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Sample Covariance

$$Cov_{XY}=\frac{\sum_{i=1}^n(X_i-\overline X)\left(Y-\overline Y\right)}{n-1}$$

Sample Correlation Coefficient

$$r_{x,y}=\frac{Cov\left(x,y\right)}{\sigma_x,\sigma_y}$$

t-test statistics with

$$d_f=n-2=\sqrt{\frac{n-2}{1-r^2}}\\H_0:\rho=0\\H_0:\rho\neq0\\t_{stat}=\sqrt{\frac{n-2}{1-r^2}}\;Accept\;H_0\;if\;-t_{\;critical}<\;t\;_{stat}<\;+t_{critical}\\$$

$$Y_i\;=b_o+b_1X_{1\;}+\varepsilon_i\\\overset\frown Y=b_o+b_1X_\;\\$$

Slope Coefficient

$$\frac{Cov(x,y)}{\sigma x^2}=\frac{r(x,y)\sigma y}{\sigma x}\\$$

Intercept Term

$$\widehat{b_0}=\overline Y-\widehat{b_1}\overline X\\$$

SSE = Squared vertical distance between estimated Y value and actual Y value. (Regression line minimizes the SSE)

$$Slope\;Term(b_1)=\frac{r\times\sigma y}{\sigma x}=\frac{Cov(x_1,y)}{\sigma^2x}$$

$$\widehat{b_0}=\overline y-\widehat{b_1}\overline x\\$$

SEE (Standard error of Estimates)

$$Degree\;of\;variability\;of\;actual{\;y}_i\;value\;relative\;to\;estimated\widehat{\;y}\;value.\\SEE=\sqrt{MSE}=\sqrt{\frac{SSE}{n-k-1}}\\\lbrack SEE\downarrow\;relationship\;strong\rbrack\\\lbrack SEE\uparrow\;relationship\;weak\rbrack\\$$

$$R^2=\frac{RSS}{TSS}=\frac{ESS}{TSS}=\frac{TSS-SSE}{TSS}\\Confidence\;Interval\;for\;b_1={\widehat b}_1\pm(t_c\times s_{{\widehat b}_1})\\Confidence\;Interval\;for\;Y=\widehat Y\pm(t_c\times s_f)\\s_f=Standard\mathit\;error\mathit\;of\mathit\;forecast\\$$

Testing of Hypothesis

$$t_{b_1}=\frac{\widehat{b_1}-b_1}{S_\widehat{b_1}}\\S_f^2={SEE}^2\left[1+\frac1n+\frac{\left(x-\overline x\right)^2}{(n-1)s_x^2}\right]\\$$

The General Multiple Linear Regression Model

$$Y_i=b_0+b_1X_{1i}+b_2X_{2i}+...+b_kX_{ki}+\varepsilon_i$$

Hypothesis Testing of Regression Coefficient

$$t\;=\frac{b_j-b_i}{S_{{\widehat b}_j}}\;\;(n\;–\;k\;–\;1)\;degrees\;of\;freedom\\$$

$$P\;value: Smallest\;level\;of\;significance\;for\;which\;the\;null\;hypothesis\;can\;be\;rejected$$

$$P\;value\;>\;a;\;H_0\;accept\;(a\;=\;Significance\;level)$$

$$P\;Value\;<\;a;\;H_0\;rejected$$

Confidence Interval

$$SSE=\sum{(y_i\;\;-\widehat y)}^2=\sum i^2\\\sigma\varepsilon^I\;=\;\sigma(y_i\;\;-\widehat y)\\SEE/SER=\sqrt{\frac{\sum{(x\;-\overline x)}^2}{n-2}}=\sqrt{\frac{\sum{(\varepsilon_i\;-\overline{\varepsilon_l})}^2}{n-2}}$$

$$If\;\overline{\varepsilon_l}=0;\;then:\sqrt{\frac{\sum\overline{\varepsilon_i}^2}{n-2}}=\sqrt{\frac{SSE}{n-2}}=\sqrt{MSE}$$

Confidence Interval for A Regression Coefficient

$${\widehat b}_j\pm(t_c\times s_{{\widehat b}_j})\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\\\;\;\;\;\;\;\;s\tan dard\;error\;of\;b_i\;\\{\widehat b}_j\rightarrow use\;(n-k-1)\;d_f\\where\;K\;=\;no\;of\;independent\;variables\\S_{bI\;}=f_n\;\lbrack SEE\rbrack\\$$

Confidence interval for forecasted variables

$$Y=\widehat Y\pm(t_c\times s_f)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\\\;\;\;\;\;\;\;S.E\;of\;\widehat Y\\TSS=SSE+ESS\\SSR\;\;\;\;\;\;\;RSS\\\sum{(y_i-\overline y)}^2=\sum{(y_i-\widehat y)}^2+\sum{(\widehat y-\overline y)}^2$$

$$ESS\;/\;RSS\;=\;the\;difference\;that\;is\;explained\;by\;independent\;variable\\R^2=\frac{ESS}{TSS}=\frac{TSS-SSR}{TSS}=1-\frac{SSE/SSR}{TSS}\;\\R^2\;is\;Always\;+ve$$

Predicting the dependent variable

$${\widehat Y}_i={\widehat b}_0+{\widehat b}_1{\widehat X}_{1i}+{\widehat b}_2{\widehat X}_{2i}+...+{\widehat b}_k{\widehat X}_{ki}$$

The F – Statistics

$$F-stat=\frac{MSR\;}{MSE}(one\;–\;tailed\;test)\\(RSS/ESS)\\Where\;MSR=mean\;regression\;sum\;of\;squares\\MSE=mean\;squared\;error\\(SSR/SSE)$$

Coefficient of Determination

$$R^2=\frac{RSS}{TSS}=\frac{ESS}{TSS}=\frac{TSS-SSE}{TSS}$$

Adjusted R2

$$R_a^2=1-\left[\left(\frac{n-1}{n-k-1}\right)\times\left(1-R^2\right)\right]$$

Anova Table

$$R^2=\frac{RSS}{TSS}\\F=\frac{MSR}{MSE}with\;k\;and\;n-k-1\;degrees\;of\;freedom\\MSR=\frac{RSS}K=\frac{RSS}1;\;K=1\;for\;simple\;linear\;regression\\MSE=\frac{SSE}{n-K-1}=\frac{SSE}{n-2}\\If\;K\;\geq\;1;\;R^2>{R_a\;}^2\;(No\;comparison\;of\;R^2\;\&\;{R_a\;}^2\;)\\{R_a\;}^2=1-\left[\left(\frac{n-1}{n-k-1}\right)\left(1-R^2\right)\right]$$

 

Linear Trend Model

$$Y_t\;=\;b_0+b_1(t)+\varepsilon_t$$

Ordinary Least Squares (OLS) regression

$${\widehat Y}_i=\;{\widehat b}_0+{\widehat b}_1(t)$$

Log - Linear Trend Model

$$Y_t\;=\;e^{b_0+b_1(t)}$$

Autoregressive Model (AR)

$$X_t\;=\;b_0+b_1X_{t-1}+\varepsilon_t$$

AR (p) Model

$$X_t\;=\;b_0+b_1X_{t-1}+b_2X_{t-2}+b_pX_{t-p}...+\varepsilon_t$$

Autocorrelation & Model Fit

$$t=\frac{\rho_{(e_{t,}e_{t-k})}}{\displaystyle\frac1{\sqrt T}}$$

Random Walk with A Drift

$$X_t\;=\;b_0+b_1X_{t-1}+\varepsilon_t$$

Covariance Stationarity

$$X_t\;=\;b_0+b_1X_{t-1}+\varepsilon_t$$

Unit Root Testing for Non-Stationary

$$X_t\;=\;b_0+b_1X_{t-1}+\varepsilon\\X_t-X_{t-1}\;=\;b_0+b_1X_{t-1}-X_{t-1}+\varepsilon$$

First Differencing

$$Y_t=X_t-X_{t-1}\Rightarrow Y_t=\varepsilon_t\\Y_t=b_0+b_1Y_{t-1}+\varepsilon_t$$

