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 + 𝑟)𝑡 𝐹𝑉 Present value of cash flow (discounted): 𝑃𝑉 = (1+𝑟)𝑡 𝐶𝐹 𝐶𝐹 𝐶𝐹 1 2 𝑡 Present value of multiple cash flows: 𝑃𝑉 = (1+𝑟) 1 + (1+𝑟)2 + ⋯ + (1+𝑟)𝑡 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 = 𝐶 𝑟 1 1 Annuity formula (value of constant but limited cash flow): 𝑃𝑉 = 𝐶 [𝑟 − 𝑟(1+𝑟)𝑡 ] Annual percentage rate (APR): Interest rate that annualized using simple interest. Calculated as 𝐴𝑃𝑅 = 12 ∗ 𝑚𝑜𝑛𝑡ℎ𝑙𝑦 𝑟𝑎𝑡𝑒 Effective annual interest rate (EAR): Interest rate that is annualized using compound interest. Calculated as 𝐸𝐴𝑅 = (1 + 𝑚𝑜𝑛𝑡ℎ𝑙𝑦 𝑟𝑎𝑡𝑒)12 − 1 Real interest rate: The nominal interest rate counted for inflation. The relationship for the two 1+𝑛𝑜𝑟𝑚𝑖𝑛𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 rates: 𝑟𝑒𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 = −1 1+𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 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 1 1 𝑓𝑎𝑐𝑒 𝑣𝑎𝑙𝑢𝑒 𝑐𝑢𝑝𝑜𝑛 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 [𝑟 − 𝑟(1+𝑟)𝑡] + (1+𝑟)𝑡 Relationships between the discount rate and the coupon rate: 𝐶𝑜𝑢𝑝𝑜𝑛 𝑟𝑎𝑡𝑒 > 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑟𝑎𝑡𝑒 → 𝐵𝑜𝑛𝑑 𝑝𝑟𝑖𝑐𝑒 > 𝐹𝑎𝑐𝑒 𝑉𝑎𝑙𝑢𝑒 (selling at a premium) 𝐶𝑜𝑢𝑝𝑜𝑛 𝑟𝑎𝑡𝑒 = 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑟𝑎𝑡𝑒 → 𝐵𝑜𝑛𝑑 𝑝𝑟𝑖𝑐𝑒 = 𝐹𝑎𝑐𝑒 𝑉𝑎𝑙𝑢𝑒 (selling at par) 𝐶𝑜𝑢𝑝𝑜𝑛 𝑟𝑎𝑡𝑒 < 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑟𝑎𝑡𝑒 → 𝐵𝑜𝑛𝑑 𝑝𝑟𝑖𝑐𝑒 < 𝐹𝑎𝑐𝑒 𝑉𝑎𝑙𝑢𝑒 (selling at a discount) Current yield: 𝑎𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑢𝑝𝑜𝑛 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 𝑏𝑜𝑛𝑑 𝑝𝑟𝑖𝑐𝑒 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 𝐶𝐹 exam, which is just about rearranging some numbers in 𝑃𝑉 = 1+𝑟 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 + 𝑃0 𝑃0 Pricing a stock (known investment horizon): 𝑃0 = 𝐷𝐼𝑉1 1+𝑟 Pricing a stock (unknown investment horizon): 𝑃0 = 𝐷𝐼𝑉 + (1+𝑟)2 2 + ⋯ + 𝐷𝐼𝑉1 1+𝑟 𝐷𝐼𝑉 𝐷𝐼𝑉𝐻 +𝑃𝐻 (1+𝑟)𝐻 𝐷𝐼𝑉 𝐷𝐼𝑉 𝑖 + (1+𝑟)2 2 + (1+𝑟)3 3 + ⋯ = ∑∞ 𝑖=1 (1+𝑟)𝑖 The Gordon growth model: Used to calculate the price of a stock if the dividends grow at a 𝐷𝐼𝑉 constant rate g. It’s calculated as 𝑃0 = 𝑟−𝑔1 . Payout ratio: Fraction of earnings paid out as dividends. Plowback ratio: Fraction of warnings retained by the firm. Lecture 4: Investment criteria 𝑁𝑃𝑉 = 𝐶0 + 𝐶1 𝐶2 𝐶3 𝐶𝑡 + + + ⋯+ 1 2 3 (1 + 𝑟) (1 + 𝑟) (1 + 𝑟) (1 + 𝑟)𝑡 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 = 𝐶0 + 𝐶1 𝐶2 𝐶𝑡 + + ⋯+ (1 + 𝑟)1 (1 + 𝑟)2 (1 + 𝑟)𝑡 𝑜𝑟 − 𝐶0 = 𝐶1 𝐶2 𝐶𝑡 + + ⋯+ (1 + 𝑟)1 (1 + 𝑟)2 (1 + 𝑟)𝑡 Profitability index: Related the value of the projects to its costs. The index is used when cash 𝑁𝑃𝑉 are not unlimited. It’s calculated as: 𝑃𝐼 = 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡. 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 NPV=0. 