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# System prepended metadata

title: ENGR 111 - Introduction to Finance and Marketing for Engineers

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# ENGR 111 - Introduction to Finance and Marketing for Engineers

## Week 1

### 1.1 What is Corporate Finance
#### Finance addresses the following 3 questions
* **Capital Budgeting**: What long-term investments should the firm choose
* **Capital Structure**: How should the firm raise funds for selected investments
* **Net Working Capital**: How should short-term assets be managed and financed
    

#### $NetWorkingCapital=CurrentAssets-CurrentLiabilities$
    

#### The Financial Manager
* The Financial Manager's primary goal is to increase the value of the firm by:
    * Selecting value creating projects
    * Making smart financing decisions

    
-----
### 1.2 The Corporate Firm
#### Forms of Business Organization
1. The Sole Proprietorship
2. The Partnership
    * General Partnership
    * Limited Partnership

3. The Corporation
    * Private
    * Public

#### Comparing Corporation vs. Partnership


|                                  | Corporation            | Partnership                                                                                 |
| -------------------------------- |:---------------------- |:------------------------------------------------------------------------------------------- |
| Liquidity                        | Easily exchange shares | Subject to substantial restrictions                                                         |
| Voting Rights                    | 1 share = 1 vote       | General Partner in charge, limited partners may have some voting rights                     |
| Taxation                         | Double                 | Partners pay taxes on distributions                                                         |
| Reinvestment and Dividend Payout | Broad latitude         | All new cash flow is distributed to partners                                                |
| Liability                        | Limitied liability     | General partners may have unlimited liability, and limited partners enjoy limited liability |
| Continuity                       | Perpetual life         | Limited life                                                                                |

#### What does it mean to "go public"?
* IPO stands for **Initial Public Offering**.
    * Before going public, a company collects venture capital from initial investors. In exchange, these investors are promised Preferred Stock.
    * When a company IPOs, a **market/financial intermediary** (e.g. Goldman Sachs) advises the company on an initial listing price. 
        * Financial advising is costly for each time a company issues new stock to the public.
        * Stock becomes **diluted** every time it is re-issued as well.
* Capital Surplus a.k.a. "paid in capital" in addition to common stock is the total money a company receives when it IPOs.
* The value of a company on the secondary market does not make the company any richer; however this convinces creditors to lend the company more money.

#### Pros and Cons of Going Public

* **Pros**
    * Gain funding that does not need to be repaid
    * More visibility
    * Market valuation
* **Cons**
    * Costs money to IPO
    * Management loses some freedom to board
    * Open to scrutiny
    * Public reporting
    * Company takeover may happen

-----
    
### 1.3 The Importance of Cash Flow
* Ultimately, the cash firm must be a *cash generating activity*
* The cash flows from the firm must **exceed** the cash flows from the financial markets

-----

### 1.4 The Goal of Financial Management
* What is the correct goal?
    * Maximize profit?
    * Minimize costs?
    * Maximize market share?
    * **Maximize shareholder wealth?**

-----

### 1.5 The Agency Problem
* Agency relationship
    * The principal hires an agent to represent their interest
    * Stockholders (principals) hire managers (agents) to run the company
* Agency Problem
    * Conflict of interest between principal and agent

#### Managerial Goals
* Managerial goals may be different from shareholder goals 
    * Expensive perquisites
    * Survival
    * Independence

* Increased growth does not necessarily lead to increased shareholder wealth

#### Managing managers
* Managerial Compensation
    * Incentives can be used to align the management and stockholder interests 
        * e.g. Stock options

* Corporate control
    * The threat of a takeover may result in better management




## Week 2

### 2.1 The Balance Sheet
* Snapshot of the firm's accounting value at a specific point in time
* Balance Sheet Identity
    * $Assets=Liabilities+StockholderEquity$


#### Balance Sheet Analysis
* When analyzing a balance sheet, the Finance Manager should be aware of three concerns:
    * Liquidity
    * Debt vs. Equity
    * Value vs. Cost

#### Liquidity 
* Refers to ease/quickness with which assets can be converted to cash
* Current assets are the most liquid
* The more liquid a firm's assets are, the easier it is to pay short-term obligations

#### Debt vs. Equity
* Creditors generally receive the first claim on firm's cash flow
* Shareholder's equity is the residual difference between assets and liabilities
    * $Equity=Assets-Liabilities$

#### Value vs. Cost
* Under Genderally Accepted Accounting Principles (GAAP), audited financial statements of firms in the US carry assets at cost
* Market value refers to price at which assets, liabilities, and equity can be bought/sold

### Breaking Down the Balance Sheet
#### Assets
|                      |     |
|:-------------------- |:--- |
| **Current Assets**   |     |
| Cash                 | The most liquid asset    |
| Accounts Receivables | Pending receivable checks from selling goods/servies    |
| Inventory            | Products waiting to be sold    |
| **Fixed Assets**     | Properties, patents, machines   |

#### Liabilities


|                     |                                                                                                                                                              |
|:------------------- |:------------------------------------------------------------------------------------------------------------------------------------------------------------ |
| **Short-term debt** |                                                                                                                                                              |
| Notes Payable       | Coupon payments for issued bonds                                                                                                                             |
| Accounts Payable    | When you've received items from a supplier but have not paid yet                                                                                             |
| Accrued Expenses    | Expenses you have not paid for but are using in production/selling i.e. utilities                                                                            |
| **Long-term debt**  | Includes bonds, which are denominated at $1000 and are promises to pay at a maturity date                                                                    |
| Deferred Taxes      | Companies can enter a deal with the IRS to defer paying their taxes. This is a good idea when expected taxable income is expected to be lower in the future.|


#### Equity

|                                   |      |
|:--------------------------------- | ---- |
| **Stockholder's Equity**          |      |
| Preferred Stock                   | Stock that is given to initial pre-IPO investors. Preferred Stock is a hybrid between common stock and a bond. Whereas dividends for common stock can be paid at the company's discretion, Preferred Stock is a promise to be paid. |
| Common Stock                      | Stock sold on the *secondary market*, or exchanged post-IPO.    |
| **Accumulated Retained Earnings** |      |
| Less Treasury Stock                  | Return from stock-buyback. This is marked as negative on the Balance Sheet. |

##### See [section 1.2](https://hackmd.io/@kaylyn-phan/engr-111#12-The-Corporate-Firm) for more information about IPO.

#### Stock Buy-Back
* Dividend payout is the share of profit amongst investors. In contrast, **Stock Buy-back** is the purchase of company shares back from investors.
    * Stock is usually bought back at a price slightly higher than the market price. This is to avoid negotiation.

* Why would a company buy back their stock?
    * This is where kaylyn is leaving off






#### More Subjective, Hard to Quantify Measures
* Customer Loyalty, Growth Potential, and other assets that are very important for small companies.
* One exception to this is when a company purchases another company. Then, the difference between the acquisition price and the increase in the greater company's balance sheet reflects the unlabeled value of the acquired company.
-----

### 2.2 The Income Statement

While the Balance Sheet reflects a firm's accounts at a specific point of time, the Income Statement measures a firm's financial performance over a **period of time**.

