A mining company wants a lease with a break clause after five years. A builder wins a government contract that may open the door to a bigger one later. A pharmaceutical company funds research that it can drop if trials fail. None of these choices appears in a simple NPV. Yet each has value, because it lets management act on what it learns.
Option thinking prices that flexibility. The same thinking also shows why the shares of a heavily indebted company can be worth something even when its assets are worth less than its debts.
This topic covers real options and how they differ from NPV, the types of real option, how they are valued with Black-Scholes (abandonment and follow-on options), equity as a call option on a company’s assets, debt as a lender’s risk-free claim less a put, the probability of default and expected loss from the model, and the valuation of warrants with dilution.
The previous topic covered Black-Scholes and put-call parity. The next covers delta hedging and the Greeks. Neither the May 2014 nor the November 2014 Pathfinder sets a stand-alone question on real options.
In 30 seconds
A real option is a choice in a real project, such as to expand (invest or grow), defer (wait and learn), or abandon and scale down (divest). NPV ignores this flexibility. Project value = NPV + value of the real options, and the value is higher when uncertainty is high.
An abandonment option is a put (Pa = PV of project cash flows, Pe = sale price). A follow-on or expansion option is a call (Pa = PV of its inflows, Pe = its cost). Weakness: no simple, settled method of valuing them.
Equity is a call option on the company’s assets (Pa = asset value, Pe = face value of debt, t = debt maturity), so equity can have value even when assets are below debt. Debt value = assets − equity. Probability of default = 1 − N(d2). Warrants: value as a call, then multiply by 1 ÷ (1 + q) for dilution, where q is warrants per share.
Key terms in plain English
| Term | What it means |
|---|---|
| Real option | A choice embedded in a capital project that management may or may not use |
| Abandonment option | The right to sell or shut a project, valued as a put |
| Follow-on (expansion) option | The right to invest more later, valued as a call |
| Warrant | A company-issued right to subscribe for new shares at a fixed price |
| Dilution | The fall in value per share when new shares are issued on exercise |
| Probability of default | 1 − N(d2): the chance that assets are below the debt at maturity |
Syllabus check
This topic is in syllabus areas A and C (investment appraisal and business valuation). It covers:
- explaining the nature of real options and giving examples;
- using Black-Scholes to value real options;
- estimating the market value of a business using option-based techniques.
CAMA 2020 and distressed companies
The Companies and Allied Matters Act 2020 (CAMA 2020) is the company law of Nigeria.
| Section | What it says | Link to this topic |
|---|---|---|
| s. 305 | A director must act in good faith in the best interests of the company | Investment choices should be made with care for the benefit of the company |
| s. 308 | A director must use the care, diligence and skill of a reasonably prudent director | Directors should consider the value of flexibility and the position of creditors when the company is near distress |
| s. 377(1) and s. 378 | Directors prepare statements that follow IFRS and give a true and fair view | Decisions later affect reported assets, profit and cash |
| s. 385 | The directors’ report covers performance and risk | Major plans and risks are reported to shareholders |
Real options
A real option is an alternative that exists within an investment opportunity: a choice, not an obligation, to (a) make a further investment, (b) abandon a failing project or (c) defer investing. A break clause in a mine lease is a real option to abandon. They are “real” because they relate to real tangible assets. They give flexibility to avoid losses and take new opportunities.
Real options theory vs NPV. NPV discounts expected cash flows and looks only at the project selection stage, summarising everything into one expected value. It ignores decisions made later as the project develops. Real options analysis also puts a value on the options embedded in the project.
ICAN’s example: Projects A and B are mutually exclusive and last ten years; A has the higher NPV, but B has an option to withdraw after five years. NPV picks A; a real options analysis values B’s option and may pick B. A cheaper rigid technology may have a higher NPV than a dearer flexible one, until the flexibility is valued.
Types
| Group | Types |
|---|---|
| Invest / grow | Scale up (sequential investments as the market grows), switch up (move to the next technology), scope up (extend into related activities) |
| Defer / delay / learn | Wait for information or skills before investing |
| Divest / withdraw / reduce | Scale down or abandon, switch down (use flexible assets more cheaply), scope down |
Valuation
For each option, identify when it is available, how it would add value, the extra investment needed to exercise it (and any needed now to create it) and whether it opens further options. Value is higher where risk and uncertainty are high. Weakness: academics have not settled a simple, satisfactory valuation approach; some use a Black-Scholes-based model.
