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L1 - Capital Allocation Line
Name
Equation
Example
E ( r C ) = r F + y ( E ( r P ) β r F ) E(r_C) = r_F + y (E(r_P)-r_F) E ( r C β ) = r F β + y ( E ( r P β ) β r F β ) or
E ( r C ) = y E ( r P ) + ( 1 β y ) r F E(r_C) = y E(r_P) + (1-y) r_F E ( r C β ) = y E ( r P β ) + ( 1 β y ) r F β
= .04 + .50*(.12-.04) = .08 or 8%
Ο c = y Ο p Ο_c = y Ο_p Ο c β = y Ο p β
= .20 * .50 = .10 or 10%
V a r = Ο 2 Var = Ο^2 Va r = Ο 2 = Ο^2
Var = 12%^2 = .0144
Ο = V a r = V a r 1 2 Ο = \sqrt{Var} = Var^{\frac{1}{2}} Ο = Va r β = Va r 2 1 β
Ο = .0144^.5 = 12%
= E ( r C ) β r F = E(r_C)-r_F = E ( r C β ) β r F β
= 12% - 2% = 10%
S = E ( r ) β r F Ο S = \frac{E(r)-r_F}{Ο} S = Ο E ( r ) β r F β β
= (.12-.04)/.20 = (8/20) = .40
E ( r C ) = r F + E ( r P ) β r F Ο P Ο C E(r_C) = r_F + \frac{E(r_P) - r_F}{Ο_P}Ο_C E ( r C β ) = r F β + Ο P β E ( r P β ) β r F β β Ο C β
Sharpe ratio = .8 C ) = rF + SΟC = 2% + .8 * 15% = 14%
Notation: r F r_F r F β = Return of Risk
F ree Assets
r P r_P r P β = Return of
P ortfolio of Risky Assets
r C r_C r C β = Return of
C omplete Portfolio
E ( r P ) / E ( r C ) E(r_P) / E(r_C) E ( r P β ) / E ( r C β ) = Expected Return of Risky/Complete Portfolio
Occasionally I use
E r P / E r C Er_P/Er_C E r P β / E r C β as shorthand for
E ( r P ) / E ( r C ) E(r_P)/E(r_C) E ( r P β ) / E ( r C β ) y y y = % of Portfolio in Risky Assets
1 β y 1-y 1 β y = % in Risk-Free Assets
Ο / Ο P / Ο C Ο/Ο_P/Ο_C Ο / Ο P β / Ο C β = Standard Deviation
S S S = Sharpe Ratio
Variance = Standard Deviation^2
Standard Deviation = SQRT of Variance
References: 1 Jun 21.pptx and L1-Capital Allocation
L2 - Optimal Risky Portfolios
Name
Equation
E ( r p ) = w 1 E ( r 1 ) + w 2 E ( r 2 ) . . . E(r_p) = w_1E(r_1) + w_2E(r_2) ... E ( r p β ) = w 1 β E ( r 1 β ) + w 2 β E ( r 2 β ) ... Ex: Erp = 60%*11% + 40%*7% = 9.4% V a r ( r p ) = w 1 2 Ο 1 2 + w 2 2 Ο 2 2 + 2 w 1 w 2 C o v 1 , 2 Var(r_p) = w_1^2 Ο_1^2 + w_2^2 Ο_2^2 + 2 w_1 w_2 \textcolor{red}{Cov_{1,2}} Va r ( r p β ) = w 1 2 β Ο 1 2 β + w 2 2 β Ο 2 2 β + 2 w 1 β w 2 β C o v 1 , 2 β Ex: Var = 60%^2 * 15%^2 + 40%^2 * 9%^2 + 2*60%*40%*.0027 = 0.010692 Using Helper Formula β£, below, it follows that: V a r ( r p ) = w 1 2 Ο 1 2 + w 2 2 Ο 2 2 + 2 w 1 w 2 Ο 1 Ο 2 C o r r 1 , 2 Var(r_p) = w_1^2 Ο_1^2 + w_2^2 Ο_2^2 + 2w_1 w_2 \textcolor{red}{Ο_1 Ο_2 Corr_{1,2}} Va r ( r p β ) = w 1 2 β Ο 1 2 β + w 2 2 β Ο 2 2 β + 2 w 1 β w 2 β Ο 1 β Ο 2 β C or r 1 , 2 β Ex: Var = 60%^2*15%^2 + 40%^2*9%^2 + 2*60%*40%*15%*9%*.2 = 0.010692
CalcTake square root of variance to get Standard Deviation/Ο (See Helper Formula β‘, below).
