Appunti di corporate finance, Appunti di Macroeconomia

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CF Contents intro .................................................................................................................................................................... 2 2 Valuing Investment Projects: Capital Budgeting Techniques......................................................................... 5 C3 Working Capital Management...................................................................................................................... 9 C4 Implementing the NPV Methodology: Cash Flows and Project Analysis ................................................. 10 CF 5 capital structure theory ............................................................................................................................ 11 CF 6 – valuing firms......................................................................................................................................... 18 7 – measuring return and risk in the stock market ........................................................................................... 24 CF 8 - Portfolio Theory .................................................................................................................................... 27 CF9 – PORTFOLIO THEORY 2 ..................................................................................................................... 31 10 - Capital Asset Pricing Model....................................................................................................................... 35 CF 11 - Estimating the Cost of Capital and the Beta ....................................................................................... 36 intro Value enhancement is what Corporate Finance is all about. 1. Capital investments 2. Capital financing 3. Dividends and return of capital - To carry on business operations, corporations need an almost endless variety of real assets. All these real assets need to be paid for. - To obtain the necessary money, the corporation sells claims on its real assets and on the cash those assets will generate. These claims are called financial assets or securities. - The financial manager stands between the firm’s operations and the financial (or capital) markets, where investors hold the financial assets issued by the firm. The financial manager’s role is illustrated tracing flow of cash from investors to the firm and back to investors again 1. The flow starts when the firm sells securities to raise cash 2. The cash is used to purchase real assets used in the firm’s operations 3. if the firm does well, the real assets generate cash inflows which more than repay the initial investment 4. Finally, the cash is either reinvested or returned to the investors who purchased the original security issuance No-Arbitrage Principle & Fisher Separation Theorem - Capital investment and financing decisions are typically separated. - Value is created only by real investment projects. - When an investment opportunity or project is identified, the financial manager first asks whether the project is worth more than the capital required to undertake it. If the answer is yes, then he considers how the project should be financed But the separation of investment and financing decisions does not mean that the financial manager can forget about investors and financial markets when analysing capital investment projects. - The fundamental objective of the firm is to maximize the value of the cash invested in Fisher Separation Theorem: Given perfect and complete capital markets, the production decision is governed solely by an objective market criterion (wealth maximization) without regard to individuals’ subjective preferences that enter their consumption decisions - An important implication of this theorem for corporate policy is that the investment decision can be delegated to managers - Given the same opportunity set, every investor will make the same production decision (P0, P1) regardless of the shape of his or her indifference curves - Production possibilities are now combined with the existence of market exchange opportunities What happens if the production/consumption decision takes place in a world where capital markets facilitate the exchange of funds at the market rate of interest? Question: Given the family of indifference curves (U1, U2, U3) and the endowment (y0, y1) at point A, what actions will we take in order to maximize our utility? - Starting at point A, we can move either along the production opportunity set or along the capital market line. Both alternatives offer a higher rate of return than our subjective time preference, but production offers the higher return (steeper slope). Therefore, we choose to invest and move along the production opportunity frontier Accounting vs. Financial View of the Firm - If capital markets are perfect (no frictions cause the borrowing rate to be different from the lending rate), then the Fisher Separation Theorem holds, meaning that individuals can delegate investment decisions to the manager of the firm in which they are owners: as regardless of the shape of the shareholders’ individual utility functions, the managers maximize the owners’ individual (and collective) wealth by choosing to invest until the rate of return on the last favourable project is equal to the capital market rate of return - If managers always make decisions that maximize the wealth of the firm’s shareholders, they must find and select the best set of investment projects to accomplish their objective - Shareholders’ wealth is the discounted value of after-tax cash flows paid out by the firm - After-tax cash flows available for consumption can be shown to be the same as the stream of dividends S0 paid to shareholders. The discounted value of the stream of dividends is: where S0 is the present value of shareholders’ wealth and ks is the market-determined required rate of return on equity capital (common stock) 2 Valuing Investment Projects: Capital Budgeting Techniques main purpose: to compare costs and benefits of a project to evaluate a long-term investment decision. A convenient choice is to use present values. - net present value (NPV) of an investment project as follows: NPV= PV (benefit) – PV (costs) - considering a standalone (or independent) decision (by undertaking this project, the firm does not constrain its ability to take other projects). - NPV rule: When making an investment decision, the alternative with the highest NPV should be chosen. Choosing this alternative is equivalent to receiving the corresponding NPV in cash today. The NPV rule states that the project should be accepted if its NPV is positive (> 0). If the NPV is negative (< 0), the project is rejected. The NPV of the project depends on the appropriate cost of capital. Often there can be uncertainty about the cost of capital of the project, and in this case, it is useful to compute an NPV profile, which graphs the project's NPV over a range of discount rates - The internal rate of return (IRR) of an investment is the discount rate that makes the NPV of the project's cash flows equal to zero. - The IRR of a project provides useful information on the sensitivity of the NPV to errors in the estimate of the cost of capital. In general, the difference between the cost of capital and the IRR is the maximum estimation error in the cost of capital that can exist without altering the original decision. - The net present value (NPV) can also be expressed in terms of rate of return, which would lead to the following rule: "Accept investment proposals that offer rates of return greater than their opportunity costs". This statement, if correctly interpreted, is absolutely true. Nevertheless, a correct interpretation of this rule is not always easy in the case of long-term investment projects. There is no ambiguity in defining the true rate of return on an investment that generates only one cash-in flow after a period: C1 indicates the cash-in-flow and -C0 the initial investment; in this way, the two equations state exactly the same thing, namely that the discount rate giving a NPV=0 is also the rate of return of the project. Unfortunately, there is no completely satisfactory way to calculate the true rate of return on a long-term asset. The best way is the so-called IRR (Internal Rate of Return). The internal rate of return is defined as the discount rate that yields a NPV = 0. The easiest way to manually calculate the IRR is to draw three or four combinations of NPV and discount rate; then join the points with a continuous line and read the discount rate for which the NPV is equal to zero. 