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Effective Spread

$$For\;Buy\;Order=2\times(Execution\;Price–Mid\;Quote)\\For\;Sell\;Order=2\times(Mid\;Quote\;–\;Execution\;Price)$$

Total Implementation Shortfall

$$Total\;Implementation\;Shortfall\;(\%)=\frac{Paper\;Gain-Real\;Gain}{Paper\;Investment}$$

Missed Trade

$$\;Missed\;Trade=\vert CP-DP\vert\times\;\#\;of\;shares\;canceled$$

Explicit Cost

$$\;Explicit\;Cost=\cos t\;per\;share\times\;\#\;of\;shares\;executed$$

Delay

$$Delay=\vert BP^\ast-DP\vert\times\#\;of\;shares\;later\;executed$$

Market Impact

$$\;Market\;Impact=\vert EP\;–\;DP\vert or\;BP^\ast\;\times\;\#\;of\;shares\;executed$$

Time – Weighted Rate of Return (TWRR)

$$R_P=(1+R_{S1})\;(1+R_{S2})\;(1+R_{S3})\;(1+R_{S4})\dots.\;(1+R_{Sk})–1$$

Money – Weighted Rate of Return (MWRR)

$$MV_1=MV_0{(1+R)}^m+\overset n{\underset{i=1}\sum}CF_i{(1+R)}^{L(i)}$$

$$P=M+S+A\\Where,\;P=portfolio\;return\\M=market\;index\;return\\S=return\;to\;style\\A=return\;due\;to\;active\;management$$

Asset Category Contribution

$$R_{AC}=\sum_{i=1}^Aw_i(R_i-R_F)$$

$$R_{IM}=\sum_{i=1}^A\;\sum_{j=1}^Mw_i\times w_{i,j}\times(R_{A,i,j}-R_{B,i,j})$$

Micro Performance Attribution

$$R_V=\overset S{\underset{j=1}{\sum(}}W_{P,j}-W_{B,j})\times(r_{B,j}-r_B)+\sum_{j=1}^SW_{B,j}\times(r_{P,j}-r_{B,j})+\overset S{\underset{j=1}{\sum(}}W_{P,j}-W_{B,j})\times(r_{P,j}-r_{B,j})$$

Ex-post Alpha

$$\alpha_A=R_{At}-{\widehat R}_A$$

Treynor’s Measure

$$T_A=\frac{{\overline R}_A-{\overline r}_f}{\beta_A}$$

Sharpe Ratio

$$Sharpe\;Ratio=\frac{{\overline R}_A-{\overline r}_f}{\sigma_A}$$

$$M^2={\overline r}_F+\left[\frac{{\overline R}_A-{\overline R}_F}{\sigma_A}\right]\sigma_M$$

Information Ratio

$$IR_A=\frac{{\overline R}_A-{\overline R}_B}{\sigma_{A-B}}$$

The Original Dietz method

$$R_{Dietz}=\frac{EMV-BMV-CF}{BMV+0.5CF}$$

Modified Dietz method

$$\frac{EMV-BMV-CF}{BMV+{\displaystyle\sum_{i=1}^n}W_i\times CF_i}$$

$$Downside\;Deviation=\sqrt{\frac{{\displaystyle\sum_{i=1}^n}\left[Min\;(Return_t-threshold,0)\right]^2}{n-1}}\\Sharpe_{Hf}=\frac{Annualized\;return-Annual\;RiskFreeRate}{Annualized\;\sigma}$$

$$Analytical\;Method\;for\;Estimating\;VaR=\left[{\widehat R}_P-(z)(\sigma)\right]V_P$$

Measuring Risk Adjusted Performance

$$Sharpe\;Ratio=\frac{{\overline R}_P-{\overline R}_F}{\sigma_P}$$

$$Return\;over\;Maximum\;Drawdown(RoMAD)=\frac{R_P}{MaximumDrawdown}\\Sortino=\frac{{\overline R}_P-MAR}{DownsideDeviation}$$

$$\beta_i=\frac{Cov(i,m)}{\sigma^2m}\\Number\;of\;Contracts=\left(\frac{\beta_T-\beta_P}{\beta_f}\right)\left(\frac{V_P}{P_f(multiplier)}\right)\\Underlying\;Portfolio+Derivative=Target\;Portfolio\\V_p\times\;\beta_p+(n\times m\times F_p)\times\beta_f=V_p\times\beta_T\\n=V_p(\beta_T–\beta_P)/m\times F_p\times\beta_F\\Effective\;Beta=\frac{Percentage\;Change\;In\;Value\;of\;Portfolio}{Percentage\;Change\;in\;Index}\\Synthetic\;Equity=Cash+Futures\\Long\;Stock+Short\;Future=Long\;risk\;free\;Bond$$

