Discounted Cash Flow (DCF) Analysis: What s Wrong With It And How To Fix It

Transcription

1 Dscounted Cash Flow (DCF Analyss: What s Wrong Wth It And How To Fx It Arturo Cfuentes (* CREM Facultad de Economa y Negocos Unversdad de Chle June 2014 (* Jont effort wth Francsco Hawas; Depto. de Ingenera Industral, U de Chle.

2 Table of Contents Introducton The Problem The Standard DCF Approach Issues wth the Standard DCF Approach A Better Approach Conclusons

3 INTRODUCTION

4 What s the Ultmate (the Most Fundamental Fnancal Queston? Answer: How Much Should I Pay for Ths? OR What s the Value of Ths?

5 THE PROBLEM

6 More Formally N Tme N A Fnancal Asset Generates Future Cash Flows ( 1, 2,, N What s the Value of ths Asset? Answer: The Present Value (PV of The Future Cash Flows But How Do We Calculate the PV of These Cash Flows?

7 THE STANDARD DCF APPROACH

8 N Tme N The Standard Approach Is To Calculate The Present Value (PV of the Usng the FUTURE CASH FLOWS APPROPRIATE DISCOUNT RATE

9 Intal Investment N Tme N PV N 1 (1 R future cash flows (? approprate dscount rate (?

10 ISSUES WITH THE STANDARD DCF APPROACH

11 A Bad Analogy: Bond Valuaton (Dean, 1951 Example: Typcal bond cash flows $ 5 $ 5 $ 5 $ 5 $ Tme N Bond Face Value = $ 100; nterest, 5% per annum BondValue PV N 1 ( 1 F R rsk free rate credt rsk spread

12 A Bad Analogy: Contnuaton Example: Typcal project (asset cash flows N Tme N PV N 1 ( 1 RF R R F R represents the approprate dscount rate: Rsk Free Rate 0 ( WACC (weghted average cost of captal or ( opportunty cost of captal

13 The Analogy (Bond Valuaton/ Project Valuaton Has Two Serous Inconsstences 1] Bond: Credt Rsk (Cash Flows are Bounded Project: No Credt Rsk (Cash Flows are Unbounded 2] Bond Valuaton: Dscount Rate s Determned by a Property of the Cash Flows (Issuer s credt rsk profle Project: Dscount Rate s Determned by a Feature of the Potental Investor (Opportunty Cost of Captal

14 What Do We Mean by Future Cash Flows? Expected Cash Flows? Typcal-Case Cash Flows? Worst-Case Scenaro Cash Flows? Base-Lne Scenaro Cash Flows?

15 Tme N N F N F F N F R R R R PV (1 (1 (1 1 ( (1 1 F F N F R R where R PV can be rewrtten as Let us dsassemble the DCF expresson

16 PV N 1 (1 R F where 1 RF 1 R F 1 In summary, the standard DCF method conssts of two steps PV N (1 R 1 F Step 2: Reduce the value of the dscounted cash flow (due to ts uncertan nature Step 1: Brng cash flow to tme = 0 (tme value of money. We dscount wth the rsk free rate

17 Thus, the standard DCF approach can be nterpreted as a specal case of the Certanty Equvalent (CE approach N F N F CE where R CE R PV ( (1 ( 1 ( 1 1

18 N F N F CE where R CE R PV ( (1 ( 1 ( 1 1 NOTE: Implct behnd the certanty equvalent (CE dea s the Utlty Functon concept. In fact, nvokng the defnton of CE, we can wrte that must satsfy the followng relatonshp ( ( ( ( ( ( 1 U E U OR U E U

19 Let Us Assume That All Cash Flows Are Postve. The Frst, of Course (t=0; Intal Investment Is Assumed To Be Negatve N Tme N PV N N 1 (1 RF 1 (1 RF CE( Implct n ths formula there are two very sgnfcant assumptons: ( The uncertanty assocated wth the cash flows ncreases as we move forward n tme; and ( Furthermore, the uncertanty ncreases accordng to a very precse (restrctve pattern

20 Suppose Now That One of the Cash Flows s Negatve (an Expendture; say 3 = -$ N Tme N R F % and 7% whch gves The standard DCF method underestmates all cash outflows (optmstc wth expendtures; whle of course t s pessmstc wth nflows In short, the conventonal DCF method ntroduces a systematc error n the valuaton process

