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# FINANCE MGMT 543

Rice University

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This 48 page Class Notes was uploaded by Jamil Bednar on Monday October 19, 2015. The Class Notes belongs to MGMT 543 at Rice University taught by Staff in Fall. Since its upload, it has received 41 views. For similar materials see /class/225049/mgmt-543-rice-university in Business, management at Rice University.

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Date Created: 10/19/15

MGMT543 HNANCE Handout 8 Basic Statistics R mkwwd bh n Portfolio Theay Evgeny Lyandres Chris Downing Fall 2005 Objectives 0 Review basic statistics 0 Calculate the return and risk of a single security 0 Understand the concept of risk premium oCalculate the return and risk on a portfolio of secun es 0 Explain the effect of diversi cation 0 Understand the concept of ef cient portfolios 2 Basic statistics 0 Random van39able a variable that takes different values with some probabilities Pmbabliity distribution a schedule of all possible realizations and respective probabilities Example tomorrow s temperature can be 90 degrees with probability 025 3995 degrees with probability 05 05 100 degrees with probability 025 22m Duicume Basic statistics Expected value mean expected realization of a random variable 1300 ZXipi 11 where Xi is the outcome in state i and p is the probability of state i Basic statistics OWhat is the expected value of tomorrow39s temperature Basic statistics Variance expected value of squared distance from the mean x 02 EltXi2pi 11 Standard dew39atiorx square root of variance 1 coo 2Xi EltXigt2pi 11 Basic statistics What are the variance and the standard deviation of tomorrow s temperature Basic statistics Covariance measure of comovement between two random variables CovltXY 2Xi EltXigtYl EltYipi 11 Correlation ooe icient standardized measure of the comovement between two random variables n Comm 2 EltXigtYi EYipi WY J xi EltXigt2priYi EltYigtzpig 1 Basic statistics Assume that there are three states of the world which characterize the temperatures tomorrow in Houston and in Tel Aviv Israel State Probability Houston TelAviv 1 025 90 85 2 05 95 80 3 025 100 90 Basic statistics oWhat are the covariance and the correlation between tomorrow s temperatures in Houston and TelAviv Basic statistics 0Covariance and correlation tell us whether two random variables tend to move in the same or the opposite direction IThe correlation between two random variables will always be between 1 and 1 If the correlation is 0 we say that the variables are uncorrelated If it is 1 then the variables are perfectly correlated and if it is 1 then the variables are perfectly negatively correlated Calculating rates of return Stock returns are made up of two components Dividends Changes in the price of stock which may be highly volatile Ending value Beginning value Beginnng value DtPtPt71 Dt Pt Pt71 Ptil Ptil Ptil Dividend yield Capital gain Calculating rates of return Suppose an investor purchases a share of IBM common stock for 100 on January 1 The stock pays 4 in dividends on December 31 and its market price on December31 is 108 What is the rate of return on this investment Calculating rates of return Rates of return on most assets are random Looking backward we can calculate realized rates of return Looking forward we characterize random events with probabilities and probability distributions Probabiity distributions are often described by summary measures such as mean and variance Calculating rates of return Suppose you have a sample of daily monthly quarterly or annual stock returns SR1R2R3 RT We want to draw inferences about the probability distribution of stock returns from this sample Calculating rates of return Historical average return may be a good approximation of expected return There are two ways to average returns over time An39thmetic mean rerun R1R2 R3 RT T used to estimate expected stock return Geometric mean leturm 1 T 1R11R21R31RT used for compounding 15 Calculating rates of return The historical variance of the return is one measure of the riskiness of a security T Variancest 22011 102 11 Standard deviation is the measure of riskiness that we will use most often St Dev 5 2 Si2 Example Given the stock prices below compute the yearly returns the average return over the entire time period and the annual variance and standard deviation of the returns The stock pays a 010 dividend each year Year 1999 2000 2001 2002 2003 Price 800 910 900 720 960 Investor preferences Would you buy a 100 bill for 90 How much would you pay for a lottery ticket that gives you a 50 chance to win 200 and a 50 chance to lose everything Investor preferences We will make two fundamental assumptions about investor preferences lnvestors prefer more wealth to less other things equal investors prefer higher expected returns lnvestors are risk averse other things equal investors prefer a lower standard deviation of their wealth 20 Risk premium Because investors are riskaverse they require a higher return for bearing risk In other words they require a risk premium We can decompose the return on a security into the riskfree component and the risk premium