CFA Level II Formula Sheet 2025 by Fabian Moa

CFA® Program Level II FORMULA SHEET (2025) Version 1.0 Prepared by: Fabian Moa, CFA, FRM, CTP, FMVA, AFM, FSA Credential FOR REFERENCE ONLY (Note: Formula Sheet is not provided in the CFA exam) Follow us on: YouTube LinkedIn Facebook Instagram NOESIS EXED SDN BHD Block VO2, Level 5, Unit 8, Lingkaran SV, Sunway Velocity, 55100 Kuala Lumpur, Malaysia Website: www.noesis.edu.sg CFA Institute does not endorse, promote, or warrant the accuracy or quality of the products or services offered by Noesis Exed. CFA Institute, CFA® and Chartered Financial Analyst® are trademarks owned by CFA Institute. CFA Level 2 (2025) Formula Sheet by Fabian Moa Contents Setting Up the Texas BA II Plus Financial Calculator ............................................................................... 4 VOLUME 1: QUANTITATIVE METHODS ................................................................................................... 4 Learning Module 1 | Basics of Multiple Regression and Underlying Assumptions ............................ 4 Learning Module 2 | Evaluating Regression Model Fit and Interpreting Model Results ................... 4 Learning Module 3 | Model Misspecification ..................................................................................... 6 Learning Module 4 | Extensions of Multiple Regression .................................................................... 6 Learning Module 5 | Time-Series Analysis.......................................................................................... 7 Learning Module 6 | Machine Learning.............................................................................................. 8 Learning Module 7 | Big Data Projects ............................................................................................... 9 VOLUME 2: ECONOMICS ....................................................................................................................... 10 Learning Module 1 | Currency Exchange Rates: Understanding Equilibrium Value ........................ 10 Learning Module 2 | Economic Growth and the Investment Decision ............................................ 12 VOLUME 3: FINANCIAL STATEMENT ANALYSIS..................................................................................... 14 Learning Module 1 | Intercorporate Investments ............................................................................ 14 Learning Module 2 | Employee Compensation - Post-Employment and Share-Based .................... 15 Learning Module 3 | Multinational Operations................................................................................ 16 Learning Module 4 | Analysis of Financial Institutions ..................................................................... 17 Learning Module 5 | Evaluating Quality of Financial Reports .......................................................... 18 Learning Module 6 | Integration of Financial Statement Analysis Techniques ................................ 18 VOLUME 4: CORPORATE ISSUERS ......................................................................................................... 19 Learning Module 1 | Analysis of Dividends and Share Repurchases ................................................ 19 Learning Module 3 | Cost of Capital: Advanced Topics .................................................................... 20 Learning Module 4 | Corporate Restructuring ................................................................................. 23 VOLUME 5: EQUITY VALUATION ........................................................................................................... 24 Learning Module 1 | Equity Valuation Applications and Processes ................................................. 24 Learning Module 2 | Discounted Dividend Valuation ...................................................................... 24 Learning Module 3 | Free Cash Flow Valuation................................................................................ 25 Learning Module 4 | Market-Based Valuation Price and Enterprise Value Multiples...................... 27 Learning Module 5 | Residual Income Valuation ............................................................................. 29 Learning Module 6 | Private Company Valuation ............................................................................ 30 VOLUME 6: FIXED INCOME ................................................................................................................... 32 Learning Module 1 | The Term Structure and Interest Rate Dynamics............................................ 32 Learning Module 2 | The Arbitrage-Free Valuation Framework ...................................................... 33 Learning Module 3 | Valuation and Analysis - Bonds with Embedded Options ............................... 34 Learning Module 4 | Credit Analysis Models .................................................................................... 36 2 CFA Level 2 (2025) Formula Sheet by Fabian Moa Learning Module 5 | Credit Default Swaps....................................................................................... 37 VOLUME 7: DERIVATIVES ...................................................................................................................... 38 Learning Module 1 | Pricing and Valuation of Forward Commitments ........................................... 38 Learning Module 2 | Valuation of Contingent Claims ...................................................................... 42 VOLUME 8: ALTERNATIVE INVESTMENTS ............................................................................................. 46 Learning Module 1 | Introduction to Commodities and Commodity Derivatives ............................ 46 Learning Module 2 | Overview of Types of Real Estate Investment ................................................ 46 Learning Module 3 | Investments in Real Estate Through Publicly Traded Securities ..................... 48 Learning Module 4 | Hedge Fund Strategies .................................................................................... 49 VOLUME 9: PORTFOLIO MANAGEMENT............................................................................................... 50 Learning Module 1 | Economics and Investment Markets............................................................... 