2 There are several problems with using P/E analysis: Earnings are historical cost accounting numbers and may be of differing quality. Business cycles may affect P/E ratios. Currently reported earnings may be quite different from your expectations of earnings in the future (E 1 ). Also, like the infinite growth model, when k < g, the model cannot be used. Explain the components of an investor’s required rate of return (i.e., the real risk-free rate, the expected rate of inflation, and a risk premium) and discuss the risk factors to be assessed in determining a country risk premium for use in estimating the required return for foreign securities. As we have discussed, the required rate of return on equity, k, is influenced by: The real risk-free rate (RFR real ), which is determined by the supply and demand for capital in the country. The real risk-free rate is the rate investors would require if there were absolutely no risk or inflation.
3 An inflation premium (IP), which investors require to compensate for their expected loss of purchasing power. A risk premium (RP) to compensate investors for the uncertainty of returns expected from an investment. Since different investments have different patterns of return and different guarantees, risk premiums can differ substantially. k = required rate of return = (1 + RFR real )(1 + IP)(1 + RP) – 1 k = required rate of return (approximate) ≈ RFR real + IP + RP The real risk-free rate and the inflation premium together comprise the nominal risk-free rate, RFR nominal. That is: RFR nominal = (1 + RFRreal)(1 + IP) – 1 This may be approximated as: RFR nominal = RFR real + IP Professor’s Note: A real rate is a rate that does not include inflation, while a nominal rate does. If a rate is not specified as being a real rate on the exam, it is safe for you to assume that it is a nominal rate.
4 The risk premium, RP, is a premium demanded for internal and external risk factors. Internal risk factors are diversifiable and include business risk, financial risk, liquidity risk, exchange-rate risk, and country risk. External risk factors, known as market risk factors, are macroeconomic in nature and are nondiversifiable. Example: Computing the nominal risk-free rate Calculate the nominal risk-free rate if the real risk-free rate is 4 percent and the expected inflation rate is 3 percent. Answer: RFR nominal = (1.04)(1.03) – 1 = 1.0712 – 1 = 7.12% Alternatively, the nominal rate is frequently approximated by summing the real rate and expected inflation: RFR nominal = 4% + 3% = 7%
5 The required rate of return on any investment is a combination of the nominal risk-free rate plus a risk premium. For equity investments, the risk premium can be determined by reference to a risk premium curve or by using the capital asset pricing model: k = RFR nominal + RP Using the capital asset pricing model (CAPM), we have: k = RFR + [E(R mkt ) – RFR] Professor’s Note: Notice here that RFR is a nominal rate. Estimating the Required Return for Foreign Securities Security valuation models and their variables are essentially the same all over the world. However, there are significant differences in the determination of these variables. To estimate the required rate of return for foreign securities, we can calculate the real risk-free rate, adjust it for the expected inflation rate, then determine the risk premium.
6 The country risk premium is estimated with consideration of five types of risk that will differ substantially from country to country. Business risk represents the variability of a country’s economic activity, along with the degree of operating leverage for firms within the country. Financial risk will be different in countries throughout the world. Liquidity risk is often found in countries with small or inactive capital markets. Exchange rate risk, the uncertainty in exchange rates, must always be taken into account when considering foreign investments. Country risk arises from unexpected economic and political events. Estimate the implied dividend growth rate, given the components of the required return on equity and incorporating the earnings retention rate and current stock price. Assuming past investments are stable and dividends are calculated to allow for maintenance of past earnings power, the firm’s earnings growth rate, g, can be defined as the firm’s earnings plowback or retention rate (RR) times the return on the equity (ROE) portion of new investments.
7 g = (RR)(ROE) Note that if RR is the earnings retention rate, (1 – RR) must be the firm’s dividend payout rate. Professor’s Note: Recall that we used the DuPont method to decompose ROE into its component parts: net profit margin × asset turnover × financial leverage = ROE. You can use these components, along with the retention rate, to calculate ROE × RR = g, the implied (sustainable) growth rate. Let’s work through an example to illustrate why g equals RR × ROE for a stable but expanding company. Example: Sustainable growth Assume ROE is constant and that new funds come solely from earnings retention. Calculate the firm’s growth rate, given that the firm earns 10 percent on equity of $100 per share and pays out 40 percent of earnings in dividends.
8 Answer: Period 1 per share earnings = EPS 1 = ROE × Equity per share = (0.10)($100) = $10 per share Period 1 dividend per share = D 1 = payout × EPS 1 = (0.40)($10) = $4.00 per share Period 1 retained earnings = RR 1 × EPS 1 = ($10)(1 – 0.4) = $6.00 per share so, Period 2 earnings per share = (0.10)($100) +(0.10)($6) = $10.60 per share Period 2 dividend per share = D2 = (0.40)($10.60) = $4.24 per share Analysis of growth: Earnings growth = (EPS1 – EPS2) / EPS1 = ($10.60 – $10) / $10 = 6% Dividend growth = ($4.24 – $4) / $4 = 6% Analysis of stock price: assume k = 10% Price at the beginning of period 1 = D 1 /(k – g c ) = $4.00/(0.10–0.06) =$100 Price at the beginning of period 2 = D 2 /(k – g c ) = $4.24/(0.10–0.06)= $106
9 The stock’s price will grow at a 6 percent rate, just as earnings and dividends will. growth = g c = (ROE) (Retention rate) = (0.1)(1 – 0.4) = 6% The growth rate here, gc = ROE × RR, is called the internal or sustainable growth rate—the rate of growth sustainable without resorting to external sources of capital (relying on retained earnings only). So, what we know about dividend growth can be summarized as follows: If a firm’s profit margin increases, ROE will increase. If ROE increases, g, which is (ROE)(RR), will increase. If g increases, the difference between k and g will decrease. If k – g decreases, the price of the stock will increase. Describe a process for developing estimated inputs to be used in the DDM, including the required rate of return and expected growth rate of dividends. As we have indicated, the DDM holds that the value of a share of stock is the present value of its cash flows.
10 Thus, the DDM requires the following three inputs: An estimate of the stock’s future cash flows, which are dividends and future price. A dividend growth rate, g. A discount rate, which is the appropriate required return on equity, k. Once the present value of the asset has been estimated, compare it to the current market price. Example: Application of DDM Assume you are analyzing the XYZ company. Its current stock price is $18.00. After reviewing XYZ’s financial data, you find that last year’s earnings were $2.00 per share. The firm’s ROE is 10 percent, and you expect it to stay that way for the foreseeable future. The firm has a stable dividend payout policy of 40 percent. The current nominal risk-free rate is 7 percent, the expected market return is 12 percent, and XYZ’s beta is 1.2. Calculate the value of XYZ and indicate whether this stock is a “buy” based on your estimate.
