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Corporate Finance formulas and definitions

Corporate Finance – Formulars and definitions
Lecture 1: Introduction
Opportunity cost of capital: Minimum acceptable rate of return on capital. Based on similar
project in the financial market with the same risk.
EBIT: Earnings before interest & tax = total revenues – cost - depreciation
NI: Net income = EBIT – net interest expense – tax
OCF: Operating cash flow = net income + depreciation + change in working capital + change
in investment capital
Lecture 2: The time value of money
Calculate future value: 𝐹𝑉 = 𝑃𝑉 ∗ (1 + π‘Ÿ)𝑑
Present value of cash flow (discounted): 𝑃𝑉 = (1+π‘Ÿ)𝑑
Present value of multiple cash flows: 𝑃𝑉 = (1+π‘Ÿ)
1 + (1+π‘Ÿ)2 + β‹― + (1+π‘Ÿ)𝑑
Perpetuity: A stream of cash flows that starts one period from now but never ends.
Annuity: A stream of cash flows that starts one period form now but is running for a limited
period of time.
Perpetuity formula (value of constant and endless cashflow): 𝑃𝑉 =
Annuity formula (value of constant but limited cash flow): 𝑃𝑉 = 𝐢 [π‘Ÿ − π‘Ÿ(1+π‘Ÿ)𝑑 ]
Annual percentage rate (APR): Interest rate that annualized using simple interest. Calculated
as 𝐴𝑃𝑅 = 12 ∗ π‘šπ‘œπ‘›π‘‘β„Žπ‘™π‘¦ π‘Ÿπ‘Žπ‘‘π‘’
Effective annual interest rate (EAR): Interest rate that is annualized using compound interest.
Calculated as 𝐸𝐴𝑅 = (1 + π‘šπ‘œπ‘›π‘‘β„Žπ‘™π‘¦ π‘Ÿπ‘Žπ‘‘π‘’)12 − 1
Real interest rate: The nominal interest rate counted for inflation. The relationship for the two
1+π‘›π‘œπ‘Ÿπ‘šπ‘–π‘›π‘Žπ‘™ π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘ π‘Ÿπ‘Žπ‘‘π‘’
rates: π‘Ÿπ‘’π‘Žπ‘™ π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ =
Lecture 3: Valuing bonds and stocks
Bond: A security that obligates the issuer to make specified payments to the bondholder.
Coupon payments: Interest payments on the bond
The price of the bond is the present value of all cash flows it generates. Calculated a
π‘“π‘Žπ‘π‘’ π‘£π‘Žπ‘™π‘’π‘’
π‘π‘’π‘π‘œπ‘› π‘π‘Žπ‘¦π‘šπ‘’π‘›π‘‘ [π‘Ÿ − π‘Ÿ(1+π‘Ÿ)𝑑] + (1+π‘Ÿ)𝑑
Relationships between the discount rate and the coupon rate:
πΆπ‘œπ‘’π‘π‘œπ‘› π‘Ÿπ‘Žπ‘‘π‘’ > π·π‘–π‘ π‘π‘œπ‘’π‘›π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ → π΅π‘œπ‘›π‘‘ π‘π‘Ÿπ‘–π‘π‘’ > πΉπ‘Žπ‘π‘’ π‘‰π‘Žπ‘™π‘’π‘’ (selling at a
πΆπ‘œπ‘’π‘π‘œπ‘› π‘Ÿπ‘Žπ‘‘π‘’ = π·π‘–π‘ π‘π‘œπ‘’π‘›π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ → π΅π‘œπ‘›π‘‘ π‘π‘Ÿπ‘–π‘π‘’ = πΉπ‘Žπ‘π‘’ π‘‰π‘Žπ‘™π‘’π‘’ (selling at par)
πΆπ‘œπ‘’π‘π‘œπ‘› π‘Ÿπ‘Žπ‘‘π‘’ < π·π‘–π‘ π‘π‘œπ‘’π‘›π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ → π΅π‘œπ‘›π‘‘ π‘π‘Ÿπ‘–π‘π‘’ < πΉπ‘Žπ‘π‘’ π‘‰π‘Žπ‘™π‘’π‘’ (selling at a discount)
Current yield:
π‘Žπ‘›π‘›π‘’π‘Žπ‘™ π‘π‘œπ‘’π‘π‘œπ‘› π‘π‘Žπ‘¦π‘šπ‘’π‘›π‘‘
π‘π‘œπ‘›π‘‘ π‘π‘Ÿπ‘–π‘π‘’
Yield to maturity (YTM): The discount rate for which the PV of the bond’s payments equals
the observed price. We are only required to make this calculation for a one-year bond for the
exam, which is just about rearranging some numbers in 𝑃𝑉 = 1+π‘Ÿ
Government bonds are assumed to be risk free.
