Copyright K. Cuthbertson, D.Nitzsche FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche Lecture Real Options Version 1/9/2001

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 investment 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 ‘dead’ that is exercise now (pay K and collect revenues)

Copyright K. Cuthbertson, D.Nitzsche OPTIONALITY Table 19.1 : Similarities Between Financial and Real Options ParamFinancial OptionReal Option SStock price Present value of expected cashflows KStrike priceInvestment cost RRiskless interest rate Riskless interest rate  Share-price volatility Volatility of project cashflows  = T-tTime-to-maturity Time until oppnity to invest disappears

Copyright K. Cuthbertson, D.Nitzsche ‘Drivers’ of NPV and real options NPV PV {Expected cash flows} PV {Fixed costs} RO PV {Expected cash flows} PV {Fixed costs}=‘strike’ Interest rate Value lost over option’s life Cash flow volatility Time to maturity Conventional NPV Real Options Payoffs = Max { PV - I 0, 0}

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 projects, without ‘optionality’, in the investment decision (ie.‘passive’) = conventional NPV Apply the (abandonment) option and get new payoffs for the ‘tree 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 S 0 =18 £ 36 =SU £ 9=SD U= 2, D = 1/U = 0.5 p u =0.5 What is the cost of capital, k ? 18 = [0.5 x x 9] / (1+k) Hence k = 25%(1+k) = [0.5 x (36) x (9)] /18 Discounting all future outcomes (SU, SU 2 etc) using k, will always ‘reproduce’ the current value of the stock S 0 (firm) p u = ‘real world probability Calculate discount rate k, from observed share price

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) = will also give the same value for S 0 (=18): S 0 = [ 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) S 0 = 90 45=S d 180= S u Baseline value of firm V 0 = ?? 100 = V d 180 = V u Abandon for K = 100 at t=1 Investment cost (at t=0 ) = 105, U=2, k = 25%, p = 0.5 Conventional NPV = = -15 Discount ‘optionality’ using k and p V 0 = [ p u V u + (1-p u ) V d ]/(1+k) = 112 NPV * = = 7 Note: Spread of outcomes with ‘optionality’ is less than ‘baseline’ Payoff = max {S T, K } S 0 = [ p u S u + (1-p u ) S d ]/(1+k) Note: ‘Baseline’ tree is perfectly correlated with ‘share price’ tree, since U=2.

Copyright K. Cuthbertson, D.Nitzsche 2) NPV Abandonment Option - using RNV Baseline value of firm ?? Abandon for K = 100 at t=1 Investment cost(at t=0 ) =110 and r = 5.25%,U= 2, q = Conventional NPV = -15Payoff = max {S T, K } V 0 = [ q V u + (1-q) V d ]/R = 123 NPV * = =18 Value of abandonment option = = 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 = ) then what is the implicit discount rate k * applicable in the ‘real world’ (ie using the real world probabitily, p) ? 123 = [ p u V u + (1-p u ) V d ]/(1+k * ) = [0.5 x 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 (S 0 ) 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 S 0 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 Time 0123 U= , q = I 0 = (or 105 rounded)

Copyright K. Cuthbertson, D.Nitzsche Fig 19.6 : Option to expand (at t=3, only) V 0 * = Time 0123 Payoff= Max{1+e)S 3 - I 3, S 3 } e=50%, I 3 = 45 European Call

Copyright K. Cuthbertson, D.Nitzsche 19.7 : Contraction option (at t=1 only) Time 0123 Payoff= Max{S 1 - I 1, (1-c%) S 1 - I * } c%=55%, I 1 = 58, I * =10 Original I 0 =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 (Baseline S d =45.1) (Baseline S u = 179.4)

Copyright K. Cuthbertson, D.Nitzsche 19.8 : American abandonment option (was 45.1) 60.0 (was 11.4) Time 0123 Payoff= Max{S i, A} A=60 At T=3, only abandon on 2 lower nodes Tree: V t-1 = max { V t rec, A } where V rec = [qV u + (1-q) V d ] /R

Copyright K. Cuthbertson, D.Nitzsche 19.9 : Investment default option (was 45.1, now max{ ,0 ) 121.4(was 179.4, now ) Time 0123 Payoff= Max{S 1 - I 1, 0 } I 1 =58and I 0 = 50 At t=1 (only) can choose whether to invest or not NPV * = = -7.4 Value of inv. Default option = (-15.4) = 8

Copyright K. Cuthbertson, D.Nitzsche : Default on debt repayment Time At t=3 (only) can default on debt of D 3 = 91 Payoff= Max{S 3 - D 3, 0 } D 3 = 91 and I 0 = 55 NPV * = = -8.5 (-8.9 in text) Value of inv. Default option = (-15.4) = 6.9 (6.5 in text)

Copyright K. Cuthbertson, D.Nitzsche :Investment default (t=1) and debt default(t=3) Time At =1, Payoff= Max{S 1 - I 1, 0 } I 1 =58 (ie.t=1 value of 55) At t=3 Payoff= Max{S 3 - D 3, 0 } D 3 =91 (ie. t=3 value of 50) NPV * = 19.6 Value of COMBINED option = (-15.4) = 35

Copyright K. Cuthbertson, D.Nitzsche 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 * ( V T ) - this is risk neutral valuation - use MCS to obtain alternative values for V T based on assessment of revenues minus costs, in a stochastic environment - key parameters: mean growth of revenues and their volatility, 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: Costs: fixed F and costs proportional to revenues, (  +  )R C t = F + (  +  )R t (  +  ) = 94% ‘baseline case’, hence profit margin= 6% The cashflow of the firm (ignoring taxes) is: Y t = R t - C t And the change in cash balances dX are: dX = -Y t 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 * ( X T ) - this is risk neutral valuation December 1999, the model gave a value for Amazon.com of $5.5bn, If the profit margin is reduced from 6% to 5% then Amazon’s value falls from $5.5bn to $4.3bn. - use MCS to obtain alternative values for V T based on assessment of revenues minus costs, in a stochastic environment - key parameters - mean growth and vol of revenues, rate of change of average growth rate, mark-up over costs

Copyright K. Cuthbertson, D.Nitzsche VALUATION OF INTERNET COMPANIES The cashflow of the firm (ignoring taxes) is: Y t = R t - C t And the change in cash balances dX are: dX = -Y t 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 * ( X T ) MCS generates values for error terms and gives different values for X T

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