11 Answer: Step 1: Determine the required rate of return: k = 0.07 + 1.2(0.12 – 0.07) = 13% Step 2: Determine the growth rate: Step 2a: RR = (1 – dividend payout) = 1 – 0.4 = 0.6 Step 2b: g = (RR)(ROE) = (0.6)(0.10) = 0.06 or 6% Step 3: Determine last year’s dividend: D 0 = E 0 (dividend payout ratio) = $2(0.4) = $0.80 Step 4: Determine next year’s dividend: D 1 = D 0 (1 + g c ) = $0.80(1 + 0.06) = $0.85 Step 5: Estimate the value: V 0 = D 1 /(k – g c ) = $0.85/(0.13 – 0.06) = $12.14 Professor’s Note: Rounding differences may occur, not unlike those you might encounter on the exam. Step 6: Compare the stock’s value to its current market price: $12.14 vs. $18.00
12 Do not buy and possibly sell this stock short. If estimated value > market price → buy If estimated value < market price → don’t buy Key Concepts 1. The top-down approach to security valuation has three steps: Forecast the influence of the general economy on the securities markets. Analyze the prospects for the various industries under your economic forecast. Analyze the individual firms in the industries under your economic forecast. 2. The returns from any investment can be measured as price change (capital gain/loss), cash income (i.e., interest, dividends, rental income, etc.), earnings, or a variety of cash flow measures for equities. 3. The preferred stock valuation model:
13 4. The calculation of the value of common stock can take different forms: One period stock valuation model: A multiple-year holding period: Infinite period model: 5. For a firm with supernormal growth (g1) over n periods followed by a constant growth rate of dividends forever (g2) can be valued as: where:D 1 = D 0 (1 + g 1 ) … D n = D 0 (1 + g 1 ) n and D n + 1 = D n (1 + g 2 )
14 6. By dividing both sides of the infinite period DDM by E 1, it can used as an earnings multiplier model: 7. The relationship between the nominal risk-free rate, the real risk-free rate, and the expected rate of inflation is: nominal risk-free rate = (1 + real risk-free rate)(1 + expected inflation) – 1. 8. The firm’s internal or sustainable growth rate, g, is equal to ROE × RR. RR is the firm’s retention rate, so (1 – RR) is the firm’s dividend payout rate.
15 INTRODUCTION TO PRICE MULTIPLES Discuss the rationales for the use of price to earnings (P/E), price to book value (P/BV), price to sales (P/S), and price to cash flow (P/CF) in equity valuation and discuss the possible drawbacks to the use of each price multiple. Calculate and interpret P/E, P/BV, P/S, and P/CF. Professor’s Note: This review is organized according to the types of price multiples. The LOS are addressed within each category. Rationales for using price-to-earnings (P/E) ratios in valuation: Earnings power, as measured by earnings per share (EPS), is the primary determinant of investment value. The P/E ratio is popular in the investment community. Empirical research shows that P/E differences are significantly related to long-run average stock returns.
16 The drawbacks of using the P/E ratio are: Earnings can be negative, which produces a useless P/E ratio. The volatile, transitory portion of earnings makes the interpretation of P/E difficult for analysts. Management discretion within allowed accounting practices can distort reported earnings and thereby lessen the comparability of P/E ratios across firms. We can define two versions of the P/E ratio: trailing and leading P/E. The difference between the two is how earnings (the denominator) are calculated. Trailing P/E ratios use earnings over the most recent 12 months in the denominator. The leading P/E ratio (also known as forward or prospective P/E) uses “next year’s expected earnings,” which is defined as either expected earnings per share (EPS) for the next four quarters or expected EPS for the next fiscal year.
17 Professor’s Note: The trailing P/E is what we see published in much of the popular financial press. The leading P/E, P 0 /E 1, is the one we calculated from the DDM. Example: Calculating a P/E ratio Byron Investments, Inc., reported EUR32 million in earnings during fiscal year 2003. An analyst forecasts an EPS over the next 12 months of EUR1.00. Byron has 40 million shares outstanding at a market price of EUR18.00 per share. Calculate Byron’s trailing and leading P/E ratios. Answer:
18 Advantages of using the price-to-book ratio (P/BV) include: Book value is a cumulative amount that is usually positive, even when the firm reports a loss and EPS is negative. Thus, a P/BV can typically be used when P/E cannot. Book value is more stable than EPS, so it may be more useful than P/E when EPS is particularly high, low, or volatile. Book value is an appropriate measure of net asset value for firms that primarily hold liquid assets. Examples include finance, investment, insurance, and banking firms. P/BV can be useful in valuing companies that are expected to go out of business. Empirical research shows that P/BV ratios help explain differences in long-run average returns. Disadvantages of using P/BV include: P/BV ratios do not recognize the value of nonphysical assets such as human capital.
19 P/BV ratios can be misleading when there are significant differences in the asset intensity of production methods among the firms under consideration. Different accounting conventions can obscure the true investment in the firm made by shareholders, which reduces the comparability of P/BV ratios across firms and countries. For example, research and development costs (R&D) are expensed in the U.S., which can understate investment and overstate income over time. Inflation and technological change can cause the book and market value of assets to differ significantly, so book value is not an accurate measure of the value of the shareholders’ investment. This makes it more difficult to compare P/BV ratios across firms. The price-to-book ratio is defined as: where:book value of equity = common shareholders' equity = (total assets – total liabilities) – preferred stock
20 We often make adjustments to book value in order to allow the P/BV ratio to more accurately measure the value of the shareholders’ investment and to create more useful comparisons across different stocks. A common adjustment is to use tangible book value, which is equal to book value of equity less intangible assets. Examples of intangible assets include goodwill from acquisitions (which makes sense because it is not really an asset) and a patent (which is more questionable since the asset and patent are separable). Furthermore, balance sheets should be adjusted for significant off-balance-sheet assets and liabilities and for differences between the fair and recorded value of assets and liabilities. Finally, book values often need to be adjusted to ensure comparability. For example, companies using the first in, first out (FIFO) inventory accounting method cannot be accurately compared with peers using the last in, first out (LIFO) method. Thus, book values should be restated on a consistent basis. Example: Calculating a P/BV ratio Based on the information in the table, calculate the current P/BV for Cisco Systems Inc. and Oracle Corp.
21 Figure: Data for Cisco Systems and Oracle Corp. Answer: Cisco Systems Inc.: Oracle Corp.: CompanyBook Value of Equity 2003 (USD millions) Sales 2003 (USD millions) Shares Outstanding 2003 (millions) Price 08/14/03 Cisco Systems Inc.28,03918,8787,001USD 17.83 Oracle Corp.6,3209,4755,233USD 12.15
22 The rationales for using the P/S ratio include: P/S is meaningful even for distressed firms, since sales revenue is always positive. This is not the case for P/E and P/BV ratios, which can be negative. Sales revenue is not as easy to manipulate or distort as EPS and book value, which are significantly affected by accounting conventions. P/S ratios are not as volatile as P/E multiples. This may make P/S ratios more reliable in valuation analysis. P/S ratios are particularly appropriate for valuing stocks in mature or cyclical industries and for start-up companies with no record of earnings. Like P/E and P/BV ratios, empirical research finds that differences in P/S are significantly related to differences in long-term average stock returns. The disadvantages of using P/S ratios are: High growth in sales does not necessarily indicate operating profits as measured by earnings and cash flow. P/S ratios do not capture differences in cost structures across companies.
