Fundamentals

Time value of money

Humans are impatient and consumption now is guaranteed, while the circumstances may make consumption impossible in the future. Some human needs (like the need for food) must be fulfilled quickly.

Unspent values can be invested and thus generate more value. One seed may generate hundreds of new seeds in a year if it sawn today.

Accounting representation

Note that the accounting representation of a firm may be unsuitable to make financial descisions. The first mostly focuses on where values are tied up, while the latter focuses on the actual cash flows.

Depriciation

Fixed assets are normally depreciatied over a their lifetime. This makes sense as their aquirement only converts value from cash to fixed assets - the real loss of value happens as the assets age.

However, when determining whether to proceed with a project, the actual cashflows are more relevant, thus the entire cost is treated as a cash flow at the time of aquirement. The loss of value is reflected in a lower cash flow the opposite way around when the asset is sold.

Do however note that the depriciation is deducted when calculating taxes, and thus affects the cash flow through taxes!

Irrelevant information

The financial statements are required to show all aspects, even those not relevant to a decision, for example money already spent. This is of course irrelevant to an investment desicion and should be disregarded.

Changes in capital

Financial statements often include working capital, but not the changes in working capital. These changes are in fact cash flows and must be included when calculating Net Present Value (NPV).

In other words, this capital is tied up in the project and thus does not generate interests (as it would otherwise do).

Utility

Assumptions made:

People are greedy, so more is always better than less

Each additional unit gives less utility than the previous

Peoples preferences are well-behaved; their preferences form a partial order

This results in utility functions $U(W)$, which express the utility given by an amount $W$ of resource (often wealth). Two common utility functions:

$U(W) = ln(W)$

$U(W) = \alpha + \beta W - \gamma W^2$

Note that $W$ may be negative, although (1) may suggest otherwise. Also, (2) breaks down with large values of $W$ (meaning it does not adhere to the assumptions).

Indifference curve

Given two resources, $W_1$ and $W_2$, the indifference curve shows which combinations gives the same utility, in other words the line defined by
$$U(W_1, W_2) = C$$
where $C$ is a constant.

Risk aversion

The utility curves from the above assumptions also leads to risk aversion.

Efficient Market Hypothesis

The Efficient Market Hypothesis states that financial markets are informationally efficient. It comes in three variants: the weak, the semi-strong and strong efficient market hypotheses.

Weak

The weak form claims that prices on traded assets reflect all past publicly available information.

Semi-strong

The semi-strong form claims that prices on traded assets reflect all publicly available information and prices instantly change to reflect new public information.

Strong

The strong form claims that prices on traded assets reflect all information (even private), and prices instantly change to reflect any new information.

Project decisions

Often, you will need to consider a project proposal where company XYZ ventures into a new area of business. To figure out whether or not to go through with a project, you need to figure out whether or not the NPV, or Net Present Value, is positive. If $ \textbf{NPV} > 0 $, go through with the project. Conversely, if $ \textbf{NPV} < 0 $, do not go through with the project. Hence, making project decisions is an exercise of calculating NPV.

Typically, projects will cost a certain one-time, up-front amount of money $ C $ to be completed, and generate a perpetual yearly revenue of some amount $ R $. If you have these two numbers, you can figure out NPV using the formula:

$$ \textbf{NPV} = -C + \frac{R}{\textbf{WACC}} $$

Now all you have to do is to calculate the WACC (Weighted Average Cost of Capital) of the project.

Calculating Weighted Average Cost of Capital (WACC)

The formula for calculating Weighted Average Cost of Capital(WACC) is:

$$ \textbf{WACC} = \frac{D}{V} r_D (1 - T_C) + \frac{E}{V} r_E $$

$ D $

Debt in dollars

$ E $

Equity in dollars

$ V $

Total value in dollars (i.e. debt + equity)

$ r_D $

Cost of debt (typically interest rate of loan)

$ r_E $

Cost of equity

$ T_C $

Corporate tax rate

Unfortunately, $ r_E $ is impossible to know. Luckily, it can be estimated by looking at previous projects in the same business domain. One way to estimate $ r_E $ is to use the Capital Asset Pricing Model, or CAPM.

Calculating $ r_E $ using Capital Asset Pricing Model (CAPM)

The formula for CAPM is

$$ r_E = r_f + \beta (r_m - r_f) $$

$ r_E $

Cost of equity

$ r_f $

Risk-free interest rate

$ r_m $

Market rate

$ \beta $

Unsystematic risk

$ (r_{m} - r_{f}) $

Market risk premium

Levering and unlevering

When doing these project calculations, we need to make sure that we don't mistakenly assume that cost of equity is the same across different leverage degrees. To account for this, we need to un-lever and re-lever as necessary. Here are some formulas that help in doing this.

Miles-Ezzell WACC calculation

When debt of a project will be rebalanced, you can use Miles-Ezzell.

$$ WACC = r - \frac{D}{V} r_D T_C \frac{1 + r}{1 + r_d} $$

$ D $

Debt in dollars

$ V $

Total value in dollars

$ T_C $

Corporate tax rate

$ r $

Opportunity cost of capital (100% equity, i.e. unlevered)

$ r_D $

Cost of debt

M&M formula

The tax part ($ 1 - T_C $) is left out when debt is continuously rebalanced.

$$ r_E = r_A + (1 - T_C) (r_A - r_D) \frac{D}{E} $$

$ D $

Debt in dollars

$ E $

Equity in dollars

$ r_E $

Cost of equity (levered)

$ r_A $

Opportunity cost of capital (100% equity, i.e. unlevered)

$ T_C $

Corporate tax rate

$ r_D $

Cost of debt

Comparison based on existing actors in market

Sometimes we're asked if an entity should go ahead with a project or not, given some numbers about the project and some numbers about existing actors in the market.