𝑅𝑎𝑡𝑒 𝑜𝑓 𝑅𝑒𝑡𝑢𝑟𝑛 = 𝐶𝑎𝑠ℎ 𝐼𝑛 𝐹𝑙𝑜𝑤 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 (𝐶𝑎𝑠ℎ 𝑂𝑢𝑡 𝐹𝑙𝑜𝑤) − 1 𝑁𝑃𝑉 = −𝐶0 + 𝐶1 1 + 𝑟 = 0 𝑟 = 𝐼𝑅𝑅 = 𝐶1 𝐶0 − 1 Based on the IRR, the rule to be applied is: accept an investment project whose opportunity cost of capital is lower than the internal rate of return. If it is equal to the internal rate of return, the project has a NPV equal to zero and, if it is finally higher, the NPV is negative. So, when we compare the opportunity cost of capital with the IRR of a project, we are actually asking if the project has a positive NPV. This way of proceeding is correct not only in the case of our example. The IRR rule will give the same answer as the net present value rule whenever the NPV of a project is a monotonous decreasing function in the discount rate. although the two criteria are "formally" equivalent, the internal rate of return rule contains several traps. 1. IRR>k, but NPV- 2. Multiple IRR 3. No IRR - The IRR implies that the reinvestment of all cash flows generated by an investment project is made at the IRR itself. This is a strong assumption which undermines its use for comparative purposes. The IRR does not represent an accurate estimate of the value creation potential of an investment project. Instead, it reflects the rate of return of a «combination» of investment projects: the original project and any further project that may be realized in the future through the reinvestment of all incremental cash flows generated by the very first (original) project at the IRR (= the implied rate of return of the original project). - The IRR cannot be used to compare the value creation potential of 2 (or multiple) distinct (stand-alone or mutually exclusive) investment projects as the «implied» IRR reinvestment rate assumption requires that, by construction, the reinvestment of cash flows from distinct projects is made at disomogenous rates. - if NPV (A) with k > NPV (B) with k, net (incremental) wealth is created and project is ACCEPTED - in general, there is a guarantee that the IRR rule works for an independent project if all negative project cash flows precede positive cash flows. If this is not the case, the IRR rule may lead to wrong decisions. Payback rule: This rule states that you should only accept a project if its cash flows pay back its initial investment within a prespecified period. To apply the payback rule, you first calculate the amount of time it takes for a project to pay back with its cash flows the initial investment, called the payback period. Then you accept the project if the payback period is equal or lower than a prespecified length of time - usually a few years. Otherwise, you reject the project. - The payback rule ignores (1) the cost of capital of the project and the time value of money, (2) cash flows after the payback and (3) is based on an ad hoc decision criterion. , the payback rule provides budgeting information about the length of time the capital will be committed to a project. Alternative projects: sometimes a company has to choose between several alternative projects. For example, a manager might consider alternative packaging solutions for a new product. When choosing any one project prevents the firm from undertaking the others, it is facing mutually exclusive investments. When projects are mutually exclusive, we need to determine which projects have a positive NPV and then rank the projects to identify the best one. In this situation, the NPV rule provides a straightforward answer: Pick the project with the highest NPV. Incremental IRR: when choosing between two mutually exclusive projects, an alternative to comparing IRRs is to calculate the incremental IRR, i.e. the IRR of the increase in cash flows that would be obtained by replacing one project with another; it is the discount rate at which it becomes optimal to switch from one project to another. The incremental IRR identifies the discount rate at which the optimal decision changes. However, when using the incremental IRR to choose between different projects you encounter the same problems described for the IRR rule: 1. Even if negative cash flows precede positive ones for individual projects, it is not certain that this will also be the case for the incremental cash flows; moreover, the incremental IRR may not even exist, or may not be unique. 2. The incremental IRR may indicate whether it is optimal to switch from one project to another, but it does not indicate whether the two projects have a positive NPV. 3. When individual projects have different opportunity costs of capital, it is not trivial to identify the opportunity cost of capital the incremental IRR should be compared to. In this case, only the NPV rule will provide a reliable answer. Project Selection with Resource Constraints In principle, a firm should make all investments with positive NPV that it can identify. In practice, there may be limits on the number of projects it can undertake. In some situations, different investment opportunities may require different amounts of a particular resource. If the availability of the resource is constrained, so that it is not possible to realize all the available opportunities, the firm must choose the best set of investments that it can make based on the resources available within it. Budget constraints force managers to choose projects that maximize NPV while fulfilling these constraints. - Suppose you consider the three projects in the table. Without budget constraints, you would invest in all projects. Suppose instead you have a budget of $100 million to invest. Project 1 has the highest NPV but uses the entire budget. With a budget of $100 million, the best choice is to carry out projects II and III with a NPV of $130 million. C4 Implementing the NPV Methodology: Cash Flows and Project Analysis 1. Forecasting Earnings a. Capital budget: lists the projects and investments that a company plans to undertake b. Capital budgeting: process used to analyse alternate investments and decide which ones to accept c. Incremental earning: the amount by which the firm’s earnings are expected to change as a result of the investment decision 2. Revenue and cost estimates 3. Revenue: price times quantity 4. Costs: up-front costs + annual overhead + per unit costs Example – Home Net 1. Capital expenditures and depreciation - Straight line depreciation: the asset’s cost is divided equally over its life 2. Interest expenses - In capital budgeting decisions, interest expense is typically not included - The rationale is that the project should be judged on its own, not on how it will be financed 3. Taxes - Marginal corporate tax rate: the tax rate on the marginal or incremental dollar of pre-tax income. (A negative tax is equal to a tax credit) - Income tax= EBITx tc - Unlevered net income calculation - EBIT x(1- tc) - (Revenue costs depreciation)x(1- tc) Example Kellogg’s Indirect Effects on Incremental Earnings 1. Opportunity cost - The value a resource could have provided in its best alternative use 1. Project externalities - Indirect effects of the project that may affect the profits of other business activities of the firm. Cannibalization is when sales of a new product displace sales of an existing product. - E.g., In the Home Net project example, 25% of sales come from customers who would have purchased an existing Linksys wireless router if Home Net were not available. Because this reduction in sales of the existing wireless router is a consequence of the decision to develop Home Net, we must include it when calculating Home Net’s incremental earnings. Sunk Costs and Incremental Earnings - Sunk costs: (Unrecoverable) costs that have been or will be paid regardless of the decision whether or not the investment is undertaken. Sunk costs should not be included in the incremental earnings analysis. - Fixed