23 While less subject to distortion than earnings or cash flows, revenue recognition practices can still distort sales forecasts. For example, analysts should look for company practices that speed up revenue recognition. An example is sales on a bill-and-hold basis, which involves selling products and delivering them at a later date. This practice accelerates sales into an earlier reporting period and distorts the P/S ratio. P/S multiples are computed by dividing a stock’s price per share by sales or revenue per share or by dividing the market value of the firm’s equity by its total sales: Example: Calculating a P/S ratio Based on the information in the table, calculate the current P/S for Cisco Systems Inc. and Oracle Corp.
24 Figure: Data for Cisco Systems and Oracle Corp. Answer: Cisco Systems Inc.: Oracle Corp.: CompanyBook Value of Equity 2003 (USD millions) Sales 2003 (USD millions) Shares Outstanding 2003 (millions) Intraday Price 08/14/03 Cisco Systems Inc.28,03918,8787,001USD 17.83 Oracle Corp.6,3209,4755,233USD 12.15
25 Rationales for using the P/CF ratio include: Cash flow is harder for managers to manipulate than earnings. Price to cash flow is more stable than price to earnings. Reliance on cash flow rather than earnings handles the problem of differences in the quality of reported earnings, which is a problem for P/E. Empirical evidence indicates that differences in price to cash flow are significantly related to differences in long-run average stock returns. There are two drawbacks to the price to cash flow ratio, both of which are related to the definition of cash flow used. We discuss the specific cash flow definitions next. Items affecting actual cash flow from operations are ignored when the EPS plus noncash charges estimate is used. For example, noncash revenue and net changes in working capital are ignored. From a theoretical perspective, free cash flow to equity (FCFE) is probably preferable to cash flow. However, FCFE is more volatile than straight cash flow.
26 Professor’s Note: Free cash flow to equity is the cash flow available to common stockholders after all operating expenses, interest and principal payments, investment in working capital, and investments in fixed assets. Calculating P/CF Ratios. There are at least four definitions of cash flow available for use in calculating the P/CF ratio: earnings-plus-noncash charges (CF), adjusted cash flow (adjusted CFO), free cash flow to equity (FCFE), and earnings before interest, taxes, depreciation, and amortization (EBITDA). Expect to see any one of them on the exam. One commonly used proxy for cash flow is earnings-plus-noncash charges (CF): CF = net income + depreciation + amortization The limitation of this definition, as we mentioned previously, is that it ignores some items that affect cash flow, such as noncash revenue and changes in net working capital.
27 Another proxy for cash flow is cash flow from operations (CFO) from the cash flow statement. The limitation of CFO, however, is that it includes items related to financing and investing activities. Therefore, analysts often adjust CFO by adding back the after-tax interest cost: Adjusted CFO = CFO + [(net cash interest outflow) (1 – tax rate)] In addition, analysts sometimes further adjust CFO for items that are not expected to persist in the future. Analysts also often use FCFE and EBITDA as proxies for cash flow. As we mentioned above, theory suggests that FCFE is the preferred way to define cash flow, but it is more volatile than straight cash flow. EBITDA is a pretax, pre-interest measure that represents a flow to both equity and debt. Thus it is better suited as an indicator of total company value than just equity value. More on this point is provided in our discussion of the Enterprise Value to EBITDA ratio. Analysts typically use trailing price to cash, which relies on the most recent four quarters of cash flow per share.
28 Given one of the four definitions of cash flow, the P/CF ratio is calculated as: where:cash flow = CF, adjusted CFO, FCFE, or EBTIDA Example: Calculating P/CF Data Management Systems, Inc. (DMS) reported net income of USD32 million, depreciation and amortization of USD41 million, net interest expense of USD12 million, and cash flow from operations of USD44 million. The tax rate is 30 percent. Calculate the P/CF ratio using CF and adjusted CFO as proxies for cash flow. DMS has 25 million shares of common stock outstanding, trading at USD47 per share. Answer: CF = USD 32 million + USD 41 million = USD 73 million adjusted CFO = USD 44 million + [(USD 12 million)(1 – 0.30)] = USD 52.4 million
29 market value of equity = 25 million shares USD47 per share = USD1,175 million KEY CONCEPTS 1. Advantages of using P/E ratios in valuation are: Earnings power is the primary determinant of investment value. The P/E ratio is popular in the investment community. Empirical research shows that P/E differences are significantly related to long-run average stock returns. 2. Disadvantages of using P/E ratios in valuation are: Earnings can be negative, which produces a useless P/E ratio. The volatile, transitory portion of earnings makes the interpretation of P/E ratios difficult for analysts. Management discretion within allowed accounting practices can distort reported earnings.
30 3. The following are advantages of using P/BV: Book value is a cumulative amount that is usually positive even when EPS is negative. Book value is more stable than EPS, so it may be more useful than P/E when EPS is particularly high, low, or volatile. Book value is an appropriate measure of net asset value for firms that primarily hold liquid assets, including finance, investment, insurance, and banking firms. P/BV can be useful in valuing companies that are expected to go out of business. Empirical research shows that P/BV ratios help explain differences in long-run average returns.
31 The discounted after-tax cash flow model links the value of a property to an investor’s specific marginal tax rate. The net present value of an investment equals the present value of after-tax cash flows, discounted at the investor’s required rate of return, minus the equity portion of the investment. Only those projects with a positive expected net present value would make financial sense. Professor’s Note: When we are calculating after-tax cash flow, after-tax refers to the investor’s marginal tax rate. Calculate the net operating income (NOI) from a real estate investment. Net operating income (NOI) is defined as the gross operating income less estimated vacancy, collections, and other operating expenses. Example: Real estate net operating income An investor is considering the purchase of a small office building and, as part of his analysis, must calculate the NOI. The information on the building is as follows:
32 Gross potential rental income: $250,000 Estimated vacancy and collection loss rate: 5% Insurance: $10,000 Taxes: $8,000 Utilities: $7,000 Maintenance: $15,000 Answer: NOI = $250,000 – ($250,000 × 0.05) – $10,000 – $8,000 – $7,000 – $15,000 = $197,500 Professor’s Note: Be aware that depreciation and financing costs are not factors in the calculation of NOI. It is assumed that maintenance will keep the property in good condition, and the value of the property is independent of any financing arrangements. Also note that the taxes that are relevant to the calculation of NOI for real estate are property taxes.
33 Calculate the value of a property using the sales comparison and income approaches. The sales comparison approach is based on sales prices of comparable properties. Valuation can then be done relative to a specific similar property or relative to a benchmark such as the mean or median home price in the area. Then additions (e.g., for more square feet) and subtractions (e.g., for poor locations) are made to estimate the value of the subject property. Another approach under the general heading of sales comparison methods (hedonic price estimation) involves regressing the sales prices of properties on the key property characteristics that influence the value of a property. The slope coefficients estimated by the regression can then be used to estimate the value of the subject property. The regression equation is used like any multiple regression model; the values of the independent (right-hand side) variables for the subject property are multiplied by the estimated slope coefficients to estimate its value.