ARCH (1) Regression Model

$${\widehat\varepsilon}_t^2=a_0+a_1{\widehat\varepsilon}_{t-1}^2+\mu_t$$

Predicting the Variance of a Time Series

$${\widehat\sigma^2}_{t+1}={\widehat a}_0+{\widehat a}_1{\widehat\varepsilon}_t^2\\$$

$$Mark\;To\;Market\;Value\;=\frac{\left(Forward\;Price\;New-F.P\;locked\;oldሻ\;ሺሺContract\;Size\right)}{\left[1+r\;\left({\displaystyle\frac n{360}}\right)\right]}\\\\$$

Covered Interest Rate Parity

$$F_{A/B}=S_{A/B}\left(\frac{1+iA}{1+iB}\right)\\F\left(disc/premium\right)=S_A+\left(\frac{\left(iA-iB\right)\left({\displaystyle\frac n{360}}\right)\rightarrow if\;Libor\;rates}{\left(1+iB\right)n/360)}\right)$$

Uncovered Interest Rate Parity

$$E{(\%\Delta S)}_{\;(A/B)}=R_A-R_B\\$$

International Fisher Relation

$$R_{nominal\;A}-R_{nominal\;A}=\;+\;E\;(Inflation_A)\;-\;E\;(Inflation_B)\\$$

Purchasing Power Parity

$$F=S\left(\frac{1+Inflation_A}{1+Inflation_B}\right)\\$$

Absolute PPP

$$S(A/B)=\frac{CPI(A)}{CPI(B)}\\$$

Relative PPP

$$\%\triangle S(A/B)\;=\;Inf_A\;-\;Inf_B\\$$

Real Exchange Rate

$$S_t\times\left(\frac{1+r_B}{1+r_A}\right)^T\\$$

$$BOP\Rightarrow Current\;A/c+Capital\;A/c+Official\;Reserve\;A/c=0\\$$

$$Real\;exchange\;rate\;A/B=(Equilibrium\;Real\;Exchange\;Rate\;A/B)\\(B\;affricates)\;\uparrow+(\uparrow Real\;Int.\;rate_B\;–\;Real\;Int.\;rate_A)\\-(Risk\;Premium_B\;-\;\uparrow Risk\;Premium_A)\\$$

Taylor’s Rule

$$R=r_n+\pi+\alpha(\pi-\pi\ast)+\beta(y-y\ast)\\$$

Real Interest Rate

$$Real\;Interest\;Rate=r_n+\pi+\alpha(\pi\;–\;\pi\ast)+\beta(\gamma-\gamma\ast)\\$$

 

$$\triangle P=\triangle GDP+\triangle(E/GDP)+\triangle(P/E)\\$$

Cobb – Douglas Function

$$Y=TK^aL^{(1-a)}\\$$

$$Output\;Per\;Worker=Y/L=T(K/L)\;a\\$$

$$Marginal\;Product\;of\;Capital=\frac{\triangle Y}{\triangle K}=\frac{\alpha Y}K\\(Cons\tan t)\\Marginal\;Productivity=\frac{\triangle Y/L}{\triangle K/L},K\uparrow L=Cons\tan t\\(Diminishing)\\$$

$$MP\;K=r(Marginal\;\cos t\;of\;K)\rightarrow rental\;price\;of\;capital\\$$

$$\downarrow\\\\$$

Growth Accounting Relation

$$\triangle Y/Y=\triangle A/A+\alpha(\triangle K/K)+(I-\alpha)(\triangle L/L)\\\\$$

$$Growth\;in\;Potential\;GDP=i)\;Long\;Term\;Growth\;of\;Techno\log y\\+\alpha(Long\;Term\;Growth\;of\;K)+(I-\alpha)\;(Long-Term\;Growth\;of\;L)\\ii)\;Long\;Term\;Growth\;of\;Labour\;Force+Long\;Term\;Growth\;in\;Labour\;Productivity\\(Output\;Per\;Worker)\;Both\;Capitals\;Depending+Techno\log y\;Process.\\\\$$

$$Labour\;Force\;Participation=\frac{Labour\;Force}{\;Working\;Age\;Population}\\Where\;Labour\;Force=Employed+Unemployed\;Available\;to\;Work\\\\$$

$$G\ast\;(Growth\;of\;Output\;Per\;Capita)=\frac\theta{\;I-\alpha}\\G\ast\;(Growth\;of\;Output)\;=\frac\theta{\;I-\alpha}+\triangle L\\\\$$

 

$$Full\;Goodwill=(Fair\;Value\;of\;Equity\;of\;Whole\;Subsidiary)\\-(Fair\;Value\;of\;Net\;Identifiable\;Assets\;of\;The\;Subsidiary)\\-Allowed\;under\;both\;IFRS\;\&\;USGAAP$$

$$Partial\;goodwill=Purchase\;Price–(\%\;owned\;\times\;\;FV\;of\;Net\;Identifiable\;Assets)\\-Allowed\;under\;only\;IFRS\\\\$$

Goodwill Impairment

US GAAP
1) CA > FV of reporting unit
2) CA of g/w implied FV of g/w

FV of unit
net identifiable asset

IFRS

1) CA> RA => loss in P/L

$$Years\;to\;Repay\;Debt\;from\;CFO=\frac{Total\;Debt}{\;Operating\;CF-Reinvestment}$$

Pension

Plan Asset  Plan Assets
FV at beginning of year PBO at beginning of year
+ Contributions
+ Actual Return
− Benefits paid

= FV at end of year

PBO at beginning of year
(+) Current service cost
(+) Interest cost
(+) PSC
(+) Actual Loss/ (-) gain
(-)Benefits

= PBO at end of year

Plan Asset > PBO Overfunded Plan
Plan Asset < PBO Underfunded Plan

$$Funded\;Status=Fair\;Value\;of\;Plan\;Asset-PBO\\TPPC=Employer\;contributions-\lbrack Ending\;Funded\;status–Beginning\;Funded\;status\rbrack\\=Employer\;contribution–\lbrack(End\;Plan\;Asset–End\;PBO)–(Beginning\;plan\;Asset–Beginning\;PBO)\rbrack\\=Employer\;contribution–\lbrack(Ending\;Plan\;Asset–Beginning\;Plan\;Asset)–(Ending\;PBO-Beginning\;PBO)\rbrack\\=Actual\;Return-\;(Current\;SC\;+\;\ln t\;\cos t\;+\;PSC\;\pm\;Actual\;gain/\;loss)\\TPPC=Current\;SC+Int\;\cos t+Past\;SC\pm Actuarial\;gain/loss–Actual\;Return$$

IFRS
i. Current Service cost in P /L
ii. Net Interest income in P/L
-income if overfunded (A > L)
-expense if underfunded (A < L)
iii. PSC in P/L
(recognized immediately)
iv. Actuarial g/l in OCI
(expected return actual return)

Not amortized over

US. GAAP
v. Current Service cos t in P/L
vi. Interest cost in P/L
Expected Return in P/L
vii. PSC in OCI
(amortized over remaining life)
viii. Actuarial g/l in OCI
(expected return-actual return)
Corridor approach
Amortize amount that is above 10% x max (A, L) over the remaining life

$$Expected\;Return=Expected\;rate\times Beginning\;plan\;Asset\\Net\;Int\;\cos t/\;income=Disc.\;Rate\times Beginning\;funded\;status\;(A-L)\\For\;IFRS,\;disc\;rate\;\&\;expected\;return\;is\;same\\Periodic\;pension\;\cos t\;in\;OCI=TPPC–periodic\;pension\;\cos t\;in\;P/L\\Or\;Actuarial\;Gains/\;losses\;+(Actual\;–\;expected)\;return$$

To reclassify

  • Op. income + Full pension exp current SC
  • Add int cost to int exp.
  • Add Actual return to other (non operating) income.

