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OUR INSTRUCTORS Basit Shajani, CFA Basit graduated Magna Cum Laude from the world-renowned Wharton School of Business at the University of Pennsylvania with majors in Finance and Legal Studies. After graduating, Basit ran his own private wealth management firm. He started teaching CFA courses more than five years ago, and upon discovering how much he enjoyed teaching, he founded Elan Guides with a view to providing CFA candidates all around the globe access to efficient and effective CFA study materials at affordable prices. Basit remains an avid follower of equity, commodities and real estate markets and thoroughly enjoys using his knowledge and real-world finance experience to bring theory to life. Peter Olinto, CPA, JD Peter has taught CPA and CFA Exam Review courses for the past ten years and is a real вЂ�celebrityвЂ™ in the CPA and CFA prep industries. Previously he worked as an auditor for Deloitte & Touche, was a tax attorney for Ernst and Young, and later spent nearly ten years teaching law, accounting, financial statement analysis, and tax at both the graduate and undergraduate levels at Fordham UniversityвЂ™s business school. He graduated Magna Cum laude from Pace University and went on to earn his JD degree from Fordham University School of Law. В© 2013 ELAN GUIDES QUANTITATIVE METHODS CORRELATION AND REGRESSION n Sample covariance = Cov (X,Y) = пЃ“ (X пЂ X)(Y пЂ Y)/(n пЂ 1) i i i=1 where: n = sample size Xi = ith observation of Variable X X = mean observation of Variable X Yi = ith observation of Variable Y Y = mean observation of Variable Y Cov (X,Y) Sample correlation coefficient = r = s Xs Y n Sample variance = 2 sX = пЃ“ (X пЂ X) /(n пЂ 1) i 2 i=1 Sample standard deviation = sX = 2 sX Test statistic r nпЂ2 Test-stat = t = 1 пЂ r2 Where: n = Number of observations r = Sample correlation Linear Regression with One Independent Variable Regression model equation = Yi = b0 + b1Xi + пЃҐi, i = 1,...., n пЂ пЂ пЂ пЂ пЂ пЂ п‚· пЂ пЂ пЂ пЂ пЂ пЂ п‚· пЂ пЂ пЂ пЂ пЂ пЂ п‚· пЂ пЂ пЂ пЂ пЂ пЂ п‚· b1 and b0 are the regression coefficients. b1 is the slope coefficient. b0 is the intercept term. пЃҐ is the error term that represents the variation in the dependent variable that is not explained by the independent variable. В© 2013 ELAN GUIDES QUANTITATIVE METHODS Regression line equation = YЛ†i = bЛ† 0 + bЛ†1Xi , i = 1,...., n Regression Residuals n пЃ“ [Y пЂ (bЛ† пЂ« bЛ† X )] i 1 i 0 2 i=1 where: Yi = Actual value of the dependent variable bЛ† 0 + bЛ† 1Xi = Predicted value of dependent variable The Standard Error of Estimate ( SEE = n n пЃ“ (Yi пЂ bЛ† 0 пЂ bЛ† 1Xi)2 i=1 ) ( ) пЃ“ (пЃҐЛ† ) 1/2 i = nпЂ2 2 1/2 i=1 nпЂ2 = ( SSE nпЂ2 ) 1/2 The Coefficient of Determination Total variation = Unexplained variation + Explained variation R2 = Explained variation = Total variation пЂ Unexplained variation Total variation Total variation Unexplained variation =1пЂ Total variation Hypothesis Tests on Regression Coefficients CAPM: RABC = RF + пЃўABC(RM вЂ“ RF) RABC вЂ“ RF = пЃЎ + пЃўABC(RM вЂ“ RF) + пЃҐ пЂ пЂ пЂ пЂ пЂ пЂ п‚· пЂ пЂ пЂ пЂ пЂ пЂ п‚· The intercept term for the regression, b0, is пЃЎ. The slope coefficient for the regression, b1, is пЃўABC The regression sum of squares (RSS) n RSS = пЃ“ (Y^ пЂ Y ) 2 i пЃ§ Explained variation i=1 The sum of squared errors or residuals (SSE) n SSE = пЃ“ (Y пЂ Y^ ) i i 2 пЃ§ Unexplained variation i=1 В© 2013 ELAN GUIDES QUANTITATIVE METHODS ANOVA Table Source of Variation Degrees of Freedom Sum of Squares k RSS n пЂпЂ пЂЁk + 1) SSE n пЂпЂ 1 SST Regression (explained) Error (unexplained) Total k = the number of slope coefficients in the regression. Prediction Intervals 2 sf = s 2 Y^ п‚± tc sf В© 2013 ELAN GUIDES [ 1пЂ« 1 n (X пЂ X)2 пЂ« 2 (n пЂ 1) sx ] Mean Sum of Squares MSR = RSS k = MSE = RSS 1 SSE n пЂпЂ 2 = RSS MULTIPLE REGRESSION AND ISSUES IN REGRESSION MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS Multiple regression equation Multiple regression equation = Yi = b0 + b1X1i + b2X2i + . . .+ bk Xki + пЃҐi, i = 1,2, . . . , n Yi Xji b0 b1, . . . , bk пЃҐi n = the ith observation of the dependent variable Y = the ith observation of the independent variable Xj , j = 1,2, . . . , k = the intercept of the equation = the slope coefficients for each of the independent variables = the error term for the ith observation = the number of observations Residual Term пЃҐЛ†i = Yi пЂ YЛ†i = Yi пЂ (bЛ†0 + bЛ† 1X1i + bЛ† 2X2i + . . .+ bЛ† k Xki) Confidence Intervals bЛ†j В± (tc п‚ґ sbЛ†j) estimated regression coefficient В± (critical t-value)(coefficient standard error) F-statistic F-stat = MSR RSS/k = MSE SSE/[n пЂпЂ пЂЁk + 1)] R2 and Adjusted R2 R2 = Total variation пЂ Unexplained variation Total variation Adjusted R2 = R2 = 1 пЂ ( n пЂпЂ 1 n пЂпЂ kпЂ пЂпЂ 1 ) = SST пЂ SSE SST = RSS SST (1 пЂпЂ R2) Testing for Heteroskedasticity- The Breusch-Pagan (BP) Test пЃЈ2 = nR2 with k degrees of freedom n = Number of observations R2 = Coefficient of determination of the second regression (the regression when the squared residuals of the original regression are regressed on the independent variables). k = Number of independent variables В© 2013 ELAN GUIDES MULTIPLE REGRESSION AND ISSUES IN REGRESSION Testing for Serial Correlation- The Durban-Watson (DW) Test DW п‚» 2(1 вЂ“ r); where r is the sample correlation between squared residuals from one period and those from the previous period. Value of Durbin-Watson Statistic (H0: No serial correlation) Reject H0, conclude Positive Serial Correlation dl 0 Do not Reject H0 Inconclusive du Inconclusive 4 пЂ du Reject H0, conclude Negative Serial Correlation 4 пЂ dl 4 Problems in Linear Regression and Solutions Problem Effect Solution Heteroskedasticity Incorrect standard errors Use robust standard errors (corrected for conditional heteroskedasticity) Serial correlation Incorrect standard errors (additional problems if a lagged value of the dependent variable is used as an independent variable) Use robust standard errors (corrected for serial correlation) Multicollinearity High R2 and low t-statistics Remove one or more independent variables; often no solution based in theory Model Specification Errors Yi = b0 + b1lnX1i + b2X2i + пЃҐ Linear Trend Models yt = b0 + b1t + пЃҐt, t = 1, 2, . . . , T where: yt = the value of the time series at time t (value of the dependent variable) b0 = the y-intercept term b1 = the slope coefficient/ trend coefficient t = time, the independent or explanatory variable пЃҐt = a random-error term В© 2013 ELAN GUIDES TIME SERIES ANALYSIS TIME-SERIES ANALYSIS Linear Trend Models yt = b0 + b1t + пЃҐt, t = 1, 2, . . . , T where: yt = the value of the time series at time t (value of the dependent variable) b0 = the y-intercept term b1 = the slope coefficient/ trend coefficient t = time, the independent or explanatory variable пЃҐt = a random-error term Log-Linear Trend Models A series that grows exponentially can be described using the following equation: yt = eb0 + b1t where: yt = the value of the time series at time t (value of the dependent variable) b0 = the y-intercept term b1 = the slope coefficient t = time = 1, 2, 3 ... T We take the natural logarithm of both sides of the equation to arrive at the equation for the loglinear model: ln yt = b0 + b1t + пЃҐt, t = 1,2, . . . , T AUTOREGRESSIVE (AR) TIME-SERIES MODELS xt = b0 + b1xt пЂ 1 + пЃҐt A pth order autoregressive model is represented as: xt = b0 + b1xt пЂ 1 + b2xt пЂ 2+ . . . + bpxt пЂ p + пЃҐt Detecting Serially Correlated Errors in an AR Model t-stat = Residual autocorrelation for lag Standard error of residual autocorrelation where: Standard error of residual autocorrelation = 1/ T T = Number of observations in the time series В© 2013 ELAN GUIDES TIME SERIES ANALYSIS Mean Reversion xt = b0 1 пЂ b1 Multiperiod Forecasts and the Chain Rule of Forecasting ^x = ^b + ^b x t+1 0 1 t Random Walks xt = xt пЂ 1 + пЃҐt , E(пЃҐt) = 0, E(пЃҐt2) = пЃі2, E(пЃҐtпЃҐs) = 0 if t п‚№пЂ s The first difference of the random walk equation is given as: yt = xt пЂ xt пЂ 1 = xt пЂ 1 + пЃҐt пЂ xt пЂ 1= пЃҐt , E(пЃҐt) = 0, E(пЃҐt2) = пЃі2, E(пЃҐtпЃҐs) = 0 for t п‚№пЂ s Random Walk with a Drift xt = b0 + b1xt пЂ 1 + пЃҐt b1 = 1, b0 п‚№пЂ 0, or xt = b0 + xt пЂ 1 + пЃҐt , E(пЃҐt) = 0 The first-difference of the random walk with a drift equation is given as: yt = xt пЂ xt пЂ 1 , yt = b0 + пЃҐt , b0 п‚№пЂ 0 The Unit Root Test of Nonstationarity xt пЂЅ b0 + b1xt пЂ 1 + пЃҐt xt пЂ xt пЂ 1 пЂЅ b0 + b1xt пЂ 1 пЂ xt пЂ 1 + пЃҐt xt пЂ xt пЂ 1 пЂЅ b0 + (b1 пЂ 1)xt пЂ 1 + пЃҐt xt пЂ xt пЂ 1 пЂЅ b0 + g1xt пЂ 1 + пЃҐt В© 2013 ELAN GUIDES TIME SERIES ANALYSIS Autoregressive Moving Average (ARMA) Models xt = b0 + b1xt пЂ 1 + . . . + bpxt пЂ p + пЃҐt + пЃ±1пЃҐt пЂ 1 +. . . + пЃ±qпЃҐt пЂ q E(пЃҐt) = 0, E(пЃҐt2) = пЃі2, E(пЃҐtпЃҐs) = 0 for t п‚№пЂ s Autoregressive Conditional Heteroskedasticity Models (ARCH Models) ^пЃҐ 2 = a + a ^пЃҐ 2 + u t 0 1 t пЂ1 t The error in period t+1 can then be predicted using the following formula: ^пЃі 2 = a^ + a^ пЃҐ^ 2 t+1 0 1 t В© 2013 ELAN GUIDES CURRENCY EXCHANGE RATES: DETERMINATION AND FORECASTING CURRENCY EXCHANGE RATES: DETERMINATION AND FORECASTING Currency Cross Rates For example, given the USD/EUR and JPY/USD exchange rates, we can calculate the cross rate between the JPY and the EUR, JPY/EUR as follows: JPY JPY USD = п‚ґ EUR USD EUR Cross Rate Calculations with Bid-Ask Spreads USD/EURask = 1.3806 USD/EURbid = 1.3802 п‚· Represents the price of EUR п‚· An investor can buy EUR with USD at this price. п‚· Represents the price of EUR (base currency). п‚· An investor can sell EUR for USD at this price (as it is the bid price quoted by the dealer). Determining the EUR/USDbid cross rate: EUR/USDbid = 1/(USD/EURask) Determining the EUR/USDask cross rate: EUR/USDask = 1 / (USD/EURbid) Forward exchange rates (F) - One year Horizom FFC/DC = SFC/DC п‚ґ (1 + iFC) (1 + iDC) FPC/BC = SPC/BC п‚ґ Forward exchange rates (F) - Any Investment Horizom FFC/DC = SFC/DC п‚ґ 1 + (iFC п‚ґпЂ Actual 360) 1 + (iDC п‚ґпЂ Actual 360) FPC/BC = SPC/BC п‚ґ 1 + (iPC п‚ґпЂ Actual 360) 1 + (iBC п‚ґпЂ Actual 360) Currencies Trading at a Forward Premium/Discount FFC/DC пЂ SFC/DC = SFC/DC FPC/BC пЂ SPC/BC = SPC/BC В© 2013 ELAN GUIDES ( ( ) ) (iFC пЂ iDC) п‚ґпЂ Actual 360 1 + (iDC п‚ґпЂ Actual 360) (iPC пЂ iBC) п‚ґпЂ Actual 360 1 + (iBC п‚ґпЂ Actual 360) (1 + iPC) (1 + iBC) CURRENCY EXCHANGE RATES: DETERMINATION AND FORECASTING Covered Interest Rate Parity 1 + (iPC п‚ґпЂ Actual 360) 1 + (iBC п‚ґпЂ Actual 360) FPC/BC = SPC/BC п‚ґ The forward premium (discount) on the base currency can be expressed as a percentage as: FPC/BC пЂ SPC/BC Forward premium (discount) as a % = SPC/BC The forward premium (discount) on the base currency can be estimated as: Forward premium (discount) as a % п‚» FPC/BC пЂ SPC/BC п‚» iPC пЂ iBC Uncovered Interest Rate Parity Expected future spot exchange rate: SeFC/DC = SFC/DC п‚ґ (1 + iFC) (1 + iDC) The expected percentage change in the spot exchange rate can be calculated as: Expected % change in spot exchange rate = пЂҐпЃ„SePC/BC = SePC/BC вЂ“ SPC/BC SPC/BC The expected percentage change in the spot exchange rate can be estimated as: Expected % change in spot exchange rate п‚» пЂҐпЃ„SePC/BC п‚»пЂ iPCпЂ пЂпЂ iBC Purchasing Power Parity (PPP) Law of one price: PXFC = PXDC п‚ґ SFC/DC Law of one price: PXPC = PXBC п‚ґ SPC/BC Absolute Purchasing Power Parity (Absolute PPP) SFC/DC = GPLFC / GPLDC SPC/BC = GPLPC / GPLBC Relative Purchasing Power Parity (Relative PPP) Relative PPP: E(S T FC/DC) = S 0 FC/DC ( T 1 пЂ« пЃ°FC 1 + пЃ°DC ) Ex Ante Version of PPP Ex ante PPP: %пЃ„SeFC/DC п‚» пЃ°eFC пЂпЂ пЃ°eDC Ex ante PPP: %пЃ„SePC/BC п‚»пЂ пЃ°ePCпЂ пЂпЂ пЃ°eBC В© 2013 ELAN GUIDES CURRENCY EXCHANGE RATES: DETERMINATION AND FORECASTING Real Exchange Rates The real exchange rate (qFC/DC) equals the ratio of the domestic price level expressed in the foreign currency to the foreign price level. qFC/DC = PDC in terms of FC PFC = PDC п‚ґ SFC/DC PFC = SFC/DC ( ) PDC PFC The Fisher Effect Fischer Effect: i = r + пЃ°e International Fisher effect: (iFC вЂ“ iDC) = (пЃ°eFC вЂ“ пЃ°eDC) Figure 1: Spot Exchange Rates, Forward Exchange Rates, and Interest Rates Ex Ante PPP Foreign-Domestic Expected Inflation Differential пЃ°eFC пЂ пЃ°eDC International Fisher Effect Expected change in Spot Exchange Rate %пЃ„SeFC/DC Forward Rate as an Unbiased Predictor Uncovered Interest Rate Parity Foreign-Domestic Interest rate Differential iFC пЂ iDC Forward Discount FFC/DC пЂпЂ SFC/DC SFC/DC Covered Interest Rate Parity Balance of Payment Current account + Capital account + Financial account = 0 Real Interest Rate Differentials, Capital Flows and the Exchange Rate qL/H вЂ“ qL/H = (iH вЂ“ iL) вЂ“ (пЃ°eH вЂ“ пЃ°eL) вЂ“ (пЃ†H вЂ“ пЃ†L) В© 2013 ELAN GUIDES CURRENCY EXCHANGE RATES: DETERMINATION AND FORECASTING