✏️ Examples (Risky Portfolio)

Click/tap for Lecture 1 & 2 Formulas

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)

$Er_P = r_F + (B × (Er_M-r_F))$

Notation

$w_1$ = 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 Table

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:
 $ρ_{12}=\frac{Cov(r_1,r_2)}{σ_1σ_2}\quad\text{ and }$
$\quad Cov(r_1,r_2)=ρ_{12}σ_1σ_2$

Click/tap for Lecture 1 Formulas

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%

Notation

$r_F$ = 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

→ Complete formula page

→ Complete formula page

✏️ Consider the following risky and risk-free assets:

	E(r)	σ	Correlation w/ Bond
T-Bill	2.40%	0	0 (risk free)
Bond fund	4.00%	9%	1
Stock Fund	10.00%	15%	-0.221

What is the Expected Return of a 60/40 portfolio with 60% stocks and 40% bonds?

What would the expected return of the 60/40 portfolio be?

✔ Click here to view answer

$$\begin{aligned} E(r_p) &= (Er_1 × w_1) + (Er_2 \times w_2) \\ &= 60\% \times 10\% + 40\% \times 4\% \\ &= 7.6\% \end{aligned}$$

✏️ What is the variance of the 60/40 portfolio? What is the standard deviation?

✔ Click here to view answer

$$\begin{aligned} Var(p) &= σ_p^2 \\ &= σ_1^2 w_1^2 + σ_2^2 w_2^2 + 2w_1 w_2 σ_1 σ_2 ρ_{(1,2)} \\ &=(9\%^2 \times 40\%^2) + (15\%^2 \times 60\%^2) + (2 \times 40\% \times 60\% \times 9\% \times 15\% \times -0.221) \\ &= 0.007964 \end{aligned}$$$$\begin{aligned} \text{Standard Deviation } &= (0.007964)^{.5} \\ &=0.089241 \\ &= 8.9\% \end{aligned}$$

Another way to calculate it:

$$SD = SQRT(9\%^2 \times 40\%^2 + 15\%^2 \times 60\%^2) + 2 \times 40\% \times 60\% \times 9\% \times 15\% \times (-0.221)$$

✏️ What is the covariance between stocks and bonds?
Using the covariance between stocks and bonds, recalculate the variance and σ of the 60/40 portfolio. Do you get the same number?

✔ Click here to view answer

$$\begin{aligned} Cov(r_1,r_2) &= σ_1 \times σ_2 \times ρ_{1,2} \\ &= 9\% \times 15\% \times 0.221 \\ &= -0.0029835 \end{aligned}$$$$\begin{aligned} Var(p) &= σ_p^2 = σ_1^2 w_1^2 + σ_2^2 w_2^2 + 2w_1 w_2 Cov_{1,2} \\ &=(9\%^2 \times 40\%^2) + (15\%^2 \times 60\%^2) + (2 \times 40\% \times 60\% \times -0.0029835) \\ &=0.00796392 \end{aligned}$$$$\begin{aligned} SD(p) &= 0.00796392^{.5} \\ &=0.0892408 \end{aligned}$$

Yes, we got the same number.

✏️ You decide that you want your risk assets to follow the classic 60/40 mix of stocks and bonds. In addition to your portfolio of risky assets, you also want to put 15% of your portfolio in a T-Bills. What is the expected return, standard deviation, and variance of your complete portfolio? What is the Sharpe Ratio? Assume you expect the following performance:

	$E(r)$	$sd$
T-Bill	2.40%	0
60/40 risky portfolio	7.6%	8.9%

✔ Click here to view answer

$$\begin{aligned} Er_c &= r_F + y(Er_P-r_F) \\ &= 2.4\% + 85\% (7.6\%-2.4\%) \\ &= 6.82\% \end{aligned}$$$$\begin{aligned} σ_c &= σ_p \times y \\ &= 8.9\% \times 85\% \\ &= 7.565\% \end{aligned}$$$$\begin{aligned} Var &= σ^2 \\ &= 7.565\%^2 \\ &= 0.0057 \end{aligned}$$$$\begin{aligned} \text{Sharpe Ratio }&= \frac{E_r-r_f}{σ} \\ &= \frac{6.82\%-2.4\%}{7.565\%} \\ &= 0.58 \end{aligned}$$

✏️ Continuing the previous exercise, assuming that you have a risk aversion coefficient of $A=2$, what would your utility be, using our standard formula for utility $U = E(r) - \frac{1}{2}Aσ^2$?

✔ Click here to view answer

$$\begin{aligned} U &= E(r)- \frac{1}{2}Aσ^2 \\ &= 6.82\% - .5 \times 2 \times 0.0057 \\ &= 0.0625 \end{aligned}$$$$\begin{aligned} U &= E(r) - \frac{1}{2}Aσ^2 \\ &= 6.82\% -.5 \times 2 \times 7.565\%^2 \\ &= 0.0625 \end{aligned}$$