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Quantitative Methods The Time Value of Money The Time Value of Money The Future Value of a Single Cash Flow FVN = PV (1 + r) N The Present Value of a Single Cash Flow PV = FV (1 + r) N PVAnnuity Due = PVOrdinary Annuity × (1 + r) FVAnnuity Due = FVOrdinary Annuity × (1 + r) Present Value of a Perpetuity PV(perpetuity) = PMT I/Y Continuous Compounding and Future Values FVN = PVe r ⋅N s Effective Annual Rates EAR = (1 + Periodic interest rate) N − 1 2 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Discounted Cash Flow Applications Discounted Cash Flow Applications Net Present Value N CFt t t=0 (1 + r ) NPV = ∑ where: CFt = the expected net cash flow at time t N = the investment’s projected life r = the discount rate or appropriate cost of capital Bank Discount Yield rBD = D 360 × F t where: rBD = the annualized yield on a bank discount basis D = the dollar discount (face value – purchase price) F = the face value of the bill t = number of days remaining until maturity Holding Period Yield HPY = P1 − P0 + D1 P1 + D1 = −1 P0 P0 where: P0 = initial price of the investment. P1 = price received from the instrument at maturity/sale. D1 = interest or dividend received from the investment. Effective Annual Yield EAY = (1 + HPY)365/ t − 1 where: HPY = holding period yield t = numbers of days remaining till maturity HPY = (1 + EAY) t /365 − 1 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 3 Discounted Cash Flow Applications Money Market Yield R MM = 360 × rBD 360 − (t × rBD ) R MM = HPY × (360/t) Bond Equivalent Yield BEY = [(1 + EAY)0.5 − 1] × 2 4 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Statistical Concepts Statistical Concepts Population Mean N µ= ∑ xi i =1 N where: xi = is the ith observation. Sample Mean n X= ∑ xi i =1 n Geometric Mean 1 + R G = T (1 + R1 ) × (1 + R 2 ) ×…× (1 + R T ) OR G = n X1X 2 X 3 … X n with X i > 0 for i = 1, 2,…, n. 1 T T R G = ∏ (1 + R t ) − 1 t =1 Harmonic Mean Harmonic mean: X H = N with X i > 0 for i = 1,2,…,N. 1 ∑x i =1 i N Percentiles Ly = ( n + 1) y 100 where: y = percentage point at which we are dividing the distribution Ly = location (L) of the percentile (Py) in the data set sorted in ascending order Range Range = Maximum value − Minimum value © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 5 Statistical Concepts Mean Absolute Deviation n MAD = ∑ Xi − X i =1 n where: n = number of items in the data set X = the arithmetic mean of the sample Population Variance N σ2 = ∑ (X i − µ)2 i =1 N where: Xi = observation i μ = population mean N = size of the population Population Standard Deviation N σ= ∑ (X i − µ)2 i =1 N Sample Variance n Sample variance = s2 = ∑ (X i − i =1 X)2 n −1 where: n = sample size. Sample Standard Deviation n s= 6 ∑ (X i − X)2 i =1 n −1 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Statistical Concepts Coefficient of Variation Coefficient of variation = s X where: s = sample standard deviation X = the sample mean. Sharpe Ratio Sharpe ratio = rp − rf sp where: rp = mean portfolio return rf = risk‐free return sp = standard deviation of portfolio returns Sample skewness, also known as sample relative skewness, is calculated as: n (X i − X)3 ∑ n i =1 SK = ( n − 1)( n − 2 ) s3 As n becomes large, the expression reduces to the mean cubed deviation. n 1 SK ≈ n ∑ (X i − X)3 i =1 s3 where: s = sample standard deviation Sample Kurtosis uses standard deviations to the fourth power. Sample excess kurtosis is calculated as: n (X i − X)4 ∑ n(n + 1) 3(n − 1)2 i =1 − KE = 4 s (n − 1)(n − 2)(n − 3) (n − 2)(n − 3) © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 7 Statistical Concepts As n becomes large the equation simplifies to: n KE ≈ (X i − X)4 ∑ 1 i=1 n s4 −3 where: s = sample standard deviation For a sample size greater than 100, a sample excess kurtosis of greater than 1.0 would be considered unusually high. Most equity return series have been found to be leptokurtic. 8 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Probability Concepts Probability Concepts Odds for an Event P (E) = a (a + b) Where the odds for are given as “a to b”, then: Odds for an Event P (E) = b (a + b) Where the odds against are given as “a to b”, then: Conditional Probabilities P(A B) = P(AB) given that P(B) ≠ 0 P(B) Multiplication Rule for Probabilities P(AB) = P(A B) × P(B) Addition Rule for Probabilities P(A or B) = P(A) + P(B) − P(AB) For Independant Events P(A B) = P(A), or equivalently, P(B A) = P(B) P(A or B) = P(A) + P(B) − P(AB) P(A and B) = P(A) × P(B) The Total Probability Rule P(A) = P(AS) + P(ASc ) P(A) = P(A S) × P(S) + P(A Sc ) × P(Sc ) The Total Probability Rule for n Possible Scenarios P(A) = P(A S1 ) × P(S1 ) + P(A S2 ) × P(S2 ) + + P(A Sn ) × P(Sn ) where the set of events {S1 , S2 ,…, Sn } is mutually exclusive and exhaustive. © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 9 Probability Concepts Expected Value E(X) = P(X1 )X1 + P(X 2 )X 2 + … P(X n )X n n E(X) = ∑ P(X i )X i i =1 where: Xi = one of n possible outcomes. Variance and Standard Deviation σ 2 (X) = E{[X − E(X)]2} n σ 2 (X) = ∑ P(X i ) [X i − E(X)]2 i =1 The Total Probability Rule for Expected Value 1. 2. E(X) = E(X | S)P(S) + E(X | Sc)P(Sc) E(X) = E(X | S1) × P(S1) + E(X | S2) × P(S2) + . . . + E(X | Sn) × P(Sn) where: E(X) = the unconditional expected value of X E(X | S1) = the expected value of X given Scenario 1 P(S1) = the probability of Scenario 1 occurring The set of events {S1, S2, . . . , Sn} is mutually exclusive and exhaustive. Covariance Cov(XY) = E{[X − E(X)][Y − E(Y)]} Cov(R A ,R B ) = E{[R A − E(R A )][R B − E(R B )]} Correlation Coefficient Corr(R A ,R B ) = ρ(R A ,R B ) = 10 Cov(R A ,R B ) (σ A )(σ B ) © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Probability Concepts Expected Return on a Portfolio N E(R p ) = ∑ wi E(R i ) = w1E(R1 ) + w2 E(R 2 ) + + w N E(R N ) i =1 where: Weight of asset i = Market value of investment i Market value of portfolio Portfolio Variance N N Var(R p ) = ∑ ∑ wi w jCov(R i ,R j ) i =1 j=1 Variance of a 2 Asset Portfolio Var(R p ) = w2A σ 2 (R A ) + w2B σ 2 (R B ) + 2w A w B Cov(R A ,R B ) Var(R p ) = w2A σ 2 (R A ) + w2B σ 2 (R B ) + 2w A w Bρ(R A ,R B )σ (R A )σ (R B ) Variance of a 3 Asset Portfolio Var(R p ) = w2A σ 2 (R A ) + w2B σ 2 (R B ) + w2C σ 2 (R C ) + 2w A w B Cov(R A ,R B ) + 2w B wC Cov(R B ,R C ) + 2wC w A Cov(R C ,R A ) Bayes’ Formula P(Event Information) = P (Information Event) × P (Event) P (Information) Counting Rules The number of different ways that the k tasks can be done equals n1 × n2 × n3 × … nk . Combinations n Cr n n! = = r ( n − r )!( r!) Remember: The combination formula is used when the order in which the items are assigned the labels is NOT important. Permutations n Pr = n! ( n − r )! © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 11 Common Probability Distributions Common Probability Distributions Discrete Uniform Distribution F(x) = n × p(x) for the nth observation. Binomial Distribution P(X=x) = n Cx (p)x (1 − p)n-x where: p = probability of success 1 − p = probability of failure nCx = number of possible combinations of having x successes in n trials. Stated differently, it is the number of ways to choose x from n when the order does not matter. Variance of a Binomial Random Variable σ 2x = n × p × (1 − p) Tracking Error Tracking error = Gross return on portfolio − Total return on benchmark index The Continuous Uniform Distribution P(X < a), P (X > b) = 0 P (x1 ≤ X ≤ x 2 ) = x 2 − x1 b−a Confidence Intervals For a random variable X that follows the normal distribution: The 90% confidence interval is x − 1.65s to x + 1.65s The 95% confidence interval is x − 1.96s to x + 1.96s The 99% confidence interval is x − 2.58s to x + 2.58s The following probability statements can be made about normal distributions • • • • 12 Approximately 50% of all observations lie in the interval Approximately 68% of all observations lie in the interval Approximately 95% of all observations lie in the interval Approximately 99% of all observations lie in the interval μ ± (2/3)σ μ ± 1σ μ ± 2σ μ ± 3σ © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Common Probability Distributions z‐Score z = (observed value − population mean)/standard deviation = (x − µ)/σ Roy’s Safety‐First Criterion Minimize P(RP< RT) where: RP = portfolio return RT = target return Shortfall Ratio Shortfall ratio (SF Ratio) or z-score = E (RP ) − RT σP Continuously Compounded Returns EAR = e r − 1 cc HPR t = e r cc ×t rcc = continuously compounded annual rate −l © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 13 Sampling and Estimation Sampling and Estimation Sampling Error Sampling error of the mean = Sample mean − Population mean = x − µ Standard Error of Sample Mean when Population Variance is known σx = σ n where: σ x = the standard error of the sample mean σ = the population standard deviation n = the sample size Standard Error of Sample Mean when Population Variance is not known sx = s n where: s x = standard error of sample mean s = sample standard deviation. Confidence Intervals Point estimate ± (reliability factor × standard error) where: Point estimate = value of the sample statistic that is used to estimate the population parameter Reliability factor = a number based on the assumed distribution of the point estimate and the level of confidence for the interval (1 − α). Standard error = the standard error of the sample statistic (point estimate) 14 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Sampling and Estimation x ± z α /2 σ n where: x = The sample mean (point estimate of population mean) zα/2 = The standard normal random variable for which the probability of an observation lying in either tail is σ / 2 (reliability factor). σ = The standard error of the sample mean. n x ± tα 2 s n where: x = sample mean (the point estimate of the population mean) tα = the t‐reliability factor 2 s = standard error of the sample mean n s = sample standard deviation © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 15 Hypothesis Testing Hypothesis Testing Test Statistic Sample statistic − Hypothesized value Standard error of sample statistic Test statistic = Power of a Test Power of a test = 1 − P(Type II error) Decision Rules for Hypothesis Tests Decision Do not reject H0 H0 is True Correct decision Incorrect decision Type I error Significance level = P(Type I error) Reject H0 H0 is False Incorrect decision Type II error Correct decision Power of the test = 1 − P(Type II error) Confidence Interval sample critical standard population sample critical standard statistic − value error ≤ parameter ≤ statistic + value error x (s n) x (s n) − (z α /2 ) ≤ µ0 ≤ + (z α /2 ) Summary H0 : μ ≤ μ0 Alternate hypothesis Ha : μ > μ0 One tailed (lower tail) test H0 : μ ≥ μ0 Ha : μ < μ0 Test statistic < critical value Test statistic ≥ critical value Probability that lies below the computed test statistic. Two‐tailed H0 : μ = μ0 Ha : μ ≠ μ0 Test statistic < lower critical value Test statistic > upper critical value Lower critical value ≤ test statistic ≤ upper critical value Probability that lies above the positive value of the computed test statistic plus the probability that lies below the negative value of the computed test statistic. Type of test One tailed (upper tail) test 16 Null hypothesis Fail to reject null if Reject null if Test statistic > critical value Test statistic ≤ critical value P‐value represents Probability that lies above the computed test statistic. © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Hypothesis Testing t‐Statistic t-stat = x − µ0 s n where: x = sample mean μ0 = hypothesized population mean s = standard deviation of the sample n = sample size z‐Statistic z-stat = x − µ0 σ n z-stat = where: x = sample mean μ = hypothesized population mean σ = standard deviation of the population n = sample size x − µ0 s n where: x = sample mean μ = hypothesized population mean s = standard deviation of the sample n = sample size Tests for Means when Population Variances are Assumed Equal t= (x1 − x2 ) − (µ1 − µ 2 ) s2p s2p n +n 1 2 1/2 where: s2p = (n1 − 1)s12 + (n 2 − 1)s22 n1 + n 2 − 2 s12 = variance of the first sample s22 = variance of the second sample n1 = number of observations in first sample n2 = number of observations in second sample degrees of freedom = n1 + n2 −2 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 17 Hypothesis Testing Tests for Means when Population Variances are Assumed Unequal t-stat = df = (x1 − x2 ) − (µ1 − µ 2 ) s12 s22 n + n 1 2 s12 s22 n + n 1 2 1/2 2 (s12 n1 )2 + (s22 n2 )2 n1 n2 where: s12 = variance of the first sample s22 = variance of the second sample n1 = number of observations in first sample n2 = number of observations in second sample Paired Comparisons Test t= d − µ dz sd where: d = sample mean difference sd s d = standard error of the mean difference = n sd = sample standard deviation n = the number of paired observations Hypothesis Tests Concerning the Mean of Two Populations ‐ Appropriate Tests 18 Population distribution Normal Relationship between samples Independent Assumption regarding variance Equal Normal Independent Unequal t‐test with variance not pooled Normal Dependent N/A t‐test with paired comparisons Type of test t‐test pooled variance © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Hypothesis Testing Chi Squared Test‐Statistic χ2 = ( n − 1) s2 σ 20 where: n = sample size s2 = sample variance σ 20 = hypothesized value for population variance Test‐Statistic for the F‐Test F= s12 s22 where: s12 = Variance of sample drawn from Population 1 s22 = Variance of sample drawn from Population 2 Hypothesis tests concerning the variance Hypothesis Test Concerning Variance of a single, normally distributed population Appropriate Test Statistic Chi‐square stat Equality of variance of two independent, normally distributed populations F‐stat © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 19 Technical Analysis Technical Analysis Setting Price Targets with Head and Shoulders Patterns Price target = Neckline - (Head − Neckline) Setting Price Targets for Inverse Head and Shoulders Patterns Price target = Neckline + (Neckline − Head) Momentum or Rate of Change Oscillator M = (V − Vx ) × 100 where: M = momentum oscillator value V = last closing price Vx = closing price x days ago, typically 10 days Relative Strength Index RSI = 100 − 100 1 + RS where: RS = Σ (Up changes for the period under consideration) Σ(| Down changes for the period under consideration|) Stochastic Oscillator C − L14 %K = 100 H14 − L14 where: C = last closing price L14 = lowest price in last 14 days H14 = highest price in last 14 days %D (signal line) = Average of the last three %K values calculated daily. Short Interest ratio Short interest ratio = Short interest Average daily trading volume Arms Index Arms index = 20 Number of advancing issues / Number of declining issues Volume of advancing issues / Volume of declining issues © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Economics Topics in Demand and Supply Analysis Topics in Demand and Supply Analysis The demand function captures the effect of all these factors on demand for a good. Demand function: QDx = f(Px, I, Py , …) … (Equation 1) Equation 1 is read as “the quantity demanded of Good X (QDX) depends on the price of Good X (PX), consumers’ incomes (I) and the price of Good Y (PY), etc.” The own‐price elasticity of demand is calculated as: EDPx = %∆QDx … (Equation 6) %∆Px If we express the percentage change in X as the change in X divided by the value of X, Equation 6 can be expanded to the following form: Slope of demand function. Coefficient on own‐price in market demand function EDPx %∆QDx = = %∆Px ∆QDx ∆Px QDx Px ∆QDx Px = … (Equation 7) ∆Px QDx Arc elasticity is calculated as: (Q 0 - Q1 ) × 100 % change in quantity demanded % ∆ Q d (Q 0 + Q1 )/2 = = EP = (P0 - P1 ) % change in price %∆P × 100 (P0 + P1 )/2 22 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Topics in Demand and Supply Analysis Income Elasticity of Demand Income elasticity of demand measures the responsiveness of demand for a particular good to a change in income, holding all other things constant. %∆QDx ED I = = %∆I EI = ∆QDx ∆I QDx I = ∆QDx I … (Equation 8) ∆I QDx Same as coefficient on I in market demand function (Equation 11) % change in quantity demanded % change in income Cross‐Price Elasticity of Demand Cross elasticity of demand measures the responsiveness of demand for a particular good to a change in price of another good, holding all other things constant. EDPy EC = %∆QDx = = %∆Py ∆QDx ∆Py QDx Py ∆QDx Py = … (Equation 9) ∆Py QDx Same as coefficient on PY in market demand function (Equation 11) % change in quantity demanded % change in price of substitute or complement © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 23 Topics in Demand and Supply Analysis Accounting Profit Accounting profit (loss) = Total revenue − Total accounting costs. Economic Profit Economic profit (also known as abnormal profit or supernormal profit) is calculated as: Economic profit = Total revenue − Total economic costs Economic profit = Total revenue − (Explicit costs + Implicit costs) Economic profit = Accounting profit − Total implicit opportunity costs Normal Profit Normal profit = Accounting profit − Economic profit Total, Average and Marginal Revenue Table: Summary of Revenue Terms2 24 Revenue Calculation Total revenue (TR) Price times quantity (P × Q), or the sum of individual units sold times their respective prices; Σ(Pi × Qi) Average revenue (AR) Total revenue divided by quantity; (TR / Q) Marginal revenue (MR) Change in total revenue divided by change in quantity; (ΔTR / ΔQ) © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Topics in Demand and Supply Analysis Total, Average, Marginal, Fixed and Variable Costs Table: Summary of Cost Terms3 Costs Calculation Total fixed cost (TFC) Sum of all fixed expenses; here defined to include all opportunity costs Total variable cost (TVC) Sum of all variable expenses, or per unit variable cost times quantity; (per unit VC × Q) Total costs (TC) Total fixed cost plus total variable cost; (TFC + TVC) Average fixed cost (AFC) Total fixed cost divided by quantity; (TFC / Q) Average variable cost (AVC) Total variable cost divided by quantity; (TVC / Q) Average total cost (ATC) Total cost divided by quantity; (TC / Q) or (AFC + AVC) Marginal cost (MC) Change in total cost divided by change in quantity; (ΔTC / ΔQ) Marginal revenue product (MRP) of labor is calculated as: MRP of labor = Change in total revenue / Change in quantity of labor For a firm in perfect competition, MRP of labor equals the MP of the last unit of labor times the price of the output unit. MRP = Marginal