Target Duration

$$Number\;of\;Contracts=(yield\;beta)\left(\frac{MD_T-MD_P}{MD_F}\right)\left(\frac{V_P}{P_f(multiplier)}\right)\\Number\;of\;Contracts=\left(\frac{D_T-D_P}{D_{CTD}P_{CTD}}\right)\;P_p$$

Covered Call

$$Covered\;Call=Long\;stock\;position+Short\;call\;position\\Value\;at\;expiration=V_T=S_T–max\;(0,\;S_T\;–\;X)\\Profit=V_T\;–S_0+c_0\\Maximum\;Profit=X\;–S_0+c_0\\Max\;loss\;(when\;S_T=0)=S_0–c_0\\Breakeven=S_T^\ast=S_0–c_0$$

Protective Put

$$Protective\;Put=Long\;stock\;position+Long\;Put\;position\\Value\;at\;expiration:V_T=S_T+max\;(0,X-S_T\;)\\Profit=V_T\;–S_0-p_0\\Maximum\;Profit=\infty\\Maximum\;loss=S_0+p_0-X\\Breakeven=S_T^\ast=S_0+p_0$$

Bull Call Spread

$$Bull\;Call\;Spread=Long\;Call\;(lower\;exercise\;price)+Short\;Call\;(higher\;exercise\;price)\\Initial\;value=V_0=c_1–c_2\\Value\;at\;expiration:VT=value\;of\;long\;call–Value\;of\;short\;call=max\;(0,\;S_T\;–\;X_1)-max\;(0,\;S_T\;–\;X_2)\\Profit=V_T-c_1+c_2\\Maximum\;Profit=X_2–X_1-c_1+c_2\\Maximum\;Loss=c_1-c_2\\Breakeven=S_T^\ast=X_1+c_1–c_2$$

Bull Put spread

$$Bull\;Put\;spread=Long\;Put\;(lower\;XP)+Short\;Put\;(higher\;XP)\\Identical\;to\;the\;sale\;of\;Bear\;Put\;Spread\;XP=exercise\;price$$

Bear Put Spread

$$Bear\;Put\;spread=Long\;Put\;(higher\;XP)+Short\;Put\;(lower\;XP)\\Initial\;value=V_0=p_2–p_1\\Value\;at\;expiration:\;V_T=value\;of\;long\;put–value\;of\;short\;put=max\;(0,\;X_2-S_T)-max\;(0,\;X_1-S_T)\\Profit=V_T–p_2+p_1\\Max\;Profit=X_2–X_1–p_2+p_1\\Max\;Loss=p_2-p_1\\Breakeven\;=S_T^\ast=X_2–p_2+p_1\\$$

Bear Call Spread

$$Bear\;Call\;Spread=Short\;Call\;(lower\;XP)+Long\;Call\;(higher\;XP)\\Identical\;to\;the\;sale\;of\;Bull\;Call\;Spread$$

Long Butterfly Spread

$$Long\;Butterfly\;Spread\;(Using\;Call)=Long\;Butterfly\;Spread\\=Long\;Bull\;call\;spread+Short\;Bull\;call\;spread\;(or\;Long\;Bear\;call\;spread)\\Long\;Butterfly\;Spread=(Buy\;the\;call\;with\;XP\;of\;X_1\;and\;sell\;the\;call\;with\;XP\;of\;X_2)\\+(Buy\;the\;call\;with\;XP\;of\;X_3\;and\;sell\;the\;call\;with\;XP\;of\;X_2)\\where,\;X_1<X_2<X_3\;and\;Cost\;of\;X_1(c_1)>Cost\;of\;X_2\;(c_2)>Cost\;of\;X_3\;(c_3)\\Value\;at\;expiration:\;V_T=max\;(0,S_T–X_1)–2\;max\;(0,\;S_T–X_2)+max\;(0,\;S_T–X_3)\\Profit=V_T–c_1+2c_2-c_3\\Max\;Profit=X_2–X_1–c_1+2c_2-c_3\\Maximum\;Loss=c_1-2c_2+c_3$$