50 Learning Module 2 | Analysis of Active Portfolio Management....................................................... 52 Learning Module 3 | Exchange-Traded Funds: Mechanics and Applications ................................... 53 Learning Module 4 | Using Multifactor Models ............................................................................... 54 Learning Module 5 | Measuring and Managing Market Risk ........................................................... 55 Learning Module 6 | Backtesting and Simulation............................................................................. 56 3 CFA Level 2 (2025) Formula Sheet by Fabian Moa CFA Level II – Formula Sheet (2025) Setting Up the Texas BA II Plus Financial Calculator Video: https://youtu.be/0MS8d8QOFmc VOLUME 1: QUANTITATIVE METHODS Learning Module 1 | Basics of Multiple Regression and Underlying Assumptions = + + + ⋯+ + + = 1, 2, 3, … , where: = dependent variable = independent variable = intercept , , … , = slope coefficients = error term = number of observations = number of independent variables , , , … , = regression coefficients = ! − " Learning Module 2 | Evaluating Regression Model Fit and Interpreting Model Results Coefficient of determination, # # = $%& $%& $%& $'% $'% $'% () + $%& $'% () #(* ()) $%& $'% () + , ,, () #(* ()) () - , $$+ = , ! " $$# = $$- = ! " − ! " = $$# $$= 1− $$+ $$+ ./ − 0 . − /0 $$- ⁄ − − 1 −1 123%) (2 # , # = 1 − 4 8 1−# 6=1−7 $$+⁄ − 1 − −1 4 CFA Level 2 (2025) Formula Sheet by Fabian Moa Akaike’s information criterion (AIC) $%& 19: = ln 7 )'% () ( where: = Sample size = Number of independent variables in the model 8+2 +1 Schwarz’s Bayesian information criterion (BIC of SBC) $%& ln 7 =9: = )'% () ( 8 + ln +1 F-distributed test statistic for jointly omitted variables >= $%& )'% () ( $%& () ? (2 & 2(, − $%& )'% () % () )'% () % () ? (2 & 2(,⁄ − − 1 ? (2 ⁄' where: ' = Number of restrictions (i.e., number of variables omitted in the restricted model compared to the unrestricted model) @ : B = BC = ⋯ = BCDE = 0 @G : At least one of the ' slopes ≠ 0 F-test for joint test of slope coefficients ANOVA Regression Residual Total >) ) ?= df − SS $$# ⁄ $$# − 1 $$- $$- ⁄ −1 $$+ J( $'% ( #(* ()) J( $'% ( - @: = =⋯= =0 @G : At least one K ≠ 0 MS = − −1 $$# ⁄ $$- ⁄ − − 1 5 I $$# ⁄ $$- ⁄ − − 1 CFA Level 2 (2025) Formula Sheet by Fabian Moa t-test statistic for slope coefficient where: /K = Regression estimate of K =K = Hypothesized value of coefficient 3 )L/M = Estimated standard error of /K = /K − =K )L/M Video (Simple Linear Regression): https://youtu.be/uR_9im2JP18 Learning Module 3 | Model Misspecification Breusch-Pagan Test +() $ χOP, = # ) ?, where: # = #-squared between squared residuals and independent variables Variance Inflation Factor (VIF) where: #K = Variation in K 9>K = 1 1 − #K explained by the other − 1 independent variables Learning Module 4 | Extensions of Multiple Regression Detecting Influential Points Sum of individual leverages for all observations = If observation’s leverage > 3 R C ! +1 S ⇒ Potentially influential observation Studentized Deleted Residual, UV∗ ∗ (∗ ( = = )X ∗ YJ$- 1 − ℎ where: ( ∗ = The residual with the th observation deleted )X ∗ = The standard deviation of the residuals = The number of independent variables J$- = Mean squared error of the regression model that deletes the th observation ℎ = The leverage value for the th observation 6 CFA Level 2 (2025) Formula Sheet by Fabian Moa Logistic Regression (Logit) [ ln 7 8= 1−[ [= + + 1 + + 1 + exp_− + ⋯+ + ` + ⋯+ [ ln 7 8 = a * 22) 1−[ b22) = ( Lc Likelihood ratio (LR) test a# = −2 a * , (, ℎ 2 () ? (2 & 2(, − a * , (, ℎ 2% () Learning Module 5 | Time-Series Analysis Linear Trend Models d = = time (independent variable) Log-Linear Trend Models Growth rate of d = ( Lf − 1 + + = ( LeCLfd ln d p-th order autoregressive model, gh i jd = + jdE + = 1,2, … , + d = + = 1,2, … , + jdE + ⋯ + k jdEk + d Test statistic for autocorrelation of residuals = where: : o mn, = #() 2% , % ? (, $ 2 2( d , dE pn = dq 2( % ? −0 (, 7 = #() 2% , % ? 1⁄√+ ℎ( () 2% , (, ? (2 & 2(, CFA Level 2 (2025) Formula Sheet by Fabian Moa Mean reverting level for AR(1) model jd = Root Mean Squared Error #J$- = r 1− $'% Dickey and Fuller Unit-Root Test jd − jdE = where: * = −1 n-Period Moving Average J1! = + * jdE + d jd + jdE + ⋯ + jdE !E qth-order Moving-Average, J1 ' jd = d + s - = 0, jd = + - d (2 ( =p , + ⋯ + sD dE ? o d, t =- dED d t =0 +s dE Autoregressive Moving-Average (ARMA) Model, ghug i, v - = 0, jdE + - d jdE + ⋯ + =p , k jdEk ? o d, t + =- d d t =0 ≠) + ⋯ + sD dED ≠) Autoregressive Conditional Heteroskedasticity (ARCH) Model ARCH(1): pwdC = + x̂d Learning Module 6 | Machine Learning Neural Networks z({ ( { {( *ℎ = b,2 {( *ℎ − a( * 8 (×} [ , 2( o o( ℎ( ,( { ℎ ()~(? • ℎ( ,2 {( *ℎ CFA Level 2 (2025) Formula Sheet by Fabian Moa Learning Module 7 | Big Data Projects Normalization of variable X !€•BG‚ ƒX„ − B! BG… − B ! = Standardization of variable X = tdG!„G•„ ƒX„ [ (? ) ,[ = #(? ,,, # = + %( [ ) o( + %( [ ) o( + > ,)( z(* 1??% ?‡ = >1 $? (= > ,)( [ ) + %( [ ) o( + %( [ ) o( + > ,)( [ ) + %( [ ) −† p o( o( + %( [ ) o( + + %( z(* o( + > ,)( [ ) o( + + %( z(* 2 × [ (? ) × #(? ,, [ (? ) + #(? ,, o( # (, >[# = > ,)( [ ) o( + %( z(* o( + > ,)( [ ) + %( [ ) o( + %( [ ) o( + > ,)( z(* + %( [ ) o( # (, +[# = ˆ ?%&( > ('%( ?‡, ˆ> = 9 o( )( ˆ ?%&( $( ( ?( : % + , z%& ( > ('%( ?‡, 9ˆ> = log 7 +>-9ˆ> = +> × 9ˆ> { ℎ‰ 2 $( ( ?() 1 8 ˆ> 9 o( o( + > ,)( z(* o( o( o( CFA Level 2 (2025) Formula Sheet by Fabian Moa VOLUME 2: ECONOMICS Learning Module 1 | Currency Exchange Rates: Understanding Equilibrium Value Cross Rates When given • and O, then O = • × O Œ • Œ Œ When given and , then = × Œ O • Œ • O Œ • • Ž R•S Bid Bid/Ask j ‡ ⁄ 1 ‡ 1⁄j Currency pair A/B B/A Video: https://youtu.be/wyDKKPkPhzw Arbitrage Opportunities Between Dealers and Interbank Video: https://youtu.be/Lqo9UZ3yyEA Covered Interest Rate Parity >•⁄„ 1? % , R 360 S = $•⁄„ ‘ “ 1? % , 1 + „ R 360 S 1+ • >•⁄„ − $•⁄„ •− = $•⁄„ 1+ 1? % , R 360 S 1? % , „ R 360 S „ Video: https://youtu.be/9jOzFA9GuHU Mark-to-Market Value of a Forward Contract Original position: Long base currency d forward at forward rate > ,%( a *> { 2= .