34 The income approach is based on taking the present value of the stream of annual NOI, assuming it is an infinite stream, using the required rate of return or “cap rate” estimated for the property. These approaches are illustrated in the following two examples. Example: Real estate valuation Continuing the previous example, the investor has obtained additional information regarding other recent sales of comparable office buildings in the vicinity. The investor can use the comparable sales information in a hedonic price model to estimate a current appraised value of the property. Assuming a current market cap rate of 10 percent, compute the value of the property using (1) the sales comparison approach and (2) the income approach. Slope Coefficient CharacteristicsUnits in $ per Unit Proximity to downtown In miles –50,000 Vacancy rate Percentage –35,000 Building size Square feet 40
35 The potential investment is half a mile from downtown, has an estimated vacancy rate of 4 percent, and is 50,000 square feet. Answer 1: Sales comparison approach Using the price model, the estimated appraised value would be: value = (–50,000 × 0.5) + (–35,000 × 0.04) + (40 × 50,000) = $1,973,600 Answer 2: Income approach Using the income approach, the appraised value of the property equals the NOI divided by the market cap rate and can be calculated as: appraisal price = NOI / market cap rate = $197,500 / 0.10 = $1,975,000 Calculate the after-tax cash flows, net present value, and yield of a real estate investment. Example: Computing after-tax cash flows, NPV, and yield for real estate Continuing the previous example, assume the investor purchases the building for $1,850,000, putting down 20 percent cash and financing
36 the remainder with a long-term mortgage at a rate of 10 percent. The annual payments on the mortgage are $156,997, and the interest portion is fully deductible for income tax purposes. The investor’s marginal income tax rate is 28 percent. Depreciation per year, using the straight-line method, is estimated to be $45,000 per year. Calculate the after-tax cash flows, net present value, and the yield of the investment. Answer: After-tax cash flow: The first year’s interest payment of $148,000 is calculated as the amount borrowed ($1,480,000) times the interest rate of the loan. After-tax net income (NOI less depreciation, less interest, net of taxes) is ($197,500 – 45,000 – 148,000) × (1 – 0.28) = $3,240. After- tax cash flow can be determined by adding depreciation back to and subtracting the principal component of the mortgage payment from the after-tax net income number. For this investment, the year 1 after-tax cash flow is $3,240 + $45,000 – $8,997 = $39,243.
37 Net present value: Three years forward, the investor plans to sell the building for $1,950,000. The remaining mortgage balance at payoff is $1,450,000. Assume that the cost of equity is 10% and the net cash flows for the investment are as follows: Year 1: $39,243 Year 2: $38,991 Year 3: $538,721 (year of sale, net of mortgage payoff, no capital gains tax) The present value of the cash flows is: ($39,243 / 1.10) + ($38,991 / 1.102) + ($538,721 / 1.103) = $472,649 The NPV is the present value of the cash flows minus the initial investment: $472,649 – $370,000 = $102,649 Yield: In summary, the cash flows of the investment are: Year 0 –$370,000 = CF 0 Year 1 $39,243 = CF 01
38 Year 2 $38,991 = CF 02 Year 3 $538,721 = CF 03 CPT → IRR = 20.18% WARM-UP: VENTURE CAPITAL Venture capital investments are private, non-exchange traded equity investments in a business venture. Investments are usually made through limited partnerships, with investors anticipating relatively high returns in exchange for the illiquidity and high-risk profile of a venture capital investment. Investments may be made at any point of the business cycle of the company, from the initial planning stages of a new venture to an established firm ready to go public. Explain the various stages in venture capital investing. The stages of venture capital investing, which overlap somewhat, are as follows: Seed stage. Investors are providing capital in the earliest stage of the business and may help fund research and development of product ideas.
39 Early stage. Early stage financing includes: Start-up financing, which typically refers to capital used to complete product development and fund initial markets. First stage financing, which refers to the funding of the transition to commercial production and sales of the product. Formative stage. Broad category which encompasses the seed stage and early stage. Later stage. Marketable goods are in production and sales efforts are underway, but the company is still privately held. Within the later stage period, second-stage investing describes investments in a company that is producing and selling a product but is not yet generating income. Third- stage financing would fund a major expansion of the company. Mezzanine or bridge financing would enable a company to take the steps necessary to go public. Broad terms, such as “expansion stage financing,” are used to describe the second and third stage, while the term “balanced stage” covers all stages, from seed through later stage.
40 Discuss venture capital investment characteristics and the challenges to venture capital valuation and performance measurement. Venture capital investment characteristics (may have some or all of the following): Illiquidity. Investors’ ability to cash out is dependent upon a successful IPO, which probably will not occur in the short term, if ever. Long-term investment horizon. Market conditions must be conducive for a public offering, and the company most likely must be at a profitable point in order for investors to recognize returns on their investment. Difficulty in valuation. Because of the uniqueness of each investment, there are few comparable assets with meaningful trading volume available for market value comparisons. Limited data. There is not much comparable historical risk and return data, nor is there much information on which to base future cash flows and earnings estimates. There also is insufficient information on what competing ideas or products other entrepreneurs may be developing. Entrepreneurial/management mismatches. Entrepreneurs with good ideas don’t always necessarily evolve into good managers as their company grows.
41 Fund manager incentive mistakes. The primary incentive for fund managers must be performance, not size or some other criteria. Timing in the business cycle. Market conditions are a primary determinant of the timing of market entrance and exit strategies. Requirement for extensive operations analysis. A successful venture capital manager must act as both a financial and operations advisor to the venture. Valuing and measuring the performance of a venture capital investment is tricky at best, due to the large probability of failure plus the overall uncertainty as to amount and timing of cash flows. The three most important factors that must be assessed are the expected payoff at exit, timing of exit, and the probability of failure. Prior to exit (or failure), evaluation of the venture’s performance must be made, although precise measurement is challenging. Difficulties include deriving accurate valuations, establishing benchmarks, and lacking reliable performance measures.
42 Calculate the net present value (NPV) of a venture capital project, given the project’s possible payoff and conditional failure probabilities. Example: Computing NPV for a venture capital opportunity A venture capital fund manager is considering investing $2,500,000 in a new project that he believes will pay $12,000,000 at the end of five years. The cost of equity for the investor is 15 percent, and the estimated probability of failure is presented in Figure. These are conditional probabilities since they represent the probability of failure in year N, given that the firm has survived to year N. Figure: Estimated Probability of Failure Calculate the NPV of the potential investment. Year12345 Failure Probability0.20 0.170.15
43 Answer: The probability that the venture survives for five years is calculated as: (1 – 0.20)(1 – 0.20)(1 – 0.17)(1 – 0.15)(1 – 0.15) = 0.3838 = 38.38% Under the original assumptions that the investment pays $12,000,000 at the end of year 5, the NPV of a successful project is – $2,500,000 + ($12,000,000 / 1.155) = $3,466,121. The NPV of the project if it fails is –$2,500,000. The expected NPV of the project is a probability- weighted average of the two possible outcomes: (0.3838 × $3,466,121) + (0.6162 × –$2,500,000) = –$210,203 The fund manager would not invest in the new project due to the negative expected NPV.
