Real Options
Real vs Financial Options
Financial options are on assets such as stocks that are traded in markets.
Real options are options on some underlying that are not traded in any market. Example of real options include valuation of a company, deciding to invest in R&D project, valuing natural resource options.
- Option to delay an investment opportunity
- Growth Option - Valuing the growth potential of a firm
- Growth Option – Option to expand a business
- Valuing Natural Resource Options
Option to Delay an Investment Opportunity
Investment
To open a restaurant in your city as part of a major chain, which can be opened immediately or after 1 year. If you do neither, you lose the right. Cost of opening the restaurant is $5 million now or after 1 year. If you open now, you will get free cash flow (FCF) of $600,000 which will grow @2% p.a. Cost of capital is 12%.
After 1 year, cost of opening is still $5 million. The risk-free interest rate is 5%. The volatility of the value of the restaurant as obtained from the return volatility of publicly traded comparable firms is 40%. If the restaurant is opened after 1 year, the FCF of the first year of $600,000 will be lost.
Should we open the restaurant now or after 1 year?
Workings
The payoff from the contract is a long call option on the value of the asset. So we can value the option using BSOPM. The relevant figures are as follows;
S = current market value of asset = 600,000/(0.12-0.02) = $6 million
K = Upfront investment = $5 million
T = 1 year, r=5%, σ=40%,
Div = FCF lost from delay = $600,000
S*= current asset value without dividend
= 6 mn – 600,000/1.12 = $5.46 million
PV(K) = $5 mn/1.05 = $4.76 million.
d1 = ln(S*/PV(K))/ σ√T + σ√T/2
= ln(5.46/4.76)/0.4 + 0.4/2 = 0.543, N(d1)=0.706,
d2 = d1 - σ√T = 0.543-0.4 = 0.143, N(d2)=0.557
c = S*N(d1) – PV(K)N(d2)
= 5.46*0.706-4.76*0.557 = $1.20 million
Conclusion
So the value today from waiting to invest in the restaurant next year and only opening if it is profitable to do so, is $1.2 million
Alternately, we could have opened today.
NPV of that decision = -Investment + PV of future FCFs = -$5 million + $600,000/(0.12-0.02) = $ 1 million.
So, one should delay the investment opportunity so that one can learn about the likely success by observing other restaurants for one year.
Growth Option – Valuing the Growth Potential of a Firm
Future growth opportunities can be thought of as a collection of real call options on potential projects. Growth options (out-of-money call options) are likely to be riskier than ongoing assets (in-the-money call options) of a firm.
Investment
A company’s only asset is a patent on a drug. If produced, the drug will generate certain profits of $1 million per annum for the 17 years of life. It will cost $10 million today to produce the drug. Assume 17-year risk-free annuity as 8% per annum and the interest rates will change in 1 year to 10% or 5%. 1-year risk-free interest rate is 6% per annum. What is the value of the patent? The interest rates are annually compounded.
Workings
We can use Binomial model to value the option.
S= value of 17-year risk-free annuity of $1 million
= $1million/0.08*[1-1.08^(-17)] = $9.122 million
Su = value of the annuity 1 year from now with 10% int. rate
= $1 million + $1mn/0.1*[1-1.1^(-16)] = $8.824 million
Sd = value of the annuity 1 year from now with 5% int. rate
= $1 million + $1million/0.05*[1-1.05^(-16)] = $11.838 million
P = risk-neutral probability = [(1+ rf)S – Sd]/(Su-Sd)
= (1.06*9.122-11.838)/(8.824-11.838) = 71.95%
Cu = NPV of the investment after 1 year with 10% interest rate
= $1million/0.10*(1-1.1^(-16))- $10 million = -$2.176 million
We take this value of Cu as nil since this option will not be exercised.
Cd = NPV of the investment after 1 year with 5% interest rate
= $1million/0.05*(1-1.05^(-16))- $10 million = $0.838 million
C =[p* Cu+ (1-p)* Cd]/ (1+ rf) =(0.7195*0+0.2805*0.838)/1.06
=$0.221 million.
Conclusion
So, the uncertainty regarding future interest rates makes the growth option valuable for the firm.
Growth Option – The Option to Expand
Investment
There is an option to grow that requires $10 million investment today. In 1 year we can find out whether the project is successful or failure. The risk-neutral probability that the project will generate $1 million per annum in perpetuity is 50%. We can double the size if successful. We assume that the annually compounded risk-free rate is constant at 6%.