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 variance. 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 away. 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 𝑠𝑡𝑜𝑐𝑘 1) ∗ (𝑏𝑒𝑡𝑎 𝑜𝑓 𝑠𝑡𝑜𝑐𝑘 1) + (% 𝑖𝑛 𝑠𝑡𝑜𝑐𝑘 2) ∗ (𝑏𝑒𝑡𝑎 𝑜𝑓 𝑠𝑡𝑜𝑐𝑘 2) CAPM: Capital asset pricing model. It’s a theory of relationship between risk and return. It’s calculated as: 𝑟𝑒 = 𝑟𝑓 + 𝛽(𝑟𝑚 − 𝑟𝑓 ). 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 information. 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. 𝐷 𝐸 WACC (weighted average cost of capital): 𝑊𝐴𝐶𝐶 = (𝑉 ∗ (1 − 𝑇𝐶 ) ∗ 𝑟𝑑 ) + (𝑉 ∗ 𝑟𝑒 ) 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 𝑟𝑒 - 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 𝐹𝐶𝐹 𝐹𝐶𝐹 𝐹𝐶𝐹 𝑃𝑉 calculation: 𝑃𝑉 = 1+𝑟11 + (1+𝑟)22 + ⋯ (1+𝑟)𝐻𝐻 + (1+𝑟)𝐻𝐻 Horizon value: Is calculated based on the next 5 years. Calculated as: 𝐹𝐶𝐹 𝑖𝑛 𝑦𝑒𝑎𝑟 6 𝑟−𝑔 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 shareholders. 𝐷 Modigliani & Miller proposition II: 𝑟𝑒 = 𝑟𝑎 + 𝐸 (𝑟𝑎 − 𝑟𝑑 ). The expected return on a common stock of a levered firm increases in proportion to the debt-equity ratio. 𝐴𝑛𝑛𝑢𝑎𝑙 𝑡𝑎𝑥 𝑠ℎ𝑖𝑒𝑙𝑑 = 𝐶𝑜𝑟𝑝𝑜𝑟𝑎𝑡𝑒 𝑡𝑎𝑥 𝑟𝑎𝑡𝑒 ∗ 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 Market value of firm = Value of all equity financed + PV tax shiled – PV of costs of financial distress. 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: 𝑚𝑎𝑟𝑔𝑖𝑛𝑎𝑙 𝑃𝑉(𝑡𝑎𝑥 𝑠ℎ𝑖𝑙𝑒𝑑) = 𝑚𝑎𝑟𝑔𝑖𝑛𝑎𝑙 𝑃𝑉(𝑑𝑖𝑠𝑡𝑟𝑒𝑠𝑠) 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(𝑒𝑐𝑜𝑛𝑜𝑚𝑖𝑐 𝑔𝑎𝑖𝑛) − 𝑃𝑉(𝑐𝑜𝑠𝑡𝑠) 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: 1+𝑟𝑓𝑜𝑟𝑒𝑖𝑔𝑛 1+𝑟$ 𝐹 = 𝑆 𝑓𝑜𝑟𝑒𝑖𝑔𝑛$$ - The differences in interest rates is reflected in the 𝑓𝑜𝑟𝑒𝑖𝑔𝑛 difference between the spot and the forward exchange rate. International Fisher effect: 1+𝑟𝑓𝑜𝑟𝑒𝑖𝑔𝑛 1+𝑟$ = 1+𝑖𝑓𝑜𝑟𝑒𝑖𝑔𝑛 1+𝑖$ - IFE says that real interest rates in all countries should be equal with differences in nominal rates reflecting differences in expected inflation. Purchasing power parity: 1+𝑖𝑓𝑜𝑟𝑒𝑖𝑔𝑛 = 1+𝑖$ 𝐸(𝑆𝑓𝑜𝑟𝑒𝑖𝑔𝑛$ ) 𝑆𝑓𝑜𝑟𝑒𝑖𝑔𝑛$ - Theory that prices of goods in all countries should be equal when translated to a common currency. The real price level is what to look for. 𝐹 Expectations theory of exchange: 𝑆 𝑓𝑜𝑟𝑒𝑖𝑔𝑛$$ = 𝑓𝑜𝑟𝑒𝑖𝑔𝑛 𝐸(𝑆𝑓𝑜𝑟𝑒𝑖𝑔𝑛$ ) 𝑆𝑓𝑜𝑟𝑒𝑖𝑔𝑛$ - 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 1+𝑟𝑓𝑜𝑟𝑒𝑖𝑔𝑛 multiple years in the future, we need to take 1+𝑟 to the power of the number of years, 1+𝑟𝑓𝑜𝑟𝑒𝑖𝑔𝑛 𝑡 e.g., ( 1+𝑟$ $ ). 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: 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑠𝑡𝑜𝑐𝑘 + 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑝𝑢𝑡 = 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑐𝑎𝑙𝑙 + 𝑃𝑉(𝑒𝑥𝑒𝑟𝑐𝑖𝑠𝑒 𝑝𝑟𝑖𝑐𝑒) 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 controlled. 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 stream. 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).