product * Product price A profit‐maximizing firm will hire more labor until: MRPLabor = PriceLabor Profits are maximized when: MRP1 MRPn =…= Price of input 1 Price of input n 2 Exhibit 3, Volume 2, CFA Program Curriculum 2012 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 25 The Firm And Market Structures The Firm And Market Structures The relationship between MR and price elasticity can be expressed as: MR = P[1 − (1/E p )] In a monopoly, MC = MR so: P[1 − (1/E p )] = MC N‐firm concentration ratio: Simply computes the aggregate market share of the N largest firms in the industry. The ratio will equal 0 for perfect competition and 100 for a monopoly. Herfindahl‐Hirschman Index (HHI): Adds up the squares of the market shares of each of the largest N companies in the market. The HHI equals 1 for a monopoly. If there are M firms in the industry with equal market shares, the HHI will equal 1/M. 26 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Aggregate Output, Price, And Economic Growth Aggregate Output, Price, And Economic Growth Nominal GDP refers to the value of goods and services included in GDP measured at current prices. Nominal GDP = Quantity produced in Year t × Prices in Year t Real GDP refers to the value of goods and services included in GDP measured at base‐year prices. Real GDP = Quantity produced in Year t × Base-year prices GDP Deflator GDP deflator = Value of current year output at current year prices × 100 Value of current year output at base year prices GDP deflator = Nominal GDP × 100 Real GDP The Components of GDP Based on the expenditure approach, GDP may be calculated as: GDP = C + I + G + (X − M) C = Consumer spending on final goods and services I = Gross private domestic investment, which includes business investment in capital goods (e.g. plant and equipment) and changes in inventory (inventory investment) G = Government spending on final goods and services X = Exports M = Imports Expenditure Approach Under the expenditure approach, GDP at market prices may be calculated as: GDP = Consumer spending on goods and services + Business gross fixed investment + Change in inventories + Government spending on goods and services + Government gross fixed investment + Exports − Imports + Statistical discrepancy © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. This equation is just a breakdown of the expression for GDP we stated in the previous LOS, i.e. GDP = C + I + G + (X − M). 27 Aggregate Output, Price, And Economic Growth Income Approach Under the income approach, GDP at market prices may be calculated as: GDP = National income + Capital consumption allowance + Statistical discrepancy … (Equation 1) National income equals the sum of incomes received by all factors of production used to generate final output. It includes: • • • • • Employee compensation Corporate and government enterprise profits before taxes, which includes: ○○ Dividends paid to households ○○ Corporate profits retained by businesses ○○ Corporate taxes paid to the government Interest income Rent and unincorporated business net income (proprietor’s income): Amounts earned by unincorporated proprietors and farm operators, who run their own businesses. Indirect business taxes less subsidies: This amount reflects taxes and subsidies that are included in the final price of a good or service, and therefore represents the portion of national income that is directly paid to the government. The capital consumption allowance (CCA) accounts for the wear and tear or depreciation that occurs in capital stock during the production process. It represents the amount that must be reinvested by the company in the business to maintain current productivity levels. You should think of profits + CCA as the amount earned by capital. Personal income = National income − Indirect business taxes − Corporate income taxes − Undistributed corporate profits + Transfer payments … (Equation 2) Personal disposable income = Personal income − Personal taxes … (Equation 3) Personal disposable income = Household consumption + Household saving 28 … (Equation 4) © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Aggregate Output, Price, And Economic Growth Household saving = Personal disposable income − Consumption expenditures − Interest paid by consumers to businesses − Personal transfer payments to foreigners … (Equation 5) Business sector saving = Undistributed corporate profits + Capital consumption allowance … (Equation 6) GDP = Household consumption + Total private sector saving + Net taxes The equality of expenditure and income S = I + (G − T) + ( X − M) … (Equation 7) The IS Curve (Relationship between Income and the Real Interest Rate) Disposable income = GDP − Business saving − Net taxes S − I = (G − T) + ( X − M) … (Equation 7) The LM Curve Quantity theory of money: MV = PY The quantity theory equation can also be written as: M/P and MD/P = kY where: k = I/V M = Nominal money supply MD = Nominal money demand MD/P is referred to as real money demand and M/P is real money supply. Equilibrium in the money market requires that money supply and money demand be equal. Money market equilibrium: M/P = RMD Solow (neoclassical) growth model Y = AF(L,K) where: Y = Aggregate output L = Quantity of labor K = Quantity of capital A = Technological knowledge or total factor productivity (TFP) © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 29 Aggregate Output, Price, And Economic Growth Growth accounting equation Growth in potential GDP = Growth in technology + WL (Growth in labor) + WK (Growth in capital) Growth in per capital potential GDP = Growth in technology + WK (Growth in capital-labor ratio) Measures of Sustainable Growth Labor productivity = Real GDP/Aggregate hours Potential GDP = Aggregate hours × Labor productivity This equation can be expressed in terms of growth rates as: Potential GDP growth rate = Long‐term growth rate of labor force + Long‐term labor productivity growth rate 30 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Understanding Business Cycles Understanding Business Cycles Unit labor cost (ULC) is calculated as: ULC = W/O where: O = Output per hour per worker W = Total labor compensation per hour per worker © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 31 Monetary And Fiscal Policy Monetary And Fiscal Policy Required reserve ratio = Required reserves / Total deposits Money multiplier = 1/ (Reserve requirement) The Fischer effect states that the nominal interest rate (RN) reflects the real interest rate (RR) and the expected rate of inflation (IIe). R N = R R + Πe The Fiscal Multiplier Ignoring taxes, the multiplier can also be calculated as: ○○ 1 1 = = 10 (1 − MPC) (1 − 0.9) Assuming taxes, the multiplier can also be calculated as: 1 [1 − MPC(1 − t)] 32 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. International Trade And Capital Flows International Trade And Capital Flows Balance of Payment Components A country’s balance of payments is composed of three main accounts: • The current account balance largely reflects trade in goods and services. • The capital account balance mainly consists of capital transfers and net sales of non‐produced, non‐financial assets. • The financial account measures net capital flows based on sales and purchases of domestic and foreign financial assets. © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 33 Currency Exchange Rates Currency Exchange Rates The real exchange rate may be calculated as: Real exchange rate DC/FC = SDC/FC × (PFC /PDC ) where: SDC/FC = Nominal spot exchange rate PFC = Foreign price level quoted in terms of the foreign currency PDC = Domestic price level quoted in terms of the domestic currency The forward rate may be calculated as: This version of the formula is perhaps easiest to remember because it contains the DC term in numerator for all three components: FDC/FC, SDC/FC and (1 + rDC) FDC/FC = 1 SFC/DC × (1 + rDC ) (1 + rDC ) or FDC/FC = SDC/FC × (1 + rFC ) (1 + rFC ) Forward rates are sometimes interpreted as expected future spot rates. Ft = St +1 (St +1 ) (r − r ) − 1 = ∆%S(DC/FC)t +1 = DC FC S (1 + rFC ) Exchange Rates and the Trade Balance The Elasticities Approach Marshall-Lerner condition: ω x ε x + ω M (ε M − 1) > 0 where: ωx = Share of exports in total trade ωM = Share of imports in total trade εx = Price elasticity of demand for exports εM = Price elasticity of demand for imports 34 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Financial Reporting and Analysis Financial Reporting Mechanics Financial Reporting Mechanics 2 Exhibit 36 10, Vol 3, CFA Program Curriculum 2012 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Understanding the Income Statement Understanding the Income Statement Basic EPS Basic EPS = Net income − Preferred dividends Weighted average number of shares outstanding Diluted EPS Diluted EPS = Preferred Net income − dividends + Weighted average + shares Shares from conversion of convertible preferred shares Convertible Convertible preferred + × (1 − t ) debt dividends interest Shares from conversion of + convertible debt Shares + issuable from stock options Comprehensive Income Net income + Other comprehensive income = Comprehensive income Ending Shareholders’ Equity Ending shareholders’ equity = Beginning shareholders’ equity + Net income + Other comprehensive income − Dividends declared © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 37 Understanding the Balance Sheet Understanding the Balance Sheet Gains and Losses on Marketable Securities Balance Sheet Items recognized on the income statement Held‐to‐Maturity Securities Reported at cost or amortized cost. Interest income. Realized gains and losses. Available‐for‐Sale Securities Reported at fair value. Trading Securities Reported at fair value. Unrealized gains or losses due to changes in market values are reported in other comprehensive income within owners’ equity. Dividend income. Dividend income. Interest income. Interest income. Realized gains and losses. Realized gains and losses. Unrealized gains and losses due to changes in market values. 