2015 CFA® EXAM REVIEW COVERS ALL TOPICS IN LEVEL I LEVEL I CFAư đ FORMULA SHEETS Copyright â 2015 by John Wiley & Sons, Inc All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey The material was previously published by Elan Guides Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the Web at www.copyright.com Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representatives or written sales materials The advice and strategies contained herein may not be suitable for your situation You should consult with a professional where appropriate Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002 Wiley publishes in a variety of print and electronic formats and by print-on-demand Some material included with standard print versions of this book may not be included in e-books or in print-on-demand If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com For more information about Wiley products, visit www.wiley.com Quantitative Methods Quantitative Methods 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 - 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 D 360 rBD = × 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 © Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright Quantitative Methods Effective Annual Yield EAY = (1 + HPY)365/ t - where: HPY = holding period yield t = numbers of days remaining till maturity HPY = (1 + EAY) t /365 - Money Market Yield R MM = 360 × rBD 360 - (t × rBD ) R MM = HPY × (360/t) Bond Equivalent Yield BEY = [(1 + EAY)0.5 - 1] ì Population Mean N à= xi i =1 N where: xi = is the ith observation Sample Mean n X= ∑ xi i =1 n Geometric Mean + R G = T (1 + R1 ) × (1 + R ) ×…× (1 + R T ) OR G = n X1X X … X n with X i > for i = 1, 2,…, n T T R G = ∏ (1 + R t ) − t =1 Harmonic Mean Harmonic mean: X H = N with X i > for i = 1,2,…,N ∑x i =1 i N © Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright Quantitative Methods 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 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 © Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright Quantitative Methods Sample Standard Deviation n s= ∑ (X i − X)2 i =1 n −1 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 - ) s3 As n becomes large, the expression reduces to the mean cubed deviation n SK ≈ (X i - X)3 ∑ i =1 n s3 where: s = sample standard deviation © Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright Quantitative Methods 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 = − s4 (n - 1)(n - 2)(n - 3) (n - 2)(n - 3) As n becomes large the equation simplifies to: n KE ≈ (X i - X)4 ∑ 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 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: © Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright Quantitative Methods Conditional Probabilities P(A B) = P(AB) given that P(B) ≠ 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 Expected Value E(X) = P(X1 )X1 + P(X )X + … P(X n )X n n E(X) = ∑ P(X i )X i i =1 where: Xi = one of n possible outcomes © Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright Quantitative Methods Variance and Standard Deviation σ (X) = E{[X - E(X)]2} n σ (X) = ∑ P(X i ) [X i - E(X)]2 i =1 The Total Probability Rule for Expected Value 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 P(S1) = the probability of Scenario 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 ) = Cov(R A ,R B ) (σ A )(σ B ) Expected Return on a Portfolio N E(R p ) = ∑ wi E(R i ) = w1E(R1 ) + w2 E(R ) + i =1 + w N E(R N ) 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 Asset Portfolio Var(R p ) = w2A σ (R A ) + w2B σ (R B ) + 2w A w B Cov(R A ,R B ) Var(R p ) = w2A σ (R A ) + w2B σ (R B ) + 2w A w Bρ(R A ,R B )σ (R A )σ (R B ) © Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright Quantitative Methods Variance of a Asset Portfolio Var(R p ) = w2A σ (R A ) + w2B σ (R B ) + w2C σ (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 )! 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 - 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 10 © Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright Equity 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)n t =1 (1 + r) 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 Price Multiples P0 D1 /E1 = E1 r−g Price to cash flow ratio = 60 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 © Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright Equity 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) © Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright 61 Fixed Income Fixed Income 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 = 62 Coupon interest − (Conversion ratio × Common stock dividend per share) Conversion ratio Market price of convertible bond −1 Straight value © Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright Fixed Income Pricing Bonds with Spot Rates PV = PMT PMT PMT + FV + +…+ (1 + Z N ) N (1 + Z1 ) (1 + Z ) z1 = Spot rate for Period z2 = Spot rate for Period 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 + 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 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright 63 Fixed Income 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 × − × DR year Year FV − PV DR = × Days FV Pricing formula for money market instruments quoted on an add‐on rate basis: PV= FV Days 1 + × AOR Year Year FV − PV AOR = × Days PV Yield Spreads over the Benchmark Yield Curve PV = PMT PMT PMT + FV + + + (1 + z N + Z) N (1 + z1 + Z)1 (1 + z + Z)2 • 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) Parties to the Securitization Party Seller Issuer/Trust Servicer SMMt = 64 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 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright Fixed Income Macaulay Duration 1 + r + 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+ ) × ( ∆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 = (PV− ) − (PV+ ) ì ( Curve) ì (PV0 ) â Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright 65 Fixed Income 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+ ) 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: %∆PV Full ≈ (− AnnModDur × ∆Yield) + × AnnConvexity × ( ∆Yield)2 2 66 © Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright Fixed Income Money convexity ∆PV Full ≈ (− MoneyDur × ∆Yield) + × MoneyCon × ( ∆Yield)2 2 Effective convexity EffCon = [(PV− ) + (PV+ )] − [2 × (PV0 )] ( ∆Curve)2 × (PV0 ) Yield Volatility %∆PV Full ≈ (− AnnModDur × ∆Yield) + × AnnConvexity × ( ∆Yield)2 2 Duration Gap Duration gap = Macaulay duration − Investment horizon © Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright 67 Fixed Income 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 ) 68 © Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright Derivatives Derivatives 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 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright 69 Derivatives 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 – (ST – X) 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 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 70 Payoff ST < X ST > X Option holder Option holder exercises the does not exercise option the option X – ST – (X – ST) © Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright Derivatives 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 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 + S0 (1 + R F )T Combining Portfolios to Make Synthetic Securities Strategy fiduciary call Consisting of Value long call + X long bond c + 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 + S0 − X (1 + R F )T c − S0 + X (1 + R F )T c0 + © Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright X − p0 (1 + R F )T p0 + S0 – c0 71 Derivatives Lowest Prices of European Calls and Puts c ≥ Max[0,S0 − p0 ≥ Max[0, X ] (1 + R F )T X − S0 ] (1 + R F )T Put‐Call Forward Parity p0 − c = [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 )] 72 © Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright Derivatives 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 and time T P0, PT = price of the put option at time and time T X = exercise price S0, ST = price of the underlying at time and time T V0, VT = value of the position at time 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 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright 73 Alternative Investments Alternative Investments Pricing of Commodity Futures Contracts Futures price = Spot price (1 + r) + Storage costs − Convenience yield r = Short‐term risk‐free rate 74 © Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright ... decision Reject H0 Incorrect decision Type I error Significance level = P(Type I error) H0 is False Incorrect decision Type II error Correct decision Power of the test = - P(Type II error) Confidence... standard deviation 12 © Wiley 2015 All Rights Reserved Any unauthorized copying or distribution will constitute an infringement of copyright Quantitative Methods Test Statistic Test statistic = Sample... constitute an infringement of copyright Quantitative Methods Conditional Probabilities P(A B) = P(AB) given that P(B) ≠ P(B) Multiplication Rule for Probabilities P(AB) = P(A B) × P(B) Addition