1. Valuing real options1 From financial options to real options 3. Real option valuations Prof. André Farber Solvay Business School ESCP March 10,2000

2. Valuing real options2 Back to Portlandia Ale Portlandia Ale had 2 different options: –the option to launch (a 2-year European call option) value can be calculated with BS –the option to abandon (a 2-year American option) How to value this American option? –No closed form solution –Numerical method: use recursive model based on binomial evolution of value –At each node, check whether to exercice or not. –Option value = Max(Option exercised, option alive)

3. Valuing real options3 Valuing a compound option (step 1) Each quaterly payment (€ 0.5 m) is a call option on the option to launch the product. This is a compound option. To value this compound option:: 1. Build the binomial tree for the value of the company 012345678 14.4617.6621.5626.3432.1739.2947.9958.62 71.60 11.8314.4617.6621.5626.3432.1739.29 47.99 9.6911.8314.4617.6621.56 26.34 32.17 7.939.6911.8314.4617.66 21.56 6.507.939.6911.83 14.46 5.326.507.93 9.69 4.355.32 6.50 3.56 4.35 2.92 u=1.22, d=0.25 up down

4. Valuing real options4 Valuing a compound option (step 2) 2. Value the option to launch at maturity 3. Move back in the tree. Option value at a node is: Max{0,[pV u +(1-p)V d ]/(1+r  t)-0.5} 012345678 1.74 4.247.8812.7118.7926.2535.2946.2759.60 0.421.954.578.3513.3019.4726.9435.99 0.000.532.164.938.8713.9920.17 0.000.000.622.365.309.56 0.000.000.000.672.46 0.000.000.000.00 0.000.000.000.00 0.00 p = 0.48, 1/(1+r  t)=0.9876 =(0.48  2.46+0.52  0.00)  0.9876

5. Valuing real options5 When to invest? Traditional NPV rule: invest if NPV>0. Is it always valid? Suppose that you have the following project: –Cost I = 100 –Present value of future cash flows V = 120 –Volatility of V = 69.31% –Possibility to mothball the project Should you start the project? If you choose to invest, the value of the project is: Traditional NPV = 120 - 100 = 20 >0 What if you wait?

6. Valuing real options6 To mothball or not to mothball Let analyse this using a binomial tree with 1 step per year. As volatility =.6931, u=2, d=0.5. Also, suppose r =.10 => p=0.40 Consider waiting one year.. V=240 =>invest NPV=140 V=120 V= 60 =>do not invest NPV=0 Value of project if started in 1 year = 0.40 x 140 / 1.10 = 51 This is greater than the value of the project if done now (20 Wait.. NB: you now have an American option

7. Valuing real options7 Waiting how long to invest? What if opportunity to mothball the project for 2 years? V = 480 C = 380 V=240 C = 180 V=120 C = 85V = 120 C = 20 V= 60 C = 9 V = 30 C = 0 This leads us to a general result: it is never optimal to exercise an American call option on a non dividend paying stock before maturity. Why? 2 reasons –better paying later than now –keep the insurance value implicit in the put alive (avoid regrets) 85>51 => wait 2 years

8. Valuing real options8 Why invest then? Up to know, we have ignored the fact that by delaying the investment, we do not receive the cash flows that the project might generate. In option’s parlance, we have a call option on a dividend paying stock. Suppose cash flow is a constant percentage per annum  of the value of the underlying asset. We can still use the binomial tree recursive valuation with: p = [(1+r  t)/(1+  t)-d]/(u-d) A (very) brief explanation: In a risk neutral world, the expected return r (say 6%) is sum of capital gains + cash payments So:1+r  t = pu(1+  t) +(1-p)d(1+  t)

9. Valuing real options9 American option: an example Cost of investment I= 100 Present value of future cash flows V = 120 Cash flow yield  = 6% per year Interest rate r = 4% per year Volatility of V = 30% Option’s maturity = 10 years   Binomial model with 1 step per year Immediate investment : NPV = 20 Value of option to invest: 35 WAIT

10. Valuing real options10 Optimal investment policy Value of future cash flows (partial binomial tree) 0 1 2 3 4 5 120.0 162.0 218.7 295.2 398.4 537.8 88.9 120.0 162.0 218.7 295.2 65.9 88.9 120.0 162.0 48.8 65.9 88.9 36.1 48.8 26.8 Investment will be delayed. It takes place in year 2 if no down in year 4 if 1 down Early investment is due to the loss of cash flows if investment delayed. Notice the large NPV required in order to invest

11. Valuing real options11 A more general model In previous example, investment opportunity limited to 10 years. What happened if their no time frame for the investment? McDonald and Siegel 1986 (see Dixit Pindyck 1994 Chap 5) Value of project follows a geometric Brownian motion in risk neutral world: dV = (r-  ) V dt +  V dz dz : Wiener process : random variable i.i.d. N(0,  dt) Investment opportunity :PERPETUAL AMERICAN CALL OPTION

12. Valuing real options12 Optimal investment rule Rule: Invest when present value reaches a critical value V* If V<V* : wait Value of project f(V) = aV  if V<V* V-I if V  V*

13. Valuing real options13 Optimal investment rule: numerical example Cost of investment I = 100 Cash flow yield  = 6% Risk-free interest rate r = 4% Volatility = 30% Critical value V*= 210 For V = 120, value of investment opportunity f(V) = 27 Sensitivity analysis  V* 2%341 4%200 6%158

14. Valuing real options14 Value of investment opportunity for different volatilities