Corporate Finance

# The Corporation and Financial Markets
### The Four Types of Firms
- Sole Proprietorships
- Partnerships
 - Limited Partnership
- Limited Liability Companies (LLC)
 - GmbH (Germany), SARL (France), SRL (Italy), SL (Spain)
- Corporations
 - C Corporations (double taxation)
 - S Corporations (no corporate tax, cannot sell preferred stock)

###Corporate Bankruptcy
- If the corporation fails to repay its debt, the control of the corporation's assets will be transferred to debt holders. 
- Thus, the end result is a change in ownership of the firm, with control passing from equity holders to debt holders.
- Importantly, bankruptcy need not result in a liquidation of the firm, debt holders can keep the business operating in the most profitable way possible.
- Corporate bankruptcy is best thought of as a ** change in ownership ** of the corporation, and not necessarily as a failure of the underlying business.

### Primary and Secondary Markets
- When a corporation issues new shares of stock and sells them to investors, it does so on the primary market.
- After this initial offering, the shares continue to trade between investors in a secondary market.

### Trading
- Bid price: the price at which market makers are willing to buy
 - the limit buy order with the highest price is the bid price
- Ask price: the price at which market makers are willing to sell
 - the limit sell order with the lowest price is the ask price
- Spread (ask price - bid price)

# Financial Statement Analysis

### The Balance Sheet
- The assets on the left side show how the firm uses its capital (investment), and the right side summarizes the sources of capital, or how a firm raise the money it needs.
- The difference between current assets and current liabilities is the firm's **net working capital**, the capital available in the short term to run the business. Firms with low (or negative) net working capital may face a shortage of funds unless they generate sufficient cash from their ongoing activities.
- Market Value vs. Book Value
 - Book value of equity is an inaccurate assessment of the true value of the firm's equity. First, many assets listed on the balance sheet are valued based on their historical cost rather than their true value today. Second, many of the firm's valuable assets are not captured on the balance sheet. Eg. the expertise of the firm's employees, the firm's reputation, the relationships with customers and suppliers, the value of future research and development innovations, and the quality of management team.
- Market-to-Book Ratio (price-to-book [P/B] ratio)

```math
Market-to-Book\,Ratio = \frac{Market\,Value\,of\,Equity}{Book\,Value\,of\,Equity}
```
- Enterprise Value = Market Value of Equity + Debt - Cash
 - Debt = interest-bearing short-term and long-term debt, excluding liabilities such as accrued expenses and accounts payable
 - the enterprise value can be interpreted as the cost to take over the business (the net cost of the business)

###Income Statement
- EPS (earnings per share)
 - diluted EPS represents earnings per share for the company calculated as though in-the-money stock options or other stock-based compensation had been exercised or dilutive convertible debt had been converted.

### The Statement of Cash Flows
- Operating/ Investment / Financing Activity

### Other Financial Statement Information
- Statement of Stockholders' Equity

```math
\begin{align}
Change\,in\,Stockholders'\,Equity &= Retained\,Earnings+Net\,sales\,of\,stock\\
&=Net\,Inome-Dividends+Sales\,of\,stock-Repurchases\,of\,stock
\end{align}
```
- Management Discussion and Analysis
 - discuss any important risks that the firm faces or issues that may affect the firm's liquidity or resources.
 - also required to disclose any off-balance sheet transactions that can impact the firm's future performance
- Notes to the Financial Statements
 - document important accounting assumptions used in preparing the statements
 - provide information specific to a firm's subsidiaries or its separate product lines
 - show details of the firm's stock-based compensation plans for employees and the different types of debt
 - details of acquisitions, spin-offs, leases, taxes, debt repayment schedules, and risk management activities

### Financial Statement Analysis

#### Profitability Ratios
Gross Margin = Gross Profit / Sales
Operating Margin = Operating Income / Sales
EBIT Margin = EBIT / Sales
Net Profit Margin = Net Income / Sales

####Liquidity Ratios
Current Ratio = Current Assets / Current Liabilities
Cash Ratio = Cash / Current Liabilities
Quick Ratio = (Cash + Near-cash(AR/short-term investments)) / Current Liabilities

#### Working Capital Ratios
Accounts Receivable Days = Accounts Receivable / Average Daily Sales
Accounts Payable Days = Accounts Payable / Average Daily Cost of Sales
Inventory Days = Inventory / Average Daily Cost of Sales
Inventory Turnover Ratio = Annual Cost of Sales / Inventory

