CFA level 1 june 2015 formula sheet

Volume 1: Quantitative Methods Formula Sheet Level I 2015 FinQuiz Formula Sheet Volume 1: Quantitative Methods Reading 5: The Time Value of Money Interest Rate • Interest Rate = Real Risk Free Interest Rate + Inflation Premium + Default Risk Premium + Liquidity Premium + Maturity Premium • Nominal Risk Free Interest rate = Real Risk Free Interest Rate + Inflation Premium • Interest rate as a growth rate = g = -1 PV and FV of Cash Flow = • PV = • PV of Perpetuity = • PV (for more than one Compounding per year) = PV= FVN + ℎ = ! "" #$ %" • FVN = &' + • FV (for more than one Compounding per year) = FVN = + • FV (for Continuous Compounding) = FVN = &' • Solving for Number of periods = N = ( )( )( *+ ,+ (where LN = natural log) -./.01 2334/5 63.0 7/.0 • Periodic Interest Rate = • Effective (or Equivalent) Annual Rate (EAR = EFF%) = • (4 80 9: ;9 =91 =3 ?30 @0/ D × ×( Stated & Effective Rates + & %A!%B C" × −1 EAR (with Continuous Compounding) = EAR = −1 Formula Sheet Volume 1: Quantitative Methods PV & FV of Ordinary Annuity • PVOA = ∑3.H • FVOA = ∑3.H O&IJ + • Size of Annuity Payment = PMT = • G = &IJ K PV of Annuity Factor = P LM ( Q L N = &IJ Q R 9: 2334=.S /T.9 M U× R U M U PV & FV of Annuity Due • PVAD = &IJ K • FVAD = &IJ Q LM N + PMT at t = PVOA + PMT R 1+ = FVOA ×(1+r) Reading 6: Discounted Cash Flow Applications Net Present Value = ∑3.H ; G G − BVW X;XXX IRR (when project’s CFs are perpetuity) = NPV = - Initial Investment + 677 = Holding Period Yield (HPR) = YZ Y[ \Z Y[ Money Weighted Rate of Return (MWR) = ∑=HW ; 677 G = (IRR represents the MWR) Time Weighted Rate of Return (TWR): ^ • TWR (when no external cash flows) = rTWR = HPR = rt = • TWR (for more than one periods) = rTWR = [(1+rt,1)ì (1+rt,2)ì (1+rt,n)] -1 Annualized TWR (when investment is for more than one year) ^ = _ + D O1 + D` … + + D3 Pbc _1 • TWR (for the year) = rTWR = [(1+R1)× (1+R2)×… (1+R365)] -1 where R1 = ^ ^ Formula Sheet Volume 1: Quantitative Methods Bank Discount Yield = BDY = rBD = Holding Period Yield = HPY = efW / / =T0 therefore Price = Par − 3 × gh efW YZ Y[ \Z Y[ 10 Effective Annual Yield = EAY = + i&j efk/ − (Rule: EAY > BDY) 11 Money Market Yield (or CD equivalent Yield) rMM: • rMM = HPY ì rMM = (rBD) ì rMM = efW /T0 /540 9: m0 Tm/ efW gh efW gh 0/ S n=55 =T0 (Rule: rMM> rBD) 12 Bond Equivalent Yield = BDY = Semiannual Yield × Reading 7: Statistical Concepts and Market Returns Range = Maximum Value – Minimum Value Class Interval = i ≥ • o ) p where i = class interval, H = highest observed value, L = lowest observed value, k = number of classes Absolute Frequency = Actual number of observations in a given class interval Relative Frequency = qrstuvwx yzx{vx|}~ •tw€u v rxz t• ‚rsxzƒ€w„t|s Cumulative Absolute Frequency = Add up the Absolute Frequencies Cumulative Relative Frequency = Add up the Relative Frequencies Arithmetic Mean = …v t• trsxzƒ€w„t|s „| w†x ‡€w€r€sx t.t• trsxzƒ€w„t|s „| w†x ‡€w€r€sx Formula Sheet Volume 1: Quantitative Methods Median = Middle number (when observations are arranged in ascending/descending order) • For Even number of observations locate median at ` • For Odd number of observations locate median at ` Mode = observation that occurs most frequently in the distribution | XXXX 10 Weighted Mean = ‰ Š = ∑„HZ „ ‰„ = (w1X1+ w2X2+….+ wnXn) where X1, X2… Xn = observed values and w1, w2… wn = corresponding weights, sum to 11 Geometric Mean = GM = c‹‰ ‰` … ‰3 with Xi≥0 for i = 1,2,…n XXXX 12 Harmonic Mean = H.M = ‰ o = ∑c Ž ‘Ž 13 Population Mean = = ( c • •Ž with ‰= > for i = 1,2,.,.,n where Xi = ith observation & N = number of observations in entire population c ∑ 14 Sample Mean = ‰X = Ž ‘Ž where n = number of observation in the sample 15 Measures of Location: • Quartiles = “= =84.