Cont^n>> TPPC -> reduction in PBO

Cont^n< TPPC => source of borrowing

From
CFO -> (+)
CFF -> (-)

The Beneish Model (M-score)

$$DSRI:\frac{Days\;Rec_{t\;}}{Days\;Rec_{t-1\;}}\uparrow X\\GMI:\frac{Gross\;Margin_{t-1\;}}{Gross\;Margin_{t\;}}\uparrow X\\AQI:\frac{Noncurrent\;asset\;except\;PP\;\&ET_t/Total\;Assets_t}{NCA\;except\;PP\;\&ET_{t-1}/Total\;Assets_t}\uparrow X\\SGI:\frac{Sales_{t\;}}{Sales_{t-1\;}}\uparrow X\\DEP:\frac{Depriciation_{t-1\;}}{Depriciation_{t\;}}\uparrow X\\SGAI:\frac{\%SGA/Sales_{t\;}}{\%SGA/Sales_{t-1\;}}\uparrow X\\Accruals:\frac{Income\;Before\;EOI-CFO}{Assets}\uparrow X\\Leverage\;Index:\frac{D/A_t}{D/A_{t-1}}(\ast higher\;the\;better\downarrow)\;X$$

Gauging Earning Persistence

$$Earnings_{(t+1)}\;=\;\alpha+\beta_1earnings_{\;t}+\varepsilon\\Earnings_{(t+1)}\;=\;\alpha+\beta_1cash\;flow_{\;t}+\beta_2accruals_{\;t}+\varepsilon$$

Sources of Earnings and ROE

$$DoPont\;Decomposition-\\ROE=NI/EBT\times EBT/EBIT\times EBIT/Revenue\times Revenue/Averagge\;Asset\times Averagge\;Asset/Averagge\;Equity$$

$$AccrualsBS=NOA\:End-NOA\:Bgn$$

$$Accruals\;Ratio\;BS=\frac{Accruals^{BS}}{\left(NOA_{End}+NOA_{Beg}\right)/2}$$

$$Accruals\;CF=NI-CFO-CFI$$

$$Accruals\;Ratio\;CF=\frac{Accruals^{CF}}{\left(NOA_{End}+NOA_{Beg}\right)/2}$$

$$CGO=EBIT+non\;cash\;changes-increase\;in\;WC$$

                         IFRS     USGAAP
Int paid           CFO/CFF   CFO
Div. Paid          CFO/ CFF   CFF
Int/Div Recd.  CFO/CFI    CFO

Market Value decomposition

$$Implied\;value=Parents\;Pro-rate\;share\;in\;associate's\;MV\\\\$$

$$Initial\;Investment\;Outlay=FC\;Investment+NWC\;Investment\;Proceeds\;from\;Sale\;of\;Earlier\;Asset\;sold\;\lbrack After\;Tax\rbrack$$

$$Include\;in\;Terminal\;Value=Initial\;NWC\;Investment$$

$$After-Tax\;operating\;CF=(EBITDA\;Dep)\;(I-Tax)+Dep^n\\Or,\;(EBIT)\;(I-tax)+Dep^n=(EBITDA)\;(I-tax)\;+\;(Dep^n\times\;tax)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Dep^n\;tax\;savings$$

Terminal Value

$$Sal_T\;+\;NWC_{INV}\;-Tax\;Marginal\;(Sal_T-B_T)$$

For Replacement Project

$$Investment\;Outlay=FC\;Investment+NWC\;Investment-Sal_0\;+T(Sal_0-B_0)\\Operating=\Delta CF=(\;\Delta S-\Delta VC)\;(1-T)+\Delta D(1-T)\\Terminal\;value=\left[Sal_{T\;new}-Sal_{T\;old}+NWC\;Investment\right]-T(Sal_{T\;new}-B_{T\;new})\\Proj.\;NPV\;without\;option\;\&\;add\;value\;later=Proj.\;NPV\;(DCF)-option\;\cos t+option\;value\\PI=I\;+\frac{NPV}{outflow}=\frac{PV\;inflow}{\;PV\;outflow}\\$$

$$Economic\;Income:\;Cash\;Flow+(Ending\;Market\;Value-Beginning\;Market\;Value)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Economic\;Dep^n\\(Firm)\;@KC\\Economic\;\mathrm\pi=\;\;\;\;NOPAT\;-\;\;\;\;\;\;\$\;WACC\;\\\;\;\;\;\;\;\;\;\downarrow\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\\\sum PV=NPV\;\;\;\;EBIT(I-t)\;\;\;\;\;\;\;\;WACC\;\times\;capital\;\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\;\\Co.\;value=NPV+initial\;investment\;(capital)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\\Residual\;Income=net\;income-equity\;charge\\\;\;\;\;\;\;\;\;\downarrow\\(Equity)\;@\;Ke\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Ke\times Bg^{\;n}\;BV_{eq}\\Co.\;value\;=NPV+BV\;of\;debt+BV\;of\;equity\\Claims\;valuation\;Approach=PV\;of\;CF\;to\;debt\;holders+PV\;of\;CF\;to\;equity\;holders\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(Principal\;+\;Int)\;\;\;\;\;\;\;\;\;\;\;(FCFE)\;(DIV+share\;repurchase)$$

Merger Acquisition

$$FCFF=NI+NCC+INT\;(I-t)-CAPEX-\Delta\;WC\;I_{NV}+\Delta\;DTL-\Delta\;DTA\\CFO=NI+NCC–WC_{INV}\\FCFE=CFO–FC\;I_{NV}+net\;borrowing\\Takeover\;Premium=\frac{Deal\;Price/Share–target\;co.Stock\;price}{Target\;co.Stock\;price}$$

Cash VS Stock Payment

Cash

$$Value\;of\;Firm=Value_{TARGET}+V_{ACQUIRER}+Synergy–Cash\;Paid\\Gain\;to\;target\_\\Price\;Paid–Price\;Merger\;Value\;of\;Target\\Gain\;to\;acquirer\\Synergy\;-\;Gain\;to\;Target$$

Stock

$$Value\;of\;Firm=V_{T\;}+V_A+synergy\\calculate\;the\;new\;price\\=\frac{V\;firm}{O/S\;shares\;of\;acquirer}\\Then\;calculate\;gain$$

Mergers & Acquisitions

Steps: -
1) DCF Method
• After calculating FCFF
• Discount FCF -> calculate PV

$$V_{firm}=\frac{FCFF_1}{\left(I+Kc\right)}+\frac{FCFF_2}{\left(I+Kc_2\right)^2}+...+\frac{FCFF_5+V_5}{\left(I+Kc\right)^5}\\Where\;V_5=\frac{FCFF_6}{Kc-g}\\Target\;V_{firm}\neq No.\;of\;shares$$

2) Comparable Company
• First calculate P/E, P/B, P/S ratios
• Take mean of all ratios
• Then multiply the mean P/E, P/B, with E, BV of the target company
• Get the mean of that = Stock price / value

Calculate Takeover Premium
Takeover Price = Stock Value X Takeover Premium
3) Comparable transaction (not needed to calculate takeover premium separately)
Take the deal price & continue to (2) process

$$Terminal\;Vfirm_5=\frac{FCF_T(1+g)}{WACC_{adj}-g}$$

MM Proposition I

$$No\;Taxes\Rightarrow V_L=V_U\;;\;EBIT/WACC\\With\;Taxes\Rightarrow V_L=V_U(t\times d)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\;\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;tax\;shield\\\boldsymbol S\boldsymbol t\boldsymbol a\boldsymbol t\boldsymbol i\boldsymbol c\boldsymbol\;\boldsymbol t\boldsymbol r\boldsymbol a\boldsymbol d\boldsymbol e\boldsymbol\;\boldsymbol o\boldsymbol f\boldsymbol f\boldsymbol\;\boldsymbol t\boldsymbol h\boldsymbol e\boldsymbol o\boldsymbol r\boldsymbol y:\;(benefit\;of\;debt)\\V_L=V_U+(t\times d)-PV(\cos ts\;of\;financial\;distress)$$

MM Proposition II

$$No\;Taxes\Rightarrow R_e=R_0+D/E(R_0-R_D)\\With\;Taxes\Rightarrow R_e=R_0+D/E(R_0-R_D)(I-T_C)$$

Dividend Share Repurchases

$$\Delta P=\frac{D(I-T_D)}{(1-T_{CG})}\\-\Delta P=when\;stock\;goes\;from\;dividend\;to\;ex-dividend\\Effective\;Tax\;Rate=Corporate\;Tax\;Rate+(I-Corporate\;Tax\;Rate)\;(IndividuaL\;Tax\;Rate)\\$$

For double taxation system we use above

For split tax
Taxes for retained ↑ as retained > distributed
Effective calculated same as above

Under Imputation Tax System
Effective Tax Rate = Shareholder’s Marginal Tax Rate
Shareholder tax bracket < company rate
Shareholder receive tax credit & vice versa.