$Income=Revenue-Expenses$

#### The U.S.C.C. Income Statemnet
|                                               |        |
| --------------------------------------------- | ------:|
| Total operating revenues                      | $2,262 |
| Cost of goods sole                            |  1,655 |
| Selling, general, and administrative expenses |    327 |
| Depreciation                                  |     90 |
| **Operating income**                          |    190 |
| Other income                                  |      29  |
| **Earnings before interest and taxes**        |      $219  |
| Interest expense                              |        |
| **Pretax income**                             |        |
| Current taxes                                 |        |
| Deferred taxes                                |        |
| **Net Income**                                |        |
| **Net Income**                                |        |
Addition to retained earnings
Dividends|     | Text     |



-----

### 2.3 Taxes

Taxes will be taxes, they're always **CHANGING**

![](https://i.imgur.com/7ynalXA.png)

Companies pay lower than what they're supposed to pay by using **INCENTIVES** like lobbying, creating jobs in neighborhoods, etc.


---

### 2.4 Net Working Capital

Represents how much of the money should be in ***liquid*** form

$NetWorkingCapital=CurrentAssets-CurrentLiabilities$

* NWC usually grows with the firm
    * If the company grows, they're going to need more NWC

-----

### 2.5 Financial Cash Flow
* Most important item that can be extracted from financial statements is the **cash flow**
* The cash flow received **from** the firm's assets must equal the cash flows **to** the firm's creditors and stockholders
    * $CF(assets)=CF(creditors)+CF(stockholders)$

-----

### 3.1 Financial Statement Analysis

-----

### 3.2 Ratio Analysis and Du Pont Idnetity
Ratios allow for better comparison through **time** or **between companies**.

#### Categories of Financial Ratios
* Short-term solvency/Liquidity Ratios
* Long-term solvency/Financial Leverage ratios
* Asset management/Turnover ratios
* Profitability Ratios
* Market Value Ratios

#### Computing Liquidity Ratios
* $CurrentRatio=\frac{CurrentAssets}{CurrentLiabilities}$
    * Similar to Net Working Capital, except it's in ratio form
* $QuickRatio=\frac{CurrentAssets-Inventory}{CurrentLiability}$
    * Take away the inventory, and then look at the rest of the assets
    * Tells how much a firm depends on their inventory

* $CashRatio=\frac{Cash}{CurrentLiability}$
* ***What does having high short-term solvency ratios imply?***
    * The ratios should be greater than 1, but not too large
    * If the current ratio is less than 1, we should check equity to make sure that the company is okay

#### Computing Leverage Ratios

* $TotalDebtRatio=\frac{Asset-Equity}{Asset}$
    * $Asset-Equity$ equals debt, so this represents "for every dollar in assets, how much debt do I have"

* $\frac{Debt}{Equity} = \frac{D}{E}$
* $EquityMultiplier (EM) = \frac{A}{E} = 1 + \frac{D}{E}$
* What does having high long-term solvency ratios imply?
    * It could be good because young company needs a lot of debt, and they are also not afraid of getting into debt
    * It could be bad because company might be financing itself with debt &rarr; less retained earnings &rarr; more debt

#### Computing Coverage Ratios
* $InterestCoverageRatio = TimesInterestEarned=\frac{EBIT}{Interest}$
* $Cash Coverage=\frac{EBIT + Depreciation + Amortization}{Interest}$

#### Computing Inventory Ratios
This is not important for companies like cloud computing services.
**Bad inventory management** is considered when your inventory sits in storage for too long, and it gets old and you keep paying for storage.
* $Inventory Turnover=\frac{Cost of Goods Sold}{Inventory}$
    * How many times you replace the inventory

* $Days'SalesInInventory=\frac{365}{InventoryTurnover}$
    * How many days does it take to sell

* What would make you under or overestimate these values
    * There may be times when your inventory is unusually high or low, so to fix this we can take the average over the years.

#### Computing Receivable Ratios
* $ReceivablesTurnover=\frac{Sales}{AccountsReceivable}$
    * The amount a company expects they won't be able to collect

* $DaysSalesInReceivables = \frac{365}{ReceivablesTurnover}$
    * How many days does it take for me to collect the credit

#### Computing Total Asset Turnover
* $TotalAssetTurnover=\frac{Sales}{TotalAssets}$
    * For every dollar you have in assets, how many sales can you get

* It is not unusual for $TAT < 1$, especailly if a firm has a large amount of fixed assets
* Having a **high $TAT$**
    * If net fixed assets are low, that makes total assets lower, which makes them look more efficent than the other company with a lower TAT

#### Computing Profitability Measures
* $ProfitMargin(PM)=\frac{NetIncome}{Sales}$
* $ReturnOnAssets(ROA)=\frac{NetIncome}{TotalAssets}$
* $ReturnOnEquity(ROE)=\frac{NetIncome}{TotalEquity}$
    * This is an important measure for investors because it's how much profit is being generated for every dollar in **equity**

* Why invest in low profit margin companies?
    * Even if they have a small profit margin, companies that have ***high volume*** will still make more

#### Computing Market Value Measures
* $EarningsPerShare(EPS)=\frac{NetIncome}{SharesOutstanding}$
* $PERatio=\frac{PricePerShare}{EPS}$
* $PlowbackRatio = \frac{RetainedEarnings}{NetIncome}$

#### Using Financial Statements
* Ratios aren't helpful by themselves. They need to be compared to something
* Time-Trend Analysis
    * See how company's performance change through time
* Peer Group Analysis
    * Compare to similar companies within industries (using SIC and NAICS codes)

#### Du-Pont Identity
* $ROE=\frac{NI}{E}$
* $ROE=PM*TAT*EM$

#### Financial Planning
* Determine your objective growth
* Use market conditions and your operational/financial capabilities to determine what increase you need in your assets to achieve the objective
* **Percentage of sales approach**
    * If sales are growing at a certain percentage, it's expected that all the numbers on the assets side should grow as well.
    * The only account on the right hand side of the Balance Sheet that grows at the same rate as assets is **Accounts Payable**
        * When you make more sales, you need to buy more supplies
    * The rate of growth in the remainder of the right hand side of the Balance Sheet is up to me and my other financial objectives (e.g. my desired maintained D/E ratio)
    * You can get external financing need without producing Pro Forma statements

#### Percent of Sales and EFN
* External Financing Needed can be also calculated as
    * $EFN=\frac{Assets}{Sales}*\Delta Sales - \frac{Spon Liab}{Sales} * \Delta Sales - (PM * ProjectedSales) * (1-d)$
    * $SponLiab$ = Spontaneous Liabilities
    * $d$ = Dividend Payout Ratio. 
    * $(1-d)$ = Retention Ratio

* **You cannot use this formula if your profit margin or amount of issued stock is going to change.**
    * The profit margin can change depending on increase in sales and the amount of interest.
    * Stock can change from buy-backs, issuing more stock, etc.







        


-----
## Week 3

### 3.4 Financial Models

-----

### 3.5 External Financing and Growth
* At low growth levels, internal financing (retained earnings) may exceed the required investment in assets.
    * $ExternalFinancingNeed = 0$ 
    * If your sales grow enough, you can get enough *internal growth* so that you don't need external financing (internal growth is rate at which you can grow without borrowing, just using retained earning)
* As the growth rate increases, internal financing will not be enough. 
    * $ExternalFinancingNeed > 0$ 
    * The company is in a *deficit.*
    * The firm will have to go to the capital markets for financing.
    * Fulfill the $EFN$ by issuing more stock ($+Equity$) or going into debt ($+Liabilities$).