Overhead Expenses: Typically, overhead costs are fixed and not incremental to the project and should not be included in the calculation of incremental earnings. - Past Research and Development Expenditures: Money that has already been spent on R&D is a sunk cost and therefore irrelevant. The decision to continue or abandon a project should be based only on the incremental costs and benefits of the product going forward. Real world complexities Typically: - Sales will change from year to year - The average selling price will vary over time - The average cost per unit will change over time E.g., product adoption and price changes Home Net 3 Calculating the free cash flow from earnings 1. Capital expenditure and depreciation - Capital expenditures are the actual cash outflows when an asset is purchased. (Included in calculating free cash flows) - Depreciation is a non-cash expense. The free cash flow estimate is adjusted for this non-cash expense 2. Net working capital (NWC) = Current assets - current liabilities = cash + inventory + receivables - payables - Most projects will require an investment in net working capital - The increase in NWC is defined as NWC1- NWC t-1 E.g. Rising Star Inc is forecasting that their sales will increase by $250,000 next year, $275,000 the following year, and $300,000 in the third year. The company estimates that additional cash requirements will be 5% of the change in sales, inventory will increase by 7% of the change in sales, receivables will increase by 10% of the change in sales, and payables will increase by 8% of the increase in sales. Forecast the increase in net working capital for Rising Star over the next three years. Calculating free cash flow directly (The term tc × Depreciation is called the depreciation tax shield) CF 5 capital structure theory The value of the firm given corporate taxes only 0 1 2 3 Sales Forecast (increase) $250,000 $275,000 $300,000 Net Working Capital Forecast Cash Requirements (5% of sales) $12,500 $13,750 $15,000 Inventory (7% of sales) $17,500 $19,250 $21,000 Receivables (10% of sales) $25,000 $27,500 $30,000 Payables (8% of sales) $20,000 $22,000 $24,000 Net Working Capital $35,000 $38,500 $42,000 Year Unlevered Net Income Free Cash Flow (Revenues Costs Depreciation) (1 ) Depreciation CapEx = − −  −  + − −  c NWC Free Cash Flow (Revenues Costs) (1 ) CapEx Depreciation = −  −  − −  +   c c NWC MM’s Assumptions Modigliani and miller wrote the seminal papers on the costs of capital, corporate valuation, and capital structure. The explicit or implicit assumptions of their model are the following: - Capital markets are frictionless - Individuals can borrow and lend at the risk-free rate - There are no costs to go bankrupt or in the case of business disruption - Firms issue only 2 types of claims: risk-free debt and (risky) equity - All firms are assumed to be in the same (operating) risk class - Corporate taxes are the only form of government levy (no wealth taxes on corporations and no personal taxes) - All cash flow streams are perpetuities (no growth is allowed) - Corporate insiders and outsiders have the same information (no signalling opportunities) - Managers always maximise shareholders’ wealth (no agency cost) - Operating cash flows are completely unaffected by changes in capital structure Many of these assumptions are unrealistic, but it can be shown that relaxing many of them does not really change the major conclusions of the model of firm behaviour that MM provide - corporate debt is risk-free will not change the results. The same information assumption rules out any signalling behaviour and shareholder wealth maximization (as management’s goal) does not allow for the existence of agency costs because managers will never seek to maximize their own benefits. - no bankruptcy costs and no personal taxes are critical and the last assumption is also crucial as the operating cash flows are not actually independent of capital structure, with the result that investment and financing decisions should be thought of as co-determinant. 1st theorem: the option or combination of options that a company chooses has no effect on its real market value Assumption: All Firms Have the Same Risk What is meant when M-M say that all firms have the same risk class? The implication is that the expected risky future cash flows from operations (CFi) vary by at most a (constant) scale factor (λ). Mathematically, this takes the following form: This implies that the expected future cash flows from any two firms (or projects) are perfectly correlated If instead of focusing on the level of cash flow, we focus on the returns, the perfect correlation becomes obvious because the returns are identical. Let us prove that: But because The Value of the Unlevered Firm Suppose the assets of a firm return the same distribution of net operating cash flows each time period for an infinite number of time periods (based upon the assumption that all cash flow streams are perpetuities). This is a “no-growth” situation because the average cash flow does not change over time. We can value this after-tax stream of cash flows by discounting its expected value at the appropriate risk-adjusted rate. The value of the unlevered firm (a firm with no debt) will be: where: Vu= the present value of an unlevered firm E (FCF)= the perpetual free cash flow after taxes P (ro)= the discount rate for an all-equity firm of equivalent risk This is the value of an unlevered firm because it represents the discounted value of a perpetual, nongrowing stream of free cash flows after taxes that would accrue to shareholders if the firm had no debt. If 2 streams of CFs differ by a scale factor, they will have the same distribution of returns, the same risk, and will require the same expected return - The figure illustrates this relationship. It shows how each dollar of pretax cash flows is divided. The firm uses some fraction to pay taxes, and it pays the rest to investors. - By increasing the amount paid to debtholders through interest payments, the amount of the pretax cash flows that must be paid as taxes decreases. The gain in total cash flows to investors is the interest tax shield. - Because the cash flows of the levered firm are equal to the sum of the cash flows from the unlevered firm plus the interest tax shield, by the Law of One Price the same must be true for the present values of these cash flows. - Thus, letting Vl and Vu represent the value of the firm with and without leverage, respectively, we have the following change to MM Proposition I in the presence of taxes: MM Proposition I with Taxes: The total value of the levered firm exceeds the value of the firm without leverage due to the present value of the tax savings from debt. The tax interest shield with permanent debt - Typically, the level of future interest payments is uncertain due to changes in the marginal tax rate, the amount of debt outstanding, the interest rate on that debt, and the risk of the firm. - For simplicity, we will consider the special case in which the above variables are kept constant. - Suppose a firm borrows debt D and keeps the debt permanently. If the firm’s marginal tax rate is c , and if the debt is riskless with a risk-free interest rate rf, then the interest tax shield each year is c × rf × D, and the tax shield can be valued as a perpetuity. PV (interest tax shield) = tc x interest/rf = tc x (rf x D)/rf = tc x D If the debt is fairly priced, no arbitrage implies that its market value must equal the present value of the future interest payments. Market value of debt = D x PV (future interest payments) If the firm’s marginal tax rate is constant, then: PV (interest tax shield) = PV (tc x future interest payments) = tc x PV(future interest payments)= tc x D The Weighted Average Cost of Capital (WACC) with Taxes - The tax benefit of leverage can also be expressed in terms of the weighted average cost of capital (WACC). When a firm uses debt financing, the cost of the interest it must pay is offset to some extent by the tax savings from the interest tax shield. With tax-deductible interest, the effective after-tax borrowing rate is r(1 −  c ). We can account for the benefit of the interest tax shield by calculating the WACC using the effective after-tax cost of