Two breakeven points

$$Breakeven=ST\ast=X_1+net\;premium=X_1+c_1–2c_2+c_3\\Breakeven=ST\ast=2X_2–X_1–Net\;premium=2X_2–X_1–(c_1–2c_2+c_3)=2X_2–X_1–c_1+2c_2-c_3$$

Short Butterfly Spread

$$Short\;Butterfly\;Spread\;(U\sin g\;Call)=Selling\;calls\;with\;XP\;of\;X_1\;and\;X_3\;and\;buying\;two\;calls\;with\;XP\;of\;X_2\\Max\;Profit=c_1+c_3–2c_2$$

Long Butterfly Spread

$$Long\;Butterfly\;Spread\;(U\sin g\;Puts)=(Buy\;put\;with\;XP\;of\;X_3\;and\;sell\;put\;with\;XP\;of\;X_2)\\+(Buy\;the\;put\;with\;XP\;of\;X_{1\;}and\;sell\;the\;put\;with\;XP\;of\;X_2)\\where,\;X_1<X_2<X_3\;and\;Cost\;of\;X_1\;(p_1)<Cost\;of\;X_2\;(p_2)<Cost\;of\;X_3\;(p_3)$$

Short Butterfly Spread

$$Short\;Butterfly\;Spread\;(U\sin g\;Puts)=Short\;butterfly\;spread\\=Selling\;puts\;with\;XPs\;of\;X_1\;and\;X_3\;and\;buying\;two\;puts\;with\;XP\;of\;X_2\\Max\;Profit=p_3\;+\;p_1\;–\;2p_2$$

For zero-cost collar

$$Initial\;value\;of\;position=V_0=S_0\\Value\;at\;expiration:\;V_T=S_T+max\;(0,\;X_1\;-\;S_T)\;–\;max\;(0,\;S_T\;–\;X_2)\\Profit=V_T–V_0=V_T–S_0\\Max\;Profit=X_2–S_0\\Max\;Loss=S_0–X_1\\Breakeven=S_T\ast=S_0$$

Straddle

$$Straddle=Buying\;a\;put\;and\;a\;call\;with\;same\;strike\;price\;on\;the\;same\;underlying\;with\;the\;same\;expiration;\\;both\;options\;are\;at\;the\;money\\Value\;at\;expiration:\;V_T=max\;(0,\;S_T-X)+max\;(0,\;X–S{{}_T)}\\Profit=V_T-p_0-c_0\\Max\;Profit=\infty\\Max\;Loss=p_0+c_0\\Breakeven=ST\ast=X\pm(p_0+c_0)\\$$

Short Straddle: Selling a put and a call with same strike price on the same underlying with the same expiration; both options are at-the-money.

  • Adding call option to a straddle “Strap”.
  • Adding put option to a straddle “Strip”.

Long Strangle = buying the put and call on the same underlying with the same expiration but with different exercise prices.

Short Strangle = selling the put and call on the same underlying with the same expiration but with different exercise prices.

Box-spread = Bull spread + Bear spread

Long Box-spread

$$Long\;Box\;spread=(buy\;call\;with\;XP\;of\;X_1\;and\;sell\;call\;with\;XP\;of\;X_2)\\+(buy\;put\;with\;XP\;of\;X_2\;and\;sell\;put\;with\;XP\;of\;X_1)\\Initial\;value\;of\;the\;box\;spread=Net\;premium=c_1–c_2+p_2–p_1\\Value\;at\;expiration:\;V_T=X_2-X_1\\Profit=X_2-X_1-\left(c_1–c_2+p_2–p_1\right)\\Max\;Profit=same\;as\;profit\\Max\;Loss=no\;loss\;is\;possible\;given\;fair\;option\;prices\\Breakeven=ST\ast=no\;break\;even;\;the\;transaction\;always\;earns\;R_f\;rate,\;given\;fair\;option\;prices\\$$

Pay-off of an interest rate Call Option

$$Pay\;off\;of\;an\;interest\;rate\;Call\;Option=(NP)\times max(0,Underlying\;rate\;at\;expiration–X\;rate)\times\frac{Days\;in\;Underlying\;Rate}{360}$$