Total return Riskfree return Risk premium Historically the market for common stocks has commanded a risk premium of approximately 8 This means that investors require on average 800 basis points above the riskfree rate in order to induce them to invest in common stocks instead of in T bills 21 Modern portfolio theory Before Harry Markowitz pioneered portfolio theory and Eugene Fama and others pioneered the ef cient markets theory investment analysis focused on picking winners in the stock market Because the investment focus was on picking winners risk was usually measured by the stock return variance or stockreturn standard deviation Modern portfolio theory One of Harry Markowitz important contributions was to recognize that rational investors hold diversi ed portfolios to minimize their risk Although stockreturn standard deviation measures the risk of a security if you hold it alone it is not very informative about how the security contributes to the riskiness of a diversi ed portfolio Portfolio weights A portfolio is uniquely defined by the portfolio weights Suppose there are N assets i1 2 N Then the portfolio weight w is Amount invested in asset 1 Total amount invested Portfolio expected returns with two assets With 2 assets the portfolio expected return is ERp w1ER1 w2ER2 Portfolio expected returns with two assets You invested 30 in IBM stock with expected return of 13 and 70 in Exxon stock with expected return of 11 What is the expected return of your portfolio Portfolio variance with two assets With 2 assets the portfolio variance is 2 2 2 2 2 op W161W2 o2 2W1W2612 2 2 2 2 2 W1 61 W2 o2 2wlwzplzolo2 Portfolio variance with two assets The standard deviation of IBM stock is 20 per annum The standard deviation of Exxon stock is 15 per annum The correlation coefficient between IBM stock return and Exxon stock return is 04 What is the variance and standard deviation of your portfolio s returns Portfolio expected return and variance with many assets Assume that you expanded your portfolio and invested 100 in Microsoft The expected return on Microsoft is 15 per year Its standard deviation is 25 Its correlation with IBM is 06 and with Exxon is 02 What is the expected return of your portfolio What are the variance and standard deviation of your portfolio s return Portfolio expected return The portfolio expected return is equal to the weighted average of the returns on the individual assets in the portfolio where the weights are given by the portfolio weights ERp iwiERi Portfolio variance The portfolio variance depends on both the variances of the individual assets in the portfolio and their cova ances o VarRp N N N Z WiZVarRi ZzwiijowRi R1 i1 11 11 j l N 2 N N zzlwi 112ZZW1WJGij 11 11 11 jail The covariance matrix The relative importance of variances and covariances Assume that you decide to invest equal amounts in all stocks that enter your portfolio Let s assume that the standard deviation of each stock is 20 and the correlation coef cient between each two stocks is 03 What is the variance of returns of each stock What is the covariance between returns of each two stocks as The relative importance of variances and covariances What is the variance of your portfolio if it includes 2 stocks 3 stocks 5 stocks 10 stocks 100 stocks The relative importance of variances and covariances As the number of stocks becomes large covariances remain the only factor affecting portfolio returns N 2 N N 2 2 op E w1 61E Ewiwjoij 11 39 391 1 111 N jVar N N 1 jCov z Cov 35 Expected returns variances and covariances Compute the expected returns and variances for portfolios consisting of the following securities Outcome Prob Return on Return on stock A stock B Boom 03 20 15 Normal 05 10 0 Bust 02 0 5 Example Portfoio 1 150 in stock A 100 in stock B Portfoio 2 50 in stock A 200 in stock B Perfect correlation Assume that stocks A and B are perfectly correlated p 1 You invest w1 in stock A and w2 in stock B What is the standard deviation of your portfolio 2 2 2 2 2 op w1 c51w2 52 2le2p12cslcs2 2 2 2 2 2 w1 61W2 52 Wlwzcloz 2 W151 W 252 Sp 2 W161 W202 Perfect correlation Stocks A and B are perfectly correlated The expected return on stock A is 10 while the expected return on stock B is 15 The standard deviation of stock A is 15 while the standard deviation of stock B is 25 How does the relation between portfolio expected return and the standard deviation of the portfolio return look like Perfect correlation Stock B Stock A a Portfo 0 expected return Portfolio standard deviation 40 20 Perfect negative correlation 0 Assume now that stocks A and B are perfectly negatively correlated p 1 You invest w1 in stock A and w2 in stock B What is the variance and standard deviation of your portfolio cs 2 W12612 W226 2W1W2p126162 W12c512 W226 2W1W26162 2 W161 W252 op W161W2G2 where indicates the absolute value Perfect negative correlation olt turns out that stocks A and B are perfectly negatively correlated The expected return on stock A is 10 while the expected return on stock B is 15 The standard deviation of stock A is 15 while the standard deviation of stock B is 25 How does the relation between portfolio expected return and the standard deviation of the portfolio return look like now 