debt. The WACC represents the effective cost of capital to the firm after including the benefits of the interest tax shield: r wacc = E/E+D x re + D/E+D x rd (1-tc) It is therefore lower than the pretax WACC, which is the average return paid to the firm’s investors. A firm’s pretax WACC equals its unlevered cost of capital and depends only on the risk of the firm’s assets. r wacc = E/E+D x re + D/E+D x rd – D/E+D x rD tc = pretax WACC – reduction due to interest tax shield Higher the debt, higher tax shield The higher the firm’s leverage, the more the firm exploits the tax advantage of debt, and the lower its WACC. This figure illustrates this decline in the WACC with the firm’s leverage ratio. The MM definition of WACC In 1963, MM provided the definition for the Weighted Average Cost of Capital (WACC) of the firm: A debated question is the correct interpretation of the leverage ratio B/I M-M (1963) provide the following interpretation: “If B*/V* denotes the firm’s long-run “target” debt ratio...then the firm can assume, to a first approximation at least, that for any particular investment dB/dI = B*/V* “ How do we determine the cost of the 2 components of the WACC, debt and equity? - The cost of debt is the risk-free rate, at least given the assumptions of the MM Proposition I - The cost of equity capital – given the undertaking of a new investment - is the change in the return to equity holders with respect to the change in their investment required by the new capital project: - The return to equity holders is the net cash flow after interest and taxes (NI). Therefore, the rate of return for equityholders is: Ks = cost of equity of the leveraged firm = ro+ tax shield x (ro-kb) x leveraged ratio Ks positively related to leveraged ratio, as debt increases risk is higher If the firm has no debt in its capital structure, the levered cost of equity capital (ks) is equal to the cost of equity for an all-equity firm ( ): Ks The MM’ II Proposition (Second Theorem) The opportunity cost of capital of shareholders (ks) increases linearly with the debt-to- equity ratio of the firm The cost of equity capital of a levered firm is a function of 4 factors 1. Cost of equity capital for an unlevered firm 2. Cost of debt 3. Debt-to-equity ratio of the firm 4. Corporate tax rate The cost of capital and its components are represented graphically as a function of the debt-to-equity ratio. - in a world without corporate taxes(a), WACC constant with respect to changes in capital structure - In a world with corporate taxes (b), the WACC decreases as more and more debt is used in the capital structure of the firm. In both cases, the cost of equity increases with higher percentages of debt (MM’s Second Theorem). This makes sense because increased leverage implies a riskier position for shareholders as their (minimum) required rate of return on capital provided to the firm becomes more variable. Thus, they will require a higher rate of return to compensate for the extra risk they take on. nSS + 0 nSS NI +  0 - In particular, the assets side approach (coordination of firm’s assets, intended to capture managerial capabilities, acquisition of control’s stake) is usually employed in valuations related to deal-making in M&A contexts, while the equity side approach is used to evaluate stocks and minority shareholdings Valuing Firms: Entity vs Equity Approach Cash flows and cost of capital must always match: - If the entity approach to valuation is applied (with the objective of valuing the firm’s total operating assets that generate the FCFO to be distributed to debtholders and equity holders), the WACC is the appropriate cost of capital; - If the equity approach to valuation is applied (with the objective of assessing the value of the equity capital in the perspective of the sole shareholders that will residually receive the FCFE), the cost of equity is the appropriate cost of capital DCF: cash flow - The cash flow (DCF’s first component) can be defined as the amount of financial resources management would be able to generate over a certain period. More specifically, a firm may produce 3 different types of cash flow: - Free Cash Flow from Operations (FCFO); - Free Cash Flow to Equity (FCFE); - Debt Cash Flow - The simplest notion of cash flow is the latter (debt cash flow), which is equal to the sum of the interests to be paid on the debt the firm has contracted and of the related principal borrowed OPERATING CAPITAL (assets) SHARE- HOLDERS EQUITY DEBTS Enterprise Value Equity Value - Such cash flow is used to determine the present value or market value of debt by discounting the interest (+ principal) payments at the cost of debt. If the cost of debt chosen as discount rate is not a market rate but the actual average cost incurred by the company, the book value of debt is determined. The book value of debt may be considered as a good proxy for its market value - To understand the other 2 types of cash flow, it is necessary to recall the logics and building blocks of a firm’s financial statements Corporate Accounting Statements - There are 3 basic accounting statements that summarize information about a firm. These financial statements provide the fundamental information we use to perform a company valuation - Accountants and evaluators will answer these questions differently. Indeed, their objectives are different. Accountants attempt to measure the current standing and immediate past performance of a firm, whereas valuation is much more forward-looking From FCFO to FCFE and Vice versa We can now provide the definitions of the Free Cash Flow from Operations and Free Cash Flow to Equity The Free Cash Flow from Operations (or Free Cash Flow to Firm) represents the cash produced by the firm’s core business, that is the cash flow remaining after operational costs, tax payments, fixed and working capital investments or disinvestments are made, but before the impact of capital structure (that is, gross of debt). It is the cash flow obtained assuming that no debt exists. Hence, it is also named as unlevered cash flow The Free Cash Flow to Equity (or Net Cash Flow Available to Shareholders) is the cash flow left over after operational costs, tax payments, fixed and working capital investments or disinvestments are made, as well as after the reimbursement of old debt or contraction of new debt (thus including interest and principal payment). This cash flow differs from the FCFO due to the consideration of the firm’s capital structure choices numerically represented by the net financial charges (interests) and debt face value (net) variations. The former identifies the monetary cost of debt; the latter account for both the cash in-flows related to contraction of new debt and cash out-flows arising from reimbursement of outstanding debt. Obviously, debt cash flows reduce the financial resources distributable to shareholders in the form of dividends. The FCFE is also named as levered cash flow The FCFE can be easily obtained by subtracting interests and principal reimbursements from the FCFO and by adding up new debt contractions to the FCFO: FCFE= FCFO – (interest payments x (1-tc)) – principal repayments + new debt As a result, the FCFE is lower than the FCFO for a levered firm, while FCFO and FCFE are identical in the absence of debt The table summarizes the procedure that should be followed to build up the FCFO and the FCFE of a firm being valued starting from both the Income Statement and the Balance Sheet (as previously reclassified according to the functional principle). This procedure involves some operations of neutralization of all those distortions that accounting rules produce on the ideal cash flow creation process within the firm The last part, if reversed, also highlights the set of enumerations due to the firm’s capital providers: shareholders and debt holders: DCF: discount rate The second component of the DCF is the discount rate. It performs 2 tasks: 1. measures the financial value of time according to the logics of discounting; (=) net EBIT (NOPAT) (+) Amortization and Depreciation (-) Net Financial Charges [OF x (1 – tc)] (+/-) Debt Variation COMPUTATION OF FIRM’S FREE CASH FLOWS Revenues (-) Operational Costs (=) EBITDA (-) Amortization and Depreciation (=) EBIT (-) Operational