Real options are used in pharmaceuticals, mining and other industries with heavy research spending.
ICAN’s abandonment option. X Plc will invest ₦300m in social housing; PV of inflows is ₦310m (NPV ₦10m). The government will buy the project back for ₦200m after four years. r = 5%, s = 30%. This is a put: Pa = 310, Pe = 200, t = 4.
- d1 = [ln(310/200) + (0.05 + 0.045)×4] ÷ (0.3×2) = (0.4383 + 0.38) ÷ 0.6 = about 1.37
d2 = 1.37 − 0.60 = 0.76. - N(d1) = 0.9147
N(d2) = 0.7764
e−0.2 = 0.8187. - Call c.
310 × 0.9147 − 200 × 0.7764 × 0.8187 = 156.44.
Put p = 156.44 − 310 + 200 × 0.8187 = ₦10.18m. - Total value.
10 + 10.18 = ₦20.18m.
ICAN’s follow-on option. If X Plc wins that contract, a further government project may follow in four years, costing ₦400m with inflows worth ₦240m now and volatility of 30%. This is a call: Pa = 240, Pe = 400, r = 5%, t = 4.
- d1 = (−0.5108 + 0.38) ÷ 0.6 = −0.22
d2 = −0.22 − 0.60 = −0.82. - N(d1) = 0.5 − 0.0871 = 0.4129
N(d2) = 0.5 − 0.2939 = 0.2061. - c = 240 × 0.4129 − 400 × 0.2061 × 0.8187 = ₦31.6m
total value = 10 + 31.6 = ₦41.6m.
Equity as a call option
When a firm borrows, the lenders effectively hold its assets and the shareholders hold a call option to buy them back by repaying the debt (the exercise price is the face value of the debt). At redemption, if assets exceed the debt, shareholders repay and keep the surplus; if not, they can walk away.
So: Pa = value of assets, Pe = face value of debt, t = time to repayment, s = volatility of assets.
ICAN’s example. A firm’s assets less current liabilities are ₦100m with volatility 0.4; debt of ₦80m, zero coupon, is due in five years; r = 5%.
- d1 = [ln(1.25) + (0.05 + 0.08)×5] ÷ (0.4×√5) = (0.2231 + 0.65) ÷ 0.8944 = 0.976
d2 = 0.976 − 0.894 = 0.082. - N(d1) = 0.8355
N(d2) = 0.5326
e−0.25 = 0.7788. - Equity = 100 × 0.8355 − 80 × 0.5326 × 0.7788 = 83.55 − 33.18 = ₦50.37m.
Firms in distress
If assets are only ₦50m against the same ₦80m debt: d1 = (−0.47 + 0.65) ÷ 0.8944 = 0.20; d2 = −0.69; N(d1) = 0.5793; N(d2) = 0.2451; equity = 50 × 0.5793 − 80 × 0.2451 × 0.7788 = 28.97 − 15.27 = ₦13.7m. The equity is worth something because volatility and time give a chance that the option goes into the money. (Company law may restrict continued trading in such a case.)
Debt, default and expected loss
Using the same firm: value of debt = assets − equity = 100 − 50.37 = ₦49.63m. The fair value links to face value: 49.63(1 + i)5 = 80, so the yield is (80 ÷ 49.63)1/5 − 1 = 10%; with a risk-free rate of 5%, the credit spread is 5%. N(d2) is the probability that the call is in the money, so the probability of default = 1 − N(d2) = 1 − 0.5326 = 46.74%.
The shareholders also hold a put on the assets written by the lenders (the right to walk away), whose value is the lenders’ expected loss. By parity: put = call + PV(debt) − assets = 50.37 + 80 × 0.7788 − 100 = 50.37 + 62.30 − 100 = ₦12.67m. Since expected loss = probability of default × loss given default, loss given default = 12.67 ÷ 0.4674 = ₦27.1m.
Warrants
A warrant is a company-issued call on its own shares, settled by issuing new shares, so it is dilutive. Method: (1) value a call as if there were no warrants, using Black-Scholes; (2) multiply by 1 ÷ (1 + q), where q is the number of warrants per share outstanding. With 0.2 warrants for each share, the factor is 1 ÷ 1.2.
