TIØ 1: Financial Engineering in Energy Markets

Transcription

1 TIØ 1: Financial Engineering in Energy Markets Afzal Siddiqui Department of Statistical Science University College London London WC1E 6BT, UK

2 COURSE OUTLINE F Introduction (Chs 1 2) F Mathematical Background (Chs 3 4) F Investment and Operational Timing (Chs 5 6) F Entry, Exit, Lay-Up, and Scrapping (Ch 7) F Recent Theoretical Work I: Capacity Sizing F Recent Theoretical Work II: Risk Aversion and Multiple Risk Factors F Applications to the Energy Sector I: Capacity Sizing, Timing, and Operational Flexibility F Applications to the Energy Sector II: Modularity and Technology Choice 1-8 September 2011 Siddiqui 2 of 49

3 LECTURE OUTLINE F Optimal stopping time problem F Risk-averse decision makers F Analytical solutions with two sources of uncertainty 1-8 September 2011 Siddiqui 3 of 49

4 TRADITIONAL NPV APPROACH F Example from McDonald (2002): oil extraction under certainty at a rate of one barrel per year forever I Current price of oil is P 0 =15,discountrateisρ =0.05, growth rate of oil is α =0.01, operating cost is c =8,andinvestmentcost is I =180 F Is it optimal to extract the oil now? I Assuming that the price of oil grows exponentially, the NPV from immediate extraction is V (P 0 ) = R e ρt P 0 0 e αt c ª dt I = P 0 ρ α c ρ I = = 35 I Since V (P 0 ) > 0, it is optimal to extract F But, would it not be better to wait longer? F Investment cost is being discounted, and the value of the oil is growing 1-8 September 2011 Siddiqui 4 of 49

5 OPTIMAL INVESTMENT TIMING F Think instead about value of perpetual investment opportunity I F (P 0 )=max T R T e ρt P 0 e αt c ρi}dt =max T P 0 ρ α e(α ρ)t c ρ e ρt Ie ρt I T = 1 α ln ³ c+ρi P 0 = I Or, invest when P T =17 I Indeed, the initial value of the investment opportunity is F (P 0 )= > 35 = V (P 0 ) F By delaying investment to the optimal time period, it is possible to maximise NPV F How does this work when the price is stochastic? 1-8 September 2011 Siddiqui 5 of 49

6 OPTIMAL INVESTMENT UNDER UNCERTAINTY F Price process evolves according to a GBM, i.e., dp t = αp t dt + σp t dz t with initial price P 0 = p I Note that (dp t ) 2 = σ 2 (P t ) 2 dt 1-8 September 2011 Siddiqui 6 of 49

7 OPTIMAL INVESTMENT UNDER UNCERTAINTY F If the project were started now, then its expected NPV is V (p) =E p R 0 e ρt {P t (c + ρi)} dt = p ρ α c ρ I F Canonical real options problem: Z F (p) =supe p e ρt {P t (c + ρi)} dt τ S τ F (p) =supe p e ρτ V (P τ ) ( µ β1 p =max V (P I )) τ S P I p I β 1 (β 2 ) is the positive (negative) root of 1 2 σ2 ζ(ζ 1) + αζ ρ =0 P I 1-8 September 2011 Siddiqui 7 of 49

8 STOCHASTIC DISCOUNT FACTOR F Proposition: The conditional expectation of the stochastic discount factor, E p [e ρτ ], is the power function, ³ p P I β1, whereτ min {t : Pt P I } F Proof: Let g(p) E p [e ρτ ] I g(p) =o(dt)e ρdt +(1 o(dt))e ρdt E p [g(p + dp )] I g(p) = o(dt)e ρdt + i (1 o(dt))e ρdt E p hg(p)+dp g 0 (p)+ 1 (dp 2 )2 g 00 (p)+o(dt) I g(p) =o(dt)+e ρdt g(p)+e ρdt αpg 0 (p)dt + e ρdt 1 2 σ2 p 2 g 00 (p)dt I g(p) = o(dt) +(1 ρdt)g(p) +(1 ρdt)αpg 0 (p)dt +(1 ρdt) 1 2 σ2 p 2 g 00 (p)dt I ρg(p)+αpg 0 (p)+ 1 2 σ2 p 2 g 00 (p) = o(dt) dt I g(p) =a 1 p β 1 + a 2 p β 2 I lim p 0 g(p) =0 a 2 =0andg(P I )=1 a 1 = 1 P β 1 I 1-8 September 2011 Siddiqui 8 of 49