The Taylor rule i = rn + пЃ° + пЃЎпЂЁпЃ°пЂ пЂпЂ пЃ°пЂЄпЂ©пЂ пЂ«пЂ пЃўпЂЁy пЂпЂ y*) where i = the Taylor rule prescribed central bank policy rate rn = the neutral real policy rate пЃ° = the current inflation rate пЃ°* = the central bankвЂ™s target inflation rate y = the log of the current level of output y* = the log of the economyвЂ™s potential/sustainable level of output qPC/BC = qPC/BC + ( rnBC пЂпЂ rnPC) + пЃЎпЃ›пЂЁпЃ°BCпЂ пЂпЂ пЃ°пЂЄBCпЂ©пЂ пЂпЂ пЂЁпЃ°PCпЂ пЂпЂ пЃ°пЂЄPCпЂ©пЃќ пЂ«пЂ пЃўпЃ›пЂЁyBC пЂпЂ y*BC) пЂпЂ пЂЁyPC пЂпЂ y*PC)] пЂпЂ пЂЁпЃ†BC пЂпЂ пЃ†PC) В© 2013 ELAN GUIDES ECONOMIC GROWTH AND THE INVESTMENT DECISION ECONOMIC GROWTH AND THE INVESTMENT DECISION Relationship between economic growth and stock prices P = GDP ( )( ) E GDP P E P = Aggregate price or value of earnings. E = Aggregate earnings This equation can also be expressed in terms of growth rates: пЃ„P = пЃ„(GDP) + пЃ„(E/GDP) + пЃ„(P/E) Production Function Y = AKпЃЎL1-пЃЎ Y = Level of aggregate output in the economy L = Quantity of labor K = Quantity of capital A = Total factor productivity. Total factor productivity (TFP) reflects the general level of productivity or technology in the economy. TFP is a scale factor i.e., an increase in TFP implies a proportionate increase in output for any combination of inputs. пЃЎ = Share of GDP paid out to capital 1 пЂ пЃЎ = Share of GDP paid out to labor y = Y/L = A(K/L)пЃЎ(L/L)1-пЃЎ = Ak пЃЎ y = Y/L = Output per worker or labor productivity. k = K/L = Capital per worker or capital-labor ratio Cobb-Douglas production function пЃ„Y/Y =пЃ„A/A + пЃЎпЃ„K/K + (1 пЂпЂ пЃЎ )пЃ„L/L Potential GDP Growth rate in potential GDP = Long-term growth rate of labor force + Long-term growth rate in labor productivity Labor Supply Total number of hours available for work = Labor force п‚ґпЂ Average hours worked per worker В© 2013 ELAN GUIDES ECONOMIC GROWTH AND THE INVESTMENT DECISION Neoclassical Model (SolowвЂ™s Model) ( )[( ) Y 1 = K s ] пЃ± + пЃ¤пЂ + n п‚єпЂ пЃ™ (1-пЃЎ) s = Fraction of income that is saved пЃ± = Growth rate of TFP пЃЎ = Elasticity of output with respect to capital y = Y/L or income per worker k = K/L or capital-labor ratio пЃ¤ = Constant rate of depreciation on physical stock n = Labor supply growth rate. Savings/Investment Equation: sy = [( ) ] пЃ± + пЃ¤пЂ + n k (1 пЂпЂ пЃЎ) Growth rates of Output Per Capita and the Capital-Labor Ratio пЃ„y пЃ± Y = + пЃЎs пЂпЃ™ y (1пЂпЃЎ) K ( ) ( пЃ„k пЃ± Y = +s пЂпЃ™ k (1пЂпЃЎ) K ) Production Function in the Endogenous Growth Model ye = f(ke) = cke В© 2013 ELAN GUIDES INVENTORIES: IMPLICATIONS FOR FINANCIAL STATEMENT AND RATIOS INVENTORIES: IMPLICATIONS FOR FINANCIAL STATEMENTS AND RATIOS Ending Inventory = Opening Inventory + Purchases - Cost of goods sold LIFO and FIFO Comparison with Rising Prices and Stable Inventory Levels LIFO FIFO COGS Higher Lower Income before taxes Lower Higher Income taxes Lower Higher Net income Lower Higher Total cash flow Higher Lower EI Lower Higher Working capital Lower Higher LIFO versus FIFO with Rising Prices and Stable Inventory Levels Type of Ratio Profitability ratios NP and GP margins Solvency ratios Debt-to-equity and debt ratio Liquidity ratios Current ratio Quick ratio Activity ratios Inventory turnover Total asset turnover В© 2013 ELAN GUIDES Effect on Numerator Effect on Denominator Income is lower under LIFO because COGS is higher Sales are the same under both Lower under LIFO Same debt levels Lower equity and assets under LIFO Higher under LIFO Current assets are lower under LIFO because EI is lower Current liabilities are the same. Lower under LIFO Quick assets are higher under LIFO as a result of lower taxes paid Current liabilities are the same Higher under LIFO COGS is higher under LIFO Average inventory is lower under LIFO Higher under LIFO Sales are the same Lower total assets under LIFO Higher under LIFO Effect on Ratio INVENTORIES: IMPLICATIONS FOR FINANCIAL STATEMENT AND RATIOS LIFO reserve (LR) EIFIFO = EILIFO + LR where LR = LIFO Reserve COGSFIFO is lower than COGSLIFO during periods of rising prices: COGSFIFO = COGSLIFO пЂ (Change in LR during the year) Net Income after tax under FIFO will be greater than LIFO net income after tax by: Change in LIFO Reserve п‚ґ (1 пЂ Tax rate) When converting from LIFO to FIFO assuming rising prices: Equity (retained earnings) increases by: LIFO Reserve п‚ґ (1 пЂпЂ Tax rate) Liabilities (deferred taxes) increase by: LIFO Reserve п‚ґ (Tax rate) Current assets (inventory) increase by: LIFO Reserve В© 2013 ELAN GUIDES INVENTORIES: IMPLICATIONS FOR FINANCIAL STATEMENT AND RATIOS Impact of an Inventory Write-Down on Various Financial Ratios Type of Ratio Profitability ratios NP and GP margins Solvency ratios Debt-to-equity and debt ratio Liquidity ratios Current ratio Activity ratios Inventory turnover Total asset turnover В© 2013 ELAN GUIDES Effect on Numerator Effect on Denominator COGS increases so profits fall Sales remain the same Lower (worsens) Debt levels remain the same Equity decreases (due to lower profits) and current assets decrease (due to lower inventory) Higher (worsens) Current assets decrease (due to lower inventory) Current liabilities remain the same. Lower (worsens) COGS increases Average inventory decreases Higher (improves) Sales remain the same Total assets decrease Higher (improves) Effect on Ratio LONG-LIVED ASSETS: IMPLICATIONS FOR FINANCIAL STATEMENTS AND RATIOS LONG-LIVED ASSETS: IMPLICATIONS FOR FINANCIAL STATEMENTS AND RATIOS Effects of Capitalization Initially when the cost is capitalized In future periods when the asset is depreciated or amortized Effects on Financial Statements пЂ п‚·пЂ пЂ пЂ Noncurrent assets increase. пЂ п‚·пЂ пЂ пЂ Cash flow from investing activities decreases. пЂ п‚·пЂ пЂ пЂ Noncurrent assets decrease. пЂ п‚·пЂ пЂ пЂ Net income decreases. пЂ п‚·пЂ пЂ пЂ Retained earnings decrease. пЂ п‚·пЂ пЂ пЂ Equity decreases. Effects of Expensing When the item is expensed Effects on Financial Statements пЂ п‚·пЂ пЂ пЂ Net income decreases by the entire after-tax amount of the cost. пЂ п‚·пЂ пЂ пЂ No related asset is recorded on the balance sheet and therefore, no depreciation or amortization expense is charged in future periods. пЂ п‚·пЂ пЂ пЂ Operating cash flow decreases. п‚·пЂ пЂ пЂ Expensed costs have no financial statement impact in future years. Financial Statement Effects of Capitalizing versus Expensing Capitalizing Net income (first year) Higher Net income (future years) Lower Total assets Higher ShareholdersвЂ™ equity Higher Cash flow from operations activities Higher Cash flow from investing activities Lower Income variability Lower Debt to equity ratio Lower Expensing Lower Higher Lower Lower Lower Higher Higher Higher Straight Line Depreciation Depreciation expense = Original cost пЂ Salvage value Depreciable life Accelerated Depreciation DDB depreciation in Year X = 2 Г— Book value at the beginning of Year X Depreciable life В© 2013 ELAN GUIDES LONG-LIVED ASSETS: IMPLICATIONS FOR FINANCIAL STATEMENTS AND RATIOS Gross investment in fixed assets Accumulated depreciation Net investment in fixed assets = + Annual depreciation expense Annual depreciation expense Annual depreciation expense Estimated useful or depreciable life The historical cost of an asset divided by its useful life equals annual depreciation expense under the straight line method. Therefore, the historical cost divided by annual depreciation expense equals the estimated useful life. Average age of asset Remaining useful life Annual depreciation expense times the number of years that the asset has been in use equals accumulated depreciation. Therefore, accumulated depreciation divided by annual depreciation equals the average age of the asset. The book value of the asset divided by annual depreciation expense equals the number of years the asset has remaining in its useful life. Income Statement Effects of Lease Classification Income Statement Item Finance Lease Operating expenses Nonoperating expenses EBIT (operating income) Total expenses- early years Total expenses- later years Net income- early years Net income- later years Lower (Depreciation) Higher (Interest expense) Higher Higher Lower Lower Higher Cash Flow Effects of Lease Classification CF Item Finance Lease CFO Higher CFF Lower Total cash flow Same В© 2013 ELAN GUIDES Operating Lease Lower Higher Same Operating Lease Higher (Lease payment) Lower (None) Lower Lower Higher Higher Lower LONG-LIVED ASSETS: IMPLICATIONS FOR FINANCIAL STATEMENTS AND RATIOS Table 9: Impact of Lease Classification on Financial Ratios Ratio Asset turnover Denominator Ratio Better or Numerator under under Finance Worse under Finance Lease Lease Effect on Ratio Finance Lease Sales- same Return on assets* Net income- lower Assets- higher Lower Worse Assets- higher Lower Worse Current ratio Current assetssame Current liabilitieshigher Lower Worse Leverage ratios (D/E and D/A**) Debt- higher Equity- same Assets- higher Higher Worse Equity- same Lower Worse Return on equity* Net income- lower **Notice that both the numerator and the denominator for the D/A ratio are higher when classifying the lease as a finance lease. Beware of such exam questions. When the numerator and the denominator of any ratio are heading in the same direction (either increasing or decreasing), determine which of the two is changing more in percentage terms. If the percentage change in the numerator is greater than the percentage change in the denominator, the numerator effect will dominate. Firms usually have lower levels of total debt compared to total assets. The increase in both debt and assets by classifying the lease as a finance lease will lead to an increase in the debt to asset ratio because the percentage increase in the numerator is greater. Financial Statement Effects of Lease Classification from LessorвЂ™s Perspective Financing Lease Operating Lease Total net income Same Same Net income (early years) Higher Lower Taxes (early years) Higher Lower Total CFO Lower Higher Total CFI Higher Lower Total cash flow Same Same В© 2013 ELAN GUIDES INTERCORPORATE INVESTMENTS INTERCORPORATE INVESTMENTS Summary of Accounting Treatment for Investments In Financial Assets In Associates Business Combinations In Joint Ventures Influence Not significant Significant Controlling Typical percentage interest Usually < 20% Usually 20% пЂ 50% Usually > 50% Varies Accounting Classified into one of Treatment four categories based on management intent and type of security. Equity method Consolidation Debt only: пЂ пЂ пЂ пЂ п‚· Held-to-maturity (amortized cost, changes in value ignored unless deemed as impaired) Debt and Equity: пЂ пЂ пЂ пЂ п‚· Held for trading (fair value, changes in value recognized in profit or loss) пЂ пЂ пЂ пЂ п‚· Available-for-sale (fair value, changes in value recognized in equity) пЂ пЂ пЂ пЂ п‚· Designated at fair value (fair value, changes in value recognized in profit or loss) Combination Merger Acquisition Consolidation В© 2013 ELAN GUIDES Description Company A + Company B = Company A Company A + Company B = (Company A + Company B) Company A + Company B = Company C Shared Control IFRS: Equity method or proportionate consolidation U.S. GAAP: Equity method (except for unincorporated ventures in specialized industries) INTERCORPORATE INVESTMENTS Adjusted Values Upon Reclassification of Sale of Receivables: CFO Lower CFF Higher Total cash flow Same Current assets Higher Current liabilities Higher Current ratio Lower (Assuming it was greater than 1) Difference between QSPE and SPE Securitized Transaction: Qualified Special Purpose Entity Securitized Transaction: Special Purpose Entity пЂ пЂ пЂ пЂ п‚· Originator of receivables sells financial assets to an SPE. пЂ пЂ пЂ пЂ п‚· The originator does not own or hold or expect to receive beneficial interest. пЂ пЂ пЂ пЂ п‚· SFAS 140 (before 2008 revision) allowed seller to derecognize the sold assets if transferred assets have been isolated from the transferor and are beyond the reach of bankruptcy, and are financial assets. пЂ пЂ пЂ пЂ п‚· Originator of receivables sells financial assets to an SPE. пЂ пЂ пЂ пЂ п‚· Seller is primary beneficiary; absorbs risks and rewards. пЂ пЂ пЂ пЂ п‚· Seller maintains some level of control. пЂ пЂ пЂ пЂ п‚· Seller is required to consolidate. пЂ пЂ пЂ пЂ п‚· SellerвЂ™s balance sheet would still show receivables as an asset. пЂ пЂ пЂ пЂ п‚· Debt of SPE would appear on sellerвЂ™s balance sheet. Impact of Different Accounting Methods on Financial Ratios Equity Method Proportionate Consolidation Acquisition Method Leverage Better (lower) as liabilities are lower and equity is the same In-between Worse (higher) as liabilities are higher and equity is the same Net Profit Margin Better (higher) as sales are lower and net income is the same In-between Worse (lower) as sales are higher and net income