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Version 1/9/2001 FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche Lecture Real Options Copyright K. Cuthbertson, D.Nitzsche Topics Basic Concepts Valuation of real options using BOPM Extension of Tree to Many Periods Valuation of Internet Company using continuous time method Copyright K. Cuthbertson, D.Nitzsche Basic Concepts Copyright K. Cuthbertson, D.Nitzsche OPTIONALITY Conventional NPV is ‘passive’ Black Gold and Crude Hole - oilfields - both have negative NPV taken separately - call option to expand, with strike= additional investme Option to Abandon - BMW purchase of Rover - put option to sell off assets Option to defer - don’t build plant today, wait - American call with strike = investment cost - Is the call worth more ‘alive’ (ie. Postpone start) or ‘d that is exercise now (pay K and collect revenues) Copyright K. Cuthbertson, D.Nitzsche OPTIONALITY Table 19.1 : Similarities Between Financial and Real Options Param Financial Option Real Option S Stock price Present value of expected cashflows K Strike price Investment cost R Riskless interest rate Riskless interest rate Share-price volatility Volatility of project cashflows = T-t Time-to-maturity Time until oppnity to invest disappears Copyright K. Cuthbertson, D.Nitzsche ‘Drivers’ of NPV and real options Conventional NPV PV {Fixed costs} NPV PV {Expected cash flows} Payoffs = Max { PV - I0 , 0} Real Options Cash flow volatility PV {Fixed costs}=‘strike’ Interest rate Time to maturity RO PV {Expected cash flows} Value lost over option’s life Copyright K. Cuthbertson, D.Nitzsche Valuation of real options using BOPM Copyright K. Cuthbertson, D.Nitzsche Steps in Valuation (BOPM) Measure the volatility of the value of the firm - from observed stock prices Assume this represents ‘outcomes’ SU or SD for existing proje without ‘optionality’, in the investment decision (ie.‘passive’) = conventional NPV Apply the (abandonment) option and get new payoffs for the ‘t including optionality’ . Discount, using risk neutral valuation to obtain ‘adjusted NPV including optionality’, then adjusted NPV* = conventional NPV + value of option Copyright K. Cuthbertson, D.Nitzsche ‘Share Price’ Tree Calculate discount rate k, from observed share price U= 2, D = 1/U = 0.5 pu =0.5 S0=18 £ 36 =SU pu = ‘real world probability £ 9=SD What is the cost of capital, k ? 18 = [0.5 x 36 + 0.5 x 9] / (1+k) (1+k) = [0.5 x (36) + 0.5 x (9)] /18 Hence k = 25% Discounting all future outcomes (SU, SU2 etc) using k, will always ‘reproduce’ the current value of the stock S0 (firm) Copyright K. Cuthbertson, D.Nitzsche ‘Baseline/existing’ projects Assume projects, without any optionality have the same risk as the firm’s existing projects, so U = 2, p = 0.5 and k is the appropriate discount rate (with no optionality/passive). Risk Neutrality Surprisingly: Discounting the above outcomes SU and SD using risk free rate r = 5.25% (ignore contin. comp) and q = (R - D)/(U-D) = 0.3693 will also give the same value for S0 (=18): S0 = [ q SU + ( 1- q) SD ]/R = 18 where R = (1+r). Thus: Instead of discounting the risky outcomes SU,SD using real probabilities p, we can use q ,and then discount using the risk free rate Copyright K. Cuthbertson, D.Nitzsche NPV of (Simple-One Period) Abandonment Option Calculate NPV of the ‘abandonment option’, by 1) using ‘real world’ probabilities p and discounting the payoffs using the cost of capital k (‘decision tree analysis’ DTA) 2) using ‘risk neutral’ probabilities q and discounting the payoffs using the risk free rate, r We find the two NPV are different The DTA approach is incorrect because it uses the discount