9 OPTIMAL INVESTMENT THRESHOLD UNDER UNCERTAINTY F Solve for optimal investment threshold, P I : ( µ β1 p F (p) =max V (P I )) P I p ³ I I First-order necessary condition yields P I = β 1 (ρ α) c + β 1 1 ρ I Note that in the case without uncertainty, β 1 = ρ P α I = c + ρi F For a level of volatility of σ =0.15, P I =25.28, and the value of the investment opportunity is F (p) = P I F Comparedtothecasewithcertainty,theinvestmentopportunity is worth more but is also less likely to be exercised 1-8 September 2011 Siddiqui 9 of 49

10 INVESTMENT THRESHOLDS AND VALUES 1-8 September 2011 Siddiqui 10 of 49

11 INVESTMENT UNDER UNCERTAINTY WITH ABANDONMENT F If the project is abandoned after investment, then the expected incremental payoff is: Z V A (p) =E p e ρt {(c ρk s ) P t } dt 0 F Solve for optimal abandonment threshold, P : ( µ β2 p F A (p) =max V A (P )) + V (p) P p 1-8 September 2011 Siddiqui 11 of 49 P = c ρ K s p ρ α ³ I First-order necessary condition yields P = β 2 (ρ α) c K β 2 1 ρ s I Solve ½ numerically for P I : F (p) = ³ β1 ³ ¾¾ β2 max p PI p P I ½V (P I )+ PI P V A (P )

12 INVESTMENT THRESHOLDS AND VALUES WITH ABANDONMENT 1-8 September 2011 Siddiqui 12 of 49

13 INVESTMENT UNDER UNCERTAINTY WITH SUSPENSION AND RESUMPTION F If the project is resumed from a suspended state, then the expected incremental payoff is: Z V R (p) =E p e ρt {P t (c + ρk r )} dt 0 F Solve for optimal resumption threshold, P : ½ ³ ¾ p β1 F R (p) =max V R (P ) P p P = p ρ α c ρ K r ³ I First-order necessary condition yields P = β 1 (ρ α) c + K β 1 1 ρ r I Substitute P back into F S (p) tosolvenumericallyforp and then repeat for F (p) toobtainp I 1-8 September 2011 Siddiqui 13 of 49

14 INVESTMENT THRESHOLDS AND VALUES WITH RESUMPTION 1-8 September 2011 Siddiqui 14 of 49

15 INVESTMENT WITH INFINITE SUSPENSION AND RESUMPTION OPTIONS F Start with the expected value of a suspended project: V c (p,, ; P,P ) = p P β1 (V o (P,, ; P,P ) K r ) F Also note the expected value of an active project: V o (p,, ; P,P p ) = c + ρ α ρ ³ β2 ³ p c K P ρ s P + V ρ α c(p,, ; P,P ) I Solve the two equations numerically, i.e., start with initial thresholds and successively iterate until convergence F Finally, solve for P I numerically: F (p,, ; P,P )= ³ β1 max p PI p {Vo P I (P I,, ; P,P ) I} 1-8 September 2011 Siddiqui 15 of 49

16 INVESTMENT THRESHOLDS AND VALUES WITH COMPLETE FLEXIBILITY 1-8 September 2011 Siddiqui 16 of 49

17 INVESTMENT THRESHOLDS WITH COMPLETE FLEXIBILITY 1-8 September 2011 Siddiqui 17 of 49

18 NUMERICAL RESULTS: Data from McDonald (2002) F P 0 = 15,c = 8, ρ = 0.05, α = 0.01,I = 180,K s = 25,K r = September 2011 Siddiqui 18 of 49

19 INCORPORATION OF RISK AVERSION F Hugonnier and Morellec (2007) take the perspective of a risk-averse decision maker with the perpetual option to invest in a project without operational flexibility F Chronopoulos, De Reyck, and Siddiqui (2011) consider a case with operational flexibility I Includes embedded options to shut down and re-start the project (infinitely) many times after initial investment I Solve for optimal investment and operational thresholds along with option value of investment opportunity F Take the approach of McDonald and Siegel (1986) to solve nested optimal stopping time problems I Specify a CRRA utility-of-wealth function I Apply result from Karatzas and Shreve (1999) concerning the discounted expected value of a function of a GBM process I Solve embedded sub-problems backwards 1-8 September 2011 Siddiqui 19 of 49

