🔎 Exam 1 Formulas

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Questions or comments? Please email robecon1452@gmail.com. Remember, your first reference is always the lectures and the homework.

L1 - Capital Allocation Line

Name

Equation

Example

Expected Return, $E(r_C)$

$E(r_C) = r_F + y (E(r_P)-r_F)$
or
$E(r_C) = y E(r_P) + (1-y) r_F$

= .04 + .50*(.12-.04) = .08 or 8%
or
= (.12 * .50) + (.04 * .50) = 8%

Standard Deviation of Complete Port, $σ_c$

$σ_c = y σ_p$

= .20 * .50 = .10 or 10%

Variance, $Var$

$Var = σ^2$ = σ^2

Var = 12%^2 = .0144

Standard Deviation, $σ$

$σ = \sqrt{Var} = Var^{\frac{1}{2}}$

σ = .0144^.5 = 12%

Risk Premium

Risk Premium $= E(r_C)-r_F$

= 12% - 2% = 10%

Sharpe Ratio, $S$

$S = \frac{E(r)-r_F}{σ}$

= (.12-.04)/.20 = (8/20) = .40

Equation for CAL

$E(r_C) = r_F + \frac{E(r_P) - r_F}{σ_P}σ_C$

Sharpe ratio = .8
E(r_C) = r_F + Sσ_C = 2% + .8 * 15% = 14%

References: 1 Jun 21.pptx and L1-Capital Allocation

$r_F$ = Return of Risk Free Assets

$r_P$ = Return of Portfolio of Risky Assets

$r_C$ = Return of Complete Portfolio

$E(r_P) / E(r_C)$ = Expected Return of Risky/Complete Portfolio

Occasionally I use $Er_P/Er_C$ as shorthand for $E(r_P)/E(r_C)$

$y$ = % of Portfolio in Risky Assets

$1-y$ = % in Risk-Free Assets

$σ/σ_P/σ_C$ = Standard Deviation

$S$ = Sharpe Ratio

Variance = Standard Deviation^2

Standard Deviation = SQRT of Variance

L2 - Optimal Risky Portfolios

Name

Equation

Expected Return of 2+ risky assets

$E(r_p) = (w_1 × Er_1) + (w_2 × Er_2) ...$
Ex: Erp = 60%*11% + 40%*7% = 9.4%

Variance with 2 risky assets from Covariance

$Var(r_p) = w_1^2 σ_1^2 + w_2^2 σ_2^2 + 2 w_1 w_2 \textcolor{red}{Cov_{1,2}}$
Ex: Var = 60%^2 * 15%^2 + 40%^2 * 9%^2 + 2*60%*40%*.0027 = 0.010692

Variance with 2 risky assets from Correlation

Using Helper Formula ④, below, it follows that:
$Var(r_p) = w_1^2 σ_1^2 + w_2^2 σ_2^2 + 2w_1 w_2 \textcolor{red}{σ_1 σ_2 Corr_{1,2}}$
Ex: Var = 60%^2*15%^2 + 40%^2*9%^2 + 2*60%*40%*15%*9%*.2 = 0.010692

SD with 2 risky assets

CalcTake square root of variance to get Standard Deviation/σ (See Helper Formula ②, below).
$σ_P=\sqrt{Var}=$ 0.010692^.5 = 10.34%

If you have a target $E(r_P$)

$w_1 = \frac{(E(r_p) - E(r_2))}{(E(r_1) - E(r_2))} \qquad w_2 = 1 - w_1$

Correlation Coefficient $(ρ_{1,2})$

$ρ_{1,2} = \frac{Cov(r_1,r_2)}{σ_1σ_2}=\frac{0.00266}{7\% × 19\%}=.2$
(ρ is always between -1 and 1)

Covariance

$\begin{aligned}Cov(r_1,r_2) &= ρ_{1,2}σ_1σ_2 \\ &= .2 × 7\% × 19\% = 0.00266\end{aligned}$
Def of Covariance $=E\{\lbrack r_1-E(r_1)\rbrack\lbrack r_2-E(r_2)\rbrack\}$

Utility Function

$U=E(r) - \frac{1}{2}Aσ^2$

Capital Asset Pricing Model (CAPM)

$Er_P = r_F + (B × (Er_M-r_F))$

References: 2 Jun 26.pptx and L2-Optimal Risky Portfolio

$w_1$ = portion invested in asset 1,

$w_2 = (1- w_1)$ = portion invested in asset 2

$E$ = "Expected "

$r$ = return for asset

$σ$ = Standard Deviation

$ρ_{1,2}$ = correlation between assets 1 and 2

$p$ = Portfolio of Risky Assets

$r_F$ = risk-free rate

$β$ = Beta

Summary of Portfolio Theory

Expected Value of Return:
CAL:
 $E(r_C) = r_F + y(E(r_P) - r_F)$
Two risky assets:
 $E(r_p) = w_1E(r1) + w_2E\,(r_2)$
Classic equation:
 $E(aX + bY) = a\,E(X) + b\,E(Y)$

