wll not exceed $1 bllon. Ths s a subjectve estmate of probablty. Cash flows as random varables A random varable s a functon that assgns real numbers to events (outcomes of an experment). E.g., the salvage value of a machne, S, s $1,500 f the market goes up (wth probablty 0.4) and $1,000 f the market goes down (wth probablty 0.6). Then, S s a random varable wth the followng probablty dstrbuton P{S = 1500} = 0.4 and P{S = 1000} = 0.6. If the random varable can take on a lmted number of values. Then, ths s a dscrete random varable. E.g., the salvage value of the machne above. If the random varable can take on an uncountable number of values. Then, ths s a contnuous random varable. E.g., you may estmate that the monthly fuel consumpton of your car s equal lkely to be between $200 and $250. Then, ths consumpton s a contnuous random varable. 6

7 Expectaton of a random varable The expectaton of a dscrete random varable X s E[ X ] = x P{ X = x } x E.g., for the machne above the expected salvage value s E[S] = 1, , = $1,200. The expectaton of a random varable s the value obtaned f the underlyng experence s repeated for a number of tmes whch s large enough and the resultng values are averaged. Usng expected value Replacng cash flows wth ther averages Here one fnds the expected value of an uncertan (random) cash flow, and then uses ths value n the economc analyss, proceedng just lke what we have been dong. Ths, often used, approach s only an approxmaton. It s subject to the flow of averages. Example of the flow of average E.g., f the weekly demand for an tem sellng at $5 s equally lkely to be 100 or 150. The stock level for ths tem at the begnnng of the week s set at 125 unts. The average demand n ths case n 125. Replacng the demand by ts average, the expected revenue s = $625. 7

8 Note however, that there s a 50% chance that the demand wll be 100, and the revenue wll be =$500. There s also a 50% chance that demand wll be 150 whch exceeds the stock level. Then, the revenue when demand s 150 s = $625. So, the actual expected revenue s = $ Usng expected value Treatng cash flows a random varables and fndng the expected PW Due to the flow of averages, a better approach for dealng wth uncertanty n cash flows s to treat these as random varables and fnd the expected value, E[PW] of the present worth. (Ths approach s more accurate but t often nvolves an nvolved use of probablty theory. In ENMG 501, you learn more about ths.) Then, an alternatve s consdered economcally vable f ts E[PW] 0. When comparng several alternatves, the alternatve wth the hghest E[PW] s selected. 8

9 Measures of varablty The varance of a dscrete random varable X s 2 ( ) 2. 2 Var[ X ] = E[( X E[ X ]) ] = x P{ X = x } E[ X ] The standard devaton of X s σ [ X ] = Var[ X ]. It measures the varablty around the expectaton. An alternatve (or an estmate) wth a large varance should be handled wth care. x 9