>d,•⁄„ − > ,•⁄„ 0 × : #(& * 2 ‡) 1+ •7 360 >d,•⁄„ = Forward rate at valuation date, t (Bid side) Video: https://youtu.be/wLqyZRrutAc 10 ,•⁄„ (Offer side) ? $ ”( & % ‡ 8 CFA Level 2 (2025) Formula Sheet by Fabian Moa Uncovered Interest Rate Parity - $•⁄„ %Δ$•X⁄„ 1? % , R 360 S = $•⁄„ ‘ “ 1? % , 1 + „ R 360 S 1+ • 1? % , R 360 S = ≈ 1? % , 1 + „ R 360 S • − „ • − „ 7 1? % , 8 360 Video (Carry Trade): https://youtu.be/_26fG3Zvzyg Absolute PPP $•⁄„ = Relative PPP %Δ$•⁄„ [• [„ 1? % , ˜• − ˜„ R 360 S 1? % , = ≈ ˜• − ˜„ 7 8 1? % , 360 1 + ˜„ R 360 S Ex ante PPP International Fisher Effect where: ˜ X = Expected inflation rate ˜ = Actual inflation rate %Δ$•X⁄„ ≈ ˜•X − ˜„X 7 • − „ 1? % , 8 360 = ˜•X − ˜„X Mundell-Fleming Model Bonus Video: https://youtu.be/xNo3GpWYgKA 11 CFA Level 2 (2025) Formula Sheet by Fabian Moa Learning Module 2 | Economic Growth and the Investment Decision Grinold-Kroner Model - #X = ˆ + Δ [⁄- + + * − Δ$ where: - #X = Expected equity return ˆ = Dividend yield Δ [⁄- = Expected repricing = Expected inflation rate * = Real economic growth rate Δ$ = Change in shares outstanding Dilution effect Δ$ = z( %‡ ? + #(, Cobb-Douglas Production Function = +™ š a Eš o( 2‡ & )& where a < 1 where: Y = Output › = Share of output allocated to capital (™) 1 − › = share of output allocated to labor (a) + = total factor productivity (+>[), represents technological progress of the economy œ•žŸ•ž Ÿ ¡ ¢£¡¤ ¡ = ™ š = +7 8 a a [( ?( *( ?ℎ *( , ~ 2%? o ‡ = [( ?( *( ?ℎ *( [( ?( + , ? ~ 2%? o ‡ ? ~ Marginal product of capital, MPK J[™ = › 7 8 ™ Amount of output that is allocated to providers of capital, a ™ ›= 12 *( ?ℎ *( , 2((~( * CFA Level 2 (2025) Formula Sheet by Fabian Moa Growth Accounting equation: ∆ = ∆+ ∆™ ∆a +› + 1−› + ™ a Growth rate in potential GDP = Long-term growth rate + Long-term growth rate of labor force in labor productivity a ?( ~ ?~ = ‰ a ?( * *( ~ ~%, Sustainable growth rate of output per capita s *∗ = 1−› Sustainable growth rate of output (Steady state growth rate) s ¦∗ = + 1−› Equilibrium output-to-capital ratio (in steady state): 1 s = § + + ¨© ™ ) 1−› where: s = growth rate of TFP › = elasticity of output with respect to capital ) = fraction of income (Y) that is saved ¨ = rate of depreciation of physical capital stock = labor supply growth = %∆a Endogeneous Growth Model Production function: ‡X = ? X Growth rate of output per capita: Δ‡X Δ = = )? − ¨ − ‡X X where: ‡X = output per worker X = capital per worker ? = marginal product of capital in the aggregate economy (constant) 13 CFA Level 2 (2025) Formula Sheet by Fabian Moa VOLUME 3: FINANCIAL STATEMENT ANALYSIS Learning Module 1 | Intercorporate Investments Investments in Associates (Equity Method) - 2 * o() &( )) ? () = =(* * $ℎ + o() &( ( )) ? () $ℎ ( 1& ” (j?()) ( − 2 o 2( 2 − ~% ?ℎ )( ~ ?( ? &( (?( o(2 Impact on Investor’s Income Statement 1& ” (j?()) $ℎ ( ª ( , ”(2 ~ & $ℎ ( = − − ~% ?ℎ )( ~ ?( 2 { ) ( & %~) ( & ) ,( ( ? &( Business Combinations (Acquisition Method) Excess purchase price = Acquisition price – %Ownership × Book value of net identifiable assets Partial Goodwill = Acquisition price – (%Ownership × Fair value of identifiable net assets) = Acquisition price – (%Ownership × Book value of identifiable net assets) – (%Ownership × Excess purchase price attributable to identifiable net assets) z ? ,, * ( () = %z:9 × > o ,%( 2( ,( ( Full Goodwill = Fair value of entity – Fair value of net identifiable assets z ? ,, * ( () = %z:9 × > o ,%( ( ‡ Video: https://youtu.be/RgxmPbx4-0o IFRS 9&~ &( , )) = : ‡ * o ,%( ? )ℎ *( ( *% US GAAP 9&~, (2 > o ,%( = (~ *% * 2{ ,, 9&~ &( , )) = : − > − #(? o( ,( & % ? )ℎ *( ( *% o ,%( 2( ‡ * o ,%( 9&~, (2 − * 2{ ,, * 2{ ,, 14 (~ ,( ( * % ′) ))( ) ))( ) CFA Level 2 (2025) Formula Sheet by Fabian Moa Learning Module 2 | Employee Compensation - Post-Employment and Share-Based Share-Based Compensation Accounting Share Award : &~( ) (j~( )( = Stock Options : &~( ) (j~( )( = > o ,%( > o ,%( { 2 () * ~( () ~ ) * ~( * 2 2 ( * 2 2 ( [ ?((2) & (j( ? )( ¬ ? o( ) = )? $ℎ () ))%(2 & ˆ ,% (2 )ℎ ( )(2 { 2) )ℎ () )ℎ () (j( ? )( − = + ? o( ) 1o( *( )ℎ ( ~ ?( %) 2 * %) 2 * )ℎ ( )(2 { 2) ℎ( (~ * ~( 2 Treasury Stock Method 1))%&(2 : )ℎ ~ ?((2) = + ~ ?((2) & (j( ? )( where: : )ℎ ~ ª ?((2) 1o( *( % (? * ”(2 )ℎ (- )(2 ? &~( ) (j~( )( & (j( ? )( = ® $ ( ~ ?( × z%& ( 0 (? * )(2 )ℎ (- )(2 ª o() (2 > = × * ? &~( ) (j~( )( { 2) ~ o ,%( 2 ( ) > > ~ ) #$ª) Forecasting Shares Outstanding With Share-Based Awards = ) ? )ℎ () = ) ? )ℎ () %) 2 *, = %) 2 *, ( 2 ~( 2 (* * ~( + $ℎ 2 + #$ª) o() (2 + $ℎ ( ~ ) (j( ? )(2 () ))%(2 & $ℎ ( )(? 2 (), − (~% ?ℎ )() ?'% ) ), ( ?. Defined Benefit Pension Plans >% 2(2 ) - 2 * ~, %) = > o ,%( o ,%( =(* = ))( ) ~, * ~, ))( ) − [( ) o ,%( +: ))( ) 15 % ,* )+ =( ( ) 1? % , ( % − ~, ))( ) ~ 2 CFA Level 2 (2025) Formula Sheet by Fabian Moa - 2 * ~( ) ,* = 9 ( () ? ) = =(* =(* * ~( ) ,* * ~( ) ,* + 1? % $( o ?( 9 ( () + + , ))⁄ * ? )) ? ) (,2 × o() &( * ~ ( 2 ? , − 2( =( ( ) ~ 2 IFRS Only In income statement, z( ? ) ⁄ #(~ (2 ~( ) =(* ( () =7 ? &( (j~( )( = * ~( ) ,* − z( :% ( 2~ ) + ? ) ⁄ )( o ?( ? ) ) =(* ~, * ( () ? &( o ,%( (,2 8× * 2( ? ))( ) ~ o() &( ( 2 Learning Module 3 | Multinational Operations Net assets = Total assets – Total liabilities Net monetary assets = Monetary assets – Monetary liabilities Current Rate Method Currency translation adjustment = Total assets of foreign subsidiary in parent currency terms − Total liabilities of foreign subsidiary in parent currency terms − Shareholder capital of foreign subsidiary in parent currency terms − Other equity items of foreign subsidiary in parent currency terms Hyperinflationary Environment IFRS #() & ( (&( ? ‡ ))( ) & , #() (&( ? -& ( ‡ ))( ) & , #() (&( ? ? &( ) (&( (&) , , = () () = = - 2 - 2 ~( ~( 2 ~ ?( 2 ~ ?( 2(j 2(j - 2 ~( 2 ~ ?( 2(j =(* * ~( 2 ~ ?( 2(j - 2 ~( 2 ~ ?( 2(j 1o( *( ~ ?( 2(j ℎ( ~( 16 2 CFA Level 2 (2025) Formula Sheet by Fabian Moa Learning Module 4 | Analysis of Financial Institutions + + , + ( 1 : && -'% ‡ 122 , = + : ~ , + ( 1: ~ , + ( 1: ~ , , #(*%, : ~ , ‡ = + ,+ ( 1 + ,+ ( 2 + : ~ , : ~ , : && -'% ‡ + ( 1 : ~ : && -'% ‡ = + ( 1# # ) ‰( *ℎ (2 1))( ) , + ,+ ( 1: ~ , +( 1 = ≥ 6.0% # # ) ‰( *ℎ (2 1))( ) + ,: ~ # , = + ≥ 4.5% , #(*%, ‡: ~ , ≥ 8.0% # ) ‰( *ℎ (2 1))( ) a '% 2 ‡ : o( *( @ *ℎ ¶% , ‡ a '% 2 1))( ) = # , a:# -j~(? (2 ? )ℎ % , {) z%& ( 2 ‡) ℎ ) ()) ,(o(, o ,%&( ? { ℎ) 2 = a:# × 30 ? )ℎ % , {) Number of days that bank can withstand a stress level volume of cash outflows for (a:# × 30 days. ,( >% 2 * 1o , ,( $ = , z$># #('% (2 $ z( $ # ,( >% 2 * ,( >% 2 * Property and Casualty Companies a )) 2 , )) 23%) &( (j~( )( ª 2( { (j~( )( : & (2 * = = = a )) (j~( )( + a )) 23%) &( (j~( )( z( ~ (& %&) ( (2 ª 2( { * (j~( )( z( ~ (& %&) { ( a )) 2 , )) 23%) &( (j~( )( + ª 2( { (j~( )( * ˆ o 2( 2) ~ , ?