For a share price of ₦180, a strike of ₦165, four years, volatility of 12% and a rate of 7%, the call is worth about ₦56.3, which becomes ₦46.9 after dilution. That is below the ₦50 investors are asked to pay, so the company may need to re-price. A quick example: if the Black-Scholes call is ₦12.72 and q = 0.2, the warrant is worth 12.72 ÷ 1.2 = ₦10.60.
The rules in more depth
Why flexibility has value
You can limit losses by abandoning and increase gains by expanding, which is the same asymmetric payoff as an option.
Why equity is worth something when assets are below debt
Limited liability caps the loss at zero, but the upside is open.
Why N(d2) matters
It is the risk-neutral chance that assets exceed the debt at maturity.
Why warrants are diluted
Exercise issues new shares at the strike price, spreading assets over more shares.
Effect on analysis
Identify whether the option is a call or a put, map the inputs correctly (what is Pa and Pe) and add the option value to the NPV.
Worked examples
Example 1: call or put? A project can be sold back to the government for a fixed price after four years.
- The right is to sell at a fixed price.
- That is a put (abandonment).
- Pa is the PV of project cash flows; Pe is the sale price.
Example 2: total project value. NPV is ₦8m and the abandonment option is worth ₦5m.
- Total = 8 + 5.
- = ₦13m.
- Accept even if NPV alone were marginal.
Example 3: debt value. Assets are ₦100m and the equity call is worth ₦50.37m.
- Debt = 100 − 50.37.
- = ₦49.63m.
- Face value is ₦80m, so the yield is 10%.
Example 4: default probability. N(d2) = 0.5326.
- P(default) = 1 − 0.5326.
- = 0.4674.
- 46.74% chance that assets are below the debt at maturity.
How to score full marks
- Name the option (call or put) and the real choice.
- Map Pa, Pe, t, r and s from the case.
- Compute with the normal table rules.
- Add the option value to the NPV.
- Comment on the weakness of the valuation.
- For equity and debt, label each result.
Common mistakes in the exam
- Treating an abandonment option as a call.
- Using the project cost as Pa instead of the PV of inflows.
- Adding the option to the NPV wrongly (check the sum).
- Forgetting the dilution factor for warrants.
- Saying equity is worthless when assets are below debt.
- Presenting the option value as certain.
Quick revision
- Real options: expand, defer, abandon; value = NPV + options.
- Abandonment = put; follow-on = call.
- Equity = call on assets, exercise price = debt face value.
- Debt = assets − equity; P(default) = 1 − N(d2).
- Expected loss = put value; LGD = put ÷ P(default).
- Warrants: Black-Scholes ÷ (1 + q).
- CAMA 2020: ss. 305, 308, 377(1), 378, 385.
Practice questions
These are original questions written in the style of the Institute of Chartered Accountants of Nigeria (ICAN) exam. Try each one before you open the answer. Full model answers follow the written questions.
Recall check: multiple choice
Recent ICAN papers are written, although older papers such as May 2014 included multiple-choice questions. These questions are for quick recall. The written questions that follow are closer to the exam.
Question 1. A real option is:
- A choice embedded in a capital project
- A traded share option
- A tax relief
- A loan covenant
Show the answer
A. Management may exercise it later.
Question 2. An option to abandon a project for a fixed price is valued as a:
- Put
- Call
- Swap
- Future
Show the answer
A. It is the right to sell.
Question 3. A follow-on project that costs more than its inflows today is valued as a:
- Call
- Put
- Bond
- Forward
Show the answer
A. Pa is the PV of its inflows, Pe its cost.
Question 4. Total project value with real options is:
- NPV plus option value
- NPV only
- Option value only
- Cost less NPV
Show the answer
A. NPV ignores flexibility.
Question 5. Real options are worth more when uncertainty is:
- High
- Low
- Zero
- Fixed
Show the answer
A. Like financial options.
Question 6. In the option view of equity, the exercise price is the:
- Face value of the debt
- Market value of equity
- Cost of equity
- Share price
Show the answer
A. Pa is the value of assets.
Question 7. Equity in a distressed firm can still have value because of:
- Volatility and time to maturity
- Tax
- Dividends
- Interest
Show the answer
A. The option may go into the money.
Question 8. The probability of default from the model is:
- 1 − N(d2)
- N(d2)
- N(d1)
- 1 − N(d1)
Show the answer
A. N(d2) is the chance of repaying.
Question 9. A warrant’s Black-Scholes call value is adjusted for dilution by multiplying by:
- 1 ÷ (1 + q)
- 1 + q
- q
- 1 − q
Show the answer
A. q is warrants per share.
Question 10. Which section of CAMA 2020 sets the director’s duty of care and skill?