Pay-off of an interest rate Put Option

$$Pay\;off\;of\;an\;interest\;rate\;Put\;Option=(NP)\times max(0,X\;rate-Underlying\;rate\;at\;expiration)\times\frac{Days\;in\;Underlying\;Rate}{360}$$

Loan Interest payment

$$Loan\;Interest\;payment=NP\times(LIBOR\;on\;previous\;reset\;date+Spread)\times\frac{Days\;in\;Settlement\;Period}{360}$$

Cap Pay-Off

$$Cap\;Pay\;Off=NP\times(0,\;LIBOR\;on\;previous\;reset\;date–X\;rate)\times\frac{Days\;in\;Settlement\;Period}{360}$$

Floorlet Pay-Off

$$Floorlet\;Pay\;Off=NP\times(0,\;X\;rate-LIBOR\;on\;previous\;reset\;date)\times\frac{Days\;in\;Settlement\;Period}{360}$$

Effective Interest

$$Effective\;Interest=Interest\;received\;on\;the\;loan+Floorlet\;pay\;off$$

Delta

$$Delta=\frac{Days\;in\;Settlement\;Period}{360}=\frac{\triangle C}{\triangle S}$$

Size of the Long position

$$Size\;of\;the\;Long\;position=N_c/N_s=-1/(\triangle C/\triangle S)=-1/Delta\\where,\;N_c=No\;of\;call\;options\\N_s=No\;of\;stocks$$

Hedging using non-identical option

  • One option has a delta of Δ1.
  • Other option has a delta of Δ2.
  • Value of the position$$V=N_1c_1+N_2c_2\\where,\;N=option\;quantity\;\&\;c=option\;price$$
  • To delta hedge:$$Desired\;Quantity\;of\;option\;1\;relative\;to\;option\;2=\frac{Delta\;of\;option\;2}{Delta\;of\;option\;1}$$

$$\;Gamma=\frac{Change\;in\;Delta}{Change\;in\;Underlying\;Price}\\Gamma\;hedge=Position\;in\;underlying+Positions\;in\;two\;options\\Vega=\frac{Change\;in\;Option\;Price}{Change\;in\;Volatility\;of\;the\;Underlying}$$

For a pay floating counterparty in a swap, the duration can be expressed as:

$$D_{pay\;floating}=D_{fixed}–D_{floating}>0$$

The duration of portfolio plus a swap position:

$$V_P(MD_T)=V_P(MD_p)+N_P(MD_{swap})$$

NP of a swap:

$$NP=V_P\left(\frac{MD_T-MD_P}{MD_{Swap}}\right)$$

Callable Bond

$$Straight\;Bond\;Buy+Receiver\;Swaption\;Sell=Callable\;Bond$$

Putable bond

$$Straight \;Bond \;Buy + Payer \;Swaption \;Buy = Putable \;bond$$

 

Risk Objectives

$$UP={\widehat R}_P-0.005(A)(\sigma_P^2)$$

Roy’s Safety-First Measure (RSF)

$$RSF=\frac{{\widehat R}_P-R_{MAR}}{\sigma_P}$$

The Investor’s Utility from Investing A Portfolio

$$U=E(R)–0.5\times\lambda\times\sigma^2$$

$$MCTR_i=\beta_{i,p}\sigma_P$$

$$ACTR_i=Wi\times MCTR_i\\\%\;ACTR\;to\;Total\;Risk=\frac{ACTR_i}{\sigma_P}$$

An Investor’s Return in Domestic Currency

$$R_{DC}=(1+R_{FC})(1+R_{FX})–1\\RDC\;=\sum_{i=1}^nw_i\left(R_{DC,i}\right)$$

Risk Variance of a Two Asset of a Portfolio

$$\sigma^2(R_{DC})\approx w^2(R_{FC})\sigma^2(R_{FC})+w^2(R_{FX})\sigma^2(R_{FX})+2w(R_{FC})\;w(R_{FX})\;\sigma(R_{FC})\;\sigma(R_{FX})\;\rho(R_{FC},R_{FX})$$

Single FC denominated Asset

$$\sigma^2(R_{DC})\approx\sigma^2(R_{FC})+\sigma^2(R_{FX})+2\sigma(R_{FC})\;\sigma(R_{FX})\;\rho(R_{FC},R_{FX})\\\sigma^2(R_{DC})\approx\sigma^2(R_{FX})\;(1+R_{FC})$$

