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Finans for teknisk-naturvitenskapelige studenter (TIØ4146)

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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.
More information on non collateralized loans: https://www.xn--forbruksln-95a.no/
Written by
marcusaf, Stian Jensen, sigveseb, cristea, aleksanb, Chaaarles, feisong, odd, Norskeaksel, mathisfo, hoanghn, jeansiss, paalch,
mads, hanskhe, Esso, Isakpisak, batherk, JacR, henrikmm, erikskaar, note, haakon8855, sigurwe, toast, loremipsum, haraldmu,
Mousa, Oys, Matias Sunde Oiesvold, madsth
Last updated:
9 months ago.
5/12/24, 8:04 AM
TIO4146: Finance for Science and Technology Students - Wikipendium
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Tags: økonomi stocks TIØ4146, wacc finance economy finans capm refinance

TIO4146: Finance for Science and Technology Students

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Tips

important pages in the book

P 118: Formelen til CAAR

P 154: Trade-off 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,. For perpetuity with growth rate ( ) is

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.

P V = A r g r > g

P V = rA−g

Wikipendium

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 well-behaved; their preferences form a partial order This results in utility functions , which express the utility given by an amount of resource (often wealth). Two common utility functions: 1. 2. Note that may be negative, although (1) may suggest otherwise. Also, (2) breaks down with large values of (meaning it does not adhere to the assumptions).

Indifference curve

Given two resources, and , the indifference curve shows which combinations gives the same utility, in other words the line defined by where is a constant.

Risk aversion

The utility curves from the above assumptions also leads to risk aversion. Example If one expects to be either 50kr or 150kr with a 50% probability of each, then however, the expected utility assuming is U (W ) W U (W ) = ln(W ) U (W ) = α + βW − γW 2 W W W 1 W 2 U ( W 1 , W 2 ) = C C W E(W ) = 0 ⋅ 50 + 0 ⋅ 150 = 100 U (W ) = ln(W ) E(U (W )) = 0 ⋅ ln(50) + 0 ⋅ ln(150) = 4.

Debt in dollars Equity in dollars Total value in dollars (i. debt + equity) Cost of debt (typically interest rate of loan) Cost of equity Corporate tax rate Cost of equity Risk-free interest rate Market rate

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 , go through with the project. Conversely, if , do not go through with the project. Hence, making project decisions is an exercise of calculating NPV. Typically, projects will cost a certain one-time, up-front amount of money to be completed, and generate a perpetual yearly revenue of some amount. If you have these two numbers, you can figure out NPV using the formula: 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: Unfortunately, is impossible to know. Luckily, it can be estimated by looking at previous projects in the same business domain. One way to estimate is to use the Capital Asset Pricing Model, or CAPM.

Calculating using Capital Asset Pricing Model (CAPM)

The formula for CAPM/Security Market Line is NPV > 0 NPV < 0 C R NPV = −C + R WACC WACC = (1 − ) + D V rD TC E V rE D E V rD rE TC rE rE

rE

rE = rf + β( rm − rf) rE rf rm

Unsystematic risk Market risk premium Debt in dollars Total value in dollars Corporate tax rate Opportunity cost of capital (100% equity, i. unlevered) Cost of debt Debt in dollars Equity in dollars Cost of equity (levered) Opportunity cost of capital (100% equity, i. unlevered) Corporate tax rate Cost of debt Total value in dollars the difference of SML (Security Market Line) and CML (Capital Market Line) 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 un-lever and re-lever as necessary. Here are some formulas that help in doing this.

Miles-Ezzell WACC calculation

When debt of a project will be periodically rebalanced, you can use Miles-Ezzell.

Modigliani-Miller (M&M) formula

The tax part ( ) is left out when debt is continuously rebalanced. β ( rm − rf) rp = rf+ rm −rf σm σp W ACC = ra− D V rD TC 1 + ra 1 + rd D V TC ra rd 1 − TC rE = rA + (1 − TC )( rA − rD) D E rA = rD (1 − TC) + D V − TCD rE E V − TCD D E rE rA TC rD V

Use Miles-Ezzel to find the WACC. If debt is permanent & fixed & predetermined Unlever using M&M (solved for ) remember to use the E and D from the company you got the from Combined version of normal WACC and normal M&M. Use the new project's D / E values. Alternatively, use normal M&M + WACC instead of this shortcut formula