is the same ROE Better (higher) as equity is lower Same as under the and net income is the same equity method Worse (lower) as equity is higher and net income is the same ROA Better (higher) as net income is the same and assets are lower Worse (lower) as net income is the same and assets are higher In-between В© 2013 ELAN GUIDES EMPLOYEE COMPENSATION: POST-EMPLOYMENT AND SHARE-BASED EMPLOYEE COMPENSATION: POST-EMPLOYMENT AND SHARE-BASED Final yearвЂ™s salary = Current salary Г— [(1 + Annual compensation increase)years until retirement] Estimated annual payment = (Estimated final salary Г— Benefit formula) Г— Years of service Annual unit credit = Value at retirement / Years of service Types of Post-Employment Benefits Amount of PostEmployment Benefit to Employee Obligation of Sponsoring Company Defined contribution pension plan Amount of future benefit is not defined. Actual future benefit will depend on investment performance of plan assets. Investment risk is borne by employee. Amount of the companyвЂ™s Not applicable. obligation (contribution) is defined in each period. The contribution, if any, is typically made on a periodic basis with no additional future obligation. Defined benefit pension plan Amount of future benefit is defined, based on the planвЂ™s formula (often a function of length of service and final yearвЂ™s compensation). Investment risk is borne by company. Amount of the future obligation, based on the planвЂ™s formula, must be estimated in the current period. Companies typically prefund the DB plans by contributing funds to a pension trust. Regulatory requirements to pre-fund vary by country. Other postemployment benefits (e.g., retireesвЂ™ health care) Amount of future benefit depends on plan specifications and type of benefit. Eventual benefits are specified. The amount of the future obligation must be estimated in the current period. Companies typically do not pre-fund other postemployment benefit obligations. Type of Benefit A companyвЂ™s pension obligation will increase as a result of: пЂ пЂ пЂ пЂ пЂ пЂ п‚· Current service costs. пЂ пЂ пЂ пЂ пЂ пЂ п‚· Interest costs. пЂ пЂ пЂ пЂ пЂ пЂ п‚· Past service costs. пЂ пЂ пЂ пЂ пЂ пЂ п‚· Actuarial losses. A companyвЂ™s pension obligation will decrease as a result of: пЂ пЂ пЂ пЂ пЂ пЂ п‚· Actuarial gains. пЂ пЂ пЂ пЂ пЂ пЂ п‚· Benefits paid. В© 2013 ELAN GUIDES Sponsoring CompanyвЂ™s Pre-funding of Its Future Obligation EMPLOYEE COMPENSATION: POST-EMPLOYMENT AND SHARE-BASED Reconciliation of the Pension Obligation: Pension obligation at the beginning of the period + Current service costs + Interest costs + Past service costs + Actuarial losses вЂ“ Actuarial gains вЂ“ Benefits paid Pension obligation at the end of the period The fair value of assets held in the pension trust (plan) will increase as a result of: пЂ пЂ пЂ пЂ пЂ пЂ п‚· A positive actual dollar return earned on plan assets; and пЂ пЂ пЂ пЂ пЂ пЂ п‚· Contributions made by the employer to the plan. The fair value of plan assets will decrease as a result of: пЂ пЂ пЂ пЂ пЂ пЂ п‚· Benefits paid to employees. Reconciliation of the Fair Value of Plan Assets: Fair value of plan assets at the beginning of the period + Actual return on plan assets + Contributions made by the employer to the plan пЂ Benefits paid to employees Fair value of plan assets at the end of the period Balance Sheet Presentation of Defined Benefit Pension Plans Funded status = Pension obligation пЂ Fair value of plan assets пЂ пЂ пЂ пЂ пЂ пЂ п‚· пЂ пЂ пЂ пЂ пЂ пЂ п‚· If Pension obligation > Fair value of plan assets: Plan is underfunded п‚® Positive funded status п‚® Net pension liability. If Pension obligation < Fair value of plan assets: Plan is overfunded п‚®пЂ Negative funded status п‚® Net pension asset. Calculating Periodic Pension Cost Priodic pension cost = Ending funded status пЂ Beginning funded status + Employer contributions Periodic pension cost = Current service costs + Interest costs + Past service costs + Actuarial losses пЂ Actuarial gains пЂ Actual return on plan assets Under the corridor method, if the net cumulative amount of unrecognized actuarial gains and losses at the beginning of the reporting period exceeds 10% of the greater of (1) the defined benefit obligation or (2) the fair value of plan assets, then the excess is amortized over the expected average remaining working lives of the employees participating in the plan and included as a component of periodic pension expense on the P&L. В© 2013 ELAN GUIDES EMPLOYEE COMPENSATION: POST-EMPLOYMENT AND SHARE-BASED Components of a CompanyвЂ™s Defined Benefit Pension Periodic Costs IFRS Component IFRS Recognition U.S. GAAP Component U.S. GAAP Recognition Service costs Recognized in P&L. Current service costs Past service costs Recognized in P&L. Recognized in OCI and subsequently amortized to P&L over the service life of employees. Net interest income/ expense Recognized in P&L as the following amount: Net pension liability or asset Г— interest rate(a) Interest expense on pension obligation Expected return on plan assets Recognized in P&L. Actuarial gains and losses including differences between the actual and expected returns on plan assets Recognized immediately in P&L or, more commonly, recognized in OCI and subsequently amortized to P&L using the corridor or faster recognition method.(b) пЂ пЂ пЂ пЂ п‚· Difference between expected and actual return on assets = Actual return пЂ (Plan assets Г— Expected return). пЂ пЂ пЂ пЂ п‚· Actuarial gains and losses = Changes in a companyвЂ™s pension obligation arising from changes in actuarial assumptions. Remeasurements: Recognized in OCI and Net return on plan not subsequently assets and actuarial amortized to P&L. gains and losses пЂ пЂ пЂ пЂ п‚· Net return on plan assets = Actual return пЂ (Plan assets Г— Interest rate). пЂ пЂ пЂ пЂ п‚· Actuarial gains and losses = Changes in a companyвЂ™s pension obligation arising from changes in actuarial assumptions. Recognized in P&L as the following amount: Plan assets Г— expected return. (a) The interest rate used is equal to the discount rate used to measure the pension liability (the yield on highquality corporate bonds.) (b) If the cumulative amount of unrecognized actuarial gains and losses exceeds 10 percent of the greater of the value of the plan assets or of the present value of the DB obligation (under U.S. GAAP, the projected benefit obligation), the difference must be amortized over the service lives of the employees. В© 2013 ELAN GUIDES EMPLOYEE COMPENSATION: POST-EMPLOYMENT AND SHARE-BASED Impact of Key Assumptions on Net Pension Liability and Periodic Pension Cost Impact of Assumption on Net Impact of Assumption on Periodic Assumption Pension Liability (Asset) Pension Cost and Pension Expense Higher discount rate Lower obligation Pension cost and pension expense will both typically be lower because of lower opening obligation and lower service costs Higher rate of compensation increase Higher obligation Higher service and interest costs will increase periodic pension cost and pension expense. Higher expected return on plan assets No effect, because fair value of plan assets are used on balance sheet Not applicable for IFRS No effect on periodic pension cost under U.S. GAAP Lower periodic pension expense under U.S. GAAP В© 2013 ELAN GUIDES MULTINATIONAL OPERATIONS MULTINATIONAL OPERATIONS пЂ пЂ пЂ пЂ пЂ пЂ п‚· The presentation currency (PC) is the currency in which the parent company reports its financial statements. It is typically the currency of the country where the parent is located. For example, U.S. companies are required to present their financial results in USD, German companies in EUR, Japanese companies in JPY, and so on. пЂ пЂ пЂ пЂ пЂ пЂ п‚· The functional currency (FC) is the currency of the primary business environment in which an entity operates. It is usually the currency in which the entity primarily generates and expends cash. пЂ пЂ пЂ пЂ пЂ пЂ п‚· The local currency (LC) is the currency of the country where the subsidiary operates. Table 1 Transaction Export sale Import purchase Type of Exposure Asset (account receivable) Liability (account payable) Foreign Currency Strengthens Weakens Gain Loss Loss Gain Methods for Translating Foreign Currency Financial Statements of Subsidiaries Current Rate/ Temporal Method Local Currency T Functional Currency CR Presentation Currency Temporal Method Local Currency T Functional Currency = Presentation Currency Current Rate Method Local Currency = Functional Currency CR Presentation Currency пЂ пЂ пЂ пЂ пЂ пЂ п‚· пЂ пЂ пЂ пЂ пЂ пЂ п‚· пЂ пЂ пЂ пЂ пЂ пЂ п‚· The current rate is the exchange rate that exists on the balance sheet date. The average rate is the average exchange rate over the reporting period. The historical rate is the actual exchange rate that existed on the original transaction date. В© 2013 ELAN GUIDES MULTINATIONAL OPERATIONS Rules for Foreign Currency Translation Current Rate Method FC = LC Income Statement Component Sales Cost of goods sold Selling expenses Depreciation expense Amortization expense Interest expense Income tax Net income before translation gain (loss) Translation gain (loss) Net income Less: Dividends Change in retained earnings Balance Sheet Component Temporal Method FC = PC Exchange Rate Used Average rate Average rate Average rate Average rate Average rate Average rate Average rate Average rate Historical rate Average rate Historical rate Historical rate Average rate Average rate Computed as Rev вЂ“ Exp N/A Computed as Rev вЂ“ Exp Historical rate Computed as NI вЂ“ Dividends Used as input for translated B/S Plug in Number Computed as пЃ„RE + Dividends Historical rate From B/S Exchange Rate Used Cash Accounts receivable Monetary assets Inventory Nonmonetary assets measured at current value Property, plant and equipment Less: Accumulated depreciation Nonmonetary assets measured at historical cost Current rate Current rate Current rate Current rate Current rate Current rate Current rate Current rate Historical rate Current rate Current rate Current rate Current rate Historical rate Historical rate Historical rate Accounts payable Long-term notes payable Monetary liabilities Nonmonetary liabilities: Measured at current value Measured at historical cost Capital stock Retained earnings Current rate Current rate Current rate Current rate Current rate Current rate Current rate Current rate Historical rate From I/S Current rate Historical rate Historical rate To balance Used as input for translated I/S N/A Cumulative translation adjustment Plug in Number В© 2013 ELAN GUIDES MULTINATIONAL OPERATIONS Balance Sheet Exposure Foreign Currency (FC) Strengthens Weakens Positive translation adjustment Negative translation adjustment Negative translation adjustment Positive translation adjustment Balance Sheet Exposure Net asset Net liability Effects of Exchange Rate Movements on Financial Statements Temporal Method, Net Monetary Liability Exposure Temporal Method, Net Monetary Asset Exposure Foreign currency strengthens relative to parentвЂ™s presentation currency пЃЁRevenues пЃЁAssets пЃЁLiabilities пЃ© Net income пЃ© ShareholdersвЂ™ equity Translation loss пЃЁRevenues пЃЁAssets пЃЁLiabilities пЃЁNet income пЃЁShareholdersвЂ™ equity Translation gain пЃЁRevenues пЃЁAssets пЃЁLiabilities пЃЁNet income пЃЁShareholdersвЂ™ equity Positive translation adjustment Foreign currency weakens relative to parentвЂ™s presentation currency пЃ© Revenues пЃ© Assets пЃ© Liabilities пЃЁNet income пЃЁShareholdersвЂ™ equity Translation gain пЃ© Revenues пЃ© Assets пЃ© Liabilities пЃ© Net income пЃ© ShareholdersвЂ™ equity Translation loss пЃ© Revenues пЃ© Assets пЃ© Liabilities пЃ© Net income пЃ© ShareholdersвЂ™ equity Negative translation adjustment Current Rate Method Measuring Earnings Quality Aggregate accruals = Accrual-basis earnings вЂ“ Cash earnings Balance Sheet Approach Net Operating Assets (NOA) NOAt = [(Total assetst пЂпЂ Casht) пЂпЂ (Total liabilitiest пЂпЂ Total debtt)] Aggregate Accruals Aggregate accrualstb/s = NOAtпЂ пЂпЂ NOAtпЂ1 Aggregate Ratio Accruals ratiotb/s = В© 2013 ELAN GUIDES (NOAt пЂпЂ NOAtпЂ1) (NOAt + NOAtпЂ1)/2 INTEGRATION OF FINANCIAL STATEMENT ANALYSIS TECHNIQUES INTEGRATION OF FINANCIAL STATEMENT ANALYSIS TECHNIQUES A Financial Statement Analysis Framework: Phase Sources of Information Examples of Output 1. Define the purpose and context of the analysis. п‚· The nature of the analystвЂ™s function, such as evaluating an equity or debt investment or issuing a credit rating. п‚· Communication with client or supervisor on needs and concerns. п‚· Institutional guidelines related to developing specific work product. п‚· Statement of the purpose or objective of analysis. п‚· A list (written or unwritten) of specific questions to be answered by the analysis. п‚· Nature and content of report to be provided. п‚· Timetable and budgeted resources for completion. 