Standard Deviation of Return (Risk):
CAL:
 $σ_C = y \;σ_P$
Two risky assets:
 $σ_P = SQRT({σ_1}^2\,{w_1}^2 + {σ_2}^2\,{w_2}^2 + 2\,w_1\,w_2\,Cov_{1,2})$
 $σ_P = SQRT({σ_1}^2\,{w_1}^2 + {σ_2}^2\,{w_2}^2 + 2\,w_1\,w_2\,σ_1\,σ_2\,Corr_{1,2})$
Classic equation:
 $Var(aX + bY) = a^2\,Var(X) + b^2\,Var(Y) + 2\,a\,b\,Cov(X,Y)$
* I wouldn't worry about using the two classic equations - but it's good to know the formulas Bruce used to come up with our equations.

Probability Helper Formulas:
Variance and Standard Deviation:
  $σ = \sqrt{Var}\quad\text{ and }\quad Var = σ^2$
Covariance and Correlation:
 $ρ_{12}=\frac{Cov(r_1,r_2)}{σ_1σ_2}\quad\text{ and }$
$\quad Cov(r_1,r_2)=ρ_{12}σ_1σ_2$

L3-L4 - CAPM and EMH

Name

Equation

Capital Asset Pricing Model (CAPM)

$Er_P = r_F + β (E(r_M)-r_F)$

In EMH, Investors value a stock as the PDV of its future cash flows: (for stocks, cash flows = dividends)

$P_{S} = \frac{E\left( D_{1} \right)}{(1 + i)} + \frac{E\left( D_{2} \right)}{(1 + i)^{2}} + \frac{E\left( D_{3} \right)}{(1 + i)^{3}} + \cdots$

CAPM Notation

Associated Jargon

$r_F = 5\%$

Risk free rate OR return on T-Bills OR return on short term government securities OR return on securities perceived to be risk-free

$E(r_S) = 12\%$

Expected (rate of) return offered by a specific security OR Expected rate of return *required by the market* for a specific portfolio (or stock) ($E(r_i)$ and $E(r_S)$ mean the same thing. 'i' just refers to a numbered security and 's' just refers to a specific stock.)

$E(r_S) - r_F = 7\%$
or $E(R_S)$

(Expected) risk premium for a specific security or portfolio
Note: I can get 5% by investing in risk-free assets, so I ignore the first 5% of the return from investing in a stock. I'm only interested in the 7% risk premium that I get above the risk-free rate.

$β$

Beta (a measure of non-diversifiable risk)

$r_M$

The actual return on the market portfolio.
Note: We imagine a portfolio consisting of every single stock, bond, and other security in existence. We call this special portfolio "the market portfolio." $r_M$ is the expected return of the market portfolio.

$E(r_M)$

Expected return of the market OR Expected market return OR Expected return of/on the market portfolio

$E(r_M) - r_F$ or $E(R_M)$

(Expected) Risk Premium of the market OR (Expected) Risk Premium of the market portfolio

$R_S = r_S - r_F$
$R_M = r_M - r_F$

Excess Return of a security
Excess Return of the market (The textbook will use this notation.)

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

$\text{Call IV} = Max (S-K, 0)$
$\text{Put IV }= Max (K-S, 0)$

$\text{Call IV} \\= Max(\$52-\$50,\$0) \\ =Max(\$2,\$0) =\$2$

P/L for Long Call or Put $= IV - Pr$

P/L from Buying a Call $= Max(S-K,0) - Pr$
P/L from Buying a Put $= Max (K-S, 0) - Pr$

$\text{Long Call} \\=\$2-\$3=-\$1$
$\text{Long Put} \\=\$0-\$3=-\$3$

P/L Short Call or Put $= - IV + Pr$

P/L from Selling a Call $= - Max (S-K, 0) + Pr$
P/L from Selling a Put $= - Max (K-S, 0) + Pr$

$\text{Short Call} \\=-\$2+\$3=\$1$
$\text{Short Put} \\=-\$0+\$3=\$3$

Premium and Time Value

Time Value = Premium - Intrinsic Value
Premium = Intrinsic Value + Time Value

$\text{TV} \\= \$3 - \$2 = \$1$

Calculate Premium

Premium = EV of the gain from an option or strategy

$= \frac{1}{4}(\$0) + \frac{1}{2}(\$2) \\ + \frac{1}{4}(\$8) = \$3$

Leverage

= (Share Price×100)/(Premium×100)

$= \frac{\$52×100}{\$3×100} = 17.3:1$

References: 5 Jul 8.pptx and L5-Options