Ο P = V a r = Ο_P=\sqrt{Var}= Ο P β = Va r β = 0.010692^.5 = 10.34%
w 1 = ( E ( r p ) β E ( r 2 ) ) ( E ( r 1 ) β E ( r 2 ) ) w 2 = 1 β w 1 w_1 = \frac{(E(r_p) - E(r_2))}{(E(r_1) - E(r_2))} \qquad w_2 = 1 - w_1 w 1 β = ( E ( r 1 β ) β E ( r 2 β )) ( E ( r p β ) β E ( r 2 β )) β w 2 β = 1 β w 1 β Ο 1 , 2 = C o v ( r 1 , r 2 ) Ο 1 Ο 2 = 0.00266 7 % Γ 19 % = . 2 Ο_{1,2} = \frac{Cov(r_1,r_2)}{Ο_1Ο_2}=\frac{0.00266}{7\% Γ 19\%}=.2 Ο 1 , 2 β = Ο 1 β Ο 2 β C o v ( r 1 β , r 2 β ) β = 7% Γ 19% 0.00266 β = .2 (Ο is always between -1 and 1) C o v ( r 1 , r 2 ) = Ο 1 , 2 Ο 1 Ο 2 = . 2 Γ 7 % Γ 19 % = 0.00266 \begin{aligned}Cov(r_1,r_2) &= Ο_{1,2}Ο_1Ο_2 \\ &= .2 Γ 7\% Γ 19\% = 0.00266\end{aligned} C o v ( r 1 β , r 2 β ) β = Ο 1 , 2 β Ο 1 β Ο 2 β = .2 Γ 7% Γ 19% = 0.00266 β Def of Covariance
= E { [ r 1 β E ( r 1 ) ] [ r 2 β E ( r 2 ) ] } =E\{\lbrack r_1-E(r_1)\rbrack\lbrack r_2-E(r_2)\rbrack\} = E {[ r 1 β β E ( r 1 β )] [ r 2 β β E ( r 2 β )]} U = E ( r ) β 1 2 A Ο 2 U=E(r) - \frac{1}{2}AΟ^2 U = E ( r ) β 2 1 β A Ο 2
Notation: w 1 w_1 w 1 β = portion invested in asset 1,
w 2 = ( 1 β w 1 ) w_2 = (1- w_1) w 2 β = ( 1 β w 1 β ) = portion invested in asset 2
E E E = "Expected "
r r r = return for asset
Ο Ο Ο = Standard Deviation
Ο 1 , 2 Ο_{1,2} Ο 1 , 2 β = correlation between assets 1 and 2
p p p =
P ortfolio of Risky Assets
r F r_F r F β = risk-
F ree rate
References: 2 Jun 26.pptx and L2-Optimal Risky Portfolio
L1, L2 - CAL/Risky Summary
Expected Value of Return
L1 - CAL
E ( r C ) = r F + y ( E ( r P ) β r F ) E(r_C) = r_F + y(E(r_P) - r_F) E ( r C β ) = r F β + y ( E ( r P β ) β r F β ) L2 - Two risky assets
E ( r p ) = w 1 E ( r 1 ) + w 2 E β ( r 2 ) E(r_p) = w_1E(r1) + w_2E\,(r_2) E ( r p β ) = w 1 β E ( r 1 ) + w 2 β E ( r 2 β )
Standard Deviation of Return (Risk)
L1 - CAL
Ο C = y β
β Ο P Ο_C = y \;Ο_P Ο C β = y Ο P β L2 - Two risky assets