‡ℎ ,2( ) ˆ o 2( 2) ~ , ?‡ℎ ,2( ) )ℎ (ℎ ,2( ) = )ℎ (ℎ ,2( ) z( ~ (& %&) ( (2 : & (2 : & = ·¸U¹º »V¼V»¹½»¾ (2 + ˆ o 2( 2) ~ , ?‡ℎ ,2( ) )ℎ (ℎ ,2( ) 17 CFA Level 2 (2025) Formula Sheet by Fabian Moa Learning Module 5 | Evaluating Quality of Financial Reports Beneish Model J-)? ( = – 4.84 + 0.920 ˆ$# + 0.528 ¦J9 + 0.404 1¶9 + 0.892 $¦9 + 0.115 ˆ-[9 – 0.172 $¦19 + 4.670 1?? % ,) – 0.327 a- 9 where: DSR day sales receivable index = #(?( o ,()d ⁄$ ,()d #(?( o ,()dE ⁄$ ,()dE ¦JdE ¦Jd _1 − [[-d + :1d /+1d ` AQI asset quality index = _1 − [[-dE + :1dE /+1dE ` $ ,()d SGI sales growth index = $ ,()dE ˆ(~ (? dE DEPI depreciation index = ˆ(~ (? d GMI gross margin index = $¦1d /$ ,()d $¦1dE /$ ,()dE ‡ (&) − : )ℎ & ~( ) , ))( ) SGAI sales, general, and administrative expenses index = Accruals = 9 ? &( ( ( (j 2 a(o( *(d LEVI leverage index = a(o( *(dE + Earnings Persistence - - *)dC = › + Þ - *)d + *)dC = › + Þ : )ℎ , {d + Þ 1?? % ,)d + Cash-flow-based accruals = NI – (CFO + CFI) Learning Module 6 | Integration of Financial Statement Analysis Techniques z( b~( * b~( * * b~( = − 1))( ) zb1 1))( ) a , () + , : )ℎ 2 + , + , =R − − S−R S 1))( ) $ℎ - ( & 9 o() &( ) a , () ˆ( = , ?(-)ℎ(( ?? % ,) : )ℎ- , {?? % ,) )(2 )(2 = = zb1d − zb1dE zb1d + zb1dE ⁄2 z9d − :>bd + :>9d zb1d + zb1dE ⁄2 18 CFA Level 2 (2025) Formula Sheet by Fabian Moa VOLUME 4: CORPORATE ISSUERS Learning Module 1 | Analysis of Dividends and Share Repurchases Dividend Payout Policies Target payout adjustment model (Lintner model) -j~(? (2 + -j~(? (2 a ) = +7 × *) 2 o 2( 2 2 o 2( 2 where: 123%) &( ? = z%& ( ‡( *( ~ ‡ % − 1 23%) &( ) Constant dividend payout ratio policy ˆ o 2( 2 ˆ o 2( 2 = ~ ‡ % × 123%) &( a ) 8× ? 2 o 2( 2 ( ~, ?( :% ( ( *) Video: https://youtu.be/hhcvNiTpZX4 EPS and BVPS After Share Repurchase -[$ ( %‡ ? = - *) ( ( %‡ $ℎ () % ) ? −1 ( 2 * ( j? ) %‡ ? % 2) Video: https://youtu.be/Pd0-QQF-VhQ = [$ ( %‡ = ? = ,%( ( ( %‡ ? − $ℎ () % ) 2 * Analysis of Dividend Safety ˆ o 2( 2 ~ ‡ % ˆ o 2( 2 ? o( >:>- ? o( *( *( = = ˆ o 2( 2) z( 9 ? &( = z( 9 ? &( ˆ o 2( 2) >:>ˆ o 2( 2) + $ℎ ( (~% ?ℎ )() 19 ,%( )ℎ ( %‡ ( %‡ ? ? CFA Level 2 (2025) Formula Sheet by Fabian Moa Learning Module 3 | Cost of Capital: Advanced Topics Weighted average cost of capital ‰1:: = {„ „ 1− + {k k + {X X where: {„ = Weight of debt in capital structure {k = Weight of preferred equity in capital structure {X = Weight of common equity in capital structure „ = Pre-tax cost of debt k = Cost of preferred equity X = Cost of common equity Cost of debt, „ Cost of equity, = X = • + : (2 )~ ( 2 • + -#[ + 9#[ =( ?ℎ& (% 9#[ = Idiosyncratic risk premium 2(j where: -#[ = Equity risk premium = [ ()( ,%( [ ()( ,%( + #() 2% , ,%( a( )( [ ‡&( ) a()) Leases = − #) ( (( > ,%( a()) + 9 a( )(2 1))( Equity Risk Premium Historical Approach (Ex-Post) 1o( -#[ = *( ( ?ℎ& 2(j ( % Gordon Growth model -#[ = ˆ − +*− 20 1o( • *( ) (( ( ) ˆ (? ,: ) ) ß CFA Level 2 (2025) Formula Sheet by Fabian Moa Grinold-Kroner Model -#[ = _ˆ + Expected repricing + - *) * { ℎ ~( )ℎ -#[ = _ˆ + Δ [⁄- + + * − Δ$` − - *) * { ℎ ~( )ℎ where: ˆ = Dividend yield of market index Δ [⁄- = Expected growth rate in P/E = Expected inflation = Càáâãäåæçèäé êëìí Càáâãîïð • (` − • ( = + * − Δ$ −1 * = Expected growth rate in real earnings per share Δ$ = Expected change in shares outstanding (Δ$ > 0 for share issuance; Δ$ < 0 for share buyback) Cost of Equity Gordon Growth Model X Two-Stage DDM [ = ! = ˆ +* [ ˆd 1+ X " d + [! 1+ X Bond Yield Plus Risk Premium Approach (BYPRP) where „ X = „ + # ) ~ (& %& = Cost of company’s long-term debt Capital Asset Pricing Model (CAPM) X = • + Þ × -#[ 21 ! CFA Level 2 (2025) Formula Sheet by Fabian Moa Fama-French model Three-factor model X Five-factor model X = • = + Þ -#[ + Þ $J= + Þò @Ja • + Þ -#[ + Þ $J= + Þò @Ja + Þó #J‰ + Þô :J1 where: $J= = Size premium @Ja = Value premium #J‰ = Profitability premium :J1 = Investment premium Expanded CAPM X = Build-Up Approach X = • + ÞkXX• -#[ + $[ + $:#[ • + -#[ + $[ + $:#[ + 9[ where: $[ = Size premium (for smaller, privately held companies) 9[ = Industry risk premium $:#[ = Company-specific risk premium Country Spread Model -#[ = -#[ 2(o(, ~(2 & ( +õ×: % ‡ where: õ = Level of exposure of the company in the local country : % ‡ ) ~ (& %& = $ o( ( * ‡ (,2 )~ ( 2 $ o( ( * ‡ (,2 = )~ ( 2 (,2 (&( * * & ( 2) 2( & (2 ℎ( ?% ( ?‡ − ℎ( 2(o(, ~(2 & ( 22 ) ~ (& %& (,2 2(o(, ~(2 & ( * o( &( 2) CFA Level 2 (2025) Formula Sheet by Fabian Moa Aswath Damodaran’s CRP : % ‡ ) ~ (& %& = $ o( ( * ‡ (,2 )~ ( 2 × pöD÷ dø pO€!„ where: pöD÷ dø = Volatility of the local country’s equity market pO€!„ = Volatility of the local country’s bond market International CAPM - X = • + Þù ú-. ûB 0 − •ü + Þ• ú- • − •ü where: -. ûB 0 − • = Risk premium of a global index • = Wealth-weighted foreign currency index return Learning Module 4 | Corporate Restructuring Evaluating Materiality Based on Size ,%( - ( ~ )( o ,%( ) ? ?'% For Acquisition/Divestiture: : ) ) o *) $ ,() For Cost Restructuring: Premium Paid Analysis * ? &~ + ( o( ~ (& %&, [#J = ˆ[ − $[ $[ ‡ where: ˆ[ = Deal price per share of the target company $[ = Unaffected stock price of the target company (i.e., pre-announcement) 23 CFA Level 2 (2025) Formula Sheet by Fabian Moa VOLUME 5: EQUITY VALUATION Learning Module 1 | Equity Valuation Applications and Processes VE – P = (V – P) + (VE – V) where: ö = Estimated intrinsic value [ = Market price = Intrinsic value Conglomerate discount = Sum-of-the-parts value – Market value Learning Module 2 | Discounted Dividend Valuation ! Discounted Dividend Valuation = = d" :>d 1+ ˆ 1+ + d ˆ 1+ + ⋯+ ˆ! 1+ ! + [! 1+ ! Gordon Growth Model ˆ ˆ 1+* = = −* −* Fixed-rate perpetual preferred stock ˆ = Value of stock = Value of a company + Present value of growth with zero-growth opportunities (PVGO) = + [ ¦b - = [ 1 [ ¦b = + - If dividend and earnings growth rate is constant, ˆ = +* [ 24 CFA Level 2 (2025) Formula Sheet by Fabian Moa Two-Stage Dividend Discount Model = ! d" ˆ 1 + *t 1+ d d + ˆ 1 + *t ! 