Taxes (EBIT x tc) (=) Current Working Capital Cash Flow (+/-) Working Capital Requirements Variation (=) Current Cash Flow (-) Net Investments (=) Free Cash Flow from Operations (=) Free Cash Flow to Equity ( (=) Free Cash Flow from Operations (+) Net Financial Charges [OF x (1 – tc)] (-) Contraction of New Debt (+) Reimbursement of Old Debt Free Cash Flow to Equity Cash Flow due to shareholders Cash Flow due to debtholders 2. incorporates the systematic riskiness (or uncertainty) affecting the expected cash flows according to the Risk- Adjusted Rate of Return Valuation formula In the firm valuation context, the discount rate is named as cost of capital as it reflects the minimum rate of return required by the company’s capital providers (shareholders and debtholders) to accept the risk of investing their own funds in the firm itself. More specifically, equity holders require a rate of return that is named as cost of equity (Ke). The cost of equity is typically determined by using the Capital Asset Pricing Model (CAPM): Where P (ro)= systematic risk - Debtholders require a rate of return that is named as cost of debt (Kd) - The “total” rate of return that can be considered as acceptable by both equity holders and debtholders for them to finance the firm’s total operating assets is the Weighted Average Cost of Capital (WACC): where: E = market value of equity; D = market value of debt; tc = tax rate The WACC is the weighted average cost of equity and debt provided to the firm, where the weights are given by the ratio of each form of capital to the total sum of capital. Weights may be market or book-based ones The Terminal Value: The Gordon Model Since one cannot estimate cash flows forever, one generally imposes closure in DCF valuation by stopping the estimation of cash flows sometime in the future and then computing a terminal value that reflects the value of the firm at that point in time: - Time is indeed the third factor to be considered in applying the DCF. The time horizon of valuation should be regarded as being infinite based upon the expectations over a firm’s lifetime. Yet, the time horizon of valuation is typically divided into 2 different periods: - explicit forecast period – a time span of definite duration (3-5 years) during which future cash flows can be easily forecasted; - synthetic forecast period – a time span of indefinite duration beyond which forecast ability of future cash flows is no longer possible. Such pattern of cash flows is replaced by a perpetual or steadily growing average free cash flow The terminal value (or continuing value) reflects the value of the firm at the end of the explicit forecast period The most common approach to assessing terminal value is based on valuation of firm as a going concern at the time of the terminal value estimation. Such approach assumes that the cash flows of the firm will grow, beyond the terminal year, at a constant rate forever: a stable growth rate. If the growth is considered stable, the terminal value can be estimated using a “perpetual growth model.” A type of perpetual growth model that is commonly used in practice is the Gordon Model (1956): The cash flow and the discount rate used in the Gordon formula will depend on whether one is valuing the firm or the equity of the firm - Under the equity approach to valuation, the terminal value of equity can be written as:  RfRERfREk Miie −+== )()(  E + D WACC = Ke x E + Kd x (1-tc) x E + D D T c T t T c t k VALUETERMINAL k CF W )1( )1(1 + + + = = Explicit Forecast Period 0 n Synthetic Forecast Period ( Once we have calculated the realized annual returns, we can compare them to see which investments performed better in a given year. - The Microsoft stock outperformed the S&P 500 in 2007, 2009 and from 2013 to 2017. Also, in 2008, Treasury Bills performed better than both Microsoft stock and the S&P 500. Note the overall tendency for Microsoft’s return to move in the same direction as the S&P 500. - The histogram in the following slide plots the annual returns for each investment. - The height of each bar represents the number of years that the annual returns were in each range indicated on the x-axis. When we plot the probability distribution in this way using historical data, we refer to it as the empirical distribution of the returns. Average Annual Returns The average annual return of an investment during some historical period is simply the average of the realized returns for each year. - Notice that the average annual return is the balancing point of the empirical distribution – in this case, the probability of a return occurring in a particular range is measured by the number of times the realized return falls in that range. Therefore, if the probability distribution of the returns is the same over time, the average return provides an estimate of the expected return. - Using the data from previous table, the average annual return for the S&P 500 from 2005 to 2017 is as follows: R = 1/13 (0.049 + 0.158 + 0.055 − 0.37 + 0.265 + 0.151+ 0.021+ 0.160+ 0.324 + 0.137 + 0.014 + 0.120 + 0.218) =10.0% The Variance and Volatility of Returns The variability of the returns is very different for each investment: - the distribution of returns in small cap shows the widest spread; - the large stocks of the S&P 500 have returns that vary less than those of small stocks, but much more than the returns of corporate bonds or US Treasury Bills. - To quantify this difference in variability, we can estimate the standard deviation of the probability distribution, by using the empirical distribution to derive this estimate. Using the same logic as we did with the mean, we estimate the variance by computing the average squared deviation from the mean. The only difficulty is that we do not actually know the mean, so indeed we use the best estimate of the mean: the average realized return: Var(R)= - If the returns used in the above equation are not annual returns, the variance is typically converted to annual terms by multiplying the number of periods per year. For example, when using monthly returns, we multiply the variance by 12 and, equivalently, the standard deviation by 12. Using Past Returns to Predict the Future: Estimation Errors To estimate the cost of capital for an investment, we need to determine the expected return that the investors will require to compensate them for that investment’s risk. If the distribution of past returns and the distribution of future returns are the same, we could look at the return investors expected to earn in the past on the same or similar investments, and assume they will require the same returns in the future. This approach implies two different difficulties: 1. We do not know what investors expected in the past; we can only observe the actual returns that were realized. If we believe that investors are neither overly optimistic or pessimistic on average, then over time, the average realized return should match investors’ expected return. 2. The average return is just an estimate of the true expected return, and is subject to estimation error. Given the volatility of stock returns, this estimation error can be large even with many years of data, as we will see next. We measure the estimation error of a statistical estimate by its standard error. The standard error is the standard deviation of the estimated value of the mean of the actual distribution around its true value; The standard error provides an indication of how far the sample average might deviate from the expected return. - If the distribution of a stock’s return is identical each year, and each year’s return is independent of prior years’ return, then we calculate the standard error of the estimate of the expected return as follows: o SD (Average of Independent, Identical Risks) = o SD (Individual Risk) Number of Observations o 95% Confidence Interval o Historical Average Return ± (2 × Standard Error) If we believe the distribution may have changed over time and we can use only more recent data to estimate the expected return, then the estimate will be even less accurate. ➢ Individual stocks tend to be even more volatile than large portfolios, and many have been in existence for only a few years, providing little data with which to estimate returns. Because