* At Zero growth rate, you will make a constant profit.
    * $ExternalFinancingNeed < 0$
    * The firm has a *surplus*.
        * One option for this surplus is expanding the dividend policy. 
            * However, investors do not like this.
        * The other option is to pay debt.
            * D/E ratio will decrease. Paying debt off interest will result in tax write-offs and an increased profit margin.
        * Another option is a stock buy-back.
            * This will keep the capital structure the same and be a good move for the company.

        * Another option is to store in cash.
            * Investors generally also do not like seeing a growing, idle cash pile.

* The **Internal Growth Rate** is the growth rate the company can achieve without relying on External Financing.
    * $InternalGrowthRate = \frac{ROA * b}{1 - ROA * b}$
    * $ROA$ = Return on Assets
    * $b$ = Retention Ratio


![](https://i.imgur.com/dnhmuna.png)

* With a growth rate $g=0$, our $Assets-SponLiab$ does not change, whereas our retained earnings continues to change by its original rate.
    * Even if we don't grow ($g=0$), the company still gets revenue, which is why $\Delta RE$ stays positive
    * Internal growth rate is represented by when $\Delta RE = \Delta (Assets-SponLiab)$. This is the intersection of the graph.
    * $SponLiab$ is just short-term debt

#### Sustainable Growth Rate
* tells us how much a company grows by using internally generated funds and **maintaining a constant D/E ratio**
* sustainable growth rate equals internal growth rate when the D/E ratio equals 0
* $SustainableGrowthRate=\frac{ROE*b}{1-ROE*b}$
    * where $b$ is the Retention Ratio and $ROE$ is the return on equity


#### Determinants of Growth
###### What determines the growth rate?
* Profit Margin
    * Operating efficiency. Profit per dollar in sales.
* Total Asset Turnover
    * Asset use efficiency. Sales per dollar in assets.
* Financial Leverage
    * Choice of optimal debt ratio. Sales per dollar invested in debt.
* Dividend Policy
    * Choice of how much to pay to shareholders vs. reinvesting in the firm. Retained earnings per dollar in net profit.

#### Caveats
* Models are just simplifications and don't always indicate which financial policies are best, but you still need a plan because otherwise you're going to be lost

-----

### 4.1 The One-Period Case

* If you invest $10,000 at 5% intrest for one year, your investment would be $10,500
* The amount due at the end of your investment is called **Future Value (FV)**

#### Future Value
* $FV = C_0 * (1- r)$
* Where $C_0$ is cash flow today (time zero), and $r$ is the appropriate interest rate

#### Present Value
* If you're promised $10,000 due in one year with $r=0.05$, your investment would be worth $9,523.81 in today's dollars
    * *"how much do I have to pay today to get it to value $10,000 next year"*

* $PV = \frac{C_1}{1 + r}$, where $C_1$ is the cash flow at date 1, and $r$ is the appropriate interest rate

### Discussion
#### Q1) If plowback ratio is 1, you can say for sure that 
* $PlowbackRatio = 0$ means that $Dividends = 0$
* $PlowbackRatio = \frac{RE}{NI} = 1$ means that $RE=NI$
* Internal growth rate is possible.

#### Q3) If a firm bases its growth projection on the rate of sustainable growth, and shows positive net income as well as positive plowback ratio then:
* Debt-Equity ratio will have to remain constant while retained earnings have to increase
    * internal growth tells how much firm grows by using retained assets as the only source

#### Q4) Citigroup's interest coverage ratio is 1.5 and the industry is 10. This implies that compared to the typical bank:
* Citi is more likely to go bankrupt if the economy worsens
* $InterestCoverageRatio=\frac{EBIT}{InterestExpense}$
    * Represents the company's ability to cover long-term debt, so if it's lower than industry standards they will go bankrupt when the economy worsens

#### Q5) Bruin software has 6 million shares of stock at the end of 2020, a ROA of 15%, Equity Multiplier of 1.2, ROE of 18%, Addition to retained earnings of $20 million, and $4 million in dividend payout. What ratios do you need to calculate the EPS?
* Don't need ROA, ROE, Equity Multiplier
* $NI = RE + Dividends$
* $EPS = \frac{NetIncome}{SharesOutstanding}$
* $NI = 20 + 4 = 24$
* $EPS = \frac{24}{6} = 4$

#### Q6) 2014 COGS are 35 million, revenue is 70 million and 5 million is the 12/31/2014 inventory. Based on year end data, what are Days Sales in Inventory?
* $DaysSalesInInventory = \frac{365Inventory}{COGS}$

#### Q7) Company renegotiated the terms of its long-term debt payment plan. Their interest payments will be lower per year but the time to pay off the debt will be extended. What kind of a short term and long term effect will this have on ROE?
* $ROE=\frac{NI}{TE}$
* In short term, as interest payments go down, NI goes up and ROE goes up
* Yet, since loan term is extended, company will make interest payments in future years that it would not have made with the original plan. So, ROE for those years will be lower in the long term

#### Q8) Tibet Corporation would like to grow by 10% from 2021 to 2022. What is the External Financing Need (assume Profit Margin and dividend payout ratio stay the same from 2021 to 2022)?

##### Income Statement
|                       |      |
|:--------------------- | ---- |
| Sales                 | 1000 |
| COGS                  | 800  |
| Taxable Income        | 200  |
| Taxes (34%)           | 68   |
| Net Income            | 132  |
| Dividends             | 44   |
| Add. to Ret. Earnings | 88   |

##### Balance Sheet


|                      |      |                              |      |
|:-------------------- |:---- |:---------------------------- | ---- |
| Current Assets       |      | Current Liabilities          |      |
| Cash                 | 160  | Accounts Payable             | 300  |
| Accounts Receivable  | 440  | Notes Payable                | 100  |
| Inventory            | 600  | Total Current Liabilities    | 400  |
| Total Current Assets | 1200 | Long-Term Debt               | 800  |
| Net Fixed Assets     | 1800 | Owners' Equity               |      |
|                      |      | Stock                        | 800  |
|                      |      | Retained Earnings            | 1000 |
|                      |      |                              |      |
| Total Assets         | 3000 | Total Liabilities and Equity | 3000 |


$EFN = \frac{Assets}{Sales}*\Delta Sales - \frac{SponLiab}{Sales} * \Delta Sales - (PM * ProjectedSales) * (1-d)$

$Assets = 3000$
$Sales = 1000$
$\Delta Sales = \frac{10}{100}*1000 = 100$
$SponLiab = Short-term liabilities = 300$
$PM = \frac{NI}{Sales} = \frac{132}{1000} = 0.132$
$ProjectedSales = 1000 + \frac{10}{1000} * 1000 = 1100$
$d = \frac{Total Dividend}{NI} = \frac{44}{132} = \frac{1}{3}$

Plug all values in to get EFN of 173.2.
 
 


---

## Week 4

#### Net Present Value

* In the one-period case:
$NPV = -Cost + PV$
* You have a cash flow that's spread over time (periodically)
* You need a metric to measure how different cash flows are spread out

Say you have $1000 today
           
|-------------------|-------------------|
$1000  --- r --- $1200

$r$:
* interest rate per year
* opportunity cost of time
* time value of money
* discount rate
* market rate
* APR (annual percentage rate)

****Calculating Net Present Value of a Cash Flow****
* What is the Present Value of $1200 in t = 1?
* a.k.a How much do I need to invest today with my best opportunity to get to $1200 in 1 year?