21 Perfect negative correlation Stock B 0 Stock A O Portfo 0 expected return Portfolio standard deviation 43 Example with two assets It is rare that two assets have either perfect correlation or perfect negative correlation Most asset returns are positively but not perfectly correlated because they all are affected by the same general economic conditions 22 Portfo 0 expected return Other correlations Stock B 0 Stock A 0 Portfolio standard deviation 45 ReturnRisk Tradeoff for World Stocks Portfolio of US and Foreign Stocks Other correlations Total return on portfolio 2 It till 139 135 lllll39t l39orcign H H llmmum 39 urizmuc i ill 4 wrll39nlw Hquot um i L Illquot 1395 Ht l l wrctgn I l mquot l illl 4039 111 00 mg I too LII Nll i 531M lnmgn 01 o no A L Wquot mm low US W7 Risk lstzuxdzn39tl lilo 39 I L lmimion ol39 ll 1 l gt quot31 33 m tl39olio s l i return 2 l 46 23 Risk in a portfolio context If your wealth were invested in a single asset then asset 2 would be riskier than asset 1 because it has a higher standard deviation However if you hold a portfolio of assets the appropriate measure of risk is not the standard deviation but the marginal contribution of the asset to the overall riskiness of your portfolio In this context the riskiness of an asset can not be measured without reference to a benchmark portfolio 47 Portfolio choices with many risky assets Portfo 0 expected return Portfolio standard deviation 24 Portfolio choices with many risky assets The minimum variance vntieris the set of portfolios that minimizes the portfolio standard deviation for a given level of expected return The ef cient ontieris the set of portfolios that maximizes the expected return for a given portfolio standard deviation The ef cient frontier is the upward sloping portion of the minimum variance frontier Portfolio choices with many risky assets lf investors prefer more to less other things equal they prefer higher expected returns and are risk averse other things equal they prefer a lower portfolio standard deviation then all investors should choose portfolios on the efficient frontier 25 The limits to diversification Since most assets are positively correlated diversi cation can reduce risk Total risk Unsystematic Systematic Diversi able Nondiversi able Firmspeci c Marketwide Firmspeci c H39scan be eliminated from a portfolio by diversi cation Marketwide risk cannot be eliminated by diversification The limits of diversification Total risk Diversifiable risk Average porlfo o r sk Marketwide risk Number of securities in a portfolio 52 26 Glossary oRandom variable Probability distribution Expected value mean Variance Standard deviation Covariance Correlation coef cient Arithmetic mean return Geometric mean return Minimum variance frontier oEf cient frontier Firmspeci c diversi able nonsystematic risk Market wide nondiversi able systematic risk 53 27 MGMT543 HNANCE Handout 4 Aden w yfhybwantcashIqbns Anathg we5 nenchy a Evgeny Lyandres Chris Downing Fall 2005 Objective 0 Learn how to calculate a firm s free cash ow to be used in valuation What to discount When accounting statements are used to estimate cash ows accounting adjustments or accruals must be undone to recapture cash ows Revenues and costs are recorded when earned or incurred rather than when cash is received or paid Capital expenditures are depreciated rather than charged against current earnings What to discount I Your rm is planning to purchase a machine in December 2005 for 500000 and incur installation costs of 20000 It is going to be depreciated for 3 years by 100000 each year and then it is going to be sold for 250000 at the end of 2008 The machine is going to produce cash ows of 200000 per year in each of the years it is going to operate What are the relevant cash ows each year 2005 2006 2007 2008 olnvestment costs and outlays installation costs as well as cash from selling past investments are cash 4 ows depreCIatIon Is not What to discount Assume now that instead of paying 500000 for the new machine you can get it by giving away the old machine and 200000 The old machine was expected to produce cash revenues of 120000 in each of the years 2006 2007 and 2008 and then could be expected to be sold for 50000 at the end of 2008 What are the relevant cash ows now 2005 2006 2007 2008 We have to take into account the value of resources on hand opportunity costs 5 What to discount You have recognized the importance of the decision whether to exchange the machine for a long time now and you have hired a consultant to help you with the decision The consultant has charged you 10000 due in December 2005 and recommended to buy the new machine How should you account for this cost in your cash ow projections oAlways ignore sunk costs What to discount Assume that in order to keep the new machine operating you need to keep inventories of 10000 at the end of 2006 15000 at the end of 2007 and you39ll sell all the inventories together with the machine at the end of 2008 Assume that no inventories were required for the old machine What is the effect of inventories on your cash ow projections 2005 2006 2007 2008 OConsider changes in inventories What to discount Assume in addition that as a result of adding new