20 RISK-AVERSE PROBLEM FORMULATION: Assumptions F Decision maker has the perpetual right to start the project at any time for deterministic investment cost, I F Price process evolves according to a GBM, i.e., dp t = αp t dt + σp t dz t with initial price P 0 = p I An active project incurs a deterministic operating cost of c F Utility-of-wealth function is U(w) = w1 γ 1 γ for 0 γ < 1 F The project may also entail (infinitely) many embedded options to shut down and re-start costlessly F Risk-free and subjective interest rates are r and ρ, respectively (both greater than α) 1-8 September 2011 Siddiqui 20 of 49

21 RISK-AVERSE PROBLEM: Timeline of Cash Flows without Operational Flexibility F Initially hold a CD of size I + c that earns the risk-free r rate of return and, at time τ1,isexchangedforastream of risky instantaneous cash flows, P t I The discounted conditional lifetime expected utility of cash R h τ flows is 1 R i e ρt U(rI + c)dt + E 0 p e ρt U(P τ1 t )dt = R h i e ρt U(rI + c)dt + E 0 p e ρτ 1 V 0 (P τ 1 ), where V 0 (p) = R E p e ρt {U(P 0 t ) U(rI + c)} dt 1-8 September 2011 Siddiqui 21 of 49

22 RISK-AVERSE PROBLEM: No Operational Flexibility F From Karatzas and Shreve (1999), the expected NPV of active project is V 0 (P τ1 ) = E Pτ1 R 0 e ρt (U (P t ) U (ri + c)) dt = β 1 β 2 p 1 γ (c+ri)1 γ ρ(1 γ)(1 β 2 γ)(1 β 1 γ) ρ(1 γ) F Value of investment opportunity: F 0 (p) = ³ β1 sup τ1 S E p [e ρτ 1 p ] V 0 (P τ1 )=max PI p V0 P I (P I ) F Optimal investment threshold is PI (γ) = h i 1 (c + ri) β2 1+γ 1 γ β September 2011 Siddiqui 22 of 49

23 RISK-AVERSE PROBLEM: Effect of Risk Aversion on Investment Threshold 1-8 September 2011 Siddiqui 23 of 49

24 RISK-AVERSE PROBLEM: Timeline of Cash Flows with Single Abandonment Option F Now, allow for abandonment at time τ 2 I The discounted conditional lifetime expected utility of cash flows is R h i h i e ρt U(rI + c)dt + E 0 p e ρτ 1 V 0 (P τ )+E 1 p e ρτ 2 V 1 (P τ 2 ), R where V 1 (p) =E p e ρt {U(c) U(P 0 t )} dt 1-8 September 2011 Siddiqui 24 of 49

25 RISK-AVERSE PROBLEM: Single Abandonment Option F Expected discounted utility of cash flows at time τ 1 is V 0 (P τ1 )+sup τ2 τ 1 E Pτ1 e ρ(τ 2 τ 1 ) V 1 (P τ2 ) F Value of investment opportunity: F 1 (p) = sup τ1 S E p e ρτ 1 V0 (P τ1 )+sup τ2 τ 1 E Pτ1 e ρ(τ 2 τ 1 ) V 1 (P τ2 ) ª I F 1 (p) =max PI p max PA P I ³ PI P A β2 V1 (P A ) ³ p P I β1 [V0 (P I )+F A (P I )], where F A (P I )= I Optimal abandonment threshold is P A(γ) =c I FONC for investment: ³ P I P A β2 ρ(β 1 β 2 ) β 1 (1 γ)v 1 (P A)=0 h i 1 β1 1+γ 1 γ β 1 β 2 (P 1 β 2 γ I ) 1 γ + (c + ri) 1 γ 1-8 September 2011 Siddiqui 25 of 49