1 + *ý 1 + ! − *ý Video: https://youtu.be/7vXWsTKiSPE The H-Model = ˆ 1 + *ý + ˆ @ *þ − *ý − *ý where: @ = half-life in years of the high-growth period *þ = Short-term growth-rate *ý = Long-term growth rate Video: https://youtu.be/IAMFZXSPKOY PRAT model Sustainable growth rate, * = × #b- Video: https://youtu.be/MnfRRRhuGpA *= z9 − ˆ o 2( 2) z9 $ ,() +1 × × × z9 $ ,() +1 +- Learning Module 3 | Free Cash Flow Valuation Free Cash Flow to the Firm (FCFF) Valuation Approach > & ,%( = d" >:>>d 1 + ‰1:: If non-operating assets = 0 d Equity Value = Firm Value – Market Value of Debt FCFE Valuation Approach -'% ‡ ,%( = d" >:>-d 1+ d 25 CFA Level 2 (2025) Formula Sheet by Fabian Moa Single-Stage (Constant Growth) FCFF and FCFE Model FCFF Valuation Approach > & ,%( = FCFE Valuation Approach -'% ‡ >:>> >:>> 1 + * = ‰1:: − * ‰1:: − * ,%( = >:>>:>- 1 + * = −* −* Free cash flow to the Firm, FCFF FCFF = NI + NCC + Int(1 – Tax Rate) – FCInv – WCInv = CFO + Int(1 – Tax Rate) – FCInv = EBIT(1 – Tax Rate) + Dep – FCInv – WCInv = EBITDA(1 – Tax Rate) + Dep(Tax Rate) – FCInv – WCInv where: NI = Net income available to common shareholders NCC = Net noncash charges (e.g. depreciation) Int = Interest expense FCInv = Fixed capital investments = Maintenance Capex + Growth Capex = Δ¦ )) [[- = Δz( [[- + ˆ(~ (? WCInv = Investment in working capital Free cash flow to the Equity, FCFE FCFE = FCFF – Int(1 – Tax Rate) + Net borrowing = z9 + z:: − >:9 o − ‰:9 o + z( = CFO – FCInv + Net borrowing where: Net borrowing = Debt issued – Debt repaid Video: https://youtu.be/rtIvIy6Fl0A If (FCInv – Dep) and WCInv funded using Debt (based on debt ratio): FCFE = NI + Dep – FCInv – WCInv + Net borrowing where: Net borrowing = DR(FCInv – Dep) + DR(WCInv) ˆ( ˆ# = ˆ( = 1))( ) 26 { * CFA Level 2 (2025) Formula Sheet by Fabian Moa If company issues preferred shares: FCFF = CFO + Int(1 – Tax Rate) + Preferred dividends – FCInv Two-Stage Free Cash Flow Models > & o ,%( = ! d" >:>>d 1 + ‰1:: -'% ‡ o ,%( = ! d" d + >:>>!C 1 § © ‰1:: − * 1 + ‰1:: ! >:>-!C >:>-d 1 + § d 1+ −* 1+ ! © Value of Firm = Value of operating assets + Value of nonoperating assets (PV of FCFF) Learning Module 4 | Market-Based Valuation Price and Enterprise Value Multiples Enterprise value, EV = Market value of common stock + Market value of preferred equity + Market value of debt + Minority interest – Cash and Short-term investments Actual Trailing P/E Leading P/E P/B P/S > J ( ~ ?( ~( )ℎ ( -[$ o( ~ (o %) 12 & ℎ) J ( ~ ?( ~( )ℎ ( (? ) (2 -[$ o( (j 12 & J = J ( ~ ?( ~( )ℎ ( o ,%( ~( )ℎ ( ( ~ ?( ~( )ℎ $ ,() ~( )ℎ ( 27 ( Justified 1− ℎ) 1+* −* 1− −* #b- − * −* Video: https://youtu.be/c0vmCUtDpZs $ = 1− 1+* × $ −* or $ = 1− × $ −* CFA Level 2 (2025) Formula Sheet by Fabian Moa Trailing D/P Leading D/P > Actual 4×J ) J Justified −* 1+* (?( '% ( ,‡ 2 o 2( 2 ( ~ ?( ~( )ℎ ( (? ) 2 o 2( 2) o( ℎ( (j ‡( J ( ~ ?( ~( )ℎ ( -[$ [ ?( ~( )ℎ Earnings yield −* −* 1− 1+* ( Underlying Earnings = Net income – non recurring gains + non recurring loss Normalized Earnings Method 1: Average EPS Approach Normalized EPS = 1 ! " -[$ Method 2: Average ROE Approach Normalized EPS = 1 ! " #b- × :% ( Price-to-Earnings Growth (PEG) Ratio [-¦ = o ,%( ~( )ℎ [/* % = Momentum Indicators Earnings surprise = Reported EPS – Expected EPS · ¹» ¹·º½V½ ¾ ¾ ºiºV¾¹ = #(~ (2 -[$ − -j~(? (2 -[$ p 1 ,‡) (? ) -[$ Standardized unexpected earnings (SUE) = Portfolio P/E ‰( *ℎ (2 ℎ & öG•! !ût þ÷•k• tX öG•! !ût þ÷•k• tX ? &( where: { = Weight of stock i in portfolio = P/E of stock i 28 = 1 ∑!" { ( CFA Level 2 (2025) Formula Sheet by Fabian Moa Learning Module 5 | Residual Income Valuation Economic Value Added (EVA) - 1d = -=9+d 1 − + − ‰1:: × 9 o() (2 : ~ Market Value Added (MVA) J 1d = J ( o ,%( #9d = -d − Residual Income Model Residual Income, RI == +§ #9 1+ ,dE > &d − 9 o() (2 : ~ × =dE + = #b- − #9 1+ + ,d × =dE #9ò 1+ ò +⋯© Video: https://youtu.be/O0KTBkEtP9M Single-stage residual income valuation model #b- − × = #9 == + == + −* −* Video: https://youtu.be/82GJu5umrB0 Tobin’s Q + ß )¶ = J ( o ,%( 2( + J #(~, ?(&( ? ) Continuing Residual Income == + áE d" = Persistence factor #9d 1+ d + 1+ − #9á 1+ ( o ,%( ('% ‡ , ))( ) áE 0≤ ≤1 If RI declines to Long-run level in mature industry, with premium over book value == + á d" #9d 1+ + d [á − =á 1+ á Video: https://youtu.be/vhRW3q70E0w Clean surplus relationship: =d = =dE + -d − ˆ od 29 CFA Level 2 (2025) Formula Sheet by Fabian Moa Learning Module 6 | Private Company Valuation Capitalized Cash Flow Method (CCM) > & o ,%( = > & o ,%( = #( o() &( -'% ‡ o ,%( = >:>> >:>> 1 + * = ‰1:: − * ‰1:: − * → -=9+ 1 − 1 − #9# ‰1:: − * (, #9# = -'% ‡ > & J = − o ,%( o ,%( ( o ,%( ˆ( * ‰1:: >:>- 1 + * −* Excess Earnings Method (EEM) ¹¾¾ z & , ”(2 = − ‰ ¹·º½V½ ¾ ( *) ,%( *: ~ ,× -j?()) ℎ( * ,( ))( ) = #() 2% , o ,%( • − > j(2 1))( ) × *) −* Value of the firm = Working capital + Fixed assets + Intangible Assets where: * = Growth of excess earnings = Required return on intangible assets Video: https://youtu.be/137ga1xgAbA Discount for Lack of Control and Marketability Discount for Lack of Control (DLOC) ˆab: = 1 − 1+: 1 , ~ (& %& Total discount = 1 – (1 – DLOC)(1 – DLOM) ˆabJ = ,%( 1+J ~ $ℎ ( ~ ?( where Strike price is based on forward price, > = $ ( • 30 ×á 1+* Œ CFA Level 2 (2025) Formula Sheet by Fabian Moa Levered and Unlevered Beta Given the beta of a comparable company (Þ‚X Þ÷!‚X X•X„ = ∗ The beta of the private company (Þ‚X ∗ Þ‚X X•X„ = Þ÷!‚X X•X„ ), §1 + .1 − X•X„ ), X•X„ Þ‚X k‚ 41 + .1 − 31 the unlevered beta: X•X„ ˆ( k‚ 0 7-'% ‡ 8© k‚ k d0 ˆ( k d 6 -'% ‡k d CFA Level 2 (2025) Formula Sheet by Fabian Moa VOLUME 6: FIXED INCOME Learning Module 1 | The Term Structure and Interest Rate Dynamics Forward Pricing Model where: ˆ>O = 1 1 + ”O >Œ,OEŒ = .1 + ˆ>O = ˆ>Œ × >Œ,OEŒ O 1 Œ,OEŒ 0 OEŒ Forward Rate Model 1 + ”O O = 1 + ”Œ Œ .1 + OEŒ Œ,OEŒ 0 where: ”O = Spot rate for period = Œ,OEŒ = = − 1 forward rate that starts in period 1 Calculating spot rate from one-period forward rates ”á = ú 1 + ” .1 + , 0.1 + , 0 … .1 + áE , 0ü ⁄á −1 Boostrapping Spot Rates From Par Rates Video: https://youtu.be/-FnweFO172Q Fixed swap rate )á = 1 − ˆ>á 1 − ˆ )? % = ∑ád" ˆ>d $%& > ? ˆ )? % a ) [ ‡&( > ? ) Swap spread = YTM of swap rate – YTM of government bond (same maturity) TED spread = LIBOR – YTM of T-bill (same maturity) LIBOR-OIS spread = LIBOR – OIS Fixed rate 32 CFA Level 2 (2025) Formula Sheet by Fabian Moa For Parallel shifts in yield curve: %∆[ = −J 2ˆ% × ∆ +J %∆[ = −- ∆[ = −J 2ˆ% × ∆ +J × [ ˆ% × ∆:% o( ∆[ = −- ˆ% × ∆:% o( × [ Non-parallel shifts (i.e. change in slope or curvature): %∆[ = −™(‡# (ˆ% Bond Risk Premium = 2 × ∆™(‡ # ( (,2 2( %, (( − , * ( & 2 ) ~ (& %& = (,2 )ℎ 2( %, (( ( & 2 Learning Module 2 | The Arbitrage-Free Valuation Framework Arbitrage-free Value of Bond = : 1+” + : 1+” + ⋯+ > +: 