$PV(1 + r) = FV$
$PV = \frac{FV}{(1+r)}$

****Example:****
    
$PV(1 + 0.10) = 1200$    
$PV = \frac{1200}{(1+0.10)}$
$NPV = -1000 + \frac{1200}{1.10}$


* $1200 in one year is worse than having $1200 today, the exact value of how much less is what we will find
* How much do I need to invest with my best opportunity to get to $1200 in one year? 


$NPV$
* NPV equal to zero means the project giving you the same as your market best opportunity
* Positive NPV is going to be better than market best opportunity. **Take positive NPV! :)**
* Reject any NPV that is negative, as it is giving you less than the best market opportunity

Multiple Periods
$C_0, C_1, C_2, C_3$, where $C_i$ represents the cost at $t=i$
$r_{01}$ represents interest rate from $t=0$ to $t=1$

$NPV = -C_0 + \frac{C_1}{(1+r_{01})} + $
How much money did I have to put aside today
* $PV(1 + r)(1 + r)=PV(1+r)^2=C_2$
* $PV = \frac{C_2}{(1 + r)^2}$

0 NPV means you make the same amount as you make in the market 
Positive means you make more, Negative means you make less

****Example****

1) An investment opportunity

|  |  |
| -------- |:-------- |
| 0        | -1000    |
| 1        | 200      |
| 2        | 200      |
| 3        | 1200     |

$NPV = 249$

Versus Investing $1000 in the Market to leave in 3 years with $1331

$FV = 1331 = 1000(1.1)^3$
$NPV$ is also 249.

****What happens if we invest the oney we get each year?****

|       |      |      |      |
|:----- |:---- | ---- | ---- |
| -1000 | 200  | 200  | 1200 |
|       | -200 | 220  | 462  |
|       |      | -420 |      |
| -1000 | 0    | 0    | 1662 |

Calculate the $NPV$ of this cash flow.

$NPV = -100 + 0 + 0 + \frac{1662}{1.1^3} = 249$

Regardless of how frequently you reinvest your money, as long as you invest at the market rate, you will still be making the same amount.

NPV is the same because you invest at the market rate. NPV discounts at the market rate, so you get the same market




|       |     |     |      |       |                        |
|:----- | --- | --- | ---- |:----- |:---------------------- |
| -1000 | 200 | 200 | 1200 |       | Investment Opportunity |
| -1000 | 100 | 100 | 100  | 1100  | Market                 |
|       | 100 | 100 | 1100 | -1100 | $Opportunity - Market$            |

The $NPV$ of the second row is zero.
$NPV$ of market investment is always zero.

What is the $NPV$ of the difference $Opportunity - Market$?

-----

### 4.2 The Multi-period Case

* Formula for the future value of an investment over many periods uses *compounded interest*.

$FV = C_0*(1+r)^T$ where $C_0$ is the cash flow at date 0.

-----

### 4.2 The Multi-period Case

Head Heavy Cash Flow vs. Tail Heavy Cash Flow
* Sometimes their values can appear symmetrical.
* But the NPV of Head Heavy Cash Flow is always greater because the value of cash is greater earlier in time.
* At any reasonable interest rate, Tail Heavy Cash Flow will lose value at a faster rate than Head Heavy Cash Flow.


-----

### 4.3 Compounding Periods

Compounding an investment $m$ times a year for $T$ year provides for future value of wealth:

$FV = C_0*(1+\frac{r}{m})^{m*T}$

Effective Rate is the equivalent annual rate of a frequently compounded rate.
* 12% annual rate, compounded monthly, is a 12.68% effective rate.

For example, we invest $50 for 3 years at 12% compounded semi-annually. Then, using the above $FV$ formula, we get $70.93.

To get the effective annual rate (EAR)
$50*(1+EAR)^3=70.93$
$EAR = 0.1236$
This means that compounding 12.36% annually is the same as compounding 12% semi-annually

#### Continuous Compounding
* calculate ftuure value of an investment compounded continuously over many periods
    * $FV = C_0 * e^{rT}$
    * $C_0$ is cash flow at date 0
    * $r$ is the stated annual interest rate
    * $T$ is the number of years

$\lim_{x \to \infty} (1+\frac{r}{m})^m - 1 = e^r - 1$ where $\frac{r}{m}$ is the effective annual rate.


-----

### 4.4 Simplifications

* **Perpetuity** - a constant stream of cash flows that lasts forever
    * Example: stock
    * $PV=\frac{C}{r}$
* **Growing perpetuity** - a stream of cash flows that grows at a constant rate forever
* **Annuity** - a 'truncated perpetuity', a stream of constant cash flows that lasts for a fixed number of periods
    * Example: bonds
* **Growing annuity** - a 'truncated perpetuity', a stream of cash flows that grows at a constant rate for a fixed number of periods
    * e.g. rent that you collect that grows at the interest rate


#### Perpetuity

* If you have a constant period, constant cash flow, you can estimate the current value with 

$PV = \frac{C}{(1+r)} + \frac{C}{(1+r)^2} + \frac{C}{(1+r)^3}...$ which converges to $PV = \frac{C}{r}$

* If you need the Present Value of $t_prev$, you would need to discount the Present Value of $t$ by multiplying $(1+r)$

#### Growing Perpetuity

$PV = \frac{C}{(1+r)} + \frac{C(1+g)}{(1+r)^2} + \frac{C(1+g)^2}{(1+r)^3}...$ which converges to $PV = \frac{C}{(r-g)}$

where $g$ is the rate of growth of the cash flow.

##### Example:
* expected dividend next year is $1.30, dividends are expected to grow at 5% forever. if discount rate is 10%, what is the value of this promised dividend stream?
    * $PV = \frac{1.30}{0.10. - 0.05} = 26$

#### Annuity

$PV = \frac{C}{(1+r)} + \frac{C}{(1+r)^2} + \frac{C}{(1+r)^3} +...+ \frac{C}{(1+r)^T}$ which converges to $PV = \frac{C}{r}[1-\frac{1}{(1+r)^T}]$

##### Example:
* If you can afford a $400 monthly car payment, how much can you afford if interest rates are 7% on 36-month loans
    * $PV = \frac{400}{0.07/12}(1-\frac{1}{(1+0.07/12)^{36}}) = 12954.59$

##### Example:
* What is the $PV$ of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%?
    * $PV = \sum_{t=1}^{4} \frac{100}{(1.09)^t}$
    * Use $PV = \frac{C}{r}[1-\frac{1}{(1+r)^T}]$

#### Growing Annuity:
$PV = \frac{C}{r-g}[1-(\frac{1+g}{(1+r)})^T]$

-----

### 5.1 Why Use Net Present Value?

* Accepting positive NPV projects benefits shareholders.
    * NPV uses cash flows
    * NPV uses all the cash flows of the project
    * NPV discounts the cash flows properly

NPV lets you compare different projects/investments that are in different formats


#### The Net Present Value (NPV) Rule
* Net Present Value = Total Present Value of Future Cash Flows + Initial Investment
* Estimating NPV:
    * Estimate future cash flows: how much? and when?
    * Estimate discount rate
    * Estimate initial costs

* Choose the highest NPV and reject if NPV < 0

-----

### 5.2 Payback Period Method
* How long does it take the project to "pay back" its initial investment
* Payback Period = number of years to recover initia lcosts
* Minimum Acceptance Criteria:
    * Set by management, completely arbitrary
* Ranking Criteria:
    * Set by management

#### Example



| Years | Project A | Project B |
|:----- |:--------- |:--------- |
| 0     | -1000     | -1000     |
| 1     | 800       | 200       |
| 2     | 200       | 200       |
| 3     | 200       | 800       |
| 4     | 200       | 800       |
| 5     | 200       | 800       |

The payback period of A is 2 years.
Payback period of B is 2 and $600/800=3/4$ years, that is 2 years and 9 months.

