BENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets

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1 BENEIT-CST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal - Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts used i ivestmet appraisal iterest rate discout factor et preset value iteral rate of retur margial productivity of capital beefit/cost ratio et beefit stream auities perpetuities cost of capital depreciatio iflatio real ad omial (moey) rates of iterest risk premium igure 2.1: Ivestmet Apprai sal Š a Private Perspectiv e How do we appraise this proposed ivestmet? ollars N ow E Compare: the world with the ivestmet (represeted by poit B, with cosumptio C 1 ad C 2 ); ad the world without the ivestmet (represeted by poit A, with cosumptio Y 1 ad Y 2 ). Y 1 C 1 Y 2 A C C2 B H ollars N ext Y ear Which do you prefer? Poit A or poit B? We ca t simply compare Y 1 +Y 2 with C 1 +C 2 because of the time value of moey (represeted by the iterest rate). Calculate preset values: PV(Y 1,Y 2 ) = ; PV(C 1,C 2 ) = E; E>, hece, prefer E i.e. udertake the ivestmet Ledig ad Borrowig We have bee assumig that if your icome stream is Y 1,Y 2, your cosumptio stream must be the same. Ad if you ivest, your cosumptio stream must be C 1,C 2. However, by ledig or borrowig at the market rate of iterest, you ca choose ay poit o the et preset value lie through A (if you do t ivest), or through B (if you do ivest). or example, if you do ivest (B) you could borrow i period 1 to fiace the cosumptio combiatio represeted by poit. Note that represets more of both commodities (dollars ow ad dollars ext year) tha A. Applyig Ivestmet ecisio Rules NPV = BC[] - AC >, hece udertake project. The relatio BC[] - AC > ca be rearraged i various ways to yield equivalet decisio rules: Beefit/Cost Ratio, BCR = BC[]/AC > 1, hece udertake the project; Margial Productivity of Capital, MP K = BC/AC > (1+r), hece udertake the project; Iteral Rate of Retur, IRR = MP K - 1 = [BC/AC] > r, hece udertake the project. To solve for IRR choose r P to set BC[1/(1+r P )] - AC = ie. IRR = [BC/AC] -1.

2 igure 2.2 : A Coutry s Iter-temporal Productio Possibilities Curve ollars Worth of Cosumptio oods Now E igure 2.3 : The Iter-temporal Effects of Iteratioal Trade ollars Worth of Cosumptio oods Now H ollars Worth of Cosumptio oods Next Year ollars Worth of Cosumptio oods Next Year igure 2.4: Beefit Stream of a TwoŠPeriod Ivestmet Project + B1 B2 Calculatig Preset NPV = -K + B 1 [1/(1+r 1 ] + B 2 [1/(1+r 1 ][1/(1+r 2 ] Assume r 1 = r 2 = r: NPV = -K + B 1 [] + B 2 [] 2 IRR: Solve a quadratic equatio for r P : = -K + B 1 [1/(1+r P )] + B 2 [1/(1+r P )] 2 - K 1 2 Year This quadratic equatio could have: oe positive solutio two positive solutios o solutio Example: K = 1.6; B 1 = 1 ; B 2 = 1 igure 2.5: Preset i Relatio to the iscout Rate Preset igure 2.6 : Calculatig Iteral Rates of Retur e Positive (x) IRR iscout Rate X (=1 + r p)

3 Aother example of a IRR calculatio: K = 1.6 ; B1 = 1 ; B2 = - 1 igure 2.7 : Calculatig Iteral Rates of Retur Two Positive s (x) Whe we solve the quadratic equatio i this case, we will get two positive IRRs What makes this example differet? There are two chages i the sig of the et beefit stream: -1.6, +1, -1 (Compare with the earlier example: -1.6, +1, +1) Xa X (=1 + rp) X2 X1 There will geerally be as may positive IRRs as there are chages i sig. igure 2.8: Preset i Relatio to the iscout Rate - the Two Positive Iteral Rates of Retur Case Preset igure 2.9: Preset i Relatio to the iscout Rate - the No Iteral Rates of Retur Case Preset iscout Rate iscout Rate Auities ad Perpetuities A auity is a stream of equal aual paymets, B, startig oe year from the preset ad termiatig after paymets. PV(A) = B/(1+r) + B/(1+r) 2 + B/(1+r) 3.+B/(1+r) A auity due is simply a auity that starts right ow: PV() = B + PV(A) - B/(1+r) Treatig PV(A) as a geometric progressio, we ca write: PV(A) = B[(1+r) - 1]/[r(1+r) ] A perpetuity is a auity that goes o for ever. The PV of a perpetuity is obtaied by lettig go to ifiity i the above expressio: PV(A) = B/r Aual Cost of Capital Suppose you have just bought a machie (e.g. a car) that cost K ad will last for years. The aual cost of capital (sometimes called its retal price) is give by C, where: C r K = t = C [( 1+ ) 1 ] = CAr (, ) ( 1 + r ) r( 1 + r) t = 1 where A(r,) is the auity factor. We also kow that aual capital cost cosists of iterest plus depreciatio: C = rk + ; assumig that depreciatio,, is treated as a costat aual cost.

