CFA L2 Formula Cheat Sheet

Fin Quiz Formula Sheet CFA Program Level I QUANTITATIVE METHODS Learning Module 1: Basics of Multiple Regression and Underlying Assumptions 1. Multiple Regression 1, 2, ... n. Y i = b_{0} + b_{1}*X_{1i} + b_{2}*X_{2i} +...+b k X ki + epsilon i ,i= 2. Prediction Equation Prediction equation = hat Y i = hat b 0 + hat b_{1}*X_{11} + hat b_{2}*X_{21} +...+ hat b k X kl + epsilon i , Learning Module 2: Evaluating Regression Model Fit and Interpreting Model Results 1. Coefficient of Determination: R² Sum of square regression Sum of square total = sum i=1 ^ n ( overline Y - overline Y )^ 2 sum i=1 ^ n (Y i - overline Y )^ 2 2. Adjusted R2 overline R ^ 2 = 1 - ((n - 1)/(n - k - 1))(1 - R ^ 2) 3. F-Statistic or F-Test MSR MSE Sum of square regression Sum of squares err n-k-1 where, df numerator = k = 1 df = n - k - 1 = n - 2 4. ANOVA ANOVA SS F MSS Regression df = 1 SSR = sum i=1 ^ n ( overline y_{i} (SSR)/k Error df = n - 2 SSE = sum t=1 ^ n ( y_{t} - hat y )^ 2 (SSE)/(n - k - 1) SSR/ k SSE / (n - k - 1) Total df = n - 1 SST = sum (i = 1)/n (y i - overline y )^ 2 5. Akaike's information criterion (AIC) AIC = n * ln((5SE)/n) + 2(k + 1) 6. Schwarz's Bayesian Information criterion (BIC or SBC) BIC = n * ln((SSE)/n) + ln(n) * (k + 1) Learning Module 3: Model Misspecification 1. Breusch-Pagan (BP) test Test statistic = n * R ^ 2 where, R ^ 2 resicfuals =R^ 2 a second regression of the squared residuals from the first regression on the independent variables n = number of observations 2. Variance Inflation Factor VIF 1 VIF = 1-R Learning Module 4: Extensions of Multiple Regression 1. Studentized residual t t l^ * = e i ^ * S e^ * = sqrt[n] n - k - 1 SSE(1-h ii )-e l ^ 2 2. Cook's distance D l = (e_{i} ^ 2)/(kMSE) [ h il (1-h il )^ 2 ] D_{1} 3. Linear regression with 3 independent Variables Y_{i} = b_{0} + b_{1}*X_{1i} + b_{2}*X_{2i} + b_{3}*X_{3i} + e_{i} 4. Logistic regression (logit) = ln(p/(1 - p)) = b_{0} + b_{1}*X_{1} + b_{2}*X_{2} + b_{3}*X_{3} + epsilon P= 1 1+ exp [-(b1X1+b2X2+b3X3+ epsilon)] Learning Module 5: Time Series Analysis 1. Linear Trend Models y_{t} = b_{0} + b_{1}*t + epsilon_{t} Predicted/fitted value of yt in period (T + 1) = hat y t + 1 = hat b 0 + hat b 1 (T+1) 2. Log-Linear Trend Models y_{t} = e ^ (h_{0} + b_{1}*t) 3. Autoregressive Time-Series Models First order autoregressive AR (1): x t =b 0 +b 1 x v - 1 + epsilon t pth-order autoregressive AR (p): x t = b_{0} + b_{1}*x_{-1} +b 2 x 1-2 +.....+b p x t-p + epsilon t 4. Mean reverting level x_{t} = b_{0}/(1 - b_{1}) 5. Chain Rule of Forecasting: One-period ahead forecast hat x t + 1 = hat b 0 + hat b t x t Two-period ahead forecast= hat x t + 2 = hat b 0 + hat b 1 x t+1 6. Random Walks and Unit Roots: Random Walk without drift: x t =x t - 1 + epsilon t where, b_{0} = 0 and b_{1} = 1 Correcting Random Walk: y_{t} =x t -x perp-1 Random walk with a drift: x t =b 0 +x t * 1 + epsilon t where, b_{0} ne0 and b_{1} = 1 By taking first difference y t =x t -x t - 1 =b 0 + epsilon t 7. Using Dickey-Fuller Test x t -x t - 1 =b 0 +(b 1 -1)x t-1 + epsilon t 8. Smoothing Past Values with n-Period Moving Average x t +x t - 1 +x t-2 +.....