To do this we calculate the opportunity cost of capital, or return on assets, for the existing projects.

Note: When estimating something for a given project only use the values belonging to that project.
I.e. the $r_E$ calculated for the existing actors/projects is not the $r_E$ you should be using to calculate the proposed project's WACC.

See also figure 6.3 on page 174 of the book for a decision tree.

1. Estimate $r_E$ for existing projects

Use the CAPM.

2. Calculate $r$ and WACC for the market

If there's several existing entities or projects in the market for which you have been given either $r_E$ or the needs to estimate it, calculate each of their $r$.
The unweighted average is typically the $r$ for the market, but make sure to look for such a statement in the problem text.

If debt is continuously rebalanced

Use either of the formulas below, depending on what information is available.

3. Calculate NPV using WACC

$$ NPV = -C + \frac{R}{WACC} $$

Option pricing

Options are financial contracts that give their holders the right, but not obligation, to buy or sell something on a future date (maturity date) at a price decided upon today. Options can be priced rationally using different models. European options (options that can only be exercised at the maturity date) should be priced using the Black-Scholes model, and American options (optinons that can be exercised at any time until the maturity date) should be priced using the Binomial options pricing model.

Put options give the right to sell. Call options give the right to buy.

Black-Scholes model

Use this when you want to calculate the price of a European-style call option.
Here are the formulas you will need. When calculating European puts, use $ -N(-d_1) $ and $ -N(-d_2) $ in the main formula instead of $ N(d_1) $ and $ N(d_2) $.

First, make a binomial lattice with $ n $ steps. $ S $ is the asset price at the current time, and $ u $ and $ d $ are the value multipliers from the formula above:

Then you want to fill in the numbers in all the nodes of the lattice with dollar (or euro, or whatever) values. Here I have just made up some numbers for the sake of example: $ S = \$100, u = 1.14, d = 0.92 $.

This is now the finished asset price lattice. Now we want to make a new lattice for the option price. It should have as many steps as the asset price lattice, but all nodes should be empty except for the right-most leaf nodes. These nodes should contain the option value at that point, calculated as $ S_n - K $, i.e. the asset price for the corresponding node in the asset lattice minus the strike price. Of course, if this expression is less than \$0, the value of the option is truncated to zero, since excercising options are, well, optional. Anyway, for our example, with $ K = \$100 $, this will yield an option price lattice that looks like this:

Now, all that is remaining is to calculate the values of the empty nodes from right to left. The value in each node, $ C_{\text{node}} $, can be calculated from its two direct child nodes (to the right) by using this nice formula:

Note it uses $r+1$ because it assumes $r$ is an actual percentage expressed as a real number (i.e. $r \in [0, 1]$).

Continuing our example, assuming $ r = 5\%, t = 1 $, we get:

The root (leftmost) node of the lattice is the binomial option price, $ \$9.26 $, which is what we wanted to calculate!

Put-Call Parity

Put-call parity expresses the relationship between the price of a European call option and a European put option. The relationship is as expressed in this equation:

$$ C - P = D(F - K) $$

$ C $

Current price of a call

$ P $

Current price of a put

$ D $

Discount rate (typically risk-free interest or similar)

$ F $

Forward price of the asset

$ K $

Strike price

Option positions

Short straddle

A good strategy when you expect the stock price to have low volatility.
The short straddle will give maximum profit if the price stays the same as today.

You sell a put option and sell a call option (short put and short call) with an at-the-money exercise price.
This option position has no upfront cost, as you only sell two options.
The downside is that if the stock price changes a lot in the future, there is a possibility for infinite losses.

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Long straddle

You buy a put option and buy a call option (long put and long call).
This is a good strategy if you expect high volatility in the stock price.

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Butterfly spread

Buy two call options – one at an in-the-money exercise price and one out-of-money.
Then sell two call options for an at-the-money exercise price.
This has an upfront cost of the cost of the two long options minus the cost of the two short options.
The advantage of a butterfly spread over a short straddle is the reduced risk in case of an increased stock price.
The downside compared to the short straddle is a lower maximum profit.

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Trade-off Theory of Capital Structure

The trade-off theory of capital structure states that as the debt/equity ratio increases, there is a trade-off between the interest tax shield and bankruptcy, causing an optimum capital structure. The theory implies that the marginal benefit of further increases in debt declines as debt increases, while the marginal cost increases as the debt increases. Informally, this means that as the cost of debt goes up, a good strategy is to decrease the debt/equity ratio, and that as the benefits go up (for instance by tax increases) the debt/equity ratio should increase.

Misc definitions

Here are some different definitions that might come in handy.

Cumulative abnormal return (CAR)

Cumulative abnormal return (CAR)
Sum of the differences between the expected return on a stock (systematic risk multiplied by the realized market return) and the actual return often used to evaluate the impact of news on a stock price.