26 RISK-AVERSE PROBLEM: Effect of Abandonment Option on Investment Threshold 1-8 September 2011 Siddiqui 26 of 49

27 RISK-AVERSE PROBLEM: Timeline of Cash Flows with Single Suspension and Resumption Option F With subsequent resumption option at τ 3 I The discounted conditional lifetime expected utility of cash flows is R h i h i e ρt U(rI + c)dt + E 0 p e ρτ 1 V 0 (P τ )+E 1 p e ρτ 2 V 1 (P τ 2 )+ i E p he ρτ 3 R 0 V 2 (P τ ), where V 3 2 (p) =E p e ρt {U(P t ) U(c)} dt 1-8 September 2011 Siddiqui 27 of 49

28 RISK-AVERSE PROBLEM: Single Suspension and Resumption Option F Time-τ 1 expected discounted utility of cash flows: V 0 (P τ1 ) + sup τ2 τ 1 E Pτ1 e ρ(τ 2 τ 1 ) V 1 (P τ2 )+sup τ3 τ 2 E Pτ2 e ρ(τ 3 τ 2 ) V 2 (P τ3 ) F Value of investment opportunity: F 2 (p) = ³ β1 p max PI p [V0 P I (P I )+F S (P I )] I F S (P I ) = max PS P I ³ PI P S β2 {V1 (P S )+F E (P S )} I F E (P S )=max PE P S ³ P S P E β1 V2 (P E ) h i 1 F Optimal resumption threshold is PE (γ) =c β 2 1+γ 1 γ β September 2011 Siddiqui 28 of 49

29 RISK-AVERSE PROBLEM: Effect of Suspension and Resumption Options on Investment Threshold 1-8 September 2011 Siddiqui 29 of 49

30 RISK-AVERSE PROBLEM: Complete Operational Flexibility F At the resumption threshold, P E, the expected utility of cash flows of an active firm is V o (P E,, ; P S,P E )= ³ β2 ³ β2 ³ P E P V1 PS (P S )+ E P S PS V 2 (P E )+ P E β1 V2 (P E )+ ½ ³ PE P S β2 ³ PS P E β1 ¾ i V 2 (P E )+ I V o (P E,, ; P S,P E ) = P i=0 ³ ½ β2 P E P ³ β2 ³ ¾ β1 i PE PS P S i=0 P S P E V 1 (P S ) = 1 PE β2 ³ PS β1 1 ³ PS P E ½ V 2 (P E )+ ³ P E P S β2 V1 (P S ) β1 I V c (P S,, ; P S,P E )=³ PS Vo P E (P E,, ; P S,P E ) I µ β1 Z p F (p) =max E PI e ρt {U(P t ) U(c + ri)} dt + P I p P I 0 β2 V c (P S,, ; P S,P E )# µ PI ¾ 1-8 September 2011 P S Siddiqui 30 of 49

31 RISK-AVERSE PROBLEM: Value Curves without Operational Flexibility 1-8 September 2011 Siddiqui 31 of 49

32 RISK-AVERSE PROBLEM: Investment Thresholds without Operational Flexibility 1-8 September 2011 Siddiqui 32 of 49

33 RISK-AVERSE PROBLEM: Results Summary with Abandonment 1-8 September 2011 Siddiqui 33 of 49

34 RISK-AVERSE PROBLEM: Abandonment Thresholds 1-8 September 2011 Siddiqui 34 of 49

35 RISK-AVERSE PROBLEM: Results Summary with Single Suspension and Resumption 1-8 September 2011 Siddiqui 35 of 49

36 RISK-AVERSE PROBLEM: Impact of Operational Flexibility and Risk Aversion on Optimal Decision Thresholds 1-8 September 2011 Siddiqui 36 of 49

37 RISK-AVERSE PROBLEM: Results Summary with Complete Flexibility 1-8 September 2011 Siddiqui 37 of 49

38 TWO SOURCES OF UNCERTAINTY: Analytical Solutions F For a perpetual investment problem with payoff of the form V (P, C) = P C, where both P ρ α ρ t and C t follow correlated GBMs, use homogeneity to convert the resulting PDE to an ODE and solve analytically for the free boundary, P (C) (Dixit and Pindyck (1994)) F But, what if the payoff is of the form V (P, C) = 1-8 September 2011 Siddiqui 38 of 49 P ρ α C ρ I? I Homogeneity no longer holds because of the I term I Pindyck (2002) examines an environmental control problem and proposes an analytical solution of the form F (P, C) =ap β C η I Adkins and Paxson (2008) formalise the proof with geometric interpretation I Heydari, Ovenden, and Siddiqui (2011) apply this technique to a problem with CCS retrofits