4 rom the precedig discussio we have: C = K/A(r,) = rk +, where A(r,) =[(1+r) - 1]/[r(1+r) ] We ca ow solve for the aual depreciatio cost: = K[(1/A(r,)) - r] = rk/[(1+r) -1] To check this calculatio, plug the expressio for ito C = rk + to get: C = K/A(r,) We have see that there are two equivalet methods of dealig with capital cost: 1. Iclude it i the et beefit stream as a cost, K, at the poit at which is occurs (usually the preset). This is what we usually do i cost-beefit aalysis; 2. Iclude it i the et beefit stream as the aual cost of iterest plus depreciatio. This is how firms usually treat capital cost (for tax reasos). Sice the preset value of the aual costs icluded uder method 2 is equal to the iitial cost accouted for uder method 1, it is importat ot to use both methods or you will double-cout capital cost. I calculatig NPV we use method 1 ad we igore ay aual iterest or depreciatio costs. The Role of Iflatio i Beefit-Cost Aalysis The iterest rates quoted i the fiacial press are omial (or moey) rates of iterest (deoted by m). The moey rate of iterest is the real rate, r, plus the expected rate of iflatio, i. m = r + i The et beefit stream is a flow of goods ad services valued at a set of prices. There are two sets of prices which are commoly used: today s prices ca be used to value commodities at all poits i time (i.e. valued at costat prices); prices at the time the commodity is produced or used ca be used (i.e. valued at curret prices). Two importat rules to remember whe discoutig to calculate et preset value: If the et beefit stream is valued at costat prices, use a real rate of iterest to compute the discout factors; If the et beefit stream is valued at curret prices, use a omial iterest rate to compute the discout factors. Suppose we wat to calculate the PV of a to of coal i year t. We ca value the coal at costat prices (today s price), P ; or we ca value it at curret prices (the price i year t), P t. The relatioship betwee these two prices is: P t = P (1+i) t where i is the aual iflatio rate. The two approaches to discoutig ad iflatio are equivalet: P [ t ] = P t [1/(1+m) t ] = P (1+i) t /(1+m) t or this equivalece to hold it must be the case that: 1 ( 1 + i) = ( 1 + r) ( 1 + m) By cross-multiplyig, it ca be see that this implies: m = r + i + ri Ad sice ri ca be igored as very small, it implies that the moey rate equals the real rate, plus the rate of iflatio. How should we deal with iflatio i beefit-cost aalysis? The easiest approach is to igore it: value costs ad beefits at costat prices ad use a real rate of iterest as the discout rate. A exceptio to this approach is whe the price of a sigificat output or iput is iflatig at a very differet rate from the rate of geeral price iflatio. I that case we would wat to value all iputs at curret prices ad use the omial iterest rate as the discout rate.

5 The Risk Premium o the iscout Rate Suppose you have built a dam which yields a aual beefit of B (valued at costat prices), but which, i ay year, may burst with probability, p. The expected beefit i year t is: E(B) = B(1-p) t, ad the preset value of expected beefit is: PV[E(B)] = B(1-p) t /(1+r) t, where r is the real rate of iterest. It is easy to show that (1-p) t /(1+r) t is approximately equal to [1/(1+r+p) t ]. I other words the risk of the dam burstig ca be accouted for by addig a risk premium to the discout rate. This procedure does ot really deal with risk, which ecoomists defie as variatio aroud the expected value.

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