38 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Understanding Cash Flow Statements Understanding Cash Flow Statements Cash Flow Classification under U.S. GAAP CFO Inflows Cash collected from customers. Interest and dividends received. Proceeds from sale of securities held for trading. CFI Inflows Sale proceeds from fixed assets. Sale proceeds from long‐term investments. Outflows Cash paid to employees. Cash paid to suppliers. Cash paid for other expenses. Cash used to purchase trading securities. Interest paid. Taxes paid. Outflows Purchase of fixed assets. Cash used to acquire LT investment securities. CFF Inflows Proceeds from debt issuance. Proceeds from issuance of equity instruments. Outflows Repayment of LT debt. Payments made to repurchase stock. Dividends payments. Cash Flow Statements under IFRS and U.S. GAAP IFRS U.S. GAAP Classification of Cash Flows Interest and dividends received Interest paid CFO or CFI CFO or CFF CFO CFO Dividend paid Dividends received Taxes paid CFO or CFF CFO or CFI CFO, but part of the tax can be categorized as CFI or CFF if it is clear that the tax arose from investing or financing activities. CFF CFO CFO Bank overdraft Included as a part of cash equivalents. Not considered a part of cash equivalents and included in CFF. Direct or indirect method. The former is preferred. Direct or indirect method. The former is preferred. However, if the direct method is used, a reconciliation of net income and CFO must be included. Taxes paid should be presented separately on the cash flow statement. If taxes and interest paid are not explicitly stated on the cash flow statement, details can be provided in footnotes. Presentation Format CFO (No difference in CFI and CFF presentation) Disclosures © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 39 Understanding Cash Flow Statements Free Cash Flow to the Firm FCFF = NI + NCC + [Int * (1 − tax rate)] − FCInv − WCInv FCFF = CFO + [Int * (1 − tax rate)] − FCInv Free Cash Flow to Equity FCFE = CFO − FCInv + Net borrowing 40 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Financial Analysis Techniques Financial Analysis Techniques Inventory Turnover Inventory turnover = Cost of goods sold Average inventory Days of Inventory on Hand Days of inventory on hand (DOH) = 365 Inventory turnover Receivables Turnover Receivables turnover = Revenue Average receivables Days of Sales Outstanding Days of sales outstanding (DSO) = 365 Receivables turnover Payables Turnover Payables turnover = Purchases Average trade payables Number of Days of Payables Number of days of payables = 365 Payables turnover Working Capital Turnover Working capital turnover = Revenue Average working capital Fixed Asset Turnover Fixed asset turnover = Revenue Average fixed assets Total Asset Turnover Total asset turnover = Revenue Average total assets © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 41 Financial Analysis Techniques Current Ratio Current ratio = Current assets Current liabilities Quick Ratio Quick ratio = Cash + Short-term marketable investments + Receivables Current liabilities Cash Ratio Cash ratio = Cash + Short-term marketable investments Current liabilities Defensive Interval Ratio Defensive interval ratio = Cash + Short-term marketable investments + Receivables Daily cash expenditures Cash Conversion Cycle Cash conversion cycle = DSO + DOH − Number of days of payables Debt‐to‐Assets Ratio Debt -to-assets ratio = Total debt Total assets Debt‐to‐Capital Ratio Debt -to-capital ratio = Total debt Total debt + Shareholders’ equity Debt‐to‐Equity Ratio Debt -to-equity ratio = Total debt Shareholders’ equity Financial Leverage Ratio Financial leverage ratio = Average total assets Average total equity Interest Coverage Ratio Interest coverage ratio = 42 EBIT Interest payments © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Financial Analysis Techniques Fixed Charge Coverage Ratio Fixed charge coverage ratio = EBIT + Lease payments Interest payments + Lease payments Gross Profit Margin Gross profit margin = Gross profit Revenue Operating Profit Margin Operating profit margin = Operating profit Revenue Pretax Margin Pretax margin = EBT (earnings before tax, but after interest) Revenue Net Profit Margin Net profit margin = Net profit Revenue Return on Assets ROA = Net income Average total assets Adjusted ROA = Net income + Interest expense (1 − Tax rate) Average total assets Operating ROA = Operating income or EBIT Average total assets Return on Total Capital Return on total capital = EBIT Short-term debt + Long-term debt + Equity Return on Equity Return on equity = Net income Average total equity Return on Common Equity Return on common equity = Net income − Preferred dividends Average common equity © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 43 Financial Analysis Techniques DuPont Decomposition of ROE Net income Average shareholders’ equity ROE = 2‐Way Dupont Decomposition Net income Average total assets × Average total assets Average shareholders’ equity ROE = ↓ ↓ ROA Leverage 3‐Way Dupont Decomposition Net income Average total assets Revenue × × Revenue Average total assets Average shareholders’ equity ROE = ↓ ↓ Net profit margin ↓ Asset turnover Leverage 5‐Way Dupont Decomposition ROE = Interest burden Asset turnover ↓ ↓ Net income EBT Average total assets EBIT Revenue × × × × EBT EBIT Revenue Average total assets Avg. shareholders’ equity ↓ ↓ ↓ Tax burden EBIT margin Leverage Price‐to‐Earnings Ratio P /E = Price per share Earnings per share Price to Cash Flow P /CE = Price per share Cash flow per share Price to Sales P /S = Price per share Sales per share Price to Book Value P /BV = 44 Price per share Book value per share © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Financial Analysis Techniques Per Share Ratios Cash flow from operations Average number of shares outstanding Cash flow per share = EBITDA per share = EBITDA Average number of shares outstanding Dividends per share = Common dividends declared Weighted average number of ordinary shares Dividend Payout Ratio Dividend payout ratio = Common share dividends Net income attributable to common shares Retention Rate Retention Rate = Net income attributable to common shares − Common share dividends Net income attributable to common shares Growth Rate Sustainable growth rate = Retention rate × ROE © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 45 Inventories Inventories LIFO versus FIFO (with rising prices and stable inventory levels.) LIFO versus FIFO when Prices are Rising LIFO COGS Higher Income before taxes Lower Income taxes Lower Net income Lower Cash flow Higher EI Lower Working capital Lower FIFO Lower Higher Higher Higher Lower Higher Higher Type of Ratio Profitability ratios NP and GP margins Effect on Numerator Income is lower under LIFO because COGS is higher Effect on Denominator Sales are the same under both Debt-to-equity Same debt levels Lower equity under LIFO Higher under LIFO Current ratio Current assets are lower under LIFO because EI is lower Current liabilities are the same Lower under LIFO Quick ratio Assets are higher as a result of lower taxes paid Current liabilities are the same Higher under LIFO Inventory turnover COGS is higher under LIFO Average inventory is Higher under LIFO lower under LIFO Total asset turnover Sales are the same Lower total assets under LIFO Effect on Ratio Lower under LIFO Higher under LIFO The LIFO Method and the LIFO Reserve: EI FIFO = EI LIFO + LR where LR = LIFO Reserve COGSfifo = COGSlifo − (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 × (1 − Tax rate) 46 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Inventories When converting from LIFO to FIFO assuming rising prices: Equity (retained earnings) increase by: LIFO Reserve × (1 − Tax rate) Liabilities (deferred taxes) increase by: LIFO Reserve × (Tax rate) Current assets (inventory) increase by: LIFO Reserve © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 47 Long-Lived Assets Long-Lived Assets Financial Statement Effects of Capitalizing versus Expensing Initially when the cost is capitalized In future periods when the asset is depreciated or amortized When the cost is expensed Effect on Financial Statements • Noncurrent assets increase. • Cash flow from investing activities decreases. • • • • • • • • Net income (first year) Net income (future years) Total assets Shareholders’ equity Cash flow from operations Cash flow from investing Income variability Debt-to-equity 48 Noncurrent assets decrease. Net income decreases. Retained earnings decrease. Equity decreases. Net income decreases by the entire after‐tax amount of the cost. No related asset