# Basic Definitions

## Value of a Firm

The value of a firm is the addition of the value its equity (the number of shareholders multiplied by the price per share) and the value of debt outstanding.

# Modigliani and Miller Propositions

## Proposition #1

$\displaystyle V_L = V_U + t_c*D_L$

## Proposition #2

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

## WACC

$\displaystyle WACC = (1-t_c)(\frac{D}{V})r_D + (\frac{E}{V})(r_E) = r_a ( 1-t_c*\frac{D}{V})$

$\displaystyle r* = r_a*(1-t_c*L)$

where $\displaystyle L = \frac{\Delta D}{APV+CF_0} = \frac{\Delta D}{PV(project)}$

and in the case where the only benefit of debt is a tax shield,

$\displaystyle L = \frac{\Delta D}{tax shield + PV(After Tax Revenue Inflows discounted at R_a)}$

The APV can then be found by calculating:

$\displaystyle APV = \frac{CF*(1-t_c)}{r^*} - Investment$

## Discounting Examples

Name of Cash Flow Yearly Cash Flow if Cash Flow is Perpetual Discounted Rate Final Value
Cash Flow to Shareholders $\displaystyle (1-t_c)(X - r_D*D)$ $\displaystyle r_E$ Value of Equity (XXX)
Cash Flow to Bondholders $\displaystyle r_D*D$ $\displaystyle r_D$ Value of Debt (XXX)
Cash Flow to Bondholders $\displaystyle r_D*D$ $\displaystyle r_D$ Value of Debt (XXX)
After-tax cash flows of assets $\displaystyle (1-t_c)X$ $\displaystyle r_A$ Value of the unleveled firm
After-tax cash flows of assets $\displaystyle (1-t_c)X$ $\displaystyle WACC_U$ Value of the unleveled firm (XXX)
After-tax cash flows of assets $\displaystyle (1-t_c)X$ $\displaystyle WACC_L$ Value of the leveled firm
After-tax cash flows of assets $\displaystyle (1-t_c)X + t_c*r_D*D$ $\displaystyle r_A and r_D$ Value of the leveled firm
Name of Cash Flow Yearly Cash Flow if Cash Flow is Perpetual Discounted Rate Final Value
Pre-Tax Cash Flow to Shareholders $\displaystyle (1-t_c)(EBIT - r_D*D_L)$ $\displaystyle r_E$ $\displaystyle E_L$
After-Tax Cash Flow to Shareholders $\displaystyle (1-t_E)(1-t_c)(EBIT - r_D*D_L)$ $\displaystyle (1-t_E)r_E$ $\displaystyle E_L$
Pre-Tax Cash Flow to Bondholders $\displaystyle (r_D)(D_L)$ $\displaystyle r_D$ $\displaystyle D_L$
After-Tax Cash Flow to Bondholders $\displaystyle (1-t_D)(r_D)(D_L)$ $\displaystyle (1-t_D)r_D$ $\displaystyle D_L$
After-corp-tax Cash Flow to assets $\displaystyle (1-t_c)(EBIT)$ $\displaystyle WACC_L$ $\displaystyle V_L$

# Options

## Type of Options

These are graphs of the payoffs of the four types of options. (The y axis is the payoff of the options)

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## Put-Call Parity

$\displaystyle P_t = C_t - S_t + \frac{X}{{(1+r_f)}^{T-t}} = C_t - S_t + XB_t$

## Replicating options with Risk Free and Stock

If a stock will be worth uS or dS after a certain time period, and the call option for the stock has a strike price of X, and the stock is worth S, the option can be replicated by lending/borrowing $\displaystyle \beta$ at the risk free rate and purchasing/selling $\displaystyle \delta$ stock.

The amounts of stock purchase and risk free lending that have to be done can be calculated by solving the following two equations simultaneously:

$\displaystyle uS*\delta + (1+r)*\beta = max( uS - X, 0)$

$\displaystyle dS*\delta + (1+r)*\beta = max( dS - X, 0)$

## Risk Neutral Valuation

$\displaystyle C_0 = \frac{q*C_u + (1-q)*C_d}{1+r}$

where $\displaystyle q = \frac{(1+r)-d}{u-d}$

## Black Scholes

This is the formula for calculating the the price of a CALL OPTION:

$\displaystyle C_0 = S_0*N(x) - \frac{X}{(1+r_f)^T}*N(x-\sigma*\sqrt{T})$

where

$\displaystyle x = \frac{ln( \frac{S_0}{\frac{X}{(1+r_f)^T}})}{ \sigma*\sqrt{T}} + \frac{1}{2}*\sigma*\sqrt{T}$