Target Pay Out Ratio Adjustment Model

$$Expected\;Dividend=Previous\;Dividend+\left[(Expected\;Increase\;in\;EPS)\times(target\;pay\;out\;ratio)\times Adjustment\;Factor\right]\\Where\;Adjustment\;factor=\frac1{Number\;of\;Years\;Over\;Which\;the\;Adjustments\;in\;Dividends\;will\;take\;place\;}\\Earnings\;yield>K_d(I-t)\Rightarrow EPS\;\uparrow\\(EPS/P_o)\\Earnings\;yield<K_d(I-t)\Rightarrow EPS\;\downarrow\\BVPS\downarrow\Rightarrow\;if\;repurchase\;price>old\;BVPS\\Repurchase\;price<old\;BVPS\Rightarrow BVPS\;\uparrow\\EPS\;After\;Buyback=\frac{Total\;Earnings-After\;Tax\;Cost\;of\;Funds\;}{No.\;of\;Shares\;O/S}$$

$$Interest\;Coverage\;ratio=EBIT/INT\\Fixed\;Interest\;Coverage\;Ratio=\frac{EBIT+lease}{Interest+lease}\\Div.\;Coverage=NI/Div\\FCFE\;Coverage=FCFE/(Div+Share\;Repurchase)\\Accounting\;income=(Operating\;Income\;Before-Interest)\;(I-t)$$

 

Equity Valuation: Application & Processes

$$IV_{analyst}-price=(IV_{actual}\;-price)+(IV_{analyst}-IV_{actual})\\Conglomerate\;Discount=Sum\;of\;individu\;al\;parts-Sum\;as\;a\;whole$$

$$Holding\;Period\;Return=\frac{P_1-P_0+CF_1}{P_0}\\Target\;Price=Price(R_e)\;D_i$$

Price Convergence

$$Expected\;return=Required\;Return+\left[\frac{V_0-P_0}{P_0}\right]\\Equity\;Risk\;Premium=Required\;Return\;on\;Equity\;Index-Risk\;Free\;Rate\\Required\;Return\;for\;Stock\;j=Risk\;free\;Return+\beta_j\times(Equity\;Risk\;Premium)+other\;risk\;premium$$

Forward Looking Estimates

Gordon Growth Model:

$$(Re-Rf)=ERP=\frac{D_I}{P_0}+g-RF\\Where\;RF=long\;term\;bond\;yield\\Equity\;Index\;Price=PV_{rapid}(r)+PV_{transition}(r)+PV_{mature}(r)$$

Supply Side Estimates

$$ERP=\left[\left(\;I+\widehat i\;\right)\times(I+r\widehat Eg)\times\left(I+P\widehat Eg\right)-1+\widehat Y\right]-\widehat{RF}$$

Capital Asset Pricing Model (CAPM)

$$R_e=R_F+(R_{equity\;index}-R_F\;)\times\beta$$

Multifactor Models

$$R_e=R_f+\beta_1F_1+\beta_2F_2+...\\Where,\;F_1=(R_{F_1}-R_F)$$

Fama French Model

$$Required\;return\;for\;stock\;j=RF+\beta_{mkt,j}\times(R_{mkt}-RF)+\beta_{SMB,j}\times(R_{small}-R_{big})\\+\beta_{HML,j}\times(R_{HBM}-R_{LBM})$$

Pastor Stambaugh Model

$$Liquidity\;+\;Fama\;French\;Model\\\downarrow\\(Liquidity\;premium)\;x\;\beta\\+ve\rightarrow\;less\;liquid\\-\;ve\rightarrow more\;liquid$$

Build Up Method

$$Required\;return=RF+equity\;risk\;premium+size\;premium+specific\;company\;premium$$

Bond yield + Risk Premium Method

$$R_e=YTM+risk\;premium\;for\;holding\;equity\\Adjusted\;\beta=(2/3)\;\times\;regression\;\beta)\;+\;(1/3\;\times\;1.0)$$

Country Spread Model

$$Premium=YTM_{Bond\;Emerging\;Market}-YTM_{Bond\;Developed\;Market}$$

Country risk rating model

$$Developed\;market\;returns\;(ERP)=a+b_1\;\times\;(ratings)\\WACC=\frac{Market\;value\;of\;debt}{Market\;value\;of\;debt\;and\;equity}\times r_d\times\left(1-Tax\;rate\right)+\\\frac{Market\;value\;of\;debt}{Market\;value\;of\;debt\;and\;equity}\times r_e$$

 

Cost of Goods Sold (COGS)

$$Forecast\;COGS=(Historical\;COGS/revenue)\;\times\;(Estimate\;of\;Future\;Revenue)$$

Financing Cost

$$Net\;debt=Gross\;debt-Cash,\;cash\;equivalents\;\&\;short\;term\;investment\\Net\;interest\;expense=Gross\;Interest\;Expense-Interest\;Income\;(on\;cash\;\&\;short\;term\;debt\;securities)$$

$$Gross(net)\;interest\;expense\;rate=gross(net)\;expense/\;gross(net)\;debt\\Yield\;on\;average\;cash=interest\;income/cash+ST\;securities\\Effective\;tax\;rate=\frac{Income\;tax\;expense}{PBT}\\Cash\;tax\;rate=\frac{Cash\;taxes\;paid}{PBT}\\Projected\;Accounts\;Receivables=Days\;Sales\;Outs\tan ding\times\left(\frac{Forecasted\;Sales}{365}\right)\\ROC=\frac{Net\;Operating\;Profit}{D+E}\;\rightarrow\;Not\;adjusted\;for\;taxes\\ROIC=\frac{NOPLAT}{D+E}\;\rightarrow\;Net\;operating\;profit\;taxes\\ROE=\frac{NI}E\;\rightarrow\;Not\;suitable\;for\;comparing\;companies\;with\;different\;capital\;structures\\Cannibalization\;rate=\frac{new\;product\;sales\;that\;replace\;existing\;product\;sales}{total\;new\;product\;sales}$$

DDM Model

One Period- $$P_0=\frac{D_1+P_1}{I+R_e}$$

Two period-

$$P_0=\frac{D_1}{I+R_e}+\frac{D_2+P_2}{\left(I+R_e\right)^2}$$

Multi period-

$$P_0=\frac{D_1}{I+R_e}+\frac{D_2}{\left(I+R_e\right)^2}+...+\frac{D_n+P_n}{\left(I+R_e\right)^n}$$

Gordon Growth Model

$$P_0=\frac{D_1}{R_e-g}$$

Present value of growth opportunities (PVGO)

$$P_0=\frac{E_1}{R_e}+PVGO$$

Justified trailing P/E

$$\frac{P_0}{E_0}=\frac{\left(I+g\right)\left(I-b\right)}{R_e-g}$$

Justified leading P/E

$$\frac{P_0}{E_1}=\frac{\left(I-b\right)}{R_e-g}$$

$$Value\;of\;perpetual\;preferred\;shares=\frac{D_P}{r_P}$$

Valuation using H Model

$$V_0=\frac{D_0(1+g_L)}{R_e-g_L}+\frac{D_0\times t/2\times(g_S-\;g_L)}{R_e-g_L}$$