Expected return on a stock is the dividend yield plus the capital gain. It’s calculated as:
𝐷𝐼𝑉1 𝑃1 − 𝑃0
Pricing a stock (known investment horizon): 𝑃0 =
Pricing a stock (unknown investment horizon): 𝑃0 =
+ (1+π‘Ÿ)2 2 + β‹― +
+ (1+π‘Ÿ)2 2 + (1+π‘Ÿ)3 3 + β‹― = ∑∞
𝑖=1 (1+π‘Ÿ)𝑖
The Gordon growth model: Used to calculate the price of a stock if the dividends grow at a
constant rate g. It’s calculated as 𝑃0 = π‘Ÿ−𝑔1 .
Payout ratio: Fraction of earnings paid out as dividends.
Plowback ratio: Fraction of warnings retained by the firm.
Lecture 4: Investment criteria
𝑁𝑃𝑉 = 𝐢0 +
+ β‹―+
(1 + π‘Ÿ)
(1 + π‘Ÿ)
(1 + π‘Ÿ)
(1 + π‘Ÿ)𝑑
Projects with a positive NPV should be accepted if the company has sufficient cash.
Payback period: Time until cash flows recover the initial investment in the project.
Dynamic payback period (discounted payback period): Same as the payback period but the
cash flows are discounted based on the time value of money principal before accumulated.
Internal rate of return (IRR): Given cash flows, the IRR is found as the discount rate that will
give NPV=0. Therefore, IRR is the minimum required internal return on the project. It’s
calculated as:
0 = 𝐢0 +
+ β‹―+
(1 + π‘Ÿ)1 (1 + π‘Ÿ)2
(1 + π‘Ÿ)𝑑
π‘œπ‘Ÿ − 𝐢0 =
+ β‹―+
(1 + π‘Ÿ)1 (1 + π‘Ÿ)2
(1 + π‘Ÿ)𝑑
Profitability index: Related the value of the projects to its costs. The index is used when cash
are not unlimited. It’s calculated as: 𝑃𝐼 = πΌπ‘›π‘£π‘’π‘ π‘‘π‘šπ‘’π‘›π‘‘.
Incremental cash flows = Cash flow with project – cash flow without project
Total cash flows = cash flow from capital investments + cash flow from investment in
working capital + cash flow from operations. It’s often calculated by adding back the
depreciation to the net income.
Sensitivity analysis: What will NPV be for many different values of an input variable.
Scenario analysis: What will NPV be for different combinations of the input variables.
Typically, best case, expected case, and worse case.
Break-even analysis: How much do our assumptions need to change such that our
calculations turn our investment decisions. We need to find the input variable that makes
Lecture 5: Risk and return
Risk premium: Average extra return relative to treasurer bills.
Variance: A measure of volatility and is the average value of squared deviations from the
mean. However, typically we use the standard deviation, which is the square root of the
Rules-of-thumb interpretations for an exact normal distribution:
- 1/3 outcomes will be more than 1 standard deviation away from the mean.
- 1/20 outcomes will be more than 2 standard deviations away from the mean.
Portfolio rate of return = fraction of portfolio in first asset * rate of return on first asset +
fraction of portfolio in second asset * rate of return on second asset.
Diversification: Strategy designed to reduce risk by spreading the portfolio across many
investments. If a portfolio includes 20 stocks, nearly all unique risk has been diversified
Lecture 6: CAPM & efficient markets
Beta: 𝛽, sensitivity of a stock’s return to the return on the market portfolio. Beta is found by
plotting market and stock returns day by day against eachother in a graph and making a linear
regression of the connection between the pots. The slope of the line is beta.
Beta of a portfolio: (% 𝑖𝑛 π‘ π‘‘π‘œπ‘π‘˜ 1) ∗ (π‘π‘’π‘‘π‘Ž π‘œπ‘“ π‘ π‘‘π‘œπ‘π‘˜ 1) + (% 𝑖𝑛 π‘ π‘‘π‘œπ‘π‘˜ 2) ∗
(π‘π‘’π‘‘π‘Ž π‘œπ‘“ π‘ π‘‘π‘œπ‘π‘˜ 2)
CAPM: Capital asset pricing model. It’s a theory of relationship between risk and return. It’s
calculated as: π‘Ÿπ‘’ = π‘Ÿπ‘“ + 𝛽(π‘Ÿπ‘š − π‘Ÿπ‘“ ).