+x t-(n-1) n 9. Correcting Seasonality in Time Series Models: For quarterly data x t =b 0 +b 1 x 1 - 1 +b 2 x 1-4 + epsilon t For monthly data x t =b 0 +b 1 x 1 - 1 +b 2 x 1-12 + epsilon t 10. ARCH model = hat epsilon^ 2 t = alpha 0 + alpha 1 hat epsilon ^ 2 t - 1 + mu_{t} where mu_{t} is an error term Predicting variance of errors in period t + 1 = hat sigma t+1 ^ 2 = hat alpha 0 + alpha 1 hat epsilon t ^ 2 Learning Module 6: Machine Learning 1. LASSO: Penalty term (when lambda > 0 )= lambda Sigma k=1 ^ K | overline b k | sum l = 1 to n (Y_{l} - Y_{l}) ^ 2 + lambda sum k=1 ^ K | tilde b k | When lambda = 0 LASSO penalized regression = OLS regression Learning Module 7: Big Data Projects 1. Normalization X l(normalixed) = X l -X min X max -X min where X_{i} = value of observation Performance Metrics: 2. Accuracy = (TP + TN)/(TP + FP + TN + FN) F1 score = (2*P*R)/(P+R) where T = true, F = 1 false, P = positive, N = negative 3. Receiver Operating Characteristic (ROC): False positive rate: FPR FPR = FP TN+FP True positive rate TPR: TPR TP TN+FP 4. Root Mean Square Error: RMSE: RMSE = -1 (Predicted-Actual() 21 ECONOMICS Learning Module 1 Currency Exchange Rates 1. Bid-offer Spread Offer price - Bid price 2. Forward Rate: Fwd rate Spot Exchange rate Forward points + 10,000 3. Forward Premium or Discount: spot exchange rate forward points 10,000 spot exchange rate -1 4. To convert spot rate into forward quote: Spot exchange rate x (1+% premium) Spot exchange ratex (1-% discount) 5. Covered interest rate parity: (1 + i_{d}) =S f/ d (1+i f )( 1 F f/d ) F f / d =S f/d ( 1+i f (1+i d ) ) Using day count convention: S fld (1+i f [ Actual 360 ])( 1 F fld ) ( 1 +i d [ Actual 360 ])= F j / d =S f/d ( 1+i s [ Acmad 360 ] 1+i s [ Acmad 360 ] ) 6. Uncovered Interest Rate Parity: i f -\% Delta S^ epsilon f / d = i_{d} \% Delta S^ e f / d = i_{f} - i_{d} Forward premium or discount: For one year horizon = S f / d ( i f -i d 1+i d ) equiv S f/d (i f -i d ) F f|d -S f|d = Using day count convention: F f, mu ^ prime -S f, mu =S f, mu [ [ Armol 360 ] 1+ij[ Armol 360 ] (i f, mu - hat t mu ) 7. Forward discount or premium as % of spot rate: Frid-Spid Sfid =(i-1) If uncovered interest rate parity holds = F fid -S fid S fid =\% 0 ^ f Delta S fid ^ c equiv(i f -i d ) 8. Purchasing Power parity (PPP) P f =S 1 / d * P d S t / d =P t /P d 9. Relative version of PPP =\% Delta S v/d = Pi*r - Pi_{d} 10. Ex ante version of PPP =\% Delta S^ e V emptyset = Pi ^ e * t - Pi^ e d 11. Real Exchange Rate SidPa P Sid qr/d= or q f / d =S f/d ( CPI d CPI f ) 12. i d =r d + Pi ^ c d i f =r f + Pi ^ 2 f i r -i d = (r_{t} - r_{d}) +( Pi^ e r - Pi^ e d ) (r_{t} - r_{d}) =(i t -i d )-( Pi^ e t - Pi^ e d ) Learning Module 1 Economic Growth 1. Economic growth: Annual % A in real GDP or in real per capita GDP 2. Relation between Stock Prices and Economic growth: P = GDP(E/(GDP))(P/E) where, P = Aggregate value (price) of equities E = Aggregate