This method does not consider the time-value of money.

#### The Payback Period Method
* Disadvantages
    * ignores the time value of money
    * ignores cash flows after the payback period
    * biased against long-term projects
    * requires an arbitrary acceptance criteria
    * a project accepted based on the payback criteria may not have a positive NPV 
        * Why is this the case
        * say you put 1000 bucks, get 200 and 800 in the second year, and your criteria is 2 years. This passes the criteria, but it doesn't have positive NPV because of the discount rate. You may end up accepting negative NPV projects with this 

* Advantages
    * Easy to understand
    * Biased towards liquidity
    * Helpful in assessing lower management

Payback period likes early cash, but doesn't look at time value of money.

This leads to...


---
### 5.3 The Discounted Payback Period
* How long does it take the project to pay back its initial investment, taking the time value of money into account?
* Decision rule: Accept the project if it pays back on a discounted basis within the specified time (only accepted if the initial investment is paid back early)
* By the time you have discounted the cash flows, you might as well calculate the NPV


##### Example: Assume that the market rate is 10%

| Years | Project A | Project B | Discounted Cash Flow of A | Discounted Cash Flow of B |
|:----- |:--------- |:--------- |:------------------------- |:------------------------- |
| 0     | -1000     | -1000     | -1000                     | -1000                     |
| 1     | 800       | 200       | 800/1.1 = 727             | 182                       |
| 2     | 200       | 200       | 165                       | 165                       |
| 3     | 200       | 800       | 150                       | 601                       |
| 4     | 200       | 800       | 137                       | 546                       |
| 5     | 200       | 800       | 124                       | 497                       |

Payback period of A is 727 + 165 = 892 2 years and 108/150=0.72 about 9 months.
Discounted payback period is always longer than payback period.



---

### 5.6 The Profitabillity Index (PI)

$PI = \frac{Total PV of FutureCashFlows}{Initial Investment}$

Minimum Acceptance Criteria: Accept if $PI > 1$

Ranking Criteria: Select alternative with highest PI


#### Using PI on Mutually Exclusive Projects


| Project      | A     | B     | A-B   |
| ------------ |:----- | ----- |:----- |
| Investment   | -100  | -70   | -30   |
| Year 1       | 120   | 20    | 100   |
| Year 2       | 80    | 140   | -60   |
| NPV( at 10%) | 75.21 | 63.88 | 11.32 |
| PI (at 10%)  | 1.75  | 1.91  | 1.38  |

Mutually exclusive means only pick one project

Comparing A to B-Market. If PI of A-B is greater than 1, pick A.

#### The Profitability Index
* Disadvantages 
    * Problems with mutually exclusive investments
    * Just because a project has a higher PI than another one, it doesn't mean it has the higher NPV

#### NPV vs. PI vs. Payback Period Method

* NPV
    * Great for ranking projects against your best market option
    * Is flawed if your data is flawed
    * Looks at all sides of cash flow

* Profitability Index

* Payback Period
    * Naive when it does not consider the time-value of money
    * Discounted Payback Period is a better option
    * If a project is accepted by the discounted payback period method, it will be accepted by the payback period method.

if a project has negative npv, can i be sure that the discounted payback period is going to reject this project what do you guys think? This means that the positive cashflow, when discounted, it is always less than the present value. So, the discounted payback period will always reject.

if a project is accepted by discounted payback period method, will it always be accepted by payback period method? Yup.

---

## Week 5

### 5.7 The Practice of Capital Budgeting
* Varies of industry
* Turning the cash flow into a rate
* NPV, Payback Period, Discounted Payback Period, Profitability Index
* The most frequently used technique for large corporations is NPV or IRR (return on investment, ROI)

Company is considering taking on a project. How would the financial statements be affected by this project?

**Sunk Costs** - You cannot include 'sunk costs' i.e. costs spent analyzing and building this project in the **past**. We only consider the current investment and future cash flow.

**Opportunity Cost** - Should always show up in your cash flow. If there's something that will be lost because of what you're doing in the future, that should be in the cash flow

**Allocated Cost** - If there's already an existing asset that you're using to produce A, B, C, and you plan to use the asset to make D, then do not include D in the cash flow.

Do not include assets that are already being used for other projects and can be reused for this new project.

**Side Effects**
* Erosion - will introducing a new project harm the sales coming from existing projects
* Synergy - will this new project boost the sales for another project
* Cannibalism - when a new product completely kills the other (e.g. iPhone killing iPod)

Costs are always easier to estimate than Sales.

Estimate depreciation of Fixed Assets.

Estimate Salvage Value = $Market Value - t(Market Value - Book Value)$
* What's the market value if you took this machine and depreciate it
* If the Book Value is higher than the market value you purchased the machine at, you get a tax refund.

Anticipate NWC Increase and recover
* You should anticipate needing more NWC to embark on a new project.
* 


#### Capital Budgeting Example

* -$16,000 Capital spending
* -$200 NWC
* The decrease in NWC over the course of the project should be recovered in the end.
* Depreciation is written down to deduct taxes paid. Depreciation is rewritten back into the Operating Cash Flow.
* After calculating the NPV
* Project financing can come from 1) Retained earnings -> Cash 2) Debt or 3) Issuing Stock
    * All 3 require financing cost, which is included in NPV
    * If my financing cost is known initially, for example, as interest from debt, I can ded


---

### 7.2 Monte Carlo Simulation

* Further attempt to model real-world uncertainty
* Simulation of capital-budgeting projects
* Step 1: Specify the Basic Model
    * what variables go into it
* Step 2: Specify a Distribution for Each Variable in the Model
    * What is the probability of getting a certain value in our model 
    * Determine a range for the variable (highest/lowest possible values), and calculate the probability of each value
* Step 3: The computer draws one outcome
* Step 4: Repeat the procedure 
* Step 5: Calculate NPV


#### Accounting & Financial Break-Even Points

* Accounting Break-even point
    * Revenue = $\overline{P}*Q$
    * $\overline{P} * Q - Deprec. - VC * Q = 0$
    * $Q$ = Number of units sold
    * When Revenue - Fixed Costs - Variable Costs - Depreciation results in a taxable income of 0.
    * $P*Q = Revenue$ to get breakeven, you need fixed costs (whatever you need to produce machine, depends on $P$) and variables costs (depends on $Q$), Depreciation, Pretax income is zero, so tax is zero. then, net income is zero. how many units should i produce per year ($Q$) so that i should achieve all these zeros. Then, $P*Q-FC$, where $FC$ is fixed costs
    * $Q=\frac{FC + Dep}{\overline{P}-VC}$
    * Covers all of our costs, *except the financing costs*
    * Does not take into account time-value of money
* Financial Break-even point
    * $(\overline{P} * Q - FC - VC * Q - Dep)(1-t)-(EAC - Dep)$
        *  $EAC$ is the equivalent annual cost
        *  $EAC = \frac{r * NPV}{1-\frac{1}{(1+r)^T}}$ (r = Discount Rate)
        *  Then, solve for $Q$ and that is your *financial break-even point*
        *  $P$ represents financial break-even *price*
* **Financial break even quantity will be higher than accounting break-even quantity** as long as the company discount rate is greater than zero.