Security market line (SML) shows the relationship between expected return and risk of
individual securities. CAPM states that fairly priced securities lie on the SML.
Efficient market hypothesis (EMH): The hypothesis states that stock prices reflect available
and relevant information. In such a market all regular investors have the same amount of
Lecture 7: WACC
Capital structure: The firm’s mix of long-term debt and equity financing.
Cost of capital: The payback the firm’s investors demand for investing money in the firm.
WACC (weighted average cost of capital): π‘Šπ΄πΆπΆ = (𝑉 ∗ (1 − 𝑇𝐢 ) ∗ π‘Ÿπ‘‘ ) + (𝑉 ∗ π‘Ÿπ‘’ )
WACC calls for the use of market values, but usually the book value of debt and equity is
given. Bond’s market value can be found by discounting with the current interest the coupons
and face value. The market price of the equity can be found by multiplying the share price
with number of outstanding shares.
The different required rate of returns (RRR):
- Bank debt (short term) = interest rate on loans
- Bond debt = Yield to maturity
- Equity financing = Use the CAPM model to calculate the π‘Ÿπ‘’
- Preferred stock = Divide dividends by price of preferred stock.
Valuating an entire busines: We can do this as treating the company as one project. We
calculate the present value of the free cash flows and the horizon value. We use the following
calculation: 𝑃𝑉 = 1+π‘Ÿ11 + (1+π‘Ÿ)22 + β‹― (1+π‘Ÿ)𝐻𝐻 + (1+π‘Ÿ)𝐻𝐻
Horizon value: Is calculated based on the next 5 years. Calculated as:
𝐹𝐢𝐹 𝑖𝑛 π‘¦π‘’π‘Žπ‘Ÿ 6
Lecture 8: Debt policy
Modigliani & Miller proposition I: In perfect markets, the market value of a company does
not depend on its capital structure. Thus, the value cannot be increased by changing the mix
of securities used to finance the company. The pizza is the same no matter how it’s sliced.
Perfect market: No taxes, no bankruptcy costs or financial distress costs, no friction and
transactional costs, no asymmetric information, and management acts only on behalf of
Modigliani & Miller proposition II: π‘Ÿπ‘’ = π‘Ÿπ‘Ž + 𝐸 (π‘Ÿπ‘Ž − π‘Ÿπ‘‘ ). The expected return on a common
stock of a levered firm increases in proportion to the debt-equity ratio.
π΄π‘›π‘›π‘’π‘Žπ‘™ π‘‘π‘Žπ‘₯ π‘ β„Žπ‘–π‘’π‘™π‘‘ = πΆπ‘œπ‘Ÿπ‘π‘œπ‘Ÿπ‘Žπ‘‘π‘’ π‘‘π‘Žπ‘₯ π‘Ÿπ‘Žπ‘‘π‘’ ∗ π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘ π‘π‘Žπ‘¦π‘šπ‘’π‘›π‘‘
Market value of firm = Value of all equity financed + PV tax shiled – PV of costs of financial
Theories for choosing debt level:
- Trade-off theory: The theory that capital structure is based on trade-off between tax
savings and distress costs of debt.
- Pecking order theory: States that firms prefer to issue debt rather than equity if
internal finance is insufficient.
- Theory of financial slack: Having ready access to cash or debt financing
- Modigliani & Miller: π‘šπ‘Žπ‘Ÿπ‘”π‘–π‘›π‘Žπ‘™ 𝑃𝑉(π‘‘π‘Žπ‘₯ π‘ β„Žπ‘–π‘™π‘’π‘‘) = π‘šπ‘Žπ‘Ÿπ‘”π‘–π‘›π‘Žπ‘™ 𝑃𝑉(π‘‘π‘–π‘ π‘‘π‘Ÿπ‘’π‘ π‘ )
Lecture 9: Payout policy
Cash dividend: The payment of cash by the firm to its shareholders.
Stock dividend: Distribution of additional shares to a firm’s stockholders.
Stock buy-back: Thereby this is an alternative to cash dividends.
Lecture 10: Mergers and acquisitions
Mergers & acquisition: Corporate combination or takeover.
Is M&A worth it for? 𝑁𝑃𝑉 = 𝑃𝑉(π‘’π‘π‘œπ‘›π‘œπ‘šπ‘–π‘ π‘”π‘Žπ‘–π‘›) − 𝑃𝑉(π‘π‘œπ‘ π‘‘π‘ )
Divestiture: When a firm sells some of the assets to another entity.