- Section 308
- Section 400
- Section 405
- Section 409
Show the answer
A. Directors should test the assumptions behind a decision.
Written questions
Question 11 (10 marks). (a) Explain the nature of real options and give three types. (5 marks) (b) Explain why NPV may undervalue a project. (3 marks) (c) State one weakness of valuing real options. (2 marks)
Question 12 (20 marks). A project costs ₦300m and the PV of its inflows is ₦310m. The government will buy it for ₦200m after four years. r = 5%, s = 30%. Given N(d1) = 0.9147, N(d2) = 0.7764 and e−0.2 = 0.8187. (a) State the type of option and the inputs. (3 marks) (b) Calculate the call value and the put value. (8 marks) (c) State the total project value.
(3 marks) (d) A firm has assets of ₦100m, debt of ₦80m due in five years, r = 5% and s = 0.4; given N(d1) = 0.8355, N(d2) = 0.5326 and e−0.25 = 0.7788, calculate the equity value, the debt value and the probability of default. (6 marks)
Question 13 (10 marks). (a) A Black-Scholes call on a company’s shares is ₦12.72 and there are 0.2 warrants per share. Calculate the warrant value. (3 marks) (b) Explain why equity of a distressed company may have value. (4 marks) (c) Explain what loss given default means. (3 marks)
Model answers
Show the model answer to Question 11
(a) Real options (5 marks). A real option is a choice, not an obligation, within an investment, for example to invest more, to abandon or to defer. (2) Types: scale up, scale down, defer, switch, scope up or scope down. Any three. (3)
(b) NPV (3 marks). NPV summarises expected outcomes at the selection stage. (1.5) It ignores later flexibility such as the right to abandon or expand, which has value. (1.5)
(c) Weakness (2 marks). No simple and satisfactory valuation approach has been established. (2)
Show the model answer to Question 12
(a) Option (3 marks). The right to sell at a fixed price is a put. (1) Pa = 310, Pe = 200, t = 4, r = 5%, s = 0.30. (2)
(b) Values (8 marks).
- Call = 310 × 0.9147 − 200 × 0.7764 × 0.8187 = 283.56 − 127.13 = 156.43. (4)
- Put = 156.43 − 310 + 200 × 0.8187 = 156.43 − 310 + 163.74 = ₦10.17m (10.18 with rounding). (4)
(c) Total (3 marks).
- NPV = 310 − 300 = 10. (1)
- Total value.
10 + 10.18 = about ₦20.2m. (2)
(d) Firm (6 marks).
- Equity = 100 × 0.8355 − 80 × 0.5326 × 0.7788 = ₦50.37m. (2)
- Debt = 100 − 50.37 = ₦49.63m. (2)
- Probability of default.
1 − 0.5326 = 46.74%. (2)
Show the model answer to Question 13
(a) Warrants (3 marks).
- Dilution factor = 1 ÷ 1.2. (1)
- Value = 12.72 ÷ 1.2 = ₦10.60. (2)
(b) Distressed equity (4 marks). Equity is a call option on the company’s assets with the debt as the exercise price. (2) Even if assets are below the debt, volatility and time give a chance the option goes into the money. (2)
(c) Loss given default (3 marks). It is the amount a lender would lose on default. (1.5) It may be less than the loan because of collateral or part repayment. (1.5)
Pathfinder on Real options, equity and debt as options, default probability and warrants
These are the ICAN Pathfinder questions that test this topic. ICAN owns the questions and the suggested solutions, so we point you to them and do not copy them. Open your Pathfinder, attempt each question under exam conditions, then check your answer against the notes above.
Neither the May 2014 nor the November 2014 Pathfinder sets a stand-alone question on real options. Work the abandonment and follow-on examples step by step.
Revise first: Real options
Case questions ask for equity, debt and default probability from the model. See the sections on equity as a call option and on debt and default.
Revise first: Equity as a call option
Source: ICAN Pathfinder, Strategic Financial Management. Diets covered: No stand-alone question in the May 2014 or November 2014 Pathfinder.
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