# 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** of the project. ## Calculating WACC The formula for calculating 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 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. Unlevering: $$ r = r_E \cdot \frac{E}{V} + r_D \frac{D}{V} $$ M&M Formula without the tax part: $$ r = \frac{r_E + r_D}{1 + \frac{D}{E}} $$ Use Miles-Ezzel or the definition of WACC to find it. #### If debt is rebalanced periodcally $$ r_E = r + (r - r_D) \frac{D}{E} (1 - \frac{T_C \cdot r_D}{1 + r_D}) $$ Rewritten for $r$: $$ r = \frac{r_E + r_D \cdot y}{1 + y}$$ Where $y = \frac{D}{E} \cdot (1 - \frac{T_C \cdot r_D}{1 + r_D})$ Use Miles-Ezzel to find the WACC. #### If debt is permanent & fixed 1: $$ r = r_D ( 1 - T_C ) \frac{D}{V - T_C D} + r_E \frac{E}{V - T_C D} $$

2: $$ \textbf{WACC} = r (1 - T_C \frac{D}{V}) $$ (combined version of normal WACC and normal M&M. Use your D / E values. You can also use normal M&M + WACC instead of this shortcut) ### 3. Use the market WACC to calculate $r_E$ for the project Use this to calculate the project WACC. Then use that to calculate the NPV. # 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 caculate the price of an 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) $. $$ C(s, t) = N(d_1) S - N(d_2) K e ^ {-rT } $$ $$ d_1 = \frac{1}{\sigma \sqrt{T}} (\ln{(\frac{S}{K})} + (r + \frac{\sigma ^ 2}{2})T) $$ $$ d_2 = \frac{1}{\sigma \sqrt{T}} (\ln{(\frac{S}{K})} + (r - \frac{\sigma ^ 2}{2})T) $$ $$ N(x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty} ^ x \mathrm{e} ^{-\frac{z ^ 2}{2}} \mathrm{d}z $$ $d_2$ can also be defined in terms of $d_1$: $d_2 = d_1 - \sigma \sqrt{T} $. $N(x)$ is not trivial to calculate. It is equivalent to the cummulative distribution function for the normal distribution, which can be looked up in a table. $ T $ : Time to maturity in fraction of interest-giving periods (e.g. 0.125 if 90 days to maturity and interest is given for one year) $ S $ : Spot price of the asset, i.e. what the asset costs now $ K $ : Strike price, i.e. what can the option be bought/sold for at maturity $ r $ : Risk-free rate (annual rate, continuously compounding) $ \sigma $ : Volatility of returns of the asset ## Binomial Options Pricing Model (BOPM) Use this when you want to calculate the price of an American-style option. Here are the formulas you will need. $$ p = \frac{e^{r t / n} - d}{u - d}, u = e ^ {\sigma ^ {\sqrt{t / n}}}, d = e ^ { - \sigma ^ {\sqrt{t / n}}} $$ $ p $ : probability of going *up* $ u $ : a value multiplier for when an asset's value goes up $ d $ : a value multiplier for when an asset's value goes down $ \sigma $ : The underlying volatility $ t $ : The time duration of a step $ n $ : The number of steps/moments. $ r $ : Risk-free interest rate 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: $$ C_{\text{node}} = e ^ {-r \Delta t } (p C_{\text{up child}} + (1 - p) C_{\text{down child}}) $$ Or you could use this formula: $$ C_{\text{node}} = \frac{p C_\text{up child} + (1 - p) C_{\text{down child}}}{(1+r) \Delta t} $$ 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 # 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.