 

$$Utility\;(U)=w(p_1)\;v(X_1)+w(p_2)\;v(X_2)\\Subjective\;expected\;U\;of\;an\;individual=\Sigma\left[u(x_i)\times Prob(x_i)\right]\;\\Bayes’\;Formula=P(A\vert B)=\left[P(B\vert A)/P(B)\right]\times P(A)\\Risk\;Premium=Certainty\;Equivalent–Expected\;Value\\Perceived\;Value\;of\;Each\;Outcome=U=w(p_1)\;v(x_1)+w(p_2)\;v(x_2)+\dots+w(p_n)\;v(x_n)\\Abnormal\;Return\;(R)=Actual\;R–Expected\;R\\$$

$$After\;tax\;(AT)Real\;Required\;return\;(RR)\;\%=\\\frac{Client's\;Required\;Expenditure\;in\;Year\;n}{Net\;Investable\;Assets}=\frac{Projected\;Needs\;in\;Year\;n}{Net\;Investable\;Assets}\\AT\;Nominal\;RR\;\%=\frac{Projected\;Needs\;in\;Year\;n}{Net\;Investable\;Assets}+Current\;Annual(Ann)\;Inflation\;(Inf)\;\%\\=AT\;real\;RR\%+Current\;Ann\;Inf\%\;\;\;\\Or,\;\;AT\;Nominal\;RR\%=(1+AT\;Real\;RR\%)\times(1+Current\;Ann\;Inf\%)–1\\Total\;Investable\;Assets=Current\;Portfolio-Current\;Year\;Cash\;Outflows+Current\;Year\;Cash\;Inflows\\Pre-tax\;income\;needed=AT\;income\;needed/(1-tax\;rate)\\Pre-tax\;Nominal\;RR=(Pre-tax\;income\;needed/Total\;investable\;assets)+Inf\%$$

If Portfolio returns are tax-deferred:

$$Pre-tax\;projected\;expenditure\;\$=AT\;projected\;expenditure\;\$/(1–tax\;rate)\\Pre-tax\;real\;RR\;\%=Pre-tax\;projected\;expenditures\;\$/Total\;investable\;assets\\Pre-tax\;nominal\;RR=(1+Pre-tax\;real\;RR\%)\times(1+Inflation\;rate\%)–1$$

If Portfolio returns are NOT tax-deferred:

$$AT\;real\;RR\%=AT\;projected\;expenditures\;\$/Total\;Investable\;assets\\AT\;nominal\;RR\%=(1+AT\;real\;RR\%)\times(1+Inf\%)–1$$

Procedure of converting nominal, pre-tax figures into real, after-tax return:

$$Real\;AT\;R=\left[Expected\;total\;R–(Expected\;total\;R\;of\;Tax-exempt\;Invs\times\;wt\;of\;Tax-exempt\;Invst)\;\right]\\\times(1–tax\;rate)+(Expected\;total\;R\;of\;Tax-exempt\;Invst\times wt\;of\;Tax-exempt\;Invst)–Inf\;rate\\$$

Or

$$Real\;AT\;R=\left[(Taxable\;R\;of\;asset\;class\;1\times wt\;of\;asset\;class\;1)+(Taxable\;R\;of\;asset\;class\;2\times wt\;of\;asset\;class\;2)\\+\dots+(Taxable\;return\;of\;asset\;class\;n\times wt\;of\;asset\;class\;n)\right]\\\times(1–tax\;rate)+(Expected\;total\;R\;of\;Tax-exempt\;Invst\times wt\;of\;Tax-exempt\;Invst)–Inf\;rate$$

 

Accrual Taxes

$$FVIF_{AT}=\left[1+r\;(1-t)\right]^n$$

Deferred Capital Gain Taxation

$$FVIF_{AT}={(1\;+\;r)}^n\;(1-t_{cg})+t_{cg}\\FVIF_{AT}={(1\;+\;r)}^n\;(1-t_{cg})+t_{cg}\left(B\right)$$

Annual Wealth Basis Taxation

$$FVIF_{AT}=\left[(1+r)\;(1-t_w)\right]^n$$

Effective Annual After-tax Return of a Blended Tax Regime

$$r^\ast=r_T\;(1-p_it_i–p_dt_d+p_{cg}t_{cg})$$

Effective Capital Gains Tax Rate

$$r^\ast=r_T\;(1-p_it_i–p_dt_d+p_{cg}t_{cg})\\T^\ast=\left[\frac{1-(p_i+p_d+p_{cg})}{1-(p_it_i+p_dt_d+p_{cg}t_{cg})}\right]$$