of the relatively large estimation error in such cases, the average return investors earned in the past is not a reliable estimate of a security’s expected return. Instead, we need to derive a different method to estimate the expected return that relies on more reliable statistical strategy. The Historical Trade-Off Between Return and Risk - Table 10.5 Volatility Versus Excess Return of U.S. Small Stocks, Large Stocks (S&P 500), Corporate Bonds, and Treasury Bills, 1926–2017 The Table shows the volatility and the excess return for each investment. The excess return is the difference between the average return for the investment and the average return for Treasury Bills, a risk-free investment, and measures the average risk premium investors earned for bearing the risk of investment. In the Figure we plot the average return versus the volatility of different investments as per the Table in the previous slides. Note the positive relationship: the investments with higher volatility have rewarded investors with higher average returns. Investors are risk averse. Riskier investments must offer investors higher average returns to compensate them for the extra risk they are taking on. The Trade-Off Between Return and Risk- Individual Stocks ➢ While volatility seems to be a reasonable measure of risk when evaluating a large portfolio, it is not adequate to explain the returns of individual securities. Firm-Specific Risk vs Systematic Risk ➢ In order to better understand firm-specific risk and systematic risk, we should consider the different types of risks our houses are exposed to. - Common risk - Independent risk Over any given time period, the risk of holding a stock is that the dividends plus the final stock price will be higher or lower than expected; What causes dividends or stock prices, and therefore returns, to be higher or lower than we expect? Usually, stock prices and dividends fluctuate due to two types of news: 1. Firm-specific news, is good or bad news about the company itself (e.g., gaining market share within its industry); 2. Market-wide news, is news about the economy as a whole and therefore affects all stocks. - Fluctuations of a stock’s return that are due to firm-specific news are independent risk. This type of risk is also referred to as firm-specific, idiosyncratic, unsystematic, unique, or diversifiable risk. - Fluctuations of a stock’s return that are due to market-wide news represent common risk. This type of risk is also called systematic, undiversifiable, or market risk. When we combine many stocks in a large portfolio, the firm-specific risks for each stock will average out and be diversified. The systematic risk, however, will affect all firms – and therefore the entire portfolio- and will not be diversified. Diversification in Stock Portfolios Firm-Specific Versus Systematic Risk. Consider two types of firms: 1. Type I firms are affected only by firm-specific risks - Their returns are equally likely to be 35% or −25%, based on factors specific to each firm’s local market. Because these risks are firm specific, if we hold a portfolio of the stocks of many types I firms, the risk is diversified. About half of the firms will have returns of 35% and half will have returns of – 25%, so that the return of the portfolio will be close to the average return of 50% (0.35) + 50% (- 0.25) = 5%. - Actual firms are affected by both market-wide risks and firm- specific risks. - When firms carry both types of risk, only the unsystematic risk will be diversified when many firm’s stocks are combined into a portfolio. - The volatility will therefore decline until only the systematic risk remains. CF 8 - Portfolio Theory Optimal Portfolio Choice - how an investor can choose an efficient portfolio demonstrate how to find the optimal portfolio for an investor who wants to earn the highest possible return given the level of volatility he (or she) is willing to accept by developing the statistical techniques of mean-variance portfolio optimization. - assumptions of the Capital assets pricing model (CAPM), the most important model of the relationship between return and risk. Under these assumptions, the efficient portfolio is the market portfolio of all stocks and securities. As a result, the expected return of any security depends upon its beta with the market portfolio. - how to calculate the expected return and volatility of the portfolio. - how an investor can create an efficient portfolio out of individual stocks, and consider the implications, if all investors attempt to do so, for an investment ’s expected return and the cost of capital of an investment project. - The CAPM is the main method used by most major corporations to calculate the cost of capital. The Return of a Portfolio To find an optimal portfolio, we need a method to define a portfolio and analyse its return. We can define a portfolio by its weights, that is the fraction of the total investment in the portfolio held in each individual investment in the portfolio: Xi = Value of investment, Total value of portfolio - These portfolio weights add up to 1 or 100% (σ𝑖 𝑥𝑖 = 1), so that they represent the way money is divided between the different individual investments in the portfolio. - Consider a portfolio with 200 shares of the Walt Disney Company worth $30 per share and 100 shares of Coca-Cola worth $40 per share. The total value of the portfolio is 200 x $30 + 100 x $40 = $10.000 and the corresponding portfolio weights are: XD =200×$30=60.00%, XC =100×$40=40.00% Given the portfolio weights, the return of the portfolio is calculated as follows. Suppose 𝑥1,𝑥2,... 𝑥𝑛 are the portfolio weights of the 𝑛 investments in a portfolio, and these investments have returns 𝑅1, 𝑅2,... 𝑅𝑛. Then, the return on the portfolio 𝑅𝑃 is the weighted average of the returns on the investments in the portfolio, where the weights correspond to portfolio weights (where the weights are the quotas in which the portfolio is divided): Eq. 11.2 Rp= The Expected Return of a Portfolio The prior equation also allows to compute the expected return of a portfolio. Using the facts that the expectation of a sum is just the sum of the expectations and that the expectation of a known multiple is just the multiple of its expectation, we arrive at the following formula for a portfolio’s expected return: E(Rp)= That is, the expected return of a portfolio is simply the weighted average of the expected returns of the investments within it, using the portfolio weights (expected returns of individual investments are weighted with the weights associated with the related investments). - In particular, the lower the correlation, the lower the volatility we can obtain. As we lower the correlation and therefore the volatility of the portfolios, the curve showing the portfolios will bend to the left to a greater degree. This effect is illustrated in the next slide’s Figure. - When the stocks are perfectly positively correlated, we can identify the set of portfolios by the straight line between them. In this extreme case (red line), the volatility of the portfolio is equal to the weighted average volatility of the two stocks: there is no diversification. - When the correlation is less than 1, instead, the volatility of the portfolios is reduced due to diversification, and the curve bends to the left. The reduction in risk (and the bending of the curve) becomes greater as the correlation decreases. At the other extreme of perfect negative correlation (blue line), the line again becomes straight, this time reflecting off the vertical axis. In particular, when the two stocks are perfectly negatively correlated, it becomes possible to hold a portfolio that bears absolutely no risk. The Volatility of a Large Portfolio The variance of a portfolio is equal to the weighted average covariance of each stock with the portfolio: Var(RP)= which reduces to Var(RP)= Diversification with General Portfolios For a portfolio with arbitrary weights, the standard deviation is calculated as follows: - Volatility of a Portfolio with Arbitrary Weights SD(RP)= Unless all of the stocks in a portfolio have a perfect positive correlation of +1 with one another, the risk of the portfolio will be lower than the weighted average volatility of the individual stocks: SD(RP)= CF9 – PORTFOLIO THEORY 2 Short Sales • Long Position o A positive investment in a security • Short Position o A negative investment in a security o In a short sale, you sell a stock that you do not own and then buy that stock back in the future. o Short selling is an advantageous strategy if you expect a stock price to decline in the future. Efficient Portfolios with Many Stocks ➢ When the set of investment opportunities increases from two to three stocks, the efficient frontier improves. Visually, the old frontier with any two stocks is located inside the new frontier. ➢ In general, adding new investment opportunities allows for greater diversification and improve the efficient frontier. The Figure in the next slides show the effect of increasing the set to 3 stocks. Even though the added stocks appear to offer inferior risk-return combinations on their own, because they allow for additional diversification, the efficient frontier improves with their inclusion. ➢ Thus, to arrive at the best possible set of risk and return opportunities, we should keep adding stocks until all investment opportunities are represented. Ultimately, based on our estimates of returns, volatilities, and correlations, we can construct the efficient frontier for all available risky investments showing the best possible risk and return combinations that we can obtain by optimal diversification. Consider adding Bore Industries to the two-stock portfolio: Although Bore has a lower return and the same volatility as Coca-Cola, it still may be beneficial to add Bore to the portfolio for the diversification benefits. Risk Versus Return: Many Stocks The efficient portfolios, those offering the highest possible expected return for a given level of volatility, are those on the northwest edge of the shaded region, which is called the efficient frontier for these three stocks. - In this case, none of the stocks, on its own, is on the efficient frontier, so it would not be efficient to put all our money in a single stock. Feasible and Efficient Portfolios The feasible set of portfolios represents all portfolios that can be constructed from a given set of stocks. An efficient portfolio is one that offers: - the most return for a given amount of risk, or – the least risk for a given amount of return. - The collection of efficient portfolios is called the efficient set or efficient frontier. Stock Expected return Volatility Correlation with Intel Coca cola Bore ind Intel 26% 50% 1.0 0.0 0.0 Coca cola 6% 25% 0.0 1.0 0.0 Bore industries 2% 25% 0.0 0.0 1.0 Optimal Investment Portfolio In the absence of risk-free borrowing and lending opportunities, any investor’s preferred investment portfolio lies on the efficient frontier of the risky investment set but the precise location depends on her/his level of risk- aversion. - The less risk-averse, the riskier the preferred portfolio of risky assets selected on the efficient frontier. Indifference Curves Indifference curves reflect an investor’s attitude toward risk as reflected in his or her risk/return trade-off function. They differ among investors because of differences in risk aversion. - An investor’s optimal portfolio is defined by the tangency point between the efficient set and the investor’s indifference curve. Risk-Free Saving and Borrowing - Thus far, we have considered the risk and return possibilities that result from combining risky investments into portfolios. By including all risky investments in the construction of the efficient frontier, we achieve maximum diversification. - There is another way besides diversification to reduce risk that we have not yet considered: we can keep some of our money in a safe, no-risk investment like Treasury Bills. Of course, doing so will reduce our expected return. Conversely, an aggressive investor, who is seeking high expected returns, might decide to borrow money to invest even more in the stock market. We will see that the ability to choose the amount to invest in risky vs. risk-free securities allows us to determine the optimal portfolio of risky stocks for an investor. Investing in Risk-Free Securities Consider an arbitrary risky portfolio P with returns 𝑅𝑃 on the best possible "efficient" frontier where the portfolios containing all existing risky securities are positioned. - Let’s look at the effect on return and risk of putting a fraction 𝑥 of our money in the portfolio, while leaving the remaining fraction (1 − 𝑥) in risk-free Treasury Bills with a yield of 𝑟. Let’s calculate the expected return and variance of this portfolio: 𝐸 [𝑅 𝑥𝑃]=(1−𝑥)𝑟𝑓 + 𝑥𝐸 [𝑅𝑃] =𝑟𝑓 +𝑥(𝐸[𝑅𝑃]−𝑟 𝑓) 1. The first equation simply states that the expected return of the portfolio is the weighted average of the expected returns of Treasury Bills and the risky portfolio (because we know upfront the current interest rate paid on Treasury Bills, we do not need to compute an expected return for them). 2. The second equation rearranges the first to give a useful interpretation: our expected return is equal to the risk-free rate plus a fraction of the portfolio’s risk premium, 𝐸[𝑅P] − 𝑟f, based on the fraction x that we invest in it. - let’s compute the volatility. The volatility of the risk-free investment is zero; the risk-free rate is known when we make the investment. Because the risk-free rate is fixed and does not move with (or against) our portfolio, its volatility and covariance with the portfolio are both zero. Thus, 10 - Capital Asset Pricing Model Portfolio Improvement: Beta and the Required Return Take an arbitrary portfolio 𝑃 and let’s consider whether we could raise its Sharpe Ratio by selling some of our risk-free assets (or borrowing money) and investing the proceeds in an investment 𝑖. If we do so, there are two consequences: - Expected Return: because we are giving up the risk-free return and replacing it with the return of stock 𝑖, our expected return will increase by 𝑖’s excess return 𝐸 [𝑅i ] – 𝑟𝑓 - Volatility: we will add the risk that 𝑖 has in common with our portfolio (the rest of 𝑖’s risk will be diversified). Incremental risk is measured by 𝑖’s volatility multiplied by its correlation with 𝑃: 𝑆𝐷(𝑅𝑖) × 𝐶𝑜𝑟𝑟(𝑅𝑖, 𝑅𝑃) Is the gain in return from investing in 𝑖 adequate to make up for the increase in risk? Another way we could have increased our risk would have been to invest more in portfolio 𝑃 itself. In that case, 𝑃’s Sharpe Ratio, 𝐸[𝑅𝑃] – 𝑟𝑓/ 𝑆𝐷(𝑅𝑃) Because the investment in 𝑖 increases risk by 𝑆𝐷(𝑅𝑖) × 𝐶𝑜𝑟𝑟(𝑅𝑖, 𝑅𝑃), it offers a larger increase in return than we could have gotten from 𝑃 alone if: - The required return is the expected return that is necessary to compensate for the risk investment 𝑖 will contribute to the risk of portfolio 𝑃 based on CAPM. - The required return for an investment 𝑖 is equal to the risk-free interest rate plus the risk premium of the current portfolio 𝑃, scaled (multiplied) by 𝑖’s sensitivity to 𝑃, 𝛽𝑃. Therefore, if 𝑖’s expected return exceeds this required return, then adding more of it will improve the performance of the portfolio. Expected Return and Efficient Portfolio As we buy shares of security 𝑖, its correlation with our portfolio will increase, ultimately raising its required return until 𝐸 𝑅 = 𝑟 . At this point, our holdings of security 𝑖 are optimal. Similarly, if security 𝑖’s expected return is less than the required return 𝑟 , we should reduce our holdings of 𝑖. As we do so, the correlation and the required return 𝑟 will fall until 𝐸 𝑅 =𝑟 - Thus, if we have no restrictions on our ability to buy or sell securities that are traded in the market, we will continue to trade until the expected return of each security equals its required return – that is, until 𝐸 𝑅 = 𝑟 holds for all 𝑖. At this point, no trade can possibly improve the risk-reward ratio of the portfolio, so our portfolio is the optimal, efficient portfolio. - So, a portfolio is efficient if and only if the expected return of every available security equals its required return. This result implies the following relationship between the expected return of any security and its beta with the efficient portfolio: Determining the Risk Premium Under the CAPM assumptions, we can identify the efficient portfolio: it is equal to the market portfolio. Thus, if we do not know the expected return of a security or