1 + ”! ! where: ”! = Spot rate for period n Backward Induction Valuation Methodology = 2 o ,%( ‡ 2( = 0.5 × + 0.5 × 1+ ý +: where: = bond’s value if the higher forward rate is realized one year hence ý = bond’s value if the lower forward rate is realized one year hence : = coupon payment that is not dependent on interest rates Video (Backward Induction Valuation): https://youtu.be/DhAVQ3hIXlQ Video (Backward Induction with Financial Calculator): https://youtu.be/FycX2UwJxCM Video (Pathwise Valuation): https://youtu.be/3oM-220oi7o Binomial Interest Rate Tree , = ,ý ( , ò, = = ,ýý ( ó ò,ýýý ( , ý ò, = ý ,ýý ( = ò,ýýý ( ó ò,ýý 33 = ò,ýýý ( CFA Level 2 (2025) Formula Sheet by Fabian Moa Equilibrium Term Structure Models Cox-Ingersoll-Ross (CIR) Model 2 = Vasicek Model s− 2 = 2 + p Y d 2” d s− d 2 + p2” where: = Speed of reversion (> 0) s = Long-run interest rate p = Interest rate volatility Arbitrage Free Models Ho-Lee Model 2 d = sd 2 + p2”d Kalotay-Williams-Fabozzi (KWF) Model 2 ln d = sd 2 + p2”d where: sd = Time-dependent drift term Learning Module 3 | Valuation and Analysis - Bonds with Embedded Options Callable and Putable Bonds Value of callable bond = Value of straight bond – Value of issuer call option Value of putable bond = Value of straight bond + Value of investor put option Video (Valuing a callable bond): https://youtu.be/lWLSodiqZaM Video (Valuing a putable bond): https://youtu.be/qmUnAtpXIAg - (? o( 2% - (? o( ? = [E − [ C 2 × Δ:% o( × [ o(j ‡ = [ + [ C −2×[ [ × Δ:% o( E 34 CFA Level 2 (2025) Formula Sheet by Fabian Moa Capped and Floored Floaters Value of capped floater = Value of straight floater – Value of cap Value of floored floater = Value of straight floater + Value of floor Video (Valuing a capped floater): https://youtu.be/d4LNMdXV9vU Video (Valuing a floored floater): https://youtu.be/YJZU0THHBNE Convertible Bonds : o( ) o ,%( uV½V ? o( J = ª 2( ,‡ * : × )ℎ ( ~ ?( ¼· ,( ( ? o( ) ~ ?( ¹ : o( ) = u· § 2 o ,%( = : J ( ? o( ) J = ~ (& %& ~( )ℎ ( J o( ) ( ? o( ) ~ (& %& : [ (& %& o( = $ *ℎ o ,%( = o( ,( : o( ) 2 ~ ?( ( ? o( ) ~ ?( J ,%( $ , − % 2( ,‡ * © *ℎ 2 ª 2( ,‡ * )ℎ ~ ?( ( ? o( ) ~ (& %& ~( )ℎ ª 2( ,‡ * )ℎ ( ~ ?( o( $ ( ( ,( 2 ~ ?( −1 *ℎ o ,%( Convertible Bond (With No Additional Options) ,%( ? ,, ,%( ,%( = + ~ *ℎ 2 ? o( ,( 2 ) ))%( ß ) ) ? Callable Convertible Bond ? ,%( o( ,( 2 = ) ,%( *ℎ Putable Convertible Bond ? ,%( o( ,( 2 = ) ,%( *ℎ ,%( ,%( ? ,, − ))%( ? ,, + ~ 2 ~ ))%( ß ) ) ? ,%( ? ,, + + ~ 2 ß ))%( ) ) ? 35 ,%( o() ~% ~ CFA Level 2 (2025) Formula Sheet by Fabian Moa Learning Module 4 | Credit Analysis Models ¦-)~ ( 2 = +J : ~ ( 2 − +J a )) * o( -j~(? (2 #(? o( ‡ = × R1 − S (j~ )% ( 2( %, ( ¦ o( &( 2 a )) #(? o( ‡ =1− )(o( ‡ ( Expected Loss = Probability of Default × Loss Given Default > ? (2 o ,%( ) ‡ : (2 o ,% 2 = > o ,%( ))%& * 2( 23%) &( , : 1= 2 : (2 ,% − 123%) &( %, ! -ad 1+ d d" d = =1 [bˆ × a¦ˆ 1+ where: -ad = Expected loss of bond at time t [bˆd = Probability of default of bond at time t a¦ˆd = Loss given default at time t = -j~(? (2 -j~ )% (d − #(? o( ‡d d = Risk-free rate at time t = Bond’s remaining tenor -ad [ (j~(? (2 , )) ~( 2 = 1+ d d [bˆd = 1 − @ ” 2 ( dE ×@ ” 2 ( Approximation of credit spread ≈ Annual hazard rate × (1 – Recovery rate) Video (Probability of Default): https://youtu.be/e7K4x48Eg4U Video (Valuing a Credit Risky Bond – Zero Interest Rate Volatility): https://youtu.be/2I9bgu-o7aI Video (YTM of Corporate Bonds – Default and Non-Default): https://youtu.be/K253Y7c2Yto Expected percentage price change of a corporate bond [ , ‡ where: %∆[ = −J 2ˆ% × ∆? (2 )~ ( 2 ? (2 & * 36 × %∆[ CFA Level 2 (2025) Formula Sheet by Fabian Moa Structural Model 1d = ˆ , + + $d In terms of… Equity Debt Call options - + = J j_1 + − ™, 0` Put options ˆ + = 1 + − J j_1 + − ™, 0` - + = 1 + − ™ + J j_™ − 1 + , 0` ˆ + = ™ − J j_™ − 1 + , 0` where: $d = -'% ‡ o ,%( &( 1á = 1))( o ,%( &( + ™ = > ?( o ,%( 2( Learning Module 5 | Credit Default Swaps :ˆ$ )~ ( 2 ≈ 1 − #(? o( ‡ # ( × [bˆ CDS payout amount = Payout ratio × Notional = (1 – Recovery rate of CTD bond) × Notional Upfront payment = PV of protection leg – PV of premium leg [ [ ª~ = − ~ (& %& : (2 $~ ( 2 > j(2 : %~ [ ?( > j(2 : (2 ≈7 − 8 × :ˆ$ ˆ% $~ ( 2 : %~ :ˆ$ ~( 100 , = 100 − ª~ :ℎ *( % :ℎ *( = × ˆ% :ˆ$ ~ ?( )~ ( 2 ~) 37 ~ (& %& CFA Level 2 (2025) Formula Sheet by Fabian Moa VOLUME 7: DERIVATIVES Learning Module 1 | Pricing and Valuation of Forward Commitments Forward Contracts Forward Pricing: > =$ 1+ á > = $ + :: − := > = $ (• á > =$ ( 1+ á • C••E•O á where: $ = Current spot price > = Forward price (set today) = Annually compounded risk-free rate = Continuously compounded risk-free rate :: = PV of Carry cost := = PV of carry benefits :: = Continuously compounded cost of carry := = Continuously compounded carry benefit Forward Valuation (Long Position): =0 d á = >d − > > = $d − áEd 1+ 1+ = $á − > áEd 38 CFA Level 2 (2025) Formula Sheet by Fabian Moa Forward Rate Agreement (FRA) ½ >#1 ~ ‡ >#1 = 7 1 + aá 1 + aq ¹ iVº·UV ½ á q − 18 7 1 B >#1 = z , × aB − >#1 1 + ˆB B 8 Valuation at time = * (prior to FRA expiration): z , × .>#1û − >#1 0 × ,%( a * >#1 * = 1 + ˆáEû áEû where: ˆB = Discount rate for m periods at t = h ℎ = FRA tenor & = Tenor of the underlying rate + = ℎ + & = Maturity of underlying instrument × B B Video (Pricing an FRA): https://youtu.be/uBmAt_z9f3Y Video (Valuing an FRA): https://youtu.be/AYKRVdaYvxY Fixed Income Forwards and Futures Pricing: ¶% (2 % % () : o( ) > = × ? ~ ?( = > = + 19 − 19á − > :9 Valuation for fixed income forward contracts: o ,%( 2 ( ( ?( { 2 ~ ?() d = [ ()( = [ _>d − > ` Valuation for fixed income futures contracts: d = Price change since previous day’s settlement where: = = Quoted bond price z%& ( ?? %(2 2 ‡) ) ?( , ) ? %~ 19 = + , 2 ‡) 2% * ℎ( ? %~ ~ ‡&( 39 ~ ‡&( ~( 2 × 1 % , ? %~ : %~ ('%( ?‡ CFA Level 2 (2025) Formula Sheet by Fabian Moa Interest Rate Swaps (IRS) Fixed Swap Rate = where: [ = 1−[ ! 1 × ! ∑" [ 1[ 1 ˆ ‡) J % ‡ S 360 1[ = Accrual period (i.e., payment frequency of swap) 1 + $~ ( R >$ = 1[ × Fixed Swap Cash Flow Pay-fixed, receive-floating IRS ,%( ${ ~ = z , × >$d − I Receive-fixed, pay-floating IRS ,%( ${ ~ = z ,× I − >$d ! " ! " [ [ Video (Pricing an Interest Rate Swap) : https://youtu.be/0QvtKZutr5E Video (Valuing an Interest Rate Swap): https://youtu.be/_A2a909etvg Currency Swap Pricing for fixed leg of currency swap in currency and 1 − [ !,G 1 × G = ! ∑ " [ ,G 1[ L Fixed Cash Flows = 1 − [ !,L 1 × ! ∑ " [ ,L 1[ >$G = 1[ × >$L = 1[ × 40 G L CFA Level 2 (2025) Formula Sheet by Fabian Moa Value of a fixed-for-fixed currency swap (Receive currency , Pay currency ) ?