Corporate Finance

Why does discounting CFs not work for options?

• We could compute CF in each period 
• The risk of an option changes every time the underlying moves, hence finding one opportunity cost of capital for an option is impossible (as with options on shares etc.)

How can the price of an option be computed?

• Black-Scholes formula (relatively easy to apply but not feasible for all kinds of options), particularly not for real options that often are of American type.
• Binomial model and risk-neutral pricing (requires more work but allows to price more types of options)

Real Options buy you flexibility (like American Options do), thus Black Scholes does not work 'cause it only works if there's a fixed maturity at which the option will be exercised

1. We need the stock price today (S), the standard deviation (sigma) and the length of one time step (h)
2. Calculate upside and downside changes (s. picture)
3. Contruct a tree with potential futures prices of the underlying
   • Assuming there exists a tradable bond and stock, the price of the option must be equal to the price of the stock/bond combination that gives the same payoff as the option (arbitrage pricing).

If the time steps used to set up the tree become infinitesimally small, the binomial tree option value goes towards the Black-Scholes option value

The more realistic a tree should be, the more steps should be taken

One easy way to get probabilities is to switch to a risk-neutral world:

• Pretend investors do not care about risk, so expected return on everything (i.e. the discount rate) is r_f, calculate expected payoff of option and discount at r_f
  • By calculating the risk-neutral probabilities, we can price options in the binomial tree relatively easily.
  • Risk-neutral probabilities are not the true (real-world) probabilities!
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Intuition? Everybody likes an arbitrage opportunity, no matter how risk-averse someone is! Everybody calculates the value of an option in the same way. Value is such as there is no arbitrage opportunity!

Calculations see picture

Insights from the Example

• Where does the value difference come from? From Flexibility
• Pure DCF approach can introduce wrong decisions
• Always try to incorporate optionalities in your investment decisions
• Difficulty: underlying parameters (you have to estimate volatility etc.

Your income might be (each with 33% probability):

• a) 440
• b) 280
• c) 120

NPV takes the expected income of 280 and makes the decsion NOW.

Real Option approach takes into account the future FLEXIBILITY: If income was 120, you simply wouldn't do the project. Thus, IF you invest, the expected income is higher (360, i.e. (440+280)/2)

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When solving a tree, you always have to start at the end and work your way backwards to the start.

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second case: higher uncertainty leads to a higher option value! Why?

Becaue on the upside, we can profit more but on the downside we don't loose more because we can opt out and simply not invest in the bad case!

Option to wait can be compared to an AMERICAN CALL OPTION.
Strike price = investment
Underlying = value of the project

If we invest, we exercise the option!

Option to wait is more valuable, the higher the volatility and less valuable, the higher the early expected cash flows from the project

Similarities to Americal Call Option:

• If the underlying pays no dividends (early CFs), the option is worth more alive than exercised and should never be exercised early
• If the underlying does pay dividends (early CFs), these payments reduce the ex-dividend price and possible payoffs at maturity, so early exercise might be rational
• Dividends to not always prompt early exercise, only if they are large enough

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If bad news arrive and cash flows are below expectations, it might be useful if companies have an option to bail out and recover part of their investment.

The option to abandon is equivalent to a PUT OPTION. A company will exercise the option if the value recovered from the project's asset is greater than the PV of continuing the project

---> Siehe auch Ausdruck und Buch

IMPORTANT:  At each node, you have to make the optimal decision!!

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MM1: "With perfect capital markets and no taxes the value of the firm is indipendent of its capital structure"

MM1: "With perfect capital markets and no taxes the value of the firm is indipendent of its capital structure"

MM2: "The expected return on equity of a levered firm increases with the D/E ratio. The rate of increase depends on the spread between expected return on assets and the return on debt"

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Both. BUT the fact that the expected ROE increases with leverage doesn't mean that shareholders are better off. Rather, the expected (or "required") ROE must increase to compensate shareholders for the increase in risk brought about by the increase in leverage.

Zum Plausibilisieren der Zahlen s. Folie 10, Chapter 4

• No taxes
• No bancruptcy costs (no costs of financial distress generally)
• Symmetric information
• Perfect and complete capital markets (no transaction costs or issuance costs, firms and investors face identical borrowing and lending costs, complete set of securities etc.)

The trade-off theory of capital structure refers to the idea that a company chooses how much debt finance and how much equity finance to use by balancing the costs and benefits. The classical version considers a balance between the dead-weight costs of bankruptcy and the tax saving benefits of debt. Often agency costs are also included in the balance. This theory is often set up as a competitor theory to the pecking order theory of capital structure.

It relaxes the assumptions of an idealized world of Modigliani and Miller.

To be precise, it relaxes the assumptions of

- no taxes
- no costs of financial distress

• Interest is regarded as a cost of doing business and is tax deductible
• Dividends are treated as a return to the firm's owners and are therefore not tax deductible

-> holding before-tax CFs fixed, debt finance reduces a firm's taxable income, resulting in higher after-tax CFs to equity and debtholders

(MV of levered firm refers to the after-tax MV of levered firm)

-> genaue Herleitung siehe Folien 29-31

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(once you take on debt, you have a probability of default)

Direct costs of financial distress (small)

• Lawyers
• Bancruptcy judges

Indirect costs of financial distress (big)

• Managers attention diverted from managing assets to managing liabilities
• Firm loses flexibility when monitored closely by creditors (covenants...)
• Assets must be sold in fire-sales. Firms with highly specific assets obtain the worst prices, especially true if assets don't have value outside the industry and the industry is cyclical
• Intangible assets may be destroyed if the firm is broken up or sold to someone else. Firms with high advertising or R&D expenditure have lover than average debt ratios (tech firms). Firms with high expenditure on PP&E have higher than average debt ratios (hotels)...

(optimal debt ratio: 1 unit more debt has the same marginal impact on the tax shield as on the cost of financial distress)