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From financial options to real options 3. Real option valuations

Learn how to value real options using binomial models and make optimal investment decisions. Explore examples and case studies from Prof. André Farber at Solvay Business School ESCP.

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From financial options to real options 3. Real option valuations

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  1. From financial options to real options3. Real option valuations Prof. André Farber Solvay Business School ESCP March 10,2000 Valuing real options

  2. 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) Valuing real options

  3. 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 0 1 2 3 4 5 6 7 8 14.46 17.66 21.56 26.34 32.17 39.29 47.99 58.62 71.60 11.83 14.46 17.66 21.56 26.34 32.17 39.29 47.99 9.69 11.83 14.46 17.66 21.56 26.34 32.17 7.93 9.69 11.83 14.46 17.66 21.56 6.50 7.93 9.69 11.83 14.46 5.32 6.50 7.93 9.69 4.35 5.32 6.50 3.56 4.35 2.92 u=1.22, d=0.25 up down Valuing real options

  4. 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,[pVu +(1-p)Vd]/(1+rt)-0.5} 0 1 2 3 4 5 6 7 8 1.744.24 7.88 12.71 18.79 26.25 35.29 46.27 59.60 0.42 1.95 4.57 8.35 13.30 19.47 26.94 35.99 0.00 0.53 2.16 4.93 8.87 13.99 20.17 0.00 0.00 0.62 2.36 5.30 9.56 0.00 0.00 0.00 0.67 2.46 0.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 p = 0.48, 1/(1+rt)=0.9876 =(0.482.46+0.520.00)  0.9876 Valuing real options

  5. 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? Valuing real options

  6. 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 Valuing real options

  7. 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 = 85 V = 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 Valuing real options

  8. 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) Valuing real options

  9. 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 Valuing real options

  10. Value of future cash flows (partial binomial tree) 0 1 2 3 4 5 120.0 162.0 218.7295.2 398.4 537.8 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 Optimal investment policy Valuing real options

  11. 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 Valuing real options

  12. 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* Optimal investment rule Valuing real options

  13. 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 Valuing real options

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

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