% ( ?‡ =z ,G × G − $d × z ,L × L •þ •þ ?% ( ?‡ = 1 ×z $d ,G × G −z ,L × L Value of a fixed-for-fixed currency swap (Receive currency , Pay currency ) ?% ( ?‡ = $d × z ,L × L − z ,G × G •þ •þ where: G L = >$G = >$L ! " ! " [ [ ,G ,L ?% ( ?‡ +[ +[ !,G !,L ×[ ×[ =z G L = = ,L × ,%( L − 1 ×z $d ?% ( ?‡ ,%( ?% ( ?‡ $d = Spot exchange rate at time (quoted as ⁄ ) ,G × G ,(* ,(* Video (Pricing a currency swap): https://youtu.be/XZlxcVByc00 Video (Valuing a currency swap): https://youtu.be/3h4mElS48aA Equity Swap Value of equity swap (receive fixed-rate, pay equity return) $d : − ×z ,−[ d [ ö!,d = $dE ,%( Cash flow for equity leg = z -'% ‡ a(* = , × [( $d ×z $dE 2 ? ('% ‡ ( % −z , , where: : = Value at time of a fixed-rate bond initiated with coupon : at Time 0 $d = Current equity index level (on the valuation date) $dE = Equity index level at last reset date 41 CFA Level 2 (2025) Formula Sheet by Fabian Moa Learning Module 2 | Valuation of Contingent Claims Hedge Ratio ℎ G‚‚ ?C − ?E = C ≥0 $ − $E ℎk÷d = ~C − ~E ≤0 $C − $E No-arbitrage Approach: ?=ℎ G‚‚ $ +[ −ℎ G‚‚ $ C + ?C = ℎ G‚‚ $ +[ −ℎ G‚‚ $ E + ?E ~ = ℎk÷d $ + [ .−ℎk÷d $ C + ~C 0 = ℎk÷d $ + [ .−ℎk÷d $ E + ~E 0 Expectations Approach: ˜= where: % = Up factor 2 = Down factor = Risk-free rate One-period Binomial Model ?= ~= 1+ −2 %−2 ˜? C + 1 − ˜ ? E 1+ ˜~C + 1 − ˜ ~E 1+ where: ˜ = Risk-neutral probability of an up-move Note: For interest rate options, ˜ = 0.5 and discount expected option payoff using the 1-period forward rates. Video (Valuing interest rate options): https://youtu.be/X4R8j_cf8SA 42 CFA Level 2 (2025) Formula Sheet by Fabian Moa Two-period Binomial Model: ?= ~= ˜ ? CC + 2˜ 1 − ˜ ? CE + 1 − ˜ ? EE 1+ ˜ ~CC + 2˜ 1 − ˜ ~CE + 1 − ˜ ~EE 1+ For 2-period American-styled call option with dividend in t = 1: $C = % × $ − [ 2 o 2( 2) $E = 2 × $ − [ 2 o 2( 2) ) ) (( (( ( ( Video: https://youtu.be/U_XkIZjJIAU Black-Scholes Option Pricing Model ? = $z 2 − ( E•á z 2 ~ = ( E•á z −2 − $z −2 $ 1 ln R S + R + p S + 2 2 = p √+ 2 = 2 − p √+ Put-call parity: • • • • ~ + $ = ? + ( E•á Hedge ratio for calls = z 2 Probability that the call option expires in the money = z 2 = [ Hedge ratio for puts = z 2 − 1 = −z −2 Probability that the put option expires in the money = 1 − z 2 [ $á < = z −2 43 $á > CFA Level 2 (2025) Formula Sheet by Fabian Moa BSM model with carry benefits ? = $( E"á z 2 − ( E•á z 2 ~ = ( E•á z −2 2 = − $( E"á z −2 $ 1 ln R S + R − # + 2 p S + p √+ 2 = 2 − p √+ ~ + $( E"á = ? + ( E•á Put-call parity: Black Option Valuation Model European Options on Futures ? = ( E•á _> + z 2 ? = ( E•á _ z −2 2 = − z 2 ` − > + z −2 ` > + 1 ln § ©+ 2p + p √+ 2 = 2 − p √+ ? = ( E•á _> + − ` + ~ Put-call parity: Interest Rate Options ? = 1[ ( E•.dM$fCd% 0 ú>#1.0, KE , ~ = 1[ ( E•.dM$f Cd%0 ú# ( E•á z −2 2 = >#1.0, ln 4 KE , B 0z B0 − # ( E•á z 2 ü − >#1.0, 1 6 + R2 p S p Y KE 2 = 2 − p Y KE 44 2 KE KE , B 0z −2 ü CFA Level 2 (2025) Formula Sheet by Fabian Moa Payer Swaption [1 þ & = 1[ × _# z 2 −# z 2 `× ! K" Receiver Swaption #-:þ & = 1[ × _# z −2 −# z −2 ` × 1 # ln R # S + R2 p S + 2 = p √+ 2 = 2 − p √+ Video (Interest Rate Options & Swaptions Equivalences: https://youtu.be/uZQO50sEzso Optimal Number of Hedging Units (for Delta Hedging) z =− [ , ˆ(, Video: https://youtu.be/v8RcvkQKFpw Option Greeks ˆ(, ˆ(, G‚‚ k÷d = ( E'á z 2 = −( E'á z −2 where: ¨ = Continuously compounded dividend yield ¦ && G‚‚ = ¦ && ? = ? + ˆ(, ~ = ~ + ˆ(, G‚‚ k÷d k÷d = ( E'á $p √+ z 2 1 × ($ + ¦ && 2 1 × ($ + ¦ && 2 G‚‚ k÷d × ($ × ($ 45 2(, [ ! K" K 1 [ K 1 CFA Level 2 (2025) Formula Sheet by Fabian Moa VOLUME 8: ALTERNATIVE INVESTMENTS Learning Module 1 | Introduction to Commodities and Commodity Derivatives $~ ~ ?( >% % () $ *( : = + − ~ ?( ~ℎ‡) ? , ? && 2 ‡ ? )) a *( ( & z( ( & : ,( 2 % % () ? ? ? = − % % () ? )~ ( 2 ?, ) * ~ ?( ?, ) * ~ ?( [ ?( ( % # ,, ( % = :% ( o( ( ?( ‡ (,2 ~ ?( − [ (o %) ~ ?( [ (o %) ~ ?( a *( ( & z( ( & ? − % % () ? ? ¬ % % () ? [( ?( *( ℎ( ~ ) ?, ) * ~ ?( ?, ) * ~ ?( ℎ( % % () ? ? = × z( ( & ( * ,,(2 % % () ? ? ?, ) * ~ ?( [ ?( + , # ,, : ,, ( = + + (% (% (% (% , + (% #( , ? * 2(j ,‡ Learning Module 2 | Overview of Types of Real Estate Investment Net Operating Income - (? o( * zb9 = ? &( b~( ¦ )) ( )) − b~( * [ − (j~( )() = = )( ( × #( ~( ‡ & ,, { ,( * (j~( )( (? o( ‡ = #(? o( ‡ × #( Loan-to-value ratio Debt service coverage ratio -'% ‡ 2 o 2( 2 a+ = ˆ$: = (= [ ( ( ?( ,( ?( ( ˆ( %) 2 * [ ~( ‡ o ,%( z( ~( ˆ( * ? &( )( o ?( [ ( j ? )ℎ , { ~( ‡ ~% ?ℎ )( ~ ?( − J 46 * *( , CFA Level 2 (2025) Formula Sheet by Fabian Moa [ ( j ? )ℎ , { = z( 1 ( ~( * ? &( − ˆ( j ? )ℎ , { = [ ( j ? )ℎ , { − + j() )( o ?( Where: + j() = × zb9 − 9 ( () (j~( )( − ˆ(~ (? Funds from Operations >>b = z( ? &( + ˆ(~ (? Gross potential rental income ¦[#9 = J + 1& ( ( ” × #( (j~( )( − z( * ) & ~ ~( ‡ ) ,() ,( )~ ?( Income Approach Direct Capitalization method [ where: ~( ‡ o ,%( = -j~(? (2 zb9 : ~ ,” ( : ~ ,” ( = −* = Required return on similar properties * = Constant growth rate Discounted Cash Flow (DCF) Methodology [ ~( ‡ o ,%( = +( & ! d" zb9d +( & , o ,%( + d 1+ 1+ ! , o ,%( = zb9! 1 + * −* Appraisal-based index #( % = zb9 − : ~ , =(* * - 2 * +R − S -j~( 2 % () ( o ,%( & ( o ,%( & =(* *& ( o ,%( Adjustment for Appraisal Lag #d = #d∗ +7 1− ∗ 8 #dE where: #d = Actual index return #d∗ = Appraisal index return ∗ #dE = Prior period’s appraisal index return = Speed at which actual returns are reflected in appraisal-based returns 0 ≤ 47 ≤1 CFA Level 2 (2025) Formula Sheet by Fabian Moa Learning Module 3 | Investments in Real Estate Through Publicly Traded Securities Net Asset Value Approach z1 ~( )ℎ (= ~( ,%( ,%( + , 2( + − * () ( 2, , () ℎ( ))( ) z%& ( )ℎ () % ) 2 * If valuation of operating real estate is not provided: ,%( ~( * () (= zb9 : ~ ( Video: https://youtu.be/WncC3BZmfs8 zb9 = ¦ )) ( , -) & (2 o ? ?‡ − (o( %( ? ,,(? ) , )) 2 − b~( * (j~( )() − z( * ) & ~ ~( ‡ ) ,() Relative Value Approach Funds from Operations: >>b = z( ? &( + ˆ(~ (? Adjusted Funds from Operations: 1>>b = >>b − z + 1& ( ? )ℎ − ” #(?% *? ~ , (j~( 2 % ( Two-Stage Dividend Discount Model ,%( #-9+ )ℎ (=[ 2 o 2( 2) + [ 48 ( & , o ,%( CFA Level 2 (2025) Formula Sheet by Fabian Moa Learning Module 4 | Hedge Fund Strategies Equity Market Neutral Pairs Trading 1& % $ℎ [ ) bo( o ,%(2 $ ? =− =( % 2( o ,%(2 ) ? × 1& % =( o( o ,%(2 ) ? 