####Interest Coverage Ratios
Interest Coverage Ratio = EBITDA (or EBIT) / Interest

####Leverage Ratios
Debt-Equity Ratio = Total Debt / Total Equity (not very useful)
Market Debt-Equity Ratio = Market Cap / Total Equity
Debt-to-Capital Ratio = Total Debt / (Total Equity + Total Debt)
Net Debt = Total Debt - Cash & Short-term Investments
Debt-to-Enterprise Value Ratio = Net Debt / (Market Value of Equity + Net Debt)

####Valuation Ratios
P/E Ratio = Market Cap / Net Income = Share Price / Earnings per Share
P/E ratio is used to assess whether a stock is over- or under-valued based on the idea that the value of a stock should be proportional to the level of earnings it can generate for its shareholders. Eg. higher growth firms have higher P/E ratios; riskier firms have lower P/E ratios.

####Operating Returns
Return on Equity (ROE) = Net Income / Book Value of Equity
Return on Assets (ROA) = (Net Income + Interest Expense) / Book Value of Assets
Return on Invested Capital (ROIC) = EBIT (1 - tax rate) / (Book Value of Equity + Net Debt)

####The DuPont Identity
```math
\begin{align}
ROE &= \frac{Net\,Income}{Sales} \times \frac{Sales}{Total\,Assets} \times \frac{Total\,Assets}{Book\,Value\,of\,Equity}\\
&= Net\,Profit\,Margin \times Asset\,Turnover \times Equity\,Multiplier!
\end{align}
```

####Key Financial Ratios
![Screen Shot 2020-06-25 at 22.50.08.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/456428e8-eb5a-5799-8746-9605bca20693.png)

###Financial Reporting in Practice
####Infamous examples of financial reporting abuses
- Enron
- WorldCom
- Bernard Madoff's Ponzi Scheme

####Legislation
- Sarbanes-Oxley Act (2002)
 - overhauling incentives and the independence in the auditing process
 - stiffening penalities for providing false information
 - forcing companies to validate their internal financial control process
- Dodd-Frank Act (2010)
 - to mitigate the compliance burden on small firms

# Financial Decision Making and the Law of One Price

###Valuation Principle
The value of an asset to the firm or its investors is determined by its competitive market price. The benefits and costs of a decision should be evaluated using these market prices, and when the value of benefits exceeds the value of the costs, the decision will increase the market value of the firm.

###Net Present Value
NPV = PV(Benefits) - PV(Costs)
- As long as the NPV is positive, the decision increases the value of the firm and is a good decision regardless of your current cash needs or preferences regarding when to spend the money.
- Regardless of our preferences for cash today versus cash in the future, we should always maximize NPV first. We can borrow or lend to shift cash flows through time and find out most preferred pattern of cash flows.

###Arbitrage and Law of One Price
If equivalent investment opportunities trade simutaneously in different markets, then they must trade for the same price in all markets.

####Separation Principle:
Security transactions in a normal market neither create nor destroy value on their own. Therefore, we can evaluate the NPV of an investment decision separately from the decision the firm makes regarding how to finance the investment or any other secutiry transcations the firm is considering.

#The Time Value of Money

###Rule of 72
```math
FV=$1\times(\,1+r\,)^N = \,$2\\
Years\,to\,double\approx72\div(interest\,rate\,in\,percent)
```

###Perpetuities
<img width="736" alt="Screen Shot 2020-07-02 at 17.11.05.png" src="https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/4471c876-a376-755d-39d1-02f7fa62a504.png">

**Present Value of a Perpetuity:**

```math
PV(C\,in\,perpetuity) = \frac{C}{r}
```
*This formula is derived by creating the same cash flow by investing PV amount of money in a bank and withdraw the interest payment every period
*Note that the first payment occurs at date 1, not date 0. (which is the end of period 0, and the beginning of period 1)

###Annuity:
<img width="821" alt="Screen Shot 2020-07-02 at 17.09.47.png" src="https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/8d560ca0-7168-87a2-dd7f-a808ee017575.png">

We derive the formula by first, invest P in the bank, and withdraw only the interest C = r x P each period. After N periods, we close the account. Thus, for an initial investment of P, we will receive an N-period annuity of C per period, plus we will get back our original P at the end.
P = PV(annuity of C for N periods) + PV(P in period N)