Should the investment be undertaken in view of the growth opportunity?
Workings
The decision tree for the investment opportunities is as follows.
Expected cash flow per annum = $1 million*0.5 = $500,000
NPV without growth option = $500,000 / 1.06 - $10 million = - $1.667 million
NPVDoubling after a year = $1 million / 1.06 - $10 million = $6.667 million and has probability of 50%.
Expected NPV after 1 year = $6.667 million *0.5 = $3.333 million since there is only 50% probability of success.
PV growth option = $3.333 million / 1.06 = $3.145 million
NPV = NPV without growth option + PV growth option
= -$1.667 million + $3.145 million = $1.478 million
NPV of the investment is positive and the firm should undertake it.
Conclusion
Here it is optimal to undertake the investment today only because of the existence of the future expansion plan.
Abandonment Option
Investment
You can open a store on lease in a recently renovated Ferry Building in NY. New store will cost $10,000 per month to operate. The traffic in new store depends on how popular the building becomes. If a tourist attraction, the revenue will be $16,000 per month; otherwise $8,000 per month. The probabilities are 50% each. The cost of set up of the store is $400,000.The risk-free interest rate is constant at 7%. You can break the lease in 2 years without cost.
Should the investment be undertaken?
Workings
Monthly discount rate = 1.07^(1/12)-1 = 0.565%
If there is no abandonment option after 2 years, you will be forced to operate the store and
Expected revenue in perpetuity = 05*16,000+0.5*8,000 = $12,000
NPV=(12,000-10,000)/1.00565–400,000 = - $46,018.
If you can abandon the store after 2 years if the store is not a tourist attraction, we calculate the NPV as the expected sum of two NPVs; 1) NPV of a perpetuity in case of tourist attraction and 2) NPV of operating just 2 years.
NPV of a perpetuity in case of tourist attraction
= (16,000-10,000)/1.00565–400,000 = $661,947
NPV of operating just 2 years and then close down = 8,000/1.00565*[1-1.00565^(-24)]- 10,000/1.00565*[1-1.00565^(-24)] – 400,000
= -$444,770
Net NPV with abandonment option = 0.5*661,947+0.5*(-$444,770) = $108,589.
Conclusion
So, with the abandonment option, the investment opportunity becomes viable.
Valuing Natural Resource Options
In a natural resource investment, the underlying asset is the resource and the value of the asset is based upon two variables - the quantity of the resource that is available in the investment and the price of the resource.
In most such investments, there is a cost associated with developing the resource, and the difference between the value of the asset extracted and the cost of the development is the profit to the owner of the resource.
Defining the cost of development as X, and the estimated value of the resource as V, the potential payoffs on a natural resource option can be written as follows:
Payoff on natural resource investment = V – X if V > X
= 0 if V≤ X
Valuing an Oil Reserve
Consider an offshore oil property with an estimated oil reserve of 50 million barrels of oil, where the present value of the development cost is $12 per barrel and the development lag is two years. The firm has the rights to exploit this reserve for the next twenty years and the marginal value per barrel of oil is $12 per barrel currently (Price per barrel - marginal cost per barrel). Once developed, the net production revenue each year will be 5% of the value of the reserves. The riskless rate is 8% and the std. dev. in ln(oil prices) is 0.03.
Current Value of the asset = So = Value of the developed reserve discounted back the length of the development lag at the dividend yield = $12 * 50 /(1.05)2 = $ 544.22 million
(If development is started today, the oil will not be available for sale until two years from now. The estimated opportunity cost of this delay is the lost production revenue over the delay period. Hence, the discounting of the reserve back at the dividend yield)
Exercise Price, K = Present Value of development cost = $12 * 50 = $600 million
Time to expiration on the option, T = 20 years
Variance in the value of the underlying asset, σ = 0.03
Risk-free interest rate, r =8%
Dividend Yield, q = Net production revenue / Value of reserve = 5%
We will use Black-Scholes-Merton model for the valuation.
Using the inputs, the model provides the following value for the call:
d1 = 3.8119 N(d1) = 0.9999
d2 = 3.6777 N(d2) = 0.9998
Call Value
= 544.22*exp(-0.05)(20)*0.9999 – 600*exp(-0.08)(20) *0.9998 = $79.07 million
This oil reserve, though not viable at current prices, still is a valuable property because of its potential to create value if oil prices go up. If you were bidding for this reserve at an auction, you would be willing to pay up to this value. The value of the reserve will increase with the variance in oil prices.
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