=93 • Quintiles = “= =84.=93 • Deciles = • Percentiles = Ly = " + ” k “= =84.=93 W , S WW where Ly is location of percentile, y = % point at which distribution is being divided and n= number of observations 16 Mean Absolute Deviation = MAD = 17 Population Variance = σ2 = X ∑c Ž• |‘G ‘ | ∑Ž• ‘Ž – — ( Formula Sheet Volume 1: Quantitative Methods 18 Population Standard Deviation = √™ ` = š 19 Sample Variance = s2 = ∑Ž• ‘Ž – — ( X — ∑c Ž• ‘Ž ‘ 20 Sample Standard Deviation (S.D) = s = š 21 Semi-variance = ∑ytz €uu ‘„ ›‘X X — ∑c Ž• ‘Ž ‘ ‘„ ‘X — 22 Semi-deviation (Semi Standard Deviation) = √ œ%• 23 Target Semi-variance = ∑ytz €uu ‘„ ›ž 24 Target Semi-Deviation = ‹ Ÿ œ%• 25 Coefficient of Variation = CV = 26 Sharpe Ratio = ‘„ n — ‘X x€| Ytzw•tu„t ¡xwvz| % "B = š∑ytz €uu ‘„›‘X ‘„ ‘X — where B = Target Value % "B = š∑ytz €uu ‘„ ›ž ‘„ n — where s= sample S.D and ‰X = sample mean x| Ăs zxx Ăxwvz| .\ t Ytzwtut Ăxwvz| 27 Excess Kurtosis = Kurtosis – 28 Geometric Mean return ≈ £ % ℎœ %B I " D # " − / =/3T0 9: 70.4 ` Reading 8: Probability Concepts Empirical Probability of an event E = P(E) = Odds for event E = Yztrruw~ t xx|w Ô twu Yztrruw~ Yztrruw~ t Ô Z Yztrruw~ t Ô Odds against event E = Z Yztrruw~ t Ô Yztrruw~ t Ô Conditional Probability of A given that B has occurred = P(A\B) = 2n n → P(B) ≠ Formula Sheet Volume 1: Quantitative Methods Multiplication Rule (Joint probability that both events will happen): P(A and B) = P(AB) = P(A\B) × P(B) P(B and A) = P(BA) = P(B\A) × P(A) Addition Rule (Probability that event A or B will occur): P(A or B) = P(A) + P(B) – P(AB) P(A or B) = P(A) + P(B) (when events are mutually exclusive because P(AB) = 0) Independent Events: • Two events are independent if: P(B\A) = P(B) or if P(A\B) = P(A) • Multiplication Rule for two independent events = P(A & B) = P(AB) = P(A)ì P(B) Multiplication Rule for three independent events = P(A and B and C) = P(ABC) = P(A) × P(B) × P(C) Complement Rule (for an event S) = P(S) + P(SC) = (where SC is the event not S) Total Probability Rule: P(A) = P(AS) + P(ASC) = P(A\S)×P(S) + P(A\SC)×P(SC) P(A) = P(AS1) + P(AS2) +….+ P(ASn) = P(A\S1)×P(S1) + P(A\S2)×P(S2)… P(A\Sn)×P(Sn) (where S1, S2, …,Sn are mutually exclusive and exhaustive scenarios) 10 Expected Return = E(wiRi) = wiE(Ri) where (wi is weight of variable i and Ri is random variable i) 11 Covariance (between two random variables Ri and Rj): Cov (Ri Rj) = 3=H OƠ D= ƯD= PODĐ ƯDĐ P Cov (Ri Rj) = Cov (Rj Ri) Cov (R, R) = σ2 (R) Formula Sheet Volume 1: Quantitative Methods 12 Portfolio Variance = σ2 (Rp) = ∑3=H ∑3§H ™ D ` ` σ2 (Rp) = e ăA + D , De + ` ` `™ ` D` + ` ` e™ D= + ` ` D` e ăA 13 Standard Deviation (S.D) = = Đ ăAOD= DĐ P ` D` , De De + + ` ăA D , D` + ` e De ;9ô O7 7ơ P 14 Correlation (between two random variables Ri, Rj) = êOD= DĐ P = 15 Bayes Formula = & Ư " \ C"VA %A" = (0 63:9 -7 ì-7ơ /.=93\ô03 (0 63:9 /.=93 ì & & %A Ơ A AV Ư " 16 Multiplication Rule of Counting = n factorial = "! = n (n-1)(n-2)(n-3)…1 17 Multinomial Formula (General formula for labeling problem) = 18 Combination Formula (Binomial Formula) = 3 ă = O3P = 3! 3! !3— !…3´ ! ! ! where n = total number of objects and r = number of objects selected 19 Permutation = 3 & = 3! ! Reading 9: Common Probability Distributions Probability Function (for a binomial random variable) p(x) = p(X=x) = O3àPƠà & = 3! !à!