Thank you for checking out this post.

I already asked a question once on this forum, and you did a great job helping me out with that topic, as I couldn’t have come across a solution myself. This time, for a totally different topic, I once again need your precious help. The thing is rather simple and might be very easy for most of you perhaps, but it has been driving me crazy lately.

I am studying real options in the context of investment project valuation. I am mainly focusing (for the sake of my diploma) on three types of real options :

- Option to expand (phase 1 project followed by phase 2 project only if phase 1 conclusive)
- Option to delay (if we wait, we will know what the CF will be, hence we will only go on with the project if its expected NPV is positive)
- Options to abandon (the extra value for the possibility to sell the investment if non conclusive in x years)

I understand the role of these methods for valuation as the standard NPV model is considered “static” because it does not value flexibility which might be available for financial managers.

However, when it comes to actual practice and examples, it gets complicated. Each book I came across had a slightly different method for calculating the value of the option, whether it is a different discount rate, different variables used, formulas, etc. The more I was looking for examples, the more I got confused because author A uses method A and it doesn’t add up with author B. Not only that, but the proposal for correction of the exam I will pass (last year’s session) on real options doesn’t make sense either.

I believe the easiest way to try to find the solution to this mess, is to present to you an example for each option type, and to tell you how I would calculate it according to what I learnt. If you could have a look and tell me what you think, it would be great!!

Option to expand :

There is a phase 1 project that is undertaken (drilling for natural gas) :

  • Investment paid in 01/01/N+1
  • CF of the project received at the end of the following 5 years, that is from 31/12/N+1 to 31/12/N+5
  • The NPV of the project as of 01/01/N+1 = -30 068,887k€ (no detail is provided, I believe it has been discounted using the discount rate applicable to the project (either the WACC or specific cost of capital if project riskier than overall activity)

The second phase will only be undertaken if the phase 1 is going well. The information is :

  • This project is supposed to take place on 01/01/N+5 (one year before the end of phase 1)
  • NPV of CF discounted as of 01/01/N+5 = 1 178 996,24k€
  • Risk-free discount rate (discrete) = 4,25%
  • Investment required on 01/01/N+5 = 1 000 000k€
  • Standard-deviation of CF = 30%

For me, we should do it like this :

  • “Static” NPV of phase 1 = - 30 068,887k€
  • Option value (using BSM model)
  • S0 = 1 178 996,24k€ discounted back to 01/01/N+1 with risk-free rate = 1 178 996,24 * (1,045)^(-4) = 988 660,67k€
  • K = 1 000 000k€
  • T = 4 years
  • Stddev = 30%
  • r = ln(1,045)= 4,40% continuous
  • d1=(ln⁡(988660.67/1000000)+(0.044+0.5*〖0.3〗^2 )*4)/(0.3√4) = 0,574326 ; d2 = -0,025673
  • N(d1) = 71,71% ; N(d2) = 48,98%
  • Hence we have : C0 = 988 660,67 * N(d1) – 1 000 000 * e^(-0.044 * 4) * N(d2) = 298 213,48k€
  • Global NPV of the project considering flexibility = -30 068,887 + 298 213,48 = 268 144,59k€

Did I make any mistakes? In particular, is it necessary to discount S0 back to 01/01/N+1? If yes, using what rate?

Option to delay with limited project life :

We are looking at a project with the following characteristics:

  • Investment in 01/01/N+1 = 50m€
  • CF of 10m€ in the end of each year from N+1 to N+5 => PV of 33,5m€ (discount rate = 15%)
  • Cost of capital (project as risky as activity) = 15%
  • “Static” NPV of project = -50 + 33,5 = -16,50m€
  • High volatility of the CF estimated : stddev = 42% with expected PV of 33,5m€
  • The project will only be possible to be undertaken for 5 years, after that the competition will be too serious and expected CF after that will be 0

We must compare the static NPV with the value of the option to delay, to see if it is worth delaying the decision.

Valuing the option :

  • S0 = 33,5m€ (CF discounted to investment date)

  • K = 50 m€ (not discounted)

  • Stddev = 42%

  • T = 5 years

  • Y = cost of delay = 1 / lifespan of the project = 1/5 = 20% (cf Damodaran’s book “Investment Valuation”)

  • R = 5% (continuous)

  • d1=(ln⁡(33,5/50)+(0.05-0,20+0.5*〖0.42〗^2 )*5)/(0.42√5) = -0,755448 ; d2 = -1,694596

  • N(d1) = 22,50% ; N(d2) = 4,51%

  • Hence the option value is = 33.5 * e^(−0.25) * (0.2250) − 50.0 * e(−0.055) * (0.0451) = $1.019 million

  • In this formula we also need to discount S0 with y% because 1 year delay is 1 year of CF “lost”

  • If the value of the option exceeds the static NPV, it means it’s best to delay the investment

My question would be : How do we know when it is best to undertake the project? We know it’s best to delay, but when exactly is the limit where we should stop delaying and start undertaking? I believe Damodaran suggests we calculate in one year, changing y to 1/(5-1) = 25% and start again the calculation, but I don’t really understand that.

Furthermore, what is the value of the overall project given that the manager can delay the investment? Is it equal to the static NPV + the value of the option, or is it equal only to the value of the option?

Option to delay without limited project life :

Same example, but no limited life of the project. We want to see if it is worth to wait 2 years before undertaking the project

  • S0 = 33,5m€ discounted two years more (because we delay the reception of CF for two years) => what rate to use? Cost of capital or risk-free rate?
  • K = 50m€ (not discounted)
  • T = 2 years
  • R = 5% (continuous)
  • Stddev = 42%
  • Calculation with BSM model

Option to abandon :

I have seen two methods which seem okay to me, and I would like to have some clarification on this. Could you please tell me if the below is right or wrong :

  • Options to abandon imply we can sell at some predetermined amount (contractual engagement) at some time, if the project isn’t as profitable as expected

If the “repurchase” date by some 3rd party is at one specific time :

  • K = Amount determined by contract
  • S0 = CF occurring after the sale, discounted to today (cost of capital) (= CF “lost” because project has been sold)
  • T = Time from investment until repurchase date

If the “repurchase” date is actually a period of x years after the start of the investment (cf Damodaran’s “Investment Valuation”) => It’s possible to sell the project at K price for t years after the investment takes place.

Here is Damodaran’s method :

  • K = Amount determined by contract (not discounted)
  • S0 = All CF of the project, discounted to start of investment at cost of capital
  • T = length of the “repurchase” period
  • Same thing as “option to differ” if the project has a limited lifespan, y = cost of delay = 1 / project lifespan (and not 1/t)

Here is some other author’s method :

  • K = Amount determined by contract, discounted with risk-free rate (meaning d1 will be calculated with a discounted K)
  • S0 = CF of the project after the end of the repurchase period (implying we sell at the end of the period, and not before), discounted to day of investment, with the cost of capital
  • Other variables unchanged

All these methods give different results, and I don’t really know which one is correct. It seems to me that we shouldn’t discount K when calculating d1, but for the rest I have no clue.

I am deeply sorry for the length of this post, but I am really lost and would appreciate any help!!

Thank you very much



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