Principles of Finance: Real Options Analysis Lecture Notes PDF

Introduction Principles of Finance Lecture 11: Multiperiod capital budgeting under uncertainty: Real options analysis (CWS ch. 9) Rikke Sejer Nielsen 1 / 20 ...

Introduction Principles of Finance Lecture 11: Multiperiod capital budgeting under uncertainty: Real options analysis (CWS ch. 9) Rikke Sejer Nielsen 1 / 20  Introduction Introduction Until now: Investment decision making under uncertainty in a one-period setting. using NPV-method as decision rule. But investments typically runs over several periods ⇒ Flexibility (Real Options) to ▶ Expand: Expand investment for a cost ▶ Contract: Receive cash for giving up the use of real assets e.g. leasing of manufacturing facility. ▶ Abandon: Receive cash from selling a real asset for a given price e.g. research and development programs. ▶ Extend: Extend the life of a project for a given cost ▶ Defer: Option to defer the start of a project. ▶... Today: Multi-period investment decisions with capital budgeting, under uncertainty. ⇒ Method: Real options analysis (ROA) 3 / 20  Introduction Assumptions for Real options analysis (ROA) ROA is based on the binomial model and the following assumptions: Marketed asset disclaimer (MAD) ⇒ Use the value of the project itself as the underlying risky asset. No arbitrage ⇒ Replicating portfolio approach for real option-valuation Efficient capital market ⇒ Model evolution of project value over time using recombining binomial trees 4 / 20  Introduction Option valuation (one period) For the project, we have the present value wo. flexibility Su = uS0 S0 Sd = dS0 and an option with value as follows Vu V0 Vd time 0 T ∆t Value of option at terminal date, t = T :  max[ST − K ; 0] if call-option VT = max[K − ST ; 0] if put-option where K is the exercise price. 5 / 20  Introduction Option valuation (one period) Continued A replicating portfolio of an option is given by m = Number of shares of underlying risky asset B = Amount invested in default-free bonds with a risk-free return of rf. where Vu =muS0 + B(1 + rf )−∆t Vd =mdS0 + B(1 + rf )−∆t So Vu − Vd uVd − dVu m= and B = (1 + rf )−∆t (u − d)S0 u−d No arbitrage ⇒ Repl. portfolio and option must have the same price. V0 = mS0 + B 6 / 20  Introduction Option valuation: Risk-neutral probabilities V0 =mS0 + B Vu − Vd uVd − dVu = S0 + (1 + rf )−∆t (u − d)S0 u−d... ((1 + rf )∆t − d)Vu + (u − (1 + rf )∆t )Vd ⇔ (1 + rf )∆t V0 = u−d Solving for the risk-neutral probability, q ⇔ (1 + rf )∆t V0 =qVu + (1 − q)Vd ⇔ V0 =(1 + rf )−∆t (qVu + (1 − q)Vd ) (1+rf )∆t −d where q = u−d. 7 / 20  Introduction Option valuation: Risk-neutral probabilities Risk neutral probability, q, since the expected return on underlying asset = risk-free return. E[ST ] = qSu + (1 − q)Sd = quS0 + (1 − q)dS0 = q(u − d)S0 + dS0 ! (1 + rf )∆t − d = (u − d)S0 + dS0 u−d   = (1 + rf )∆t − d S0 + dS0 = (1 + rf )∆t S0 One-year (∆t = 1) expected return on underlying asset E[S1 ] − S0 = rf S0 8 / 20  Introduction Option valuation (Two periods) Assume u and d are constant Decision trees are recombining u 2 S0 uS0 The present value of the project wo. flexibility: S0 udS0 = duS0 dS0 d 2 S0 Vuu Vu and the value of the option: V0 Vud = Vdu Vd Vdd time 0 t T ∆t ∆t ⇒ Determine option value by using backwards induction. 9 / 20  Introduction Option valuation (Two periods) European options Assume rf is constant over time ⇒ q is constant. Value of option at terminal date, t = T :  max[ST − K ; 0] if call-option VT = max[K − ST ; 0] if put-option ⇒ for all scenarios at time T : Vuu , Vud and Vdd. Value of option after j ups, at time s = t for t ̸= T :   Vs,j = (1 + rf )−∆t qVs+1,j+1 + (1 − q)Vs+1,j 10 / 20  Introduction Option valuation (Two periods) European options, cont. So in our simple case with only two periods and ∆t = 1, we have Value of the option, t = 1:   Vu =(1 + rf )−1 qVuu + (1 − q)Vud   Vd =(1 + rf )−1 qVud + (1 − q)Vdd Value of the option today, t = 0:   V0 =(1 + rf )−1 qVu + (1 − q)Vd   =(1 + rf )T q 2 Vuu + 2(1 − q)qVud + (1 − q)2 Vdd 11 / 20  Introduction Option valuation (Two periods) American options Assume rf is constant over time ⇒ q is constant. American options with option to exercise until T: ⇒ Need to check if optimal to exercise or continue at every t. Value of option at terminal date, t = T :  max[ST − K ; 0] if call-option VT = max[K − ST ; 0] if put-option ⇒ for all scenarios at time t: Vuu , Vud and Vdd. Value of option after j ups, at time s = t for t ̸= T :  h  i −∆t  max Ss,j − K ; (1 + rf )   qVs+1,j+1 + (1 − q)Vs+1,j if call-option  h  i Vs,j = max K − Ss,j ; (1 + rf )−∆t qVs+1,j+1 + (1 − q)Vs+1,j if put-option     | {z } | {z } Exercise value Continuation value 12 / 20  Introduction Option valuation (Two periods) American options, cont. So in our simple case with only two periods and ∆t = 1, we have Value of option at time t = 1:  h  i max Su − K ; (1 + rf )−1 qVuu + (1 − q)Vud  if call-option Vu = h  i max K − Su ; (1 + rf )−1 qVuu + (1 − q)Vud  if put-option ⇒ Do the same for Vd. Value of the option today, t = 0:  h  i max S0 − K ; (1 + rf )−1 qVu + (1 − q)Vd  if call-option V0 = h  i max K − S0 ; (1 + rf )−1 qVu + (1 − q)Vd  if put-option 13 / 20  Introduction Real option analysis (one real options) Example - deferral option with K = $125. From Figure 3 in CWS: and risk-free return, rf = 5%. Example on the board. 14 / 20  Introduction Real option analysis (several real options) Example For a project, we have and three real options to 1 Abandon the project for an abandonment value of $190, 2 Reduce the value of the project by 20% and get $50 in cash (contraction option), 3 Expand the value of the project by 30% at a cost of $70. It is assumed that none of the real options affect each other (for simplicity). (1+rf )∆t −d (Note, in same cases they use the exact q = u−d = 0.555556 and in other cases q = 0.56.) 15 / 20  Introduction Real option analysis (several real options) Example From Figure 8 in CWS: 16 / 20  Introduction Real option analysis (several real options) Example From Figure 8 in CWS: 17 / 20  Introduction Real option analysis (several real options) Example From Figure 8 in CWS: 18 / 20  Introduction Real option analysis (several real options) Example 19 / 20  Introduction References CWS, ch. 9 20 / 20