Ο P = S Q R T ( Ο 1 2 β w 1 2 + Ο 2 2 β w 2 2 + 2 β w 1 β w 2 β C o v 1 , 2 ) Ο_P = SQRT({Ο_1}^2\,{w_1}^2 + {Ο_2}^2\,{w_2}^2 + 2\,w_1\,w_2\,Cov_{1,2}) Ο P β = SQRT ( Ο 1 β 2 w 1 β 2 + Ο 2 β 2 w 2 β 2 + 2 w 1 β w 2 β C o v 1 , 2 β ) Ο P = S Q R T ( Ο 1 2 β w 1 2 + Ο 2 2 β w 2 2 + 2 β w 1 β w 2 β Ο 1 β Ο 2 β C o r r 1 , 2 ) Ο_P = SQRT({Ο_1}^2\,{w_1}^2 + {Ο_2}^2\,{w_2}^2 + 2\,w_1\,w_2\,Ο_1\,Ο_2\,Corr_{1,2}) Ο P β = SQRT ( Ο 1 β 2 w 1 β 2 + Ο 2 β 2 w 2 β 2 + 2 w 1 β w 2 β Ο 1 β Ο 2 β C or r 1 , 2 β )
Foundational Formulas from Probability Bruce derived the previous four formulas from the following two classic formulas.
Expected Value of Return
E ( a X + b Y ) = a β E ( X ) + b β E ( Y ) E(aX + bY) = a\,E(X) + b\,E(Y) E ( a X + bY ) = a E ( X ) + b E ( Y )
Standard Deviation of Return (Risk)
V a r ( a X + b Y ) = a 2 β V a r ( X ) + b 2 β V a r ( Y ) + 2 β a β b β C o v ( X , Y ) Var(aX + bY) = a^2\,Var(X) + b^2\,Var(Y) + 2\,a\,b\,Cov(X,Y) Va r ( a X + bY ) = a 2 Va r ( X ) + b 2 Va r ( Y ) + 2 a b C o v ( X , Y )
Probability Helper Formulas These are worth knowing well!
Variance and Standard Deviation
Ο = V a r Β andΒ V a r = Ο 2 Ο = \sqrt{Var} \quad\text{ and } \\
Var = Ο^2 Ο = Va r β Β andΒ Va r = Ο 2
Covariance and Correlation
Ο 12 = C o v ( r 1 , r 2 ) Ο 1 Ο 2 Β andΒ Ο_{12}=\frac{Cov(r_1,r_2)}{Ο_1Ο_2}\quad\text{ and } Ο 12 β = Ο 1 β Ο 2 β C o v ( r 1 β , r 2 β ) β Β andΒ C o v ( r 1 , r 2 ) = Ο 12 Ο 1 Ο 2 Cov(r_1,r_2)=Ο_{12}Ο_1Ο_2 C o v ( r 1 β , r 2 β ) = Ο 12 β Ο 1 β Ο 2 β
L3-L4 - CAPM and EMH
Name
Equation
Example
Capital Asset Pricing Model (CAPM)
E ( r S ) = r F + Ξ² [ E ( r M ) β r F ] E(r_S) = r_F + Ξ² [E(r_M)-r_F] E ( r S β ) = r F β + Ξ² [ E ( r M β ) β r F β ] = 3 % + . 3 Γ ( 16 % β 3 % ) = 3\% + .3 Γ (16\%-3\%) = 3% + .3 Γ ( 16% β 3% )
Holding Period Return (HPR)
= ( P 1 β P 0 + D 1 ) P 0 = \frac{(P_1 - P_0 + D_1)}{P_0} = P 0 β ( P 1 β β P 0 β + D 1 β ) β = ( $ 103 β $ 97 + $ 2 ) $ 97 = \frac{(\$103 - \$97 + \$2)}{\$97} = $97 ( $103 β $97 + $2 ) β
In EMH, Investors value a stock as the PDV of its future cash flows: (for stocks, cash flows = dividends)
P S = E ( D 1 ) ( 1 + i ) + E ( D 2 ) ( 1 + i ) 2 + E ( D 3 ) ( 1 + i ) 3 + β¦ P_S = \frac{E(D_1)}{(1+i)} + \frac{E(D_2)}{(1+i)^2} + \frac{E(D_3)}{(1+i)^3} + \ldots P S β = ( 1 + i ) E ( D 1 β ) β + ( 1 + i ) 2 E ( D 2 β ) β + ( 1 + i ) 3 E ( D 3 β ) β + β¦