2. Collect input data. п‚· Financial statements, other financial data, questionnaires, and industry/economic data. п‚· Discussions with management, suppliers, customers, and competitors. п‚· Company site visits (e.g., to production facilities or retail stores) п‚· Organized financial statements. п‚· Financial data tables. п‚· Completed questionnaires, if applicable. 3. Process input data, as required, into analytically useful data. п‚· Data from the previous phase. п‚· Adjusted financial statements. п‚· Common-size statements. п‚· Forecasts. 4. Analyze/interpret the data. п‚· Input data and processed data п‚· Analytical results 5. Develop and communicate conclusions and recommendations (e.g., with an analysis report). п‚· Analytical results and previous reports Institutional guidelines for п‚· published reports п‚· Analytical report answering questions posed in Phase 1 п‚· Recommendations regarding the purpose of the analysis, such as whether to make an investment or grant credit. 6. Follow-up. п‚· Information gathered by periodically repeating above steps as necessary to determine whether changes to holdings or recommendations are necessary п‚· Update reports and recommendations DuPont Analysis ROE = Tax Burden Г— Interest burden Г— EBIT margin Г— Total asset turnover Г— Financial leverage ROE = NI EBT Г— EBT EBIT Г— EBIT Revenue Г— Revenue Average Asset Г— Average Asset Average Equity В© 2013 ELAN GUIDES CAPITAL BUDGETING CAPITAL BUDGETING Expansion Project Initial investment outlay for a new investment = FCInv + NWCInv NWCInv = пЃ„Non-cash current assets вЂ“ пЃ„Non-debt current liabilities Annual after-tax operating cash flows (CF) CF = (S вЂ“ C вЂ“ D) (1 вЂ“ t) + D or CF = (S вЂ“ C) (1 вЂ“ t) + tD Terminal year after-tax non-operating cash flow (TNOCF): TNOCF = SalT + NWCInv вЂ“ t(SalT вЂ“ BVT) Replacement Project Investment outlays: Initial investment for a replacement project = FCInv + NWCInv вЂ“ Sal0 + t(Sal0 вЂ“ BV0) Annual after-tax operating cash flow: пЃ„CF = (пЃ„S вЂ“ пЃ„C) (1 вЂ“ t) + tпЃ„D Terminal year after-tax non-operating cash flow: TNOCF = SalT + NWCInv вЂ“ t(SalT вЂ“ BT) Mutually Exclusive Projects with Unequal Lives пЂ пЂ пЂ пЂ пЂ пЂ±пЂ® Least Common Multiple of Lives Approach In this approach, both projects are repeated until their вЂ�chainsвЂ™ extend over the same time horizon. Given equal time horizons, the NPVs of the two project chains are compared and the project with the higher chain NPV is chosen. пЂ пЂ пЂ пЂ пЂ пЂ пЂІпЂ® Equivalent Annual Annuity Approach (EAA) This approach calculates the annuity payment (equal annual payment) over the projectвЂ™s life that is equivalent in present value (PV) to the projectвЂ™s NPV. The project with the higher EAA is chosen. SML Ri = RF + Гџi[E(RM) вЂ“ RF] Ri = Required return for project or asset i RF = Risk-free rate of return Гџi = Beta of project or asset i [E(RM) вЂ“ RF] = Market risk premium В© 2013 ELAN GUIDES CAPITAL BUDGETING Economic Income Economic income = After-tax operating cash flow + Increase in market value Economic income = After-tax operating cash flow + (Ending market value вЂ“ Beginning market value) Economic income = After-tax operating cash flow вЂ“ (Beginning market value вЂ“ Ending market value) Economic income = After-tax cash flows вЂ“ Economic depreciation Economic Profit Economic profit = [EBIT (1 вЂ“ Tax rate)] вЂ“ $WACC Economic profit = NOPAT вЂ“ $WACC NOPAT = Net operating profit after tax $WACC = Dollar cost of capital = Cost of capital (%) Г— Invested capital Under this approach, a projectвЂ™s NPV is calculated as the sum of the present values of economic profit earned over its life discounted at the cost of capital. п‚Ґ NPV = MVA = EPt пЃ“ (1 + WACC) t Residual Income Residual income = Net income for the period вЂ“ Equity charge for the period Equity charge for the period = Required return on equity Г— Beginning-of-period book value of equity The RI approach calculates value from the perspective of equity holders only. Therefore, future residual income is discounted at the required rate of return on equity to calculate NPV. п‚Ґ NPV = RIt пЃ“ (1 + r ) E t Claims Valuation пЂ пЂ пЂ пЂ пЂ пЂ п‚· пЂ пЂ пЂ пЂ пЂ пЂ п‚· пЂ пЂ пЂ пЂ пЂ пЂ п‚· First, we separate the cash flows available to debt and equity holders Then we discount them at their respective required rates of return. пЂ пЂ пЂ пЂ пЂ пЂ o Cash flows available to debt holders are discounted at the cost of debt, пЂ пЂ пЂ пЂ пЂ пЂ o Cash flows available to equity holders are discounted at the cost of equity. The present values of the two cash flow streams are added to calculate the total value of the company/asset. В© 2013 ELAN GUIDES CAPITAL STRUCTURE CAPITAL STRUCTURE The Capital Structure Decision rWACC = rD(1 пЂ t) + rE rD = Marginal cost of debt rE = Marginal cost of equity t = Marginal tax rate D = Market value of the companyвЂ™s outstanding debt E = Market value of shareholdersвЂ™ equity V = D + E = Value of the company MM Proposition II without Taxes: Higher Financial Leverage Raises the Cost of Equity rWACC = () ( ) rD + rE = r0 CompanyвЂ™s cost of equity (rE) under MM Proposition II without taxs is calculated as: Intercept Independent variable rE = r0 + (r0 пЂ rD) Dependent variable Slope The total value of the company is calculated as: V= Interest EBIT пЂ Interest + rD rE The systematic risk (Гџ) of the companyвЂ™s assets can be expressed as the weighted average of the systematic risk of the companyвЂ™s debt and equity. пЃўA = () () + This formula can also be expressed as: пЃўE = + (пЃўA пЂ пЃўD) В© 2013 ELAN GUIDES () CAPITAL STRUCTURE Relaxing the Assumption of no Taxes = + tD The WACC is then calculated as: rWACC = rD(1 пЂ t) + rE And the cost of equity is calculated as: rE = r0 + (r0 пЂ rD) (1 пЂ t) Modigilani and Miller Propositions Without Taxes Proposition I Proposition II With Taxes = rE = r0 + (r0 пЂ rD) = + tD rE = r0 + (r0 пЂ rD) (1 пЂ t) The Optimal Capital Structure: The Static Trade-Off Theory VL = VU + tD вЂ“ PV(Costs of financial distress) В© 2013 ELAN GUIDES DIVIDENDS AND SHARE REPURCHASE DIVIDENDS AND SHARE REPURCHASE The expected decrease in share price when it goes ex-dividend can be calculated using the following equation: PW PX = п‚ґD Pw = Share price with the right to receive the dividend PX = Share price without the right to receive the dividend D = Amount of dividend TD = Tax rate on dividends TCG = Tax rate on capital gains Double Taxation System ETR = CTR + [(1 вЂ“ CTR) Г— MTRD] ETR = Effective tax rate CTR = Corporate tax rate MTRD = InvestorвЂ™s marginal tax rate on dividends Split-Rate Tax System ETR = CTRD + [(1 вЂ“ CTRD) Г— MTRD] CTRD = Corporate tax rate on earnings distributed as dividends. Stable Dividend Policy The expected increase in dividends is calculated as: Expected dividend increase = Increase in earnings Г— Target payout ratio Г— Adjustment factor Adjustment factor = 1/N N = Number of years over which the adjustment is expected to occur Analysis of Dividend Safety Dividend payout ratio = (dividends / net income) Dividend coverage ratio = (net income / dividends) FCFE coverage ratio = FCFE / [Dividends + Share repurchases] В© 2013 ELAN GUIDES MERGERS AND ACQUISITION MERGERS AND ACQUISITION Mergers and the Industry Life Cycle Industry Life Cycle Stage Industry Description Motives for Merger Types of Merger Pioneering development п‚· Low but slowly increasing sales growth. п‚· Substantial development costs. п‚· Younger, smaller companies may sell themselves to larger firms in mature or declining industries to enter into a new growth industry. п‚· Young companies may merge with firms that allow them to pool management and capital resources. п‚· Conglomerate п‚· Horizontal Rapid accelerating growth п‚· High profit margins. п‚· Low competition. п‚· To meet substantial capital requirements for expansion. п‚· Conglomerate п‚· Horizontal Mature growth п‚· Decrease in the entry of new competitors. п‚· Growth potential remains. п‚· To achieve economies of scale, savings, and operational efficiencies. п‚· Horizontal п‚· Vertical Stabilization and market maturity п‚· Increasing capacity constraints п‚· Increasing competition. п‚· To achieve economies of scale in research, production, and marketing to match low costs and prices of competitors. п‚· Large companies may buy smaller companies to improve management and provide a broader financial base. п‚· Horizontal Deceleration of growth and decline п‚· Overcapacity. п‚· Eroding profit margins. п‚· Horizontal mergers to ensure survival. п‚· Vertical mergers to increase efficiency and profit margins. п‚· Conglomerate mergers to exploit synergy. п‚· Companies in the industry may acquire companies in young industries. п‚· Horizontal п‚· Vertical п‚· Conglomerate Source: Adapted from J. Fred Weston, Kwang S. Chung, and Susan E. Hoag, Mergers, Restructuring, and Corporate Control (New York: Prentice Hall, 1990, p.102) and Bruno Solnik and Dennis McLeavy, International Investments, 5th edition (Boston: Addison Wesley, 2004, p. 264 вЂ“ 265). В© 2013 ELAN GUIDES MERGERS AND ACQUISITION Major Differences of Stock versus Asset Purchases Approval Stock Purchase Target shareholders receive compensation in exchange for their shares. Shareholder approval required. Tax: Corporate No corporate-level taxes. Payment Tax: Shareholder Target companyвЂ™s shareholders are taxed on their capital gain. Liabilities Acquirer assumes the targetвЂ™s liabilities. Asset Purchase Payment is made to the selling company rather than directly to shareholders. Shareholder approval might not be required. Target company pays taxes on any capital gains. No direct tax consequence for target companyвЂ™s shareholders. Acquirer generally avoids the assumption of liabilities. Herfindahl-Hirschman Index (HHI) n пЃ“ i ( Sales or output of firm i Total sales or output of market п‚ґ 100 ) 2 HHI Concentration Levels and Possible Government Response Post-Merger HHI Less than 1,000 Between 1,000 and 1,800 More than 1,800 Concentration Not concentrated Moderately concentrated Highly concentrated Change in HHI Any amount 100 or more 50 or more Government Action No action Possible challenge Challenge FCFF is estimated by: Net income + Net interest after tax = Unlevered income + Changes in deferred taxes = NOPLAT (net operating profit less adjusted taxes) + Net noncash charges вЂ“ Change in net working capital вЂ“ Capital expenditures (capex) Free cash flow to the firm (FCFF) Net interest after tax = (Interest expense вЂ“ Interest income) (1 вЂ“ tax rate) Working capital = Current assets (excl. cash and equivalents) вЂ“ Current liabilities (excl. short-term debt) В© 2013 ELAN GUIDES MERGERS AND ACQUISITION Comparable Company Analysis TP = (DP пЂ SP) SP TP = Takeover premium DP = Deal price per share SP = TargetвЂ™s stock price per share Bid Evaluation Target shareholdersвЂ™ gain = Takeover premium = PT вЂ“ VT AcquirerвЂ™s gain = Synergies вЂ“ Premium = S вЂ“ (PT вЂ“ VT) S = Synergies created by the merger transaction The post-merger value of the combined company is composed of the pre-merger value of the acquirer, the pre-merger value of the target, and the synergies created by the merger. These sources of value are adjusted for the cash paid to target shareholders to determine the value of the combined post-merger company. VA* = VA + VT + S вЂ“ C VA* = Value of combined company C = Cash paid to target shareholders В© 2013 ELAN GUIDES EQUITY VALUATION: APPLICATIONS AND PROCESSES EQUITY VALUATION: APPLICATIONS AND PROCESSES Perceived mispricing: Perceived mispricing = True mispricing + Error in the estimate of intrinsic value. VE вЂ“ P = (V вЂ“ P) + (VE вЂ“ V) VE = Estimate of intrinsic value P = Market price V = True (unobservable) intrinsic value В© 2013 ELAN GUIDES RETURN CONCEPTS RETURN CONCEPTS Holding Period Return Holding period return = PH вЂ“ P0 + DH P0 PH = Price at the end of the holding period P0 = Price at the beginning of the period DH = Dividend Required Return пЂ пЂ пЂ пЂ пЂ пЂ п‚· пЂ пЂ пЂ пЂ пЂ пЂ п‚· The difference between an assetвЂ™s expected return and its required return is known as expected alpha, ex ante alpha or expected abnormal return. пЂ пЂ пЂ пЂ пЂ пЂ o Expected alpha = Expected return вЂ“ Required return The difference between the actual (realized) return on an asset and its required return is known as realized alpha or ex post alpha. пЂ пЂ пЂ пЂ пЂ пЂ o Realized alpha = Actual HPR вЂ“ Required return for the period When the investorвЂ™s estimate of intrinsic value (V0) is different from the current market price (P0), the investorвЂ™s expected return has two components: 1. 