9 o() (2 Merger Arbitrage Strategy For a stock-for-stock deal: [ ‡ &( *( ) )%??()) %, = zŒ × [Œ − zá × [á where: zŒ = Number of acquirer’s shares to short sell [Œ = Share price of acquirer post announcement zá = Number of target’s shares to buy [á = Share price of target post announcement Convertible Arbitrage Strategy : : o( ) o( ) ~ ?( = : o ,%( = $ℎ o( ,( : o( ) ( ~ ?( × : 2 ~ ?( o( ) Conditional Factor Risk Model # ,d = › + Þ , > ? 1 d+Þ, > ? 2 d + ⋯ + Þ ,) > ? ™ d+ +ˆd Þ , > ? 1 d + ˆd Þ , > ? 2 d + ⋯ + ˆd Þ ,) > ? ™ d + ( where: # ,d = Return of hedge fund in period Þ ,) > ? ™ d = Exposure to risk factor ™ for hedge fund in period during normal times ™ d = Incremental exposure to risk factor ™ for hedge fund in period ˆd Þ ,) > ? during financial crisis periods ˆd = Dummy variable that equals 1 during financial crisis periods (0 otherwise) › = Intercept for hedge fund ( ,d = Random error with zero mean and standard deviation p 49 ,d CFA Level 2 (2025) Formula Sheet by Fabian Moa VOLUME 9: PORTFOLIO MANAGEMENT Learning Module 1 | Economics and Investment Markets One-period real-risk free rate: ,d, = 1 * d, ü -d ú& where: * d, ü = Inter-temporal rate of substitution -d ú& [ ?( where: öúP+,-f,ç$f ü C‚,,f ? od ú[+dC ) ‡ ))( = −1 -ú[+dC ,tE ü + ? od ú[+dC 1 + ,d, ,tE * d, ü ,& = risk neutral present value * d, ü = covariance between investor’s inter-temporal rate of substitution ,& and the random future price the investment at t + 1, based on the information available to investor today. ) = time to maturity of investment ,tE Default-free nominal coupon-paying bond [d = & t" :>dCt 1 + ,d,t + sd,t + ˜d,t t where: ,d,t = Real-risk free rate sd,t = Expected inflation rate ˜d,t = Uncertainty in future inflation rate sd,t + ˜d,t = Breakeven rate of inflaton Short-dated nominal zero-coupon government bonds (e.g., T-bills) [d = :>dCt 1 + ,d,t + sd,t 50 t CFA Level 2 (2025) Formula Sheet by Fabian Moa ~ Taylor Rule = 9d + ˜d + 0.5 ˜d − ˜d∗ + 0.5 d d − ∗ d where: ~ d = policy rate at time t 9d = level of real short-term interest rates that balance long-term savings and borrowing in the economy ˜d = rate of inflation ˜d∗ = target rate of inflation d = logarithmic level of actual GDP ∗ d = logarithmic level of potential real GDP ∗ d − d = output gap Corporate bond [d = & t" . dCt ü -d ú:> 1 + ,d,t + sd,t + ˜d,t + #d,t t where: #d,t = Credit premium Equity [d = t" [d = . dCt ü -d ú:> 1 + ,d,t + sd,t + ˜d,t + #d,t + /d,t t" . dCt ü -d ú:> 1 + ,d,t + sd,t + ˜d,t + õd,t t t where: /d,t = Equity premium relative to risky bonds õd,t = #d,t + /d,t = Equity risk premium Commercial Real Estate [d = & t" where: 0d,t = liquidity risk premium . dCt ü -d ú:> 1 + ,d,t + sd,t + ˜d,t + #d,t + /d,t + 0d,t 51 t CFA Level 2 (2025) Formula Sheet by Fabian Moa Learning Module 2 | Analysis of Active Portfolio Management Active return, #Œ = #P − #O 1,~ℎ , ›k = #k − Þk #O #Œ = ,%( 22(2, $#P = $#O + 9# â K" ∆{K #O,K + â K" {P,K #Œ,K p #P = p #O + p #Œ For optimal Sharpe ratio, + ) ( : ( 9 & 9: ≈ 2 [ ?( , : ( , ‡ p #Œ = +: = : ?( , 9: = : 9# p #O $#O † 7 , ∆{ p 8 p *ℎ ? ,, − 1 #Œ † , 8 7 p p Forecasted active return, † = 9: × p × $ where: $ is set of standardized forecasts of expected returns across securities Mean-variance optimal weights Full Fundamental Law ∆{ ∗ = † pŒ p 9: √=# - #Œ = +: × 9: √=#pŒ 9# = +: × 9: √=# p #Œ = +: × 9# ∗ p #O $#O $#P = $#O + +: 52 9#∗ CFA Level 2 (2025) Formula Sheet by Fabian Moa Performance Measurement #Œ = - #Œ |9: + z )( +: = Proportion of variation in realized performance attributed to realized information coefficient where: 9: = realized information coefficient Ex-ante measurement of skill - #Œ = Independence of Investment Decision =# = 9: p p• Œ z 1+ z−1 m Learning Module 3 | Exchange-Traded Funds: Mechanics and Applications End-of-day ETF premium or discount (%) -+> ~ ?( − z1 ~( )ℎ z1 ~( )ℎ ( ( Intraday ETF premium or discount (%) @ ,2 * ~( # % 2 ~ 2? ) -+> ~ ?( − 9 2 ? (2 z1 ~( )ℎ 9 2 ? (2 z1 ~( )ℎ ( % =# % 2 ~ 2( ? ) 2( ? ) % = b ( { ‡ ? && )) 53 ( % +J *(&( (( % % × 2 + = 2 ) )~ ( 2 % CFA Level 2 (2025) Formula Sheet by Fabian Moa Learning Module 4 | Using Multifactor Models Arbitrage Pricing Theory (APT) - #k = # + õ Þk, + ⋯ + õ) Þk,) where: - #k = the expected return to portfolio p # = the risk-free rate Þk,K = the sensitivity of the portfolio to factor j õK = the expected reward for bearing the risk of factor j ™ = the number of factors Carhart Four-Factor Model -.#k 0 = # + Þk, #J#> + Þk, $J= + Þk,ò @Ja + Þk,ó ‰Ja where: #J#> = Return on a value-weighted equity index minus one-month T-bill rate SMB = small minus big; average return on three small-cap portfolios minus the average return on three large-cap portfolios HML = high minus low; average return on two high book-to-market portfolios minus average return on two low book-to-market portfolios WML = winners minus losers, a momentum factor; return on a portfolio of past year’s winners minus return on a portfolio of past year’s losers. Macroeconomic Factor Model # = + > + > +⋯+ ) >) + where: > = the surprise in the factor k = the sensitivity of the return on asset i to a surprise in factor k, k = 1, 2, …, = Expected return on the portfolio Fundamental Factor Model = Value of attribute for asset − Average value of attribute p Values of attribute 54 CFA Level 2 (2025) Formula Sheet by Fabian Moa Return Attribution 1? o( ( % + 9 ? & = #P − #O = *( , ) " , [ 47 )( ) , =( ?ℎ& > ? 8 −7 8 6 ×R )( ) o ‡ o ‡ (% +- = ) #P − #O 9# = S + $(?% )(,(? ‡ #P − #O ) #P − #O Active risk squared = Active factor risk + Active specific risk Learning Module 5 | Measuring and Managing Market Risk Parametric VaR (Using Normal Distribution) ,%( # = −ú-.#k 0 − ” × pk ü × #) , [ , ,%( where: -.#k 0 = Portfolio expected return pk = Portfolio standard deviation Two-asset portfolio: -.#k 0 = { - # +{ - # pk = { p + { p + 2{ { m , p p Scaling from daily returns to annual returns (Assuming 1 year = 250 trading days): #„G ‚ø × 250 2 * 2 ‡) Scaling from daily standard deviation to annual standard deviaton: 9 ? (&( , # 9 # = # Percentage change in bond price: ΔB ≈ −ˆ% = p„G ‚ø × √250 ( ?ℎ *( − Δ‡ 1 + : 1+‡ 2 55 # ( ( ?ℎ Δy o(j ‡ 7 8 1+‡ *( CFA Level 2 (2025) Formula Sheet by Fabian Moa New call price: ? + Δ? ≈ ? + Delta Δ$ + Gamma Δ$ New put price: ~ + Δ~ ≈ ~ + Deltak Δ$ + Gammak Δ$ Learning Module 6 | Backtesting and Simulation No formula. 56 + (* + (* Δp k Δp