the cost of capital of an investment, we can use the CAPM to find it by using the market portfolio as a benchmark. The expected return of an investment is given by its beta with the efficient portfolio. But if the market portfolio is efficient, we can rewrite the previous equation as: The CAPM Equation for the Expected Return The beta of a stock measures its volatility due to market risk relative to the market, and thus captures the security’s sensitivity to market risk. We can interpret the CAPM equation as follows. Following the law of one price, in a competitive market, investments with similar risk should have the same expected return. Because investors can eliminate firm-specific risk by diversifying their portfolios, the right measure of risk is the investment’s beta with the market portfolio, 𝛽𝑖. The Security Market Line There exists a linear relationship between a stock’s beta and its expected return - Panel (b) of next slide’s Figure graphs this line through the risk-free investment (with a beta of 0) and the market portfolio (with a beta of 1); it is called the security market line (SML) and, under the CAPM assumptions, is the line along which all individual securities should lie when plotted according to their expected return and beta - This result contrasts the capital market line shown in Panel (a), where there is no clear relationship between an individual stock’s volatility and its expected return. As we showed for McDonald’s (MCD), a stock’s expected return is due only to the fraction of its volatility that is common with the market 𝐶𝑜𝑟𝑟 𝑅𝑀𝐶𝐷,𝑅𝑀𝑘𝑡 ×𝑆𝐷(𝑅𝑀𝐶𝐷); The distance of each stock to the right of the capital market line is due to its diversifiable risk. The relationship between risk and return for individual securities becomes evident only when we measure market risk (that is, systematic risk) rather than total risk. Summary of the Capital Asset Pricing Model The CAPM leads to two major conclusions. 1. The market portfolio is the efficient portfolio. Therefore, the highest expected return for any given level of volatility is obtained by a portfolio on the capital market line, which combines the market portfolio with risk- free saving (investment) or borrowing. 2. The risk premium for any investment is proportional to its beta with the market. Therefore, the relationship between risk and the required return is given by the security market line. - The CAPM model is based on strong assumptions; because some of these assumptions do not fully describe investors’ behavior, some of the model’s conclusions are not completely accurate; for example, it is certainly not the case that every investor holds the market portfolio. - Nevertheless, financial economists find the qualitative intuition underlying the CAPM compelling and persuasive, so it is still the most important model of return and risk; that is the most common model for pricing risk. While not perfect, it is widely regarded as a very useful approximation of the return-risk relationship and is used by firms and practitioners/investment bankers as a practical tool to estimate a stock’s expected return and an investment (project)’s cost of capital. CF 11 - Estimating the Cost of Capital and the Beta The Equity Cost of Capital The Capital Asset Pricing Model (CAPM) is a practical way to estimate. The cost of capital of any investment opportunity equals the expected return of available investments with the same beta. The estimate is provided by the Security Market Line equation: The Market Portfolio Market Capitalization: The total market value of a firm’s outstanding shares MV = (Number of Shares of i Outstanding) × (Price of i per Share) = N × P Value-Weighted Portfolio: A portfolio in which each security is held in proportion to its market capitalization Market indexes -> E.g.s: S&P, Wiltshire 5000, Dow Jones Industrial Average (DJIA) Investing in a Market Index - Index funds – mutual funds that invest in the S&P 500, the Wilshire 5000, or some other index - Exchange-traded funds (ETFs) – trade directly on an exchange but represent ownership in a portfolio of stocks o E.g.: SPDRS (Standard and Poor’s Depository Receipts) represent ownership in the S&P 500 o Most practitioners use the S&P 500 as the market proxy, even though it is not actually the market portfolio The Market Risk Premium - Determining the Risk-Free Rate o The yield on U.S. Treasury securities o Surveys suggest most practitioners use 10- to 30-year treasuries - The Historical Risk Premium o Estimate the risk premium (E[RMkt] − rf) using the historical average excess return of the market over the risk-free interest rate - Using historical data has two drawbacks: o Standard errors of the estimates are large o Backward looking, so may not represent current expectations - A Fundamental Approach o One alternative is to solve for the discount rate that is consistent with the current level of the index - r = Div + g = Dividend Yield + Expected Dividend Growth Rate Beta Estimation - Using Historical Returns: Recall, beta is the expected percent change in the excess return of the security for a 1% change in the excess return of the market portfolio - Beta corresponds to the slope of the best-fitting line in the plot of the security’s excess returns versus the market excess return - Using Linear Regression (The statistical technique that identifies the best-fitting line through a set of points. - (R −r )=α +β(R −r )+ε f Mkt f o αi is the intercept term of the regression o Βi RMkt − rf) represents the sensitivity of the stock to market risk. When the market’s return increases by 1%, the security’s return increases by βi% o εi is the error term and represents the deviation from the best-fitting line and is zero on average o αi represents a risk-adjusted performance measure for the historical returns. o If αi is positive, the stock has performed better than predicted by the CAPM. o If αi is negative, the stock’s historical return is below the SML Linear Regression: Given data for rf, Ri, and RMkt, statistical packages for linear regression can estimate βi. Interpreting beta - If beta = 1.0, stock is average risk. - If beta > 1.0, stock is riskier than average. - If beta < 1.0, stock is less risky than average. Interpreting Regression Results - The R2 measures the percent of a stock’s variance that is explained by the market. The typical R2 is: 0.3 for an individual stock, over 0.9 for a well-diversified portfolio. - The 95% confidence interval shows the range in which we are 95% sure that the true value of beta lies. The typical range is: o from about 0.5 to 1.5 for an individual stock. o from about .92 to 1.08 for a well-diversified portfolio. Competition and Capital Markets Identifying a Stock’s Alpha - To improve the performance of their portfolios, investors will compare the expected return of a security with its required return from the security market line. - The difference between a stock’s expected return and its required return according to the security market line is called the stock’s alpha. αs = E[Rs] – rs - When the market portfolio is efficient, all stocks are on the security market line and have an alpha of zero. - Profiting from Non-Zero Alpha Stocks: Investors can improve the performance of their portfolios by buying stocks with positive alphas and by selling stocks with negative alphas. If E Rs > rs then UNDERVALUED (𝛼s > 0) If E Rs = rs then FAIRLY VALUED (𝛼s = 0) If E Rs < rs then OVERVALUED (𝛼s < 0) The Debt Cost of Capital Debt Yields Versus Returns - Yield to maturity is the IRR an investor will earn from holding the bond to maturity and receiving its promised payments. - If there is little risk the firm will default, yield to maturity is a reasonable estimate of investors’ expected rate of return. - If there is significant risk of default, yield to maturity will overstate investors’ expected return. - Consider a one-year bond with YTM of y. For each $1 invested in the bond today, the issuer promises to pay $(1 + y) in one year. Suppose the bond will default with probability p, in which case bond holders receive only $(1 + y − L), where L is the expected loss per $1 of debt in the event of default. So the expected return of the bond is rd =(1−p)y+p(y−L)=y−pL = Yield to Maturity − Prob (default ) Expected Loss Rate