customers you will have accounts receivable of 5000 at the end of 2006 and 10000 at the end of 2007 and 2008 each The accounts receivable will be liquidated at the end of 2009 How would accounts receivable affect your cash ow projections 2005 2006 2007 2008 2009 Consider changes in accounts receivable What to discount Assume that the suppliers of inventories allow you to postpone some payments have accounts payable The accounts payable will amount to 3000 in 2006 and 2007 and will be liquidated in 2008 What is the effect on your cash ows 2005 2006 2007 2008 OConsider changes in accounts payable What to discount Inventories accounts receivable accounts payable is called net walking capital In other words net working capital is the difference between current assets and current liabilities An increase in net working capital is a cash out ow A decrease in net working capital is a cash in ow To summarize we should always account for changes in the net working capital What to discount Assume now that you decided to keep the old machine and also to buy a new one The cash ows from the old machine are 120000 per year the cash ows from the new one is 200000 per year and the total cash ows are 320000 Which of these number we should consider while evaluating the purchase of the new machine Always consider only incremental cash ows n What to discount It turns out that if you buy the new machine in addition to keeping the old one the cash ows from the old machine are going to be reduced by 20000 per year How should we take this information into account oAlways consider incidental effects What to discount We are interested in aftertax cash ows Thus we should always consider taxes OAftertax operating cash ows assume for now that all revenues and expenses except for depreciation are in cash CF Revenues Expenses Taxes Taxes can be written as Taxes TC Revenues Expenses Depreciation where TC is the corporate tax rate 13 What to discount Substituting the rst expression into the second yields CF Rev Exp TC Rev Exp Dep or CF RevExp 1 To Dep TC CF Aftertax revenue net of expenses Depreciation tax shield What to discount What is the effect of depreciation on the cash ows from the new machine No depreciation With depreciation Gross income Depreciation Taxable income Taxes 35 Net income Aftertax cash flow What to discount Assume that the prices are going to increase by 4 in each of years 2006 2007 and 2008 How should we account for that 0 Inflation it does not matter whether you use real or nominal cash flows in your analysis The only important thing is that you are consistent Nominal cash ows must be discounted at the nominal rate while real cash ows must be discounted at the real rate 0 However discounting nominal cash ows allows the exibility for different cash ow streams to grow at different rates of in ation eg you might project wage rates to grow faster or slower than the cost of raw materials 5 What to discount Assume that in order to buy the new machine you need to take a loan of 200000 which has to be repaid in equal installments of 75000 at the end of years 2006 2007 and 2008 How should we account for these nancing cash ows 0 Do not include nancing costs in your cash ows These should be reflected in the discount rate 7 Deriving free cash flow from earnings 0 Note that Eamings befone interest and taxes EB77 is Reduced by depreciation and amortization Not affected by changes in net working capital Not affected by the purchase or sale of capital assets Computed before taxes Deriving free cash ow from earnings 0 Free cash ow EBIT depreciation and amortization change in net working capital capital expenditures sale of capital assets realized capital gains realized capital losses EBIT tax rate Deriving free cash flow from earnings Assume that a rm faces a project that has cash ows for the next 4 years The initial investment in year 0 is 40000000 The investment is going to be depreciated entirely using a linear schedule EBIT before depreciation and amortization EBITDA is 12000000 in each of years 14 The working capital at the end of year 1 is 2000000 It is expected to increase by 1000000 per year and to be sold at the end of year 4 The estimated selling price of the project at the end of year 4 is 20000000 0 Taxes are paid at a constant rate of 34 Deriving free cash flows from earnings How do we calculate EBIT What is the depreciation and amortization What is the change in net working capital What are the capital expenditures What are the proceeds from selling capital assets and what are the capital gainslosses How much taxes to we pay 21 Deriving free cash flow from earnings in 000s 0 1 2 3 4 EBITDA Depreciation EBIT Depreciation Change in net WC Capital expenditures Sale of capital assets Realized capital gains Taxes Free cash flow 22 Cash flow estimation main rules Investment costs and sale of investment are relevant not depreciation Include opportunity costs Forget sunk costs Don39t forget changes in net working capital Don39t confuse total and incremental payoffs Include all incidental effects Estimate cash ows on an aftertax basis 23 Analyzing investment projects Should you get an MBA You currently have a job that pays 55400 a year Yourjob is secure and you can