2. The required return (rT) earned on the assetвЂ™s current market price; and The return from convergence of price to value [(V0 вЂ“ P0)/P0]. Internal Rate of Return Next yearвЂ™s expected dividend Intrinsic Value = Required return вЂ“ Expected dividend growth rate V0 = D1 ke вЂ“ g If the asset is assumed to be efficiently-priced (i.e. the market price equals its intrinsic value), the IRR would equal the required return on equity. Therefore, the IRR can be estimated as: Required return (IRR) = ke (IRR) = D1 P0 Next yearвЂ™s dividend Market price + Expected dividend growth rate +g В© 2013 ELAN GUIDES RETURN CONCEPTS Equity Risk Premium The required rate of return on a particular stock can be computed using either of the following two approaches. Both these approaches require the equity risk premium to be estimated first. 1. Required return on share i = Current expected risk-free return + Гџi(Equity risk premium) пЂ пЂ пЂ пЂ пЂ пЂ пЂ п‚· 2. пЂ пЂ пЂ пЂ пЂ пЂ A beta greater (lower) than 1 indicates that the security has greater-than-average (lower-thanaverage) systematic risk. Required return on share i = Current expected risk-free return + Equity risk premium В± Other risk premia/discounts appropriate for i пЂ пЂ пЂ пЂ пЂ пЂ пЂ п‚· This method of estimating the required return is known as the build-up method. It is discussed later in the reading and is primarily used for valuations of private businesses. Gordon Growth Model (GGM) Estimates GCM equity risk premium estimate = D1 P0 + g вЂ“ rLTGD Macroeconomic Model Estimates Equity risk premium = {[(1 + EINFL) (1 + EGREPS) (1 + EGPE) вЂ“ 1] + EINC} вЂ“ Expected RF Expected inflation = 1 + YTM of 20-year maturity T-bonds 1 + YTM of 20-year maturity TIPS вЂ“ 1 The Captial Asset Pricing Model (CAPM) Required return on i = Expected risk-free rate + Betai (Equity risk premium) The Fama-French Model ri = RF + пЃўimktRMRF + пЃўisizeSMB + пЃўivalueHML Гџmkt = Market beta Гџsize = Size beta Гџvalue = Value beta The Pastor-Stambaugh model (PSM) ri = RF + пЃўimktRMRF + пЃўisizeSMB + пЃўivalueHML + пЃўiliqLIQ Гџliq = Liquidity beta В© 2013 ELAN GUIDES RETURN CONCEPTS BIRR model ri = T-bill rate + (Sensitivity to confidence risk Г— Confidence risk) + (Sensitivity to time horizon risk Г— Time horizon risk) + (Sensitivity to inflation risk Г— Inflation risk) + (Sensitivity to business cycle risk Г— Business cycle risk) + (Sensitivity to market timing risk Г— Market timing risk) Build-up method ri = Risk-free rate + Equity risk premium + Size premium + Specific-company premium For companies with publicly-traded debt, the bond-yield plus risk premium approach can be used to calculate the cost of equity: BYPRP cost of equity = YTM on the companyвЂ™s long-term debt + Risk premium Adjusting Beta for Beta Drift Adjusted beta = (2/3) (Unadjusted beta) + (1/3) (1.0) Estimating the Asset Beta for the Comparable Publicly Traded Firm: BASSET reflects only business risk of the comparable company. Therefore it is used as a proxy for business risk of the project being studied. 1 ГџASSET = ГџEQUITY ( 1 + (1 - t) D E ) BEQUITY reflects business and financial risk of comparable company. where: D/E = debt-to-equity ratio of the comparable company. t = marginal tax rate of the comparable company. To adjust the asset beta of the comparable for the capital structure (financial risk) of the project or company being evaluated, we use the following formula: BPROJECT reflects business and financial risk of the project. ГџPROJECT = ГџASSET 1 + (1 - t) D E BASSET reflects business risk of project. where: D/E = debt-to-equity ratio of the subject company. t = marginal tax rate of the subject company. Country Spread Model ERP estimate = ERP for a developed market + Country premium В© 2013 ELAN GUIDES RETURN CONCEPTS Weighted Average Cost of Capital (WACC) WACC = MVD MVD + MVCE rd (1 вЂ“ Tax rate ) + MVCE MVD + MVCE MVD = Market value of the companyвЂ™s debt rd = Required rate of return on debt MVCE = Market value of the companyвЂ™s common equity r = Required rate of return on equity В© 2013 ELAN GUIDES r DISCOUNTED DIVIDEND VALUATION DISCOUNTED DIVIDEND VALUATION One-Period DDM V0 = D1 (1 + r)1 + P1 (1 + r)1 = D1 + P1 (1 + r)1 V0 = The value of the stock today (t = 0) P1 = Expected price of the stock after one year (t = 1) D1 = Expected dividend for Year 1, assuming it will be paid at the end of Year 1 (t = 1) r = Required return on the stock Multiple-Period DDM V0 = D1 Dn Pn 1 + ... + n+ (1 + r) (1 + r) (1 + r)n n V0 = Dt пЃ“ (1 + r) t + t=1 Pn (1 + r)n Expression for calculating Value of a share of stock п‚Ґ V0 = Dt пЃ“ (1 + r) t t=1 Gordon Growth Model V0 = D0 (1 + g) D1 , or V0 = (r вЂ“ g) (r вЂ“ g) Present value of Growth Opportunities V0 = E1 + PVGO r P/E ratio Justified leading P/E ratio = P0 = D1/E1 E1 Justified trailing P/E = P0 E0 = D1/E0 rпЂg rпЂg = = (1 пЂпЂ b) rпЂg D0 (1 пЂ«пЂ g) / E0 rпЂg = (1 пЂпЂ b)(1 пЂ«пЂ g) rпЂg В© 2013 ELAN GUIDES DISCOUNTED DIVIDEND VALUATION Value of Fixed-Rate Perpetual Preferred Stock D r V0 = Two-Stage Dividend Discount Model n V0 = пЃ“ t=1 D0 (1 + gS)t D0 (1 + gS)n(1 + gL) + (1 + r)t (1 + r)n(r вЂ“ gL) gS = Short term supernormal growth rate gL = Long-term sustainable growth rate r = required return n = Length of the supernormal growth period The H-Model V0 = D0 (1 + gL) D0H (gs вЂ“ gL) + r вЂ“ gL r вЂ“ gL gS = Short term high growth rate gL = Long-term sustainable growth rate r = required return H = Half-life = 0.5 times the length of the high growth period The H-model equation can be rearranged to calculate the required rate of return as follows: r= ( ) D0 [(1 + gL) + H(gs вЂ“ gL)] + gL P0 The Gordon growth formula can be rearranged to calculate the required rate of return given the other variables. r= D1 +g P0 Sustainable growth rate (SGR) g = b Г— ROE b = Earnings retention rate, calculated as 1 вЂ“ Dividend payout ratio В© 2013 ELAN GUIDES DISCOUNTED DIVIDEND VALUATION ROE can be calculated as: ROE = Net income Sales Total assets Г— Г— Sales Total assets ShareholdersвЂ™ equity PRAT model g = Profit margin Г— Retention rate Г— Asset turnover Г— Financial leverage g= Net income - Dividends Net income Sales Total assets Г— Г— Г— Net income Sales Total assets ShareholdersвЂ™ equity В© 2013 ELAN GUIDES FREE CASH FLOW VALUATION FREE CASH FLOW VALUATION FCFF/FCFE п‚Ґ Firm Value = FCFFt пЃ“ (1+WACC) t t=1 WACC = MV(Equity) MV(Debt) r rd (1 пЂ Tax Rate) + MV(Debt) + MV(Equity) MV(Debt) + MV(Equity) Equity Value = Firm Value пЂ Market value of debt п‚Ґ Equity Value = FCFEt пЃ“ (1 + r) t t=1 Computing FCFF from Net Income FCFF = NI + NCC + Int(1 пЂ Tax Rate) пЂ FCInv пЂ WCInv Investment in fixed capital (FCInv) FCInv = Capital expenditures пЂ Proceeds from sale of long-term assets Investment in working capital (WCInv) WCInv = Change in working capital over the year Working capital = Current assets (exc. cash) пЂ Current liabilities (exc. short-term debt) Table: Noncash Items and FCFF Noncash Item Depreciation Amortization and impairment of intangibles Restructuring charges (expense) Restructuring charges (income resulting from reversal) Losses Gains Amortization of long-term bond discounts Amortization of long-term bond premiums Deferred taxes В© 2013 ELAN GUIDES Adjustment to NI to Arrive at FCFF Added back Added back Added back Subtracted Added back Subtracted Added back Subtracted Added back but requires special attention FREE CASH FLOW VALUATION Computing FCFF from CFO Table: IFRS versus U.S. GAAP Treatment of Interest and Dividends IFRS U.S. GAAP Interest received CFO or CFI CFO Interest paid CFO or CFF CFO Dividend received Dividends paid CFO or CFI CFO or CFF CFO CFF FCFF = CFO + Int(1 пЂ Tax rate) пЂ FCInv Computing FCFF from EBIT FCFF = EBIT(1 вЂ“ Tax rate) + Dep вЂ“ FCInv вЂ“ WCInv Computing FCFF from EBITDA FCFF = EBITDA(1 вЂ“ Tax rate) + Dep(Tax rate) вЂ“ FCInv вЂ“ WCInv Computing FCFE from FCFF FCFE = FCFF вЂ“ Int(1вЂ“ Tax rate) + Net borrowing Computing FCFE from Net Income FCFE = NI + NCC вЂ“ FCInv вЂ“ WCInv + Net Borrowing Computing FCFE from CFO FCFE = CFO + FCInv вЂ“ Net borrowing Computing FCFE from EBIT FCFE = EBIT(1 вЂ“ Tax rate) вЂ“ Int(1 вЂ“ Tax rate) + Dep вЂ“ FCInv вЂ“ WCInv + Net borrowing Computing FCFE from EBITDA FCFE = EBITDA(1 вЂ“ Tax rate) вЂ“ Int(1 вЂ“ Tax rate) + Dep(Tax rate) вЂ“ FCInv вЂ“ WCInv + Net borrowing В© 2013 ELAN GUIDES FREE CASH FLOW VALUATION Uses of FCFF Increases in cash balances Plus: Net payments to providers of debt capital + Interest expense (1 вЂ“ tax rate) + Repayment of principal пЂ New borrowings Plus: Net payments to providers of equity capital + Cash dividends + Share repurchases пЂ New equity issues = Uses of FCFF Uses of FCFE Increases in cash balances Plus: Net payments to providers of equity capital + Cash dividends + Share repurchases пЂ New equity issues = Uses of FCFE Constant Growth FCFF Valuation Model Value of the firm = FCFF1 FCFF0 (1 + g) = WACC - g WACC - g WACC = Weighted average cost of capital g = Long-term constant growth rate in FCFF Constant Growth FCFE Valuation Model Value of equity = FCFE1 FCFE0 (1 + g) = r-g r-g r = Required rate of return on equity g = Long-term constant growth rate in FCFE An International Application of the Single-Stage Model Value of equity = В© 2013 ELAN GUIDES FCFE0 (1 + greal) rreal пЂ greal FREE CASH FLOW VALUATION General expression for the two-stage FCFF model: n Firm value = FCFFn+1 FCFFt 1 пЃ“ (1 + WACC) + (WACC пЂ g) (1 + WACC) t n t=1 Firm value = PV of FCFF in Stage 1 + Terminal value Г— Discount Factor General expression for the two-stage FCFE model: n Equity value = FCFEt пЃ“ (1 + r) + FCFFn+1 1 rпЂg (1 + r)n t t=1 Equity value = PV of FCFE in Stage 1 + Terminal value Г— Discount Factor Determining Terminal Value Terminal value in year n = Justified Trailing P/E Г— Forecasted Earnings in Year n Terminal value in year n = Justified Leading P/E Г— Forecasted Earnings in Year n + 1 Non-operating Assets and Firm Value Value of the firm = Value of operating assets + Value of non-operating assets В© 2013 ELAN GUIDES MARKET-BASED VALUATION: PRICE AND ENTERPRISE VALUE MULTIPLES MARKET-BASED VALUATION: PRICE AND ENTERPRISE VALUE MULTIPLES Price to Earnings Ratio Trailing P/E ratio = Current Stock Price Last yearвЂ™s EPS Forward P/E ratio = Current Stock Price Expected EPS Price to Book Ratio P/B ratio = Market price per share Book value per share P/B ratio = Market value of common shareholdersвЂ™ equity Book value of common shareholdersвЂ™ equity Book value of equity = Common shareholdersвЂ™ equity = ShareholdersвЂ™ equity вЂ“ Total value of equity claims that are senior to common stock Book value of equity = Total assets вЂ“ Total liabilities вЂ“ Preferred stock Price to Sales Ratio P/S ratio = Market price per share Sales per share Relationship between the P/E ratio and the P/S ratio P/E Г— Net profit margin = (P / E) Г— (E / S) = P/S Price to Cash Ratio P/CF ratio = Market price per share Free cash flow per share Dividend Yield Justified trailing dividend yield Trailing dividend yield = Last yearвЂ™s dividend / Current price per share Justified leading dividend yield Leading dividend yield = Next yearвЂ™s dividend / Current price per share В© 2013 ELAN GUIDES MARKET-BASED VALUATION: PRICE AND ENTERPRISE VALUE MULTIPLES Justified P/E Multiple Based on Fundamentals D1 V0 = (r пЂ g) Justified leading P/E multiple Justified leading P/E = P0 = E1 D1/E1 rпЂg = (1 пЂпЂ b) rпЂg (1 вЂ“ b) is the payout ratio. Justified trailing P/E multiple Justified trailing P/E = P0 E0 = D1/E0 rпЂg = D0 (1 пЂ«пЂ g) / E0 rпЂg = (1 пЂпЂ b)(1 пЂ«пЂ g) rпЂg Justified P/B Multiple Based on Fundamentals P0 B0 = ROE пЂпЂ g rпЂg ROE = Return on equity r = required return on equity g = Sustainable growth rate Justified P/S Multiple Based on Fundamentals P0 S0 = (E0/S0)(1 пЂпЂ b)(1 пЂ«пЂ g) rпЂg E0/S0 = Net profit margin 1 вЂ“ b = Payout ratio Justified P/CF Multiple Based on Fundamentals FCFE0 (1 пЂ« g) V0 = (r пЂ g) Justified Dividend Yield D0 P0 = rпЂg 1пЂ«g В© 2013 ELAN GUIDES MARKET-BASED VALUATION: PRICE AND ENTERPRISE VALUE MULTIPLES P/E-to-growth (PEG) ratio PEG = P/E Growth (%) Terminal price based on fundamentals TVn = Justified leading P/E п‚ґ Forecasted earningsn +1 TVn = Justified trailing P/E п‚ґ Forecasted earningsn Terminal price based on comparables TVn = Benchmark leading P/E п‚ґ Forecasted earningsn +1 TVn = Benchmark trailing P/E п‚ґ Forecasted earningsn EV/EBITDA Multiple Enterprise value = Market value of common equity + Market value of preferred stock + Market value of debt вЂ“ Value of cash and short-term investments EBITDA = Net income + Interest + Taxes + Depreciation and amortization Alternative Denominators in Enterprise Value Multiples Free Cash Net plus minus plus plus less less Flow to the Income Interest Tax Savings Depreciation Amortization Investment in Investment in Firm = Expense on Interest Working Capital Fixed Capital EBITDA= Net plus plus Income Interest Taxes Expense plus plus Depreciation Amortization EBITA = Net plus plus Income Interest Taxes Expense plus Amortization