**Present Value of an Annuity:**

```math
\begin{align}
PV(annuity\,of\,C\,for\,N\,periods)&=P-PV(P\,in\,period\,N)\\
&=P-\frac{P}{(1+r)^N}\\
&=P\bigl(1-\frac{1}{(1+r)^N}\bigr)\\
&=C\times\frac{1}{r}\bigl(1-\frac{1}{(1+r)^N}\bigr)
\end{align}
```

**Future Value of an Annuity:**

```math
\begin{align}
FV(annuity) &= PV\times(1+r)^N\\
&= \frac{C}{r}\bigl(1-\frac{1}{(1+r)^N}\bigr)\times(1+r)^N\\
&= C\times\frac{1}{r}\bigl((1+r)^N-1\bigr)
\end{align}
```
*Note that the first payment occurs at date 1, not date 0. (which is the end of period 0, and the beginning of period 1)

###Growing Cash Flows

Conventions:
*The first payment occurs at date 1, not date 0. (which is the end of period 0, and the beginning of period 1)
*The first payment does not include growth.


**Present Value of a Growing Perpetuity**

<img width="817" alt="Screen Shot 2020-07-02 at 17.11.53.png" src="https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/41bb04d4-20a0-4a09-cb15-cbbc9bb1f77b.png">

```math
PV(growing\,perpetuity)=\frac{C}{r-g}
```

*we assume g < r for a growing perpetuity (it's unlikely to pay an amount that forever grew faster than the interest rate - the sum is infinite!)

**Present Value of a Growing Annuity**
<img width="709" alt="Screen Shot 2020-07-02 at 17.34.23.png" src="https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/a9c10f70-892d-b080-925d-a7b1e3231b8c.png">

```math
PV = C\times\frac{1}{r-g}\Bigl(1-\bigl(\frac{1+g}{1+r}\bigr)^N\Bigr)
```

#Interest Rates

###The Effective Annual Rate (EAR)
EAR is the actual amount of interest with compounding.

**For Discount Rate Period Conversion:**
n-Period Discount Rate = $(1+r)^n-1$

Eg. 1yr -> 2yr = $(1+r)^2 - 1$

1yr -> 6month = $(1+r)^{\frac{1}{2}}-1$

###Annual Percentage Rates (APR)
APR is the simple interest earned in one year **without** compounding.
IMPORTANT: we cannot use APR itself as a discount rate because it doesn't reflect the true amount. 
Instead, the APR with k compounding periods is a way of quoting the actual interest rate for each compounding period:

```math
Actual\,Interest\,Rate\,per\,Period =\frac{APR}{k\,(periods/year)}
```

Converting APR to EAR:

```math
1+EAR=\Bigl(1+\frac{APR}{k}\Bigr)^k
```
The EAR increases with the frequency of compounding:
<img width="824" alt="Screen Shot 2020-07-07 at 23.25.54.png" src="https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/5d93e7c3-5721-5e2b-29f4-0d284b57b9ef.png">

###Computing the Outstanding Loan and Interest Paid
<img width="813" alt="Screen Shot 2020-07-07 at 23.53.16.png" src="https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/0d9d67fe-5900-2ca1-5e19-02901507c7ce.png">

# Valuing Stocks
### The Dividend-Discount Model
One Year:
![Screen Shot 2020-08-24 at 22.12.40.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/e4ba3ffb-3576-5dd0-d240-613eb6686751.png)
Multiyear (eventually same value as one-year investment):
![Screen Shot 2020-08-24 at 22.14.40.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/2d7b229d-d2f8-e9c8-f129-511dea0eca3d.png)
Then, if N goes to infinite:
![Screen Shot 2020-08-24 at 22.16.45.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/6caabb7d-4f41-8918-74ce-fe338ffcda3f.png)
That is, the price of the stock is equal to the present value of the expected future dividends it will pay.

####Dividend Yields, Capital Gains, and Total Returns

![Screen Shot 2020-08-24 at 22.13.20.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/e5ea58ba-ea35-3066-e341-3b99ab672e71.png)

###Applying the Dividend-Discount Model

####Constant Dividend Growth
(Gordan Growth Model)
Same as the growing perpetuity:
![Screen Shot 2020-08-24 at 22.21.46.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/d51fc530-dc20-4fa9-4982-1e708ae7af98.png)
$r_E$ = Equity Cost of Capital
g = expected growth rate of dividends

Rearrange: we see that g equals the expected capital gain rate. In other words, with constant expected dividend growth, the expected growth rate of the share price matches the growth rate of dividends.
![Screen Shot 2020-08-24 at 22.22.15.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/a68766c2-4e7a-f1f8-99c8-1e10d83f3475.png)
![Screen Shot 2020-08-24 at 22.23.54.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/8c400b17-b55e-5ef7-c6cf-807c812ee371.png)

####Dividends Versus Investment and Growth
- The firm’s share price increases with the current dividend level, Div1, and the expected growth rate, g.
- The firm faces a trade-off: Increasing growth may require investment, and money spent on investment cannot be used to pay dividends. 

A simple model of growth:
![Screen Shot 2020-08-24 at 22.37.35.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/ea4edf0d-6796-4e13-90eb-a58028705bd1.png)
The firm can increase its dividend in 3 ways:
1. It can increase its earnings (net income).
2. It can increase its dividend payout rate.
3. It can decrease its shares outstanding.

To achieve new earnings:
![Screen Shot 2020-08-24 at 22.39.10.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/535ad6c6-e0dc-a3af-878d-d0f4bae7611a.png)
![Screen Shot 2020-08-24 at 22.39.38.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/5c489da4-0c4f-ddb5-c88d-37020ef936af.png)
![Screen Shot 2020-08-24 at 22.40.14.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/4f7b5f0f-d759-9e82-87b8-ddfcbb75644a.png)
This growth rate is sometimes referred to as the firm’s sustainable growth rate, the rate at which it can grow using only retained earnings.

- Q: If a firm wants to increase its share price, should it cut its dividend and invest more, or should it cut investment and increase its dividend?
- A: Cutting the firm’s dividend to increase investment will raise the stock price if, and only if, the new investments have a positive NPV. (ROI > Cost of capital)

#### Changing Growth Rates:
If the firm is expected to grow at a long-term rate g after year N + 1
![Screen Shot 2020-08-24 at 22.48.50.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/257faf71-0a97-c9a9-c4e8-3f31d6d3bca4.png)

0 -> N year: changing growth rates (or no dividends)
N -> year: constant growth rate (terminal value at year N)
![Screen Shot 2020-08-24 at 22.48.07.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/7bf3eae8-40e0-a622-abc3-b2acf458f9d8.png)

####Limitations of the Dividend-Discount Model - Uncertainty
- small changes in the assumed dividend growth rate can lead to large changes in the estimated stock price.
- difficult to know which estimate of the dividend growth rate is more reasonable

### Total Payout Model
![Screen Shot 2020-08-29 at 10.53.40.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/1c8a39ea-3139-166d-6170-c510138969c1.png)

![Screen Shot 2020-08-24 at 23.54.39.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/c64e32b7-c4d8-2832-c390-a27b2bae3d7b.png)

### The Discounted Free Cash Flow Model
1. Calculate the FCF
![Screen Shot 2020-08-25 at 0.12.33.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/31cb56a1-28a6-27b4-bee7-44c08e5afe22.png)
2. Calculate the Terminal Value ($V_N$)
![Screen Shot 2020-08-25 at 0.13.17.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/62f52d8f-26d6-d473-3d0d-552aff168b3d.png)
3. Find the Current Enterprise Value (PV of FCF+TV)
![Screen Shot 2020-08-25 at 0.15.25.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/7647fb14-b33d-a2d2-685e-db8927d9ed02.png)
4. Estimate the stock price
![Screen Shot 2020-08-25 at 0.16.10.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/c6012154-7feb-bb76-d7d2-9182a29b97f8.png)
Example:
![Screen Shot 2020-08-25 at 0.16.39.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/fbe860c4-c1e5-f21c-6724-eebde020c2fb.png)

### Valuation Multiples

#### The P/E Ratios
![Screen Shot 2020-08-25 at 0.37.49.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/ff9215f1-0e08-5e5e-b2ce-9f528d9757d9.png)
$P_0 = EPS_1 * P/E\,(of\,comparable\,firms)$
$EPS_1$ is based on its forward earnings.

