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, $ {PV} = \frac{A}{r} $. For perpetuity with growth rate $g$ ($r>g$) is $ {PV} = \frac{A}{r - g} $

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.

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 $U(W)$, which express the utility given by an amount $W$ of resource (often wealth). Two common utility functions:

1. $U(W) = ln(W)$
2. $U(W) = \alpha + \beta W - \gamma W^2$

Note that $W$ may be negative, although (1) may suggest otherwise. Also, (2) breaks down with large values of $W$ (meaning it does not adhere to the assumptions).

Weak - historical

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.

Semi-strong - public

Under the semi-strong 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.

Mean-Variance Efficiency

Mean Variance Efficiency is a way of evaluating an alternative based on expected return (mean) and risk (variance).

The expected return $E(X)$ is defined as:

$$ E(X) = \Sigma (Probability \times Outcome) $$

Then variance is:

$$ Var(X) = \Sigma (Probability \times (Outcome - Mean)^2) $$

Thus we can prioritize between two or more choices by utilizing these numbers. The combination of these two measures can tell us whether an alternative with a higher risk has a subsequently high enough reward, in comparison to its less risky alternative.

Calculating Weighted Average Cost of Capital (WACC)

The formula for calculating Weighted Average Cost of Capital(WACC) is:

$$ \textbf{WACC} = \frac{D}{V} r_D (1 - T_C) + \frac{E}{V} r_E $$

$ 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, $ r_E $ is impossible to know. Luckily, it can be estimated by looking at previous projects in the same business domain. One way to estimate $ r_E $ is to use the Capital Asset Pricing Model, or CAPM.

Calculating $ r_E $ using Capital Asset Pricing Model (CAPM)

The formula for CAPM/Security Market Line is

$$ r_E = r_f + \beta (r_m - r_f) $$

$ r_E $

Cost of equity

$ r_f $

Risk-free interest rate

$ r_m $

Market rate

$ \beta $

Unsystematic risk

$ (r_{m} - r_{f}) $

Market risk premium

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 $ r_p = r_f + \frac{r_m - r_f}{\sigma_m} \sigma_p $

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.

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 ($r_a$), for the existing projects.

Note: When estimating something for a given project only use the values belonging to that project. I.e. the $r_E$ calculated for the existing actors/projects is not the $r_E$ you should be using to calculate the proposed project's WACC.

See also figure 6.3 on page 174 of the book for a decision tree.

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 $r_a$.

• 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 $r_d$) and issue cost at $t_0$.

• In the end, the APV = PV of base case + PV of tax advantage – issue cost.

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 $ -N(-d_1) $ and $ -N(-d_2) $ in the main formula instead of $ N(d_1) $ and $ N(d_2) $.

$$ C(s, t) = N(d_1) S_0 - N(d_2) X e ^ {-rT } $$ $$ d_1 = \frac{1}{\sigma \sqrt{T}} (\ln{(\frac{S_0}{X})} + (r + \frac{\sigma ^ 2}{2})T) $$ $$ d_2 = \frac{1}{\sigma \sqrt{T}} (\ln{(\frac{S_0}{X})} + (r - \frac{\sigma ^ 2}{2})T) $$ $$ N(x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty} ^ x \mathrm{e} ^{-\frac{z ^ 2}{2}} \mathrm{d}z $$

$d_2$ can also be defined in terms of $d_1$: $d_2 = d_1 - \sigma \sqrt{T} $.

$N(x)$ 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.

$ T $

Time to maturity in fraction of interest-giving 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 $

Risk-free rate (annual rate, continuously compounding)

$ \sigma $

Volatility of returns of the asset

To macth discrete and continuous time volatility

$$ u = e ^ {\sigma {\sqrt{\delta t}}}, d = e ^ { - \sigma {\sqrt{\delta t}}} $$ $\delta t $ is small time interval.

to replicate a hedging portfolio with a option,portfolio=a fraction $\Delta$ of the stock + a risk free loan of D,

$$ \Delta = \frac{O_u - O_d}{(u - d)S}$$ $$ D=\frac{u O_d - d O_u}{(u - d)r}$$

Binomial Options Pricing Model (BOPM):discrete time

Use this when you want to calculate the price of an American-style option. Here are the formulas you will need.

$$ p = \frac{e^{r t / n} - d}{u - d}, u = e ^ {\sigma ^ {\sqrt{t / n}}}, d = e ^ { - \sigma ^ {\sqrt{t / n}}} $$

$ 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 $

Risk-free interest rate

$e^{r t / n}$ can be replaced with $r$

to replicate a hedging portfolio with a option,portfolio=a fraction ∆ of the stock + a risk free loan of D,

$$ \Delta = \frac{O_u - O_d}{(u - d)S}$$ $$ D=\frac{u O_d - d O_u}{(u - d)r}$$

First, make a binomial lattice with $ n $ steps. $ S $ is the asset price at the current time, and $ u $ and $ d $ are the value multipliers from the formula above:

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 = 1.14, d = 0.92 $.

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 $ S_n - K $, i.e. 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 $ K = \$100 $, this will yield an option price lattice that looks like this:

Now, all that is remaining is to calculate the values of the empty nodes from right to left. The value in each node, $ C_{\text{node}} $, can be calculated from its two direct child nodes (to the right) by using this nice formula:

$$ C_{\text{node}} = e ^ {-r \Delta t } (p C_{\text{up child}} + (1 - p) C_{\text{down child}}) $$

Or you could use this formula:

$$ C_{\text{node}} = \frac{p C_\text{up child} + (1 - p) C_{\text{down child}}}{(1+r) \Delta t} $$

Note it uses $r+1$ because it assumes $r$ is an actual percentage expressed as a real number (i.e. $r \in [0, 1]$).

Continuing our example, assuming $ r = 5\%, t = 1 $, we get:

The root (leftmost) node of the lattice is the binomial option price, $ \$9.26 $, which is what we wanted to calculate!

Matching discrete and continuous time volatility

$$u = e ^ {\sigma {\sqrt{ \delta t }}}, d = e ^ { - \sigma {\sqrt{ \delta t }}},p = \frac{e^{r \delta t} - d}{u - d},$$

small time interval of american step $\delta t$. Like 1 year with 2 step then $\delta t = 0.5$

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:

$$ C - P = D(F - X) $$

$ C $

Current price of a call

$ P $

Current price of a put

$ D $

Discount rate (typically risk-free interest or similar)

$ F $

Forward price of the asset

$ X $

Strike price

Option positions

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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.

More information on non collateralized loans: https://www.xn--forbruksln-95a.no/