is recorded on the balance sheet and therefore, no depreciation or amortization expense is charged in future periods. Operating cash flow decreases. Expensed costs have no financial statement impact in future years. Capitalizing Higher Lower Higher Higher Higher Lower Lower Lower Expensing Lower Higher Lower Lower Lower Higher Higher Higher © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Long-Lived Assets Straight Line Depreciation Original cost − Salvage value Depreciable life Depreciation expense = Accelerated Depreciation DDB depreciation in Year X = 2 × Book value at the beginning of Year X Depreciable life Estimated Useful Life Estimated useful life = Gross investment in fixed assets Annual depreciation expense Average Cost of Asset Average age of asset = Accumulated depreciation Annual depreciation expense Remaining Useful Life Remaining useful life = Net investment in fixed assets Annual depreciation expense 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. © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 49 Income Taxes Income Taxes Effective Tax rate Effective tax rate = Income tax expense Pretax income Income Tax Expense Income tax expense = Taxes Payable + Change in DTL − Change in DTA Treatment of Temporary Differences Balance Sheet Item Asset Asset Liability Liability Carrying Value versus Tax Base Carrying amount is greater. Tax base is greater. Carrying amount is greater. Tax base is greater. Results in… DTL DTA DTA DTL Income Tax Accounting under IFRS versus U.S. GAAP IFRS ISSUE SPECIFIC TREATMENTS Revaluation of fixed assets Recognized in equity as and intangible assets. deferred taxes. 50 U.S. GAAP Revaluation is prohibited. Treatment of undistributed profit from investment in subsidiaries. Recognized as deferred taxes except when the parent company is able to control the distribution of profits and it is probable that temporary differences will not reverse in future. No recognition of deferred taxes for foreign subsidiaries that fulfill indefinite reversal criteria. No recognition of deferred taxes for domestic subsidiaries when amounts are tax‐free. Treatment of undistributed profit from investments in joint ventures. Recognized as deferred taxes except when the investor controls the sharing of profits and it is probable that there will be no reversal of temporary differences in future. No recognition of deferred taxes for foreign corporate joint ventures that fulfill indefinite reversal criteria. Treatment of undistributed profit from investments in associates. Recognized as deferred Deferred taxes are taxes except when the recognized from temporary investor controls the sharing differences. of profits and it is probable that there will be no reversal of temporary differences in future. © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Income Taxes IFRS DEFERRED TAX MEASUREMENT Tax rates. Tax rates and tax laws enacted or substantively enacted. Deferred tax asset recognition. Recognized if it is probable that sufficient taxable profit will be available in the future. U.S. GAAP Only enacted tax rates and tax laws are used. Deferred tax assets are recognized in full and then reduced by a valuation allowance if it is likely that they will not be realized. DEFERRED TAX PRESENTATION Offsetting allowed only if Offsetting of deferred tax Same as in IFRS. assets and liabilities. the entity has right to legally enforce it and the balance is related to a tax levied by the same authority. Balance sheet classification. Classified on balance sheet as net noncurrent with supplementary disclosures. Classified as either current or noncurrent based on classification of underlying asset and liability. © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 51 Non-Current (Long-Term) Liabilities Non-Current (Long-Term) Liabilities Income Statement Effects of Lease Classification Income Statement Item Operating expenses Nonoperating expenses EBIT (operating income) Total expenses‐ early years Total expenses‐ later years Net income‐ early years Net income‐ later years Finance Lease Lower Higher Higher Higher Lower Lower Higher Operating Lease Higher Lower Lower Lower Higher Higher Lower Balance Sheet Effects of Lease Classification Balance Sheet Item Assets Current liabilities Long term liabilities Total cash Capital Lease Higher Higher Higher Same Operating Lease Lower Lower Lower Same Cash Flow Effects of Lease Classification CF Item CFO CFF Total cash flow Capital Lease Higher Lower Same Operating Lease Lower Higher Same Impact of Lease Classification on Financial Ratios Ratio Asset turnover Numerator under Finance Lease Sales‐ same Denominator Ratio Better or under Finance Worse under Lease Effect on Ratio Finance Lease Assets‐ higher Lower Worse Return on assets* Net income lower in early years 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 Return on equity* Net income lower in early years Equity same Lower Worse * In early years of the lease agreement. 52 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Non-Current (Long-Term) Liabilities Financial Statement Effects of Lease Classification from Lessor’s Perspective Total net income Net income (early years) Taxes (early years) Total CFO Total CFI Total cash flow Financing Lease Same Higher Higher Lower Higher Same Operating Lease Same Lower Lower Higher Lower Same Definitions of Commonly Used Solvency Ratios Solvency Ratios Description Numerator Denominator Debt‐to‐assets ratio Expresses the percentage of total assets financed by debt Total debt Total assets Debt‐to‐capital ratio Measures the percentage of a company’s total capital (debt + equity) financed by debt. Total debt Total debt + Total shareholders’ equity Debt‐to‐equity ratio Measures the amount of debt financing relative to equity financing Total debt Total shareholders’ equity Financial leverage ratio Measures the amount of total assets supported by one money unit of equity Average total assets Average shareholders’ equity Leverage Ratios Coverage Ratios Interest coverage ratio Measures the number of times a EBIT company’s EBIT could cover its interest payments. Interest payments Fixed charge coverage ratio Measures the number of times a company’s earnings (before interest, taxes and lease payments) can cover the company’s interest and lease payments. Interest payments + Lease payments EBIT + Lease payments © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 53 Financial Reporting Quality Financial Reporting Quality Relationship between Financial Reporting Quality and Earnings Quality Financial Reporting Quality Low Earnings High (Results) Quality LOW financial reporting quality impedes assessment of earnings quality and Low impedes valuation. 54 High HIGH financial reporting quality enables assessment. HIGH earnings quality increases company value. HIGH financial reporting quality enables assessment. LOW earnings quality decreases company value. © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Corporate Finance Capital Budgeting Capital Budgeting Net Present Value (NPV) n CFt − Outlay (1 + r) t t =1 NPV = ∑ where: CFt = after‐tax cash flow at time, t. r = required rate of return for the investment. This is the firm’s cost of capital adjusted for the risk inherent in the project. Outlay = investment cash outflow at t = 0. Internal Rate of Return (IRR) n CFt ∑ (1 + IRR) t = Outlay t =1 n CF t ∑ (1 + IRR) t − Outlay = 0 t =1 Average Accounting Rate of Return (AAR) AAR = Average net income Average book value Profitability Index PI = 56 PV of future cash flows NPV = 1+ Initial investment Initial investment © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Cost of Capital Cost of Capital Weighted Average Cost of Capital WACC = (wd )(rd )(1 − t) + (wp )(rp ) + (we )(re ) where: wd = Proportion of debt that the company uses when it raises new funds rd = Before‐tax marginal cost of debt t = Company’s marginal tax rate wp = Proportion of preferred stock that the company uses when it raises new funds rp = Marginal cost of preferred stock we = Proportion of equity that the company uses when it raises new funds re = Marginal cost of equity To Transform Debt‐to‐equity Ratio into a Component’s Weight D E = D =w d D 1+ E D + E wd + we = 1 Valuation of Bonds n PMT FV P0 = ∑ t + n t =1 rd rd 1 + 1 + 2 2 where: P0 = current market price of the bond. PMTt = interest payment in period t. rd = yield to maturity on BEY basis. n = number of periods remaining to maturity. FV = Par or maturity value of the bond. Valuation of Preferred Stock Vp = Dp rp where: Vp = current value (price) of preferred stock. Dp = preferred stock dividend per share. rp = cost of preferred stock. © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 57 Cost of Capital Required Return on a Stock Capital Asset Pricing Model re = R F + β i [E(R M ) − R F ] where: [E(RM) − Rf] = Equity risk premium. RM = Expected return on the market. βi = Beta of stock. Beta measures the sensitivity of the stock’s returns to changes in market returns. RF = Risk‐free rate. re = Expected return on stock (cost of equity). Dividend Discount Model P0 = D1 re − g where: P0 = current market value of the security. D1 = next year’s dividend. re = required rate of return on common equity. g = the firm’s expected constant growth rate of dividends. Rearranging the above equation gives us a formula to calculate the required return on equity: re = D1 +g P0 Sustainable Growth Rate D g = 1 − × ( ROE ) EPS Where (1 − (D/EPS)) = Earnings retention rate Bond Yield plus Risk Premium Approach re = rd + risk premium 58 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Cost of Capital To Unlever the beta β ASSET 1 = β EQUITY 1 + (1 − t ) D E To Lever the beta D β PROJECT = β ASSET 1 + (1 − t ) E Country Risk Premium re = R F + β [E(R M ) − R F + CRP] Country risk = premium Break point = Sovereign yield × spread Annualized standard deviation of equity index Annualized standard deviation of sovereign bond market in terms of the developed market currency Amount of capital at which a component’s cost of capital changes Proportion of new capital raised from the component © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 59 Measures of Leverage Measures of Leverage Degree of Operating Leverage DOL = Percentage change in operating income Percentage change in units sold DOL = Q × (P − V) Q × (P − V) − F where: Q = Number of units sold P = Price per unit V = Variable operating cost per unit F = Fixed operating cost Q × (P − V) = Contribution margin (the amount that units sold contribute to covering fixed costs) (P − V) = Contribution margin per unit Degree of Financial Leverage DFL = Percentage change in net income Percentage change in operating income DFL = [Q(P − V) − F](1 − t) [Q(P − V) − F] = [Q(P − V) − F − C](1 − t) [Q(P − V) − F − C] where: Q = Number of units sold P = Price per unit V = Variable operating cost per unit F = Fixed operating cost C = Fixed financial cost t = Tax rate 60 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Measures of Leverage Degree of Total Leverage Percentage change in net income Percentage change in the number of units sold DTL = DTL = DOL × DFL DTL = Q × (P − V) [Q(P − V) − F − C] where: Q = Number of units produced and sold P = Price per unit V = Variable operating cost per unit F = Fixed operating cost C = Fixed financial cost Break point PQ = VQ + F + C where: P = Price per unit Q = Number of units produced and sold V = Variable cost per unit F = Fixed operating costs C = Fixed financial cost The breakeven number of units can be calculated as: Q BE = F+C P−V Operating breakeven point PQ OBE = VQ OBE + F Q OBE = F P−V © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 61 Working Capital Management Working Capital Management Current Ratio = Quick Ratio = Current assets Current liabilities Cash + Short term marketable investments + Receivables Current liabilities Accounts receivable turnover = Credit sales Average receivables Number of days of receivables = = Inventory turnover = Accounts receivable Sales on credit / 365 Cost of goods sold Average inventory Number of days of inventory = = Payables turnover = Accounts receivable Average days sales on credit Inventory Average day’s cost of goods sold Inventory Cost of goods sold / 365 Purchases Average trade payables Number of days of payables = = Accounts payables Average day’s purchases Accounts payables Purchases / 365 Purchases = Ending inventory + COGS − Beginning inventory Operating cycle = Number of days of inventory + Number of days of receivables Net operating cycle = Number of days of inventory + Number of days of receivables − Number of days of payables 62 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Working Capital Management 360 Face value − price 360 Money market yield = = Holding period yield × × Days Days Price 365 Face value − price 365 Bond equivalent yield = = Holding period yield × × Days Days Price 360 Face value − price 360 Discount basis yield = = % discount × × Days Days Face value % Discount = Face value − Price Price Inventory turnover = Cost of goods sold Average inventory Number of days of inventory = Inventory Average days cost of goods sold = Inventory Cost of goods sold / 365 = 365 Inventory turnover 365 Discount Implicit rate = Cost of trade credit = 1 + 1 − Discount Number of days beyond discount period −1 Accounts payable Average day’s purchases Accounts payable 365 = Purchases / 365 Payables turnover Number of days of payables = Line of credit cost = Interest + Commitment fee Loan amount Banker’s acceptance cost = Interest Interest = Net proceeds Loan amount − Interest Interest + Dealer’s commission + Backup costs Loan amount − Interest © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 63 Portfolio Management Risk Management: An Introduction Risk Management: An Introduction Figure 1-1: The Risk Management Framework in an Enterprise Risk Governance Risk Drivers Board Management Goals Strategies Risk Tolerance Establish Risk Management Infrastructure MODIFY Risk Mitigation (Allocate to) Risky Activities & Management Identify Risks Measure Risks NO Risk Budgeting Risk Exposures Policies & Processes Risks in Line? Montitor Risks YES Reports (Communications) Strategic Analysis © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 65 Portfolio Risk and Return: Part I Portfolio Risk and Return: Part I Holding Period Return Pt − Pt −1 + D t Pt − Pt −1 D t = + = Capital gain + Dividend yield Pt −1 Pt −1 Pt −1 P + DT = T −1 P0 R= where: Pt = Price at the end of the period Pt-1 = Price at the beginning of the period Dt = Dividend for the period Holding Period Returns for more than One Period R = [(1 + R1 ) × (1 + R 2 ) × . . . × (1 + R n )] − 1 where: R1, R2, . . . , Rn are sub‐period returns Geometric Mean Return R = {[(1 + R1 ) × (1 + R 2 ) × . . . × (1 + R n )]1/n} − 1 Annualized Return rannual = (1 + rperiod )n − 1 where: r = Return on investment n = Number of periods in a year 66 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Portfolio Risk and Return: Part I Portfolio Return R p = w1R1 + w2 R 2 where: Rp = Portfolio return w1 = Weight of Asset 1 w2 = Weight of Asset 2 R1 = Return of Asset 1 R2 = Return of Asset 2 Variance of a Single Asset T ∑ (R t − µ)2 t =1 σ2 = T where: Rt = Return for the period t T = Total number of periods μ = Mean of T returns Variance of a Representative Sample of the Population T s2 = ∑ (R t − R)2 t =1 T −1 where: R = mean return of the sample observations s2 = sample variance Standard Deviation of an Asset T σ= T ∑ (R t − µ)2 t =1 s= T ∑ (R t − R)2 t =1 T −1 Variance of a Portfolio of Assets σ 2P = N ∑ wi w jCov(R i ,R j ) i, j=1 N σ 2P = ∑ w2i Var(R i ) + i =1 N ∑ i, j=1, i ≠ j wi w jCov(R i ,R j ) © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 67 Portfolio Risk and Return: Part I Standard Deviation of a Portfolio of Two Risky Assets σ p = w12 σ12 + w22 σ 22 + 2w1w2 σ1σ 2ρ1,2 or w12 σ12 + w22 σ 22 + 2w1w2 Cov1,2 Utility Function 1 U = E(R) − Aσ 2 2 where: U = Utility of an investment E(R) = Expected return σ2 = Variance of returns A = Additional return required by the investor to accept an additional unit of risk. 68 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Portfolio Risk and Return: Part II Portfolio Risk and Return: Part II Capital Allocation Line The CAL has an intercept of RFR and a constant slope that equals: [E(R i ) − RFR] σi Expected Return on portfolios that lie on CML E(R p ) = w1R f + (1 − w1 ) E(R m ) Variance of portfolios that lie on CML σ 2 = w12 σ 2f + (1 − w1 )2 σ 2m + 2w1 (1 − w1 )Cov(R f , R m ) Equation of CML E(R p ) = R f + E(R m ) − R f × σp σm where: y‐intercept = Rf = risk‐free rate slope = E(R m ) − R f = market price of risk. σm Systematic and Nonsystematic Risk Total Risk = Systematic risk + Unsystematic risk Return‐Generating Models k k j=1 j= 2 E(R i ) − R f = ∑ β ij E(Fj ) = β i1[E(R m ) − R f ] + ∑ β ij E(Fj ) The Market Model R i = α i + βi R m + ei Calculation of Beta βi = Cov(R i ,R m ) ρi,m σ i σ m ρi,m σ i = = σm σ 2m σ 2m © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 69 Portfolio Risk and Return: Part II The Capital Asset Pricing Model E(R i ) = R f + β i [E(R m ) − R f ] Sharpe ratio Sharpe ratio = Rp − Rf σp Treynor ratio Treynor ratio = Rp − Rf βp M‐squared (M2) M2 = (R p − R f ) σm − (R m − R f ) σp Jensen’s alpha α p = R p − [R f + β p (R m − R f )] Security Characteristic Line R i − R f = α i + β i (R m − R f ) 70 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Equity Market Organization and Structure Market Organization and Structure The price at which an investor who goes long on a stock receives a margin call is calculated as: P0 × 72 (1 − Initial margin) (1 − Maintenance margin) © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Security Market Indices Security Market Indices The value of a price return index is calculated as follows: N VPRI = ∑ ni Pi i =1 D where: VPRI = Value of the price return index ni = Number of units of constituent security i held in the index portfolio N = Number of constituent securities in the index Pi = Unit price of constituent security i D = Value of the divisor Price Return The price return of an index can be calculated as: PR I = VPRI1 − VPRI0 VPRI0 where: PRI = Price return of the index portfolio (as a decimal number) VPRI1 = Value of the price return index at the end of the period VPRI0 = Value of the price return index at the beginning of the period The price return of each constituent security is calculated as: PR i = Pi1 − Pi0 Pi0 where: PRi = Price return of constituent security i (as a decimal number) Pi1 = Price of the constituent security i at the end of the period Pi0 = Price of the constituent security i at the beginning of the period The price return of the index equals