39 PROBLEM FORMULATION: Assumptions F Long-term electricity (E t in $/MWh e ), coal (F t in $/MWh), and CO 2 (C t in $/t) prices are exogenous and evolve according to correlated GBMs, i.e., I de t = α E E t dt + σ E E t dz E, df t = α F F t dt + σ F F t dz F, dc t = α C C t dt + σ C C t dz C,andE[dz i dz j ]=ρ ij dt i, j F In response to CO 2 emissions restrictions, the plant owner may retrofit withccsforaninvestmentcostof I ccs (in$)toobtainareductionintheemissionsrate,² C (in t/mwh e ),alongwithanincreaseintheheatrate,² F (in MWh/MWh e ) 1-8 September 2011 Siddiqui 39 of 49

40 PROBLEM FORMULATION: Assumptions (continued) F Annual electricity production of plant, Q (in MWh e ), is unaffected by retrofit decision F Retrofit occurs instantaneously upon decision F Infinite lifetime for the plant regardless of retrofit option F The exogenous discount rate is μ 1-8 September 2011 Siddiqui 40 of 49

41 PROBLEM FORMULATION: CCS Retrofit Decision F First, determine the PV of benefits from the CCS retrofit: I V pc (E,F,C) =QE R (E 0 t e μt ² F F t e μt ² C C t e μt )dt E, F,C h i V pc E (E,F,C) =Q μ α E ² F F μ α F ² C C μ α C h i I V pc (E,F,C)+V ccs E (F, C) =Q μ α E ²ccs F F μ α F ²ccs C C μ α i C V ccs (F, C) =Q h (²F ² ccs F )F μ α F + (² C ² ccs C )C μ α C 1-8 September 2011 Siddiqui 41 of 49

42 PROBLEM FORMULATION: CCS Retrofit Option Value F Use the Bellman Equation to solve for the option value to retrofit to CCS: I μw ccs dt = E[dW ccs ] 1 2 σ2 F F 2 WFF ccs α C CWC ccs μw ccs = σ2 CC 2 W ccs CC + ρσ F σ C FCW ccs FC + α F FW ccs 0 F + F Guess W ccs (F, C) =af β C η I H(β, η) = 1 2 σ2 F β(β 1)+ 1 2 σ2 Cη(η 1)+ρσ F σ C βη+α F β+α C η μ = 0 I The roots of H fall on an ellipse that passes through all four axes 5 (Adkins and Paxson (2008)) 5 β η 1-8 September 2011 Siddiqui 42 of 49

43 PROBLEM FORMULATION: CCS Retrofit Option Value F Value-matching and smooth-pasting conditions I W ccs (F, C (F )) = V ccs (F, C (F )) I ccs I WF ccs (F, C (F )) = VF ccs (F, C (F )) I WC ccs (F, C (F )) = VC ccs (F, C (F )) I This system gives us a linear relationship between β and η: β = Q(² F ² ccs F )(η 1)F (μ α F )I ccs Q(² F ² ccs F )F F Impose this line on the ellipse H(β, η) =0 I Two sets of solutions: β 1 < 0andη 1 > 0 β 2 > 0andη 2 < 0 I Hence, W ccs (F, C) =a 1 F β 1 C η 1 + a 2 F β 2 C η 2 I For low values of C, the option value is worthless, i.e., a 2 =0, which implies W ccs (F, C) =a 1 F β 1 C η September 2011 Siddiqui 43 of 49

44 PROBLEM FORMULATION: CCS Retrofit Option Value 1-8 September 2011 Siddiqui 44 of 49

45 NUMERICAL EXAMPLE: Data μ α F α C σ F σ C ρ ² F ² C Q GWh e ² ccs F ² ccs C $1.3 billion I ccs F 0 C 0 $15.50/MW h $31.81/t 1-8 September 2011 Siddiqui 45 of 49

46 NUMERICAL EXAMPLE: CCS Retrofit Option Values 1-8 September 2011 Siddiqui 46 of 49

47 NUMERICAL EXAMPLE: CCS Retrofit Thresholds 1-8 September 2011 Siddiqui 47 of 49

48 NUMERICAL EXAMPLE: Sensitivity Analysis 1-8 September 2011 Siddiqui 48 of 49

49 QUESTIONS 1-8 September 2011 Siddiqui 49 of 49