Sustainable growth rate

$$SGR(g)=b\times ROE\\Where,\;ROE=\frac{NI}{Stockholders'\;Equity}=\frac{NP(NI)}{Sales}\times\frac{Sales}{Total\;Assets}\times\frac{Total\;Assets}{Stockholders'\;Equity}\\So,\;g=\left(\frac{Net\;Income-Dividends}{Net\;Income}\right)\times\frac{Net\;Income}{Sales}\times\frac{Sales}{Total\;Assets}\times\frac{Total\;Assets}{Stockholders'\;Equity}$$

$$Intrinsic\;value>market\;value\rightarrow undervalued\\Intrinsic\;value=market\;value\rightarrow fairly\;valued\\Intrinsic\;value<market\;value\rightarrow overvalued$$

FCFF 4 Approaches

$$FCFF=NI+NCC+Interest(1-TAX)-fCInvestment-WCInvestment\\FCFF=NI+NCC-WC\;Investment+Interest(1-TAX)-fC\;investment=CFO+Interest(1-TAX)-FCInvestment\\FCFF=EBIT(1-TAX)+Depreciation-fC\;investment-WC\;investment\\FCFF=EBITDA(1-TAX)+(Depreciation\times TAX)-fC\;investment-WC\;investment$$

FCFE 4 Approaches

$$FCFF=FCFF-Interest(1-TAX)+NET\;Borrowings\\FCFF=NI+NCC-WC\;Investment-FC\;investment+NET\;Borrowings\\FCFF=CFO-FC\;Investment+NET\;Borrowings\\FCFF=NI-(1-DR)\left[(FC\;Investment-Depreciation\;)+WC\;Investment\right]\\therefore,1-DR=1-\frac DA=\frac{A-D}A=\frac EA$$

Single Stage FCFF / FCFE Model

$$FCFF:\;Value\;of\;Firm=\frac{FCFF_1}{WACC-g}=\frac{FCFF_0\times(1+g)}{WACC-g}\\FCFF:\;Value\;of\;Equity=\frac{FCFF_1}{r-g}=\frac{FCFE_0\times(1+g)}{r-g}\\Terminal\;Value\;in\;year\;n=(trailing\;P/E)\times(earnings\;i\;n\;years\;n)\\Terminal\;Value\;in\;year\;n=(leading\;P/E)\times(forecasted\;earnings\;in\;year\;n+1)=\frac PE\times E_0(1+g)$$

$$Trailing\;P/E=\frac{Market\;price\;per\;share}{EPS\;over\;previous\;12\;months}\\Leading\;P/E=\frac{Market\;price\;per\;share}{Forecasted\;EPS\;over\;next\;12\;months}\\P/B\;Ratio=\frac{Market\;price\;of\;Equity}{Book\;value\;of\;Equity}\\P/S\;Ratio=\frac{Market\;price\;of\;Equity}{Total\;Sales}\\Dividend\;Yield:\;D/P\\Trailing\;D/P=D_o/P_o\\Leading\;D/P=D_I/P_o$$

Justified P/E Multiple

$$P_o=\frac{D_1}{R_e-g}\\Justified\;trailing\;P/E=\frac{P_0}{E_0}=\frac{\left(1-b\right)\left(1+g\right)}{r-g}\\Leading\;P/E=\frac{P_0}{E_0}=\frac{1-b}{r-g}$$

Justified P/B Multiple

$$Justified\;P/B\;Ratio=\frac{ROE-g}{r-g}$$

Justified P/S Multiple

$$Justified\;\frac{P_0}{S_0}=\frac{\left(\frac{E_0}{S_0}\right)\times\left(1-b\right)\left(1+g\right)}{r-g}\\$$

Justified P/CF Multiple

$$V_0=\frac{FCFE_0(1+g)}{r-g}\\$$

Justified Dividend Yield

$$\frac{D_0}{P_0}=\frac{r-g}{1+g}\\$$

Fed & Yardeni Model

Fed model:

$${(E/P)}_{S\&P}>{(E/P)}_{10\;yr\;T-Bond}\;\Rightarrow undervalued\\{(E/P)}_{S\&P}<{(E/P)}_{10\;yr\;T-Bond}\;\Rightarrow overvalued\\$$

Yardeni model:

$$Earnings\;yield\;of\;market\;(E/P)=yield\;on\;‘A’\;rated\;bond-k\times(Long\;term\;growth\;rate)\\$$

Peg Ratio

$$Peg\;ratio=\frac{P/E\;ratio}g\\CF=Net\;Income+depreciation+amortization\\FCFE=CFO-FC\;Inv+Net\;borrowing\\P/CF=\frac{MV\:of\;equity}{CF}=\frac{Market\;price\;per\;share}{CF\:per\;share}\\EV/EBITDA_{ratio}=\frac{enterprise\;value}{EBITDA}\\\\$$

Momentum indicator

$$Earnings\;Surprise=Reported\;EPS-Expected\;EPS\\S\tan dardized\;Unexpected\;Earnings\;(SUE)=\frac{earnings\;surprise}{SD\;of\;earnings\;surprise}\\$$

$$RI=Net\;Income-Cost\;of\;Equity\times Equity\;Capital(equity\;charge)\\=\;(ROE-r)\;BV_{equity\;(t-I)}\;where\;BV=beginning\;BV\\EVA=NOPAT-(WACC\times TOTAL\;CAPITAL)\rightarrow Beginning\;invested\;capital\;(D\;+\;E)\\=\left[EBIT-(1-tax)\right]-WACC\\MVA=Market\;Value-Total\;Capital\\RI_t\;=\;E_t-(r\;\times B_{t-1})\;=(ROE-r)\times B_{t-1}\\$$

Intrinsic Value

$$P_0=B_0+\left\{\frac{RI_1}{\left(1+r\right)^1}+\frac{RI_2}{\left(1+r\right)^2}+\frac{RI_3}{\left(1+r\right)^3}+...\right\}\\$$

Single stage Residual Model

$$P_0=B_0+\left[\frac{(ROE-r)\times\;B_0}{r-g}\right]\\$$

The growth rate implied by the market price in a single- stage residual income

$$g=r-\left[\frac{\;B_0\times(ROE-r)}{P_0-B_0}\right]\\$$

$$Tobin’s\;Q=\frac{market\;value\;of\;debt+market\;value\;of\;equity}{replacement\;\cos t\;of\;total\;asset}\\P_0=B_0+(PV\;of\;interim\;high\;growth\;RI)+(PV\;of\;continuing\;residual\;income)\\PV\;of\;Continuing\;Residual\;Income\;in\;year\;T-1=\frac{RI_T}{1+r-\omega}\\$$

 

$$V_F=\frac{FCFF_1}{WACC-g}\\V_E=\frac{FCFE_1}{K_e-g}\\Control\;premium=pro\;rata\;value\;of\;controlling\;interest-pro\;rata\;value\;of\;non\;controlling\;interest.\\Adjusted\;control\;premium\;(applicable\;for\;MVIC\;multiple)=(control\;premium\;on\;equity)\times(1-DR)\\\left[DR\;=\;Debt\;to\;asset\;ratio\right]\\DLOC=1-\left[1/(1+control\;premium)\right]\\Total\;discount\;for\;lack\;of\;market\;ability=1-\left[(1-DLOC)\;(1-DLOM)\right]\\$$