corporate earnings 3. Expressing in terms of logarithmic rates: (1/Τ) % ΔΡ = (1/T) % AGDP + (1/T) %A (E / G * DP) + (1 / T)% * Delta(P / E) % Delta in stock MV = % A in GDP + % A in share of earnings (profit) in GDP + % Delta in the P/E multiple 4. A two-factor aggregate production function: Y = AF(K, L) Y = Level of aggregate output in the economy L = 1 Quantity of labor or number of workers or hours in the economy K = Stock of capital used to produce goods and services A = Total Factor Productivity (TFP) 5. Cobb-Douglas Production Function F(K, L) = K ^ alpha * L ^ (1 - alpha) 6. Under Cobb-Douglas production function: Marginal product of capital = MPK = alpha AK alpha * 1L * 1a = alpha*Y / K aY/K=ra= r (K) / Y = Capital income/Output or GDP 7. Output per worker or Average labor productivity ( Y / L or y): GDP/Labor input = TFP capital-to-labor ratio x share of capital in GDP Or y = Y / L = A * K ^ alpha 8. Contribution of Capital Deepening = Labor productivity growth rate - TFP 9. Contribution of Improvement in technology Labor productivity growth rate - Capital Deepening 10. Growth Accounting based on Solow Approach ΔΥ/Υ - ΔΑ/Α + α ΔΚ/Κ + (1 - α) AL/L 11. Labor productivity growth accounting equation Growth rate in potential GDP = long-term growth rate of labor force + long-term growth rate in labor productivity 12. Balanced or Steady State Rate of Growth in Neoclassical Growth Theory: Growth in physical capital stock = Delta*K = SY - delta*K 13. In the steady state: Growth rate of capital per worker = Ak/k=Ay/y = ΔΑ/Α + α Ak / k = (TFP)/(1 - alpha) = Steady state growth rate of labor productivity. Growth rate of Total output = Delta*Y / Y = 1 Growth rate of TFP scaled by labor force share + Growth rate in the labor force = theta/(1 - a) + n Steady state Output-to-capital ratio = Y/K * (1/s)[(theta/(1 - alpha)) + delta + n] = Psi Gross investment per worker = [(theta/(1 - alpha)) + delta + n] * k Slope of straight line = [8+n+0/(1 - α)] 14. During the transition to the steady state growth path: Growth rates of output per capita =Ay/y=[()+as(-)] = Ψ (theta/(1 - a)) + as(y / k - Psi) Capital-to-labor ratio s =Ak/k=[()+(-)] = (theta/(1 - a)) + s(y / k - Psi) 15. Proportional impact of the saving rate change on the capital-to-labor ratio and per capita income over time: k new K new Less: Kold Y Kold y nav y old =[ k nav k old ]^ alpha 16. Production function in the endogenous growth model y_{e} = f(k_{e}) = c*k_{e} Growth rate of output per capita Aye/Ye Ake/ke = sc-6-n Learning Module 3 Economics of Regulation FINANCIAL STATEMENT ANALYSIS (1) Learning Module 1 Interoperate Investments 1. Goodwill: = Cost of acquisition investor's share of the FV of the net identifiable assets 1 a-1 Y Purchase Price Xxx (XXX) (% of Ownership Interest x BV of Investee's Net Assets) = Excess Purchase Price Xxx Less: Attributable to Net Assets P& E (% of Ownership Interest x difference b/w BV & FV) Land (% of Ownership Interest difference b/w BV & FV) = Residual Amount (Treated as Goodwill) Xxx where P & E = Plant and Equipment