the weighted average price return of the constituent securities. It is calculated as: PRI = w1PR1 + w2PR2 + . . . + wNPRN where: PRI = Price return of the index portfolio (as a decimal number) PRi = Price return of constituent security i (as a decimal number) wi = Weight of security i in the index portfolio N = Number of securities in the index © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 73 Security Market Indices Total Return The total return of an index can be calculated as: TR I = VPRI1 − VPRI0 + Inc I VPRI0 where: TRI = Total return of the index portfolio (as a decimal number) VPRI1 = Value of the total return index at the end of the period VPRI0 = Value of the total return index at the beginning of the period IncI = Total income from all securities in the index held over the period The total return of each constituent security is calculated as: TR i = P1i − P0i + Inc i P0i where: TRi = Total return of constituent security i (as a decimal number) P1i = Price of constituent security i at the end of the period P0i = Price of constituent security i at the beginning of the period Inci = Total income from security i over the period The total return of the index equals the weighted average total return of the constituent securities. It is calculated as: TRI = w1TR1 + w2TR2 + . . . + wNTRN where: TRI = Total return of the index portfolio (as a decimal number) TRi = Total return of constituent security i (as a decimal number) wi = Weight of security i in the index portfolio N = Number of securities in the index Calculation of Index Returns over Multiple Time Periods Given a series of price returns for an index, the value of a price return index can be calculated as: VPRIT = VPRI0 (1 + PR I1 ) (1 + PR I2 ) . . . (1 + PR IT ) where: VPRI0 = Value of the price return index at inception VPRIT = Value of the price return index at time t PRIT = Price return (as a decimal number) on the index over the period 74 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Security Market Indices Similarly, the value of a total return index may be calculated as: VTRIT = VTRI0 (1 + TR I1 ) (1 + TR I2 ) . . . (1 + TR IT ) where: VTRI0 = Value of the index at inception VTRIT = Value of the index at time t TRIT = Total return (as a decimal number) on the index over the period Price Weighting wPi = Pi N ∑ Pi i =1 Equal Weighting wEi = 1 N where: wi = Fraction of the portfolio that is allocated to security i or weight of security i N = Number of securities in the index Market‐Capitalization Weighting wiM = Q i Pi N ∑ Q jPj j=1 where: wi = Fraction of the portfolio that is allocated to security i or weight of security i Qi = Number of shares outstanding of security i Pi = Share price of security i N = Number of securities in the index The float‐adjusted market‐capitalization weight of each constituent security is calculated as: wiM = fi Q i Pi N ∑ f jQ jPj j=1 where: fi = Fraction of shares outstanding in the market float wi = Fraction of the portfolio that is allocated to security i or weight of security i Qi = Number of shares outstanding of security i Pi = Share price of security i N = Number of securities in the index © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 75 Security Market Indices Fundamental Weighting wFi = Fi N ∑ Fj j=1 where: Fi = A given fundamental size measure of company i 76 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Overview of Equity Securities Overview of Equity Securities Return Characteristics of Equity Securities Total Return, Rt = (Pt – Pt−1 + Dt) / Pt−1 where: Pt‐1 = Purchase price at time t − 1 Pt = Selling price at time t Dt = Dividends paid by the company during the period Accounting Return on Equity ROE t = NI t NI t = Average BVE t (BVE t + BVE t −1 )/2 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 77 Equity Valuation Concepts and Basic Tools Equity Valuation Concepts and Basic Tools Dividend Discount Model (DDM) Value = D1 D2 D∞ 1 + 2 ++ (1 + k e )∞ (1 + k e ) (1 + k e ) n Dt t t =1 (1 + k e ) Value = ∑ One year holding period: Value = dividend to be received year-end price + (1 + k e )1 (1 + k e )1 Multiple‐Year Holding Period DDM V= D1 D2 Pn 1 + 2 ++ (1 + k e )n (1 + K e ) (1 + k e ) where: Pn = Price at the end of n years. Infinite Period DDM (Gordon Growth Model) PV0 = D0 (1 + gc )1 D0 (1 + gc )2 D 0 (1 + gc )3 D0 (1 + gc )∞ + + + + (1 + k e )∞ (1 + k e )1 (1 + k e )2 (1 + k e )3 This equation simplifies to: PV = D0 (1 + gc )1 D1 1 = k e − gc (k e − gc ) The long‐term (constant) growth rate is usually calculated as: gc = RR × ROE 78 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Equity Valuation Concepts and Basic Tools Multi‐Stage Dividend Discount Model Value = D1 D2 Dn Pn n + 1 + 2 ++ (1 + k e ) (1 + k e )n (1 + k e ) (1 + k e ) where: Dn +1 Pn = k e − gc Dn = Last dividend of the supernormal growth period Dn+1 = First dividend of the constant growth period The Free‐Cash‐Flow‐to‐Equity (FCFE) Model FCFE = CFO − FC Inv + Net borrowing Analysts may calculate the intrinsic value of the company’s stock by discounting their projections of future FCFE at the required rate of return on equity. ∞ FCFE t t t =1 (1 + k e ) V0 = ∑ Value of a Preferred Stock When preferred stock is non‐callable, non‐convertible, has no maturity date and pays dividends at a fixed rate, the value of the preferred stock can be calculated using the perpetuity formula: V0 = D0 r For a non‐callable, non‐convertible preferred stock with maturity at time, n, the value of the stock can be calculated using the following formula: n Dt F t + (1 + r) (1 + r)n t =1 V0 = ∑ where: V0 = value of preferred stock today (t = 0) Dt = expected dividend in year t, assumed to be paid at the end of the year r = required rate of return on the stock F = par value of preferred stock © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 79 Equity Valuation Concepts and Basic Tools Price Multiples P0 D1 /E1 = E1 r−g Price to cash flow ratio = Market price of share Cash flow per share Price to sales ratio = Market price per share Net sales per share Price to sales ratio = Market value of equity Total net sales P/BV = Current market price of share Book value per share P/BV = Market value of common shareholders’ equity Book value of common shareholders’ equity where: Book value of common shareholders’ equity = (Total assets – Total liabilities) – Preferred stock Enterprise Value Multiples EV/EBITDA where: EV = Enterprise value and is calculated as the market value of the company’s common stock plus the market value of outstanding preferred stock if any, plus the market value of debt, less cash and short term investments (cash equivalents). 80 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Fixed Income Fixed-Income Securities: Defining Elements Fixed-Income Securities: Defining Elements Bond Coupon Coupon = Coupon rate × Par value Coupon Rate (Floating) Coupon Rate = Reference rate + Quoted margin Coupon Rate (Inverse Floaters) Coupon rate = K − L × (Reference rate) Callable Bonds Value of callable bond = Value of non‐callable bond − Value of embedded call option Value of embedded call option = Value of non‐callable bond − Value of callable bond Putable Bonds Value of putable bond = Value of non‐putable bond + Value of embedded put option Value of embedded put option = Value of putable bond − Value of non‐putable bond Traditional Analysis of Convertible Securities Conversion value = Market price of common stock × Conversion ratio Market conversion price = Market price of convertible security Conversion ratio Market conversion premium per share = Market conversion price − Current market price Market conversion premium ratio = Premium payback period = Market conversion premium per share Market price of common stock Market conversion premium per share Favorable income differential per share Favorable income differential per share = Premium over straight value = 82 Coupon interest − (Conversion ratio × Common stock dividend per share) Conversion ratio Market price of convertible bond −1 Straight value © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Introduction to Fixed-Income Valuation Introduction to Fixed-Income Valuation Pricing Bonds with Spot Rates PV = PMT PMT PMT + FV 1 + 2 +…+ (1 + Z N ) N (1 + Z1 ) (1 + Z 2 ) z1 = Spot rate for Period 1 z2 = Spot rate for Period 2 zN = Spot rate for Period N Flat Price, Accrued Interest and the Full Price Figure: Valuing a Bond between Coupon‐Payment Dates PV Full = PV Flat + AI AI = t/T × PMT PV Full = PV × (1 + r) t/T Semiannual bond basis yield or semiannual bond equivalent yield 1 + SAR M M M SAR N = 1 + N N Important: What we refer to as stated annual rate (SAR) is referred to in the curriculum as APR or annual percentage rate. We stick to SAR to keep your focus on a stated annual rate versus the effective annual rate. Just remember that if you see an annual percentage rate on the exam, it refers to the stated annual rate. Current yield Current yield = Annual cash coupon payment Bond price © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 83 Introduction to Fixed-Income Valuation Option‐adjusted price Value of non‐callable bond (option‐adjusted price) = F lat price of callable bond + Value of embedded call option Pricing formula for money market instruments quoted on a discount rate basis: Days PV = FV × 1 − × DR year Year FV − PV DR = × Days FV Pricing formula for money market instruments quoted on an add‐on rate basis: PV= FV 1 + Days × AOR Year Year FV − PV AOR = × Days PV Yield Spreads over the Benchmark Yield Curve PV = • • PMT PMT PMT + FV 1 + 2 + ...