The Relationship Between the Discount Factor and the Spot Rate

$$P_T=\frac1{{(1+S_T)}^T}$$

Forward Rates

$$F_{(j,k)}=\frac1{\left[1+f(j,k)\right]^k}$$

The Forward Pricing Model

$$F_{(j,k)}=\frac{P_{(j,k)}}{P_j}$$

The Forward Rate Model

$$\left[1+S_{(j,k)}\right]^{(j+k)}=\left(1+S_j\right)^j\left[1+f(j,k)\right]^k$$

Swap Rate

$$Swap\;Rate=\frac{1-d_L}{\sum d_L}\\Swap\;Spread_t\;:\;=\;Swap\;Rates_t-Treasury\;yield_{t\;}(Same\;Maturity)\\I\;spread=Riskiness\;of\;Corporate\;Bond\;Over\;Banks\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\\comp^n\;credit\;liquidity\;risk=corp\;Bond-swap\;rate\\TED\;Spread=(3months\;LIBOR\;Rate)-(3months\;T\;bill\;Rate)\\LIBOR-\;O/S\;Spread\;=\uparrow=banks\;unwilling\;to\;lend;\;\downarrow=liquidity\\LIBOR\rightarrow Includes\;credit\;risk\\O/S\;Spread\rightarrow minimal\;credit\;risk\\O/S\;Spread\rightarrow measure\;of\;money\;market\;securities\;risk\\\\$$
Cox Ingersoll Ross Model (CIR)

$$dr=\left[a(b-r)dt+\left(\sigma\sqrt r.dz\right)\right]$$

Vasicek Model

$$dr=\left[a(b-r)dt+\left(\sigma dz\right)\right]$$

Ho Lee Model

$$dr_t=\theta_t\;dt+\sigma dz_t$$

Sensitivity to Parallel, Steepness, and Curvature Movements

$$\frac{\Delta P}P=D_L\Delta X_L-D_S\Delta X_S-D_C\Delta X_C$$

Value of option embedded in a bond

$$V_{Call}\;=\;V_{straight}\;\;bond-V_{callable}\;bond\\V_{Put}\;=\;V_{Putable}\;\;bond-V_{Straight}\;bond\\OAS=Z-Call\;Risk\\OAS=Z+Put\;Risk\\Effective\;Duration=\frac{P_2-P_1}{2P_0\triangle y}\\Effective\;Convexity=\frac{P_2-P_1-2P_0}{P_0\left(\triangle y\right)^2}\\Market\;Conversion\;Premium\;Ratio=\frac{Conversion\;Premium\times Market\;Per\;Share}{Market\;Price\;of\;Convertible\;Stock}\\Conversion\;Value=Market\;Price\;of\;Stock\times Conversion\;Ratio\\Market\;Conversion\;Price=\frac{Market\;Price\;of\;Convertible\;bond}{Convertion\;Ratio}\\Market\;Conversion\;Premium\;Per\;Share=Market\;Conversion\;Price-Stock’s\;Market\;Price\\Premium\;Over\;Straight\;Value=\frac{Market\;Price\;of\;Convertible\;bond}{Straight\;Value}-1$$

Put Call Parity

$$C-P=PV\;(Forward\;price\;of\;the\;bond\;on\;exercise\;date)-PV(Exercise\;price)$$

$$Present\;Value\;of\;Expected\;Losses=Expected\;Loss+Risk\;Premium-Time\;Value\;Discount\\Value\;of\;Stock_{T\;}=\;Max\;(0,\;A_T-K)\\Value\;of\;Debt_{T\;}=\;Min\;(A_T,K)\\Probability\;of\;Default=I-N\;(e_2)\\e_1=\frac{\ln\left({\displaystyle\frac{A_t}K}\right)+\mu(T-t)+{\displaystyle\frac12}\sigma^2(T-t)}{\sigma\sqrt{T-t}}$$

Key rate duration total duration -> same effect if parallel shift

$$Present\;Value\;of\;Expected\;Losses=Expected\;Loss+Risk\;Premium-Time\;Value\;Discount\\Value\;of\;Stock_{T\;}=\;Max\;(0,\;A_T-K)\\Value\;of\;Debt_{T\;}=\;Min\;(A_T,K)\\Probability\;of\;Default=I-N\;(e_2)\\e_1=\frac{\ln\left({\displaystyle\frac{A_t}K}\right)+\mu(T-t)+{\displaystyle\frac12}\sigma^2(T-t)}{\sigma\sqrt{T-t}}\\Key\;rate\;duration\;total\;duration\;\rightarrow same\;effect\;if\;parallel\;shift\\Duration\;exposure=Add\;the\;duration\\Effective\;Duration=\frac{P_2-P_1}{2P_0\triangle y}\\Effective\;Convexity=\frac{P_2-P_1}{P_0\left(\triangle y\right)^2}\\\%\;\Delta\;Bond\;Price=-\Delta y\times ED=\frac12\times EC\times\left(\triangle y\right)^2\\VCB=VNCB-Call\;Price\\VPB=VN_{PB}\;+Put\;Price$$

$$Pay\;out\;Amount=pay\;out\;Ratio\times NP\\Pay\;out\;Ratio=I-(Recovery\;Rate)\;\%\\Hazard\;rate/conditional\;Prob.\;Of\;default=Prob.\;(PD/Default\;has\;not\;occurred)\\Expected\;Loss=Hazard\;Rate\times LGD\;(\%\;terms)\\Upfront\;payment=PV\;(protection\;leg)-PV\;(premium\;leg)\\\downarrow\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\;\\Based\;on\;CDS\;spread\;\;Based\;on\;coupon\;rate\\Upfront\;Premium=(CDS\;spread-CDS\;coupon)\times duration\;of\;spread\\Price\;of\;CDS=100-Upfront\;Premium\;(\%)$$

Valuation After Inception of CDS

$$Profit\;for\;protection\;buyer\approx(\triangle spread\times duration)\times Notional\;Principal\;Or,\\Profit\;for\;protection\;buyer\;(\%)\approx change\;in\;spread\;(\%)\times duration$$

$$Forward\;Price=Price\;That\;Prevents\;Profitable\;Riskless\;Arbitrage\;in\;Frictionless\;Markets$$

The Forward Contract Price

$$FP=S_0\times{(I+R_f)}^t$$

Forward Contracts with Discrete Dividends

$$FP(on\;An\;Equity\;Security)=(S0-PVD)\times(1+Rf)T\\FP(on\;an\;Equity\;Security)=\left[S_0\times{(1+R_f)}^T\right]-FVD\\$$

Value of the Long Position in A Forward Contract on A Dividend Paying Stock

$$FP(on\;An\;Equity\;Security)=(S0-PVD)\times(1+Rf)T\\FP(on\;an\;Equity\;Security)=\left[S_0\times{(1+R_f)}^T\right]-FVD\\V_t(long\;position)=(S_t-PVD_t)-\left[\frac{FP}{\left(I+R_f\right)^{T-t}}\right]\\$$

Equi ty Forward Contracts with A Continuous Dividends

$$FP(on\;An\;Equity\;Index)=S_0\times e^{(R_f^c-\delta^c)\times T}=(S_0\times e^{\delta^c\times T})\times e^{R_f^c\times T}\\F=\frac{Spot\times e^{interest\times t}}{e^{Dividends\times t}}\\$$

Forward Price on A Coupon Paying Bond

$$FP=\left[\left(Full\;price\right)\left(1+R_f\right)^T-AI_T-FVC\right]\\QFP=FP/CF=\left[\left(Full\;price\right)\left(1+R_f\right)^T-AI_T-FVC\right]\left(\frac1{CF}\right)\\$$

Covered Interest Rate Parity

$$FT(Currency\;Forward\;Contract)=S0\times\frac{\left(1+R_{PC}\right)^T}{\left(1+R_{BC}\right)^T}\\$$

Valuing Currency Forward Contracts After Initiation

$$FT(Currency\;Forward\;Contract)=S0\times\frac{\left(1+R_{PC}\right)^T}{\left(1+R_{BC}\right)^T}\\V_{t\;}=\frac{\left[FP_t-FP\right]\times\left(contract\;size\right)}{\left(1+r_{PC}\right)^{(T-t)}}=\left[\frac{S_t}{\left(1+R_{BC}\right)^{(T-t)}}\right]-\left[\frac{F_T}{\left(1+R_{PC}\right)^{(T-t)}}\right]\\Value\;of\;Futures\;Contract=Current\;Futures\;Price-Previous\;Mark\;to\;Market\;Price\\$$

Discount factors (zs)

$$Z=\frac1{\left[1+\left(LIBOR\times\frac{days}{360}\right)\right]}\\SFR\;(Periodic)=\frac{1-last\;discount\;factor}{sum\;of\;discount\;factors}$$

Market Value of An Interest Rate Swap

$$Value\;to\;The\;Payer=\sum Z\;\times\;(SFR_{New}-SFR_{Old}\;)\times\frac{days}{360}\times Notional\;Principal$$