44 Discuss the descriptive accuracy of the term “hedge fund,” define hedge fund in terms of objectives, legal structure, and fee structure, and describe the various classifications of hedge funds. Hedge funds today utilize a wide variety of strategies, which may or may not utilize hedging techniques to reduce or eliminate risk. The term “hedge fund” does not begin to describe this broad asset class that has evolved over the past two decades. The common objective of hedge funds is that they strive for absolute returns. That is to say that hedge funds are not constrained by the fact that they must perform relative to some specific benchmark or index and simply seek to maximize returns in all market scenarios. Most hedge funds are in the form of either a limited partnership, a limited liability corporation, or an offshore corporation. In the U.S., limited partnerships that abide by certain guidelines (regarding the maximum allowable number of investors, the “qualifications” of the investors, and the prohibition of advertising) are exempt from most SEC regulations. Because the number of investors is limited, the amount of their individual investments is relatively large, usually $200,000 or more.
45 The manager of the fund receives compensation that is comprised of two components. The base fee is typically around 1 percent of assets, and the manager receives this fee regardless of performance of the hedge fund. The second component, the incentive fee, is paid based on the actual returns of the fund. Some structures allow the manager to participate in all returns, while other structures pay the manager only if performance exceeds a target return, such as the risk-free rate. Classifications of Hedge Funds Hedge funds can usually be classified by investment strategy; however, there is a great deal of overlap among categories. Some hedge fund classifications are: Long/short funds make up the largest category of hedge funds in terms of asset size. These funds take long and short common stock positions, use leverage, and are invested in markets worldwide. By definition they are not market-neutral but seek to profit from greater returns on the long positions than on the short positions.
46 Market-neutral funds are a type of long/short fund that strives to hedge against general market moves. Managers may try to achieve this through any of several strategies, some involving derivatives. The fund may still have long and short positions, but the positions will offset each other so that the effect is a net zero exposure to the market. Global macro funds make bets on the direction of a market, currency, interest rate, or some other factor. Global macro funds are typically highly leveraged and rely heavily upon derivatives. Event-driven funds strive to capitalize on some unique opportunity in the market. This may involve investing in a distressed company or in a potential merger and acquisition situation. Discuss the benefits and drawbacks to fund of funds investing. Fund of funds investing involves creating a fund open to both individuals and institutional investors, which in turn invests in hedge funds.
47 Benefits. Funds of funds enable investors with limited capital to invest in a portfolio of hedge funds. Likewise, investors with more capital can diversify their holdings by investing in several hedge funds via a fund of funds for roughly the same amount required for directly investing in a single hedge fund. Fund of funds investing may grant new investors access to hedge funds that might otherwise be closed to them due to limitations on the number of investors. A fund of funds manager will have the expertise necessary to choose high-quality hedge funds and will also perform the due diligence required by investing in hedge funds. Drawbacks. Fund of funds managers charge a management fee in addition to those fees already charged by the hedge fund manager. Diversification among hedge funds will decrease the investor’s risk but most likely his return as well, from which additional fees must be subtracted. Fund of funds managers may or may not deliver returns superior to what an investor might achieve by selecting her own hedge funds.
48 Discuss the leverage and unique risks of hedge funds. The majority of hedge fund managers utilize some form of leverage in order to enhance returns. Some arbitrage opportunities may have such a small return that leverage is necessary to make the strategy meaningful. However, leveraged positions can sometimes backfire and cause losses to be magnified. Hedge funds typically limit the amount of leverage that can be used, and fund managers are legally required to operate within the limit. One way a hedge fund can increase its leverage is by borrowing through a margin account. Also, a hedge fund manager could borrow external funds to either buy more assets or sell short more than the equity in the fund. A third way is for hedge fund managers to utilize those securities that only require posting margin versus trading in cash securities requiring full payment. Risks associated with hedge funds include: Illiquidity. Investing in those markets with little liquidity, such as derivatives, decreases a hedge fund’s trading flexibility.
49 Potential for mispricing. Investments in esoteric securities that trade infrequently may lead to difficulty in determining true current market value. Broker-dealers who are financing such securities tend to be conservative in their valuations, thereby increasing the amount of cash that is required to be posted by the hedge fund. Counterparty credit risk. A broker-dealer is involved in almost every transaction a hedge fund enters into, thereby creating significant counterparty risk to the hedge fund. Settlement errors. Hedge funds bear the risk that the counterparty will fail to deliver a security as agreed on settlement day. Short covering. Short selling is a component of many common hedge fund strategies. Hedge fund managers run the risk that they will have to cover their shorts and repurchase securities at a price higher than where they originally sold. Margin calls. Margin calls on an already highly leveraged position can result in forced selling of assets, possibly at a loss.
50 Discuss the performance of hedge funds, the biases present in hedge fund performance measurement, and explain the effect of survivorship bias on the reported return and risk measures for a hedge fund data base. There are numerous hedge fund indexes designed to measure historical performance; however, they may not provide much meaningful information on hedge funds as an asset class because each hedge fund’s structure is so unique. Since hedge funds are not legally required to publicly disclose performance, only those hedge funds that elect to disclose performance information are included in the indexes. Some general conclusions regarding hedge funds can be derived. Hedge funds, as a class, have historically outperformed both U.S. equity and bond market benchmarks (S&P 500 and Lehman Brothers Government/Corporate Bond Index). Hedge funds have demonstrated a lower risk profile than traditional equity investments as measured by standard deviation.
51 In recent years, the Sharpe ratio, which is a reward-to-risk ratio, has been consistently higher for hedge funds than for most equity investments and has been comparable to that of fixed-income investments. There is a low correlation between the performance of hedge funds and conventional investments. As with the evaluation of historical data for any investment instrument, certain biases may exist that should be considered. It is common investment knowledge that past performance is not a reliable indicator of future performance. Some significant biases are: “Cherry picking” by managers. The only information available to be included in the indexes is what the fund managers submit. Managers may be unwilling to disclose poor performance and choose to offer information only on those funds with successful track records. The index may overestimate returns for the hedge fund industry as a whole. Incomplete historical data. Again, because disclosure is voluntary, only fund managers with a respectable track record would be willing to be included in an index. Past performance of the industry is inflated by an index because poor past performance is excluded.
52 Survival of the fittest. As with any industry, only the fittest hedge funds survive. By design, an index only includes ongoing funds and excludes those that have failed. The index in effect is biased toward only the “success stories” of the industry. Smoothed pricing. Because many of the assets held in hedge funds are not actively traded, managers rely upon broker-dealers to mark their positions and estimate “market” value. Because they are estimates and not based upon actual transactions, values tend to be more stable over time, thereby reducing reported volatility. Asymmetrical returns. Some investment strategies used by hedge funds may have a limited upside potential but unlimited downside potential. Traditional risk measures, such as standard deviation or value at risk, do not fully account for this asymmetric return profile. Fee structures and incentives. A typical hedge fund fee structure pays the manager a small fixed fee and then a substantial percentage of gains. This structure may cause fund managers to take big risks, especially if past performance is bad and they have “nothing to lose.”