![](https://i.imgur.com/EoaEECH.jpg)


---

### Bond Market

* Unlike stocks, Bonds have a promised cash flow at a certain date
* A **bond** is a document that tells you 3 things:
    * Face Value - the amount of money that is promised to be paid at the bond's maturity date. In the U.S., the face value is $1000
    * Maturity Date - this is fixed
    * Coupon - some fixed amount.
        * $Coupon Rate = \frac{CouponValue}{FaceValue}$
        * Coupon Rate is not the same as Interest Rate. This is because Bonds can be bought and sold before its maturity date, and its face value might not be the same as the amount you profit from it.
* The owner of a bond can change, and whoever holds it will receive its promised payments.
* **Pure discount bonds** do not pay coupons.
* **Arbitrage** is the purchase of an undervalued asset to immediately sell it at a higher price
    * Arbitrage promises that assets are always fairly valued in the market

A **deep market** is required for us to find a value that is going to represent what's happening out there. 
* With a lot of buyers and sellers on either side, the price reflects all pieces of information on the market.

Yield a.k.a. Yield to Maturity: When I buy a bond, at that point, if I were to keep it until maturity, what will be the rate on average per year that I make on this bond?



Example:
$BondPrice=982=\frac{80}{1+r}+\frac{1080}{(1+r)^2}$
$r=9\%$ 
$r$ is called Yield (Yield to maturity)


A **par bond** is sold at face value.

A **premium bond** is sold above face value. In this case, the yield is less than the market rate.

A **discount bond** is sold below face value. In this case, the yield is higher than the market rate.

A bond **overshoots** when the bond value climbs above $1000.

$Bond Price (Value) = \frac{C}{r}(1-\frac{1}{(1+r)^T})+\frac{FV}{(1+r)^T}$
$C$ = coupon/annuity
$r$ = market rate. It will match the bond's yield when the bond is sold at its value
* In this class, we will refer to Bond Value and Price as the same thing.

Ex. Suppose we have a bond issued at time $t=0$ with Face Value of $1000 for 10 years with coupon payments of $80 per year. The market rate is 8%. What is the price of this bond?

$BondPrice = \frac{80}{1.08} + \frac{80}{1.08^2} + ... + \frac{1080}{1.08^{10}}$

#### Comparing Bonds

Bond 1: 1-year, FV = $1000, CR = 10%
Bond 2: 30-year, FV = $1000, CR = 10%

Market Rate is 10%.

The price of both bonds is $1000.
But if the market rate goes down, the long-term bond's value goes up by much more. We calculate the price of a bond by discounting its coupons and face value by the market rate. When the market rate drops, the price of a bond becomes much higher.

**The price of the bond and its yield are negatively correlated.**

**Long-term bonds are more risky than short-term bonds because they are more heavily impacted by changing interest rates.**

So, long term bonds sell at a lower price.

The **yield curve** plots the price vs. maturity period relationship of U.S. Treasury bonds.

The yield curve shows that long-term bonds have a higher yield. The yield curve always flattens or inverts before a recession (At least in the U.S., which is a domestic-driven economy).

Promised Yield - the yield assuming the risk of default is zero
Expected Yield - the yield taking into account the risk of default and the haircut
Haircut - the amount of money you lose when a bond goes into default
$ExpectedYield = p*PromisedYield + (1-p)*(PromisedYield-Haircut)$


----


with bonds, you have certainty of company um promises and committing to a certain payment plan. well, if the company goes under, things are not going to happen. but if the company is large, like ibm, then you're going to have certainty. This is why the bond market is considered safer than the stock market

bond tells you face value, (amount of money that is promised to be paid at a certain date, this value is fixed), maturity date (fixed value), coupon (amount that is fixed)
* Coupon Rate is $\frac{coupon}{facevalue}$

if the company is offering the bond, it's called a corporate bond. 
Example: Promises to pay $1000 in two years. It promises to pay 80 bucks along the way. Bonds that don't pay any coupon are called pure discount bonds. they are selling at a discounted bonds.

no coupon bonds are called pure discount bonds

but let's focus on the ones that have coupons, they will usually have coupons. Coupons are going to be given every year. Everything in bonds is fixed. it's not a movign target. stocks are a moving target. 

so, when you're first issuing the bond, it's really important to choose your coupon rate. 

for the buyer, the amount of money they are making. if you do not hold the bond until maturity, then you have the potential to get more/less money.

Coupon rate is not the same as interest rate. not the same.

Difference between bond price and bond value. the price that it converges to is what we're trying to calculate.

Oooh this is underpriced, this is underpriced, let's jump on it because it's gonna g oup. if the arbitrage: makes sure that things are accurately priced. They buy and sell it at the same time so that they get a little bit of profit

here's a picasso. when i first went to the us i tried to buy a mural. 

alright im selling a picasso. how much u wanna pay. let's say im giving you 10000 each year lets start the bidding at five thousand, hah, 0000. let's mae it a game of throns. you wanna get that. ohhh okay fine ill watch it

the deep market is required for us to find a value that is going to represent what's happening out there. a lot of buyers and sellers on either side, for the price to reflect all pieces of information on the market.

Yield of the bond: when i buy a bond, at that point, if i were to keep it until maturity, what will be the rate on average per year that i make on this bond

Example:
$BondPrice=982=\frac{80}{1+r}+\frac{1080}{(1+r)^2}$
$r=9\%$ 
$r$ is called Yield (Yield to maturity)

i say on average because when i invest by buying this quality, in the first year im gonna get 80 bucks. in the second year, im gonna get 1080 bucks. instead of getting a specific rate every year, let's find a rate that can be averaged to every year.

what's gonna happen to the bond price if the market is offering 8%. The price will go up. when will the price stop increasing? exactly at 1000. Because at 1000 dollars this bond will yield 8%

$1000=\frac{80}{1+r}+\frac{1080}{1+r}^2$
$r=8\%$
Par-bond: sold at face value
usually, in the economy, things overshoot until they find equilibrium. So if the price goes up to $1018, what is this bond yielding now?
$1018=\frac{80}{1+r}+\frac{1080}{(1+r)^2}$
$r=7\%$

$CR=yield$ when it's a par bond

if bond is selling a ta higher price than the face value, it is called a premium bond. and the yield will be less than the coupon rate

discount bond, when it's selling less than face value. in this case, the yield is greater than the coupon rate

$BondPrice (value) = \frac{C}{r}(1-\frac{1}{(1+r)^T})+\frac{FV}{(1+r)^T}$, where $r$ is market rate, then yield will match the market rate if the bond price is selling at its value

For a bond will FV=$1000, matures in T years with a coupon of C per year

assume you have a bond issued at time = 0 and gives 80 dollars per year, 10 year bond. 2-10 year bonds are considered medium-term. The market is 8%. face value at the end is $1000. what is the price of the bond. 