Equity Swaps

$$SFR(Periodic)=\frac{1-Last\;Discount\;Factor}{Sum\;of\;Discount\;Factors}$$

Put Call Parity

$$C_0+\frac X{{(1+\;R_F)}^T}=P_0+S_D\\H=\frac{C^+-C^-}{S^+-S^-}$$

Black Scholes Model

$$C_0=\frac X{{(1+\;R_F)}^T}=P_0+S_D\\H=\frac{C^+-C^-}{S^+-S^-}\\C_o=S_0\;N(d_1)-e^{-rt}XN(d_2)\\P_o=e^{rt}XN(-d_2)-S_0\;N(-d_1)\\Where\\d_1=\frac{ln\lbrack{\displaystyle\frac SX}\rbrack+(r+{\displaystyle\frac{\sigma^2}2})T}{\sigma\sqrt T}\\d_2=d_1-\sigma\sqrt T$$

Options on Dividend Paying Stocks

$$C_o=S_0\;e^{-\delta t}N(d_1)-e^{-rt}N(d_2)\\P_o=e^{-rt}N(-d_2)-S_0\;e^{-\delta t}N(-d_1)\\Where\;\delta=\;Continuously\;Compounded\;Dividend\;Yield\\d_1=\frac{ln({\displaystyle\frac SX})+(r-\delta+{\displaystyle\frac{\sigma^2}2})T}{\sigma\sqrt T}\\d_2=d_1-\sigma\sqrt T$$

Options on Currencies

$$C_o=S_0\;e^{-rBT}N(d_1)-e^{-rPT}N(d_2)\\P_o=e^{-rPT}N(-d_2)-S_0\;e^{-rBT}N(-d_1)$$

The Black Model

$$C_o=\frac{F_T}{e^{rt}}N(d_1)-\frac X{e^{rt}}N(d_2)\\Where,\;d_1=\frac{\ln\left({\displaystyle\frac{F_T}X}\right)+\left({\displaystyle\frac{\sigma^2}2}\right)t}{\sigma\sqrt t}\\\;d_2=d_1-\sigma\sqrt t$$

Interest Rate Options

$$C_o=\frac{AP}{e^{r(N\times{\displaystyle\frac{30}{360}})}}\lbrack FRA(M\ast N)N(d_1)-XN(d_2)\rbrack\times NP\\Where,\;AP=Accrual\;Period=\frac{Actual}{365}=\left[\frac{\left(N-M\right)\times30}{360}\right]\\NP=Notional\;Principal\;on\;the\;FRA$$

Swaptions

$$V_{payer\;swaption}=PVA\;\lbrack SFR.\;N(d_1)-XN(d_2)\rbrack\times NP\times AP\\\Delta C=Call\;Delta\times\Delta s+\frac12Gamma\times\Delta s^2\\\Delta P=Put\;Delta\times\Delta s+\frac12Gamma\times\Delta s^2\\$$

$$V_{payer\;swap}=V_{floating\;rate\;note}-V_{fixed\;rate\;bond}\\Duration\;of\;Payer\;Swap=Duration\;of\;Floating\;Rate\;Bond-Duration\;of\;Fixed\;Rate\;Bond\\Long\;Futures\;+Risk\;Free\;Asset=Long\;Stock\\$$

Covered Call

$$Initial\;Investment=S_0-C_0\\Max\;Profit=C_0-(S_0-X)\\Max\;loss=Initial\;Inv\\Breakeven\;price=Initial\;Inv\\$$

Bull Call

$$Initial\;Investment=C_L-C_H\\Max\;loss=Initial\;Inv\\Breakeven=X_L+(C_L-C_H)\\Max\;\mathrm\pi=(X_H\;–\;X_L)\;–\;(C_L\;–\;C_H)\\$$

Bear Call

$$Initial\;Inflow=C_L-C_H\\Max\;\mathrm\pi=Initial\;Inflow\\Breakeven=X_L+(C_L-C_H)\\Max\;loss=(X_H\;–\;X_L)\;–\;(C_L\;–\;C_H)\\$$

Collar

$$Breakeven=S_0+P_0+C_0\\Max\;\mathrm\pi=(X_H-S_0)+C_0-P_0\\Max\;loss=(S_0\;–\;X_L)\;+(P_0-C_0)\\$$

Long Straddle

$$Max\;Profit=S_T\;–\;X-(C_0+P_0)\\Max\;Loss=C_0+P_0\\Breakeven=X-(C_0\;+\;P_0)\;and\;\;X+(C_0+P_0)\\$$

Breakeven Price Analytics

$$\sigma_{annual}=\%\triangle PX\sqrt{\frac{252}{trading\;days\;until\;maturity}}\\Where\;\%\;\triangle P\;=\frac{\vert breakeven\;price-current\;price\vert}{current\;price}\\$$

 

$$Capitalization\;Rate=\frac{Net\;Operating\;Income}{Property\;Value}\\Property\;Value=\frac{Net\;Operating\;Income}{Capitalization\;Rate}\\NAVPS\;(u\sin g\;current\;market\;values)=\frac{A-L}{Number\;of\;Shares}(here\;property\;value\;is\;added)\\$$

FFO and AFFO

Funds from Operations:

$$Accounting\;net\;earnings+Depreciation\;expense+Deferred\;tax\;expenses\;(i.e.\;deferred\;tax\;liabilities)\\-Gains\;from\;sale\;of\;property\;and\;debt\;restructuring+Losses\;from\;sale\;of\;property\;and\;debt\;restructure\\=Funds\;from\;Operations$$

Adjusted Funds from Operations:

$$Fund \;from \;Operations (FFO)- Non \;cash \;rent\; adjustment- Recurring\; maintenance \;type \;capital \;expenditure \;and \;leasing \;commissions\\
=Adjusted \;funds \;from \;operations$$

Income Approach

Net Operating Income:

$$Rental \;income \;(if \;fully \;occupied)+ other \;income\\= Potential \;gross \;income\;- vacancy \;and \;collection \;loss\\
= Effective \;gross \;income \;- Operating expense\;
= Net operating income$$

The Capitalization Rate:

$$Cap\;Rate=\frac{NOI_I}{Value}\\or,Value=V_0=\frac{NOI}{Cap.Rate}\\or,\;Cap.Rate=\frac{NOI}{Comparable\;Sales\;Price}\\P_0=\frac{D_1}{R_e-g}\\or,\;P_0=\frac{NOI}{Cap.Rate}\\or,\;Cap.Rate=\frac{NOI}{P_0}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\;\\Value\;of\;property\;today$$

$$Value=\frac{Rent}{ARY}\\or,\;ARY=\frac{Rent}{Comparable\;Sale\;P_x}\\Gross\;Income\;Multiplier=\frac{Sales\;Price}{Gross\;Income}\\Value=Gross\;Income\times Gross\;Income\;Multiplier$$

Discounted Cash Flow Method:

$$Cap\;Rate=Discounted\;Rate-Growth\;Rate=R_e-g\\Discounted\;Rate=Cap\;Rate+Growth\;Rate\\Value=V_0=\frac{NOI}{(r-g)}=\frac{NOI}{Cap\;Rate}$$

Cost Approach

$$V_{property}=(Replacement\;\cos t-Cost\;of\;fixing\;curable\;items)-Depreciation-Incurable\;expense\;capitalised+Market\;value\;of\;land\\Where\;Depreciation=\frac{Effective\;Age}{Economic\;Life}\times\left[Replacement\;Cost-Curable\;Items\right]$$

Appraisal Based indices

$$Return=\frac{NOI–Capital\;Expenditure+End\;Market\;Value–Beginning\;Market\;Value}{Beginning\;Market\;Value}\;$$