53 Effect of Survivorship Bias The effect of survivorship bias is greater for a hedge fund database than for other asset classes because of the lack of required reporting standards in the industry. Hedge funds are normally exempt from SEC regulations regarding reporting and only publicly disclose performance information on a voluntary basis. Fund managers tend to “cherry pick” the information they choose to release, reporting on their more successful funds while not providing information on poorly performing or defunct funds. Reported returns for a hedge fund database are therefore overstating performance because of survivorship bias. Survivorship bias has the opposite effect on the risk measures of a hedge fund database. Hedge funds with highly volatile returns tend to fail more frequently, and defunct funds are not generally included in the database. Because the database would only include the more stable funds that have survived, the risk measure of hedge funds as an asset class would be understated.
54 Explain how the legal environment affects the valuation of closely held companies. Legal issues. The equity shares of closely held companies are not publicly traded and are not subject to the same SEC regulations as public companies regarding reporting and disclosure. As the name implies, closely held companies are held by a relatively small group of owners. The companies may be in the form of any number of legal entities: corporations, partnerships, or sole proprietorships. Some corporations’ legal structures are designed to take advantage of current tax codes, such as subchapter S corporations. Other corporations may elect to operate as general or limited partnerships, which dictate the extent of a partner’s liability for the corporation. The choice of structure affects the investors’ rights and responsibilities and, ultimately, the value of their investments. When litigation situations arise, there can be questions as to the “value” of the corporation. The legal definitions of intrinsic value, fundamental value, and fair value can differ among jurisdictions. There are not frequent transactions in the open market upon which to estimate
55 value. Valuation, therefore, is based upon either a forecast of future cash flows, actual past cash flows, or a combination of both. Both the purpose of the valuation and the legal jurisdiction affect the factors on which value is based and how it is calculated. Describe alternative valuation methods for closely held companies and distinguish among the bases for the discounts and premiums for these companies. Valuation. There are three different valuation methods for closely held companies: The cost approach. What is the cost today to replace the company’s assets in their present state? The comparables approach. What is the value relative to an appropriate benchmark value? The benchmark would be based upon market prices of similar companies, adjusted for such factors as transaction date and any unique characteristics of the company. The benchmark may be difficult to establish if no comparable companies have been sold recently.
56 The income approach. What is the net present value of the company based upon discounted future cash flows? When valuing closely held companies, lack of liquidity and lack of marketability can both be important factors. Value can be determined by analyzing operationally similar publicly traded companies to establish a base value, to which the liquidity and marketability discounts can be applied. Another factor to be considered is which party has control of the company. A discount for minority interest may be necessary for valuing a position that lacks the ability to influence corporate decision-making. Likewise, a premium would be appropriate for the valuation of a controlling ownership position. The application of premiums and discounts depends on the characteristics of the subject securities relative to the characteristics of the securities used in estimating the base value. If the base value used is the market price of publicly traded shares, the valuation of a majority interest in a closely held company may require the application of discounts for lack of liquidity and marketability and of a premium for control, for example.
57 Discuss distressed securities investing and the similarities between venture capital investing and distressed securities investing. When companies are on the brink of bankruptcy or have already filed for bankruptcy protection, their securities are considered “distressed.” Also included in the group of distressed securities are those companies attempting to avoid bankruptcy by pursuing an out-of-court debt restructuring. In a typical bankruptcy scenario, the original holders of the company’s debt negotiate for an equity position in the new, restructured corporation. The original equity shareholders then receive a somewhat diluted equity position in the reorganized company. A typical distressed security investment strategy would be to purchase the debt of the struggling company, prereorganization, in the hopes of ultimately owning an equity position in a new, revitalized operation. Pursuing a distressed security strategy is somewhat similar to venture capital investing. Both asset classes are illiquid, have a long expected investment horizon, and require heavy involvement by investors in order to be successful. Both situations mandate extensive analytical work in order to avoid pricing or valuation mistakes.
58 Discuss the role of commodities as a vehicle for investing in production and consumption. Investing in commodities gives an investor exposure to an economy’s production and consumption growth. When the economy experiences growth, the demand for commodities increases, and price increases are likely. When housing starts to increase, the demand for lumber will increase; when automobile sales are high, the demand for steel is likely high as well. During recessions, commodity prices are likely to fall with decreased demand. Overall, swings in commodity prices are likely to be larger than changes in finished goods prices. Discuss the motivation for investing in commodities, commodities derivatives, and commodity-linked securities. The motivation for investing in commodities may be as an inflation hedge for hedging purposes or for speculation on the direction of commodity prices over the near term. Most investors do not invest directly in commodities which need to be transported and stored. Passive investors who hold commodities as an asset class for diversification or those who
59 hold commodities as a long-term inflation hedge are more likely to invest in a collateralized futures position. A collateralized futures position or collateralized futures fund is a combination of an investment in commodity futures and an investment in Treasury securities equal in value to the value of the futures position. Active investors may invest in commodity futures in an attempt to profit from economic growth which is associated with higher commodity prices. Commodity-linked equity investments also provide exposure to commodity price changes. Small commodity-producing companies are likely to experience share price returns which are strongly tied to the prices of the commodities they produce. Commodity-linked bonds provide income as well as exposure to commodity price changes since the overall return is based on the price of a single commodity such as gold or oil. Other commodity-linked bonds are linked to inflation through payments based on inflation or a commodity price index. These bonds may be attractive to a fixed-income portfolio manager who wants exposure to commodity price changes but cannot invest either directly in commodities or in derivative securities.
60 Discuss the sources of return on a collateralized commodity futures position. Establishing a collateralized position in commodities futures requires simultaneously purchasing a specific futures contract and the amount of government securities equal to the value (not the purchase price) of the futures contract. The total return on this strategy will equal the change in price of the futures contract plus the interest earned on the government securities. Example: Commodity futures returns A passive manager purchases a position worth $50 million in underlying value of a futures contract. The manager also buys $50 million worth of 10-year notes that pay an interest rate of 5 percent. Compute the gain in the value of the position if, at the end of one month, the futures contract position is worth $51 million and the price of the 10-year notes is unchanged.
61 Answer: Gain on the futures position = $1,000,000 Interest earned on the notes = $50,000,000 × 0.05 × (30 / 360) = $208,333.33 Total gain = $1,000,000 + $208,333.33 = $1,208,333.33
62 AN INTRODUCTION TO PORTFOLIO MANAGEMENT Define risk aversion and cite evidence that suggests that individuals are generally risk averse. Risk aversion refers to the fact that individuals prefer less risk to more risk. Risk-averse investors: Prefer lower to higher risk for a given level of expected returns. Will only accept a riskier investment if they are compensated in the form of greater expected return. In Figure, we examine the concept of risk aversion using indifference curves.
