TIO4146: Finance
Tips
important pages in the book
P 118: Formelen til CAAR
P 154: Tradeoff theories
P 174: Formelen til Project decisions's oppgavene
P 254: Option pricing in continuous time
Fundamentals
Time value of money
The value of money decreases as time passes by. Main two reasons:

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

Unspent values can be invested and thus generate more value. One seed may generate hundreds of new seeds in a year if sown today.
Perpetuity
perpetuity = annuity with infinite number of payments,
Accounting representation
Note that the accounting representation of a firm may be unsuitable to make financial descisions. The first mostly focuses on where values are tied up, while the latter focuses on the actual cash flows.
Depreciation
Fixed assets are normally depreciatied over a their lifetime. This makes sense as their aquirement only converts value from cash to fixed assets  the real loss of value happens as the assets age.
However, when determining whether to proceed with a project, the actual cashflows are more relevant, thus the entire cost is treated as a cash flow at the time of aquirement. The loss of value is reflected in a lower cash flow the opposite way around when the asset is sold.
Do however note that the depreciation is deducted when calculating taxes, and thus affects the cash flow through taxes!
Irrelevant information
The financial statements are required to show all aspects, even those not relevant to a decision, for example money already spent. This is of course irrelevant to an investment decision and should be disregarded.
Changes in capital
Financial statements often include working capital, but not the changes in working capital. These changes are in fact cash flows and must be included when calculating Net Present Value (NPV).
In other words, this capital is tied up in the project and thus does not generate interests (as it would otherwise do).
Utility
Assumptions made:
 People are greedy, so more is always better than less
 Each additional unit gives less utility than the previous
 Peoples preferences are wellbehaved; their preferences form a partial order
This results in utility functions
$U(W) = ln(W)$ $U(W) = \alpha + \beta W  \gamma W^2$
Note that
Indifference curve
Given two resources,
Risk aversion
The utility curves from the above assumptions also leads to risk aversion.
Example
If one expects
This follows from the nonlinearity of the utility function. The amount
The difference
Coefficients
The risk is dependent on the curvature (second derivative) of the utility function.
ArrowPratt absolute risk aversion coefficient:
Corresponding relative risk aversion coefficient:
Efficient Market Hypothesis (EMH)
The Efficient Market Hypothesis states that financial markets are informationally efficient.
 Significant underperformance and insignificant overperformance don't contradict the EMH while the underreaction or overreaction to news do.
 The preevent CAAR drifts do not test the EMH.
 The EMHs predicts that funds cannot consistently outperform the market, i.e. deviations from riskadjusted expected returns are random.
It comes in three variants: the weak, the semistrong and strong efficient market hypotheses.
Weak  histrical
Market efficiency occurs when all past price histories are fully reflected in current prices.
Tests of technical analyses are often weak form if only using perceived patterns in plotted past prices.
Semistrong  public
Under the semistrong form of market efficiency current prices fully reflect all publicly available information; in addition to price histories this includes financial statements, surveys of investor sentiment, announcemnt, articles in the (financial) press, product, industry and macroeconomic data, etc.
Strong  private
The strong form claims that prices on traded assets reflect all information (even private), and prices instantly change to reflect any new information. Fund performance is generally considered a test of strong form efficiency, because the models, databases and strategies that funds use are not public 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
Typically, projects will cost a certain onetime, upfront amount of money
Now all you have to do is to calculate the WACC (Weighted Average Cost of Capital) of the project.
Alternatively, the APV (Adjusted present value) could also be used for NPV, which is preferred to WACC in the case that the leverage ratio of the leveraged company is constantly changing during the duration of the investment.
Calculating Weighted Average Cost of Capital (WACC)
The formula for calculating Weighted Average Cost of Capital(WACC) is:
$ 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,
Calculating $ r_E $ using Capital Asset Pricing Model (CAPM)
The formula for CAPM/Security Market Line is
$ r_E $  Cost of equity
$ r_f $  Riskfree interest rate
$ r_m $  Market rate
$ \beta $  Unsystematic risk
$ (r_{m}  r_{f}) $  Market risk premium
the difference of SML and CML lies in that: CML is pricing relation for efficient portfolios; while SML valid for all investments, incl. inefficient portfolios and individual stocks; CML prices all risks while SML only prices the systematic risk.
The formula for CML/Capital Market Line is
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 unlever and relever as necessary. Here are some formulas that help in doing this.
MilesEzzell WACC calculation
When debt of a project will be periodically rebalanced, you can use MilesEzzell.
$ D $  Debt in dollars
$ V $  Total value in dollars
$ T_C $  Corporate tax rate
$ r_a $  Opportunity cost of capital (100% equity, i.e. unlevered)
$ r_d $  Cost of debt
M&M formula
The tax part (
$ 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
M&M formula with
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 (OCC), or return on assets (
Note: When estimating something for a given project only use the values belonging to that project.
I.e. the
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 $r_a$ and WACC for the market
If there's several existing entities or projects in the market for which you have been given either
If debt is continuously rebalanced
Use either of the formulas below, depending on what information is available.
Unlevering:
M&M Formula without the tax part:
Use MilesEzzel or the definition of WACC to find it.
If debt is rebalanced periodically
Rewritten for
Where
Use MilesEzzel to find the WACC.
If debt is permanent & fixed & predetermined
3. Calculate NPV using WACC
Adjusted present value (APV)