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 (Un-levered) and without side-effects, the present base case value is dicsounted at OCC. Step 2. Then, calculate all the side-effects, include tax advantage (from a given level of debt by discounting tax shield at the cost of debt ) and issue cost at. 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 Black-Scholes 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 american option without exercising is alive while it turns to dead after exercise. Put:the right to sell at exercise price, expecting the price decrease. Call:the right to buy at exercise price, expecting the price increases. Moneyness Put Call In the money At the money Out of the money Payoff = Value of the Option = Profit = Payoff - Future value of option cost =. position & option payoff at maturity ra = rD (1 − TC) + D V − TCD rE E V − TCD ra rE WACC = ra(1 − TC ) D V N P V = −Cost + CashF low W ACC ra rd t 0 S 0 < X S 0 > X S 0 = X S 0 = X S 0 > X S 0 < X Vt Vt −V 0 ert

Time to maturity in fraction of interest-giving periods (e. T=90/365=0. if 90 days to maturity and interest is given for one year) Spot price of the asset, i. what the asset costs now Strike price, i. what can the option be bought/sold for at maturity Risk-free rate (annual rate, continuously compounding) Volatility of returns of the asset long a put long a call short a put short a call

Black-Scholes model:contious time

Use this when you want to calculate the price of a European-style call option. Here are the formulas you will need. When calculating European puts, use and in the main formula instead of and . can also be defined in terms of :. is not trivial to calculate. It is equivalent to the cumulative distribution function for the normal distribution, which can be looked up in a table. To macth discrete and continuous time volatility is small time interval. to replicate a hedging portfolio with a option,portfolio=a fraction of the stock + a risk free loan of D, max[X − S 0 , 0] max[ S 0 − X, 0] min[X − S 0 , 0] min[ S 0 − X, 0] −N (− d 1 ) −N (− d 2 ) N ( d 1 ) N ( d 2 ) C(s, t) = N ( d 1 ) S 0 − N ( d 2 ) Xe−rT d 1 = (ln ( ) + (r + )T ) 1 σ T −− √ S 0 X σ 2 2 d 2 = (ln ( ) + (r − )T ) 1 σ T √ −− S 0 X σ 2 2 N (x) = dz 1 2 π −− √ ∫ x −∞ e− z 2 2 d 2 d 1 d 2 = d 1 − σ √ −T− N (x) T S 0 X r σ u = eσ √δt , d =e−σ √δt δt Δ Δ = Ou −Od (u − d)S

Then you want to fill in the numbers in all the nodes of the lattice with dollar (or euro, or whatever) values. Here I have just made up some numbers for the sake of example: S = $100, u = 1, d = 0.

This is now the finished asset price lattice. Now we want to make a new lattice for the option price. It should have as many steps as the asset price lattice, but all nodes should be empty except for the right-most leaf nodes. These nodes should contain the option value at that point, calculated as , i. the asset price for the corresponding node in the asset lattice minus the strike price. Of course, if this expression is less than $0, the value of the option is truncated to zero, since excercising options are, well, optional. Anyway, for our example, with , this will yield an option price lattice that looks like this: Sn − K K = $

Current price of a call The root (leftmost) node of the lattice is the binomial option price, , which is what we wanted to calculate!

Matching discrete and continuous time volatility

small time interval of american step. Like 1 year with 2 step then

Put-Call Parity

Put-call parity expresses the relationship between the price of a European call option and a European put option. The relationship is as expressed in this equation: $9. u = eσ √δt , d = e−σ √δt , p = , erδt− d u − d δt δt = 0. C − P = D(F − X) C

Current price of a put Discount rate (typically risk-free interest or similar) Forward price of the asset 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 at-the-money 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. /
/ \

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. \ / /

Butterfly spread

Buy two call options – one at an in-the-money exercise price and one out-of-money. Then sell two call options for an at-the-money 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. /
_/ _

Collar

Also called "Split-Strike Conversion" Buy an out-of-the-money put option and sell an out-of-the-money call option (long put and short call). It reduces volatility because it ‘cuts off’ the high and low tails of the stock’s returns 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. P D F X

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. More information on non collateralized loans: xn--forbruksln-95a/

####### Written by

marcusaf, Stian Jensen, sigveseb, cristea, aleksanb, Chaaarles, feisong, odd, Norskeaksel, mathisfo, hoanghn, jeansiss, paalch, mads, hanskhe, Esso, Isakpisak, batherk, JacR, henrikmm, erikskaar, note, haakon8855, sigurwe, toast, loremipsum, haraldmu, Mousa, Oys, Matias Sunde Oiesvold, madsth Last updated: 9 months ago.