64 The curved lines, I1, I2, and I3, represent indifference curves because all investments (combinations of risk and expected return) that lie along each curve are equally preferred. Note that these indifference curves have a positive slope, whereas the indifference curves we examined in the economics section were negatively sloped. This is because in the economics analysis we had two “goods,” and now we have a “good” (expected return) and a “bad” (risk). A higher or more preferred indifference curve now lies in the northwest direction (more expected return and less risk). Focusing on indifference curve I1, a risk-averse investor whose preferences are represented by these curves will be equally happy with or indifferent among any risk/return combinations on this curve. Notice that as risk increases, a risk-averse investor demands an increasingly higher rate of return as compensation. While an investor would be equally happy with any point on I1, she prefers all risk/return combinations on I2 to any combination on I1. In reality, there are an infinite number of indifference curves, and the indifference curves for any given investor can never cross.
65 The fact that most individuals buy some sort of insurance, whether auto, health, or homeowners, indicates that they are generally risk averse. It is interesting to note, however, that an individual may exhibit risk-averse tendencies in one area and not in others. For example, many people buy auto insurance to protect against the costs associated with auto accidents but will not buy health insurance or will buy lottery tickets or participate in other forms of gambling. (See Exam Flashback #1.) List the assumptions about individuals’ investment behavior of the Markowitz Portfolio Theory. In the investment framework developed by Nobel laureate Harry Markowitz, the following assumptions about investor behavior are made: Returns distribution. Investors look at each investment opportunity as a probability distribution of expected returns over a given investment horizon. Utility maximization. Investors behave such that they maximize their expected utility over a given investment horizon, and their indifference curves exhibit diminishing marginal utility of wealth (i.e., they are convex).
66 Risk is variability. Investors measure risk as the variance (standard deviation) of expected returns. Risk/return. Investors make all investment decisions by considering only the risk and return of an investment opportunity. This means that their utility (indifference) curves are a function of the expected return (mean) and the variance of the returns distribution they envision for each investment. Risk aversion. Given two investments with equal expected returns, investors prefer the one with the lower risk. Likewise, given two investments with equal risk, investors prefer the one with the greater expected return. Professor’s Note: Make sure you understand each of the Markowitz assumptions—it will make asset pricing models easier to grasp.
67 Compute expected return for an individual investment and for a portfolio. Professor’s Note: It’s not obvious whether the exam will require that you describe and calculate expected returns using expectational (probabilistic) data or historical data, so we will do it both ways here and throughout this review wherever appropriate, just to be safe. Expected Return for an Individual Investment The expected rate of return from expectational data for a single risky asset can be calculated as: where: P i = probability that state i will occur R i = asset return if the economy is in state i
68 The expected return, based on expectational data, is simply the weighted mean of the distribution of all possible returns. The expected rate of return from historical data for a single risky asset can be calculated as: where : R t = the return in time period t n = the number of time periods (using historical returns) Professor’s Note: The expected return with historical data is simply the average return over n years. Example: Calculating expected return from expectational data The first three columns of Figure contain the probability of outcomes (states of the world) and the returns for a security in each state of the world. Calculate the expected return on the security.
69 Answer: The computation of expected return is illustrated in the fourth column of Figure. Figure: Computing Expected Return Example: Calculating expected return from historical data Assume that the returns on a stock over the first six months of the year are +10%, –15%, +20%, +25%, –30%, and +20%. Compute the expected (average) return. State of the World Probability (P i )Return (R t )Expected Return (P i R t ) Expansion0.255.0%(0.25) (5.0) = 1.25% Normal0.5015.0%(0.50) (15.0) = 7.50% Recession0.2525.0%(0.25) (25.0) = 6.25%
70 Answer: Expected Return for a Portfolio of Risky Assets The expected return on a portfolio of assets is simply the weighted average of the returns on the individual assets, weighted by their portfolio weights. Thus, for a two-asset portfolio, the expected return is: E(R p ) = w 1 E(R 1 ) + w 2 E(R 2 ) where: E(R 1 ) = expected return on asset 1 E(R 2 ) = expected return on asset 2 w 1 = percentage of the total portfolio value invested in asset 1 w 2 = percentage of the total portfolio value invested in asset 2
71 Compute the variance and standard deviation for an individual investment. Professor’s Note: It’s not obvious whether the exam will require you to calculate the variance and standard deviation of returns using expectational (probabilistic) data or historical data, so we will do it both ways just to be safe. In finance, the variance and standard deviation of expected returns are common measures of investment risk. Both of these related measures determine the variability of a distribution of returns about its mean. The variance and standard deviation of rates of return from expectational data for an individual investment are calculated as:
72 where: R i = return in state of the world i P i = probability of state i occurring E(R) = expected return Example: Calculating variance with expectational data The returns expectations from the previous example are reproduced in the first three columns of Figure. Using this expectational data, calculate the variance and standard deviation of expected returns. Recall that the expected return is 15 percent. Answer: Based upon the computations illustrated in Figure, the variance and standard deviation are 0.0050 and 7.07 percent, respectively.
73 Figure: Variance and Standard Deviation Computation The variance and standard deviation of returns from historical data are calculated as: State iProbability P i Return R i Expected Return E(R) [(R i ) – E(R)] 2 P i [(R i ) – E(R)] 2 Expansion0.250.050.150.01(0.25) (0.01) = 0.0025 Normal0.500.15 0.00(0.50) (0.00) = 0.0000 Recession0.25 0.150.01(0.25) (0.01) = 0.0025 Variance = P i [(R i ) – E(R)] 2 = 0.0025 + 0.0000 + 0.0025 = 0.0050 Standard deviation = (0.0050) 1/2 = 0.0707 = 7.07%
74 where: R t = return in period t = average return (expected return) n = number of returns Example: Calculating variance with historical data The historical returns from the previous example are reported in the first column of Figure. Compute the variance and standard deviation. Answer: As shown in Figure, the computations of the variance and standard deviation result in 0.0417 and 20.41 percent, respectively.
75 Figure: Variance and Standard Deviation Using Historical Data RtRt (R t – )(R t – ) 2 +0.1000+0.05000.0025 – 0.1500– 0.20000.0400 +0.2000+0.15000.0225 +0.2500+0.20000.0400 – 0.3000– 0.35000.1225 +0.2000+0.15000.0225 = 0.0500 = 0.2500 2 = 0.2500 / 6 = 0.0417 = 0.2041 = 20.41%
76 Compute the covariance of rates of return, and show how it is related to the correlation coefficient. Covariance measures the extent to which two variables move together over time. A positive covariance means that the variables (e.g., rates of return on two stocks) tend to move together. Negative covariance means that the two variables tend to move in opposite directions. A covariance of zero means there is no relationship between the two variables. The covariance between two assets computed from expectational data is equal to: where: R t,1 = return on asset 1 in state i R t,2 = return on asset 2 in state i P = probability of state i occurring E(R 1 ) = expected return on asset 1 E(R 2 )= expected return on asset 2
77 Example: Calculating covariance with expectational data Calculate the covariance between Asset 1 and Asset 2 with the returns distribution described in the first three columns of Figure. Answer: First, we must compute the expected return for each of the assets as follows: Armed with the asset’s expected returns, we can compute the covariance following the procedure illustrated in Figure.