Ratios to consider for evaluation

$$Debt\;Service\;Coverage\;Ratio.=\frac{First\;year\;NOI}{Debt\;Service}\left[higher\;the\;better\right]\\Loan\;to\;Value\;Ratio=\frac{Loan\;Amt}{\;Appraisal\;Value}\left[lower\;the\;better\right]\\Equity\;Dividend\;Rate=\frac{First\;year\;CF}{Equity}$$

$$Valuation\;After\;Renovation=\frac{Stabilised\;NOI}{Cap.Rate}\\Valuation\;After\;Renovation-PV\;of\;loss=Total\;Value$$

Mostly talking about LBO

$$\frac DE\rightarrow Public\\\frac D{EBITDA}\rightarrow P/E\;firm$$

LBO’s Exit Value

$$Investment\;Cost+Earning\;Growth+Increase\;in\;Price\;Multiple+Reduction\;in\;Debt=Exit\;Value$$

Valuation Issues in Venture Capital Investments

$$The\;Post\;Money\;Valuation\;of\;The\;Investment\;Company\;is:\\PRE+INV=POST\\The\;Ownership\;Proportion\;of\;The\;Venture(VC)Investor\;is:\\=INV/POST$$

Paid in capital (PIC) $$Paid\;in\;Capital(PIC)=\frac{Paid\;in\;Capital\;(Cumulative)}{Committed\;Capital}\\Distributed\;to\;paid\;in\;capital(DPI)=\frac{Total\;Distribution\;(Cumulative)}{Total\;Invested\;Capital}\\Residual\;Value\;to\;paid\;in\;capital(RVPI)=\frac{Value\;of\;fund}{Total\;Invested\;capital}\\Total\;Value\;to\;paid\;in\;capital(TVPI)=DPI+RVPI$$

NAV Before Distribution

$$NAV\;Before\;Distribution=NAV\;After\;Distributions\;in\;Prior\;Year+Capital\;Called\;Down-Management\;Fees+Operating\;Results$$

NAV After Distribution

$$NAV\;After\;Distribution=NAV\;Before\;Distributions-Carried\;Interest-Distributions$$

The Fraction of VC Ownership ( f ) for the VC Investment can be computed as

$$NPV\;Method:\;f=\frac{INV}{POST}\\IRR\;Method:\;f=\frac{FV(INV)}{Exit\;Value}$$

 

 

$$Total\;Return=Collateral\;Return(HPY\;on\;T\;bill)+Price\;Return\left(\frac{P_I-P_O}{P_O}\right)+Roll\;Return\\Price\;Return=\frac{Current\;Price-Previous\;Price}{Previous\;Price}\\Roll\;Return=\frac{Price\;of\;Expiring\;Futures\;Contract\;-\;Price\;of\;New\;Futures\;Contract}{Price\;of\;Expiring\;Futures\;Contract}$$

APT

$$E(R_A)=R_f+B_{A,1}\;\lambda_1+B_{A,2}\;\lambda_2+...$$

Multifactor

$$Macro-economic:\;R_i=E\left(R_i\right)+b_1F_1+b_2F_2+...+b_nF_n+e_t\\Fundamental\;Factor:\;R_i=a_i+b_{i,1}F_{P/E}+b_{j,2}F_{Size}+e_t$$

Standardized P/E Sensitivity

$$b_{i1}=\frac{\left({\displaystyle\frac PE}\right)i-\overline{P/E}}{\sigma P/E}$$

Active Return

$$Active\;Return=(R_P)-(R_B)\\Active\;Risk=\sigma_{(R_P-R_B)}\\IR=\frac{{\overline R}_P-{\overline R}_B}{\sigma_{(R_P-R_B)}}$$

Return Attribution

$$Active\;Return=Factor\;Return+Security\;Selection\;Return$$

Factor Return

$$Factor\;Return=\sum_{i=1}^k\left(\beta_{Pi}-\beta_{bi}\right)\times\lambda_i$$

Active Specific Risk

$$Active\;Specific\;Risk=\sum_{i=1}^n\left(W_{Pi}-W_{bi}\right)^2\times\sigma_{\varepsilon i^2}$$

Carhart Model

$$E(R)=R_F+\beta_1RMRF+\beta_2SMB+\beta_3HML+\beta_4WML$$

$$\sigma_{Portfolio}^2=W_A^2\sigma_A^2+W_B^2\sigma_B^2+2W_AW_BCov_{AB}\\E(R_i)=R_f+Beta_i\;\left[E(R_{MKT})-R_f\right]\\Change\;in\;Price=-Duration\;(\triangle Y)+\frac12Convexity\;{(\triangle Y)}^2\;Change\;in\;Call\;Price=delta\;(\triangle S)+\frac12gamma\;{(\triangle S)}^2+vega(\triangle V)\\Discount\;Rate=R+\pi+\theta+\gamma+K+\phi\\Inter\;temporal\;Rate\;of\;Substitution=\frac{U_t}{U_o}=\frac{Future}{Current}\\P_O=E(m_t)$$

Real Risk free rate of Return

$$R=\frac{1-\;P_0}{P_0}=\left(\frac I{E(m_t)}\right)-1$$

$$R=\frac{1-\;P_0}{P_0}=\left(\frac I{E(m_t)}\right)-1\\P_0=\frac{E\left(P_1\right)}{I+R}+cov(P_1,M_1)\\P_0\;is\;lower;\;Return\;\uparrow\;(Since\;Risk\;Taken)\\\frac{E\left(P_1\right)}{I+R}\;;When\;no\;risk=P_0\\Disc.\;Rate=R+\pi\;(short\;term)\\R+\pi+\theta\;(long\;term)$$

Taylor Rule

$$r=R_n+\pi\;+0.5(\pi-\pi\ast)+0.5(y-y\ast)$$

$$BEI=Yield\;on\;Non\;Inflation\;Indexed\;Bond-Yield\;on\;Non\;Inflation\;Indexed\;Bond\\BEI=\pi+\theta$$

$$Active\;Return\;E(R_A)=E(R_P)-E(R_B)$$

For an Active Portfolio of N Securities

$$E(R_A)==\sum\triangle w_iE(R_i)$$

Weighted Average of Securities Returns

$$E(R_P)=\sum w_{P,i}E(R_{P,j})\;and\;E(R_P)=\sum w_{B,i}E(R_{B,j})\;$$

Ex ante Active Return

$$E(R_A)=\sum\triangle w_{P,i}E(R_{P,j})-\sum w_{B,i}E(R_{B,j})\;$$

Security Selection Return

$$E(R_A)=\sum\triangle w_iE(R_{B,j})+\sum w_{P,i}E(R_{A,j})\;$$

Sharpe Ratio

$$Sharpe\;Ratio=\frac{R_P-R_F}{\sigma_P}$$

IR

$$IR=\frac{R_P-R_B}{\sigma_{(R_P-R_B)}}$$

With Optimal Level of Active Risk

$$SR_P=\sqrt{SR_B^2+IR^2}\\Total\;Risk\;of\;The\;Portfolio:\;\sigma_P^2\;=\sigma_B^2+\sigma_A^2$$

Unconstrained

$$IR\ast=IC\times\sqrt{BR}\\E(R_A)\ast=IC\sqrt{BR\sigma_A}$$

Constrained

$$IR=IC\times\sqrt{BR}\times TC\\E(R_A)=IC\times\sqrt{BR}\times TC\times\sigma_A\\SR_{pc}=\sqrt{SR_\beta^2+(IR^2\times TC)}\\\sigma_{CA}=\frac{TC.IR\ast}{SR_B}\times\sigma_B$$

Ex post Performance Measurement

$$E(R_A\vert IC_R)=TC\times IC_R\sqrt{BR}\sigma_A\\R_A=E(R_A\vert IC_R)+noise$$

The Expected Active Return for A Given Target Level of Active Risk

$$E(R_A)=IR\times\sigma_A\\IC=2(\%\;correct)-1\\\sigma_c=\left[\sigma_x^2+\sigma_y^2-2\sigma_x\sigma_yr_{x,y}\right]^{1/2}$$

Annualized Active Risk

$$\sigma_A=\sigma_c\times\sqrt{BR}$$

Annualized Active Return

$$E\left(R_A\right)=IC\sqrt{BR}\times\sigma_A\\BR=\frac N{1+(N-I)r}$$