+ (1 + z N + Z) N (1 + z1 + Z) (1 + z 2 + Z) The benchmark spot rates z1, z2, zN are derived from the government yield curve (or from fixed rates on interest rate swaps). Z refers to the z‐spread per period. It is constant for all time periods. Option‐adjusted Spread (OAS) OAS = z‐spread − Option value (bps per year) 84 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Introduction to Asset-Backed Securities Introduction to Asset-Backed Securities Parties to the Securitization Party Seller Issuer/Trust Servicer SMMt = Description Originates the loans and sells loans to the SPV The SPV that buys the loans from the seller and issues the asset-backed securities Services the loans Party in Illustration ABC Company SPV Servicer Prepayment in month t Beginning mortgage balance for month t − Scheduled principal payment in month t © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 85 Understanding Fixed-Income Risk and Return Understanding Fixed-Income Risk and Return Macaulay Duration 1 + r 1 + r + [N × (c − r)] MacDur = − − (t/T) c × [(1 + r) N − 1] + r r c = Coupon rate per period (PMT/FV) Modified Duration ModDur = MacDur 1+ r Modified duration has a very important application in risk management. It can be used to estimate the percentage price change for a bond in response to a change in its yield‐to‐ maturity. %∆PV Full ≈ − AnnModDur × ∆Yield If Macaulay duration is not already known, annual modified duration can be estimated using the following formula: ApproxModDur = (PV− ) − (PV+ ) 2 × ( ∆Yield) × (PV0 ) We can also use the approximate modified duration (ApproxModDur) to estimate Macaulay duration (ApproxMacDur) by applying the following formula: ApproxMacDur = ApproxModDur × (1 + r) Effective Duration EffDur = 86 (PV− ) − (PV+ ) 2 × ( ∆Curve) × (PV0 ) © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Understanding Fixed-Income Risk and Return Duration of a Bond Portfolio Portfolio duration = w1D1 + w2 D2 +…+ w N D N Annual ModDur = Annual MacDur 1+ r Money Duration MoneyDur = AnnModDur × PVFull The estimated (dollar) change in the price of the bond is calculated as: ΔPVFull = – MoneyDur × ΔYield Price Value of a Basis Point PVBP = (PV− ) − (PV+ ) 2 A related statistic is basis point value (BPV), which is calculated as: BPV = MoneyDur × 0.0001 (1 bps expressed as a decimal) Annual Convexity ApproxCon = (PV− ) + (PV+ ) − [2 × (PV0 )] ( ∆Yield)2 × (PV0 ) Once we have an estimate for convexity, we can estimate the percentage change in a bond’s full price as: 1 %∆PV Full ≈ (− AnnModDur × ∆Yield) + × AnnConvexity × ( ∆Yield)2 2 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 87 Understanding Fixed-Income Risk and Return Money convexity 1 ∆PV Full ≈ (− MoneyDur × ∆Yield) + × MoneyCon × ( ∆Yield)2 2 Effective convexity EffCon = [(PV− ) + (PV+ )] − [2 × (PV0 )] ( ∆Curve)2 × (PV0 ) Yield Volatility 1 %∆PV Full ≈ (− AnnModDur × ∆Yield) + × AnnConvexity × ( ∆Yield)2 2 Duration Gap Duration gap = Macaulay duration − Investment horizon 88 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Fundamentals of Credit Analysis Fundamentals of Credit Analysis Expected Loss Expected loss = Default probability × 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 ≈ − Modified duration × ∆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 ≈ −(MDur × ∆Spread) + (1/2 × Convexity × ∆Spread 2 ) © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 89 Derivatives Basics of Derivative Pricing and Valuation Basics of Derivative Pricing and Valuation Fundamental value of an asset (S0) that incurs costs (θ) and generates benefits (γ): E(ST ) S0 = T −θ+ γ (1 + r + λ) Arbitrage and Replication: Asset + Derivative = Risk-free asset Asset − Risk-free asset = − Derivative Derivative − Risk-free asset = − Asset Forward Contract Payoffs: Long position Short position ST > F(0,T) ST – F(0,T) (Positive payoff) ST < F(0,T) ST – F(0,T) (Negative payoff) –[ST – F(0,T)] (Negative payoff) –[ST – F(0,T)] (Positive payoff) Forward price: F(0,T) = S0 (1 + r)T F(0,T) = (S0 − γ + θ)(1 + r)T or F(0,T) = S0 (1 + r)T − ( γ − θ)(1 + r)T * Note that benefits (γ) and costs (θ) are expressed in terms of present value. Value of a forward contract: Vt (0,T) = St − [F(0,T) / (1 + r)T− t ] Vt (0,T) = St − ( γ − θ)(1 + r) t − [F(0,T) / (1 + r)T− t ] Time At initiation Long Position Value Zero, as the contract is priced to prevent arbitrage Short Position Value Zero, as the contract is priced to prevent arbitrage During life of the contract At expiration F(0,T) St − T− t (1+r ) F(0,T) T − t − St (1+r ) ST – F(0,T) F(0,T) – ST © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 91 Basics of Derivative Pricing and Valuation Net payment made (received) by the fixed‐rate payer on a swap: Net fixed rate payment t = [Swap fixed rate − (LIBOR t −1 + spread)]* (No. of days/360)* NP Call Option Payoffs Option Position Call option holder Call option writer Descriptions Choice to buy the underlying asset for X Obligation to sell the underlying asset for X if the option holder chooses to exercise the option Payoff ST > X Option holder exercises the option ST – X ST < X Option holder does not exercise the option 0 – (ST – X) 0 Moneyness and Exercise Value of a Call Option Moneyness In‐the‐money At‐the‐money Out‐of‐the‐money Current Market Price (St) versus Exercise Price (X) St is greater than X St equals X St is less than X Intrinsic Value Max [0, (St – X)] St – X 0 0 Put Option Payoffs Option Position Descriptions Put option holder Choice to sell the underlying asset for X Obligation to buy the underlying asset for X if the option holder chooses to exercise the option Put option writer 92 Payoff ST > X ST < X Option holder Option holder exercises the does not exercise option the option 0 X – ST – (X – ST) 0 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Basics of Derivative Pricing and Valuation Moneyness and Exercise Value of a Put Option Moneyness In‐the‐money At‐the‐money Out‐of‐the‐money Current Market Price (St) versus Exercise Price (X) St is less than X St equals X St is greater than X Intrinsic Value Max [0, (X – St)] X – St 0 0 Fiduciary Call and Protective Put Payoffs Security Call option Zero coupon bond Fiduciary call payoff Value if ST > X ST – X X ST Value if ST < X Zero X X Put option Stock Protective put payoff Zero ST ST X – ST ST X Put‐Call Parity c0 + X = p 0 + S0 (1 + R F )T Combining Portfolios to Make Synthetic Securities Strategy fiduciary call Consisting of Value long call + X long bond c 0 + long call long call c0 = long put long put p0 = long underlying asset long bond long underlying asset long bond S0 = X (1 + R F )T = (1 + R F )T Equals Strategy Consisting of = Protective put long put + long underlying asset Synthetic call long put + long underlying asset + short bond Synthetic put long call + short underlying asset + long bond Synthetic long call + long underlying bond + short put asset Synthetic long put + long bond underlying asset + short call Value p0 + S0 p 0 + S0 − X (1 + R F )T c 0 − S0 + X (1 + R F )T c0 + © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. X − p0 (1 + R F )T p0 + S0 – c0 93 Basics of Derivative Pricing and Valuation Lowest Prices of European Calls and Puts c 0 ≥ Max[0,S0 − p0 ≥ Max[0, X ] (1 + R F )T X − S0 ] (1 + R F )T Put‐Call Forward Parity p0 − c 0 = [X − F(0,T)] (1 + R F )T Binomial Option Pricing c= πc + + (1 − π)c − (1 + r) π= (1 + r − d) (u − d) Hedge ratio n= c+ − c − S+ − S− Lowest Prices of American Calls and Puts C0 ≥ Max[0, S0 − X/(1 + RFR)T ] P0 ≥ Max[0, (X − S0 )] 94 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. Basics of Derivative Pricing and Valuation Summary of Options Strategies Holder Writer Call CT = max(0,ST − X) Value at expiration = CT Profit: Π = CT − C0 Maximum profit = ∞ Maximum loss = C0 Breakeven: ST* = X + C0 CT = max(0,ST − X) Value at expiration = –CT Profit: Π = –CT − C0 Maximum profit = C0 Maximum loss = ∞ Breakeven: ST* = X + C0 Put PT = max(0,X − ST) Value at expiration = PT Profit: Π = PT − P0 Maximum profit = X − P0 Maximum loss = P0 Breakeven: ST* = X − P0 PT = max(0,X − ST) Value at expiration = –PT Profit: Π = –PT − P0 Maximum profit = P0 Maximum loss = X − P0 Breakeven: ST* = X − P0 Where: C0, CT = price of the call option at time 0 and time T P0, PT = price of the put option at time 0 and time T X = exercise price S0, ST = price of the underlying at time 0 and time T V0, VT = value of the position at time 0 and time T Π = profit from the transaction: VT − V0 r = risk‐free rate Covered Call Value at expiration: VT = ST − max(0,ST − X) Profit: Π = VT − S0 + C0 Maximum profit = X − S0 + C0 Maximum loss = S0 − C0 Breakeven: ST* = S0 − C0 Protective Put Value at expiration: VT = ST + max(0,X − ST) Profit: Π = VT − S0 − P0 Maximum profit = ∞ Maximum loss = S0 + P0 − X Breakeven: ST* = S0 + P0 © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 95 Alternative Investments Introduction to Alternative Investments Introduction to Alternative Investments Pricing of Commodity Futures Contracts Futures price = Spot price (1 + r) + Storage costs − Convenience yield r = Short‐term risk‐free rate © Wiley 2017 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright. 97