78 Figure: Computing Covariance The covariance between two asset returns using historical data is computed as: where: R i,1 = return on asset 1 in period t R i,2 = return on asset 2 in period t R 1 = mean return asset 1 R 2 = mean return on asset 2 n = number of returns(using historical returns) PiPi R i,1 R i,2 (R i,1 ) – E(R 1 )(R i,2 ) – E(R 2 )P i [(R i,1 ) – E(R 1 )] [(R i,2 ) – E(R 2 )] 0.250.050.32– 0.10+0.16–0.004 0.500.150.14+0.00– 0.020.000 0.25 0.04+0.10– 0.12– 0.003 Cov 1,2 = P i [(R i,1 ) – E(R 1 )] [(R i,2 ) – E(R 2 )] = – 0.007
79 Professor’s Note: There is a similar formula for the covariance in the quantitative methods material that has n – 1 in the denominator. The difference is that the formula we are working with here is a population measure, whereas in the quant material, n – 1 is used because the covariance there is a sample statistic. Example: Calculating covariance with historical data Calculate the covariance for the returns of Stock 1 and Stock 2 given the six months of historical returns presented in the first three columns of Figure. Answer: The covariance calculation is demonstrated in the right side of Figure.
80 Figure: Calculating Covariance From Historical Returns Correlation. The magnitude of the covariance depends on the magnitude of the individual stocks’ standard deviations and the relationship between their co-movements. The covariance is an absolute measure and is measured in return units squared. YearStock 1Stock 2(R t – 1 )(R t – 2 )(R t – 1 ) (R t – 2 ) 1998+0.10+0.20+0.05+0.10+0.005 1999–0.15–0.20 –0.30+0.060 2000+0.20–0.10+0.15–-0.20–0.030 2001+0.25+0.30+0.20 +0.040 2002–0.30–0.20–0.35–0.30–0.105 2003+0.20+0.60+0.15+0.50+0.075 1 = 0.05 2 = 0.10 = 0.255 Cov = 0.255/6 = 0.0425
81 Covariance can be standardized by dividing by the product of the standard deviations of the two securities being compared. This standardized measure of co-movement is called correlation and is computed as: Professor’s Note: The calculation of correlation is the same whether we are using expectational or historical data. The term ρ 1,2 is called the correlation coefficient between the returns of securities 1 and 2. The correlation coefficient has no units. It is a pure measure of the co- movement of the two stocks’ returns and is bounded by –1 and +1. How should you interpret the correlation coefficient? A correlation coefficient of +1 means that returns always move together in the same direction. They are perfectly positively correlated. A correlation coefficient of –1 means that returns always move in the exact opposite direction. They are perfectly negatively correlated.
82 A correlation coefficient of zero means that there is no relationship between the two stocks’ returns. They are uncorrelated. One way to interpret a correlation (or covariance) of zero is that, in any period, knowing the actual value of one variable tells you nothing about the other. Example: Computing correlation The covariance between the returns on two stocks is 0.0425. The standard deviations of stocks 1 and 2 are 0.2041 and 0.2944, respectively. Calculate and interpret the correlation between the two assets. Answer: The returns from the two stocks are positively correlated, meaning they tend to move in the same direction at the same time. However, the correlation is not perfect because the correlation coefficient is less than one.
83 Example: Computing covariance The correlation between the returns on two stocks is 0.56. The standard deviations of the returns from Stock 1 and Stock 2 are 0.1544 and 0.0892, respectively. Calculate and interpret the covariance between the two assets. Answer: Cov 1,2 = 0.56 × 0.1544 × 0.0892 = 0.0077 The covariance between the returns from Stock 1 and Stock 2 shows that the two securities’ returns tend to move together. However, the strength of this tendency cannot be measured using the covariance—we must rely on the correlation to provide us with an indication of the relative strength of the relationship.
84 List the components of the portfolio standard deviation formula, and explain which component is more important to consider when adding an investment to a portfolio. Earlier in this review, we showed that the expected return for a portfolio of risky assets is the weighted average of the expected returns of the individual assets in the portfolio. This is not the case for the variance and standard deviation of a portfolio of risky assets. The variance and, by association, the standard deviation of a portfolio of two assets are not simple weighted averages of the asset variances (standard deviations). Portfolio variance (standard deviation) is not only a function of the variance (standard deviation) of the returns of the individual assets in the portfolio. It is also a function of the correlation (covariance) among the returns of the assets in the portfolio. The general formula for the standard deviation for a portfolio of n risky assets is as follows:
85 where : P 2 = portfolio variance w i = the market weight of asset i i 2 = variance of returns for asset i Cov i,j = the covariance between the returns of assets i and j For a portfolio of two risky assets this is equivalent to: For a portfolio of three risky assets, the expanded form is: Note that in the first formula for a two-asset portfolio we have substituted 1 2 1,2 for Cov 1,2 (using the definition of 1,2 ) because the formula is often written this way as well to emphasize the role of correlation in portfolio risk.
86 The first part of the formula is intuitive—the risk of a portfolio of risky assets depends on the risk of the assets in the portfolio and how much of each asset is in the portfolio (the ’s and the w’s). The second part of the formula is there because the risk (standard deviation) of a portfolio of risky assets also depends on how the returns on the assets move in relation to each other (the covariance or correlation of their returns). Note that if the asset returns are negatively correlated, the final term in the formula for a two-asset portfolio is negative and reduces the portfolio standard deviation. If the correlation is zero, the final term is zero, and the portfolio standard deviation is greater than when the correlation is negative. If the correlation is positive, the final term is positive, and portfolio standard deviation is greater still. The maximum portfolio standard deviation for a portfolio of two assets with given portfolio weights will result when the correlation coefficient is +1 (perfect positive correlation). When assets are perfectly positively correlated, there is no diversification benefit.
87 Other things equal, the higher (lower) the correlation between asset returns, the higher (lower) the portfolio standard deviation. WARM-UP: PORTFOLIO RISK AND RETURN FOR A TWO- ASSET PORTFOLIO Before we move on to the next LOS, let’s take a minute to show graphically the risk-return combinations from varying the proportions of two risky assets and then to show how the graph of these combinations is affected by changes in the correlation coefficient for the returns on the two assets. Figure provides the risk and return characteristics for two stocks, Sparklin’ and Caffeine Plus. Figure shows the calculation of portfolio risk and expected return for portfolios with different proportions of each stock. Figure: Risk/Return Characteristics for Two Individual Assets Caffeine PlusSparklin’ Expected return (%)11%25% Standard Deviation (%)15%20% Correlation0.3
88 Figure: Possible Combinations of Caffeine Plus and Sparklin’ The plot in Figure represents all possible expected return and standard deviation combinations attainable by investing in varying amounts of Caffeine Plus and Sparklin’. We’ll call it the risk-return tradeoff curve. W Caffeine Plus 100%80%60%40%20%0% W Sparklin’ 0%20%40%60%80%100% E(R P )11.0%13.8%16.6%19.4%22.2%25.0% PP 15.0%13.7% 14.9%17.1%20.0%