Step 1. Calculate the base case value of the project as if it is all equity financed (Unlevered) and without sideeffects, the present base case value is dicsounted at OCC
$r_a$ . 
Step 2. Then, calculate all the sideeffects, include tax advantage (from a given level of debt by discounting tax shield at the cost of debt
$r_d$ ) and issue cost at$t_0$ . 
In the end, the APV = PV of base case + PV of tax advantage – issue cost.
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 BlackScholes model, and American options (options that can be exercised at any time until the maturity date) should be priced using the Binomial options pricing model.The american option without exercising is alive while it turns to dead after exercise.
 Put:the right to sell at exercuse price, expecting the price decrease.
 Call:the right to buy at exercuse price, expecting the price increases.
Moneyness  Put  Call 
In the money  
At the money  
Out of the money 
 Payoff = Value of the Option =
$V_t$  Profit = Payoff  Future value of option cost =
$V_t  V_0 e^ {rt} $ .
position & option  payoff at maturity 
long a put  
long a call  
short a put  
short a call 
BlackScholes model:contious time
Use this when you want to calculate the price of a Europeanstyle call option.
Here are the formulas you will need. When calculating European puts, use
$ T $  Time to maturity in fraction of interestgiving periods (e.g. T=90/365=0.247 if 90 days to maturity and interest is given for one year)
$ S_0 $  Spot price of the asset, i.e. what the asset costs now
$ X $  Strike price, i.e. what can the option be bought/sold for at maturity
$ r $  Riskfree rate (annual rate, continuously compounding)
$ \sigma $  Volatility of returns of the asset
To macth discrete and continuous time volatility
to replicate a hedging portfolio with a option,portfolio=a fraction
Binomial Options Pricing Model (BOPM):discrete time
Use this when you want to calculate the price of an Americanstyle option. Here are the formulas you will need.
$ p $  probability of going up, also called risk neutral probability
$ 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 all steps
$ n $  The number of steps/moments.
$ r $  Riskfree interest rate
to replicate a hedging portfolio with a option,portfolio=a fraction ∆ of the stock + a risk free loan of D,
First, make a binomial lattice with
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:
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 rightmost leaf nodes. These nodes should contain the option value at that point, calculated as
Now, all that is remaining is to calculate the values of the empty nodes from right to left. The value in each node,
Or you could use this formula:
Note it uses
Continuing our example, assuming
The root (leftmost) node of the lattice is the binomial option price,
Matching discrete and continuous time volatility
small time interval of american step
PutCall Parity
Putcall 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 $  Current price of a call
$ P $  Current price of a put
$ D $  Discount rate (typically riskfree interest or similar)
$ F $  Forward price of the asset
$ X $  Strike price
Option positions
Short straddle
A good strategy when you expect the stock price to have low volatility. The short straddle will give maximum profit if the price stays the same as today.
You sell a put option and sell a call option (short put and short call) with an atthemoney exercise price. This option position has no upfront cost, as you only sell two options. The downside is that if the stock price changes a lot in the future, there is a possibility for infinite losses.
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Long straddle
You buy a put option and buy a call option (long put and long call). This is a good strategy if you expect high volatility in the stock price.
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Butterfly spread
Buy two call options – one at an inthemoney exercise price and one outofmoney. Then sell two call options for an atthemoney exercise price. This has an upfront cost of the cost of the two long options minus the cost of the two short options. The advantage of a butterfly spread over a short straddle is the reduced risk in case of an increased stock price. The downside compared to the short straddle is a lower maximum profit.
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Collar
Also called "SplitStrike Conversion"
Buy an outofthemoney put option and sale an outofthemoney call option (long put and short call). It reduces volatility because it ‘cuts off’ the high and low tails of the stock’s returns.The short call caps the possible gains from the stock (provides a ceiling), but it generates cash. The cash is used to increase the return of the position and to finance the put, which insures against a (large) loss of stock (provides a floor). So a part of the upware potential is given up to reduce the downside risk, resulting is lower volatility.
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Tradeoff Theory of Capital Structure
The tradeoff theory of capital structure states that as the debt/equity ratio increases, there is a tradeoff 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.
According to Tradeoff Theory of optimal structure,
 All changes should be in the direction of the optimun and lead to an increase in firm value, e.g. a positive CAAR (Cumulative Average Abnormal Return), a positive price reaction.
 General purpose assets can easily be redeployed by other companies (a high resale value) and lose little value in a bankruptcy/financial distress, they have low bankruptcy costs and such assets can carry much debt. Firmspecific assets have limited or no value to other firms, and they have high bankruptcy costs and can carry little debt.
 Rented and leased equipment offer no security for loans.
 Firms should use some debt because of its tax benefits, thus zero leverage firms have a suboptimal capital structure.
Real options analysis
An investment project in real assets is valued as an option if
 the exercise price of the option is the investment cost.
 the underlying value of the option is the project's revenue.
American Put Option: The sellback option to abandon a project at any point of its lifetime and sell its assets in the second hand market
European Call Option: In the binomial model, the option to defer an investment project with one period.
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.
Annual percentage rates on loans (APRs)
Annual percentage rates indicate the rate of return set by the bank (lender) on a financial instrument. The borrower (debtor) agrees to pay a set interest rate over time. It is calculated on a yearly basis, with 12 months and 365 days as the standard timeframe. APR rates are a point of reference when seeking to calculate the total cost of borrowing money. Common examples are mortages and consumer loans.
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