EBIT = Net plus plus Income Interest Taxes Expense Justified forward P/E after accounting for Inflation P0 E1 = 1 пЃІпЂ пЂ«пЂ (1 пЂ пЃ¬) I пЃ¬пЂ = The percentage of inflation in costs that the company can pass through to revenue. пЃІпЂ = Real rate of return I = Rate of inflation В© 2013 ELAN GUIDES MARKET-BASED VALUATION: PRICE AND ENTERPRISE VALUE MULTIPLES Unexpected earnings (UE) UEt = EPSt вЂ“ E(EPSt) Standardized unexpected earnings (SUE) SUEt = EPSt пЂ E(EPSt) пЃі[EPSt пЂ E(EPSt)] EPSt = Actual EPS for time t E(EPSt) = Expected EPS for time t пЃі[EPSt пЂ E(EPSt)] = Standard deviation of [EPSt пЂ E(EPSt)] В© 2013 ELAN GUIDES RESIDUAL INCOME VALUATION RESIDUAL INCOME VALUATION The Residual Income Residual income = Net income вЂ“ Equity charge Equity charge = Cost of equity capital Г— Equity capital Residual income = After-tax operating profit пЂ Capital charge Capital charge = Equity charge + Debt charge Debt charge = Cost of debt Г— (1 вЂ“ Tax rate) Г— Debt capital Economic Value Added EVA = NOPAT вЂ“ (C% Г— TC) NOPAT = Net operating profit after tax = EBIT (1 вЂ“ Tax rate) C% = Cost of capital (WACC) TC = Total capital Market Value Added MVA = Market value of the company вЂ“ Accounting book value of total capital Market value of company = Market value of debt + Market value of equity. The Residual Income Model RIt = Et вЂ“ (r Г— Bt-1) RIt = Residual income at time t Et = Earnings at time t r = Required rate of return on equity Bt-1 = Book value at time t-1 Intrinsic value of a stock: п‚Ґ V0 = B0 + п‚Ґ RIt пЃ“ (1 + r) t i=1 Et пЂ rBt-1 пЃ“ (1 + r) = B0 + t i=1 V0 = Intrinsic value of the stock today B0 = Current book value per share of equity Bt = Expected book value per share of equity at any time t r = Required rate of return on equity Et = Expected EPS for period t RIt = Expected residual income per share В© 2013 ELAN GUIDES RESIDUAL INCOME VALUATION Residual Income Model (Alternative Approach) RIt = EPSt - (R Г— Bt-1) RIt = (ROE - r)Bt-1 п‚Ґ V0 = B0 + пЃ“ (ROEt пЂ r)Bt-1 t=1 V0 = B0 + (1 + r)t ROE пЂ r B0 rпЂg TobinвЂ™s q TobinвЂ™s q = Market value of debt and equity Replacement cost of total assets Multi-Stage Residual Income Valuation T V0 = B0 + пЃ“ t=1 (Et пЂ rBt пЂ1) t (1 + r) + PT пЂ BT (1 + r)T When residual income fades over time as ROE declines towards the required return on equity, the intrinsic value of a stock is calculated using the following formula: T-1 V0 = B0 + пЃ“ t=1 (Et пЂ rBt пЂ1) t (1 + r) + ET пЂ rBT-1 (1 + r пЂ пЃ·)(1 + r)TпЂ1 пЃ·пЂ = Persistence factor. Implied Growth Rate g=rпЂ [ (ROE пЂ r) Г— B0 V0 пЂ B0 ] В© 2013 ELAN GUIDES PRIVATE COMPANY VALUATION PRIVATE COMPANY VALUATION The Capitalized Cash Flow Method Vf = FCFF1 WACC пЂ gf Vf = Value of the firm FCFF1 = Free cash flow to the firm for next twelve months WACC = Weighted average cost of capital gf = Sustainable growth rate of free cash flow to the firm V= FCFE1 rпЂg V = Value of the equity FCFE1 = Free cash flow to the equity for next twelve months r = Required return on equity g = Sustainable growth rate of free cash flow to the equity Methods Used to Estimate the Required Rate of Return for a Private Company Capital Asset Pricing Model Required return on equity = Risk-free rate + (Beta Г— Market risk premium) Expanded CAPM Required return on equity = Risk-free rate + (Beta Г— Market risk premium) + Small stock premium + Company-specific risk premium Build-Up Approach Required return on equity = Risk-free rate + Equity risk premium + Small stock premium + Company-specific risk premium + Industry risk premium Discount for Lack of Control (DLOC) DLOC = 1 - В© 2013 ELAN GUIDES 1 1 + Control Premium PRIVATE REAL ESTATE INVESTMENTS PRIVATE REAL ESTATE INVESTMENTS Net Operating Income Rental income at full occupancy + Other income (such as parking) = Potential gross income (PGI) пЂ Vacancy and collection loss = Effective gross income (EGI) пЂ Operating expenses (OE) = Net operating income (NOI) The Direct Capitalization Method Cap rate = Discount rate вЂ“ Growth rate The cap rate can be defined as the current yield on an investment: Capitalization rate = NOI1 Value Rearranging the above equation, we can estimate the value of a property by dividing its firstyear NOI by the cap rate. Value = NOI1 Cap rate An estimate of the appropriate cap rate for a property can be obtained from the selling price of similar or comparable properties. Cap rate = NOI Sale price of comparable property The cap rate derived by dividing rent by recent sales prices of comparables is known as the all risks yield (ARY). The value of a property is then calculated as: Market value = Rent1 ARY Other Forms of the Income Approach Gross income multiplier = Selling price Gross income Value of subject property = Gross income multiplier п‚ґ Gross income of subject property В© 2013 ELAN GUIDES PRIVATE REAL ESTATE INVESTMENTS The Discounted Cash Flow Method (DCF) Value = NOI1 (r вЂ“ g) The Terminal Capitalization Rate Terminal value = NOI for the first year of ownership for the next investor Terminal cap rate Appraisal-Based Indices Return = NOI пЂ Capital expenditures + (Ending market value пЂ Beginning market value) Beginning market value Loan to Value ratio LTV ratio = Loan amount Appraised value Debt Service Coverage ratio DSCR = NOI Debt service Equity dividend rate/Cash-on-cash return Equity dividend rate = First year cash flow Equity investment В© 2013 ELAN GUIDES PUBLICLY TRADED REAL ESTATE SECURITIES PUBLICLY TRADED REAL ESTATE SECURITIES VALUATION: NET ASSET VALUE APPROACH Capitalization rate Capitalization rate = NOI of a comparable property Total value of comparable property Net Asset Value per Share NAVPS = Net Asset Value Shares outstanding VALUATION: RELATIVE VALUATION (PRICE MULTIPLE) APPROACH Funds from operations (FFO) Accounting net earnings Add: Depreciation charges on real estate Add: Deferred tax charges Add (Less): Losses (gains) from sales of property and debt restructuring Funds from operations Adjusted funds from operations (AFFO) Funds from operations Less: Non-cash rent Less: Maintenance-type capital expenditures and leasing costs Adjusted funds from operations AFFO is preferred over FFO as it takes into account the capital expenditures necessary to maintain the economic income of a property portfolio. В© 2013 ELAN GUIDES PRIVATE EQUITY VALUATION PRIVATE EQUITY VALUATION Quantitative Measures of Return пЂ пЂ пЂ пЂ пЂ пЂ п‚· PIC (paid in capital): Ratio of paid in capital to date to committed capital. пЂ пЂ пЂ пЂ пЂ пЂ п‚· DPI (distributed to paid-in) or cash-on-cash return: Value of cumulative distributions paid to LPs as a proportion of cumulative invested capital. пЂ пЂ пЂ пЂ пЂ пЂ o (DPI = Cumulative distributions / PIC) пЂ пЂ пЂ пЂ пЂ пЂ п‚· RVPI (residual value to paid-in): Value of LPsвЂ™ shareholdings held with the fund as a proportion of cumulative invested capital. пЂ пЂ пЂ пЂ пЂ пЂ o RVPI = NAV after distributions / PIC пЂ пЂ пЂ пЂ пЂ пЂ п‚· TVPI (total value to paid-in): Value of portfolio companiesвЂ™ distributed (realized) and undistributed (unrealized) value as a proportion of cumulative invested capital. пЂ пЂ пЂ пЂ пЂ пЂ o TVPI = DPI + RVPI NAV before distributions = Prior yearвЂ™s NAV after distributions + Capital called down вЂ“ Management Fees + Operating results NAV after distributions = NAV before distributions вЂ“ Carried interest вЂ“ Distributions Total Exit Value Exit value = Initial cost + Earnings growth + Multiple expansion + Debt reduction Post-money valuation (POST) POST = PRE + I Proportionate ownership of the VC investor = I / POST Post-money value Post-money value = Exit value (1 + Required rate of return )Number of years to exists Required wealth Required wealth = Investment п‚ґ (1 + IRR) Number of years to exit Ownership propotion Ownership proportion = Required wealth / Exit value В© 2013 ELAN GUIDES PRIVATE EQUITY VALUATION Shares to be issued Shares to be issued = Proportion of venture capitalist investment п‚ґпЂ Shares held by company founders Proportion of investment of company founders Price per share Price per share = Amount of venture capital investment Number of shares issued to venture capital investment Adjusted discount rate Adjusted discount rate = 1+r вЂ“1 1вЂ“q r = Discount rate unadjusted for probability of failure. q = Probability of failure. В© 2013 ELAN GUIDES FUNDAMENTALS OF CREDIT ANALYSIS FUNDAMENTALS OF CREDIT ANALYSIS Expected Loss Expected loss = Default probability п‚ґ Loss severity given default Yield on a corporate bond: Yield on a corporate bond = Real risk-free interest rate + Expected inflation rate + Maturity premium + Liquidity premium + Credit spread Yield Spread: Yield spread = Liquidity premium + Credit spread For small, instantaneous changes in the yield spread, the return impact (i.e. the percentage change in price, including accrued interest) can be estimated using the following formula: Return impact п‚» вЂ“ Modified duration Г— пЃ„Spread For larger changes in the yield spread, we must also incorporate the (positive) impact of convexity into our estimate of the return impact: Return impact п‚» вЂ“ (MDur Г— пЃ„Spread) + (1/2 Г— Convexity Г— пЃ„Spread2) В© 2013 ELAN GUIDES TERM STRUCTURE AND VOLATILITY OF INTEREST RATES TERM STRUCTURE AND VOLATILITY OF INTEREST RATES Measuring Historical Yield Volatility Xt = 100 п‚ґ ln yt ( ) ytпЂ1 where yt = yield on day t Annualizing the Standard Deviation Annualized standard deviation = Daily standard deviation п‚ґпЂ пѓ–пЂ пЃЋпЃЇпЂ®пЂ of days in a year Calculating Variance of Daily Yield Changes T Xt2 Variance = t = 1 T пЂпЂ 1 пЃ“ ... assigns an equal weight to all observations T Wt Xt2 Variance = t = 1 T пЂпЂ 1 пЃ“ ... attaches a greater weight to more recent information where: Wt = the weight assigned to each daily yield change observation such that the sum of the weights equals 1. В© 2013 ELAN GUIDES VALUING BONDS WITH EMBEDDED OPTIONS VALUING BONDS WITH EMBEDDED OPTIONS Treasury Market Benchmark Spread Measure Nominal Zero-volatility Option-adjusted Benchmark Treasury yield curve Treasury spot rate curve Treasury spot rate curve Reflects Compensation For Credit risk, liquidity risk and option risk Credit risk, liquidity risk and option risk Credit risk and liquidity risk Specific Bond Sector with a Given Credit Rating Benchmark Spread Measure Nominal Zero-volatility Option-adjusted Benchmark Sector yield curve Sector spot rate curve Sector spot rate curve Reflects Compensation For Credit risk, liquidity risk and option risk Credit risk, liquidity risk and option risk Credit risk and liquidity risk Issuer-Specific Benchmark Spread Measure Nominal Zero-volatility Option-adjusted Benchmark Issuer yield curve Issuer spot rate curve Issuer spot rate curve Reflects Compensation For Liquidity risk and option risk Liquidity risk and option risk Liquidity risk Summary of Relationships between Benchmark, OAS and Relative Value Benchmark Treasury market Negative OAS Overpriced (rich) security Bond sector with a Overpriced (rich) security given credit rating (assumes credit rating higher (assumes credit rating than security being analyzed) higher than security being analyzed) В© 2013 ELAN GUIDES Zero OAS Overpriced (rich) security Positive OAS Comparison must be made between security OAS and OAS of comparable securities (required OAS): If security OAS > required OAS, security is cheap If security OAS < required OAS, security is rich If security OAS = required OAS, security is fairly priced Overpriced (rich) security (assumes credit rating higher than security being analyzed) Comparison must be made between security OAS and OAS of comparable securities (required OAS): If security OAS > required OAS, security is cheap If security OAS < required OAS, security is rich If security OAS = required OAS, security is fairly priced VALUING BONDS WITH EMBEDDED OPTIONS Summary of Relationships between Benchmark, OAS and Relative Value (Contd.) Benchmark Positive OAS Zero OAS Negative OAS Bond sector with a given credit rating (assumes credit rating lower than security being analyzed) Comparison must be made between security OAS and OAS of comparable securities (required OAS): If security OAS > required OAS, security is cheap If security OAS < required OAS, security is rich If security OAS = required OAS, security is fairly priced IssuerвЂ™s own securities Overpriced (rich) security Underpriced (cheap) Underpriced (cheap) security security (assumes credit rating lower than (assumes credit rating security being analyzed) lower than security being analyzed) Fairly valued Under priced (cheap) security Determining Bond Value at a Node Applying Backward Induction Bond's value in higher-rate state 1-year forward 1-year rate at the node at which we are calculating the bond's value, VHHL п‚° VHHHL пЂ« C r4,HHHL NHHHL Cash flow in higher rate state NHHL r3,HHL VHHL NHHLL r4,HHLL VHHLLпЂ« C Cash flow in lower rate state Bond's value in lower-rate state 1-year forward The present values of the these two cash flows discounted at the 1-year rate (r3,HHL) at Node NHHL are: 1. 