Spin off: The process where a business separates the ongoing operations of a specific unit
into two parts and gives the shareholders of the original parent firm shares in the two parts.
The new unit and parent function now as separate entities.
Carve outs: Like a spin off, but the carve out issues shares of the new firm to the public
instead of existing shareholder.
LBO: Leveraged buyout of a firm. A group of shareholders (or management – MBO) buy the
firm and it is to a large extent financed with debt.
Lecture 11: International financial management
Direct quote: Amount of domestic currency that you pay for one foreign currency.
Indirect quote: Amount of foreign currency that you get for one domestic currency.
Spot rate = S = Exchange rate for an immediate transaction
Forward rate = F = Exchange rate for a future transaction
Interest rate parity:
= 𝑆 π‘“π‘œπ‘Ÿπ‘’π‘–π‘”π‘›$$ - The differences in interest rates is reflected in the
difference between the spot and the forward exchange rate.
International Fisher effect:
- IFE says that real interest rates in all
countries should be equal with differences in nominal rates reflecting differences in expected
Purchasing power parity:
𝐸(π‘†π‘“π‘œπ‘Ÿπ‘’π‘–π‘”π‘›$ )
- Theory that prices of goods in all countries
should be equal when translated to a common currency. The real price level is what to look
Expectations theory of exchange: 𝑆 π‘“π‘œπ‘Ÿπ‘’π‘–π‘”π‘›$$ =
𝐸(π‘†π‘“π‘œπ‘Ÿπ‘’π‘–π‘”π‘›$ )
- Th expected spot rate equals the
forward rate.
If we are in an international setting, we can find the relevant cash flows in the currency of
foreign, use the interest rate parity to translate the cash flows back to local currency and then
discount them using the NPV formula.
Forward rates multiple years in the future. In case you need to predict the forward rate
multiple years in the future, we need to take 1+π‘Ÿ
to the power of the number of years,
1+π‘Ÿπ‘“π‘œπ‘Ÿπ‘’π‘–π‘”π‘› 𝑑
e.g., (
CAPM in an international setting: We need to use a global beta for the calculations, as it
reflects the risk more precisely than a local beta. The local beta is often greater than the
global beta due to diversification.
Lecture 12: Options
Derivatives: Securities whose payoff are determined by the values of other financial variables
(such as the price of a stock).
Option: The buyer has a tight but not an obligation. Call option: the right to buy an asset at a
specified exercise price (strike price). Put option: the right to sell an asset at a specified
exercise price (strike price).
Option: The seller has the obligation. Call option: the obligation to sell an asset at a specified
exercise price (strike price). Put option: the obligation to buy an asset at a specified exercise
price (strike price).
Protective put: You buy a stock and a put option on the stock to protect yourself against
losses when owning a stock.
Straddle: Buying a call and put option on the same underlying asset.
Bull spread: Buying a call in a stock at exercise price X and issuing (selling) a call in the
same stock with the same expiration data, but a higher exercise price.
Butterfly spread: Buying a call with a low exercise price, buying another call with higher
exercise price, selling two calls with exercise price in between the two bought call options.
Put-call parity: π‘£π‘Žπ‘™π‘’π‘’ π‘œπ‘“ π‘ π‘‘π‘œπ‘π‘˜ + π‘£π‘Žπ‘™π‘’π‘’ π‘œπ‘“ 𝑝𝑒𝑑 = π‘£π‘Žπ‘™π‘’π‘’ π‘œπ‘“ π‘π‘Žπ‘™π‘™ + 𝑃𝑉(𝑒π‘₯π‘’π‘Ÿπ‘π‘–π‘ π‘’ π‘π‘Ÿπ‘–π‘π‘’)
Real options: Options embedded in real assets. E.g., expansion option, contraction or
abandonment option, timing option, alter option, shut down option etc. An real option can be
priced using a decision three.
Lecture 13: Risk management
Risk management: The process by which various risk exposures are identified, measured, and
Forward contracts: Contracts (an obligation, not an option) between two parties for delivery
(buy or sell) of an asset at a negotiated price (forward price) on a set date in the future.
Futures contracts: Contract similar to a forward, except there is an intermediary that creates a
standardized contract (typically exchange traded contracts).
Swaps: Arrangement where two counterparties exchange on stream of cash flows for another
Value at risk (VaR) is a statistical way of measuring risk. VaR returns an estimated $ value
(or % of portfolio). The definition: The worst loss we can have for a given time horizon and
for a given level of significance (probability).