1000->price if market is 8% @ t=0
$BondPrice=\frac{80}{1.08}+\frac{80}{1.08^2}+...+\frac{1080}{1.08^{10}} = 1000$

if market reate goes up, can you sell the bond at 1000? you get a lower price, because only

Market is higher, yield is lower. therefore, you have to sell bond at a lower price

Two Bonds:


1. 1-year, FV = $1000, CR=10%
2. 30-year, FV=$1000, CR=10%

Market is 10%. What's the price of the first bond

**The yield and the price are negatively correlated**

the bond price will change so that it matches the market

but the yield of long term bonds are higher (yield curve: the yields of long-term short-term government treasury bonds) 

Every investor looks at the yield curve every morning to see which long/short-term bond is in demand more. 

ir the yield curve flattens/inverts, then there is a high probability that there is going to be a recession 

---

### Discussion Week 5
Annuity is a constant period cash-flow. We can calculate the Future Value of an Annuity using
$FV = \frac{C}{r}((1+r)^T-1)$

A $1000 annuity over 5 years has a FV of
$1000(1+0.05)^1+1000(1+0.05)^2+1000(1+0.05)^3+1000(1+0.05)^4+1000(1+0.05)^5=5525.64$

We can obtain the Present Value of an Annuity with
$PV = \frac{C}{r}[1-\frac{1}{(1+r)^T}]$

A Perpetuity is an annuity with no termination date.

We can obtain the Present Value of a Perpetuit with
$PV = \frac{C}{r}$

---
## Week 6

long term bonds are affected more by unknowns (like how the market changes)
this is why investors only buy long-term bonds if they are premium bonds (extra yield)

Determinants of Bond Value
1. term structure: longterm bonds will offer a premium yield
2. Inflation. whenever you have a fixed cash flow (bondn), inflation will affect the purchasing bond. so by investing long-term bond, the purchasing power will be lower in the future because of inflation. So, longer-term bonds will offer an inflation premium
3. municipalities bond. state bond will give more yields. state bond are not taxed. federal bonds are taxed.
4. Liquidity

#### Default Risk / Premium Example
Market rate: 9%
FV: 1000
CR: 80/1000
The bond price

965 in year 0, 1080 year 1, market price is 9%
we expect the price to be 1080/1.09=990, so there's probably a default
965(1.09)=(1-p)1080+p540 => p=10%, where p is the probability that we go into default

rating agencies: they try to get a probability of default for each bond
investment grade bonds: As Bs 
Junk bonds: B- and below

#### Current yield & capital gains yield
10 year bond, coupons are 100, face value is 1000
CR=10%, market=10%
$BP_{t=0}=1000$(selling at par)
after the first coupon collection, $BP_{t=1}=1000$

Yield (YTM)
1. current yield: coupon is the price of the bond
at t=0, current yield is 100/1000=10%. at t=1, current yield is 100/1000=10%. as long as the market stays the same, current yield will be the same
3. capital gains yield
$cgy = \frac{P_t-P_{t-1}}{P_{t-1}}$
$\frac{1000 - 1000}{1000}=0\%$



---

### Week 6
#### Stock Valuation
price and yield are negatively correlated for bonds. bonds are also more stable, less risks

Case 1: Zero Growth
- assume that dividends will remain at the same level forever $Div_1 = Div_2 = Div_3 = ...$
- value of zero growth stock is just hte present value of a perpetuity 
- $P_0 = \frac{Div}{R}$

Case 2: Constant Growth
- assume dividends grow at constant rate forever
- value of this stock equals present value of a growing perpetuity
- $P_0=\frac{Div_1}{R-g}$




---

## Week 7

There's two diff approaches for stock valuation 

market is well known, product is well known, so the company knows what the sales will be and how fast they grow. 

#### Case 2: Constant Growth

Assume that dividends will grow at a constant rate forever

$Div_i=Div_0(1+g)^i$

Since future cash flows grow at a constnat rate forever, the value of a constant growth stock is the PV of a growing perpetuity
$P_0=\frac{Div_1}{R-g}$

#### Case 3: Differential Growth
* if company is young, the growth is not constant yet
* assume dividends grow at different rates in future and then grow at a constant rate thereafter
* To value a Differential Growth stock, we need to:
    * estimate future dividends in the foreseeable future
    * estimate future stock price when the stock becomes a Constant Growth Stock
    * Compute total present value of the estimated future dividends and future stock price at the appropriate discount rate
* Assume dividends will grow at rate $g_1$ for $N$ years and grow at $g_2$ thereafter
* Present value of growing perpetuity of T years growing at rate $g_1$
    * $P = \frac{C}{R-g_1} * [1- \frac{(1+g_1)^T}{(1+R)^T}] + \frac{\frac{a}{b}}{c} $

#### 9.2 Estimates of Parameters
* value of a firm depends upon its growth rate $g$ and its discount rate $R$
    * $g=RetentionRatio*ReturnOnRetainedEarnings$
    * $Earnings Next Year = Earnings This Year + ROE * (b)(EarningsThisYear)$
        * ROE is a good approximation for return on Retained Earnings
    * $1+g = 1 + b * ROE$
    * Recall $P_0 = \frac{Div_1}{R-g}$
    * Once we approximate g, we can estimate the return on this perpetuity


#### 9.3 Growth Opportunities
* opportunities to invest in positive NPV projects
* Value of a firm can be conceptualized as the sum of the value of a firm that pays out all of its earnings as dividends plus the NPV of the growth opportunities
* $P=\frac{EPS}{R}+NPVGO$


#### Retention Rate and Firm Value
* increase in the retention rate will:
    * reduce the divided paid to shareholders
    * 

if $ROE > R$, then $\frac{\partial P}{\partial b} > 0$

#### 9.4 Price-Earnings Ratio
* Many analysis frequently relate earnings per share to price
* $PE Ratio=\frac{PricePerShare}{EPS}$
* A firm's PE Ratio is positively related to growth opportunities and negatively related to risk $(R)$


---

## Chapter 10: Risk and Return

how to calculate return on investment, stdev of investment (how much it jiggles)

### 10.1 Returns
* Dollar Returns
* sum of the cash received and the change in value of the asset, in dollars
* $DollarReturn=Dividend + \Delta MarketValue$

### 10.2 Holding Period Return
* $(1+R_1) * (1+R_2) * (1+R_3) *...$
* geometric average will always be less than the arithmetic average

#### Inflation
* real interest: r (increase in purchasing power)
* Nominal interest: R (increase in dollar amount)
* Inflation rate: f
* Fisher Formula: $1+r=\frac{1+R}{1+f}$
* Approximation for real return: $r=R-f$

### 10.3 Return Statistics
* The history of capital market returns can be summarized by describing the:
    * average return (mean, average, expected value are used interchangeably): $\overline{R}=\frac{R_1+...+R_T}{T}$
    * standard deviation of these returns: $SD=\sqrt{VAR}=\sqrt{\frac{(R_1-\overline{R})^2+...+(R_T-\overline{R})^2}{T-1}}$

### 10.4 Average Stock Returns and Risk-Free Returns
* The *Risk Premium* is the added return resulting from bearing risk
* one of the most significant observations of stock market data is the long-run excess of stock return ovre the risk-free return
* The Risk-Return Tradeoff
    * with higher *Annual Return Average*, you get higher *annual return standard deviation* (this is the risk). (higher return means higher risk)

### 10.5 Risk Statistics
* Measures of risk that we discuss are variance and standard deviation 
* if you get a large enough sample drawn from a normal distribution, it looks like a bell-shaped curve
* 68.26% of values are 1 standard deviation away, 95.44% of values are 2 standard deviations away, 99.74% that the values are 3 standard deviations away