Modified Version of The Standard Deferred Capital Gains FV Formula

Standard Formula:$$FVIF_{AT}={(1\;+\;r)}^n\;(1-t_{cg})+t_{cg}(B)\;$$

Modified Formula:$$Modified\;FVIF_{AT}={(1+r^\ast)}^n\times(1-T^\ast)+T^\ast-(1-B)t_{cg}$$

Tax – Deferred Accounts (TDA)

$$FVIF_{AT}={(1+r)}^n(1-t_n)$$

Tax – Exempt Account (TEA):

$$FVIF_{AT}={(1+r)}^n$$

After – Tax Return and Risk are calculated as:

$$r_{AT}=r(1-t)\\\sigma_{AT}=\sigma(1-t)\\$$

 

$$Core\;Capital\;(CC)\;Spending\;Needs=\sum_{j=1}^N\frac{p(Survival_j)\times Spending_j}{(1+r)^j}\\Expected\;Real\;Spending=Real\;Annual\;Spending\times Combined\;Probability\\CC\;needed\;to\;maintain\;given\;spending\;pattern=Annual\;Spending\;Needs\;/\;Sustainable\;Spending\;Rate$$

Tax-Free Gifts

$$\;RV_{Tax-free\;gift\;}=\frac{\left[1+r_g(1-t_{ig})\right]^n}{\left[1+r_e(1-t_{ie})\right]^n(1-T_e)}$$

Taxable Gifts

$$\;RV_{Taxable\;gift\;}=\frac{\left[1+T_g\right]\left[(1-r_g)(1-t_{ig})\right]^n}{\left[1+r_e(1-t_{ie})\right]^n(1-T_e)}$$

The relative after-tax value of the when the donor pays gift tax and when the recipient’s estate will not be taxable (assuming rg = re and tig = tie)

$$RV_{Taxable\;gift\;}=\frac{FV_{Gift}}{FV_{Bequest}}\;\\\frac{\left[1+r_g\left(1-t_{ig}\right)^n\right](1-T_g+T_gT_e)}{\left[1+r_e(1-t_{ie})\right]^n(1-T_e)}\\FV_{no\;skipping}=PV\left[{(1+r)}^{n1}\right]\left[(1-t)\right]\left[{(1+r)}^{n2}(1-t)\right]\\FV_{skipping}=PV\left[{(1+r)}^N(1-T_e)\right]$$

Charitable Gratuitous Transfers

$$RV_{Charitable\;Donation\;}=\frac{FV_{Charitable\;Gift}}{FV_{Bequest}}\;\\\frac{\left(1+r_g\right)^n+T_a\left[1+r_e(1-t_e)\right]^n(1-T_e)}{\left[1+r_e(1-t_{ie})\right]^n(1-T_e)}$$

 

Human Capital

$$Human\;Capital\;HC_O=\sum_{t=1}^N\frac{W_t}{\left(1+r\right)^t}$$

extended model

$$Extended\;Model\;HC_O\;=\sum_{t=1}^N\frac{p\left(S_t\right)W_{t-1}\left(1+g_t\right)}{\left(1+r_f+y\right)^t}$$

Income Yield

$$Income\;Yield\;(payout)=\frac{Total\;Ongoing\;Annual\;Income}{Initial\;Purchase\;Price}\\$$

Mortality Weighted Net Present Value of the Pension

$$mNPV_0=\sum_{t=1}^N\frac{p\left(S_t\right)b_t}{\left(1+r\right)^t}\\\\$$

 

Simple Spending Rule

$$\;Spending_t=S_{market\;value\;asset\;(t-1)}$$

Rolling 3 years Average Spending Rule

$$\;Spending_t=SpendingRate\left(\frac{MV_{t-1}+MV_{t-2}+MV_{t-3}}3\right)$$

Geometric Spending Rate

$$\;Spending_t=R(Spending_{t-1})\;(1+inflation_{t-1})+(1-\;R)\;S\;(MV_{t-1})\;\;\;\;\;\;\;$$