References: 3 Jul 1.pptx and L3-CAPM | 4 Jul 3.pptx and L4-EMH
L5 - Options
Name
Equation
Example
Intrinsic Value of a Call
CallΒ IV = M a x ( S β K , 0 ) \text{Call IV} = Max (S-K, 0) CallΒ IV = M a x ( S β K , 0 ) PutΒ IVΒ = M a x ( K β S , 0 ) \text{Put IV }= Max (K-S, 0) PutΒ IVΒ = M a x ( K β S , 0 ) CallΒ IV = M a x ( $ 52 β $ 50 , $ 0 ) = M a x ( $ 2 , $ 0 ) = $ 2 \text{Call IV} \\= Max(\$52-\$50,\$0) \\ =Max(\$2,\$0) =\$2 CallΒ IV = M a x ( $52 β $50 , $0 ) = M a x ( $2 , $0 ) = $2
P/L for Long Call or Put
= I V β P r = IV - Pr = I V β P r
P/L from
Buying a Call = M a x ( S β K , 0 ) β P r = Max(S-K,0) - Pr = M a x ( S β K , 0 ) β P r P/L from
Buying a Put = M a x ( K β S , 0 ) β P r = Max (K-S, 0) - Pr = M a x ( K β S , 0 ) β P r LongΒ Call = $ 2 β $ 3 = β $ 1 \text{Long Call} \\=\$2-\$3=-\$1 LongΒ Call = $2 β $3 = β $1 LongΒ Put = $ 0 β $ 3 = β $ 3 \text{Long Put} \\=\$0-\$3=-\$3 LongΒ Put = $0 β $3 = β $3
P/L Short Call or Put
= β I V + P r = - IV + Pr = β I V + P r
P/L from
Selling a Call = β M a x ( S β K , 0 ) + P r = - Max (S-K, 0) + Pr = β M a x ( S β K , 0 ) + P r P/L from
Selling a Put = β M a x ( K β S , 0 ) + P r = - Max (K-S, 0) + Pr = β M a x ( K β S , 0 ) + P r ShortΒ Call = β $ 2 + $ 3 = $ 1 \text{Short Call} \\=-\$2+\$3=\$1 ShortΒ Call = β $2 + $3 = $1 ShortΒ Put = β $ 0 + $ 3 = $ 3 \text{Short Put} \\=-\$0+\$3=\$3 ShortΒ Put = β $0 + $3 = $3
Premium and Time Value
Time Value = Premium - Intrinsic Value
TV = $ 3 β $ 2 = $ 1 \text{TV} \\= \$3 - \$2 = \$1 TV = $3 β $2 = $1
Calculate Premium
Premium = EV of the gain from an option or strategy
= 1 4 ( $ 0 ) + 1 2 ( $ 2 ) + 1 4 ( $ 8 ) = $ 3 = \frac{1}{4}(\$0) + \frac{1}{2}(\$2) \\ + \frac{1}{4}(\$8) = \$3 = 4 1 β ( $0 ) + 2 1 β ( $2 ) + 4 1 β ( $8 ) = $3
Leverage
= (Share PriceΓ100)/(PremiumΓ100)
= $ 52 Γ 100 $ 3 Γ 100 = 17.3 : 1 = \frac{\$52Γ100}{\$3Γ100} = 17.3:1 = $3 Γ 100 $52 Γ 100 β = 17.3 : 1
References: 5 Jul 8.pptx and L5-Options