rate k, which is calculated from the payoffs without any optionality. This discount rate is too large, given the lower riskiness of outcomes with the abandonment option The risk neutral approach gives the correct answer for the NPV (with optionality) Copyright K. Cuthbertson, D.Nitzsche 1) One period Abandonment Option- using DTA (ie. k and p) Investment cost (at t=0 ) = 105, U=2, k = 25%, p = 0.5 Note: ‘Baseline’ tree is perfectly correlated with ‘share price’ tree, since U=2. Baseline value of firm 180= Su S0= 90 Abandon for K = 100 at t=1 180 = Vu V0 = ?? 45=Sd S0 = [ pu Su + (1-pu) Sd ]/(1+k) Conventional NPV = 90 - 105 = -15 100 = Vd Payoff = max {ST , K } Discount ‘optionality’ using k and p V0 = [ pu Vu + (1-pu) Vd ]/(1+k) = 112 NPV* = 112 - 105 = 7 Note: Spread of outcomes with ‘optionality’ is less than ‘baseline’ Copyright K. Cuthbertson, D.Nitzsche 2) NPV Abandonment Option - using RNV Investment cost(at t=0 ) =110 and r = 5.25%,U= 2, q = 0.3693 Baseline value of firm Abandon for K = 100 at t=1 180 90 180 ?? 45 Conventional NPV = -15 100 Payoff = max {ST , K } V0 = [ q Vu + (1-q) Vd ]/R = 123 NPV* = 123 - 105 =18 Value of abandonment option = 123 - 90 = 33 Copyright K. Cuthbertson, D.Nitzsche NPV Abandonment Option - using RNV If, using RNV gives the correct value for the project (with optionality) , (ie. V = 122.89) then what is the implicit discount rate k* applicable in the ‘real world’ (ie using the real world probabitily, p) ? 123 = [ pu Vu + (1-pu) Vd ]/(1+k* ) = [0.5 x 180 + 0.5 x 100] / (1+k* ) Hence k* = 13.8% Why is k* = 13.8% less than k = 25% ? And which is correct ? Copyright K. Cuthbertson, D.Nitzsche NPV Abandonment Option - Comparison Baseline/existing projects (no optionality), has V = 90 Abandonment using k=25%, V = 112 Abandonment using options theory V= 123 (and k*=13.8%) Valuation using k=25% understates the value of the project with optionality, since the latter has less risky outcomes and should be discounted at a lower rate than ‘existing’ projects RNV implicitly uses the ‘correct’ (lower) discount rate of k*=13.68% Copyright K. Cuthbertson, D.Nitzsche Where does the ‘baseline’ tree come from? 1) Assume the ‘baseline’ project is ‘scale enhancing’ so its risk is the same as the riskiness of all the firms existing projects 2) Then we can use the (annual) volatility of the firms stock returns as a measure of ‘risk’ of the baseline project. 3) Then take U = exp[ x sqrt(dt) ] and D = 1/U 4) The risk neutral probabilities q can be shown to be: q = (R - D) / ( U-D) - these are use to value the project with optionality Copyright K. Cuthbertson, D.Nitzsche Summary: Optionality and NPV 1) Work out the ‘tree’ for project (ie. SU, SD etc) without optionality Value of project without optionality (S0) equals the PV of value in ‘up’and ‘down’ states(SU, SD etc), using either p and k or RNV (ie. q and r ) ~ both give S0 2) Reconstruct the tree including optionality Obtain the ‘adjusted’ PV of project by working back through this tree using ‘risk neutral valuation’ Copyright K. Cuthbertson, D.Nitzsche When is it worth using real options? When conventional NPV is close to zero When there is great uncertainty in future outcomes -options have more value the higher is When project has a long life - options have higher value as T increases Copyright K. Cuthbertson, D.Nitzsche Extension of Tree to Many Periods 3 Copyright K. Cuthbertson, D.Nitzsche Figure 19.5 : Evolution of ‘baseline’ valuation U= 1.9937, q = 0.3693 I0 = 105.4 (or 105 rounded) 713.2 357.7 179.4 179.4 90 90 45.1 45.1 22.6 11.4 0 1 2 Copyright K. Cuthbertson, D.Nitzsche 3 Time Fig 19.6 : Option to expand (at t=3, only) Payoff= Max{1+e)S3 - I3, S3 } e=50%, I3 = 45 European Call 1024.8 493.9 224.1 236.6 105.7 V0* = 113.4 45.1 50.7 22.6 11.4 0 1 2 Copyright K. Cuthbertson, D.Nitzsche 3 Time 19.7 : Contraction option (at t=1 only) Original I0 =105, is invested 50 at t=0 and 58 at t=1- same PV Option is to invest smaller amount I*=10 at t=1, with 55% lower revenues Payoff= Max{S1 - I1, (1-c%) S1- I* } c%=55%, I1 = 58, I*=10 121.4 (Baseline Su= 179.4) 10.3 (Baseline Sd =45.1) 48.8 0 1 2 Copyright K. Cuthbertson, D.Nitzsche 3 Time 19.8 : American abandonment option Payoff= Max{Si , A} A=60 At T=3, only abandon on 2 lower nodes Tree: Vt-1 = max { Vt rec, A } where Vrec = [qVu + (1-q) Vd] /R 713.2 357.7 179.4 184.7 98.9 107.1 60.0 (was 45.1) 70.6 60 60.0 (was 11.4) 0 1 2 Copyright K. Cuthbertson, D.Nitzsche 3 Time 19.9 : Investment default option At t=1 (only) can choose whether to invest or not Payoff= Max{S1 - I1 , 0 } I1 =58 and I0 = 50 121.4(was 179.4, now 179.4-58) 42.6 0 0 (was 45.1, now max{45.1-58,0 ) 1 2 3 NPV* = 42.6 - 50 = -7.4 Value of inv. Default option = -7.4 - (-15.4) = 8 Copyright K. Cuthbertson, D.Nitzsche Time 19.10 : Default on debt repayment At t=3 (only) can default on debt of D3 = 91 Payoff= Max{S3 - D3 , 0 } D3 = 91 and I0 = 55 622.2 271.3 88.3 113.8 31.0 46.5 0 10.9 0 0 0 1 2 3 * NPV = 46.5 - 55 = -8.5 (-8.9 in text) Time Value of inv. Default option = -8.5 - (-15.4) = 6.9 (6.5 in te Copyright K. Cuthbertson, D.Nitzsche 19.11 :Investment default (t=1) and debt default(t=3) At =1, Payoff= Max{S1 - I1 , 0 } At t=3 Payoff= Max{S3 - D3 , 0 } I1 =58 (ie.t=1 value of 55) D3 =91 (ie. t=3 value of 50 622.1 271.3 88.3 55.8 31.0 19.6 0 0 0 0 NPV* = 19.6 Value of COMBINED option = 19.6- (-15.4) = 35 Copyright K. Cuthbertson, D.Nitzsche Time COMBINED OPTIONS Value of investment default option = 8 Value of debt default option = 6.9 Value of combined option = 35 Option values are not additive Copyright K. Cuthbertson, D.Nitzsche VALUATION OF INTERNET COMPANIES Copyright K. Cuthbertson, D.Nitzsche VALUATION OF INTERNET COMPANIES Have value because of expansion option and option to set up allied sites (option on an option) Given the terminal value of the firm, the value today is V = (1/R) E*( VT) - this is risk neutral valuation - use MCS to obtain alternative values for VT based on assessment of revenues minus costs, in a stochastic enviro - key parameters: mean growth of revenues and their volatili rate of change of average growth rate, mark-up over costs Copyright K. Cuthbertson, D.Nitzsche VALUATION OF INTERNET COMPANIES Stochastic representation of revenues and costs Actual changes in revenues Change in expected growth in revenues (Risk-adjusted) process for revenue R is mean reverting: dRt ( t 1 t )dt t dz1* Rt dt [k ( ) 2t ]dt t dz2* Costs: fixed F and costs proportional to revenues, (+)R Ct = F + ( + )Rt ( + ) = 94% ‘baseline case’, hence profit margin= 6% The cashflow of the firm (ignoring taxes) is: Copyright K. Cuthbertson, D.Nitzsche VALUATION OF INTERNET COMPANIES The cashflow of the firm (ignoring taxes) is: Yt = Rt - Ct And the change in cash balances dX are: dX = -Yt dt All cash is ‘retained’ and earns the risk-free rate, r If cash balances fall to zero, firm is assumed to go bankrupt Given the terminal value of the firm, the value today is V = (1/R) E*( XT) MCS generates values for error terms and gives different val for XT Copyright K. Cuthbertson, D.Nitzsche VALUATION OF INTERNET COMPANIES December 1999: Amazon.com = $5.5bn Sensitivity Analysis If the profit margin is reduced from 6% to 5% then Amazon’s value falls from $5.5bn to $4.3bn. Copyright K. Cuthbertson, D.Nitzsche END OF SLIDES Copyright K. Cuthbertson, D.Nitzsche