Leverage – Adjusted Duration Gap

$$\;LADG=D_{asset}–(L/A)\;D_{liabilities\;}\;$$

Market Volatility

$$\sigma_t^2=\beta\sigma_{t-1}^2+(1-\beta)\sigma_t^2$$

Multifactor Models

$$R_i=\alpha_i+\beta_{i,1}\;F_1+\beta_{i,2\;}F_2+\varepsilon_i$$

Gordon Growth Model

$$P_0=\frac{Div_1}{{\widehat R}_i-g}\Rightarrow{\widehat R}_i=\frac{Div_1}{P_0}+g$$

Grinold – Kroner Model

$${\widehat R}_i=\frac{D_1}{P_0}+i+g-\triangle S\triangle\left(\frac PE\right)$$

Expected Income Return

$$Expected\;Income\;Return=\left(\frac{D_1}{P_0}-\triangle S\right)$$

Expected Nominal Earnings Growth 

$$Expected\;Nominal\;Earnings\;Growth=i+g$$

Expected Repricing Return

$$\;Expected\;Repricing\;Return=\triangle\left(\frac PE\right)$$

Estimating Fixed Income Returns: Risk Premium Approach

$$R_B=Real\;Risk-Free\;Rate+Inflation\;Risk\;Premium+Default\;Risk\;Premium\\+Illiquidity\;Risk\;Premium+Maturity\;Risk\;Premium+Tax\;Premium$$

The equation for the CAPM

$${\widehat R}_i=R_F+\beta_i({\widehat R}_M-R_F)$$

The Covariance Between Two Markets Given Two Factors

$$The\;covariance\;between\;two\;markets\;given\;two\;factors\;is:\\Cov(i,j)=\beta_{i,1}\;\beta_{j,1}\;\sigma_{F_1}^2+\beta_{i,2}\;\beta_{j,2}\;\sigma_{F_2}^2+\left(\beta_{i,1}\;\beta_{j,2}+\beta_{i,2}\;\beta_{j,1}\right)Cov(F_1,F_2)\\If\;there\;is\;only\;one\;factor\;driving\;returns\\Cov(i,j)=\beta_i\;\beta_j\;\sigma_M^2\\$$

Target Interest Rate

$$R_{target}=r_{neutral}+\left[0.5\;(GDP_{expected}\;–\;GDP_{trend})+0.5\;(i_{expected}\;–\;i_{target})\right]\\$$

 

Cobb-Douglas Production Function

$$Y=A\times K^\alpha\times L^\beta\\\frac{\triangle Y}Y\approx\frac{\triangle A}A+\alpha\frac{\triangle K}K+(1-\alpha)\frac{\triangle L}L$$

Solow Residual 

$$Solow\;Residual=\%\triangle TEP=\;\%\triangle Y–\alpha\;(\%\triangle K)–(1–\alpha)\%\triangle L$$

Gordon Growth Model for Mature Economies

$$V_0=\frac{D_1}{r-\overline g}=\frac{D_0\left(1+\overline g\right)}{r-\overline g}$$

H – Model for Emerging Economies

$$V_0=\frac{D_0}{r-g_L}\left[(1+g_L)\frac N2(g_s-g_L)\right]\\$$

Fed Model Ratio

$$Fed\;Model\;Ratio=\frac{S\;\&\;P\;Earnings\;Yield}{10\;year\;Treasury\;Yield}\\$$

The Yardeni Model

$$P_0=\frac{E_1}{r-g}\\\frac{E_1}{r-g}=Y_B-d\;\left(LTEG\right)$$

P/10-Year MA

$$\;P/10-Year\;MA\;(E)=\frac{Market\;Price\;of\;S\&P\;500}{10\;year\;Average\;Real\;Reported\;Earnings}$$

Tobin’s q

$$Tobin’s\;q=\frac{Market\;Value\;of\;Debt+Equity}{Asset\;Replacement\;Cost}$$

Equity q

$$Equity\;q=\frac{Market\;Value\;of\;Equity}{Replacement\;Value\;of\;Assets-Liabilities}$$

Modelling Expected Return

$$Income\;Yield=\frac{Annual\;Coupon\;Payment}{Current\;Bond\;Portfolio\;Price}$$

Roll Down Return

$$Roll\;Down\;Return=\frac{\left(Bond\;Price_{End}-Bond\;Price_{Beginning}\right)}{Bond\;Price_{Beginning}}$$