21 Assume We Have to Choose Between Two Projects: (A and (B. Moreover, Assume That the Level of Uncertanty n the Cash Flows of (A [ 1, 2, ] and (B [Y 1, Y 2, ] Is Dfferent. For Example, In Case (A We Mght Have Better Estmates of the Cash Flows. The Standard DCF Method Cannot Deal Wth Ths Stuaton: It Treats The Cash Flows From (A and (B As If They Were Subject To the Same Uncertanty PV ( A (1 R F PV ( B Y (1 R F Same Level of Uncertanty Wll Lead to the Wrong Decson

22 A More Phlosophcal (However Practcal Consderaton [1] The problem n queston s nterestng because the cash flows (numerator are uncertan (stochastc. However, the standard DCF approach has chosen to deal wth ths uncertanty n the NUMERATOR by INCREASING THE VALUE OF THE DENOMINATOR (that s, manpulatng the wrong target [2] As a result, all the efforts have been placed on tryng to determne the correct dscount rate (denomnator nstead of payng attenton to the probablstc nature of the numerator In Summary: the DCF method s based on a strange repudaton of statstcs

23 A BETTER APPROACH

24 A New Approach Must Be Cleaner (And Honest Conceptually Must Deal Openly and Explctly (and Separately Wth the Two Elements of Any Valuaton [1] Tme Value of Money (Dscount wth the Rsk Free Rate [2] Uncertanty n the Cash Flows (Descrbe them Probablstcally, for example, Usng ther Means and Standard Devatons

25 Problem Statement Let be the vector of cash flows assocated wth a project, =(x 0, x 1,, x n Assume that follows a multvarate dstrbuton, F, that s, F(μ, C where μ denotes the expected value of (μ = E(x, for =0, 1,,n and C represents the correspondng (n+1 x (n+1 varancecovarance matrx. Assumng that the analyst has estmated the values of μ and σ (for =0, 1,, n and ρ,j (for, j= 0, 1,, n and j, and that the structure of the multvarate functon (F s known The problem conssts of developng a method to estmate the expected value and the standard devaton of the usual metrcs (net present value (NPV, nternal rate of return (IRR, payback perod (PBP, etc. plus ther probablstc dstrbutons.

26 In the case of the NPV, we can obtan analytcal expressons for both ts expected value and ts varance, regardless of the dstrbuton assumed for. where R desgnates the dscount rate (the rsk-free rate snce the uncertanty n the cash flows s captured va ts varance-covarance matrx.

27 Smulaton Approach The dea s to treat the problem at hand usng a Monte Carlo smulaton algorthm. The dffculty, of course, resdes on the ablty to generate sample cash flow vectors accordng to the specfed dstrbuton, namely, F(μ, C.

28 General Case Algorthm No assumptons are made regardng the structure of F, or, alternatvely, the nature of the margnal dstrbutons of the x s (=0, 1,, n. The Gaussan copula technque can be used to generate a famly of cash flow vectors ( [0] Fnd the Cholesky decomposton (factorzaton of the correlaton matrx, that s, fnd a matrx L such that, Cρ= L L [1] Generate Y = (y 0, y 1,, y n where the y s are random draws from d N(0, 1; [2] Compute Z=LY; [3] Let Φ denote the cumulatve dstrbuton functon of the standard normal dstrbuton; determne the vector U = (u 0, u 1,, u n, wth u = Φ(z for =0, 1,, n (thus, 0 u 1; [4] Determne, the desred sample vector, usng x = F -1 (u for =0, 1,, n where the functon F -1 represents the nverse cumulatve dstrbuton of the desred margnal dstrbutons (of x. Repeatng ths loop (steps [1] through [4] several tmes, we can generate suffcent cash flow vectors (, wth the desred propertes, to estmate whatever fgures of mert (and ther dstrbutons we need.

29 Specal Case Algorthm Ths case refers to the stuaton n whch the cash flows follow a multvarate normal dstrbuton, that s, MN(μ, C. In ths case, we can stll rely on the Gaussan copula to generate the random sample vectors. However, the algorthm becomes much smpler. Proceed as follows: [0] Fnd the Cholesky decomposton of the correlaton matrx, that s, fnd L such that, Cρ= L L ; [1] Generate Y = (y 0, y 1,, y n where the y s are random draws from d N(0, 1; [2] Compute Z=LY; [3] Determne, the desred sample vector, usng x = μ + σ z (for =0, 1,, n. Repeatng ths loop (steps [1] through [3] several tmes, we can get enough vectors 's to estmate the desred quanttes by averagng across all samples.