### 10.6 More on Average Returns
* Arithmetic average - return earned in an average period over multiple periods
* Geometric average - average compound return per period over multiple periods
* geometric average is always less than arithmetic unless all the returns are the same 

Geometric Average Example:
$(1+R_g)^4=(1+R_1)*(1+R_2)*(1+R_3)*(1+R_4)$, then solve for $R_g$



---
## Portfolio Analysis

#### Two Stock Case

Portfolio size = $100


|                          A                           | B                |
|:----------------------------------------------------:|:---------------- |
|                   $\overline{r_A}$                   | $\overline{r_B}$ |
| $\sigma_A$                                           | $\sigma_B$       |
| $w_A = \frac{Dollars Invested In A}{Portfolio Size}$ | $w_B=1-w_A$      |
|                                                      |                 |
|                                                      |                  |

$ExpectedReturnOfPortfolio = weightedaverageofindividialreturns$

Higher return (y-axis) -> Higher risk (x-axis)
* A has lower return and risk than B
* If P had $w_A=0.5$ and $w_B=0.5$, then $\overline{r_P}=0.5\overline{r_A}+0.5\overline{r_B}$
* Margin Investing (weight can be less than 0 and greater than 1)

Risk Level of the Portfolio we just formed:
* when you combine different assets, the return pattern is very predictable. However, the **risk level** can do some unexpected things
* $\sigma_P^2=w_A^2\sigma_A^2+2w_Aw_BCov(A,B)+w_B^2\sigma_B^2=w_A^2\sigma_A^2+2w_Aw_B\sigma_A\sigma_B\rho_{A,B}+w_B^2\sigma_B^2$ (this will be from -1 to 1)
* Covariance $Cov(A,B)$ $\sigma_{A,B}$
* $Cov(A,B)=\frac{(R_i^A-\overline{r_A})(R_i^B-\overline{r_B})}{r-1k}$
* To make it unit-dependent, we use **correlation** $\rho_{A,B}=Corr(A,B)=\frac{\sigma_{A,B}}{\sigma_A\sigma_B}$
* $-1\le Corr(A,B)\le 1$
* If A and B are positively correlated, then they are going in the same direction (knowledge of A gives you information on B)

Risk is represented by $\sigma$
$\sigma_p^2=(w_A\sigma_A+w_B\sigma_B)^2$

Combining two assets, you can reduce the risk of both of them. We do this because the two assets have low correlation (With $\rho=-1$, you can have 0 risk. Obviously, this is impossible because you can't find two assets that are perfectly negatively correlated. Market risk prevents this from happening)
* Opportunity set, different lines representing the risk
* MVP: Minimum variance portfolio, the correlation with lowest risk
* **Try to get low correlation when picking your assets**
* **Go to the NorthWest**

![](https://i.imgur.com/fbigqsd.jpg)


The minimum standard deviation we can get will be less than the individual standard deviations when $\rho < 0$. But it can also happen with low positive $\rho$.


#### Calculation of Minimum Variance Portfolio

A: $\overline{r_A}, \sigma_A, w_B$

B: $\overline{r_B}, \sigma_B, w_B$

$w_A+w_B=1$
$w_B=1-w_A$

$Risk=\sigma_p^2=w_A^2\sigma_A^2+2w_A(1-w_B)\sigma_{A,B}+(1-w_A)^2\sigma_B^2$

*Minimize the risk to find the weights of A & B in theportfolio &rarr; MVP*

$\frac{\partial \sigma_P^2}{\partial w_A}=2w_A\sigma_A^2+2\sigma_{A,B}(1-2w_A)-2(1-w_A)\sigma_B^2$

Set this equal to zero and solve for $w_A$, and then plug that into $w_B=1-w_A$. Now you know the weights of the portfolio.

### What if we have more than 2 stocks in the portfolio?

![](https://i.imgur.com/bDRavPv.jpg)

Expected Return of a portfolio that is formed by more than 2 stocks:
* $E(r_p)=\overline{r_p}=w_A\overline{r_A}+w_B\overline{r_B}+...+w_Z\overline{r_Z}$



|     | A   | B                    | C   | ... | ZB  |
|:--- |:--- |:-------------------- | --- | --- |:--- |
| A    |  $w_A^2 \sigma_A^2$   | $w_Aw_B\sigma_{A,B}$ |     |     |     |
|  B   | 
   |                      |     |     |     |
|  C   |     |                      |     |     |     |

Everything besides the diagonal will be the covriances
The diagonal represents the individual risk values (not that important)

#### Example

![](https://i.imgur.com/wWlbcEu.jpg)



|                | A   | B   |
|:-------------- | --- |:--- |
| $\overline{r}$ | 5%  | 10% |
| $\sigma$       | 10% | 20% |
| w              | 80% | 20% |


$\sigma_{A,B}=-0.005$
$\rho_{A,B} = \frac{\sigma_{A,B}}{\sigma_A * \sigma_B}=-0.25$
$E(r_p)=(0.8)(0.05)+(0.2)(0.1)=6\%$

$\sigma_p^2=(0.8^2)(0.1^2)+2*0.8*0.2*(-0.005)+(0.2^2)(0.2^2)$ &rarr; $\sigma_p=8\%$

We can see that by combining these two stocks, we reduced the risk level to 8% and increased return by 6%

#### The Source of Risk
1. Systematic Risk &rarr; Market Risk
2. Idiosyncratic Risk &rarr; Company-specific risk

Market risk always stays constant. The more assets you have in your portfolio, idiosyncratic risk goes down

![](https://i.imgur.com/d4JMtb0.jpg)


Beta ($\beta$) Risk: The risk of an individual stock compared to the market.

Let's define the market: 

![](https://i.imgur.com/H5RjM0r.jpg)

#### Two Rules of Investment
1. Diversification: Combine stocks that have low correlation
2. Separation: separate the decision to pick "which risky portfolio" from the "final portfolio" that you will invest in

![](https://i.imgur.com/EaMYHUZ.jpg)


#### Go Back to Beta

Graph market return (x-axis) vs. stock i return (y-axis) for different data points (each data point represents a different date)

Oh, i wonder what is the relationship between the market return and the stock return. when i look at the scatterplot, it's all over the place. but obviously this is a sample, i wonder how these numbers will congregate when we fit a line through it (linear or non-linear). Use least squares to calculate a linear line by making the points closest to the line that you pick.

$slope=\frac{Cov(r_i, r_m)}{\sigma_m^2} = \beta$

We are looking at the risk, in **comparison** to the market.


A high Beta ($\beta > 1$) implies that the stock amplifies the motion of the market.
A low Beta ($\beta < 1$) implies that the stock dampens the motion of the market.
If the Beta is 1 then it's the same as the market

$\beta_p=w_1\beta_1+...+w_n\beta_n$ 

#### Capital Asset Pricing Model (CAPM)
$\bar r_i = r_f + \beta_i(\bar r_m - r_f)$
$\bar r_i$ = expected return of stock $i$
$\beta_i$ = Beta of stock $i$
$\bar r_m$ = expected market return
$r_f$ = risk-free rate


---

## Chapter 11

systematic risk is the entire portfolio, whereas unsystematic risk is represented by individual assets

unsystematic risk can be diversified away, meaning increasing the size of your portfolio

Total Risk = systematic risk + unsystematic risk

Capital Market Line with the steepest slope is the best