Leverage Portfolio’s Return

$$r_p=\frac{Portfolio\;Return}{Portfolio\;Equity}$$

Leveraged Portfolio’s Return

$$\;(RoE)=R_A+\frac DE\left[R_A-K_d\right]$$

Leverage

$$\;Leverage=\frac{Notional\;Value\;of\;Contract-Margin\;Amount}{Margin\;Amount}$$

Rebate Rate

$$Rebate\;Rate=Collateral\;Earnings\;Rate–Security\;Lending\;Rate$$

Convexity

$$Convexity=\frac{Mac.Duration^2+Mac.Duration+Dispersion}{(1+periodic\;IRR)^2}$$

Basis Point Value

$$Basis\;Point\;Value\;(BPV)=MD\times V\times0.0001$$

Future BPV

$$FutureBPV\approx\frac{BPV_{CTD}}{CF_{CTD}}$$

Future Contracts

$$N_f=\frac{Liability\;Portfolio\;BPV-Asset\;Portfolio\;BPV}{Futures\;BPV}$$

Effective Duration

$$Effective Duration=\frac{(PV_+)-(PV_-)}{2\times\triangle Curve\times(PV_0)}$$

Projected Holding Period Return for each bond

$$Projected\;Holding\;Period\;Return\;for\;each\;bond\;can\;be\;approximated\;as:\\\left[ending\;MD\times(Manager’s\;Forecasted\;Yield–Implied\;Forward\;Yield)\right]+Coupon\;Yield$$

Portfolio PVBP

$$Portfolio\;PVBP=0.0001\times modified\;duration\times portfolio\;value$$

Excess Return

$$XR=(s\times t)–\triangle s\times SD$$

Expected XR

$$EXR=(s\times t)–\left(\triangle s\times SD\right)-\left(t\times p\times L\right)$$

Spread risk as the spread bond will underperform in relative price change when spread widens

$$\;\%\triangle\;Value=-MD\triangle y\\\;\%\triangle\;Relative\;Value=-D_S\triangle y\\Spread=y_{\;higher\;yield}–y_{\;government}$$

Information Ratio

$$\;Information\;Ratio\approx Information\;Coefficient\times\sqrt{Investor's\;Breadth}$$

Maximize Utility Through Manager Selection

$$U_A=r_A-\lambda_A\times\sigma_A^2$$

Portfolio Active Risk

$$Portfolio\;Active\;Risk=\sqrt{\sum_{i=1}^n{w^2}_{a,i}\;{\sigma^2}_{a,i}}$$

Total Active Risk

$$Total\;Active\;Risk=\sqrt{\left(True\;Active\;Risk^2\right)+\left(Misfit\;Active\;Risk^2\right)}$$

True Information Ratio

$$\;True\;Information\;Ratio=\frac{True\;Active\;Return}{True\;Active\;Risk}$$

$$Effective\;No.\;of\;Stocks=\frac1{HHI}\\HHI=\sum_{i=1}^nW_i^2$$

Return Based Style Analysis

$$R_P=a+b_1SCG+b_2\;LCG+b_3\;SCV+b_4\;LCV+\varepsilon$$

Ex-Post RA

$$Ex-Post\;R_A\;=\sum_{i=1}^n\triangle W_iR_i$$

Decomposition of Ex Post Active Return

$$R_A\;=\sum\left(\beta_{Pk}-\beta_{bk}\right)\times F_k+(\alpha+\varepsilon)\\E(R_A)=IC\sqrt{BR}\sigma_{R_A}TC$$

Active Share

$$Active\;Share=\frac12\sum_{i=1}^n\left|W_{pi}W_{bi}\right|$$

Active Risk

$$Active\;Risk=\sqrt{\frac{\sum_{t=1}^T\left(R_P-R_B\right)^2}{T-1}}$$

Decompostion of Active Risk

$$\sigma_{R_A}=\sqrt{\sigma^2((\beta_{Pk}-\beta_{bk})\times F_k)+\sigma^2e}$$

CVi

$$CV_i=\beta_i\sum_{j=1}^n\beta_jCV_{ij}=\beta_jCV_{ij}$$

CAVi

$$CAV_i=\left(W_{pi}-W_{bi}\right)\sum_{j=1}^n\left(W_{pj}-W_{bj}\right)RC_{ij}=\left(W_{pi}-W_{bi}\right)\times RC_{iP}$$