30 EAMPLE

31 Project Cash Flows: Intal Statement Year Phase Cash Flow (US$ Mllons 0 Constructon Constructon A 2 Constructon Constructon Operaton Operaton Operaton Operaton Operaton B Operaton Operaton Operaton Operaton Operaton Operaton Operaton Operaton 28.0 The cash flows follow a normal dstrbuton wth means (μ s accordng to the data shown n Table 1. Phase A: λ A = 10%. σ = λ A μ for =0, 1, 2 and 3. We also make the assumpton that durng ths phase the cash flows are uncorrelated. Phase B: λ B = 40% σ = λ B μ for =4, 5,, 16. We also assume a correlaton submatrx B wth a tr-dagonal structure and wth dentcal values ρ = 20% along the upper- and lower-sub-dagonals. Fnally, the rsk free rate, R s 6.2 %.

32 Project Cash Flows: Now the Complcaton!!!! Year Phase Cash Flow (US$ Mllons 0 Constructon Constructon Constructon Constructon Operaton Operaton Operaton Operaton Operaton Operaton Operaton Operaton Operaton Operaton Operaton Operaton Operaton 28.0 The cash flows follow a normal dstrbuton wth means (μ s accordng to the data shown n Table 1. Durng the constructon phase (phase A the uncertanty n the cash flows s gven by a coeffcent of varaton, λ A = 10%. That s, we have σ = λ A μ for =0, 1, 2 and 3. We also make the assumpton that durng ths phase the cash flows are uncorrelated. For the operatons phase (phase B we assume the cash flows to be more volatle and thus we take λ B, the coeffcent of varaton, to be equal to 40%. Thus, σ = λ B μ for =4, 5,, 16. We also assume a correlaton sub-matrx B wth a trdagonal structure and wth dentcal values ρ = 20% along the upper- and lower-sub-dagonals. Fnally, the rsk free rate, R,s 6.2 %. We ntroduce now a modfcaton: a government subsdy n the form of a mnmum revenue guaranty, $ 24 mllon, durng the operaton phase. Thus, f durng the operaton phase the cash flows, at any tme, fall below $ 24 mllon, the government makes up the dfference. In essence, the effect of the government nterventon s to change the dstrbuton of the cash flows x (=4,, 16 from normal to censored normal dstrbutons. In these nstances, the correspondng probablty densty functons wll exhbt a dscontnuty at x = $ 24 mllon, to account for the probablty mass concentraton on the left sde of the dstrbuton.

33 The E(NPV and St-Dev(NPV, can be computed drectly nvokng Equatons (1 and (2. E(NPV = $ mllon St-Dev(NPV = $ mllon. Consstent wth those obtaned from the Monte Carlo smulaton ($ ; $ A 95%-confdence nterval for the NPV estmated from the smulaton gves [$ mllon, $ mllon]. Strctly speakng, the NPV does not follow a normal dstrbuton; however, the central lmt theorem suggests that n ths case the normalty assumpton mght render a good approxmaton. And ths s ndeed the case: a confdence nterval assumng normalty yelds [$ mllon, $ mllon]. Fnally, a senstvty analyss to nvestgate the response of the St-Dev(NPV to varatons n λ B, ρ and R ndcates that the St-Dev(NPV s most senstve to changes n λ B, then R, and then ρ. Specfcally, a 10% change n any of these parameters (holdng the others constant results n a 8.13%, and 0.99% change n the value of St-Dev(NPV respectvely.

34 Smulaton Results (IRR, Project Wth Government Support Metrc IRR (% Mean Std-Dev 1.97 Skewness 0.52 Kurtoss % Confdence Interval Lower Bound Upper Bound Normal Emprcal Senstvty Analyss (% Varaton E(IRR Std-Dev(IRR Skewness Kurtoss ρ (10% Up 0.00% 0.75% 1.70% 0.39% λ B (10% Up 1.56% 6.41% 5.66% 1.24%

35 CONCLUSIONS

36 Valuatons Usng The Dscounted Cash Flow Method? JUST SAY NO!!!!