2. ( ( VHHHL + C 1 + r3,HHL VHHLL + C 1 + r3,HHL ) ) пѓў Present value in the higher one-year rate scenario пѓў Present value in the lower one-year rate scenario В© 2013 ELAN GUIDES VALUING BONDS WITH EMBEDDED OPTIONS Finally, the expected value of the bond, VHHL at Node NHHLis calculated as: 1 2 ( VHHHL + C 1 + r3,HHL )( + VHHLL + C 1 + r3,HHL ) Determining Call Option Value Value of call option = Value of option-free bond вЂ“ Value of callable bond. Determining Put Option Value Value of put option = Value of putable bond пЂ Value of option-free bond Effective Duration and Effective Convexity Duration = VпЂпЂ пЂпЂ V+ 2V0 (пЃ„y) Convexity = VпЂпЂ пЂпЂ V+ пЂ 2V0 2V0 (пЃ„y)2 Traditional Analysis of a Convertible Security Conversion value = Market price of common stock п‚ґпЂ Conversion ratio Market conversion price = Market price of convertible security Conversion ratio Market conversion premium per share = Market conversion price пЂпЂ Current market price Market conversion premium ratio = Premium payback period = Market conversion premium per share Favorable income differential per share Favorable income differential per share = Premium over straight value = В© 2013 ELAN GUIDES Market conversion premium per share Market price of common stock Coupon interest пЂ (Conversion ratio п‚ґпЂ Common stock dividend per share) Conversion ratio Market price of convertible bond Straight value пЂпЂ пЂ± VALUING BONDS WITH EMBEDDED OPTIONS An Option-Based Valuation Approach Covertible security value = Straight value пЂ«пЂ Value of the call option on the stock Covertible callable bond value = Straight value пЂ«пЂ Value of the call option on the stock пЂпЂ Value of the call option on the bond Covertible callable and putable bond value = Straight value пЂ«пЂ Value of the call option on the stock пЂпЂ Value of the call option on the bond пЂ«пЂ Value of the put option on the bond В© 2013 ELAN GUIDES MORTGAGE-BACKED SECTOR OF THE BOND MARKET MORTGAGE-BACKED SECTOR OF THE BOND MARKET Single Monthly Mortality Rate (SMM) SMMt = Prepayment in month t Beginning mortgage balance for month t пЂ Scheduled principal payment in month t Prepayment in month t = SMM Г— (Beginning mortgage balance for month t пЂ Scheduled principal payment in month t) Conditional Prepayment Rate (CPR) CPR = 1 пЂпЂ (1 пЂ SMM)12 Given the CPR, the SMM can be computed as: SMM = 1 пЂпЂ (1 пЂ CPR)1/12 Average Life Average life = T t п‚ґпЂ ProjectedпЂ principal recieved at time t t=1 12пЂ п‚ґпЂ TotalпЂ principal пЃ“ t = Number of months Distribution of Prepayment Risk in a Sequential-Pay CMO Tranche Contraction Risk Extension Risk A (sequential pay) HIGH LOW B (sequential pay) C (sequential pay) Z (accrual pay) LOW HIGH п‚› п‚њ Prepayment Risk in Different PAC Tranches Tranche Prepayment Risk PAC I - Senior LOW PAC I - Junior PAC II Support HIGH п‚њ В© 2013 ELAN GUIDES ASSET-BACKED SECTOR OF THE BOND MARKET ASSET-BACKED SECTOR OF THE BOND MARKET Parties to the Securitization Party Description Party in Illustration Seller Originates the loans and sells loans to the SPV ABC Company Issuer/Trust The SPV that buys the loans from the seller and issues the assetbacked securities SPV Servicer Services the loans Servicer Manufactured Housing-Backed Securities SMM = ABS 1 вЂ“ [ABS Г— (M вЂ“ 1)] ABS = SMM 1 + [SMM Г— (M вЂ“ 1)] VALUING MORTGAGE-BACKED AND ASSET-BACKED SECURITIES Cash Flow Yield ABS and MBS typically have monthly cash flows, so the cash flow yield on these securities is compared to the yield on Treasury coupon securities based on their bond equivalent yields. The bond equivalent yield for MBS/ABS is calculated as: Bond equivalent yield = 2 [(1 + monthly cash flow yield)6 вЂ“ 1] Option Cost Option cost = Zero-volatility spread вЂ“ Option-adjusted spread Duration Duration = VпЂпЂ пЂ V+ 2V0 (пЃ„ y) В© 2013 ELAN GUIDES DERIVATES DERIVATIVES FORWARD MARKETS AND CONTRACTS Value of a Forward Contract Time Long Position Value Zero, as the contract is priced to prevent arbitrage At initiation During life of the contract F(0,T) St пЂ At expiration Short Position Value Zero, as the contract is priced to prevent arbitrage F(0,T) T-t (1 + r)T-t (1 + r) ST пЂ F(0,T) пЂпЂ St F(0,T) пЂ ST Price of an Equity Forward with Discrete Dividends n PV(D,0,T) = Di пЃ“ (1 + r) i=1 ti ... Approach I T F(0,T) = [S0 вЂ“ PV(D,0,T)] (1 + r) n FV(D,0,T) = пЃ“ D (1 + r) TпЂti i i=1 ... Approach II T F(0,T) = S0 (1 + r) вЂ“ FV(D,0,T) Price of an Equity Forward with Continuous Dividends c c F(0,T) = (S0eпЂпЃ¤ T)er T c c F(0,T) = S0 п‚ґпЂ e(r пЂпЃ¤ )T rc = Continuously compounded risk-free rate пЃ¤c = Continuously compounded dividend yield Value of an Equity Forward Vt(0,T) = [St вЂ“ PV(D,t,T)] вЂ“ [F(0,T) / (1 + r)T вЂ“ t] PV(D,t,T) = PV of dividends expected to be received over the remainder of the contract term (between t and T). Assuming continuous compounding, the value of a forward contract on a stock index or portfolio can be calculated as: Vt(0,T) = SteвЂ“пЃ¤c(T вЂ“ t) вЂ“ F(0,T)eвЂ“rc(T вЂ“ t) St Vt(0,T) = пЃ¤c(T вЂ“ t) e В© 2013 ELAN GUIDES вЂ“ F(0,T) erc(T вЂ“ t) DERIVATES Calculating the No-Arbitrage Forward Price for a Forward Contract on a Coupon Bond F(0,T) = [B0C(T+Y) вЂ“ PV(CI,0,T)] Г— (1 + r)T Or F(0,T) = [B0C(T+Y)] (1 + r)T вЂ“ FV(CI,0,T) BC = Price of coupon bond T = Time of forward contract expiration Y = Remaining maturity of bond upon forward contract expiration T+Y = Time to maturity of the bond at forward contract initiation. PV(CI,0,T) = Present value of coupon interest expected to be received between time 0 (contract initiation) and time T (contract expiration). FV(CI,0,T) = Future value of coupon interest expected to be received between time 0 (contract initiation) and time T (contract expiration). Valuing a Forward Contract on a Coupon Bond The value of the long position in a forward contract on a fixed income security prior to expiration can be calculated as: Vt(0,T) = BtC(T+Y) вЂ“ PV(CI,t,T) вЂ“ F(0,T) / (1 + r)T вЂ“ t PV(CI,t,T) = Present value of coupon payments that are expected to be received between time t and time T. C Bt (T+Y) = Current value of coupon bond with time T+Y remaining until maturity Pricing a Forward Rate Agreement 1 + L0(h + m) FRA(0,h,m) = ( ) ( ) ( ) h+m 360 пЂ1 1 + L0( h ) h 360 360 m FRA(0,h,m) = The annualized rate on an FRA initiated at Day 0, expiring on Day h, and based on m-day LIBOR. h = Number of days until FRA expiration m = Number of days in underlying hypothetical loan h+m = Number of days from FRA initiation until end of term of underlying hypothetical loan. L0 = (Unannualized) LIBOR rate today В© 2013 ELAN GUIDES DERIVATES FRA Payoff NP Г— [(Market LIBOR вЂ“ FRA rate) Г— No. of days in the loan term / 360] FRA payoff = 1 + [Market LIBOR Г— (No. of days in the loan term / 360)] Valuing FRA prior to expiration NP Г— [(Current forward rate вЂ“ FRA rate) Г— No. of days in the loan term / 360] 1 + {Current LIBOR Г— [(No. of days in loan term + No. of days till contract expiration) / 360]} Or: 1 + FRA(0,h,m) 1 Vg (0,h,m) = пЂ hпЂg 1 + Lg (h пЂ g) 360 ( ) 1 + Lg (h + m пЂ g) ( m 360 ( ) h+mпЂg 360 ) g = Number of days since FRA initiation. Pricing a Currency Forward Contracts F(0,T) = S0 Г— (1 + RDC)T (1 + RFC)T F and S are quoted in terms of DC per unit of FC RDC = Domestic risk-free rate RFC = Foreign risk-free rate T = Length of the contract in years. Remember to use a 365-day basis to calculate T if the term is given in days. Valuing a Currency Forward Contract The value of the long position in a currency forward contract at any time prior to maturity can be calculated as follows: Vt (0,T) = St пЂ (1 + RFC)(TпЂt) F (0,T) (1 + RDC)(TпЂt) Assuming continuous compounding, the price and value of a currency forward contract can be calculated by applying the formulas below: F(0,T) = (S0eвЂ“ r cFC Г— T Vt(0,T) = [St / er В© 2013 ELAN GUIDES ) Г— er cFC Г— (T вЂ“ t) cDC Г— T or F(0,T) = S0 Г— e(r cDC Г— (T вЂ“ t) ] вЂ“ [F(0,T) / er ] cDC вЂ“ rcFC) Г— T rc here represents a continuously compounded riskfree rate in these formulas. FUTURES MARKETS AND CONTRACTS FUTURES MARKETS AND CONTRACTS If we ignore the effects of the mark-to-market adjustment on futures contracts, we can make the simplifying assumption that the futures price and forward price are the same. f0(T) = F(0,T) = S0 Г— (1 + r)T f0(T) = Futures price today of a futures contract that expires at time T. F(0,T) = Forward price of a forward contract that expires at time T. S0 = Spot price of underlying asset today r = Annual risk-free rate The Effect of Storage or Carrying Costs on the Futures Price f0(T) = S0 (1 + r)T + FV(SC,0,T) The Effect of Monetary Benefits on the Futures Price f0(T) = S0 (1 + r)T пЂ FV(CF,0,T) The Effect of Non-Monetary Benefits on the Futures Price FV(CB,0,T) = Costs of storage вЂ“ Nonmonetary benefits (Convenience yield) If costs exceed benefits, FV(CB,0,T) is a positive number and is known as cost of carry. In this case, the general futures pricing formula is given as: f0(T) = S0 (1 + r)T + FV(CB,0,T) Pricing Treasury Bond Futures f0(T) = B0C(T+Y) [(1 + r0(T)]T вЂ“ FV(CI,0,T) BC = Price of coupon bond T = Time of futures contract expiration Y = Remaining maturity of bond upon futures contract expiration T+Y = Time to maturity of the bond at futures contract initiation. r0(T) = Interest rate at time 0 for period until time T. FV(CI,0,T) = Future value of coupon interest expected to be received between time 0 (contract initiation) and time T (contract expiration). The adjusted futures price of a t-bond futures contract is calculated as: f0(T) = B0C (T + Y) [1 + r0 (T)]T пЂ FV (CI,0,T) CF(T) CF(T) = Conversion factor on CTD bond В© 2013 ELAN GUIDES FUTURES MARKETS AND CONTRACTS Pricing Stock Index Futures f0(T) = S0 (1 + r)T вЂ“ FV(D,0,T) f0(T) = S0 п‚ґ e(r cвЂ“ пЃ¤c)T Pricing Currency Futures f0(T) = S0 п‚ґ (1 + rDC)T (1 + rFC)T F and S are quoted in terms of DC/FC rDC = Domestic currency interest rate rFC = Foreign currency interest rate T = Length of the contract in years. Remember to use a 365-day year if maturity is given in days. If interest rates are assumed to be continuously compounded, then the no-arbitrage futures price is calculated as: f0(T) = S0 Г— e(r cDC вЂ“ rcFC)Г—T rc represents the continuously compounded risk-free rate. В© 2013 ELAN GUIDES OPTION MARKETS AND CONTRACTS OPTION MARKETS AND CONTRACTS Put-Call Parity C0 + X = P0 + S0 (1 + RF)T Synthetic Securities Strategy Consisting of fiduciary call long call + long bond long call long call long put long put long underlying asset long bond Value Equals Strategy Consisting of Value X (1 + RF)T = Protective put long put + long underlying asset P0 + S0 C0 = long put + long Synthetic call underlying asset + short bond P0 + S0 пЂ X (1 + RF)T P0 = Synthetic put long call + short underlying asset + long bond C0 пЂ S0 + X (1 + RF)T long underlying asset S0 = Synthetic underlying asset long call + long bond + short put C0 + long bond X (1 + RF)T = Synthetic bond long put + long underlying asset + short call C0 + X пЂ P0 (1 + RF)T P 0 + S 0 пЂ C0 В© 2013 ELAN GUIDES OPTION MARKETS AND CONTRACTS One-Period Binomial Model Computing the two possible values of the stock: S+ = Su S- = Sd Binomia Call Option Pricing Call payoff = Max(0, S+ вЂ“ X) Binomial Put Option Pricing Put payoff = Max (0, X вЂ“ ST) Compute the risk-neutral probabilities: пЃ°= (1 + r пЂ dпЂ© (u пЂ d) Calculating the value of the call option: пЃ° c+ + (1 вЂ“ пЃ°) c- c= 1+r Calculating the value of the put option: p= пЃ° p+ + (1 вЂ“ пЃ°) p1+r Hedge Ratio n= c+ пЂ cS+ пЂ S- Intrinsic value of caplet at expiration: Caplet value = Max {0, [(One-year rate вЂ“ Cap rate) п‚ґ Notional principal]} 1 + One-year rate Intrinsic value of floorlet at expiration: Floorlet value = В© 2013 ELAN GUIDES max {0, [(Floor rate вЂ“ One-year rate) п‚ґ Notional principal]} 1 + One-year rate OPTION MARKETS AND CONTRACTS The Black-Scholes-Merton Formula c c = S0N(d1) пЂ XeпЂr TN(d2) c p = Xe-r T[1 пЂ N(d2)] пЂ S0[1 пЂ N(d1)] Where: d1 = ln(S0 X) + [rc + (пЃіпЂІпЂ пЂІпЂ©пЃќпЃ” пЃіпЂ пЂ пЂ пЃ” d2 = d1 пЂ пЃіпЂ пЂ пЂ пЃ” пЃі= the annualized standard deviation of the continuously compounded return on the stock rc = the continuously compounded risk-free rate of return N(d1) = Cumulative normal probability of d1. Delta Delta = Change in option price Change in underlying price Change in option price = Delta п‚ґ Change in underlying price An approximate measure for option delta can be obtained from the BSM model: пЂ пЂ пЂ пЂ пЂ пЂ п‚· N(d1) from the BSM model approximately equals call option delta. пЂ пЂ пЂ пЂ пЂ пЂ п‚· N(d1) вЂ“ 1 approximately equals put option delta. Therefore: пЃ„ c п‚» N(d1) п‚ґ пЃ„ S пЃ„ p п‚» N(d1) вЂ“ 1] п‚ґ пЃ„ S Put-Call Parity for Forward Contracts Value at Expiration ST п‚Ј X Transaction Current Value Call and Bond Buy call Buy bond Total c0 [X вЂ“ F(0,T)]/(1 + r)T c0 + [X вЂ“ F(0,T)]/(1 + r)T 0 X вЂ“ F(0,T) X вЂ“ F(0,T) ST вЂ“ X X вЂ“ F(0,T) ST вЂ“ F(0,T) p X вЂ“ ST ST вЂ“ F(0,T) X вЂ“ F(0,T) 0 ST вЂ“ F(0,T) ST вЂ“ F(0,T) Put and Forward Buy put Buy forward contract Total 0 0 p0 ST > X В© 2013 ELAN GUIDES OPTION MARKETS AND CONTRACTS Put-call-forward parity c0 + X вЂ“ F(0,T) = p0 (1 + r)T Forward Contract and Synthetic Forward Contract Value at Expiration Transaction ST п‚Ј X Current Value Forward Contract Long forward contract Synthetic Forward Contract Buy call Sell put Buy (or sell) bond Total ST > X 0 ST вЂ“ F(0,T) ST вЂ“ F(0,T) c0 вЂ“ p0 0 вЂ“ ( X вЂ“ S T) X вЂ“ F(0,T) ST вЂ“ F(0,T) ST вЂ“ X 0 X вЂ“ F(0,T) ST вЂ“ F(0,T) T [X вЂ“ F(0,T)]/(1 + r) c0 вЂ“ p0 + [X вЂ“ F(0,T)]/(1 + r)T The Black Model The Black model is used to price European options on futures. c c = eпЂr T [f0(T)N(d1) пЂ XN(d2)] c p = eпЂr T (X[1 пЂ N(d2)] пЂ f0(T)[1 пЂ N(d1)]) Where: d1 = ln(f0(T) X) + (пЃіпЂІпЂ пЂІпЂ©пЃќпЃ” пЃіпЂ пЂ пЂ пЃ” d2 = d1 пЂ пЃіпЂ пЂ пЂ пЃ” f0(T) = the futures price c Notice that the Black model is similar to the BSM model except that eпЂr T f(T) is substituted for S0. In fact, the price of a European option on a forward or futures would be the same as the price of a European option on the underlying asset if the options and the forward/futures contract expire at the same point in time. В© 2013 ELAN GUIDES SWAP MARKETS AND CONTRACTS SWAP MARKETS AND CONTRACTS The Swap Fixed Rate Swap fixed rate = ( 1 пЂпЂ B0(N) B0(1) + B0(2) + B0(3) + ... + B0(N) ) п‚ґпЂ 100 Valuing a Swap Value of pay-fixed side of plain-vanilla interest rate swap: Present value of floating-rate payments пЂ Present value of fixed-rate payments Value of pay-floating side of plain-vanilla interest rate swap: Present value of fixed-rate payments пЂ Present value of floating-rate payments Valuing Equity Swaps вЂ�Pay a fixed rate and receive the return on equityвЂ™ swap [(1 + Return on equity) п‚ґ Notional principal] пЂ PV of the remaining fixed-rate payments вЂ�Pay a floating rate and receive the return on equityвЂ™ swap [(1 + Return on equity) п‚ґ Notional principal] пЂ PV (Next coupon payment + Par value) The value of a вЂ�pay the return on one equity instrument and receive the return on another equity instrumentвЂ™ swap is calculated as the difference between the values of the two (hypothetical) equity portfolios: [(1 + Return on Index 2) п‚ґ NP] вЂ“ [(1 + Return on Index 1) п‚ґ NP] Payer swaption (Market fixed-rate вЂ“ Exercise rate) п‚ґ No. of days in the payment period 360 п‚ґпЂ Notional principal Receiver swaption (Exercise rate вЂ“ Market fixed-rate) п‚ґ No. of days in the payment period 360 п‚ґпЂ Notional principal В© 2013 ELAN GUIDES INTEREST RATE DERIVATIVE INSTRUMENTS INTEREST RATE DERIVATIVE INSTRUMENTS Table: Caps, Floors, Interest Rate Options, Bond Options and Interest Rates Security Benefits whenвЂ¦ Long cap (floor) Interest rates rise (fall) Long call (put) option on interest rates Interest rates rise (fall) Long call (put) option on a fixed income instrument Interest rates fall (rise) Payoff to the buyer of an interest rate cap Payoff = Max [0,(Market interest-rate вЂ“ Cap rate) п‚ґ No. of days п‚ґпЂ Notional principal] 360 Payoff to the buyer of an interest rate floor Payoff = Max [0,(Floor rate вЂ“ Market interest-rate) п‚ґ В© 2013 ELAN GUIDES No. of days 360 п‚ґпЂ Notional principal] PORTFOLIO CONCEPTS PORTFOLIO CONCEPTS Expected return on Two-Asset Portfolio E(RP) = w1E(R1) + w2E(R2) E(R1) = expected return on Asset 1 E(R2) = expected return on Asset 2 w1 = weight of Asset 1 in the portfolio w2 = weight of Asset 2 in the portfolio Variance of 2-asset portfolio: пЃі2P = w12пЃі21 + w22пЃі22 + 2w1w2пЃІпЂ±пЂ¬пЂ пЂІпЃі1пЃі2 пЃі1= the standard deviation of return on Asset 1 пЃі2= the standard deviation of return on Asset 2 пЃІпЂ±пЂ¬пЂ пЂІ= the correlation between the two assetsвЂ™ returns Variance of 2-asset portfolio: пЃіP2 = w12пЃі21 + w22пЃі22 + 2w1w2Cov1,2 Cov1,2 = пЃІпЂ±пЂ¬пЂ пЂІпЃі1пЃі2 Expected Return and Standard Deviation for a Three-Asset Portfolio Expected return on 3-asset portfolio: E(RP) = w1E(R1) + w2E(R2) + w3E(R3) Variance of 3-asset portfolio: пЃіP2 = w12пЃі12 + w22пЃі22 + w32пЃі23 + 2w1w2пЃІпЂ±пЂ¬пЂ пЂІпЃі1пЃі2 + 2w1w3пЃІпЂ±пЂ¬пЂ пЂіпЃі1пЃі3 + 2w2w3пЃІпЂІпЂ¬пЂ пЂіпЃі2пЃі3 Variance of 3-asset portfolio: пЃіP2 = w12пЃі12 + w22пЃі22 + w32пЃі23 + 2w1w2CovпЂ±пЂ¬пЂ пЂІ + 2w1w3CovпЂ±пЂ¬пЂ пЂі + 2w2w3CovпЂІпЂ¬пЂ пЂі В© 2013 ELAN GUIDES PORTFOLIO CONCEPTS Expected Return and Variance of the Portfolio For a portfolio of n assets, the expected return on the portfolio is calculated as: n E(RP) = пЃ“ w E(R ) j j j=1 The variance of the portfolio is calculated as: n пЃі2P = n пЃ“ пЃ“ w w Cov(R ,R ) i i=1 j i j j=1 Variance of an Equally-weighted Portfolio пЃіP2 = 1 2 nпЂ1 пЃі + Cov n n пЃі2P = пЃі2 ( 1 пЂпЂ пЃІ n +пЃІ ) Expected Return for a Portfolio Containing a Risky Asset and the Risk-Free Asset E(RP) = RFR + пЃіP [E(Ri) пЂпЂ RFR] пЃіi Standard Deviation of a Portfolio Containing a Risky Asset and the Risk-Free Asset пЃіP = wiпЃіi В© 2013 ELAN GUIDES PORTFOLIO CONCEPTS CML Expected return on portfolios that lie on CML: E(RP) = w1Rf + (1 пЂ w1)E(Rm) Variance of portfolios that lie on CML: пЃі2 = w12пЃіf2 + (1 пЂ w1)2пЃіm2 + 2w1(1 пЂ w1)Cov(Rf , Rm) Equation of CML: E(RP) = Rf + E(Rm) пЂ Rf п‚ґ пЃіP пЃіm Calculation and Interpretation of Beta пЃўiпЂ пЂЅ Cov(Ri,Rm) пЃіm2 пЂЅ пЃІi,mпЃіi,пЃіm пЃіm2 пЂЅ пЃІi,mпЃіi пЃіm The Capital Asset Pricing Model E(Ri) пЂЅпЂ Rf +пЂ пЃўi[E(Rm) вЂ“ Rf ] The Decision to Add an Investment to an Existing Portfolio пЂЁ пЂ© E(Rnew) пЂ RF E(Rp) пЂ RF пЂѕ Corr(Rnew,Rp) пЃіp пЃіnew Market Model Estimates Ri = пЃЎi + пЃўi RM + пЃҐi Ri = Return on asset i RM = Return on the market portfolio пЃЎi = Average return on asset i unrelated to the market return пЃўi = Sensitivity of the return on asset i to the return on the market portfolio пЃҐi = An error term пЂ пЂ пЂ пЂ пЂ пЂ п‚· пЂ пЂ пЂ пЂ пЂ пЂ п‚· пЃўi is the slope in the market model. It represents the increase in the return on asset i if the market return increases by one percentage point. пЃЎi is the intercept term. It represents the predicted return on asset i if the return on the market equals 0. Expected return on asset i E(Ri) = пЃЎi + пЃўiE(RM) В© 2013 ELAN GUIDES PORTFOLIO CONCEPTS Variance of the return on asset i Var(Ri) = пЃў2i пЃі2M+ пЃіпЃҐпЂ 2i Covariance of the returns on asset i and asset j Cov(Ri,Rj) = пЃўiпЃўjпЃі2 M Correlation of returns between assets i and j Corr(Ri,Rj) = 2 пЃўiпЃўjпЃіM 2 (пЃў2i пЃіM2 + пЃіпЃҐпЂ 2i )1/2 (пЃў2j пЃіM + пЃіпЃҐпЂ 2j )1/2 Market Model Estimates: Adjusted Beta Adjusted beta = 0.333 + 0.667 (Historical beta) Macroeconomic Factor Models Ri = ai + bi1FINT + bi2FGDP + пЃҐi Ri = the return to stock i ai = the expected return to stock i FINT = the surprise in interest rates FGDP = the surprise in GDP growth bi1 = the sensitivity of the return on stock i to surprises in interest rates. bi2 = the sensitivity of the return on stock i to surprises in GDP growth. пЃҐi = an error term with a zero mean that represents the portion of the return to stock i that is not explained by the factor model. Fundamental Factor Models Ri = ai + bi1FDY + bi2FPE + пЃҐi Ri = the return to stock i ai = intercept FDY = return associated with the dividend yield factor FPE = return associated with the P-E factor bi1 = the sensitivity of the return on stock i to the dividend yield factor. bi2 = the sensitivity of the return on stock i to the P-E factor. пЃҐi = an error term Standardized sensitivities are computed as follows: bij = Assets iвЂ™s attribute value пЂ Average attribute value пЃіпЂЁAttribute values) В© 2013 ELAN GUIDES PORTFOLIO CONCEPTS Arbitrage Pricing Theory and the Factor Model E(RP) = RF + пЃ¬1пЃўp,пЂ±пЂ пЂ«пЂ пЂ®пЂ®пЂ®пЂ пЂ« пЃ¬пЃ‹пЃўp,пЃ‹ E(Rp) = Expected return on the portfolio p RF = Risk-free rate пЃ¬ j = Risk premium for factor j пЃўp,j = Sensitivity of the portfolio to factor j K = Number of factors Active Risk TE = s(Rp пЂ RB) Active risk squared = s2(Rp пЂ RB) Active risk squared = Active factor risk + Active specific risk n Active specific risk = пЃ“w пЃі i=1 a пЂІ i пЃҐi Where: wai = The ith assetвЂ™s active weight in the portfolio (i.e., the difference between the assetвЂ™s weight in the portfolio and its weight in the benchmark). пЃіпЃҐпЂІ = The residual risk of the ith asset (i.e., the variance of the ith assetвЂ™s returns that is not explained i by the factors). Active factor risk = Active risk squared вЂ“ Active specific risk. Active Return Active return = Rp вЂ“ RB Active return = Return from fctor tilts + Return from asset selection K Active return = пЃ“[(Portfolio sensitivity) пЂ (Benchmark sensitivity) ] п‚ґ (Factor return) + Asset selection j j j j=1 В© 2013 ELAN GUIDES PORTFOLIO CONCEPTS FactorвЂ™s Marginal Contribution to Active Risk Squared (FMCAR) K baj FMCARj = FMCARj = пЃ“ b Cov(F ,F ) a i j i i=1 Active risk squared Active factor risk Active risk squared where: baj = The portfolioвЂ™s active exposure to factor j K baj пЃ“ b Cov(F ,F ) = The active factor risk for factor j a i j i i=1 The Information Ratio IR = Rp пЂ RB s(Rp пЂ RB) В© 2013 ELAN GUIDES THE THEORY OF ACTIVE PORTFOLIO MANAGEMENT THE THEORY OF ACTIVE PORTFOLIO MANAGEMENT Weight of security k in the active portfolio (Portfolio A) wk = пЃЎk / пЃіпЂІпЂЁekпЂ© n пЃ“ пЃЎi / пЃіпЂІпЂЁeiпЂ© i=1 The expression for the optimal weight, w*, of the active portfolio (Portfolio A) in the optimal risky portfolio (Portfolio P) is given as: пЃЎA w* = пЃі2(eA) пЃЎA(1пЂпЃўA) + RM пЃі2 M Assuming (for simplicity) that the beta of Portfolio A equals 1, the optimal weight, w0, of Portfolio A in Portfolio P is calculated as: пЃЎA w0 = RM = пЃі2(eA) пЃЎA /пЃі2(eA) RM /пЃі2M пЃі2M If the beta of Portfolio A does not equal 1, we can use the following equation to determine the optimal weight, w*, of Portfolio A in Portfolio P. w* = w0 пЂ±пЂ пЂ«пЂ (1пЂпЃўA)w0 Evaluation of Performance Sharpe Ratio The Sharpe ratio of the optimal risky portfolio (Portfolio P) can be separated into contributions from the market and active portfolio as follows: 2 2 SP = SM пЃі2A + пЃі2(eA) 2 = пЃ› пЃќпЃ› пЃќ RM пЃіM + 2 пЃіA пЃі(eA) Information Ratio 2 пЃ› пЃќ пЃ› пЃќ пЃЎA пЃі(eA) n =пЃ“ i=1 2 пЃЎi пЃі(ei) В© 2013 ELAN GUIDES THE THEORY OF ACTIVE PORTFOLIO MANAGEMENT Imperfect Forecasts of Alpha Values Actual (realized) alpha: пЃЎпЂ пЂЅпЂ RпЂ пЂпЂ пЃўRM To measure the forecasting accuracy of the analyst, we can regress alpha forecasts (пЃЎf) on realized alpha (пЃЎ). пЃЎf пЂЅ aпЂ°пЂ пЂ«пЂ aпЂ±пЃЎпЂ пЂ«пЂ пЃҐ For simplicity, we assume that aпЂ° and aпЂ± equal 0 and 1 respectively. Given that forecast errors (пЃҐ) are uncorrelated with true alpha (пЃЎ) i.e., CovпЃҐ,пЃЎ equals 0, the variance of the forecast is given as: пЃіпЃЎfпЂІ пЂЅ пЃіпЃЎпЂІ + пЃіпЃҐпЂІ We can evaluate the quality of the analystвЂ™s forecasts by calculating the coefficient of determination of the regression described above. пЂІпЂ R пЂЅ пЃіпЃЎпЂІ пЃіпЃЎfпЂІ пЂЅ пЃіпЃЎпЂІ пЃіпЃЎпЂІ + пЃіпЃҐпЂІ This estimate of R2 is used as a shrinking factor to adjust the analystвЂ™s forecasts of alpha. В© 2013 ELAN GUIDES THE PORTFOLIO MANAGEMENT PROCESS AND THE INVESTMENT POLICY STATEMENT THE PORTFOLIO MANAGEMENT PROCESS AND THE INVESTMENT POLICY STATEMENT Risk Tolerance Willingness to Take Risk Below Average Above Average Ability to Take Risk Below Average Above Average Below-average risk tolerance Resolution needed Resolution needed Above-average risk tolerance Return Requirements and Risk Tolerances of Various Investors Type of Investor Return Requirement Risk Tolerance Individual Depends on stage of life, circumstances, and obligations Varies Pension Plans (Defined Benefit) The return that will adequately fund liabilities on an inflationadjusted basis Depends on plan and sponsor characteristics, plan features, funding status, and workforce characteristics Pension Plans (Defined Contribution) Depends on stage of life of individual participants Varies with the risk tolerance of individual participants Foundations and Endowments The return that will cover annual spending, investment expenses, and expected inflation Determined by amount of assets relative to needs, but generally above- average or average Life Insurance Companies Determined by rates used to determine policyholder reserves Below average due to factors such as regulatory constraints Non-Life- Insurance Companies Determined by the need to price policies competitively and by financial needs Below average due to factors such as regulatory constraints Banks Determined by cost of funds Varies В© 2013 ELAN GUIDES

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