#### Enterpise Value Multiples
(advantageous if we want to compare firms with different amounts of leverage)
![Screen Shot 2020-08-25 at 0.42.20.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/e252ccad-7ce9-4eb5-7224-d90336a2a680.png)
$V_0 = EBITDA_1 * Enterpise\,Value\,Multiples\,(of\,comparable\,firms)$

$P(share\,price)=\frac{V_0-Debt}{Number\,of\,shares}$
* $V_0$ is the Enterprise Value

#Risk and Return
###Common Measures of Risk and Return
####Expected Return
![Screen Shot 2020-08-26 at 22.00.55.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/343274c7-f8c0-f434-bf2f-a2d52e0ce675.png)
![Screen Shot 2020-08-26 at 22.01.29.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/038d291a-56fc-b5ca-9452-46cc737e51b8.png)
![Screen Shot 2020-08-26 at 22.01.45.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/1c876aae-c64c-3c2a-471a-5b76f275d726.png)

####Variance and Standard Deviation
![Screen Shot 2020-08-26 at 22.02.22.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/85214550-b328-9189-ab73-f1144da7ca34.png)
![Screen Shot 2020-08-26 at 22.02.40.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/0faa9e36-620d-56c6-88e3-aa70949e00c5.png)
![Screen Shot 2020-08-26 at 22.02.57.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/0004dfd2-113c-e487-4276-e19e5ab9e4e7.png)
We refer to the standard deviation of a return as its ***volatility***.

###Historical Returns of Stocks and Bonds
####Realized Annual Returns
![Screen Shot 2020-08-26 at 22.05.17.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/331b4919-8a91-e71d-43a8-6d623085b817.png)
####Empirical Distribution of the Returns
![Screen Shot 2020-08-26 at 22.07.08.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/bb9a2871-2262-4865-4c8a-9c0140d8158f.png)
####Average Annual Return
![Screen Shot 2020-08-26 at 22.09.26.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/a87665a5-1de1-e60f-0083-1b0a14c8bd6f.png)

- Arithmetic Average Returns Versus Compound Annual Returns:
 - $(\frac{Final\,Value}{Initial\,Investment})^{\frac{1}{T}} - 1$
 - The compound annual return is a better description of the long-run historical performance of an investment.
 - the arithmetic average return is a better estimate of investment’s expected return over a future horizon based on its past performance.

####The Variance and Volatility of Returns
![Screen Shot 2020-08-26 at 22.13.40.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/d4d9e263-18c8-0f74-8f02-0471629ac895.png)
####Estimation Error
when risks are independent and identical, the standard deviation of the average is known as the standard error
![Screen Shot 2020-08-26 at 22.23.01.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/c6bd0053-2b2f-ed07-2a22-dbd5963d39b1.png)
- The 95% confidence interval for the expected return is:
![Screen Shot 2020-08-26 at 22.23.28.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/d151d6d2-1ba3-074e-7351-a8c543eab000.png)

###Firm-Specific Versus Systematic Risk, and the Risk Premium
- The risk premium for diversifiable risk is zero, so investors are not compensated for holding firm-specific risk.
- The risk premium of a security is determined by its systematic risk and does not depend on its diversifiable risk.

###Measuring Systematic Risk
####Identifying Systematic Risk: The Market Portfolio
- Efficient portfolio. An efficient portfolio cannot be diversified further—that is, there is no way to reduce the risk of the portfolio without lowering its expected return.
- The best way to identify an efficient portfolio is one of the key questions in modern finance. 
- it is common in practice to use the S&P 500 portfolio as an approximation for the market portfolio, under the assumption that the S&P 500 is large enough to be essentially fully diversified.

####Sensitivity to Systematic Risk: Beta
![Screen Shot 2020-08-27 at 0.01.04.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/b89b2204-ea3a-eaeb-982b-b2ff57645e20.png)
- The beta of a security is the expected % change in its return given a 1% change in the return of the market portfolio.
- Beta measures the sensitivity of a security to market-wide risk factors.
- Utilities, drug and food companies tend to be stable and highly regulated, and thus are insensitive to fluctuations in the overall market - lower betas
- technology, furniture, and luxury retailers stocks tend to have higher betas

###Beta and the Cost of Capital
####Capital Asset Pricing Model (CAPM)
![Screen Shot 2020-08-27 at 0.09.58.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/0be66b2c-09ac-0dce-07f8-78a0fb9f9977.png)
![Screen Shot 2020-08-27 at 0.10.29.png](https://qiita-image-store.s3.ap-northeast-1.amazonaws.com/0/620849/f740d476-d02b-dc6d-22bf-365bc1047fd9.png)