reasonably expect to get a raise of 45 per year for the next 25 years when you intend to retire You have an opportunity to go back to school and get an MBA starting right now You have considered a number of good schools including Rice and have spent 1500 visiting your top 3 choices Finally you chose Rice where the cost is 32000 per year and it takes exactly 2 years to get a degree Analyzing investment projects Should you get an MBA Your advisors at Rice tell you that your starting base salary after business school will be 91000 and that the salary should grow by about 5 for the next 23 years you ll retire at that age remember Your living expenses at school would be 5000 more per year than if you were working moving books etc but your utility would be about the same either way you39d be having just as much fun You believe that you can make about 75 after tax on your money throughout your career if you invest your savings Taxes on income are a at 25 Analyzing investment projects Should you get an MBA If you get an MBA If you don t get an MBA Year Pretax CF Aftertax Pretax CF Aftertax CF CF b0lL Analyzing investment projects Should you get an MBA NPV MBA oNPV No MBA Analyzing investment projects Producing keyboards Your rm is considering investing in a machine to produce computer keyboards The price of the machine is 400000 and its economic life is 5 years when it will become worthless The machine will be fully depreciated by the straightline method The machine will produce 10000 keyboards each year The price of the keyboard will be 40 in the rst year and will increase by 5 each year Analyzing investment projects Producing keyboards The production cost per keyboard will be 20 in the rst year and will increase by 10 each year The corporate tax rate is 34 The appropriate discount rate is 15 0 Determine the NPV of this project Analyzing investment projects Producing keyboards 0 1 2 3 Taxes Free cash flow NPV Analyzing investment projects GAP GAP is considering buying an online cash register software from IBM so it can effectively deal with its retail sales The software package costs 750000 and will be depreciated down to zero using the straightline method over its threeyear economic life The marketing department predicts that the sales will increase by 600000 per year as a result of the new machine for the next 3 years after which the incremental change will be zero Analyzmg Investment prOJects Example GAP After 3 years the software can be sold for 40000 GAP also needs to add working capital of 25000 immediately This additional working capital will be recovered in full at the end of the software s life GAP s corporate tax rate is 35 The appropriate discount rate is 17 0 Determine the NPV of the new software Analyzing investment projects GAP in 000s O 1 2 3 Revenues Realized capital gains EBITDA Depreciation EBIT Depreciation Change in WC Capital expenditures Sale of capital assets Realized capital gains Tax rateEBT Free cash flow Analyzing investment projects ExmnpbGAP NPV Analyzing investment projects Example Tiny Air Inc Tiny Air Inc is a small pro table onejet airline that ies four commuter ights daily between Houston and Dallas Tiny Air is considering a purchase of an additional jet that costs 1000000 Tiny Air believes that there is a shortlived opportunity that will last exactly six years to provide service between Houston and San Antonio At the end of year 6 all major airlines will begin ying that route making it unpro table for Tiny Air For tax purposes the jet can be depreciated over a ten year life by the straight line method The CFO of Tiny Air is sure that the plane will be worth 700000 at the end of sixth yeag Analyzing investment projects Example Tiny Air Inc The tax rate for all taxes that Tiny Air pays is a at 40 Running a small airline is a risky business so Tiny Air uses 20 as its cost of capital Revenues associated with operating the route will be 900000 per year and operating expenses crew fuel etc will be 500000 per year oWhat is the NPV of the proposed new route Analyzing investment projects Example Tiny Air Inc in 0005 0 1 2 3 4 Revenues Expenses Realized capital gains EBITDA Depreciation EBIT Depreciation Change in WC Capital expenditures Sale of capital assets Realized capital gains Tax rateEBT Free cash flow Analyzing investment projects Example Tiny Air Inc NPV new project Projects with different lives 0Suppose a rm has to choose between the following two machines Machine C0 C1 02 C3 NPV at 10 A 100 20 20 B 120 30 30 20 OWhat machine should the firm choose Projects with different lives The NPV of machine B is higher But is the NPV a fair criterion in this case What if instead of the two machines that generate NPVA and NPVB respectively we would have two annuities that would have the same respective N PVS 20 Projects with different lives oWhen mutually exclusive projects have different lives you should convert present values to equivalent annual costs The equivalent annual cost is an annuity which has exactly the same life and present value as the project you are analyzing Projects with different lives Annuity payment A oAnnuity payment B OWhat machine should the firm choose 21

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